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abstract: 'The first extrasolar planets were discovered in 1992 around the millisecond pulsar PSR 1257+12. We show that recent developments in the study of accretion onto magnetized stars, plus the existence of the innermost, moon-sized planet in the PSR 1257+12 system, suggest that the pulsar was born with approximately its current rotation frequency and magnetic moment. If so, this has important implications for the formation and evolution of neutron star magnetic fields as well as for the formation of planets around pulsars. In particular, it suggests that some and perhaps all isolated millisecond pulsars may have been born with high spin rates and low magnetic fields instead of having been recycled by accretion.'
author:
- 'M. Coleman Miller and Douglas P. Hamilton'
title: Implications of the PSR 1257+12 Planetary System for Isolated Millisecond Pulsars
---
Introduction
============
The remarkably stable rotation frequencies of millisecond pulsars (MSP), with typical period derivatives of $10^{-21}-10^{-19}$ s s$^{-1}$, make them extremely sensitive to periodic perturbations such as those produced by orbiting companions. This sensitivity led to the first discovery of extrasolar planets, around the Galactic disk pulsar PSR 1257+12 (Wolszczan & Frail 1992). This pulsar has a period of $P=6.219\times 10^{-3}$ s and a period derivative of ${\dot P}=1.2\times 10^{-19}$, which in a standard magnetic dipole spindown model implies a dipole magnetic field of $B=3\times 10^{19}(P{\dot P})^{1/2}\,{\rm G} \approx 8.8\times 10^8$ G and a characteristic age $\tau_c=P/2{\dot P}=8\times 10^8$ yr. Timing residuals from PSR 1257+12 suggested that there were two Earth-mass planets around this pulsar (Wolszczan & Frail 1992). The planetary origin of the timing residuals was confirmed by observations (Wolszczan 1994) of the expected secular perturbations due to interaction between these planets (Rasio et al. 1993; Malhotra 1993; Peale 1993). These observations also revealed the presence of a third, moon-sized planet closer in (Wolszczan 1994). In order of increasing distance from the pulsar, the masses and semimajor axes of the three planets are $M_1=0.015/\sin i_1\,M_\oplus$ at 0.19 AU, $M_2=3.4/\sin i_2\,M_\oplus$ at 0.36 AU, and $M_3=2.8/\sin i_3\,M_\oplus$ at 0.47 AU, where $i_1$, $i_2$, and $i_3$ are the orbital inclination angles ($i=0$ is face-on, $i=90^\circ$ is edge-on). All three planets are in nearly circular orbits, with eccentricities $<$0.02. There is some evidence for a fourth planet (Wolszczan 1996) which, if it exists, has a mass $M_4\sim
0.05-81\,M_\oplus$ and a semimajor axis $\sim$6-29 AU, where the mass uncertainty is largely due to uncertainty in the fractional contribution of such a planet to the observed spindown of the pulsar (Wolszczan et al. 2000a). Recently, some doubt was cast on the existence of the innermost planet by Scherer et al. (1997), who pointed out that its 25.3 day orbital period is close to the solar rotation period at the 17$^\circ$ solar latitude of PSR 1257+12, and suggested that the modulation might actually be due to modulation in the electron density of the solar wind in that direction. However, if this effect is important it would also be expected to be observed in other millisecond pulsars. More importantly, the oscillation amplitude does not depend on the radio frequency (Wolszczan et al. 2000b), contrary to what is expected for a plasma effect. Hence, the 25 day modulation of the frequency from PSR 1257+12 is due to a planet.
The existence of this system has produced much speculation about its origin. As discussed by Phinney & Hansen (1993), the proposed formation mechanisms can be divided into presupernova scenarios, in which the planets existed before there was a neutron star in the system, and postsupernova scenarios, in which the planets formed after the supernova. Podsiadlowski (1993) reviewed a large number of these proposed mechanisms. Presupernova scenarios include those in which planets survive the supernova (Bailes, Lyne, & Shemar 1991) or are captured into orbit around the neutron star by a direct stellar collision (Podsiadlowski, Pringle, & Rees 1991) or are formed in orbit around a massive binary (Wijers et al. 1992). As reviewed by Podsiadlowski (1993), all of these mechanisms are met with serious objections. For example, direct stellar collisions in the Galactic disk are expected to be exceedingly rare, and a supernova explosion in a single-star system would be highly likely to unbind any planets initially in orbit around it and any remaining planets would have high eccentricities.
For these reasons, more attention has focused on postsupernova scenarios. Some models propose that PSR 1257+12 is a “recycled" pulsar which has been spun up by accretion, in analogy with other millisecond pulsars. In these models the star might have accreted matter from a remnant disk, for example from the disrupted remains of a merger between two white dwarfs or a white dwarf and a neutron star (Podsiadlowski et al. 1991), or from a massive disk left over from a phase of Be binary mass transfer (Fabian & Podsiadlowski 1991), or by deflation of a Thorne-Zytkow object (Podsiadlowski et al. 1991). Alternately, the accretion could have been from a stellar companion, which was then removed or disguised as a planet. One picture, which was motivated by a report of a single planet around PSR 1829–10 that was later retracted (Bailes et al. 1991), is that the companion was evaporated by flux from the neutron star until it had planetary mass (Bailes et al. 1991; Krolik 1991; Rasio, Shapiro, & Teukolsky 1992). A variant of this model is that as the companion is ablated it expands, eventually being disrupted and forming a $\sim 0.1\,M_\odot$ disk around the neutron star, from which the planets eventually form (Stevens, Rees, & Podsiadlowski 1992). Another possibility is that the ablated matter may not escape the system, instead forming a circumbinary disk from which planets form (Tavani & Brookshaw 1992; Banit et al. 1993). The stellar companion would, in this scenario, be evaporated completely by the neutron star.
A third class of postsupernova models, distinct from those involving disks or disrupted companions, suggests that the planets formed from fallback matter from the supernova (Bailes et al. 1991; Lin, Woosley, & Bodenheimer 1991), or from matter that had been ablated from a binary companion prior to the supernova (Nakamura & Piran 1991). The matter from which the planets formed might also have been accreted from the companion if supernova recoil sent the neutron star through the companion (suggested by C. Thompson; summarized in Phinney & Hansen 1993). In such models the neutron star was born with approximately its current spin rate and magnetic moment, and was therefore not spun up by accretion, in contrast to the standard formation scenario for millisecond pulsars.
Here we argue in favor of this third class of models. We therefore suggest that many or all isolated MSP may have simply been born as they are now. This alleviates potential problems with the birthrate of MSP versus the birthrate of low-mass X-ray binaries (LMXB; these are usually considered the progenitors of all MSP in the recycling scenario, and in our model are still the progenitors of binary MSP). This picture also suggests, but does not require, that supernovae may produce a bimodal distribution of neutron star magnetic fields and possibly spin rates.
We begin our argument in § 2 by showing that in most scenarios the planets must have formed in approximately their current location. We then show that models requiring the neutron star to be spun up by accretion subsequent to the supernova typically run into at least one of the following problems: (1) if the planets form before or during the spinup, they will be evaporated by the accretion luminosity, and (2) if the planets form after spinup, they must form from some remnant disk, but the particle luminosity from the neutron star is sufficient to disperse a tenuous disk of material faster than it can be supplied by, e.g., evaporated material from a companion. We also argue that the lack of planetary bodies with masses greater than Ceres around other isolated millisecond pulsars (Wolszczan 1999) strongly constrains the formation of this system, and in particular suggests a probabilistic scenario in which isolated MSP either capture enough mass to form planets or capture virtually no mass, rather than a smooth distribution in between. In § 3 we summarize the allowed formation histories, and we discuss the implications of such a formation scenario for the MSP population in § 4. We present our conclusions in § 5.
Physical Constraints on Models
==============================
It is useful to consider first the evolutionary path leading to the current high rate of spin of the pulsar. Clearly, it was either born with and has sustained a high spin frequency, or it was spun up by accretion at some point in its evolution. The accretion scenarios can be further subdivided according to whether the planets formed before, during, or after the spin-up phase, and whether they formed at their current locations or they formed farther out and later migrated inwards. In this section we therefore first consider dynamical migration, then investigate the effects of photon and particle luminosity on gas and planetesimals in a disk.
Dynamical Migration
-------------------
In the section following this one, we will put strong constraints on possible planetary formation mechanisms by assuming that the planets formed at their current distances from the pulsar. Here, we examine the validity of this assumption by considering possible mechanisms by which the planets may have formed further from the pulsar and subsequently migrated inward to their present positions. In order for such migration to have occurred, the planets must have interacted with an amount of mass roughly comparable to their own. There are three main possibilities: 1) gravitational scattering by other planets or protoplanets, 2) interaction with a disk of planetesimals, and 3) interaction with a gas disk.
Gravitational scattering of protoplanets can be quickly ruled out as a substantial source of radial migration for the pulsar planets under consideration. In this scenario, we model interactions between planets as randomly-oriented velocity impulses, and assume that several to several tens of scattering events have happened during the formation of the system. Large stochastic eccentricities and inclinations are expected in systems in which significant gravitational scattering has occurred. For example, if cumulative scattering events are capable of changing planetary semimajor axes by a few tens of percent, they will also be strong enough to induce orbital eccentricities of order 0.2 and inclinations above 10 degrees. While large inclinations cannot be ruled out in the PSR 1257+12 system, the near circular orbits of all three planets argue strongly against significant gravitational scattering.
Interactions with a disk of gas or planetesimals are required to attain the nearly circular orbits that are observed. Can these interactions also cause planets to migrate significantly? In the case of planetesimals in the PSR 1257+12 system, the answer is again no. Escape velocities from Earth-sized planets are of the order of 15 km/s while the orbital velocity of even the outermost pulsar planet is of order 50 km/s. Since orbital velocities dominate escape velocities, gravitational scattering of planetesimals is relatively weak. Even the outermost planet cannot eject planetesimals from the system unless an unlikely sequence of multiple favorable close approaches occurs; collisions between planets and planetesimals are much more likely. The planets, therefore, absorb the vast majority of the planetesimals while nearly conserving the total angular momentum of the system. The most that can happen is that the inner and outer planets separate somewhat; there is no systematic inward migration. In this scenario, conservation of angular momentum implies that the three planets could not all have formed further from the pulsar than they are now, thereby avoiding the destructive mechanisms that we discuss below.
Finally, interactions with a gas disk can cause solid objects to move inward toward the pulsar. So that angular momentum is conserved, an equivalent amount of gas must be offset outward. If the composition of the gas is roughly solar, the solids that condense to form the planets account for at most a few percent of the total mass. The rest of the mass remains in gaseous form in an extended thickened disk encircling the pulsar. Planet-sized objects raise waves in the disk, and the gravitational perturbations of these waves put torques back on the planets which can systematically change their orbits. The details of how this occurs depend most strongly on the mass of the planet, and the radial density profile of the disk. Given a large disk mass, planetary migration by this mechanism could be substantial. However, if the pulsar is to be spun up by accretion, the $\sim 10^7 - 10^8$ year spinup timescale is long compared to the expected $\sim
10^6-10^7$ yr survival time of protoplanetary disks (Bachiller 1996). We expect that the lifetime of a protoplanetary disk will be especially short in the high-luminosity environment of the pulsar. Therefore, for most of the spinup time, the planets must have been unshielded by a disk and close to their present distances.
Accretion Luminosity and Ablation
---------------------------------
If the planets formed at approximately their current locations, they could be affected by the photon or particle luminosity from the neutron star, either during accretion or after. Here we consider ablation of the planets by the photon luminosity produced by accretion, and in the next section we discuss ablation of a protoplanetary disk and planetesimals by high-energy particles produced by pulsar spindown.
Higher luminosity during accretion means more rapid and effective ablation of the planets, so in order to be conservative we will calculate the minimum luminosity required for spinup. The luminosity, in turn, is related to the accretion rate, which may be estimated by magnetic torque balance arguments (e.g., Ghosh & Lamb 1979). These arguments show that if the star is spun up entirely by accretion (the standard assumption in LMXB recycling scenarios), then the equilibrium spin frequency, at which the net torque vanishes, can be characterized by the orbital frequency at some radius $r_t$. The required accretion rate increases rapidly with decreasing $r_t$ and hence with increasing stellar spin frequency, so the most conservative assumption is that the neutron star is currently spinning at the highest frequency it has ever had. In reality, substantial spindown via magnetic dipole braking has likely taken place.
To calculate the luminosity required for a given $r_t$, we write $r_t=\omega_c r_A$, where $\omega_c$ is the “fastness parameter" and $r_A$, the Alfvén radius, is (see, e.g., Shapiro & Teukolsky 1983, pg. 451) $$r_A=3.5\times 10^8 L_{37}^{-2/7}\mu_{30}^{4/7}
\left(M\over{M_\odot}\right)^{1/7}R_6^{-2/7}\,{\rm cm.}$$ Several recent analyses (e.g., Li & Wang 1999; Psaltis et al. 1999) have concluded that $\omega_c>0.8$. Therefore, $$r_t=2.8\times 10^8 (\omega_c/0.8)L_{37}^{-2/7}\mu_{30}^{4/7}
\left(M\over{M_\odot}\right)^{1/7}R_6^{-2/7}\,{\rm cm.}$$ Here $L=10^{37}L_{37}$ erg s$^{-1}$, $\mu=10^{30}\mu_{30}$ G cm$^3$, and $R=10^6R_6$ cm. From this equation we see that higher $\omega_c$ means larger $r_t$ for a given luminosity, implying a lower stellar spin frequency. Therefore, the required luminosity increases with increasing $\omega_c$. The equilibrium period is then $P_{\rm eq}=2\pi(r_t^3/GM)^{1/2}$ (valid in Schwarzschild spacetime). This period cannot exceed the current rotational period of PSR 1257+12, $P=6.2\times 10^{-3}$ s. Moreover, models and observations of neutron-star low-mass X-ray binaries suggest that accretion may cause the field to decay (see, e.g., Bhattacharya et al. 1992), so the field strength during accretion was at least as large as it is now. Therefore, we can solve for the minimum luminosity during accretion by setting $P_{\rm eq}=P$ and substituting in $\mu=8.8\times 10^{26}$ G cm$^3$ and $R_6=1$. The result is $L_{37}>0.9(M/M_\odot)^{-2/3}(\omega_c/0.8)^{7/2}$. Even for a relatively massive neutron star with $M=2\,M_\odot$, this is $5\times 10^{36}$ erg s$^{-1}\approx 10^3L_\odot$.
At the distance of the inner planet, $r=3\times 10^{12}$ cm, the equivalent blackbody temperature for a perfectly absorbing surface is $T=[L/(\sigma 4\pi r^2)]^{1/4}=5300$ K even for $M=2\,M_\odot$. At the $\sim$keV energies typical of accretion emission from a neutron star surface, the absorption cross section is much greater than the scattering cross section, especially if heavy elements are present, so the assumption of nearly perfect absorption is likely to be good. Furthermore, the dependence on albedo is very weak (only the 1/4 power), so the temperature estimate above is robust. This temperature is sufficient to boil any element likely to be in abundance around the star. Therefore, either direct evaporation or ablation due to the impinging X-rays can proceed efficiently.
Consider evaporation first, assuming that the innermost planet is formed of an element of atomic weight $A$. We also conservatively assume a density $\rho\approx 10$ g cm$^{-3}$; a more realistic, less dense planet would be more easily evaporated. The radius of the planet is then $1.3\times 10^8$ cm, so the scale height of the atmosphere created is $h=kT/mg=1.2\times 10^9A^{-1}$ cm. If the planet is composed of elements significantly lighter than iron ($A=56$), then the scale height is comparable to $R$, so the illumination will cause the planet to swell up and disperse on about a sound crossing time. Even for iron, $h\approx 0.15\,R$. The weakening of gravity with increasing radius means that a substantial fraction of the gaseous iron will be at large enough distances to escape; for example, a calculation including the $r^{-2}$ dependence of gravitational acceleration shows that $\approx 1$% of the planet will be at radii $>4\,R$, and $>$0.1% will be unbound. Therefore, even if the innermost planet is pure iron, it will be evaporated in a few hundred sound crossing times, a matter of only days. Note that even if there is an optically thick disk present, the blackbody temperature at the orbital radius of the planet is unchanged, so evaporation will proceed efficiently.
A separate argument, which is also important for the two outer planets, is that the X-ray illumination can ablate the planets directly. Numerical analyses (e.g., Phinney et al. 1988; van den Heuvel et al. 1988) suggest that a fraction $\sim$10% of the luminosity intercepted by a gaseous companion to a neutron star ablates that companion, and that the material leaving the companion does so at approximately the escape velocity. A rough estimate of the time scale for ablation therefore is obtained by dividing the gravitational binding energy of the companion by 0.1 times the X-ray luminosity intercepted by the companion. If an optically thick exists at this time it will be able to shield the planets from ablation. However, as we show in the next section, the required mass of such a disk is comparable to the mass of the planets, and hence a disk massive enough to shield the planets will drag them rapidly inwards as the disk accretes.
At a distance of $3\times 10^{12}$ cm, about $10^{-9}$ of the total luminosity falls on the surface of a planet of radius $\sim 10^8$ cm, so $10^{-10}$ of the total energy released by accretion is used for ablation of the innermost planet. This total energy may be estimated from the amount of angular momentum necessary to spin up the star to its current rotation frequency. That frequency is $\omega\approx 10^3$ rad s$^{-1}$, so for a neutron star with a typical moment of inertia $I\approx 10^{45}$ g cm$^2$ the angular momentum is $10^{48}$ g cm$^2$ s$^{-1}$. Assuming that the star was rotating much more slowly than this prior to accretion, this is the amount of angular momentum that must be accreted. At the $\sim 5\times 10^6$ cm radius at which the torque is exerted, the specific angular momentum is close to its Newtonian form $\ell\approx\sqrt{GMr}$, or $\ell\approx 3\times 10^{16}$ cm$^2$ s$^{-1}$ for $M=1.4\,M_\odot$. The neutron star must therefore accrete $3\times 10^{31}$ g$\approx
0.01\,M_\odot$ to spin up to its current frequency. Assuming that the accretion efficiency is $L/{\dot M}\sim 0.2\,c^2$, typical for accretion onto the surface of a neutron star, this will release a total of $6\times 10^{51}$ ergs, taking $4\times 10^7$ yr at $5\times 10^{36}$ erg s$^{-1}$. Multiplying by $\sim 10^{-10}$ yields $6\times 10^{41}$ erg. If we again conservatively assume that the companion is pure iron, the binding energy at a radius $1.3\times 10^8$ cm is about $GM^2/R=4\times 10^{36}$ erg. This is only $10^{-5}$ of the ablation energy. Therefore, even if evaporation were somehow suppressed, the inner planet would be ablated within a few hundred years, much shorter than the $\sim
10^{7-8}$ yr spent in the luminous LMXB phase.
The outer planets would also be in danger. Their semimajor axes are approximately twice that of the inner planet, so the temperature is down by a factor $2^{1/2}$, to $\sim$3800 K. This will boil any abundant element except pure carbon, so again ablation would be highly efficient unless the outer planets were truly diamonds in the roughest of environments. If the densities of the outer planets are the same as that of the inner planet, their radii scale like $M^{1/3}$, so their binding energies scale like $M^2/R=M^{5/3}$. Their surface area goes like $M^{2/3}$, so the ratio of binding energy to intercepted luminosity, assuming the same radiation flux, goes like $M$. At twice the distance and $\sim$200 times the mass of the inner planet, the ratio of binding energy to intercepted luminosity is a factor $\sim$1000 larger for the outer planets, which means that they will be evaporated in a time $\sim 10^5$ yr, still a factor $\sim 100$ shorter than the accretion time. Therefore, all three planets would be destroyed if they were in their current positions and the pulsar was spun up by accretion. This shows that if the pulsar was spun up by accretion, that accretion had to take place before planets existed in the system.
These constraints are summarized in Figure 1. Here we plot contours of constant destruction time (from the combined effects of evaporation and ablation) against the mass of the object and the photon flux received, from $10^0$ years (leftmost contour) to $10^7$ years (bottom right contour). To give a conservative upper limit to the destruction time we assume planets made of pure iron with densities of 10 g cm$^{-3}$. The locations in this plot of the three planets in the PSR 1257+12 system are indicated with dots. For high fluxes, direct evaporation destroys the planet quickly, but for lower fluxes ablation becomes more important, hence the change in the slopes of the lines around a destruction time of $\sim 10^5$ years. At a flux less than $F\approx 5\times 10^9$ erg cm$^{-2}$ s$^{-1}$, so that iron remains liquid, destruction by either evaporation or ablation is highly inefficient.
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Effect of pulsar particle luminosity
------------------------------------
The previous section concentrated on the effect of photon luminosity on already-existing planets. Now consider the effect of particle luminosity on a protoplanetary disk. This is of relevance to models of the PSR 1257+12 system such as that of Banit et al. (1993), in which planets form from matter ablated from a companion star. It is also important in projecting whether any millisecond pulsars are expected to have planetary systems consisting solely of small, asteroid-sized, objects. We show that the particle luminosity is great enough that if there is a time in the evolution of the system when a massive disk does not exist, then a massive disk cannot form because it would be ablated efficiently by the particle luminosity.
The spindown energy of isolated pulsars is thought to be released primarily in the form of highly relativistic particles. These particles interact with matter in a fundamentally different way than do X-rays. X-rays interact with electrons, and are not energetic enough to eject nuclei in one collision. In contrast, the relativistic particles are typically nuclei, which interact with other nuclei. Their energy is enormous compared to any relevant binding energy (gravitational or chemical), and hence can eject a particle in a single collision. Therefore, whereas X-ray ablation has a threshold in intensity, ablation by relativistic particles always occurs, albeit at negligible rates in some cases.
If the disk is optically thin to these particles, then only a small fraction of the spindown energy is imparted to the disk. Any impacts will deposit far more than the specific binding energy, and hence any nucleus in the disk hit by a high-energy particle will be sent to infinity, but the energy per particle will be so high that the resulting mass flux from the disk will be small. If we now imagine that the optical depth of the disk is increased, then a larger and larger fraction of the spindown energy is transferred to the disk. If the disk is still optically thin to the high-energy particles then each high-energy particle only interacts with one disk nucleus, and the mass flux will scale roughly with the optical depth of the disk. However, when the optical depth to the high-energy particles exceeds unity, then each high-energy particle from the neutron star will interact with more than one target nucleus, and the collisions will spawn further high-energy particles that interact in their turn. The energy imparted per particle is therefore diminished, meaning that the mass flux from the disk (which depends on the total energy deposited in the disk divided by the average energy per ejected particle) is increased more rapidly than the optical depth. If, however, the optical depth is very large, then the average energy per particle drops below the gravitational binding energy, and the mass flux drops quickly. The maximum mass flux therefore occurs when the average energy per particle is comparable to the binding energy.
Similar considerations apply to the formation of a planet from grains in such an environment, assuming that the grains are unshielded from the pulsar particles. Small grains are affected little, because they are optically thin. By contrast, larger rocks (with optical depth slightly larger than unity) are exposed to the full flux of the particles, and again the mass flux produced by the particles peaks when the average energy per particle is comparable to the binding energy, although here the binding energy is chemical instead of gravitational.
If there is a supply of matter to a nascent disk (as in the picture of Banit et al. 1993, in which the supply comes from the ablation of the stellar companion), then the mass of the disk depends on whether the supply rate ${\dot M}_{\rm supply}$ is greater than or less than the maximum rate ${\dot M}_{\rm max}$ at which ablation can remove mass. If ${\dot M}_{\rm supply}>{\dot M}_{\rm max}$ then the disk increases steadily in mass, indefinitely in principle. If ${\dot M}_{\rm supply}<{\dot M}_{\rm max}$ then the disk will reach an equilibrium mass when ablation balances supply.
Let us now quantify this picture. Ultrarelativistic particles interacting with protons have a stopping column depth of about $\sigma_p$=100 g cm$^{-2}$ (e.g., Slane & Fry 1989). At the distance $R=3\times 10^{12}$ cm of the inner planet in the PSR 1257+12 system, the gravitational binding energy is about $U=10^{14}$ erg g$^{-1}$, and the area of a sphere at radius $R$ is $A\approx 10^{26}$ cm$^2$. The current spindown luminosity of the pulsar is ${\dot E}\approx 10^{34}$ erg s$^{-1}$. If the disk subtends a solid angle that is $\epsilon=0.1$ of the whole sphere, this means that the maximum mass flux in the wind is ${\dot M}_{\rm max}=
\epsilon{\dot E}/U=10^{19}$ g s$^{-1}$. To estimate ${\dot M}_{\rm supply}$ we assume a supply of matter due to evaporation of a stellar companion (as in the “black widow" pulsar PSR 1957+20). We also make the generous assumption that the evaporation rate is equal to that in the black widow pulsar even though the spindown luminosity of PSR 1957+12 is ten times less than that of PSR 1957+20. Then ${\dot M}_{\rm supply}=10^{17}$ g s$^{-1}$. This is less than ${\dot M}_{\rm max}$ by two orders of magnitude, and hence an equilibrium disk mass will be reached. Assuming that equilibrium occurs at an optical depth of a few, the total mass in equilibrium is about $10^{27}$ g. In addition, since the mass is coming from the companion, about 97% of this mass is expected to be in hydrogen or helium, leaving no more than $10^{26}$ g and probably a factor of a few less in metals, compared with the $\sim 4\times 10^{28}$ g in the two largest planets in the system. Moreover, relativistic particles would shatter complex nuclei and reduce the metal fraction even more. The remaining mass in metals would be far less than needed to form the current planets.
We now consider whether planets can form in this environment from small grains, if the grains are not shielded from the pulsar radiation. The relevant binding energy $U$ is molecular, which we assume is $\sim$1 eV per molecule or about $10^{11}$ erg g$^{-1}$. The particle flux at the distance of the innermost planet is $F\approx 10^{34}$ erg s$^{-1}/10^{26}$ cm$^2
=10^8$ erg cm$^{-2}$ s$^{-1}$. The mass loss rate from the rock due to radiation therefore has a maximum ${\dot m}_{\rm
max}=FA/U$, where $A=\pi a^2$ is the cross-sectional area of the rock. For a stopping column depth of 100 g cm$^{-2}$, the radius of a grain with optical depth $\tau\sim 10$ (so that the average energy per particle is close to the molecular binding energy) is $a\sim 100$ cm. The timescale for evaporation of a particle of mass $m={4\over 3}\pi
a^3\rho$, where $\rho$ is the density, is then $t_{\rm evap}=m/{\dot
m}={4\over 3} \rho aUF^{-1}$. This is $<10^6$ s for $a<100$ cm and any reasonable density. Therefore, in this environment rocks cannot grow from small grains to radii greater than a few tens of centimeters. If larger planetesimals are present in the system prior to exposure to the particle radiation, then the radiation will tend to ablate the planetesimals. We may estimate the rate of ablation by computing the area strongly affected by the high-energy particles. This area is the region with a path length through the planetesimal such that the energy per particle at exit exceeds the chemical binding energy. Calling this distance $d$ and the planetesimal radius $R$, the cross-sectional area with a path length less than $d$ turns out to be simply ${1\over 4}\pi d^2$, independent of $R$ if $R>d/2$. The mass loss rate ${\dot m}$ is therefore also independent of $R$, and hence the evaporation time scales as $m\sim R^3$. Hence, within $10^{7-8}$ yr all planetesimals of radius less than $\sim 1$ km will be evaporated by the $10^{34}$ erg s $^{-1}$ flux from PSR 1257+12. The lower limit to the survival radius may be even larger, because the main face of a planetesimal with radius 1 km has an area a factor of a million larger than $\pi d^2$, with $d\approx 100$ cm. A small fraction of this incident radiation may ablate additional matter. The lower limit to planetesimal radius is thus probably $\sim 1-10$ km.
Particle radiation therefore prevents the formation of planets when the disk is not very optically thick to the high-energy particles. Thus, for planets to form around a pulsar, the disk must have sufficient mass to shield itself from the radiation. That is, the energy of the relativistic particles must be degraded sufficiently that their average energy is less than the binding energy. In a single collision, the most efficient reduction of energy occurs when both the primary nucleus and the target nucleus receive half of the original energy. One can therefore conservatively assume that in each interaction, the maximum energy per particle is reduced by a factor of two, so that after $n$ scatterings the typical energy after propagation through the medium is reduced by a factor $\sim 2^n$. As long as the particles are relativistic, forward beaming means that the number of scatterings after traversing an optical depth $\tau$ is $n\sim\tau$. When the particles are nonrelativistic then beaming is minor, and hence the number of scatterings increases more rapidly, as $n\sim \tau^2$. Assuming an initial Lorentz factor of $\sim 10^{3-5}$, the minimum optical depth required to reduce the particles to nonrelativistic energies is therefore conservatively $\tau\sim 10-20$. Assuming a disk covering fraction of $\epsilon\sim$10%, the required disk mass is then at least $M_{\rm tot}=\epsilon \pi r^2\tau\sigma_p$, or about $1-2\times 10^{28}$ g at a radius $r=3\times 10^{12}$ cm. If the initial disk mass is less than this, we expect no planets to form. Given that this is already greater than one Earth mass, this may mean that isolated millisecond pulsars either have planetary-mass objects around them or nothing, and hence that we should not expect systems with just asteroids. If so, it suggests that microsecond timing noise in millisecond pulsars is not dominated by asteroids, as was discussed as a concern by Wolszczan (1999).
Constraints from the lack of planets around other isolated MSP
--------------------------------------------------------------
In addition to the constraints just listed, formation mechanisms for planets around millisecond pulsars have another constraint, common to scenarios proposed for any rare object: the mechanism cannot be [*too*]{} good, or else more examples would be seen. In this case, why are there no other planets around isolated MSP, given the extreme sensitivity of MSP timing to such perturbations? Of course, it could be that the PSR 1257+12 system had a unique history, but here we make the Ockham’s razor assumption that all isolated millisecond pulsars are formed in the same way.
Of the nine isolated MSP in the Galactic disk, only PSR 1257+12 has confirmed planets around it, even though asteroid-sized objects with orbital periods of a few years or less could be detected with current techniques (Wolszczan 1999). The gap of at least three orders of magnitude between these mass upper limits and the mass of the planets around PSR 1257+12 suggests that the pulsar planetary system is not simply in the high-mass tail of a distribution, but is instead the result of a rare event. This argues against disrupted companion scenarios (Podsiadlowski et al. 1991; Fabian & Podsiadlowski 1991), in which there is always $0.01-0.1\,M_\odot$ in the disk after disruption. Such a high disk mass is expected to be extremely favorable for the formation of planets, because the column depth is high enough to shield the matter from the pulsar flux (see above). It would therefore be surprising that only one MSP has planetary-mass objects around it. Instead, an idea such as one in which supernova recoil kicks the neutron star through the companion (Phinney & Hansen 1993), has many desirable properties. In this picture, only if the neutron star intersects the companion will it accrete mass and potentially form planets. Otherwise, virtually no mass is accreted and the star simply spins down in isolation. We explore this idea further in the next section.
Allowed Formation Histories
===========================
The physical constraints in § 2 may be summarized as follows. (1) If PSR 1257+12 was spun up by accretion from a companion, then ablation makes formation of the planets before or during this accretion highly implausible. (2) Particle radiation from the pulsar will destroy a disk if the disk has too low a mass. It will also prevent large grains from forming in an unshielded environment. Therefore, either supply of mass to the disk must exceed the mass loss rate or the disk mass must initially be well in excess of $\sim 10^{28}$ g. (3) The formation mechanism cannot be inevitable for isolated millisecond pulsars, or other examples would be seen.
The most plausible mechanisms are therefore those in which an isolated neutron star sometimes (in $\sim$10% of cases) obtains a disk of mass $>10^{28}$ g from which planets form, but in most cases does not acquire significant mass. This favors ideas such as the supernova recoil scenario (Phinney & Hansen 1993). Given that 1 out of 9 isolated MSP have planets, then prior to the supernova the stellar companion must subtend a few percent to a few tens of percent of the sky, assuming that the direction of the kick delivered to the neutron star is not correlated with the direction to the companion. This would require a separation of a few times the radius of the presupernova star, if the stellar companion is also a massive star, implying a separation of $\sim 10^{12}-10^{13}$ cm. The observed distribution of initial orbital separations $a_{\rm init}$ of massive stars is $\sim 1/a_{\rm init}$ (e.g., Kraicheva et al. 1979), so tens of percent of massive binaries are expected to be in this range of separations.
If the neutron star receives a kick in the direction of the companion, then in order to eventually form planets it needs to capture at least $10^{28}$ g from the companion. We can make a very rough estimate of the mass captured by making a Bondi-Hoyle type assumption that the matter captured from the companion star has an impact parameter $b$ relative to the neutron star that is less than $b_{\rm max}$, where $b_{\rm max}$ is defined such that the effective orbital velocity $v_{\rm orb}=(GM_{\rm NS}/b_{\rm max})^{1/2}$ is equal to the kick velocity $v_{\rm kick}$ of the neutron star. The proper motion of PSR 1257+12 is measured at 300 km s$^{-1}$ (Wolszczan 1999); Identifying this as the kick velocity yields $b_{\rm max}\approx
10^{11}$ cm. If this matter is captured with an efficiency $\epsilon\approx 10^{-3}-10^{-1}$ then the amount of matter captured from the companion is typically $M_{\rm cap}\sim\epsilon b_{\rm max}^2
R_c{\bar\rho}_c$, where $R_c$ and ${\bar\rho}_c$ are the radius and average density of the companion. For a companion of mass $10\,M_\odot$, radius $R_c=10^{12}$ cm, and average density $\rho_c=10^{-2}$ g cm$^{-3}$, the captured mass is $\approx
10^{29-31}$ g, which is in the needed range. Although the initial scale of the disk, $b_{\rm max}\approx 10^{11}$ cm, is two orders of magnitude smaller than the current planetary system, conservation of angular momentum implies that the disk will spread significantly as it evolves. Planets will form only after the outermost portions of the disk have spread and cooled enough to allow efficient condensation of solids (Lissauer 1988). Thus planetary systems formed by this mechanism should consist of terrestrial mass planets confined within a few AU of the central pulsar.
Such a disk mass would effectively shield the protoplanets from the radiation of the neutron star. In addition, when particle radiation dominates the emission from the pulsar (as opposed to accretion radiation) the luminosity of $\sim 10^{34}$ erg s$^{-1}$ would produce a blackbody temperature of only a few hundred Kelvin at the distances of the planets. At such temperatures, ionization is low and molecule and grain formation is thought to proceed efficiently (Lissauer 1993). This is therefore an environment supportive of planet formation, particularly given that the characteristic age $\tau_c\sim 10^9$ yr of the pulsar is much longer than the $\sim 10^7$ yr required for planet formation. The allowed mass range of the disk would also accommodate fourth planet with a mass $\simless 100\,M_\oplus$ if the existence of this planet is confirmed by a long baseline of observations.
This scenario implies that many isolated neutron stars may have planets in orbit around them. Why, then, are there no other known planets around isolated pulsars, millisecond or otherwise? There is a reported detection of a planet in the PSR B1620-26 system, but this is a triple system in a globular cluster and probably formed by a different mechanism, such as an exchange interaction (Sigurdsson 1993; Ford et al. 2000). We attribute the paucity of detected pulsar planets to the stringent conditions for such planets to be observable. Young radio pulsars such as the Crab or Vela are extremely powerful emitters of particle radiation. For example, the spindown luminosity of the Crab pulsar, which is thought to emerge primarily as high-energy particles, is $\sim
5\times 10^{38}$ erg s$^{-1}$. This will prevent planetary formation within several AU, and will continue to do so until the spindown luminosity drops by more than an order of magnitude. In addition, the formation of planetary-mass objects is thought to require $\sim
10^7$ yr, by which point any pulsar with a magnetic field $\sim
10^{12}$ G or higher will have spun down past the death line, and will therefore not be detectable as a pulsar. As a consequence, planets would not be detectable around the neutron star. Finally, even if planets are present around a young, high-field pulsar, the timing noise makes difficult the detection of sub-Earth mass planets (Wolszczan 1999).
Hence, if an isolated neutron star has planets around it, the only chance for their presence to be detected is for the neutron star to have a weak magnetic field, so that it remains a pulsar long enough for planets to form. For the pulses to be detected, the spin frequency then has to be high, otherwise the total spindown energy is too low. This means that the star is born with a high frequency, and that accretion from any remnant disk does not spin the star down significantly. With these constraints on observability, it is not surprising that there is only one known planetary system around an isolated pulsar.
Implications for Millisecond Pulsars
====================================
Our picture requires that the neutron star in PSR 1257+12 was born with approximately its current magnetic field strength of $B\approx
10^9$ G. If instead it were born with a $\sim 10^{12}$ G magnetic field then it would have spun down rapidly to a period of order seconds, either because of magnetic dipole spindown or because of accretion torques. To be spun up by accretion to millisecond periods, the accretion would have had to proceed at near-Eddington rates after the field had decayed to $B\sim 10^9$ G. The decay time is at least $10^7-10^8$ years for solitary pulsars (Bhattacharya et al. 1992). Moreover, the existence of neutron stars with $B\sim 10^{12-13}$ G in high-mass X-ray binaries, which often accrete at near-Eddington rates and have accretion lifetimes of millions of years, suggests that even active accretion does not cause the field to decay in less than $\sim
10^6$ yr. Therefore, the accretion rate would have had to be close to Eddington after several million years. Studies of accretion from a remnant disk (Cannizzo, Lee, & Goodman 1990) show that the system becomes self-similar quickly and that the accretion rate drops like a power law in time, ${\dot M}\propto t^{-7/6}$, so such a high accretion rate millions of years after the onset of accretion would imply an unrealistically high initial accretion rate.
We also suggest that the initial spin frequency of the neutron star was close to its current value. Otherwise, the star would have to be spun up by accretion. Even if the initial field was weak, for accretion from the disk to spin the star up to its current rate would again require near-Eddington accretion for several million years, and the low accretion rate tail of the accretion from the disk is likely to slow the star down below its current spin rate. Therefore, we propose that the pulsar in this system, and by extension perhaps all isolated millisecond pulsars, was born with approximately its current spin rate and magnetic field strength. We emphasize that these considerations need not apply to isolated millisecond pulsars in globular clusters, in which there are other possible formation channels for isolated MSP (e.g., exchange interactions; Sigurdsson 1993).
The link between millisecond pulsars and neutron star low-mass X-ray binaries is well established. Some 80% of millisecond pulsars in the Galactic disk are in binaries, compared to only 1% of slower pulsars (Lorimer 2000). In addition, the recent discovery of 401 Hz pulsations from the LMXB SAX J1808–3658 (Wijnands & van der Klis 1998) is the strongest evidence yet that neutron stars in LMXBs can have spin frequencies comparable to those of MSP. It was thought a decade ago (Kulkarni & Narayan 1988) that there was a significant discrepancy between the birthrate of short orbital period ($<$25 day) MSP and their presumed progenitor LMXBs, in that the pulsar birth rate was two orders of magnitude too large. However, better statistics have been collected and the factor is now less than four (Lorimer 2000).
Isolated millisecond pulsars may pose a different problem, however. The small number of isolated MSP makes sample variance a significant concern, but analysis of the current data suggests that isolated MSP have a lower luminosity than MSP in binaries, and may have other distinct properties as well (Bailes et al. 1997). In addition, the best estimates of the birthrate of isolated MSP in the Galaxy, $2\times 10^{-5}$ yr$^{-1}$ (Lorimer 2000), are at least ten times the birthrate estimates for binary MSP (note, however, that the isolated MSP estimate is strongly influenced by the large weights attached to a few low-luminosity sources). If this rate and these discrepancies are confirmed by future surveys with larger samples, it suggests that isolated millisecond pulsars are formed by a different channel, which does not involve recycling (Bailes et al. 1997).
Our analysis of PSR 1257+12 suggests that the different channel may simply be that some neutron stars are born with fast spins and weak magnetic fields. Birth with rapid spin is compatible with the probable origin of the Crab pulsar and other similar pulsars, which could have been born with millisecond periods. Birth with a weak field is not directly comparable to the known population of young pulsars, although with a birth rate of $2\times 10^{-5}$ yr$^{-1}$ in the Galaxy and assuming a supernova remnant lifetime of $\sim 10^5$ yr it is not surprising that no known supernova remnant harbors a millisecond pulsar. It is therefore possible that pulsars born with weak fields are simply at the tail end of a distribution peaked at $B_{\rm init}\sim 10^{12}$ G, and it is equally possible that weak-field isolated pulsars are the product of a completely different set of processes in the supernova, or of rare types of supernovae.
Conclusions
===========
We have argued that the physical constraints on the PSR 1257+12 system, in particular the existence of the small innermost planet, in addition to the lack of planets around other isolated millisecond pulsars, points strongly towards an evolutionary history in which the neutron star had a high initial spin rate and weak initial magnetic field and formed planets from a disk of captured matter. We also predict that in the absence of other planets, objects of asteroid size or smaller will not form around millisecond pulsars, due to ablation by the flux of high-energy particles.
We dedicate this paper to the memory of John Wang, a valued friend and colleague and an outstanding astrophysicist. This work was supported in part by NSF Career Grant AST9733789 (DPH) and NASA grant NAG 5-9756 (MCM).
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|
{
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---
abstract: 'Longitudinal data are common in clinical trials and observational studies, [where missing outcomes due to dropouts are always encountered]{}. Under such context with the assumption of missing at random, [the weighted generalized estimating equations (WGEE) approach is widely adopted for marginal analysis]{}. [Model selection on marginal mean regression]{} is a crucial aspect of data analysis, and identifying an appropriate correlation structure for model fitting may also be of interest and importance. However, [the existing information criteria for model selection in WGEE have limitations]{}, such as separate criteria for the selection of marginal mean and correlation structures, unsatisfactory selection performance in small-sample set-ups and so on. In particular, [there are few studies to develop joint information criteria for selection of both marginal mean and correlation structures]{}. In this work, by embedding empirical likelihood into the WGEE framework, we propose two innovative information criteria named a joint empirical Akaike information criterion (JEAIC) and a joint empirical Bayesian information criterion (JEBIC), which can simultaneously select the variables for marginal mean regression and also correlation structure. [Through extensive simulation studies,]{} these empirical-likelihood-based criteria exhibit robustness, flexibility, and outperformance compared to the other criteria including the weighted quasi-likelihood under the independence model criterion, the missing longitudinal information criterion and the joint longitudinal information criterion. [In addition, we provide a theoretical justification of our proposed criteria, and present two real data examples in practice for further illustration.]{}'
author:
- |
Chixiang Chen$^{1}$, Biyi Shen$^{1}$, Lijun Zhang$^{2}$, Yuan Xue$^{3}$, Ming Wang$^{*1}$\
$^{1}$Division of Biostatistics and Bioinformatics, Department of Public Health Sciences\
Penn State College of Medicine, Hershey, PA, U.S.A\
$^{2}$Institute for Personalized Medicine, Penn State College of Medicine, Hershey, PA, U.S.A\
$^{3}$ School of Statistics, University of International Business and Economics, Beijing, China\
[$^{*}$Contact Email: [email protected]]{}
bibliography:
- 'biomsample\_bib.bib'
title: 'Empirical-likelihood-based criteria for model selection on marginal analysis of longitudinal data with dropout missingness'
---
\[firstpage\]
Akaike information criterion; Bayesian information criterion; Empirical likelihood; Longitudinal data; Missing at random; Model selection; Weighted generalized estimating equation.
Introduction {#s:intro}
============
Longitudinal data are common in clinical trials and observational studies. Due to the research interest in conducting inference on the population-level parameter estimates, generalized estimating equations (GEE) has been widely employed for marginal regression analysis, where the correlations among the observations within subjects are treated as nuisance parameters [@liang1986; @ming2014]. [In longitudinal studies, missing data is typically encountered,]{} which poses challenges for model fitting and model selection. [There are three types of missing data: missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR), depending on whether the factors related to missing probability are observed or not [@little2014].]{} For instance, subjects may drop out of the study or are lost to follow-up due to several reasons such as drug resistance or side effects. Under such context, MAR is commonly and reasonably assumed for statistical inference. Literature has shown that [the estimates based on regular GEE]{} are biased for longitudinal data under MAR [@laird1988]. [@robins1995] first proposed the weighted GEE (WGEE) method for bias correction by incorporating an inverse probability weight matrix. Given the correctly specified model for missing data, the consistency of WGEE estimates [still holds even when the “working" correlation structure is misspecified]{}.
[Model selection is a crucial aspect of longitudinal data analysis. Without a doubt, identifying the variables for the marginal mean structure is always essential. Also, an improper correlation structure may lead to loss of efficiency of parameter estimates. This problem has been exclusively investigated for complete longitudinal data; however, when the missing data exist, the efficiency improvement is still under exploration, but several works have shown that selecting a proper correlation structure for WGEE is somewhat promising and important [@gosho2014; @gosho2016; @shardell2008; @preisser2002].]{} [To accomplish these selection goals, development of model information criteria has gained substantial attention by researchers.]{} [@pan2001] first proposed one of the most popularly used information criteria, the quasi-likelihood under the independence model criterion (QIC), but it does not accommodate missing data. For longitudinal data with dropout missingness under MAR, [@shen2012] proposed two separate measures based on the quadratic loss function, the missing longitudinal information criterion (MLIC) and the MLIC for correlation (MLICC), for selection of marginal mean regression and correlation structures in WGEE, respectively. Another option for marginal model selection under this scenario is the weighted quasi-likelihood information criterion (QICW$_p$) by accommodating the weight matrix into QIC [@platt2013]. Later on, [@gosho2016] proposed QICW$_r$ by modifying the penalty term of QICW$_p$ for selection of both marginal mean and correlation structures. Most recently, [@shen2017] proposed the joint longitudinal information criterion (JLIC) with regards to the joint selection of marginal mean and correlation structures for longitudinal data with missing outcomes and covariates. However, [the aforementioned criteria have the following limitations: 1) ignoring missing data; 2) losing model selection power when different criteria for either marginal mean structure selection or correlation structure selection are implemented; 3) leading to unsatisfactory results in selection rates, particularly when [the sample size is small]{}]{} [@shen2012; @shen2017; @gosho2016].
On the other hand, the empirical likelihood approach by adopting a purely observation-based technique has recently gained more attention due [to the relaxing]{} of parametric [distributional]{} assumption, and literature has already shown its outperformance in regression analysis especially on confidence interval construction [@owen1988; @qin1994; @qin2009]. However, empirical-likelihood-based model selection criteria have not been widely investigated yet. [@eic1995] first proposed the empirical information criterion (EIC), [but pointed out that convergence to a proper solution was not reached in estimation, particularly when the number of estimating equations is larger than the number of parameters.]{} Later, [[@variyath2010] introduced adjusted empirical likelihood criteria, the empirical Akaike information criterion (EAIC) and the empirical Bayesian information criterion (EBIC), to guarantee the existence of a solution.]{} [However]{}, [the computational issue remains if the estimators have bounded support (e.g., a correlation coefficient). [@chen2012] applied empirical likelihood for only the correlation structure selection in GEE under complete longitudinal data and proposed to use plug-in estimators obtained from GEE; however, no theoretical justification of plug-in estimators was provided in their work]{}. To our knowledge, there is little work on empirical-likelihood-based model selection criteria accommodating missing data under the longitudinal framework.
In this paper, [two motivated data applications are provided.]{} One is a large epidemiological study, the Atherosclerosis Risk in Communities (ARIC) study. Systolic blood pressure (SBP), a crucial risk factor for cardiovascular disease (CVD), is of clinical and research interest, and characterizing its longitudinal patterns over time can help for CVD risk prediction and determine relatively more effective treatment or medication [@muntner2015; @parati2013]. [The other one is a study of Schizophrenia disorder.]{} The mean level, as well as visit-to-visit variability on severity measurements, is associated with deficits in emotional processing and functional impairment [@simon2007; @bilderbeck2016], which could reflect drug effectiveness and indicate a strategy for prevention of disease progression. To achieve these clinical objectives, we need to identify the best fitting model among different candidates. Here, we propose two information criteria named a joint empirical Akaike information criterion (JEAIC) and a joint empirical Bayesian information criterion (JEBIC), which can simultaneously select marginal mean and correlation structures in WGEE for longitudinal data with dropout missingness under MAR. The basic strategy is that the empirical-likelihood-based criteria are first established by utilizing parameter estimates from WGEE together with the proposed empirical likelihood, and thus JEAIC and JEBIC can be constructed by incorporating extra penalty terms. [These criteria are easy to implement in statistical software]{}, and [potential computational issues]{} can be avoided because the parameter estimates are obtained directly from WGEE. Also, this work can be extended to accommodate more general missing patterns (i.e., intermittent missingness). For simplicity, we mainly focus on monotone dropout missingness here.
The paper is organized as follows. In Section \[s:method\], we formulate the problem, introduce WGEE and the existing model selection criteria, and then provide the proposed information criteria of JEAIC and JEBIC based on the empirical likelihood. The theoretical justification [for]{} our proposal is granted under certain conditions with detailed proof in [the Supporting Information]{}. In Section \[s:simulation\], we conduct extensive simulations under a variety of scenarios with continuous and categorical outcomes [to evaluate the performance of the two proposed criteria when compared with the current existing alternatives]{}. Lastly, we illustrate the application of our scheme by utilizing two real data examples in Section \[s:example\], and conclude with a discussion in Section \[s:discussion\].
Methodology {#s:method}
===========
Notation
--------
Let $\bfY_i = (Y_{i1},\dots, Y_{iT})^\prime$ and $\bfX_i = (\bfX_{i1},\dots, \bfX_{iT})^\prime$ [denote the outcomes and covariates]{} collected from subject $i, i=1, \dots, n$, respectively, [where $Y_{ij}$ is the $j^{th}$ outcome and a $p \times 1$ vector of covariates $\bfX_{ij}$ includes the intercept,]{} $j = 1,\dots, T$. For simplicity, we assume balanced data with equal numbers of observations for all subjects. Let $\bfmu_i = E(\bfY_i |\bfX_i)$ and $\bfV_i = Var(\bfY_i |\bfX_i)$ be [the conditional mean and variance of $\bfY_i$]{}. Note that $\bfmu_i$ is usually modeled [as]{} $\xi(\bfmu_i)=\bfX_i \bfbeta$ with $\xi$ as a known and pre-specified link function depending on the type of outcomes and $\bfbeta$ as a $p \times 1$ vector of regression parameters [@mccullagh1989]. In addition, $\bfV_i$ can be written by $\bfA^{1/2}_i \bfC_i(\bfrho) \bfA^{1/2}_i$, where the matrix $\bfA_i$ is a $T \times T$ diagonal matrix with diagonal elements $var(Y_{it} |\bfX_{it})= \phi \nu (\mu_{it})$, where $\nu$ is a known function, and $\phi$ is a dispersion parameter which could be known or has to be estimated if unknown; $\bfC_i (\bfrho)$ is a pre-specified “working" correlation matrix depending on a set of parameters $\bfrho$. Here, we consider the outcomes subject to missingness under the assumption of MAR, [where the indicator $R_{ij}=1$ for the observed $Y_{ij}$ and $R_{ij}=0$, otherwise]{}. For simplicity, we focus on dropout missingness, but it can be straightforwardly extended to accommodate other general missing patterns [@robins1995; @shen2017].
WGEE {#section 2.2}
----
For longitudinal data with dropouts under MAR, WGEE has been proposed by incorporating a weight matrix based on the inverse probability of observing the outcomes to [adjust for the missing mechanism]{} [@robins1995]. Let the probability of observing the outcome for the $i^{th}$ subject as $\bfomega_{i}=(\omega_{i1}, \ldots, \omega_{iT})^\prime$, where $\omega_{ij}=Pr(R_{ij}=1| \bfY_i, \bfH_i)$ [with $\bfH_{i}$ including potential predictors which could be overlapped with $\bfX_i$.]{} Note that $\omega_{ij}=\lambda_{i1} \times \lambda_{i2}\times \dots \times \lambda_{ij}$ where $\lambda_{i1}=1$ (the outcomes at baseline are all observed) and $\lambda_{ij}=Pr(R_{ij}=1|R_{i,j-1}=1, \bfY_i, \bfH_{i}), j=2, \dots, T$. Given the data $(R_{ij}, \bfY_i, \bfH_{i})$, $\lambda_{ij}$ can be estimated based on the partial likelihood from a logistic regression, $\sum_{i=1}^{n} \sum_{j=2}^{T} R_{i,j-1}log[\lambda_{ij}(\bftheta)^{R_{ij}} \{1-\lambda_{ij}(\bftheta)\}^{1-R_{ij}} ]$, where $\bftheta$ is a $q \times 1$ vector of regression parameters with consistent estimates obtained by $$\label{ees}
\bfS_{n\bftheta}=\frac{1}{n}\sum_{i=1}^{n}\bfs_i(\bftheta)=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=2}^{T}R_{i,j-1}\Big\{R_{i,j}-\lambda_{ij}(\bftheta)\Big\}\bfH_{ij},$$ with $\text{logit}\big(\lambda_{ij}(\bftheta)\big)=\bfH_{ij}^\prime\bftheta$. Thus, the predicted probability $\widehat{\lambda}_{ij}$ and thereafter $\widehat{\omega}_{ij}$ can be calculated. After plugging $\widehat{\bfomega}$ into $\bfW_i$, the estimating equations for the parameters $\bfbeta$ are $$\label{eebeta}
g(\bfbeta)=\sum_{i=1}^{n} g(\bfX_i, \bfY_i, \bfbeta; \widehat{\bfomega})=\sum_{i=1}^{n}\bfD_i^{\prime} \bfV_i^{-1} \bfW_i(\bfY_i-\bfmu_i)=0,$$ where $\bfD_i=\partial \bfmu_i/\partial \bfbeta^{\prime}$ which is a $T \times p$ matrix, $\bfV_i=\bfA_i^{1/2}\bfC_i \bfA_i^{1/2}$, and $\bfW_i$ is the weight matrix with diagonal elements $R_{ij}/\widehat{\omega}_{ij},j=1, \dots, T$. The estimate $\widehat{\bfbeta}$ is consistent even if the “working" correlation matrix is misspecified, and $\sqrt{n}(\widehat{\bfbeta}-\bfbeta)$ is asymptotically normal distributed under mild regulatory conditions, given that the dropout model is correctly specified (i.e., $E \bfW_i=\bfI_T$, with $\bfW_i$ evaluated at the true value $\bfomega_0$) [@robins1995].
Note that given any pre-specified “working" correlation matrix $\bfC$ other than an independent correlation structure, the correlation coefficient $\bfrho$ needs to be estimated. Usually, the correlation estimates can be obtained based on an iterative process by utilizing the Pearson residuals [@wedderburn1974]. But, the correlation coefficient estimate for the longitudinal data with missing outcomes could be biased, while the unbiased estimate for $\rho_{jk}$ is $\widehat{\rho}_{jk}(\widehat{\bfbeta})=[\{1/\{(n-p)\phi\}]\sum_{i=1}^{n}{e_{ij}(\widehat{\bfbeta})e_{ik}(\widehat{\bfbeta})R_{ij}R_{ik}/\widehat{\omega}_{i,jk}} $ where $\widehat{\omega}_{i,jk}$ is the estimate of $\omega_{i,jk}=Pr(R_{ij}=1, R_{ik}=1| \bfY_i, \bfH_{ij}, \bfH_{ik})$ and $e_{ij}(\bfbeta)$ is the residual $(Y_{ij}-\mu_{ij})/\sqrt{\nu(\mu_{ij}})$ ($1\leq j<k\leq T$). Because of dropout missingness, the weights can be simplified as $\omega_{i,jk}=\omega_{ik}=Pr(R_{ik}=1| \bfY_i, \bfH_{ik})$ and then $\widehat{\rho}_{jk}(\widehat{\bfbeta})=[1/\{(n-p)\phi\}]\sum_{i=1}^{n}{e_{ij}(\widehat{\bfbeta})e_{ik}(\widehat{\bfbeta})R_{ik}/\widehat{\omega}_{ik}}$; [For other missing patterns (i.e., intermittent), the estimation would become more complicated [@robins1995; @chen2010].]{} In addition, $\phi$ is assumed to be known or estimated as $\widehat{\phi}(\widehat{\bfbeta})=\{1/(nT-p)\}\sum_{i=1}^{n}\sum_{j=1}^{T}e^2_{ij}(\widehat{\bfbeta})R_{ij}/\widehat{\omega}_{ij}$ (released afterwards for mathematical simplicity). For convenient notation, we stack the estimating equations by subject $i$ for the parameters $\bfgamma=(\bfbeta^\prime, \bfrho^\prime)^{\prime}$ as follows $$\begin{aligned}
\label{wgee}
\bfg\Big(\bfX_i, \bfY_i, \bfgamma; \widehat{\bfomega}_i\Big)=&
\begin{pmatrix}
\bfD_i^\prime \bfV_i^{-1}\bfW_i\Big\{\bfY_i-\bfmu_i(\bfbeta)\Big\}\\
\bfzeta(\bfX_i,\bfY_i, \bfrho; \widehat{\bfomega}_i)
\end{pmatrix},
\end{aligned}$$ where $\bfzeta(\bfX_i,\bfY_i, \bfrho; \widehat{\bfomega}_i)$ is some estimating equation for the correlation coefficients $\bfrho$ based on weighted Pearson residuals. Taking an unstructured case for example, $\bfzeta(\bfX_i,\bfY_i, \bfrho; \widehat{\bfomega}_i)$ could be $\bfkappa_i(\bfbeta)-\bfrho\phi(1-p/n)$, where $\bfkappa_i(\bfbeta)=\big(\widehat{\rho}_{i12}(\bfbeta), \ldots, \widehat{\rho}_{i1T}(\bfbeta), \ldots, \widehat{\rho}_{i(T-1)T}(\bfbeta)\big)^{\prime}$ with $\widehat{\rho}_{ijk}(\bfbeta)=e_{ij}(\bfbeta)e_{ik}(\bfbeta)R_{ik}/\widehat{\omega}_{ik}, 1\leq j<k\leq T$, and $\bfrho=\big(\rho_{12},\cdots,\rho_{1T},\cdots,\rho_{(T-1)T}\big)^\prime$.
Model selection criteria {#section 2.3}
------------------------
### Overview of Existing Criteria {#section2.3.1}
Before introducing our proposed information criteria, we first [conduct a literature review of several key criteria on model selection for WGEE in longitudinal data analysis, with dropout missingness under MAR.]{} One called MLIC was proposed for the selection on marginal mean regression by [@shen2012], which is based on the expected quadratic loss function and modifies Mallows’s $C{_p}$ statistics (in linear regression). Given the estimates $\widehat{\bfgamma}=(\widehat{\bfbeta}^\prime, \widehat{\bfrho}^\prime)^{\prime}$ and $\widehat{\bfomega}$, MLIC is calculated by $$\begin{aligned}
MLIC=\sum_{i=1}^{n} (\bfY_i-\widehat{\bfmu}_i)^\prime \bfW_i (\bfY_i-\widehat{\bfmu}_i)+2Tr(\bfE_n^{-1} \bfJ_n),
\end{aligned}$$ where $\bfE_n=\sum_{i=1}^{n}\bfD_i^\prime \bfV_i^{-1} \bfW_i \bfD_i$ and $\bfJ_n=\sum_{i=1}^{n}(\bfD_i^\prime \bfV_i^{-1} \bfepsilon_i \bfepsilon_i^\prime -\bfG_i \bfepsilon_i^\prime)\bfD_i$ with $\bfepsilon_i=\bfW_i (\bfY_i-\bfmu_i^{0})$ and $\bfG_i=(\sum_{m=1}^{n}\bfQ_m \bfs_m^\prime)(\sum_{m=1}^{n}\bfs_m \bfs_m^\prime)^{-1} \bfs_i$ where $\bfQ_i=\bfD_i^\prime \bfV_i^{-1} \bfW_i(\bfY_i-\widehat{\bfmu}_i)$ and $\bfs_i$ is the score component of the $i^{th}$ individual in the partial likelihood for the dropout model in (\[ees\]). [Note that $\bfmu_i^{0}$ is estimated]{} by the largest candidate model based on the collected information, and numerical studies via simulation have shown that the misspecification of this model has mild or negligible influence on the performance of MLIC. In addition, [[@shen2012] also provided]{} MLICC for correlation structure selection by modifying the penalty term.
Another commonly used criterion for such context is QICW$_r$ [@gosho2016], which is extended from regular QIC by incorporating the inverse probability weight matrix. Given the estimates $\widehat{\bfgamma}=(\widehat{\bfbeta}^\prime, \widehat{\bfrho}^\prime)^{\prime}$ and $\widehat{\bfomega}$, the QICW$_r$ statistic is provided as $$\begin{aligned}
QICW_r=-2 \sum_{i=1}^{n} \sum_{j=1}^{T} Q_w(\widehat{\bfbeta}, \widehat{\bfomega}; \bfY_{i}, \bfX_{i}, \bfH_{i})+2 Tr(\widehat{\bfPhi}_{I}\widehat{\bfV}_w),
\end{aligned}$$ where $Q_w(\widehat{\bfbeta}, \widehat{\bfomega}; \bfY_{ij}, \bfX_{i}, \bfH_{i})$ is the weighted log quasi-likelihood function under an independence correlation structure, and $\widehat{\bfPhi}_{I}=-\sum_{i=1}^{n} \sum_{j=1}^{T}(\partial^2 Q_w/\partial \bfbeta \partial \bfbeta ^\prime) \mid_{\bfbeta=\widehat{\bfbeta}}$.
### Proposed Criteria of JEAIC and JEBIC {#section2.3.2}
To begin with, we first propose the full weighted estimating equation $\bfG_F$ by accommodating a stationary correlation structure for the empirical likelihood, which is given by $$\begin{aligned}
\label{full}
\bfG_F\Big(\bfX_{Fi},\bfY_i,\widetilde{\bfbeta},\bfrho^c, \bftheta \Big)=&
\begin{pmatrix}
\bfD_i^\prime \bfV_i^{-1}\bfW_i\big\{\bfY_i-\bfmu_i(\widetilde{\bfbeta})\big\}\\
\bfU_i(\widetilde{\bfbeta})-\boldsymbol{h}(\bfrho^c)\phi\\
\bfs_{i}(\bftheta)
\end{pmatrix},
\end{aligned}$$ where $\bfs_i(\bftheta)$ is the estimating equation for $\bftheta$ in (\[ees\]). Notation $\widetilde{\bfbeta}\in {{\mathcal R}}^L$ in $\bfG_F$ denotes a vector of parameters with the same dimensionality as $\bfbeta_F\in {{\mathcal R}}^L$ from our proposed full mean structure with $\bfX_{Fi}$ as the covariates for the $i^{th}$ subject. [Without loss of generality]{}, we can always rearrange the covariate matrix $\bfX_{Fi}$ so that the first $p-$dimensional vector in $\widetilde{\bfbeta}$ [equals the parameter]{} vector $\bfbeta$ from the candidate model, and the remaining elements in $\widetilde{\bfbeta}$ [equal zeros]{}, thus $\widetilde{\bfbeta}=(\bfbeta^\prime, {\bf 0}^\prime)^{\prime}$. In addition, a stationary correlation structure is proposed for the full WGEE to estimate correlation coefficients, i.e., $\bfrho_F^{ST}=(\rho_1^{\text{ST}},\ldots,\rho_{T-1}^{ST})^{\prime}$, $\bfU_i(\widetilde{\bfbeta})=\big(U_{i1}(\widetilde{\bfbeta}), U_{i2}(\widetilde{\bfbeta}),\ldots, U_{i(T-1)}(\widetilde{\bfbeta})\big)^\prime$ with $U_{im}(\widetilde{\bfbeta})=\sum_{j=1}^{T-m}(R_{i,j+m}/\omega_{i,j+m})e_{ij}(\widetilde{\bfbeta})e_{i,j+m}(\widetilde{\bfbeta}).$ Also, for any pre-specified correlation structure denoted by the superscript $c$ (nested within a stationary correlation structure), $\boldsymbol{h}(\bfrho^c)=\Big(\rho_1^c\big(T-1-p/n\big),\ldots,\rho_{T-1}^c(1-p/n)\Big)^\prime$ with $\bfrho^c=(\rho^c_1,...,\rho^c_{T-1})^\prime\in {{\mathcal R}}^{T-1}$. For instance, $\bfrho^{EXC}=(\rho^{\text{EXC}},...,\rho^{\text{EXC}})^{\prime}$ when [an exchangeable (EXC) correlation structure]{} is fitted. Here, we consider a stationary correlation structure for the proposed full model; however, it [can]{} be extended to a more general case (i.e., unstructured), [which may substantially increase the number of parameters needing estimation, and thus likely lead to convergence issues particularly for small $n$ and relatively large $T$]{}.
Combining all the information above, we thus have the following empirical likelihood ratio, which is the key component to select marginal mean and correlation structures: $$\label{el}
R^F(\bfbeta,\bfrho^c,\bftheta)=\sup_{\bfbeta,\bfrho^c,\bftheta}\left\{ \prod_{i=1}^{n} np_i; p_i>0, \sum_{i=1}^{n}p_i=1, \sum_{i=1}^{n}p_i\bfG_F\big(\bfX_{Fi},\bfY_i,\widetilde{\bfbeta},\bfrho^c, \bftheta\big)=0 \right\},$$ where $p_i=P(\bfY=\bfy_i,\bfX=\bfx_i)$. Here, we assume that only the distributions with an atom of probability on each $\bfy_i$ and $\bfx_i$ have nonzero likelihood. [Therefore, $\{p_i\}$’s will follow the rule of traditional probability with the sum equal to one.]{} [Without imposing constraints defined by the estimating equations]{}, $\prod_{i=1}^{n} p_i$ is maximized as $\prod_{i=1}^{n} (1/n)$. Thus, the empirical likelihood ratio is defined as $\prod_{i=1}^{n} np_i$. [More basic properties about empirical likelihood can be found in [@owen2001]]{}. An intuitive rationale of model selection based on proposed empirical likelihood ratio is as follows: when the estimators $\widehat{\bfbeta}_F$, $\widehat{\bfrho}^{ST}_F=(\widehat{\rho}_1^{\text{ST}},...,\widehat{\rho}_{T-1}^{ST})^{\prime}$ are obtained from [the WGEE method]{} with $\bfX_{Fi}$ and a stationary correlation structure from (\[wgee\]), and $\widehat{\bftheta}$ is calculated from (\[ees\]), we will have $R^F(\widehat{\bfbeta}_F, \widehat{\bfrho}^{ST}_F, \widehat{\bftheta})=1$, which achieves the upper limit of the empirical likelihood ratio. However, the estimators $\widehat{\bfbeta}$ and $\widehat{\bfrho}^c$ other than $\widehat{\bfbeta}_F$ and $\widehat{\bfrho}^{ST}_F$ [will lead to]{} $R^F(\widehat{\bfbeta}, \widehat{\bfrho}^c, \widehat{\bftheta})<1$. The departure from 1 indicates the misspecification of the model to the degree reflected by the magnitude of the deviation. In other words, [the closer the mean and correlation structures approach the underlying true values, the closer $R^F$ will approach $1$]{}, which ensures the potential for joint selection of marginal mean and correlation structures.
Thereafter, by plugging the parameter estimates $(\widehat{\bfbeta}^\prime,\widehat{\bfrho}^{c\prime})^\prime$ from a candidate model in WGEE (\[wgee\]) and $\widehat{\bftheta}_{ML}$ obtained based on the estimating equation (\[ees\]) into $R^F(\widehat{\bfbeta},\widehat{\bfrho}^c, \widehat{\bftheta}_{ML})$, the empirical likelihood ratio is [the solution of the following equation by utilizing the Lagrange multiplier method [@owen2001],]{} $$\label{elr}
-2\log R^F(\widehat{\bfbeta},\widehat{\bfrho}^c,\widehat{\bftheta}_{ML})=2\sum_{i=1}^{n}\log\big\{1+\boldsymbol{\lambda}^\prime\bfG_F(\bfX_i,\bfY_i,\widehat{\widetilde{\bfbeta}},\widehat{\bfrho}^c, \widehat{\bftheta}_{ML})\big\},$$ where the parameter $\boldsymbol{\lambda}$ can be solved by applying the Newton-Raphson method based on $$\label{lambda}
\sum_{i=1}^{n}\frac{\bfG_F(\bfX_i,\bfY_i,\widehat{\widetilde{\bfbeta}},\widehat{\bfrho}^c, \widehat{\bftheta}_{ML})}{1+\boldsymbol{\lambda}^\prime\bfG_F(\bfX_i,\bfY_i,\widehat{\widetilde{\bfbeta}},\widehat{\bfrho}^c, \widehat{\bftheta}_{ML})}=0.$$
Thus, for longitudinal data with dropout missingness under MAR, our proposed information criteria are defined by $$\begin{split}
\text{JEAIC}&=-2\log R^F(\widehat{\bfbeta},\widehat{\bfrho}^c, \widehat{\bftheta}_{ML})+2\widetilde{p},\\
\text{JEBIC}&=-2\log R^F(\widehat{\bfbeta},\widehat{\bfrho}^c, \widehat{\bftheta}_{ML})+\widetilde{p}\log n,
\end{split}$$ where $\widetilde{p}$ denotes the total number of parameters. The asymptotic property of our proposed information criteria can be evaluated based on the existing work. In particular, in the work by [@eic1995], EIC has been proved to be an asymptotically unbiased estimate that is proportional to the expected Kullback-Leibler distance [between two discrete empirical distributions]{}. Also, [@variyath2010] evaluated the consistency of EBIC. [In both of their works, general estimating equations are considered, but it is straightforward to embed our proposed full estimating equations (\[full\]) into their theoretical framework when the empirical likelihood estimators are utilized.]{} [However, our proposed approach is built upon the plug-in estimators, thus, it is important to assess the asymptotic proprieties of these plug-in estimators and their relationship with the empirical likelihood estimators. ]{}
### Asymptotic Properties of Plug-in Estimators
In this section, we will investigate the asymptotic properties of our plug-in estimators under MAR, and explain why we advocate [such an alternative]{}. First, we investigate the asymptotic behavior of estimators $\widehat{\bfbeta}_{EL}$, $\widehat{\bfrho}^c_{EL}$, and $\widehat{\bftheta}_{EL}$ from maximizing the profile empirical likelihood ratio. Inspired by [@qin1994] and [@qin2009], we derive the asymptotic properties of the estimator shown in Theorem \[theorem 1\] with the proof sketched in [the Supporting Information]{}.
\[theorem 1\] Let us denote $$\begin{split}
&\bfg_F(\bfX_i, \bfY_i, \widetilde{\bfbeta},\bfrho^c, \bftheta)=
\begin{pmatrix}
\bfD_i^\prime \bfV_i^{-1}\bfW_i\Big\{\bfY_i-\bfmu_i(\widetilde{\bfbeta})\Big\}\\
\bfU_i(\bfbeta)-\boldsymbol{h}(\bfrho^c)\phi
\end{pmatrix}, \\
&\big(\widehat{\bfbeta}^\prime_{EL}, \widehat{\bfrho}^{c\prime}_{EL}, \widehat{\bftheta}^\prime_{EL}\big)^\prime =\operatorname*{arg\,max}_{\bfbeta,\bfrho^c,\bftheta} R^F(\bfbeta,\bfrho^c,\bftheta), ~\text{and}~ \bfs=\bfs_i(\bftheta).
\end{split}$$ Under the conditions specified in [the Supporting Information]{} and given $\bfgamma=(\bfbeta^\prime,\bfrho^{c\prime})^\prime$ and $\bftheta$ with corresponding true values $\bfgamma_0$ and $\bftheta_0$, we have
(1) $$\label{taylor}
\left(\begin{array}{c}
\widehat{\bfgamma}_{EL}-\bfgamma_0\\ \widehat{\bftheta}_{EL}-\bftheta_0 \end{array}\right)=
\left(\begin{array}{c}
-\bfV_{\ast}\bfA_{\ast}\bfQ_{n}^\ast\\ \bfOmega\bfS_{n\bftheta} \end{array}\right)
+o_p(\textbf{n}^{-\frac{1}{2}}),$$
where $\bfS_{n\bftheta}$ is defined in (\[ees\]), and $$\begin{split}
\bfV_\ast=&\Bigg[E\big(\frac{\partial \bfg_{F} }{\partial\bfgamma^\prime} \big)^\prime \Big\{E\bfg_{F}\bfg_{F}^\prime-E\big(\frac{\partial \bfg_{F} }{\partial\bftheta^\prime} \big) \big(E\bfs\bfs^\prime\big)^{-1}E\big(\frac{\partial \bfg_F }{\partial\bftheta^\prime}\big)^\prime\Big\}^{-1}E\big(\frac{\partial \bfg_F }{\partial\bfgamma^\prime} \big)\Bigg]^{-1},\\
\bfA_\ast=&E\big(\frac{\partial \bfg_{F} }{\partial\bfgamma^\prime} \big)^\prime \Big\{E\bfg_{F}\bfg_{F}^\prime-E\big(\frac{\partial \bfg_{F} }{\partial\bftheta^\prime} \big) \big(E\bfs\bfs^\prime\big)^{-1}E\big(\frac{\partial \bfg_F }{\partial\bftheta^\prime}\big)^\prime\Big\}^{-1}, \\
\bfQ^\ast_{n}=&\frac{1}{n} \sum_{i=1}^{n} \bfg_F(\bfX_i, \bfY_i, \widetilde{\bfbeta},\bfrho^c, \bftheta)+E\big(\frac{\partial \bfg_F }{\partial\bftheta^\prime} \big)^\prime E(\bfs\bfs^\prime)^{-1}\bfS_{n\bftheta}, ~ \bfOmega=\big(E\bfs\bfs^\prime\big)^{-1}.
\end{split}$$
(2) Furthermore, the asymptotic normality can be derived from (\[taylor\]) $$\sqrt{n}\left(\begin{array}{c}
\widehat{\bfgamma}_{EL}-\bfgamma_0\\ \widehat{\bftheta}_{EL}-\bftheta_0 \end{array}\right) \overset{d}{\to}
\bfN\left(\left(\begin{array}{c}
\bf0\\ \bf0 \end{array}\right), \left(\begin{array}{cc}
\bfSigma_{11} & \bf0\\ \bf0 & \bfSigma_{22} \end{array}\right)\right),$$ with $\bfSigma_{11}=\bfV_{\ast}\bfA_{\ast}\textbf{Cov}(\bfQ_{n}^\ast)\bfA_{\ast}^\prime\bfV_{\ast}^\prime$, $\bfSigma_{22}=\bfOmega\textbf{Cov}(\bfS_{n\bftheta})\bfOmega^\prime$.
(3) [$-2\log R^F(\widehat{\bfbeta}_{EL}, \widehat{\bfrho}^c_{EL}, \widehat{\bftheta}_{EL})$ follows a $\chi^2$ distribution with $\widetilde{L}-\widetilde{p}$ degrees of freedom where $\widetilde{L}$ is the number of estimating equations in (\[full\])]{} and $\widetilde{p}$ as the total number of parameters.
An interesting finding from Theorem \[theorem 1\] is that the empirical-likelihood-based estimator $\widehat{\bftheta}_{EL}$ is asymptotically equivalent to the estimator $\widehat{\bftheta}_{ML}$ from partial likelihood in (\[ees\]) since they have the same influence function. Also, the estimator $\widehat{\bftheta}_{EL}$ [is asymptotically independent of]{} the estimator $\widehat{\bfgamma}_{EL}$ by Theorem \[theorem 1\] (II). Thus, we can substitute $\widehat{\bftheta}_{ML}$ [in]{} $R^F(\bfbeta,\bfrho^c,\bftheta)$ first and then estimate $\bfgamma$ [by maximizing]{} $R^F(\bfgamma;\widehat{\bftheta}_{ML})$, [by which means, the estimator is asymptotically equivalent to the estimator $\widehat{\bfgamma}_{EL}$, thus we keep this notation for this context.]{} Such plug-in method can definitely decrease the dimensionality of parameters for estimation by only focusing on $\bfgamma$, and thus [reducing the computational]{} burden in particular when the dimension of $\bftheta$ is relatively large.
However, maximizing $R^F(\bfgamma;\widehat{\bftheta}_{ML})$ to estimate $\bfgamma$ still raises [computational issues]{} since the number of the estimating equations may exceed the number of parameters, which requires 0 to be inside the convex hull of data to guarantee the existence of solution [@chen2012; @variyath2010]. Furthermore, the bounded support of correlation coefficients also increases the difficulty among the existing algorithms. Instead, we advocate to substitute the empirical likelihood estimators $\widehat{\bfgamma}_{EL}$ in $R^F(\widehat{\bfgamma}_{EL}; \widehat{\bftheta}_{ML})$ with the estimators [from a candidate model fitting in WGEE]{} (\[wgee\]), which can [avoid computational issues]{} and ensure convenient application. Here, we investigate the asymptotic relationship between the WGEE and empirical-likelihood-based estimators, which is summarized in the following theorem:
\[theorem 2\] Under Theorem \[theorem 1\] and the conditions provided in [the Supporting Information]{}, the estimates $\widehat{\bfgamma}_{EL}=(\widehat{\bfbeta}^\prime_{EL},\widehat{\bfrho}^{c\prime}_{EL})^\prime$ from empirical likelihood based on (\[el\]) and $\widehat{\bfgamma}=(\widehat{\bfbeta}^\prime,\widehat{\bfrho}^{c\prime})^{\prime}$ based on WGEE (\[wgee\]) are asymptotically equivalent.
The proofs for exchangeable and AR1 scenarios are provided in [the Supporting Information]{}. Theorem \[theorem 2\] implies that the WGEE estimator is a reasonable approximation of the empirical likelihood estimator under certain conditions, indicating that any asymptotic properties induced by the empirical likelihood estimator would be reasonably invoked by the WGEE estimator. More discussion on conditions is referred to [the Supporting Information]{}.
Simulation studies {#s:simulation}
==================
In this section, we investigate the numerical performance of our proposed criteria under various settings, and compare with several existing criteria such as MLIC and QICW$_r$ as well as the most recent work of JLIC. We expect better performance of the two proposed criteria compared to the existing alternatives. [In addition]{}, JEBIC might have better control of false positive rates than JEAIC under relatively large sample sizes [@variyath2010].
Our first scenario considers binary outcomes, and the true marginal mean structure is $$\log\big(\frac{\mu_{ij}}{1-\mu_{ij}}\big)=\beta_0+x_{i1}\beta_1+x_{ij2}\beta_2, ~\text{for } i=1,...,n, j=1,...,T,$$ where $x_{i1}$ is the subject (cluster) level covariate generated from [the uniform distribution over $[0,1]$]{} and $x_{ij2}=j-1$ is a time-dependent covariate. The number of observations (i.e., cluster size) is $T=3$. The true parameter vector $\bfbeta=(\beta_0, \beta_1, \beta_2)^{\prime}$ in the marginal mean is $(-1, 1, 0.4)^{\prime}$. The true correlation structure is exchangeable with a correlation coefficient $\rho_0=0.5$. The dropout model is $$\log\big(\frac{\lambda_{ij}}{1-\lambda_{ij}}\big)=\theta_0+y_{i(j-1)}\theta_1+h_{ij}\theta_2, ~\text{for } i=1,...,n, j=2,...,T,$$ [where the covariate $h_{ij}$ is uniformly distributed over $[-0.5,0.5]$]{}. Different choices for the parameters $\bftheta=(\theta_0, \theta_1, \theta_2)^{\prime}$ can ensure the missing probability (denoted by $m$) around $0.2$ and $0.3$, i.e., $\bftheta=(1.74, 0.5, -0.8)^{\prime}$ is for $m=0.2$ and $\bftheta=(1.05, 0.5, -0.8)^{\prime}$ is for $m=0.3$.
[In the first scenario, we consider a correctly specified dropout model.]{} Then, we also evaluate the robustness of our proposal when [the dropout model is misspecified because of the left out variable $h_{ij}$ in the regression [@shen2017].]{} In addition, we generate one redundant variable [$x_{ij3} \sim N(0,1)$]{}. The full model considered for our proposed criteria as well as MLIC/MLICC includes three variables, $x_{i1}$, $x_{ij2}$ and $x_{ij3}$. Six potential marginal mean structures are considered with three types of “working" correlation structures (i.e., exchangeable (EXC), AR1 and Independence (IND)) for model fitting. To [summarize the simulation results]{}, 500 Monte Carlo data sets with sample size $n=100, 200$ are generated for each scenario, and the selection rate for each combination of marginal mean and correlation structures is reported. Moreover, we also consider the scenarios with Gaussian outcomes, the ones [where]{} the assumption of MAR is violated, and also the ones with redundant variables. Due to limited space, we cannot show all these results here, but provide them in [the Supporting Information]{}.
[On the other hand]{}, to compare our proposal with JLIC, we consider the same set-ups (with binary and Gaussian outcomes) in [@shen2017] by utilizing their supporting program functions for simulations. The detailed information on parameter set-ups is not provided here but can be referred to [@shen2017]. All the simulations are conducted in R and MATLAB software.
Setups Method $\bfC(\bfrho)$ $x_1$ $x_3$ $\bfx_1,\bfx_2$ $x_1,x_3$ $x_2,x_3$ $x_1,x_2,x_3$ Total
-------- ---------- ---------------- ------- ------- ----------------- ----------- ----------- --------------- -----------
n=100 JEAIC AR1 0.004 0 0.082 0 0.016 0.006 0.108
m=0.2 **EXC** 0.026 0.008 **0.578** 0.002 0.186 0.092 **0.892**
IND 0 0 0 0 0 0 0
Total 0.03 0.008 **0.66** 0.002 0.202 0.098 1
JEBIC AR1 0.02 0.004 0.072 0 0.014 0 0.11
**EXC** 0.09 0.028 **0.566** 0.002 0.2 0.004 **0.89**
IND 0 0 0 0 0 0 0
Total 0.11 0.032 **0.638** 0.002 0.214 0.004 1
MLIC AR1 0.008 0.008 0.2 0.002 0.14 0.06 0.418
**EXC** 0.008 0.008 **0.28** 0.004 0.168 0.068 **0.536**
IND 0.004 0 0.018 0 0.016 0.008 0.046
Total 0.02 0.016 **0.498** 0.006 0.324 0.136 1
QICW$_r$ AR1 0 0 0.062 0 0.038 0.04 0.14
**EXC** 0.006 0.004 **0.436** 0.002 0.236 0.112 **0.796**
IND 0 0 0.03 0 0.02 0.014 0.064
Total 0.006 0.004 **0.528** 0.002 0.294 0.166 1
n=100 JEAIC AR1 0.01 0.002 0.102 0 0.03 0.016 0.16
m=0.3 **EXC** 0.042 0.026 **0.472** 0.004 0.198 0.098 **0.84**
IND 0 0 0 0 0 0 0
Total 0.052 0.028 **0.574** 0.004 0.228 0.114 1
JEBIC AR1 0.038 0.014 0.082 0 0.028 0.002 0.164
**EXC** 0.126 0.066 **0.44** 0.002 0.188 0.014 **0.836**
IND 0 0 0 0 0 0 0
Total 0.164 0.08 **0.522** 0.002 0.216 0.016 1
MLIC AR1 0.01 0.01 0.164 0 0.106 0.064 0.354
**EXC** 0.036 0.026 **0.29** 0.002 0.174 0.06 **0.588**
IND 0.002 0.004 0.028 0.002 0.014 0.008 0.058
Total 0.048 0.04 **0.482** 0.004 0.294 0.132 1
QICW$_r$ AR1 0.002 0.002 0.05 0.002 0.028 0.03 0.114
**EXC** 0.008 0.006 **0.452** 0.002 0.232 0.136 **0.836**
IND 0 0 0.026 0 0.012 0.012 0.05
Total 0.01 0.008 **0.528** 0.004 0.272 0.178 1
n=200 JEAIC AR1 0 0 0.034 0 0.008 0.008 0.05
m=0.2 **EXC** 0 0 **0.73** 0 0.096 0.124 **0.95**
IND 0 0 0 0 0 0 0
Total 0 0 **0.764** 0 0.104 0.132 1
JEBIC AR1 0 0 0.042 0 0.012 0 0.054
**EXC** 0.01 0 **0.806** 0 0.114 0.016 **0.946**
IND 0 0 0 0 0 0 0
Total 0.01 0 **0.848** 0 0.126 0.016 1
MLIC AR1 0 0 0.23 0 0.064 0.068 0.362
**EXC** 0.002 0 **0.392** 0 0.082 0.098 **0.574**
IND 0.002 0 0.036 0 0.006 0.02 0.064
Total 0.004 0 **0.658** 0 0.152 0.186 1
QICW$_r$ AR1 0 0 0.056 0 0.012 0.02 0.088
**EXC** 0 0 **0.56** 0 0.114 0.168 **0.842**
IND 0 0 0.04 0 0.002 0.028 0.07
Total 0 0 **0.656** 0 0.128 0.216 1
n=200 JEAIC AR1 0 0 0.066 0 0.014 0.008 0.088
m=0.3 **EXC** 0.006 0 **0.646** 0 0.132 0.128 **0.912**
IND 0 0 0 0 0 0 0
Total 0.006 0 **0.712** 0 0.146 0.136 1
JEBIC AR1 0.002 0 0.074 0 0.014 0 0.09
**EXC** 0.038 0.004 **0.704** 0.002 0.152 0.01 **0.91**
IND 0 0 0 0 0 0 0
Total 0.04 0.004 **0.778** 0.002 0.166 0.01 1
MLIC AR1 0.002 0 0.214 0.002 0.056 0.056 0.33
**EXC** 0.002 0.002 **0.386** 0 0.124 0.098 **0.612**
IND 0 0 0.03 0 0.01 0.018 0.058
Total 0.004 0.002 **0.63** 0.002 0.19 0.172 1
QICW$_r$ AR1 0 0 0.066 0 0.006 0.03 0.102
**EXC** 0 0 **0.554** 0 0.118 0.18 **0.852**
IND 0 0 0.018 0 0.004 0.024 0.046
Total 0 0 **0.638** 0 0.128 0.234 1
: Performance of JEAIC and JEBIC compared with MLIC and QICW$_r$: Percentage of selecting six candidate logistic models across 500 Monte Carlo datasets; $T=3$, $\rho=0.5$. The model with $\{x_1,x_2\}$ and [an EXC]{} correlation structure is the true model. Notation $n$ and $m$ [denote the sample size and the missing probability]{}, respectively.[]{data-label="table1"}
\[table2\]
Setups Method $\bfC(\bfrho)$ $x_1$ $x_3$ $\bfx_1,\bfx_2$ $x_1,x_3$ $x_2,x_3$ $x_1,x_2,x_3$ Total
-------- ---------- ---------------- ------- ------- ----------------- ----------- ----------- --------------- -----------
n=100 JEAIC AR1 0.002 0.002 0.092 0 0.012 0.008 0.116
m=0.2 **EXC** 0.024 0.01 **0.566** 0.002 0.191 0.09 **0.884**
IND 0 0 0 0 0 0 0
Total 0.026 0.012 **0.659** 0.002 0.203 0.098 1
JEBIC AR1 0.022 0.004 0.084 0 0.012 0 0.122
**EXC** 0.096 0.022 **0.557** 0.002 0.195 0.006 **0.878**
IND 0 0 0 0 0 0 0
Total 0.118 0.026 **0.641** 0.002 0.207 0.006 1
MLIC AR1 0.006 0.012 0.212 0 0.126 0.054 0.41
**EXC** 0.012 0.006 **0.258** 0.006 0.184 0.074 **0.54**
IND 0 0 0.028 0 0.016 0.006 0.05
Total 0.018 0.018 **0.498** 0.006 0.326 0.134 1
QICW$_r$ AR1 0 0 0.056 0 0.042 0.04 0.138
**EXC** 0.008 0.004 **0.436** 0.002 0.232 0.114 **0.796**
IND 0 0 0.034 0 0.02 0.012 0.066
Total 0.008 0.004 **0.526** 0.002 0.294 0.166 1
n=100 JEAIC AR1 0.01 0.002 0.094 0.002 0.032 0.02 0.16
m=0.3 **EXC** 0.046 0.026 **0.484** 0.004 0.194 0.086 **0.84**
IND 0 0 0 0 0 0 0
Total 0.056 0.028 **0.578** 0.006 0.226 0.106 1
JEBIC AR1 0.042 0.012 0.078 0 0.028 0.002 0.162
**EXC** 0.142 0.066 **0.436** 0 0.184 0.01 **0.838**
IND 0 0 0 0 0 0 0
Total 0.184 0.078 **0.514** 0 0.212 0.012 1
MLIC AR1 0.01 0.008 0.154 0.002 0.098 0.046 0.318
**EXC** 0.038 0.032 **0.296** 0.002 0.184 0.066 **0.618**
IND 0.002 0.004 0.028 0 0.016 0.014 0.064
Total 0.05 0.044 **0.478** 0.004 0.298 0.126 1
QICW$_r$ AR1 0.002 0 0.048 0 0.03 0.034 0.114
**EXC** 0.008 0.004 **0.456** 0.004 0.232 0.132 **0.836**
IND 0 0 0.022 0 0.012 0.016 0.05
Total 0.01 0.004 **0.526** 0.004 0.274 0.182 1
n=200 JEAIC AR1 0 0 0.036 0 0.008 0.008 0.052
m=0.2 **EXC** 0 0 **0.726** 0 0.092 0.13 **0.948**
IND 0 0 0 0 0 0 0
Total 0 0 **0.762** 0 0.1 0.138 1
JEBIC AR1 0 0 0.042 0 0.012 0.002 0.056
**EXC** 0.01 0 **0.806** 0 0.11 0.018 **0.944**
IND 0 0 0 0 0 0 0
Total 0.01 0 **0.848** 0 0.122 0.02 1
MLIC AR1 0 0 0.216 0 0.05 0.072 0.338
**EXC** 0 0 **0.404** 0 0.098 0.098 **0.6**
IND 0.002 0 0.032 0 0.006 0.022 0.062
Total 0.002 0 **0.652** 0 0.154 0.192 1
QICW$_r$ AR1 0 0 0.056 0 0.01 0.028 0.094
**EXC** 0 0 **0.558** 0 0.108 0.168 **0.834**
IND 0 0 0.04 0 0.002 0.03 0.072
Total 0 0 **0.654** 0 0.12 0.226 1
n=200 JEAIC AR1 0 0 0.068 0 0.012 0.008 0.088
m=0.3 **EXC** 0.008 0 **0.662** 0 0.132 0.11 **0.912**
IND 0 0 0 0 0 0 0
Total 0.008 0 0.73 0 0.144 0.118 1
JEBIC AR1 0.004 0.002 0.078 0 0.012 0 0.096
**EXC** 0.04 0 **0.698** 0.002 0.154 0.01 **0.904**
IND 0 0 0 0 0 0 0
Total 0.044 0.002 **0.776** 0.002 0.166 0.01 1
MLIC AR1 0.002 0 0.206 0.002 0.054 0.072 0.336
**EXC** 0.004 0.002 **0.374** 0 0.13 0.094 **0.604**
IND 0 0 0.032 0 0.012 0.016 0.06
Total 0.006 0.002 **0.612** 0.002 0.196 0.182 1
QICW$_r$ AR1 0 0 0.064 0 0.004 0.028 0.096
**EXC** 0 0 **0.542** 0 0.124 0.192 **0.858**
IND 0 0 0.018 0 0.004 0.024 0.046
Total 0 0 **0.624** 0 0.132 0.244 1
: Performance of JEAIC and JEBIC compared with MLIC and QICW$_r$ when the dropout model is misspecified: Percentage of selecting six candidate logistic models across 500 Monte Carlo datasets; $T=3$, $\rho=0.3$. The model with $\{x_1,x_2\}$ and [an EXC]{} correlation structure is the true model. Notation $n$ and $m$ denote [denote the sample size and the missing probability]{}, respectively.[]{data-label="mis_dropout"}
In Table \[table1\], We find out that both JEAIC and JEBIC outperform two-stage MLIC/MLICC and QICW$_r$ across different settings. In general, all methods exhibit better selection behaviors if sample size increases or missing probability decreases, but the superiority of our proposal becomes more apparent compared to the other alternatives regarding higher improvement in selection rates. Under relatively small sample size, JEAIC and JEBIC behave similarly on joint model selection, while JEBIC seems more promising under relatively large sample size by imposing more penalty on both parameter number and sample size, which agrees with our expectation [@variyath2010]. On the other hand, the performances of MLIC/MLICC and QICW$_r$ are not satisfactory and consistently stable across different setups despite having slightly better performance as the sample size increases. Similar patterns and selection rates can be found in Table \[table2\] [, which indicates]{} that misspecification of [the dropout model]{} does not have much influence on the performance of our proposed criteria when the MAR assumption still holds.
Moreover, using the same set-ups in the first scenario, we conduct further investigation by only considering marginal mean selection given a pre-specified correlation structure according to the editor’s suggestion. The results, in [the Supporting Information]{}, imply that the misspecified correlation structure would worsen the selection performance. More interestingly, in Table \[table1\], the marginal selection rates, for mean structures (column total) regardless of the correlation structure selection, is comparable or even slightly higher than the Oracle one under which the true correlation structure is specified and fixed for the marginal mean selection. These findings provide further evidence of our joint selection’s advantages; thus, even though the marginal mean structure is the sole interest, the implementation of the joint selection would promise a satisfactory selection rate. Also, the additional simulations provided in [the Supporting Information]{} further indicate the robustness of our proposal when the MAR assumption is violated, and also show the generalization into the cases with different types of outcomes or a relatively large number of redundant predictors in candidate models. Even for the scenarios with relatively higher missing proportions (i.e., $m=0.5$), our proposal is still applicable (results not shown). Overall, our proposed JEAIC and JEBIC outperform the other existing criteria, and JEBIC is highly recommended when the sample size is relatively large in real applications.
Setups Method $\bfC(\bfrho)$ 1 2 **3** 4 5 6 7 8 9 10 total
-------- -------- ---------------- ------- --- ----------- ------- --- ------- ------- --- ------- ------- -----------
m=0.1 JLIC AR1 0 0 0.03 0 0 0.007 0.007 0 0 0.003 0.047
**EXC** 0.006 0 **0.645** 0 0 0.132 0.147 0 0 0.023 **0.953**
IND 0 0 0 0 0 0 0 0 0 0 0
Total 0.006 0 **0.675** 0 0 0.139 0.154 0 0 0.026 1
JEAIC AR1 0 0 0.006 0 0 0.001 0.003 0 0 0.001 0.011
**EXC** 0.002 0 **0.698** 0 0 0.138 0.128 0 0 0.023 **0.989**
IND 0 0 0 0 0 0 0 0 0 0 0
total 0.002 0 **0.704** 0 0 0.139 0.131 0 0 0.024 1
JEBIC AR1 0 0 0.011 0 0 0 0 0 0 0 0.011
**EXC** 0.011 0 **0.952** 0 0 0.017 0.009 0 0 0 **0.989**
IND 0 0 0 0 0 0 0 0 0 0 0
total 0.011 0 **0.963** 0 0 0.017 0.009 0 0 0 1
m=0.2 JLIC AR1 0 0 0.057 0 0 0.011 0.009 0 0 0.001 0.078
**EXC** 0.008 0 **0.63** 0.002 0 0.12 0.137 0 0.025 0 **0.922**
IND 0 0 0 0 0 0 0 0 0 0 0
Total 0.008 0 **0.687** 0.002 0 0.131 0.146 0 0.025 0.001 1
JEAIC AR1 0.001 0 0.016 0 0 0.002 0.004 0 0 0.001 0.024
**EXC** 0.001 0 **0.687** 0.001 0 0.146 0.12 0 0 0.021 **0.976**
IND 0 0 0 0 0 0 0 0 0 0 0
total 0.002 0 **0.703** 0.001 0 0.148 0.124 0 0 0.022 1
JEBIC AR1 0.001 0 0.022 0 0 0 0 0 0 0 0.023
**EXC** 0.026 0 **0.922** 0 0 0.015 0.014 0 0 0 **0.977**
IND 0 0 0 0 0 0 0 0 0 0 0
total 0.027 0 **0.944** 0 0 0.015 0.014 0 0 0 1
: Performance of JEAIC and JEBIC compared with JLIC for scenarios with binary outcomes. The sample size $n=500$, $T=3$, $\rho=0.3$ across 1000 Monte Carlo datasets. Ten candidate models are considered: $\{1\}=\{x_1\}$, $\{2\}=\{x_3\}$, $\{3\}=\{x_1, x_2\}$, $\{4\}=\{x_1,x_3\}$, $\{5\}=\{x_3,x_4\}$, $\{6\}=\{x_1,x_2,x_4\}$, $\{7\}=\{x_1,x_2,x_3\}$, $\{8\}=\{x_1,x_3,x_4\}$, $\{9\}=\{x_2,x_3,x_4\}$, $\{10\}=\{x_1,x_2,x_3,x_4\}$. Note that Model {3}=$\{x_1,x_2\}$ with [an EXC]{} correlation structure is the true model. The variables $x_3$ and $x_4$ are redundant.[]{data-label="table3"}
Setups Method $\bfC(\bfrho)$ 1 2 3 4 5 6 **7** 8 9 10 total
-------- -------- ---------------- --- --- --- ------- --- --- ----------- ------- ------- ------- -----------
m=0.1 JLIC AR1 0 0 0 0 0 0 0.082 0 0.027 0.012 0.121
**EXC** 0 0 0 0 0 0 **0.654** 0 0.141 0.083 **0.878**
IND 0 0 0 0 0 0 0 0 0 0.001 0.001
Total 0 0 0 0 0 0 **0.736** 0 0.168 0.096 1
JEAIC AR1 0 0 0 0 0 0 0.002 0 0 0 0.002
**EXC** 0 0 0 0 0 0 **0.802** 0 0.124 0.072 **0.998**
IND 0 0 0 0 0 0 0 0 0 0 0
Total 0 0 0 0 0 0 **0.804** 0 0.124 0.072 1
JEBIC AR1 0 0 0 0 0 0 0.002 0 0 0 0.002
**EXC** 0 0 0 0 0 0 **0.991** 0 0.005 0.002 **0.998**
IND 0 0 0 0 0 0 0 0 0 0 0
Total 0 0 0 0 0 0 **0.993** 0 0.005 0.002 1
m=0.2 JLIC AR1 0 0 0 0 0 0 0.163 0 0.034 0.027 0.224
**EXC** 0 0 0 0.001 0 0 **0.542** 0.002 0.136 0.091 **0.772**
IND 0 0 0 0 0 0 0 0 0.001 0.003 0.004
Total 0 0 0 0.001 0 0 **0.705** 0.002 0.171 0.121 1
JEAIC AR1 0 0 0 0 0 0 0.01 0 0.006 0.001 0.017
**EXC** 0 0 0 0 0 0 **0.744** 0 0.156 0.083 **0.983**
IND 0 0 0 0 0 0 0 0 0 0 0
Total 0 0 0 0 0 0 **0.754** 0 0.162 0.084 1
JEBIC AR1 0 0 0 0 0 0 0.016 0 0.001 0 0.017
**EXC** 0 0 0 0 0 0 **0.975** 0 0.007 0.001 **0.983**
IND 0 0 0 0 0 0 0 0 0 0 0
Total 0 0 0 0 0 0 **0.991** 0 0.008 0.001 1
: Performance of JEAIC and JEBIC compared with JLIC for scenarios with Gaussian outcomes. The sample size $n=500$, $T=3$, $\rho=0.3$ across 1000 Monte Carlo datasets. Ten candidate models are considered: $\{1\}=\{x_1\}$, $\{2\}=\{x_2\}$, $\{3\}=\{x_1, x_2\}$, $\{4\}=\{x_1,x_3\}$, $\{5\}=\{x_1,x_3,x_{1,3}\}$, $\{6\}=\{x_1,x_2,x_{1,2}\}$, $\{7\}=\{x_1,x_2,x_3\}$, $\{8\}=\{x_2,x_3,x_{2,3}\}$, $\{9\}=\{x_1,x_2,x_3,x_{1,2},x_{1,3}\}$, $\{10\}=\{x_1,x_2,x_3,x_{1,2},x_{1,3},x_{2,3}\}$. Note that Model $\{7\}$=$\{x_1,x_2,x_3\}$ with [an EXC]{} correlation structure is the true model. The variable $x_4$ is redundant.[]{data-label="table4"}
Tables \[table3\] and \[table4\] summarize the comparison between our proposal and JLIC on joint selection performance when the missing probability is 0.1 or 0.2 under binary and Gaussian scenarios. All results show that JEAIC and JEBIC outperform JLIC with [higher selection rates for the true underlying model.]{} The improvement becomes more substantial when the outcomes are in continuous scale. In addition, with relatively larger sample size, JEBIC performs even better, [which suggests a possible advantage in controlling false positive rates]{}.
Real Data Applications {#s:example}
======================
Case 1: the Atherosclerosis Risk in Communities (ARIC) study
------------------------------------------------------------
The ARIC study was designed to investigate the causes of atherosclerosis and its clinical outcomes, the trends in rates of hospitalized myocardial infarction (MI) and [coronary heart disease]{} (CHD) in aged 45-64 years men and women from four US communities. We select Forsyth County to identify a total of 1,036 white patients who were diagnosed with hypertension at the first examination in 1987-1989 for analysis [@kim2012]. [The existing literature has shown that SBP is an important risk factor for CVD risk prediction; however, the findings on its longitudinal pattern vary across studies due to several factors such as small sample size, lack of model diagnosis, limiting factors and so on [@muntner2015]. Here, we utilize the large epidemiological ARIC study for more exploration.]{} During the study period, longitudinal SBP measures were collected at approximately three-year intervals (1987-1989, 1990-1992, 1993-1995, and 1996-1998). There exist 355 dropout subjects, [leading to a monotone missing pattern]{}. The baseline covariates of interest are considered for exploration: age (in years), gender(1=female; 0=male), diabetes (1=fasting glucose $\geq$ 126mg/dL; 0=fasting glucose $<$ 126mg/dL), ever smoker (1=yes; 0=no), and also the examination times are coded as 1, 2, 3 and 4 for four time intervals. Before modeling, data processing is conducted, where the age variable is centered at the mean age of 54 and divided by 10 to represent a decade, and also SBP is standardized [@kim2012]. Also, the dropout probability $\lambda_{ij}$ is estimated from a logistic model with independent variables including all baseline covariates aforementioned and $Y_{i,j-1}$, $Y_{i,j-2}$, and $Y_{i,j-3}$.
$\bfC(\bfrho)$ Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8
---------- ---------------- ------------------------- ------------------------- ------------------------ ------------------------- ------------------------- ------------------------- ------------------------ -------------------------
time 0.05$^\dagger$ (0.012) 0.05$^\dagger$ (0.012) 0.05$^\dagger$ (0.012) 0.05$^\dagger$ (0.012) 0.05$^\dagger$ (0.012) 0.05$^\dagger$ (0.012) 0.05$^\dagger$ (0.012) 0.05$^\dagger$ (0.012)
gender -0.15$^\dagger$ (0.054) -0.10$^\dagger$ (0.050) -0.11$^\dagger$ (0.052) -0.14$^\dagger$ (0.052) -0.10$^\dagger$ (0.050) -0.13$^\dagger$ (0.052)
smoke -0.10 (0.057) -0.07 (0.052) -0.11$^\dagger$ (0.055) -0.07 (0.052) -0.11$^\dagger$ (0.055)
age 0.36$^\dagger$ (0.045) 0.37$^\dagger$ (0.045) 0.37$^\dagger$ (0.045) 0.36$^\dagger$ (0.045) 0.36$^\dagger$ (0.046) 0.36$^\dagger$ (0.045)
diabetes 0.14 (0.076) 0.09 (0.078) 0.10 (0.076) 0.09 (0.077)
JEAIC AR1 123.46 **55.24** 59.58 117.78 56.26 58.04 62.51 58.95
EXC 129.37 71.02 73.19 121.8 70.37 70.51 72.66 69.68
IND 922.75 789.34 781.46 896.88 798.1 780.41 785.85 791.26
JEBIC AR1 148.18 **79.96** 84.29 142.5 85.92 87.7 92.17 93.55
EXC 154.09 95.73 97.91 146.52 100.03 100.17 102.32 104.28
IND 942.52 809.11 801.23 916.66 822.81 805.13 810.57 820.91
MLIC EXC 5118 **5037.9** 5040.7 5117.2 5040.1 5042.5 5044.9 5044.8
QICW$_r$ EXC 5114.7 **5035.1** 5037.9 5113.5 5036.6 5038.6 5041 5040.1
: Analysis of the ARIC study based on eight candidate marginal mean regressions and three potential correlation structures. Summary results include WGEE estimates with standard errors in parentheses under [[an AR1]{}]{} “working" correlation structure, and JEAIC, JEBIC, MLIC and QICW$_r$ for model selection. Note that for MLIC and QICW$_r$, [an EXC]{} correlation structure is selected based on MLICC and QICW$_r$, respectively. Notation $^\dagger$ denotes the corresponding p-value$<0.05$. []{data-label="tablesbp"}
Table \[tablesbp\] summarizes the results with [the boldface values]{} indicating that the information criterion is the smallest among possible candidate models. From Table \[tablesbp\], [Model 2 with [an AR1]{} correlation structure is selected by JEAIC, JEBIC, while Model 2 with [an EXC]{} correlation structure is selected by MLIC/MLICC and QICW$_r$.]{} Thus, marginal mean regression is selected consistently; however, the discrepancy in the selected correlation structures based on different criteria shows the necessity and importance to utilize more robust and reliable information criteria. Furthermore, we check the empirical pairwise correlations between times, and a decreasing trend is shown when time gap becomes larger, indicating our selection is reasonable and valid. The final selected model, Model 2, includes three variables: time, gender, and age, which all have significant effects on SBP.
Case 2: the National Institute of the Mental Health Schizophrenia (IMPS) Study
------------------------------------------------------------------------------
To [further evaluate]{} our proposal for categorical outcomes, we consider the data from the IMPS study [that includes 293 patients in the treatment group who were given drugs chlorprom azine, fluphenazine, or thioridazine as treatment]{} and 93 patients in placebo group [@gibbons1994]. For each patient, the severity of schizophrenia disorder (IMPS79) was measured (range: 0-7) at week 0, 1, 3, 6 (time=$\sqrt{\text{week}}$). Here, we define $Y = 1$ if IMPS $\geq 4$; otherwise, $Y = 0$. The goal is to investigate treatment effect (drug=1 for treatment; 0 for placebo) and sex (1=male; 0=female) on $Y$. The dropout probability $\lambda_{ij}$ is estimated from a logistic regression with the predictors $drug_{ij}$, $sex_{ij}$, $time_{ij}$, $Y_{i,j-1}$, $Y_{i,j-2}$, and $Y_{i,j-3}$.
$\bfC(\bfrho)$ Model 1 Model 2 Model 3 Model 4 Model 5 Model 6
------------ ---------------- -------------------------- -------------------------- -------------------------- -------------------------- -------------------------- --------------------------
time -1.339 (0.081)$^\dagger$ -1.372 (0.084)$^\dagger$ -1.166 (0.208)$^\dagger$ -1.372 (0.084)$^\dagger$ -1.180 (0.239)$^\dagger$
drug -0.618 (0.182)$^\dagger$ -0.854 (0.236)$^\dagger$ -0.357 (0.438) -0.860 (0.237)$^\dagger$ -0.524 (0.492)
sex 0.116 (0.184) -0.188 (0.494)
time\*drug -0.256 (0.271) -0.252 (0.229)
time\*sex 0.023 (0.171)
sex\*drug 0.345 (0.460)
JEAIC AR1 27.55 398.50 **16.08** 17.55 17.64 22.63
EXC 94.52 491.44 90.70 91.87 94.14 101.78
IND 223.56 496.46 209.77 210.76 209.69 212.86
JEBIC AR1 39.42 410.37 **31.91** 37.33 37.42 54.28
EXC 106.38 503.31 106.53 111.65 113.92 133.43
IND 231.48 504.37 221.64 226.59 225.51 240.56
MLIC AR1 261.9 321.5 **255.8** 256 256.5 257.5
QICW$_r$ AR1 1554.8 1872.2 1529.6 **1529.5** 1532.7 1537.1
: Analysis of the IMPS study based on six candidate marginal mean regressions and three correlation structures. Summary results include WGEE estimates with standard errors in parentheses under [an AR1]{} “working" correlation structure, and JEAIC, JEBIC, MLIC and QICW$_r$ for model selection. Note that for MLIC and QICW$_r$, [an AR1]{} correlation structure is selected based on MLICC and QICW$_r$, respectively. Notation $^\dagger$ denotes the corresponding p-value$<0.05$. []{data-label="tableexa"}
Table \[tableexa\] summarizes the results of model fitting and comparisons. Note that previous work [has shown that [an AR1]{} correlation structure]{} is preferred based on MLICC; thus MLIC and QICW$_r$ are calculated given this AR1 selection. Table \[tableexa\] shows that Model 3 is selected as the best candidate model based on JEAIC, JEBIC, and MLIC because of the minimum values among all six candidate models. However, QICW$_r$ selects Model 4 as the best one even though the value is slightly lower than that [of Model 3]{}. Lastly, the final selected model, Model 3, includes two variables, time and drug, which both have significant effects on the risk of severe schizophrenia disorder.
Discussion {#s:discussion}
==========
In this paper, we heuristically introduce two innovative information criteria, JEAIC and JEBIC, for longitudinal data with dropout missingness under MAR. [The proposed criteria are evaluated in both theoretical and numerical studies with better performance compared to MLIC, QICW$_r$ and JLIC under a variety of scenarios]{}. In particular, the expected quadratic loss distance based upon which MLIC and JLIC are derived is [a model-free criterion]{}, which only measures how well the estimated means approximate to the population means but without identifying [the true mean structure [@ye1998].]{} [Thus, it might not be easy to distinguish two mean structures, which are both close to the true mean under finite samples]{}. On the other hand, QICW$_r$ modifies QIC and implements correlation structure selection based on so-called “more informative" penalty term [@gosho2016]. [However, it is unclear in theory whether and how correctly]{} specifying a “working" correlation structure will intrinsically minimize the penalty term in QICW$_r$. [In contrast, our proposed JEAIC and JEBIC]{} are based on empirical likelihood, which are distribution-free and efficiently driven by observed data and informative estimating equations. This accordingly provides scientific sense why our empirical-likelihood-based criteria would have outperformance, assuming that the true underlying model is nested within the full estimating equations. [Our approach is easy to be implemented in software with the code available in the Supporting Information]{}. Also, extensive simulations show that our proposed criteria perform computationally efficient and are flexible to be extended for more complicated scenarios, indicating the potential for wide application.
Despite the aforementioned advantages brought up from JEAIC and JEBIC, [there is still substantial work]{} for further evaluation or improvement, for instance, selection stability to account for sampling variability may need more check via extensive simulation studies using a bootstrap approach. Also, two other potential extensions may include: 1) to accommodate more general missing patterns such as intermittent missingness; 2) to consider the missingness on some time-dependent covariates or high-dimensional predictors (i.e., gene expression data) [@chen2010], which is also commonly encountered in practice nowadays. Therefore, how to generalize our proposal and accurately perform joint model selection under these scenarios [still needs to be explored]{}.
Acknowledgments {#ack .unnumbered}
===============
Wang’s research was partially supported by Grant UL1 TR002014 and KL2 TR002015 from the National Center for Advancing Transnational Sciences (NCATS). The content is solely the responsibility of the authors and does not represent the official views of the National Institute of Health, the National Science Foundation and other research sponsors.
Supporting information {#supporting-information .unnumbered}
======================
[The Web Appendices of proofs, additional tables for the simulation studies, the IMPS data example analyzed in Section 4.2 and R codes implementing our method are available with this article at the Biometrics website on Wiley Online Library.]{}
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---
author:
- |
Jorge Alfaro and Pablo González.\
*Facultad de Física, Pontificia Universidad Católica de Chile.*\
*Casilla 306, Santiago 22, Chile.*\
[email protected], [email protected]
title: 'VELOCITY AND DISTRIBUTION OF PRIMORDIAL NEUTRINOS.'
---
ABSTRACT {#abstract .unnumbered}
========
The Cosmic Neutrinos Background (**CNB**) are Primordial Neutrinos decoupled when the Universe was very young. Its detection is complicated, especially if we take into account neutrino mass and a possible breaking of Lorentz Invariance at high energy, but has a fundamental relevance to study the Big-Bang. In this paper, we will see that a Lorentz Violation does not produce important modification, but the mass does. We will show how the neutrinos current velocity, with respect to comobile system to Universe expansion, is of the order of $1065$ $\left[\frac{km}{s}\right]$, much less than light velocity. Besides, we will see that the neutrinos distribution is complex due to Planetary motion. This prediction differs totally from the usual massless case, where we would get a correction similar to the Dipolar Moment of the **CMB**.\
INTRODUCTION {#introduction .unnumbered}
============
From the beginning, the photons and all particles were coupled forming a plasma that was evolving under the influence of the Universe expansion. In such a moment, when the photons were dominating the expansion, the neutrinos were decoupling from the plasma and evolved in an independent way. One of the last discoveries about neutrino is its mass. This has relevant effects in the Standard Model and in some of its characteristics, distinguishing it from the photons. One of them, which we will study, is its velocity. Thus, we will analyze the evolution of the neutrinos’s kinetic energy since its decoupling till today.\
Other phenomenon that we will study, that is directly related with the first one, is the neutrinos distribution. The detection of the Cosmic Microwave Background of Photons (**CMB**) is the best proof of the Big-Bang scenario [@abg] that helped to check or refute models that describe it, and study the composition of the Universe. Because of this, it is important to study the Cosmic Neutrinos Background (**CNB**), especially the form of their Distribution Function to consider the effect of the peculiar velocity of the planet, named Dipolar Moment in the **CMB**, and optimize the detection. This is already complicated due to the low interaction that the neutrinos have with ordinary matter. The calculation will be done for photons and neutrinos in parallel.\
Finally, we will include a Lorentz Invariance Violation (LIV) represented by an alteration to the Dispersion Relation of energy given by [@liv; @Alfaro1; @Alfaro2; @Alfaro3]:
$$E^2 = v_{max}^2p^2 + m^2c^4$$
Where $v_{max} = c(1-\alpha)$ is the maximum attainable particle velocity with $\alpha \sim (10^{-22}-10^{-23})$ . The motivation to use this LIV comes from the possibility that, at the high energies available in the Big Bang there take place some LIV due to Quantum Gravity [@liv; @Alfaro2; @Alfaro3]. If such a LIV exists, the first problem is the appearance of a privileged reference system, but fortunately exists a natural candidate, the one where the **CMB** is isotropic. A LIV without a preferred frame as in Double Special Relativity [@dsr], will not be considered here.\
ENERGY AND VELOCITY OF PRIMORDIAL NEUTRINOS.
============================================
Initially the neutrinos were in thermal equilibrium with the rest of matter. For this, is necessary that $\Gamma_{i} \gg H$, where $\Gamma_{i}$ is the rate of interactions of the species $i$, $H
\propto T^2$ is Hubble’s constant and $T$ the temperature. While the neutrinos are kept in equilibrium, its distribution will be given by Fermi-Dirac’s statistics:
$$f_{eq}(E,T) = \frac{1}{e^{\frac{E-\mu}{k_BT}}+1}$$
During cosmic expansion, the temperature will be diminishing down to a point where $\Gamma_{\nu} \lesssim H$ and $\Gamma_{i \neq
\nu} \gg H$. This means that the neutrinos lost the equilibrium and are decoupled from the rest of the matter. We will name $T_{\nu,D}$ the neutrinos decoupling temperature that are obtained when we impose $\Gamma_{\nu} \simeq H(T) $. To see what is happening with its distribution we will do the following analysis. For a time $t_0$, an observer sees in any direction a quantity $dN
= f d^3r d^3p$ of neutrinos in an volume $d^3r$ and with momentum between $\vec{p}$ and $\vec{p} +d\vec{p}$. After a $dt$ time, the neutrinos have not interacted, so $dN$ remains constant, but the volume in which they are, have increased in a factor $\left(\frac{R(t_0+dt)}{R(t_0)} \right)^3$ and the momentum has diminished in $\frac{R(t_0)}{R(t_0+dt)}$, because of the expansion of the Universe. This means that $f(E,T_{\nu}) $ is constant in time. Therefore, for $t> t_D$ (or $T_{\nu} < T_{\nu,D}$) with $t_D$ the moment in which is produced the decoupling, the distribution function is given by [@Early; @uni]:
$$\label{feq} f[E(p(t)),T_{\nu}(t)] = f_{eq}[E(p_D),T_{\nu,D}] =
f_{eq}\left[E\left(p(t)\frac{R(t)}{R_D}\right),T_{\nu,D}\right]$$
When the subscript $D$ refers to the age of decoupling. In addition, we know that the number of neutrinos, the total energy and the energy per neutrino are given by:
$$\label{Nf} N_{\nu} = \frac{gV}{(2\pi\hbar)^3} \int
f(p,T_{\nu})d^3p$$
$$\label{Ef} E_{\nu} = \frac{gV}{(2\pi\hbar)^3} \int E(p)
f(p,T_{\nu})d^3p$$
$$\label{ef} \varepsilon_{\nu} = \frac{E_{\nu}}{N_{\nu}}$$
Where:
$$E^2(p) = v_{max}^2 p^2 + m^2c^4$$
to allow for a small LIV in the dispersion relation.\
Now we can determine the Distribution Function that they will have after being decoupled. It is possible to express the energy of the neutrinos (high energies and small masses) as $E(t) = v_{max,\nu}
p(t)$ during the decoupling (We use an expansion with zero order in the mass because $f$ depends exponentially on $E$), and as $p_D
= p(t) \frac{R(t)}{R_D}$ we obtain using (\[feq\]):
$$\label{f_Rel} f[p,T_{\nu}] =
\frac{1}{e^{\frac{v_{max,\nu}p}{k_BT_{\nu}}}+1}$$
With $T _{\nu} = T _{\nu, D} \frac{R_D}{R (t)}$ and $\mu _{\nu}
=0$ because of the low interaction that they have with matter. This means that the distribution of neutrinos after decoupling is Fermi’s with temperature $T_{\nu}$, therefore $RT_{\nu} = cte$. Replacing it in (\[Nf\]) and (\[Ef\]):
$$N_{\nu} = \frac{gV}{(2\pi\hbar)^3} \int \frac{1}{e^{\frac{v_{max,\nu}p}{k_BT_{\nu}}}+1} d^3p$$ $$E_{\nu} = \frac{gV}{(2\pi\hbar)^3} \int \frac{E(p)}{e^{\frac{v_{max,\nu}p}{k_BT_{\nu}}}+1} d^3p$$
Naturally, $N_{\nu}$ will be constant in time. Using the change of variable $x = \frac{v _{max,\nu} p}{k_BT _{\nu}}$, we obtain:
$$\label{N} N_{\nu} =
\frac{3gV\zeta(3)(k_BT_{\nu})^3}{4\pi^2\hbar^3v_{max,\nu}^3}$$
Where $\zeta (3) = 1.2021$ is the Riemann’s Zeta function. We see that, in fact, $N_{\nu}$ keeps constant in time because $V \propto
R^{3} (t) $ and $T_{\nu} \propto R^{-1}(t) $. To determine $E_{\nu}$, we must compute the integral, which is complicated for the general case. Thus , we will analyze the extreme cases where the neutrinos continue being relativistic and when they do not. Due to the spherical symmetry, the velocity is only radial, therefore we just must determine its modulus. The modulus of the velocity of a particle is given by:
$$v = \frac{\partial \varepsilon}{\partial p}$$
Being $\varepsilon$ and $p$ the energy and the momentum of a particle, related by our dispersion relation:
$$\varepsilon^2 = v_{max}^2 p^2 + m^2c^4$$
While the particle continues being relativistic, developing the derivative till the second order in the mass ($\varepsilon \gg
mc^2$), we obtain:
$$\label{vrel} v_{\nu} \simeq
v_{max,\nu}\left(1-\frac{1}{2}\left(\frac{mc^2}{\varepsilon}\right)^2\right)$$
Notice that we must use $E (p) = v_{max} p$ to calculate $E_{\nu}$, to the order of approximation in the mass that we are considering.\
Now, if the particle becomes Non-Relativistic, we have that the energy and the velocity of a particle to second order in the momentum ($pv_{max, \nu} \ll mc^2$) will be:
$$\varepsilon_{\nu} \simeq mc^2 + \left(\frac{v_{max,\nu}}{c}\right)^2\frac{p^2}{2m}$$
$$\label{vnorel} v_{\nu} = \frac{\partial \varepsilon}{\partial p}
\simeq \left(\frac{v_{max,\nu}}{c}\right)^2\frac{p}{m} =
v_{max,\nu}\sqrt{2\left(\frac{\varepsilon}{mc^2}-1\right)}$$
Where we see that, to keep the order in the momentum, the calculation must be up to second order in the expression of $E(p)$, therefore $E(p) = mc^2 +
\left(\frac{v_{max,\nu}}{c}\right)^2\frac{p^2}{2m}$.\
Relativistic Neutrinos
----------------------
As we said, to determine $E _{\nu}$ we must use $E (p) = v _{max} p$. With this, we obtain the expression:\
$$\label{Erelaun} E_{\nu} =
\frac{7\pi^2gV(k_BT_{\nu})^4}{240\hbar^3v_{max,\nu}^3}$$
Using (\[N\]) and (\[Erelaun\]) in (\[ef\]) and (\[vrel\]), we obtain:
$$\label{ener_rel} \varepsilon_{\nu} =
\frac{7\pi^4}{180\zeta(3)}k_BT_{\nu}$$
$$\label{veldesrel_rel} v_{\nu} =
v_{max,\nu}\left(1-\frac{1}{2}\left(\frac{180\zeta(3)m_{\nu}c^2}{7\pi^4k_BT_{\nu}}\right)^2\right)$$
If we define the relative velocity between the neutrinos and the photons as $\Delta v = c - v_{\nu}$, result:
$$\Delta v = \Delta
v_{max}+\frac{v_{max,\nu}}{2}\left(\frac{180\zeta(3)m_{\nu}c^2}{7\pi^4k_BT_{\nu}}\right)^2$$
Where $\Delta v_{max} = c - v_{max, \nu} = c\alpha_{\nu}$. We can see that this factor vanishes if the violation does not exist. Evaluating numerically:
$$\label{c-veldesrel_rel} \frac{\Delta v}{c} =
\alpha_{\nu}\left(1-5.04 \times
10^{-2}\left(\frac{M_{\nu}}{k_BT_{\nu}}\right)^2\right)+5.04
\times 10^{-2}\left(\frac{M_{\nu}}{k_BT_{\nu}}\right)^2$$
Where we have separated the LIV dependent part from the rest.\
Non-Relativistic Neutrinos
--------------------------
In this case we have that $E (p) = m _{\nu} c^2 + \left (\frac{v _{max, \nu}}{c} \right)
^2\frac{p^2}{2m _{\nu}}$, therefore, when we evaluate in $E_{\nu}$ using (\[N\]), we obtain:\
$$\label{Eyanorel}E_{\nu} = N_{\nu}m_{\nu}c^2 \left(1 +
\frac{1}{2}\left(\frac{k_BT_{\nu}}{m_{\nu}c^2}\right)^2
\frac{I_4}{I_2}\right)$$
With $I_n = \int_0^{\infty} \frac{x^n}{e^x +1} dx = \left
(1-\frac{1}{2^n} \right) n! \zeta (n+1) $. Then, evaluating in (\[ef\]) and (\[vnorel\]), we have:
$$\label{eneryanorel}
\varepsilon_{\nu} = m_{\nu}c^2 \left(1 +
15\frac{\zeta(5)}{2\zeta(3)}\left(\frac{k_BT_{\nu}}{m_{\nu}c^2}\right)^2
\right)$$
$$\label{veldesrel_norel} v_{\nu} =
v_{max,\nu}\sqrt{15\frac{\zeta(5)}{\zeta(3)}}\frac{k_BT_{\nu}}{M_{\nu}}$$
Giving a relative velocity:
$$\label{c-veldesrel_norel}\frac{\Delta v}{c} =
\alpha_{\nu}\sqrt{15\frac{\zeta(5)}{\zeta(3)}}\frac{k_BT_{\nu}}{M_{\nu}}
+
\left(1-\sqrt{15\frac{\zeta(5)}{\zeta(3)}}\frac{k_BT_{\nu}}{M_{\nu}}\right)$$
Where we have separated the LIV part from the rest and $\zeta (5)
= 1.0369$. Since the neutrino velocity cannot be higher than its maximum velocity, we must see to what temperatures this approximation is valid. We have that $v_{\nu}> v_{max, \nu}$ if $k_BT_{\nu}> \sqrt{\frac{\zeta(3)}{15\zeta(5)}} M_{\nu}$. It means that the approximation is valid if $k_BT_{\nu} \ll \sqrt
{\frac{\zeta (3)}{15\zeta(5)}} M_{\nu} \sim 0.28 M_{\nu}$.\
Numerical Results and Analysis
------------------------------
In the age of decoupling of the neutrinos, we know that $k_BT_{\nu,D} \simeq (2 - 4)$ \[MeV\] and, currently, $k_BT_{\nu, 0}
= 1.68 \times 10^{-4}$ \[eV\]. In addition to this, for cosmological parameters, we know [@masa]:
$$\sum_{i} m_{\nu_i} \leq 0.17 [eV]$$
That clearly indicates that they are relativistic in the moment of the decoupling. There exist many estimations of the masses of the neutrinos, but none of them are very precise. Thus, we will use $m_{\nu} \simeq 0.17 [eV]$. This way, we are sure of being inside the correct limits and we will find the maximum effect that the mass could have in the velocity of neutrinos. This way, none of these estimations is below $k_BT_{\nu,0}$, therefore they are Non-Relativistic nowadays.\
Before discussing the results, we will analyze the effect of the LIV. Thus, we will compare our relativistic expressions with our Non-Relativistic ones. If we observe these expressions, we can see that both are proportional to $v_{max, \nu}$, which is the only thing that depends on $\alpha_{\nu}$. It means that the difference in percentage between the case with and without LIV is always:
$$\frac{v_{\nu}(0) - v_{\nu}(\alpha_{\nu})}{v_{\nu}(0)}100\% = \alpha_{\nu} 100\% = 1 \times 10^{-20} \%$$
Therefore, it is not possible that this LIV has an important effect in the neutrinos, then we will continue our calculations using $\alpha=0$.\
Previously we mentioned that our Non-Relativistic approximation is valid if $\frac{k_BT_{\nu}}{M_{\nu}} \leq 0.28$. At present we have that $\frac{k_BT_{\nu}}{M_{\nu}} \simeq 10^{-3}$ fulfilling the Non-Relativistic bound, but with a mass $100$ times minor ($\sim 2 \times 10^{-3}$ \[eV\]) the bound is not respected. However it does not correspond to the relativistic case either. To be kept relativistic, we need a mass $10000$ times smaller or less ($\sim
2 \times 10^{-5}$ \[eV\]).\
In Figure \[graf\_vel\_rel\] the evolution of the velocity of the neutrinos due to the expansion of the Universe is represented graphically . We define the adimensional quantities $z =
\frac{M_{\nu}}{k_BT _{\nu}}$, $y =\frac{v _{\nu}}{c}$. It is indicated in the graph that the time grows towards bigger values in $z$. Clearly, we see that the neutrinos suffer a rapid deceleration from the time of decoupling . Then, this deceleration begins to diminish slowly, approaching a zero velocity.\
All the estimations of $M _{\nu}$ indicate that we are in a zone dominated by the Non-Relativistic approximation. The estimation for the smallest masses ranges between $10^{-4}$ and $10^{-3}$ \[eV\]. Remembering that our top limit is $0.17$ \[eV\], we see that we are currently in the region $0.6 < z <1012$, which is a very wide range. Evaluating numerically in (\[veldesrel\_norel\]), we obtain $v_{\nu} = 3.55 \times 10^{-3} c = 1065$ \[$\frac{km}{s}$\], with a mass of $0.17$ \[eV\]. This velocity will be bigger if we use smaller neutrino masses.\
Up to now, we have assumed that the neutrinos are not affected by the galactic potential, they are free particles and are not relics from the Milky Way [@white1]-[@white2]. To check this point, we consider the relation between kinetic and potential energy of the neutrino in the Milky Way. That is:
$$\frac{m_{\nu}v^2}{2} = \frac{GMm_{\nu}}{R}$$ $$v = \sqrt{\frac{2GM}{R}}$$
Where $v$ would be the limit velocity where the potential energy is comparable with the kinetic energy. Evaluating in $M \simeq 2
\times 10^{42}$ \[kg\] and $R \simeq 4.7 \times 10^{20}$ \[m\], mass and radius of the Milky Way respectively, we obtain $v \simeq 754$ $\left [\frac{km}{s} \right]$. Since $v_{\nu} \gtrsim 1000$ $\left
[\frac{km}{s} \right]$, our supposition is correct.\
THE CNB DISTRIBUTION.
=====================
To determine the effective neutrinos distribution (distribution from Earth), we need to use equation (\[f\_Rel\]) in the comobile system of the Universe. In addition to this, we will use the photons distribution when they are decoupled, that is:
$$f(p,T) = \frac{1}{e^{\frac{pc}{kT}}-1}$$
That corresponds to an ultra-relativistic Bose-Einstein’s distribution with $RT = cte$. Currently, it has a temperature of $2.73$ \[K\]. In addition, we saw in the previous chapter that a LIV of the form:
$$E^2 = v_{max}^2p^2 + m^2c^4$$
Is not markedly different in its energy and velocity in comparison to the usual Dispersion Relation ($v _{max} = c$). Because of this the usual special relativity rules are valid. For instance, Lorentz’s Transformations for the coordinates of space-time and of energy - momentum. Then, we can compare the neutrinos distribution in the comobile system and Earth. It is possible to demonstrate (See Appendix):
$$\label{f=f'} f'(p',T') = f(p,T)$$
$$\label{E(E')} E = \gamma (E' - v_t p'\cos(\theta'))$$
Where the primed elements refer to the reference system of Earth and the non primed to the comobile. $\theta'$ is the angle that is formed between the vision line and the direction of Earth motion and $v_t$ is the Earth’s velocity [@vel; @pec]. We can see that the distribution function is invariant under Lorentz’s Transformation and the energy changes with the angle of vision.\
Now we will analyze some cases. First, the photons of the CMB to guide us because they are already very well known , and secondly, the neutrinos. Using expression (\[E(E’)\]) we will determine $p'$ as a function of $p$.\
Photons
-------
In this case, we have that $E = cp$, therefore the expression (\[E(E’)\]) is reduced to:\
$$p = \frac{1 - \frac{v_t}{c}\cos(\theta')}{\sqrt{1-\left(\frac{v_t}{c}\right)^2}}p'$$
Replacing in (\[f=f’\]), we obtain:
$$f'(p',T'_{\gamma}) = f\left(\frac{1-\frac{v_t}{c}\cos(\theta')}{\sqrt{1-\left(\frac{v_t}{c}\right)^2}}p',T_{\gamma}\right)$$
As the photons, after being decoupled, continue with a distribution of the form:
$$f_{\gamma} = \frac{1}{e^{\frac{pc}{k_BT_{\gamma}}}-1}$$
We can leave our expression as:
$$f'(p',T'_{\gamma}) = f\left(p',T_{\gamma}\frac{\sqrt{1-\left(\frac{v_t}{c}\right)^2}}{1
- \frac{v_t}{c}\cos(\theta')}\right)$$
Therefore, the photons distribution detected from Earth, $f'$, in a specific direction, will be of the same form that the one detected in the comobile system to the Universe, but with a different temperature given by:
$$T'_{\gamma} = T_{\gamma}\frac{\sqrt{1-\left(\frac{v_t}{c}\right)^2}}{1
- \frac{v_t}{c}\cos(\theta')}$$
If we consider that $v_t \ll c$, we have:
$$T'_{\gamma} \simeq T_{\gamma}\left(1 +
\frac{v_t}{c}\cos(\theta')\right)$$
$$\label{Dipol} \frac{\Delta T_{\gamma}}{T_{\gamma}} \simeq
\frac{v_t}{c}\cos(\theta')$$
that is known as the Dipolar Moment, and is of the order of $10^{-4}$.\
Neutrinos
---------
Now, we have particles with mass. Currently, the neutrinos are Non-Relativistic, therefore $E = m_{\nu} c^2 + \frac{p^2}{2m_{\nu}}$ for both the comobile and Earth systems. Evaluating in (\[E(E’)\]) and using the approximation $v_t \ll c$ up to second order in $p' $ and $v_t$, we obtain:
$$p^2 = p'^2 - 2m_{\nu}v_tp'\cos(\theta') + m_{\nu}^2v_t^2$$
Evaluating in (\[f=f’\]), we have:
$$\label{f'_neutrin} f'(p',T'_{\nu}) = f\left(\sqrt{p'^2 -
2m_{\nu}v_tp'\cos(\theta') + m_{\nu}^2v_t^2},T_{\nu}\right)$$
In this case is impossible to find a relation between $T'_{\nu}$ and $T_{\nu}$, but we know that the distribution is given by (\[f\_Rel\]). Seemingly, we can only notice the effects graphically. To facilitate our analysis, it will be helpful to define the number of neutrinos per solid angle $d\Omega'$ of momentum as:
$$\frac{dN}{d\Omega'} = \frac{gV}{(2\pi
\hbar)^3}f'(p',T'_{\nu})p'^2dp$$
With this, we can obtain the distribution function of the number of particles:
$$\label{F'} F'(p',T'_{\nu}) = \frac{gV}{(2\pi
\hbar)^3}f'(p',T'_{\nu})p'^2$$
In our case, the distribution function $F'$ will be:
$$F'(p',T'_{\nu}) \propto \frac{p'^2}{e^{\frac{\sqrt{p'^2 -
2m_{\nu}v_tp'\cos(\theta') + m_{\nu}^2v_t^2}c}{k_BT_{\nu}}}+1}$$
ANALYSIS
--------
To do our analysis, it is useful to introduce the adimensional variables $x = \frac{p'c}{k_BT_{\nu}}$, $a(\theta') =
\frac{m_{\nu} v_tc}{k_BT_{\nu}} \cos (\theta') $ and $b = a(0)$. With these parameters, our distribution is:
$$\label{F_rel}F' \propto \frac{x^2}{e^{\sqrt{x^2 - 2ax + b^2}}+1}$$
Considering a terrestrial velocity $v_t \simeq 300$ \[$\frac{km}{s}$\], we can see that $b \simeq 1$ for $M_{\nu} =
0.17$. It means that $-1 \leq a(\theta') \leq 1$. This range will be smaller if we use a smaller mass, but then the Non-Relativistic approximation is less precise.\
In Figure \[grafF-rel\] it is shown $F'$; here we have used a value of $b\sim 1$ and some representative values of $a
(\theta') $ (See Table \[direc\]).\
**$a(\theta')$** **Direction of Observation**
------------------ ----------------------------------------------- --
$1$ In favour of the Terrestrial Movement
$0.5$ $60^0$ deflected to the Terrestrial Movement
$0$ Perpendicular to the Terrestrial Movement
$-0.5$ $120^0$ deflected to the Terrestrial Movement
$-1$ Against the Terrestrial Movement
: Directions of Observation.
[[Values of $a
(\theta') $ used in Figure \[grafF-rel\] with the corresponding direction of observation.]{}]{} \[direc\]
Let’s remind that the distribution $F'$ represents the particles number that come from certain direction and momentum. We see in Figure \[grafF-rel\] that the distribution suffers a loss of homogeneity, which is translated in more neutrinos observed in favour of the Earth’s movement, but simultaneously the form of the distribution function is altered much more with regard to the distribution of the comobile system. If we move away from this direction, the neutrino number detected diminishes considerably and the small momentum are favored.\
The distribution maximum must fulfill the equation:
$$\label{n_eq}\left(2\sqrt{x_{max}^2 - 2ax_{max} + b^2} - x_{max}^2
+ ax_{max}\right)e^{\sqrt{x_{max}^2 - 2ax_{max} + b^2}} +
2\sqrt{x_{max}^2 - 2ax_{max} + b^2}=0$$
It is complicated to find a general expression for $x_{Max}$, although a numerical treatment is readily available. As an example, we can study the extreme cases $a (\theta) = b$ and $a
(\theta) =-b$ where $b \simeq 1$ for $M _ {\nu} = 0.17$ \[eV\]. Evaluating in (\[n\_eq\]), we obtain:
$$x_{max}(a=b) = 2.463$$ $$x_{max}(a=-b) = 2.091$$
This means that the momentum of the majority of detected neutrinos will be:
$$2.091 \leq \frac{p'c}{k_BT_{\nu}} \leq 2.463$$
This will be a useful information to plan the detectors.\
Now, if we use a smaller mass, the differences between different directions in the distributions diminish. In Figures \[grafF-rel\_10\] and \[grafF-rel\_100\] we can see the distributions with a mass $10$ and $100$ times smaller, where the Non-Relativistic approximation can be still valid. To compare, the photons distribution appears in Figure \[grafF-fot\] for different observation angles. Comparing this with Figure \[grafF-rel\_100\], we see that the effect produced in the neutrinos distribution is bigger always than the produced in the photons.\
This happens because the photons always go to a bigger velocity than the terrestrial ($c \gg v_t$), therefore Earth would be almost still with regard to the comobile system, doing that the isotropy almost does not change. On the other hand, if the neutrinos acquire mass, they will be found submitted to a deceleration as the Universe expands (See Figure \[graf\_vel\_rel\]) so that that currently the neutrinos are Non-Relativistic, with a velocity not much higher than $v_t$. This means that the effect of the terrestrial movement begins being important in the velocities addition and will be increased with the time due to the constant cooling of the neutrinos. This will reach the point in which the neutrino velocity will be much smaller than $v_t$ and, practically, the planetary movement will predominate. This will be reflected in an increase of the distribution in the direction of the terrestrial movement.\
CONCLUSION. {#conclusion. .unnumbered}
===========
The mass of the neutrinos brought important modifications to its velocity. Without mass, the neutrinos would have supported a constant velocity and equal to the light velocity, $c$. On the other hand, with non zero masses, its velocity is affected by a strong deceleration (See Figures \[graf\_vel\_rel\]), therefore they are Non-Relativistic nowadays. As we have developed an expression for the velocity with regard to the comobile system to the expansion of the Universe, it is necessary to use the addition of velocities to determine the mean neutrino velocity relative to Earth. Thus, we use Lorentz’s Transformations since the LIV did not bring any important effect. The difference that is produced in its velocity with and without LIV is of $\sim 10^{-20}$ %, which is totally negligible. Then, we can use the invariance of the distribution function to relate the comobile system to the terrestrial.\
In the same way, the mass of the neutrinos brought important changes to the distribution. Unlike the photons, it was not possible to introduce a similar term to the Dipolar Moment because the temperature would depend on $p'$. Greater the mass greater the effect. In addition, the distribution is widely favored in the Earth’s direction, but if we move away from this direction, the neutrinos number diminishes. In spite of that the variation depends greatly on the mass; as time goes by, the neutrinos will be cooling diminishing little by little its velocity. This means that in some moment the velocity of the neutrinos will be less than the terrestrial speed. In the future, the neutrinos will be almost still in comparison to the Earth’s velocity. In this moment, we will only detect the neutrino that “crash” with Earth when it advances.\
To sum up, we see that the existence of the neutrino mass produces a relatively important effect in its evolution, which is reflected in the perception that we have of them especially in the loss of homogeneity in the distribution function. Thus, for its detection is advisable to use detectors of neutrinos directed in favour to the terrestrial movement or to use a satellite located in someone of Lagrange’s points of the Solar System to keep the isotropic distribution of the comobile system, as the satellite **Planck Surveyor** that will observe **CMB** [@planck].\
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors want to thank A. Reisenegger for an interesting discussion. The work of JA and PG was partially supported by Fondecyt \# 1060646.
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Tomozawa, Y. *Cosmic microwave background dipole, peculiar velocity and Hubble flow.* eprint arXiv:0705.1317.
Hobson M.P; Barreiro R.B; Toffolatti L; Lasenby A.N; Sanz J.L; Jones A.W; Bouchet F.R. *The effect of point sources on satellite observations of the cosmic microwave background.* arXiv:astro-ph/9810241v1.
Debbasch, F; Rivet, J. P; van Leeuwen, W. A. *Invariance of the relativistic one-particle distribution function.* Physica A, Volume 301,(2001) Issue 1-4, p. 181-195.
APPENDIX. {#appendix. .unnumbered}
=========
The special relativity rules say us that two reference systems can be related by:
$$\vec{x} = \vec{x'}_{\perp} + \gamma (\vec{x'}_{\parallel} + \vec{v}_t t')$$ $$t = \gamma \left(t' + \frac{\vec{v}_t\cdot\vec{x'}}{c^2}\right)$$
and
$$\vec{p} = \vec{p'}_{\perp} + \gamma \left(\vec{p'}_{\parallel} + \frac{\vec{v}_t}{c^2} E'\right)$$ $$E = \gamma (E' + \vec{v}_t\cdot\vec{p'})$$
Where the primed reference system is moving away from the non primed to a velocity $\vec{v}$. The coefficients with the subscripts $\parallel$ and $\perp$ represent the parallel and perpendicular components of the velocity $\vec{v}$ respectively. Since we are considering a particle in the universe, our primed and non primed reference systems will be, respectively, Earth and comobile system to the Universe expansion, therefore $\vec{v}$ is the planet velocity. Thus, from now, we will call its $\vec{v}_t$.\
If particles go to Earth along the vision line (See Figure \[sist\_ref\]), the Lorentz’s transformation can be written as:
$$\label{Trans_Lor_esp} x_{\perp,i} = x'_{\perp,i}~~~~x_{\parallel}
= \gamma(x'_{\parallel} + v_t t')~~~~t = \gamma \left(t' +
\frac{v_t x'_{\parallel}}{c^2}\right)$$
$$\label{Trans_Lor_mom} p_{\perp,i} = p'_{\perp,i}~~~~p_{\parallel}
= \gamma \left(p'_{\parallel} - \frac{v_t}{c^2}E'\right)~~~~E =
\gamma(E' - v_t p'_{\parallel})$$
![[Description of both reference systems. $S$: Comobile reference system to the Universe expansion. Earth has a velocity $v_t$ and the neutrino has momentum $p$. Between both we have the vision angle $\theta$. $S'$: Earth reference system. Earth is still and the neutrino has momentum $p'$. We have the vision angle $\theta' $ measured from Earth. The coordinates system of $S$ and $S'$ are related by the Lorentz’s Transformation.]{}[]{data-label="sist_ref"}](dipolo1.png){width="80.00000%"}
Where $i=1\textrm{, } 2$ label both perpendicular coordinates to $\vec{v}_t$. Its differential form considering an instantaneous measurement from Earth, is $t' =cte$ or $dt' =0$, giving:
$$\label{Trans_Lor_difesp} dx_{\perp,i} =
dx'_{\perp,i}~~~~dx_{\parallel} = \gamma dx'_{\parallel}~~~~dt =
\gamma \frac{v_t dx'_{\parallel}}{c^2}$$
$$\label{Trans_Lor_difmom} dp_{\perp,i} =
dp'_{\perp,i}~~~~dp_{\parallel} = \gamma \left(dp'_{\parallel} -
\frac{v_t}{c^2}dE'\right)~~~~dE = \gamma(dE' - v_t
dp'_{\parallel})$$
Now, when we count the particles number from Earth in a specific direction instantaneously ($dt' =0$), inside a volume $d^3r'$ we have $dN$ particles with momentum between $\vec{p'}$ and $\vec{p'}
+ d\vec{p'}$. In addition, we know that $dN$ is given by:
$$\label{dNf'} dN = f'(p',T')d^3p'd^3r'$$
Where $f' $ is the distribution function on Earth. In the comobile system, the particles are in a volume $d^3r$ and with values of momentum between $\vec{p}$ and $\vec{p} + d\vec{p}$, but in a $dt$ time, given by (\[Trans\_Lor\_difesp\]) (different from zero because $dt' =0$) some particles enter or exit of $d^3r$. Thus, the particle number, in this system, is given by:
$$\label{dNf} dN = f(p,T)d^3pd^3r + f(p,T)d^3p d\vec{S} \cdot
\vec{u} dt$$
Where $\vec{u} = c^2\frac{\vec{p}}{E}$ is the particle’s velocity and $d\vec{S}$ is the differential area, with normal direction. Both expressions for $dN$ are, simply, a variation of the continuity equation:
$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0$$
With $\rho = f(p, T) d^3p$. Since $dN$ must be the same in both systems, we must equal (\[dNf’\]) and (\[dNf\]). Then:
$$\label{relacion_f}f'(p',T')d^3p'd^3r' = f(p,T)d^3p\left(d^3r +
c^2dt\frac{\vec{p} \cdot d\vec{S}}{E}\right)$$
With (See Figure \[d3r\]):
$$d^3r = dx_{\parallel} \wedge dx_{\perp,1} \wedge dx_{\perp,2}$$ $$d\vec{S} = -(dx_{\parallel} \wedge dx_{\perp,1}\hat{x}_{\perp,2} + dx_{\parallel} \wedge dx_{\perp,2}\hat{x}_{\perp,1} + dx_{\perp,1} \wedge dx_{\perp,2}\hat{x}_{\parallel})$$ $$\vec{p} = -(p_{\parallel}\hat{x}_{\parallel} + p_{\perp,1}\hat{x}_{\perp,1} +
p_{\perp,2}\hat{x}_{\perp,2})$$
![[Representation of the volume element $d^3r$, where we see the surface elements. We can see, clearly, the vectorial direction of $\vec{p}$ and $d\vec{S}$.]{}[]{data-label="d3r"}](dipolo2.png){width="80.00000%"}
Where $ \wedge $ represents the anti-commutative product between the differentials. Evaluating in (\[relacion\_f\]), using (\[Trans\_Lor\_mom\]) and (\[Trans\_Lor\_difesp\]), we have:
$$\label{rel_f_f'} f'(p',T')d^3p'd^3r' = f(p,T)d^3pd^3r'\frac{E'}{E}$$
Replacing (\[Trans\_Lor\_difmom\]) in $d^3p = dp_{\perp,1} \wedge
dp_{\perp,2} \wedge dp_{\parallel}$, we obtain:
$$d^3p = dp'_{\perp,1} \wedge dp'_{\perp,2} \wedge \gamma\left(dp'_{\parallel} - \frac{v_t}{c^2} dE'\right)$$
But we know that ${E'}^2 = c^2({p'}_{\perp,1}^2 +{p'}_{\perp,2}^2
+{p'}_{\parallel}^2) + m^2c^4$. Deriving, we obtain the relation $E'dE' = c^2(p'_{\perp,1}dp'_{\perp,1} + p'_{\perp,2}dp'_{\perp,2}
+ p'_{\parallel}dp'_{\parallel})$. Evaluating:
$$d^3p = dp'_{\perp,1} \wedge dp'_{\perp,2} \wedge dp'_{\parallel}\gamma\left(1 - \frac{p'_{\parallel}}{E'}v_t\right)$$
Where we have used that $dp'_{\perp, i} \wedge dp'_{\perp, i} =
0$ for anti-conmutativity. Using (\[Trans\_Lor\_mom\]), $d^3p$ stays:
$$d^3p = d^3p'\frac{E}{E'}$$
Then, (\[rel\_f\_f’\]) is reduced to:
$$f'(p',T') = f(p,T)$$
This means that the distribution function is Lorentz invariant. In reference [@f_inv], this has been discussed differently. They used:
$$dN = f(p,T)d^3pd^3r$$
Naturally, they obtained that $f$ is not Lorentz invariant.\
![[Neutrino Velocity Representation (Ecs. \[veldesrel\_rel\] y \[veldesrel\_norel\]). it is had being dominated by the relativistic expression (Blue) and then for the Non-Relativistic (Red). The general expression would be a composition of both.]{}[]{data-label="graf_vel_rel"}](vel_rel.png){width="80.00000%"}
![[Primordial Neutrinos Distribution in the current age (Ec \[F\_rel\]) for different values of $a$ and $b \simeq 1$ that corresponds to $M_{\nu} = 0.17$ \[eV\]. The black curve represents to the distribution in the comobile system.]{}[]{data-label="grafF-rel"}](F_rel.png){width="55.00000%"}
![[Primordial Neutrinos Distribution in the current age (Ec \[F\_rel\]) for different values of $a$ and $b \simeq 0.1$ that corresponds to $M_{\nu} = 0.017$ \[eV\]]{}[]{data-label="grafF-rel_10"}](F_rel_10.png){width="55.00000%"}
![[Primordial Neutrinos Distribution in the current age (Ec \[F\_rel\]) for different values of $a$ and $b \simeq 0.01$ that corresponds to $M_{\nu} = 0.0017$ \[eV\]]{}[]{data-label="grafF-rel_100"}](F_rel_100.png){width="55.00000%"}
![[Primordial Photons Distribution in the current age (Ec \[F\_rel\]) for different values of the observation angle $\theta'$.]{}[]{data-label="grafF-fot"}](F_foton.png){width="55.00000%"}
|
{
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}
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Composite super-moiré lattices in double aligned graphene heterostructures {#composite-super-moiré-lattices-in-double-aligned-graphene-heterostructures .unnumbered}
==========================================================================
**Authors:** Zihao Wang$^{1\dagger}$, Yi Bo Wang$^{1\dagger}$, J. Yin$^{1,2}$, E. Tóvári$^{3}$, Y. Yang$^{1,3}$, L. Lin$^{3}$, M. Holwill$^{3}$, J. Birkbeck$^{3}$, D. J. Perello$^{1,3}$, Shuigang Xu$^{1,3}$, J. Zultak$^{3}$, R. V. Gorbachev$^{1,3,4}$, A. V. Kretinin$^{3,5}$, T. Taniguchi$^{6}$, K. Watanabe$^{6}$, S. V. Morozov$^{7}$, M. Anelković$^{8}$, S.P. Milovanović$^{8}$, L. Covaci$^{8}$, F.M. Peeters$^{8}$, A. Mishchenko$^{1,3}$, A. K. Geim$^{1,3}$, K. S. Novoselov$^{1,3,9,10}$, Vladimir I. Fal’ko$^{1,3,4}$, Angelika Knothe$^{3*}$, C. R. Woods$^{1,3*}$\
Supplementary Information {#supplementary-information .unnumbered}
--------------------------
### Contents: {#contents .unnumbered}
Details of the super-moiré superlattice perturbation theory\
### Details of the super-moiré superlattice perturbation theory {#details-of-the-super-moiré-superlattice-perturbation-theory .unnumbered}
We study the long-range periodic “super”-moire pattern which appears due to beatings between the two moires at the top and bottom interfaces. The six reciprocal lattice vectors of the top (bottom) moire pattern are given by $\mathbf{b}^{\alpha}_m=\mathbf{G}_m-\mathbf{g}^{\alpha}_m$ ($\mathbf{b}^{\beta}_m=\mathbf{G}_m-\mathbf{g}^{\beta}_m$) for $m=0,\dots,5$, where $\mathbf{G}_m$ denote the reciprocal lattice vectors of graphene, and $\mathbf{g}^{\alpha}_m$ ($\mathbf{g}^{\beta}_m$) are the reciprocal lattice vectors of the top (bottom) hBN. From these, we construct the combinations $\mathbf{d}_{m,k}= \mathbf{b}^{\alpha}_{m}-\mathbf{b}^{\beta}_k$. For twist different angles $\theta^{\alpha}$ ($\theta^{\beta}$) of the top (bottom) hBN layer, these become very small or vanish completely and hence constitute the shortest reciprocal lattice vectors of the “super”-moire pattern. These cases are studied below.
### Derivation of the Hamiltonians of shortest period {#derivation-of-the-hamiltonians-of-shortest-period .unnumbered}
The low-energy contribution (for the shortest effective Bragg vectors $\mathbf{d}_{m,k}=\mathbf{b^{\alpha}}_{m}-\mathbf{b}_{k}^{\beta}$) of the superlattice Hamiltonian which originates from interference reads in second order perturbation theory $$\begin{aligned}
\nonumber{\text{H}}_{n,m}^{int} = \delta{\text{H}}^{(2)}_{\alpha\beta} + \delta{\text{H}}^{(2)}_{\beta\alpha},\end{aligned}$$ with\
(S1)
while the corresponding contribution due to strain caused by reconstruction is given by $(40)$\
(S2)
We consider the terms of the Hamiltonians above with $m = k$, under the assumption of very small angles (for which $U_i^{\beta}\approx U_i^{\alpha }=: U_i$):\
(S3)
For the contribution due to strain, the cases in which the two hBN layers are either parallel, or antiparallel with respect to each other must be distinguished:\
Parallel case: $$\begin{aligned}
\nonumber {\text{H}}_{m,m}^{P}=& - U_0 ( w_s^{\alpha} + w_s^{\beta}) f^{(m,m)}_1 (\mathbf{r}) -U_3 ( w_s^{\alpha} - w_s^{\beta}) f^{(m,m)}_2 (\mathbf{r}) \sigma_3 \\
\nonumber&+U_1 ( w_s^{\alpha} + w_s^{\beta})\frac{\sqrt{3}a}{4\pi}\frac{1}{(\theta^{\beta}-\theta^{\prime})} \boldsymbol{\nabla}f^{(m,m)}_2 (\mathbf{r}) \boldsymbol{\sigma} \\
\nonumber&+U_0 ( w_{as}^{\alpha} - w_{as}^{\beta}) f^{(m,m)}_2 (\mathbf{r}) -U_3 ( w_{as}^{\alpha} + w_{as}^{\beta}) f^{(m,m)}_1 (\mathbf{r}) \sigma_3 \\
&+U_1 ( w_{as}^{\alpha} - w_{as}^{\beta})\frac{\sqrt{3}a}{4\pi}\frac{1}{(\theta^{\beta}-\theta^{\alpha})} \boldsymbol{\nabla}f^{(m,m)}_1 (\mathbf{r}) \boldsymbol{\sigma},
$$
(S4)
Antiparallel case: $$\begin{aligned}
\nonumber {\text{H}}_{m,m}^{AP}=& - U_0 ( w_s^{\alpha} - w_s^{\beta}) f^{(m,m)}_1 (\mathbf{r}) -U_3 ( w_s^{\alpha} + w_s^{\beta}) f^{(m,m)}_2 (\mathbf{r}) \sigma_3 \\
\nonumber &+U_1 ( w_s^{\alpha} - w_s^{\beta})\frac{\sqrt{3}a}{4\pi}\frac{1}{(\theta^{\beta}-\theta^{\alpha})} \boldsymbol{\nabla}f^{(m,m)}_2 (\mathbf{r}) \boldsymbol{\sigma} \\
&\nonumber+U_0 ( w_{as}^{\alpha} + w_{as}^{\beta}) f^{(m,m)}_2 (\mathbf{r}) -U_3 ( w_{as}^{\alpha} - w_{as}^{\beta}) f^{(m,m)}_1 (\mathbf{r}) \sigma_3 \\
&+U_1 ( w_{as}^{\alpha} + w_{as}^{\beta})\frac{\sqrt{3}a}{4\pi}\frac{1}{(\theta^{\beta}-\theta^{\alpha})} \boldsymbol{\nabla}f^{(m,m)}_1 (\mathbf{r}) \boldsymbol{\sigma},
$$
(S5)
in terms of the functions $f^{(m,m)}_1(\mathbf{r})=\sum_m e^{i (\mathbf{b}_{m}^{\alpha}+\mathbf{b}_{m}^{ \beta})\cdot\frac{\mathbf{R}}{2} } e^{i\mathbf{d}_{m^{ },m}\cdot\mathbf{r}}$, $f^{(m,m)}_2(\mathbf{r})=i\sum_m (-1)^m e^{i (\mathbf{b}_{m}^{\alpha}+\mathbf{b}_{m}^{ \beta})\cdot\frac{\mathbf{R}}{2} } e^{i\mathbf{d}_{m^{ },m}\cdot\mathbf{r}}$ and $\Re[\mathbf{G}_m\cdot \mathbf{u}_{\mathbf{b}^{\alpha}_m} ] = \Re[\mathbf{G}_m\cdot \mathbf{u}_{-\mathbf{b}^{\alpha}_m} ] = -(-1)^n w_{as}^{\alpha}$, $\Im[\mathbf{G}_m\cdot \mathbf{u}_{\mathbf{b}^{\alpha}_m}]= -\Im[\mathbf{G}_m\cdot \mathbf{u}_{-\mathbf{b}^{\alpha}_m}] =w_s^{\alpha}$.\
Under the assumption that for small and almost equal angles $w_i^{\alpha} \approx w_i^{\beta}\approx w_i$, and keeping in mind that the gradient terms in [equations ]{}(S3), (S4), (S5) can be removed by a gauge transformation $(28)$ we arrive at the superlattice Hamiltonians ${\text{H}}_{m,m}$ of [equation ]{}(2) in the main text.
### All combinations of short effective moire Bragg vectors {#all-combinations-of-short-effective-moire-bragg-vectors .unnumbered}
For all the possible shortest effective Bragg vectors $\mathbf{d}_{m,k}$ we list the functions that describe the period and density of the corresponding super-moire structures.\
$\bigstar\;\mathbf{d}_{m,m}=\mathbf{b}^{\alpha}_{m}-\mathbf{b}^{\beta}_{m}=\frac{4\pi}{\sqrt{3}\text{a}(1+\delta)}
\begin{pmatrix}
\sin\theta^{\alpha}-\sin\theta^{\beta}\\
\cos\theta^{\beta}-\cos\theta^{\alpha}
\end{pmatrix},
$
$$\begin{aligned}
\nonumber A_{m,m}= \text{a} ({\delta}+1) \frac{1}{\sqrt{2-2 \cos (\theta^{\alpha}-\theta^{\beta})}},\end{aligned}$$
$$\begin{aligned}
\nonumber n_{m,m}= -\frac{16 [\cos (\theta^{\alpha}-\theta^{\beta})-1]}{\sqrt{3} \text{a}^2 ({\delta}+1)^2},\end{aligned}$$
Divergence of $A_{m,m}$ and zero $n_{m,m}$ for all twist angles with $\theta^{\alpha}=\theta^{\beta}$.\
$\bigstar\;\mathbf{d}_{m,m+1}=\mathbf{b}^{\alpha}_{m}-\mathbf{b}^{\beta}_{m+1}=\frac{2\pi}{\text{a}(1+\delta)}
\begin{pmatrix}
\frac{ 3 {\delta}+2 \sqrt{3} \sin\theta^{\alpha}-\sqrt{3} \sin \theta^{\beta}-3 \cos\theta^{\beta}+3}{3} \\
\frac{{\delta}-2 \cos \theta^{\alpha}-\sqrt{3} \sin \theta^{\beta}+\cos \theta^{\beta}+1 }{\sqrt{3}}
\end{pmatrix},
$
Angles of the critical points (Divergence of $A$ and zero $n$):\
\
$\bigstar\;\mathbf{d}_{m,m+2}=\mathbf{b}^{\alpha}_{m}-\mathbf{b}^{\beta}_{m+2}=\frac{2\pi}{\text{a}(1+\delta)}
\begin{pmatrix}
\frac{3 {\delta}+2 \sqrt{3} \sin \theta^{\alpha}+\sqrt{3} \sin \theta^{\beta}-3 \theta^{\beta}+3 }{3} \\
\frac{ -3 {\delta}+2 \cos\theta^{\alpha}+\sqrt{3} \sin \theta^{\beta}+\cos \theta^{\beta}-3}{\sqrt{3}}
\end{pmatrix},
$
\
Angles of the critical points (Divergence of $A$ and zero $n$): $$\theta^{\alpha}=-\theta^{\beta}=\tan ^{-1}\left[ \frac{\sqrt{3} \left(-{\delta}+\sqrt{1-3 {\delta} ({\delta}+2)}-1\right)}{3 {\delta}+\sqrt{1-3 {\delta} ({\delta}+2)}+3} \right]\approx -\sqrt{3} \delta.$$
$\bigstar\;\mathbf{d}_{m,m+3}=\mathbf{b}^{\alpha}_{m}-\mathbf{b}^{\beta}_{m+3}=\frac{4\pi}{\sqrt{3}\text{a}}
\begin{pmatrix}
\frac{\sin \theta^{\alpha}+\sin \theta^{\beta}}{1+\delta} \\
2-\frac{\cos \theta^{\alpha}+\cos \theta^{\beta}}{{\delta}+1}
\end{pmatrix},
$
$$\begin{aligned}
\nonumber &A_{m,m+3}\\
\nonumber &= \text{a} \frac{({\delta}+1)}{\sqrt{2}} \frac{1}{ \sqrt{-2 ({\delta}+1) \cos\theta^{\alpha}-2 ({\delta}+1) \cos \theta^{\beta}+2 {\delta} ({\delta}+2)+\cos (\theta^{\alpha}- \theta^{\beta})+3}}\end{aligned}$$
$$\begin{aligned}
\nonumber &n_{m,m+3}\\
\nonumber &= -\frac{16 [ -2 ({\delta}+1) \cos\theta^{\alpha}-2 ({\delta}+1) \cos \theta^{\beta}+2 {\delta} ({\delta}+2)+\cos (\theta^{\alpha}- \theta^{\beta})+3]}{\sqrt{3} \text{a}^2 ({\delta}+1)^2},\end{aligned}$$
with maximum of $A_{m,m+3}$ and minimum of $ n_{m,m+3}$ at $\theta^{\alpha}=\theta^{\beta}=0$.
$\bigstar\;\mathbf{d}_{m,m+4}=\mathbf{b}^{\alpha}_{m}-\mathbf{b}^{\beta}_{m+4}=\frac{2\pi}{\text{a}(1+\delta)}
\begin{pmatrix}
\frac{-3 {\delta}+2 \sqrt{3} \sin \theta+\sqrt{3} \sin \theta^{\prime}+3 \cos\theta^{\prime}-3}{3} \\
\frac{3 {\delta}-2 \cos\theta+\sqrt{3} \sin \theta^{\prime}-\cos\theta^{\prime}+3}{\sqrt{3}}
\end{pmatrix},
$
\
Angles of the critical points (Divergence of $A$ and zero $n$): $$\theta^{\alpha}=-\theta^{\beta}=\tan ^{-1}\left[\frac{\sqrt{3} \left({\delta}-\sqrt{1-3 {\delta} ({\delta}+2)}+1\right)}{3 {\delta}+\sqrt{1-3 {\delta} ({\delta}+2)}+3}\right]\approx \sqrt{3} \delta.$$
$\bigstar\;\mathbf{d}_{m,m+5}=\mathbf{b}^{\alpha}_{m}-\mathbf{b}^{\beta}_{m+5}=\frac{2\pi}{\text{a}(1+\delta)}
\begin{pmatrix}
\frac{ 3 {\delta}-2 \sqrt{3} \sin\theta^{\alpha}+\sqrt{3} \sin\theta^{\beta}-3 \cos \theta^{\beta}+3}{3} \\
\frac{{\delta}-2 \cos\theta^{\alpha}+\sqrt{3} \sin \theta^{\beta}+\cos\theta^{\beta}+1 }{\sqrt{3}}
\end{pmatrix},
$
\
Angles of the critical points (Divergence of $A$ and zero $n$): $$\theta^{\alpha}=-\theta^{\beta}= \tan ^{-1}\left[\frac{-\sqrt{-({\delta}-1) ({\delta}+3)} {\delta}-\sqrt{-({\delta}-1) ({\delta}+3)}+\sqrt{3}}{{\delta} ({\delta}+2)-2}\right] \approx \frac{{\delta}}{\sqrt{3}}.$$
|
{
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---
abstract: |
Present day bio-medical research is pointing towards the fact that virtually almost all diseases are manifestations of complex interactions of genetic susceptibility factors and modifiable environmental conditions. Cognizance of gene-environment interactions may help prevent or detain the onset of complex diseases like cardiovascular disease, cancer, type2 diabetes, autism or asthma by adjustments to lifestyle.
In this regard, we extend the Bayesian semiparametric gene-gene interaction model of [Bhattacharya16]{} to detect not only the roles of genes and their interactions, but also the possible influence of environmental variables on the genes in case-control studies. Our model also accounts for the unknown number of genetic sub-populations via finite mixtures composed of Dirichlet processes, which are related to each other through a hierarchical matrix-normal structure, incorporating gene-gene and gene-environment interactions. An effective parallel computing methodology, developed by us harnesses the power of parallel processing technology to increase the efficiencies of our conditionally independent Gibbs sampling and Transformation based MCMC (TMCMC) methods.
Applications of our model and methods to simulation studies with biologically realistic case-control genotype datasets obtained under five distinct set-ups of gene-environment interactions action yield encouraging results in each case. We followed these up by application of our ideas to a real, case-control based genotype dataset on early onset of myocardial infarction. Beside being in broad agreement with the reported literature on this dataset, the results obtained give some interesting insights to the differential effect of gender on MI.\
[[**Keywords**]{}: *Case-control study; Dirichlet process; Gene-gene and gene-environment interaction; Matrix normal; Parallel processing; Transformation based MCMC.* ]{}
author:
- 'Durba Bhattacharya and Sourabh Bhattacharya[^1]'
bibliography:
- 'irmcmc.bib'
title: ' Effects of Gene-Environment and Gene-Gene Interactions in Case-Control Studies: A Novel Bayesian Semiparametric Approach'
---
[**Introduction**]{} {#sec:intro}
====================
Although many people tend to classify the cause of a disease as either genetic or environmental, only a few diseases like Huntington’s Disease(HD) or GM2 gangliosidosis have so far been identified as purely genetic disorders. As indicated by many epidemiological studies, a different effect of a genotype is often observed on disease risk in persons with different environmental exposures (See [Mapp03]{}, [Khouri05]{}). Also there may be multiple genes which interact with each other to cause a disease only when an environmental factor passes a given threshold, implying thereby that presence of a risk allele may not be exposing all individuals to the same risk.
[Hunter05]{} and [Mather76]{}, point out that estimation of only the separate contributions of genes and environment to a disease, ignoring their interactions, will lead to incorrect estimation of the proportion of the disease (the “population attributable fraction”) that is explained by the genes, the environment, and their joint effect.
Study of gene-environment interaction is important to the field of pharmacogenetics also, since the efficacy and side-effects of some medications can vary depending on an individual’s genotype (see [Scott11]{}). Hence, extensive study of gene-environment interactions through sophisticated statistical modelling is necessary to devise new methods of disease prevention, detection and intervention.
Gene-environment interaction is often conceptualized as genetic control of sensitivity to different environments ([Purcell02]{}). According to [Mather76]{} (see also [Ottman10]{}) gene-environment interaction is defined as “a different effect of an environmental exposure on disease risk in persons with different genotypes”. As genes are the fundamental units of change in an environmental response system, in order to model the gene-environment interaction effectively, it is important to understand the mechanism through which genes and environment interact together to bring about a physiological change in an individual. An environmental exposure could trigger a physiological change in a number of ways. Exposure to certain environmental stimuli may directly or indirectly alter the epigenome of an individual. Exposure to mutagens like high doses of x-ray or nuclear radiation, smoking etc. can enter into the body through tissues and directly interfere with the DNA sequence or replication mechanism. Some environmental stimuli may affect DNA indirectly by altering transcription factors and hence changing the expressions of certain genes. Many gene-gene interactions have been shown to be started by some environmental exposure. For example, excessive alcohol intake has been shown to suppress TACE gene, which then activates less MTHFR, resulting in reduced folate metabolism, causing depression.
Although the study of gene-environment interaction has become essential to the understanding of the aetiology of almost every disease, very little success has so far been achieved in this field. This want of success may be attributed to many causes like inadequacy of models incorporating the complex mechanism through which genes and environment may affect a disease risk ([Wang10]{}). Indeed, given the complexity involved in the gene-environment interactions, no simple linear or additive relationship alone can model the relationship effectively. According to [Wright02]{} and [Wang10]{}, although statistical definition of gene environment interaction may lack clear biological interpretations, quantification of biological interaction should be based on statistical concepts of interaction. Furthermore, inadequacy of data regarding environmental exposure of individuals and stratified population structure are also important factors impeding success of the existing methods in this field. Association tests based on a pooled set of genetically diverse subpopulations (i.e., having differences in allele frequencies across subpopulations) may result in extremely inflated rates of false positives (see [Bhattacharjee10]{}).
The above discussion points towards the fact that the widely-used log-linear models (see, for example, [Mukherjee08a]{}, [Mukherjee08b]{}, [Mukherjee10]{}, [Mukherjee12]{}, [Sanchez12]{}, [Ahn13]{}, [Ko13]{}) are perhaps not quite adequate for modeling complex gene-gene and gene-environment interactions. Moreover, such models consider quite restrictive and ad-hoc association structures for simplifying computation and only attempt to test whether or not the interaction is present without being able to quantify the strength of the interaction. Uncertainty regarding unknown number of subpopulations are also not generally accounted for in the existing interaction models.
Our Bayesian hierarchical mixture model framework is aimed at incorporating all the aforementioned desirable mechanisms through which gene-environment interaction, along with the isolated effects of genes and their interactions may affect an individual’s risk of being affected by a disease, taking into account the fact that the underlying population may be stratified in nature. Since the number of sub-populations is not usually known, one must coherently and carefully account for the uncertainty associated with the unknown number of sub-populations. An additional feature of our model is learning about the number of underlying genetic sub-populations. Because of dependence on environmental variables, our Bayesian semiparametric model comprises Dirichlet process based finite mixture models even at the individual subject level, in addition to genetic and case-control status. The mixtures share a complex dependence structure between themselves through suitable hierarchical matrix-normal distributions, suitably taking account of the dependence induced by the environmental variable. To detect the roles of genes, environment, gene-gene and gene-environment interactions, we extend the gene-gene interaction model and the associated Bayesian hypotheses testing methods of [Bhattacharya16]{} (henceforth, BB), and for the purpose of computation we develop a powerful parallel Markov chain Monte Carlo (MCMC) algorithm which exploits the conditional independence structures inherent in our Bayesian model, and combines the efficiencies of our Gibbs sampling method associated with the mixtures and Transformation based MCMC (TMCMC) of [Dutta14]{}.
The rest of our paper is structured as follows. We introduce our proposed Bayesian semiparametric gene-environment interaction model in Section \[sec:proposal\]. In Section \[sec:detection\] we extend the Bayesian hypothesis testing procedures proposed in BB to learn about the roles of genes, environmental variables and their interactions in case-control studies. In Section \[sec:simulation\_study\] we demonstrate the validity of our model and methods with successful applications to five biologically realistic simulated data sets associated with five different set-ups. We also analysed a case-control type myocardial infarction data set obtained from dbGap with our model and methods, the results of which we report and discuss in detail in Section \[sec:realdata\]. As we point out, our results broadly agree with and in some cases contrast the existing results on this data set. Finally, we summarize our work with concluding remarks in Section \[sec:conclusion\]. Further details are provided in the supplement, whose sections and figures have the prefix “S-" when referred to in this paper.
[**A new Bayesian semiparametric model for gene-gene and gene-environment interactions**]{} {#sec:proposal}
===========================================================================================
[**Case-control genotype data**]{} {#subsec:data}
----------------------------------
For $s=1,2$ denoting the two chromosomes, let $x^s_{ijkr}=1/0$ indicate respectively the presence and absence of the minor allele at $r$-th locus of the $j$-th gene for the $i$-th individual belonging to the $k$-th group of case/control, where $k=0,1$, with $k=1$ denoting case; $i=1,\ldots,N_k$; $r=1,\ldots,L_j$ and $j=1,\ldots,J$; let $N=N_1+N_2$. Let ${\boldsymbol{E}}_i$ denote a set of environmental variables associated with the $i$-th individual. In what follows, we model this case-control genotype data, along with the information on the environmental variables using our Bayesian semiparametric model, described in the next few sections.
[**Mixture models based on Dirichlet processes**]{} {#subsec:mixtures}
---------------------------------------------------
Let ${{\mathbf}{x}}_{ijkr}=(x^1_{ijkr},x^2_{ijkr})$ represent the genotype at the $r$-th locus of the $j$-th gene for the $i$-th individual belonging to the $k$-th group of case/control, and let ${\boldsymbol{X}}_{ijk}=({{\mathbf}{x}}_{ijk1},{{\mathbf}{x}}_{ijk2},\ldots,{{\mathbf}{x}}_{ijkL_j})$ denote the genotype information of the $i$-th individual of the $k$-th group at all the $L_j$ loci corresponding to the $j$-th gene. We assume that for every triplet $(i,j,k)$, ${\boldsymbol{X}}_{ijk}$ have the mixture distribution $$[{\boldsymbol{X}}_{ijk}]=\sum_{m=1}^M\pi_{m ijk}\prod_{r=1}^{L_j}f\left({{\mathbf}{x}}_{ijkr}\vert p_{m ijkr}\right),
\label{eq:mixture1}$$ where $f\left(\cdot\vert p_{m ijkr}\right)$ is the Bernoulli mass function given by $$f\left({{\mathbf}{x}}_{ijkr}\vert p_{m ijkr}\right)=
\left\{p_{m ijkr}\right\}^{x^1_{ijkr}+x^2_{ijkr}}
\left\{1-p_{m ijkr}\right\}^{2-(x^1_{ijkr}+x^2_{ijkr})},
\label{eq:pmf1}$$ and $M$ denotes the [*maximum*]{} number of mixture components possible.
Allocation variables $z_{ijk}$, with probability distribution $$[z_{ijk}=m]=\pi_{m ijk},
\label{eq:alloc_z}$$ for $i=1,\ldots,N_k$ and $m=1,\ldots,M$, allow representation of (\[eq:mixture1\]) as $$[{\boldsymbol{X}}_{ijk}|z_{ijk}]=\prod_{r=1}^{L_j}f\left({{\mathbf}{x}}_{ijkr}\vert p_{z_{ijk} ijkr}\right).$$ Following [Majumdar13]{}, BB, we set $\pi_{m ijk}=1/M$, for $m=1,\ldots,M$, and for all $(j,k)$. Letting ${\boldsymbol{p}}_{m ijk}=\left(p_{m ijk1},p_{m ijk2},\ldots,p_{m ijkL_j}\right)$ denote the vector of minor allele frequencies at the $L_j$ loci of the $j$-th gene for the $i$-th individual of the $k$-th group of case/control corresponding to the $m$-th subpopulation (note that the vector depends upon the chromosomes through the respective genes), we next assume that $$\begin{aligned}
{\boldsymbol{p}}_{1ijk},{\boldsymbol{p}}_{2ijk},\ldots,{\boldsymbol{p}}_{Mijk}&\stackrel{iid}{\sim} {\boldsymbol{G}}_{ijk};\label{eq:dp1}\\
{\boldsymbol{G}}_{ijk}&\sim \mbox{DP}\left(\alpha_{ijk}{\boldsymbol{G}}_{0,ijk}\right),\label{eq:dp2}\end{aligned}$$ where $\mbox{DP}\left(\alpha_{ijk}{\boldsymbol{G}}_{0,ijk}\right)$ stands for Dirichlet process with expected probability measure ${\boldsymbol{G}}_{0,ijk}$ having precision parameter $\alpha_{ijk}$. We specify the base probability measure ${\boldsymbol{G}}_{0,ijk}$ as follows: for $m=1,\ldots,M$ and $r=1,\ldots,L_j$, $$p_{mijkr}\stackrel{iid}{\sim} \mbox{Beta}\left(\nu_{1ijkr},\nu_{2ijkr}\right),
\label{eq:dp3}$$ under ${\boldsymbol{G}}_{0,ijk}$. Coincidences among ${\boldsymbol{P}}_{Mijk}=\left\{ {\boldsymbol{p}}_{1ijk},{\boldsymbol{p}}_{2ijk},\ldots,{\boldsymbol{p}}_{Mijk}\right\}$, which occur with positive probability, is the property of the DP based mixture models that we exploit to learn about the actual number of mixture components. The associated Polya urn distribution of ${\boldsymbol{P}}_{Mijk}$ can be derived by marginalizing over ${\boldsymbol{G}}_{ijk}$: $$\left[{\boldsymbol{p}}_{mijk}|{\boldsymbol{P}}_{Mijk}\backslash \{{\boldsymbol{p}}_{mijk}\}\right]
\sim\frac{\alpha_{ijk}}{\alpha_{ijk}+M-1}{\boldsymbol{G}}_{0,ijk}\left({\boldsymbol{p}}_{mijk}\right)
+\frac{1}{\alpha_{ijk}+M-1}\sum_{m'\neq m=1}^M\delta_{{\boldsymbol{p}}_{m'ijk}}\left({\boldsymbol{p}}_{mijk}\right),
\label{eq:polya}$$ where $\delta_{{\boldsymbol{p}}_{m'ijk}}(\cdot)$ denotes point mass at ${\boldsymbol{p}}_{m'ijk}$. This scheme is useful for constructing an efficient Gibbs sampling strategy for simulating the mixtures conditional on the other parameters, embedded in a parallel MCMC strategy that we devise, bypassing the infinite-dimensional random measure ${\boldsymbol{G}}_{ijk}$. Coincidences among the mixture components associate the triplets $(i,j,k)$ to different mixtures with varying number of components. Indeed, the genotype distributions of any two individuals $i$ and $i'$ arising from a given sub-population with the same gene indexed by $j$ but with different case-control status, are likely to be different, so that $(i,j,k=0)$ and $(i',j,k=1)$ may correspond to different mixtures. Also, for any two genes indexed by $j$ and $j'$, $(i,j,k)$ and $(i,j',k)$ may correspond to different mixtures because of differences in the distribution of genotypes of genes $j$ and $j'$ for the $i$-th individual. Furthermore, for any two individuals indexed by $i$ and $i'$, $(i,j,k)$ and $(i',j,k)$ are likely to be associated with different mixtures because the genotype distribution of the $j$-th gene may be affected by different environmental exposures ${\boldsymbol{E}}_i$ and ${\boldsymbol{E}}_{i'}$. Thus, it seems that the Dirichlet process based mixtures realistically take account of the various genotypic sub-populations and the number of such sub-populations the data arise from.
The above ideas are similar in essence to those in BB, but note that in their case, since the environmental effect ${\boldsymbol{E}}_i$ is not considered, the mixtures were with respect to $(j,k)$ only, not with respect to $(i,j,k)$ as in our current scenario influenced by ${\boldsymbol{E}}_i$.
Following BB, we set $M$, the maximum possible number of sub-populations to be $30$ and $\alpha_{ijk}=10$ in our applications. These choices are not affected by the presence of environmental variables, and performed adequately in our Bayesian analyses.
[**Modeling the complex dependence structure with appropriate modeling of the parameters of ${\boldsymbol{G}}_{0,ijk}$**]{} {#subsec:dependence_structure}
---------------------------------------------------------------------------------------------------------------------------
We specify the dependence structure between the genes and the environment by primarily seeing to it that the environment may act upon gene-gene interaction without affecting the marginal distributions of the genotypes of the individual genes. However, we also take into account the fact that in some cases the environmental variables may cause changes in the distributions of the genotypes. Modelling the parameters of the expected probability measure ${\boldsymbol{G}}_{0,ijk}$ through a relevant hierarchical matrix-normal prior helps us incorporate the complex G$\times$E, G$\times$G and also the SNP$\times$SNP effects appropriately.
### [**Modeling the parameters of ${\boldsymbol{G}}_{0,ijk}$**]{} {#subsubsec:model_G_0}
We model $\nu_{1ijkr}$ and $\nu_{2ijkr}$, for each loci $r=1,\ldots,L_j$, in $j$-th gene, of every individual $i$, having case or control status $k$, that is for every $(i,j,k)$, as the following: $$\begin{aligned}
\nu_{1ijkr}&=\exp\left(u_{jr}+\lambda_{ijk}+\mu_{jk}+{\boldsymbol{\beta}}'_{jk}{\boldsymbol{E}}_i\right);\label{eq:nu_1}\\
\nu_{2ijkr}&=\exp\left(v_{jr}+\lambda_{ijk}+\mu_{jk}+{\boldsymbol{\beta}}'_{jk}{\boldsymbol{E}}_i\right).\label{eq:nu_2}\end{aligned}$$ The complex dependence structure that may exist between the SNPs within a gene and between the genes has been incorporated in our model by the parameters $u_{jr}$ , $v_{jr}$ and $\lambda_{ijk}$, $\mu_{jk}$ respectively (see BB for details). Here ${\boldsymbol{E}}_{i}$ is the $d$-dimensional vector of continuous environmental variables for the $i$th individual. The model can be easily extended to include categorical environmental variables along with the continuous ones.
Note that, non-null ${\boldsymbol{\beta}}_{jk}$ indicates significant marginal effect of the environmental variable ${\boldsymbol{E}}$ on the $j$-th gene. In Section \[subsubsec:matrix\_normal\] we introduce a modeling strategy that accounts for the complex phenomenon through which gene-gene interaction gets modified under the environmental effect, even though the marginal effects of the genes remain unchanged.
### [**Matrix normal prior for $\lambda_{ijk}$’s**]{} {#subsubsec:matrix_normal}
Let ${\boldsymbol{\lambda}}=({\boldsymbol{\lambda}}_1,\ldots,{\boldsymbol{\lambda}}_J)$, where ${\boldsymbol{\lambda}}_j=(\lambda_{1j0},\ldots,\lambda_{n_0j0},\lambda_{1j1},\ldots,\lambda_{n_1j1})$, for $j=1,\ldots,J$. Note that $\lambda_{ijk}$ is shared by every locus of the $j$-th gene of the individual indexed by $(i,k)$.
We consider the following model for ${\boldsymbol{\lambda}}$: $${\boldsymbol{\lambda}}\sim \mathcal N\left({\boldsymbol{\xi}},{\boldsymbol{A}}\otimes\tilde{\boldsymbol{\Sigma}}\right),
\label{eq:mn1}$$ where ${\boldsymbol{A}}$ is the $J\times J$ left covariance matrix, indicating gene-gene interaction in the absence of environmental effect, and $\tilde{\boldsymbol{\Sigma}}={\boldsymbol{\Sigma}}+\phi\mathcal E$ is the right covariance matrix under the effect of the environmental variable $E$. Here $\phi\geq 0$, ${\boldsymbol{\Sigma}}$ is some positive definite matrix, and the $(i,j)$-th element of the positive definite matrix $\mathcal E$, associated with the environmental variable ${\boldsymbol{E}}$, is given by $$\mathcal E_{ij}=\exp\left(-b\|{\boldsymbol{E}}_i-{\boldsymbol{E}}_j\|^2\right),
\label{eq:E}$$ where $b>0$ is a smoothness parameter.
Note that $\phi=0$ indicates absence of environmental effects on gene-gene interaction. It is quite important to observe that, because of the above Gaussian assumption, even for non-zero $\phi$, which points towards indirect effect of environmental factors on the epigenome, triggering genetic interactions, the marginal genotypic distributions associated with the $J$ genes of our model remain unaffected by ${\boldsymbol{E}}$. For convenience, we represent the $JN$-dimensional vector ${\boldsymbol{\lambda}}$ as a $J\times N$ matrix ${\boldsymbol{\Lambda}}$, which has the following probability density function: $$\pi({\boldsymbol{\Lambda}})=\frac{\exp\left[-tr\left\{\tilde{\boldsymbol{\Sigma}}^{-1}\left({\boldsymbol{\Lambda}}-{\boldsymbol{\xi}}\right)^T{\boldsymbol{A}}^{-1}
\left({\boldsymbol{\Lambda}}-{\boldsymbol{\xi}}\right)\right\}\right]}
{\left(2\pi\right)^J\left|{\boldsymbol{A}}\right|^N\left|{\boldsymbol{\Lambda}}\right|^J}.
\label{eq:pi_Lambda}$$ It follows that $${\boldsymbol{\Lambda}}^{col,k}\sim \mathcal N_J\left({\boldsymbol{\xi}}^{col,k},\tilde\sigma_{kk}{\boldsymbol{A}}\right),
\label{eq:mvn_col}$$ where ${\boldsymbol{\Lambda}}^{col,k}$ and ${\boldsymbol{\xi}}^{col,k}$ are the $k$-th columns of ${\boldsymbol{\Lambda}}$ and ${\boldsymbol{\xi}}$, respectively. The covariance matrix between ${\boldsymbol{\Lambda}}^{col,k_1}$ and ${\boldsymbol{\Lambda}}^{col,k_2}$ is given by $$cov\left({\boldsymbol{\Lambda}}^{col,k_1},{\boldsymbol{\Lambda}}^{col,k_2}\right)=\tilde\sigma_{k_1k_2}{\boldsymbol{A}},
\label{eq:mvn_cov}$$ where $\tilde\sigma_{k_1k_2}$ denotes the $(k_1,k_2)$-the element of $\tilde{\boldsymbol{\Sigma}}$. Also, $${\boldsymbol{\Lambda}}^{row,j}\sim \mathcal N_{N}\left({\boldsymbol{\xi}}^{row,j},a_{jj}\tilde{\boldsymbol{\Sigma}}\right),
\label{eq:mvn_row}$$ where ${\boldsymbol{\Lambda}}^{row,j}$ and ${\boldsymbol{\xi}}^{row,j}$ are the $j$-th rows of ${\boldsymbol{\Lambda}}$ and ${\boldsymbol{\xi}}$, respectively. Further, $$cov\left({\boldsymbol{\Lambda}}^{row,j_1},{\boldsymbol{\Lambda}}^{row,j_2}\right)=a_{j_1j_2}\tilde{\boldsymbol{\Sigma}}.
\label{eq:mvn_cov2}$$ In our applications, following BB, we choose ${\boldsymbol{\xi}}={\boldsymbol{0}}$.
To summarize, the matrix-normal prior imposes a dependence structure between the genes through the gene-gene interaction matrix ${\boldsymbol{A}}$, and $\tilde{\boldsymbol{\Sigma}}$ features the direct or indirect effect of the environmental factors, on the epigenome of the individuals. The randomness associated with the matrix-normal prior on ${\boldsymbol{\Lambda}}$ incorporates dependence between the SNPs within a gene.
Further discussion regarding the effect of environmental variables on gene-gene interaction is provided in Section S-1 of the supplement.
### [**Priors for $u_{jr}$ and $v_{jr}$**]{} {#subsubsec:u_v_prior}
We follow BB in setting, for $j=1,\ldots,J$, $u_{jr'}=u_{r'}$ and $v_{jr'}=v_{r'}$ for $r'=1,\ldots,L$, where $L=\max\{L_j;~j=1,\ldots,J\}$, and assuming for $r'=1,\ldots,L$, $$\begin{aligned}
u_{r'} &\stackrel{iid}{\sim} N(0,1);\label{eq:u_r}\\
v_{r'} &\stackrel{iid}{\sim} N(0,1).\label{eq:v_r}\end{aligned}$$
See BB for the details regarding the choice of $u_{jr}$ and $v_{jr}$.
### [**Priors on $\mu_{jk}$, ${\boldsymbol{\beta}}_{jk}$, ${\boldsymbol{A}}$, ${\boldsymbol{\Sigma}}$, $b$ and $\phi$**]{} {#subsubsec:prior_alpha_beta}
We put the following hierarchical priors on ${\boldsymbol{\mu}}=(\mu_{jk};~j=1,\ldots,J;~k=0,1)$ and ${\boldsymbol{\beta}}=({\boldsymbol{\beta}}_{\ell};~\ell=1,\ldots,D)$, where ${\boldsymbol{\beta}}_{\ell}=(\beta_{\ell jk};~j=1,\ldots,J;~k=0,1)$: $$\begin{aligned}
{\boldsymbol{\mu}}&\sim \mathcal N\left({\boldsymbol{0}},{\boldsymbol{A}}_{\alpha}\otimes{\boldsymbol{\Sigma}}_{\alpha}\right)\label{eq:alpha_prior}\\
{\boldsymbol{\beta}}_{\ell}&\stackrel{iid}{\sim}
\mathcal N\left({\boldsymbol{0}},{\boldsymbol{A}}_{\beta}\otimes{\boldsymbol{\Sigma}}_{\beta}\right);~\ell=1,\ldots,D.\label{eq:beta_prior}\end{aligned}$$ For priors on ${\boldsymbol{A}}_{\alpha}$, ${\boldsymbol{A}}_{\beta}$, ${\boldsymbol{\Sigma}}_{\alpha}$ and ${\boldsymbol{\Sigma}}_{\beta}$, we first consider their respective Cholesky decompositions: ${\boldsymbol{A}}_{\alpha}={\boldsymbol{C}}_{\alpha}{\boldsymbol{C}}_{\alpha}'$, ${\boldsymbol{A}}_{\beta}={\boldsymbol{C}}_{\beta}{\boldsymbol{C}}_{\beta}'$, ${\boldsymbol{\Sigma}}_{\alpha}={\boldsymbol{D}}_{\beta}{\boldsymbol{D}}_{\beta}'$ and ${\boldsymbol{\Sigma}}_{\beta}={\boldsymbol{D}}_{\beta}{\boldsymbol{D}}_{\beta}'$. We assume that the diagonal elements of the above Cholesky factors are $iid$ $Gamma(0.01,0.01)$, that is, gamma distribution with mean 1 and variance 100. We assume the non-zero off-diagonal elements of the Cholesky factors to be $iid$ $\mathcal N(0,10^2)$.
Using the same Cholesky decomposition idea, we assume that the off-diagonal elements of the Cholesky factors of ${\boldsymbol{A}}$ and ${\boldsymbol{\Sigma}}$ to be $iid$ $\mathcal N(0,10^2)$, and the diagonal elements to be $iid$ $Gamma(0.01,0.01)$.
We put log-normal priors on $b$ and $\phi$, so that both $\log (b)$ and $\log (\phi)$ are normally distributed with mean zero and variance 100.
Recall that the mixtures associated with gene $j\in\{1,\ldots,J\}$, and individual $i\in\{1,\ldots,N_k\}$ and case-control status $k\in\{0,1\}$, are conditionally independent of each other, given the interaction parameters. This allows us to update the mixture components in separate parallel processors, conditionally on the interaction parameters. Once the mixture components are updated, we update the interaction parameters using a specialized form of TMCMC, in a single processor. A schematic representation of our model and the parallel processing algorithm is provided in Figures \[fig:schematic1\]. Details of our parallel processing algorithm are provided in Section S-2 of the supplement.
![[**Schematic diagram for our model and parallel processing idea:**]{} The arrows in the diagram represent dependence between the variables. The ranks of the processors updating the sets of parameters in parallel using Gibbs sampling are also shown. Once the other parameters are updated in parallel, the interaction parameters are updated using TMCMC by the processor with rank zero.[]{data-label="fig:schematic1"}](plots/Picture1-crop.pdf){width="10cm" height="10cm"}
[**Detection of the roles of environment, genes and their interactions in case-control studies**]{} {#sec:detection}
===================================================================================================
[**Formulation of appropriate Bayesian hypothesis testing procedures**]{} {#subsec:test_formulation}
-------------------------------------------------------------------------
In order to investigate if genes have any effect on case-control, it is pertinent to test $$H_{01}: h_{0j}=h_{1j}; ~j=1,\ldots,J,
\label{eq:H_0}$$ versus $$H_{11}:\mbox{not}~H_{01},
\label{eq:H_1}$$ where $$\begin{aligned}
h_{0j}(\cdot)&=\prod_{i=1}^{N_0}\left\{\sum_{m=1}^M\pi_{m ijk=0}
\prod_{r=1}^{L_j}f\left(\cdot\vert p^r_{m ijk=0}\right)\right\};\label{eq:h_0}\\
h_{1j}(\cdot)&=\prod_{i=1}^{N_1}\left\{\sum_{m=1}^M\pi_{m ijk=1}
\prod_{r=1}^{L_j}f\left(\cdot\vert p^r_{m ijk=1}\right)\right\}.\label{eq:h_1}\end{aligned}$$ We shall also test, for $\ell=1,\ldots,D$; $j=1,\ldots,J$, and $k=0,1$: $$H_{02}:\beta_{\ell j k}=0~\mbox{versus}~H_{12}:\beta_{\ell j k}\neq 0,
\label{eq:beta_test}$$ and $$H_{03}:\phi=0~\mbox{versus}~H_{13}:\phi\neq 0.
\label{eq:phi_test}$$ The cases that can possibly arise and the respective conclusions are the following:
- If $\underset{1\leq j\leq J}{\max}~d(h_{0j},h_{1j})$ is significantly small with high posterior probability, then $H_{01}$ is to be accepted. If $h_{0j}$ and $h_{1j}$ are not significantly different, then it is plausible to conclude that the $j$-th gene is not marginally significant in the case-control study.
- Suppose that $H_{01}$ is accepted (so that genes have no significant role) and that $\beta_{\ell jk}$ is significant, at least for some $\ell$, $j$ and $k$, but $\phi$ is insignificant. This may be interpreted as the environmental variable ${\boldsymbol{E}}$ having some altering effect on the $j$-th gene, that doesn’t affect the disease status. If $\phi$ turns out to be significant, then this would additionally imply that the environmental variable ${\boldsymbol{E}}$ influences gene-gene interaction, but not in a way that causes the disease.
- If $H_{01}$ is rejected, indicating that the genes have significant roles to play in causing the disease, but none of the $\beta_{\ell jk}$ or $\phi$ turn out to be significant, then only genes, not ${\boldsymbol{E}}$, are responsible for causing the disease. In that case, the disease may be thought to be of purely genetic in nature.
- Suppose $H_{01}$ is rejected, $\beta_{\ell j0}$ and $\beta_{\ell j1}$ turn out to be significant, but that $H_{0\ell j}:\beta_{\ell j0}=\beta_{\ell j1}$ is accepted.Then although ${\boldsymbol{E}}$ is insignificant with respect to the marginal effect of gene $j$, it affects the disease status by triggering gene-gene interaction in some genes if $\phi$ turns out to be significant.
- If $H_{01}$ is rejected, $\beta_{\ell jk}$ is significant for some $\ell$, $j$, $k$, and $\phi$ is insignificant, then the presence of ${\boldsymbol{E}}$ has altering effect on some genes, which, in turn, cause the disease. In this case, since $\phi$ is insignificant, ${\boldsymbol{E}}$ does not seem to influence gene-gene interaction.
- If $H_{01}$ is rejected, $\beta_{\ell jk}$ is insignificant for all $\ell$, $j$, $k$, but $\phi$ is significant, then significant effect of ${\boldsymbol{E}}$ on altering the marginal effect of genes is to be ruled out, and one may conclude that the underlying cause of the disease is gene-gene interaction, which has been adversely affected by the environmental variable.
- If $H_{01}$ is rejected, $\beta_{\ell jk}$ is significant for some $\ell$, $j$, $k$, and $\phi$ is also significant, then the environmental variable has possibly significantly affected both the marginal and also gene-gene interaction adversely to cause the disease.
[**Hypothesis testing based on clustering modes**]{} {#subsec:clustering}
----------------------------------------------------
For $k=0,1$, let $i_k$ denote the index of the “central" clusterings of ${\boldsymbol{P}}_{Mijk}=\left\{ {\boldsymbol{p}}_{1ijk},{\boldsymbol{p}}_{2ijk},\ldots,{\boldsymbol{p}}_{Mijk}\right\}$, $i=1,\ldots,N_k$. The concept of central clustering has been introduced by [Sabya11]{}. Significant divergence between the two clusterings of ${\boldsymbol{P}}_{Mi_0jk=0}=\left\{ {\boldsymbol{p}}_{1i_0jk=0},{\boldsymbol{p}}_{2i_0jk=0},\ldots,{\boldsymbol{p}}_{Mi_0jk=0}\right\}$ and ${\boldsymbol{P}}_{Mi_1jk=1}=\left\{ {\boldsymbol{p}}_{1i_1jk=1},{\boldsymbol{p}}_{2i_1jk=1},\ldots,{\boldsymbol{p}}_{Mi_1jk=1}\right\}$, for $j=1,\ldots,J$. clearly indicates that the $j$-th gene is marginally significant. Once $i_0$ and $i_1$ are determined, we shall consider the clustering distance between ${\boldsymbol{P}}_{Mi_0jk=0}$ and ${\boldsymbol{P}}_{Mi_1jk=1}$, denoted by $\hat d\left({\boldsymbol{P}}_{Mi_0jk=0},{\boldsymbol{P}}_{Mi_1jk=1}\right)$, as a suitable measure of divergence. We shall be particularly interested in $$d^*=\max_{1\leq j\leq J}\hat d\left({\boldsymbol{P}}_{Mi_0jk=0},{\boldsymbol{P}}_{Mi_1jk=1}\right);
\label{eq:d_star}$$ In Section S-3 of the supplement we include a brief discussion of the aforementioned methodology. BB point out that although significantly large divergence between clusterings indicate rejection of the null hypothesis, insignificant clustering distance need not necessarily provide strong enough evidence in favour of the null. In other words, even if the clustering distance is insignificant, it is important to check if the parameter vectors being compared are significantly different. In this regard, BB propose an appropriate divergence measure based on Euclidean distances of the logit transformations of the minor allele frequencies. The necessary ideas in our current context are discussed in Section S-3.1 of the supplement. In our case, in order to compute the Euclidean distance, we first compute the averages $\bar{p}_{mijk}=\sum_{r=1}^{L_j}p_{m,ijkr}/L_j$, then consider their logit transformations $\mbox{logit}\left(\bar{p}_{mijk}\right)=\log\left\{\bar{p}_{mijk}/(1-\bar{p}_{mijk})\right\}$. Then, we compute the Euclidean distance between the vectors $$\mbox{logit}\left(\bar{{\boldsymbol{P}}}_{Mi_0jk=0}\right)=\left\{\mbox{logit}\left(\bar{p}_{1i_0jk=0}\right),
\mbox{logit}\left(\bar{p}_{2i_0jk=0}\right),
\ldots, \mbox{logit}\left(\bar{p}_{Mi_0jk=0}\right)\right\}$$ and $$\mbox{logit}\left(\bar{{\boldsymbol{P}}}_{Mi_1jk=1}\right)=\left\{\mbox{logit}\left(\bar{p}_{1i_1jk=1}\right),
\mbox{logit}\left(\bar{p}_{2i_1jk=1}\right),
\ldots, \mbox{logit}\left(\bar{p}_{Mi_1jk=1}\right)\right\}.$$ We denote the Euclidean distance associated with the $j$-th gene by $$d_{E,j}=d_{E,j}\left(\mbox{logit}\left(\bar{{\boldsymbol{P}}}_{Mi_0jk=0}\right),
\mbox{logit}\left(\bar{{\boldsymbol{P}}}_{Mi_1jk=1}\right)\right),$$ and denote $\underset{1\leq j\leq J}{\max}~d_{E,j}$ by $d^*_E$.
[**Formal Bayesian hypothesis testing procedure integrating the above developments**]{} {#subsec:testing}
---------------------------------------------------------------------------------------
In our problem, we need to test the following for reasonably small choices of $\varepsilon$’s: $$H_{0,d^*}:~d^*< \varepsilon_{d^*}\hspace{2mm}\mbox{versus}\hspace{2mm}H_{1,d^*}:~d^*\geq\varepsilon_{d^*};
\label{eq:hypothesis_d_star}$$ $$H_{0,d^*_E}:~d^*_E< \varepsilon_{d^*_E}\hspace{2mm}\mbox{versus}\hspace{2mm}H_{1,d^*_E}:~d^*_E\geq\varepsilon_{d^*_E};
\label{eq:hypothesis_d_star_E}$$ $$H_{0,\beta_{\ell jk}}:~\left|\beta_{\ell jk}\right|< \varepsilon_{\ell jk}\hspace{2mm}
\mbox{versus}\hspace{2mm}H_{1,\beta_{\ell jk}}:~\left|\beta_{\ell jk}\right|\geq\varepsilon_{\ell jk},
\label{eq:hypothesis_beta}$$ $$\mbox{for}~\ell=1,\ldots,D;~j=1,\ldots,J;~k=0,1;$$ $$H_{0,\phi}:~\phi< \varepsilon_{\phi}\hspace{2mm}
\mbox{versus}\hspace{2mm}H_{1,\phi}:~\phi\geq\varepsilon_{\phi}.
\label{eq:hypothesis_phi}$$ If $H_0$ is rejected in (\[eq:hypothesis\_d\_star\]) or in (\[eq:hypothesis\_d\_star\_E\]), we could also test if the $j$-th gene is influential by testing, for $j=1,\ldots,J$, $H_{0,\hat d_j}:~\hat d_j< \varepsilon_{\hat d_j}\hspace{2mm}\mbox{versus}\hspace{2mm}
H_{1,\hat d_j}:~\hat d_j\geq\varepsilon_{\hat d_j}$, where $\hat d_j=\hat d\left({\boldsymbol{P}}_{Mi_0jk=0},{\boldsymbol{P}}_{Mi_1jk=0}\right)$; we could also test $H_{0,d_{E,j}}:~d_{E,j}< \varepsilon_{d_{E,j}}\hspace{2mm}\mbox{versus}\hspace{2mm}H_{1,d_{E,j}}:~d_{E,j}\geq\varepsilon_{d_{E,j}}$.
To test if gene-gene interactions are significant, one may test, following BB, $H_{0,j,j^*}:~\left|{\boldsymbol{A}}_{jj^*}\right|<\varepsilon_{A_{jj^*}}$ versus $H_{1,j,j^*}:~\left|{\boldsymbol{A}}_{jj^*}\right|\geq\varepsilon_{A_{jj^*}}$, for $j^*\neq j$, ${\boldsymbol{A}}_{jj^*}$ being the $(j,j^*)$-th element of ${\boldsymbol{A}}$. If $H_{1,j,j^*}$ is accepted for some (or many) $j^*\neq j$, then this would indicate significant interaction between the $j^*$-th and the $j$-th genes.
As argued in BB, here also it is easily seen that our testing procecure is equivalent to Bayesian multiple testing procedures that minimize the Bayes risk of additive “0-1" and “$0-1-c$" loss functions (see BB for the details; see also [Berger85]{}). Since it is well-known that Bayesian multiple tetsing methods automatically provide multiplicity control through the inherent hierarchy (see, for example, [Scott10]{}), separate error control is not necessary. A brief, schematic representation of the hierarchy of the hypothesis tests is shown in Figure \[fig:schematic3\].
![[**Schematic diagram for our Bayesian testing idea.**]{}[]{data-label="fig:schematic3"}](plots/schematic_testing.pdf){width="10cm" height="10cm"}
Our choices of the $\varepsilon$’s are based on the idea of null model introduced in BB. In a nutshell, we first specify an appropriate null model, which, for example, is the same model as ours but with ${\boldsymbol{A}}$ and $\tilde{\boldsymbol{\Sigma}}$ set to identity matrices to reflect the null hypotheses of “no interaction" and the same mixture distributions under cases and controls for each gene for no genetic effect. From the null model thus specified, we then generate case-control genotype data and fit our general Bayesian model to this “null data" and set $\varepsilon$ to be the $55$-th percentile of the relevant posterior distribution. The rationale and details of this procedure are provided in BB (particularly in Section S-7 of their supplement)
[**Simulation studies**]{} {#sec:simulation_study}
==========================
For simulation studies, we first generate biologically realistic genotype data sets under stratified population with known G$\times$G and G$\times$E set ups from the GENS2 software of [Pinelli12]{}. We consider simulation studies in $5$ different true model set-ups: (a) presence of gene-gene and gene-environment interaction, (b) absence of genetic or gene-environmental interaction effect, (c) absence of genetic and gene-gene interaction effects but presence of environmental effect, (d) presence of genetic and gene-gene interaction effects but absence of environmental effect, and (e) independent and additive genetic and environmental effects.
As we demonstrate, our model and methodologies successfully identify the marginal effects of the genes, along with the G$\times$G and G$\times$E, and the number of sub-populations. Details are provided in Section S-4 of the supplement.
[**Application of our model and methodologies to a real, case-control dataset on Myocardial Infarction**]{} {#sec:realdata}
===========================================================================================================
MI (more commonly, heart attack), has been subjected to much investigation for detecting the underlying genetic causes, the possible environmental factors and their interactions. Application of our ideas to a case-control genotype dataset on early-onset of myocardial infarction (MI) from MI Gen study, obtained from the dbGaP database ([**http://www.ncbi.nlm.nih.gov/gap**]{}), led to some interesting insights into gene-environment and gene-gene interactions on incorporating sex as the environmental factor.
[**Data description**]{} {#subsec:myo_data}
------------------------
The MI Gen data obtained from dbGaP consists of observations on presence/absence of minor alleles at $727478$ SNP markers associated with 22 autosomes and the sex chromosomes of $2967$ cases of early-onset myocardial infarction, $3075$ age and sex matched controls. The average age at the time of MI was 41 years among the male cases and 47 years among the female cases. The data also consists of the sex information of the individuals, which we incorporate in our Bayesian model. The data broadly represents a mixture of four sub-populations: Caucasian, Han Chinese, Japanese and Yoruban. SNPs were mapped on to the corresponding genes using the Ensembl human genome database ([**http://www.ensembl.org/**]{}). However, technical glitches prevented us from obtaining information on the genes associated with all the markers. As such, we could categorize $446765$ markers out of $727478$ with respect to $37233$ genes.
For our analysis, we considered a set of SNPs that are found to be individually associated with different cardiovascular end points like LDL cholesterol, smoking, blood pressure, body mass etc. in various GWA studies published in NHGRI catalogue and augmented this set further with another set of SNPs found to be marginally associated with MI in the MIGen study (see [LucasG12]{}). Our study also includes SNPs that are reported to be associated with MI in various other studies; see [Erdmann10]{}, [LuQi11]{} and [Wang04]{}. In all, we obtained 271 SNPs. Unfortunately, only 33 of them turned out to be common to the SNPs of our original MI dataset on genotypes, which has been mapped on to the genes using the Ensembl human genome database. However, we included in our study all the SNPs associated with the genes containing the 33 common SNPs. Specifically, our study involves the genotypic information on 32 genes covering 1251 loci, including the 33 previously identified loci for $200$ individuals. We chose this relatively small number of individuals to ensure computational feasibility. However, even this data set, along with our model and prior, yielded results that are not only compatible with, but also complement the results established in the literature.
Categorization of the case-control genotype data into the four sub-populations, each of which are likely to represent several further and rather varied sub-populations genetically, implies that the maximum number of mixture components must be fixed at some value much higher than $4$. As before, we set $M=30$ and $\alpha_{jk}=10$ for every $(j,k)$, to facilitate data-driven inference.
We chose a similar set-up for the null model. That is, we chose the same number of genes and the same number of loci for each gene, the same number of cases and controls, the same value $M=30$, but $\alpha_{jk}=1.5$ for every $(j,k)$, as in our simulation studies. We use the same priors as in the real data set-up except that we set ${\boldsymbol{A}}$ and ${\boldsymbol{\Sigma}}$ to be identity matrices to ensure that the genetic interaction is not present and set the same mixture distribution under cases and controls for each gene to ensure the absence of genetic effects.
**Remarks on incorporation of the sex variable in our model** {#subsec:sex_incorporation}
-------------------------------------------------------------
In our case, ${\boldsymbol{E}}_i=E_i$, a one-dimensional binary variable, where $E_i=1$ if the $i$-th individual is male and $E_i=0$ if female. Hence, ${\boldsymbol{\beta}}_{jk}=\beta_{jk}$ is a scalar quantity. In (\[eq:nu\_1\]) and (\[eq:nu\_2\]) we considered the environmental variable to be continuous, but remarked that the model can be easily extended to include categorical variables. Indeed, in this case the exponentials of (\[eq:nu\_1\]) and (\[eq:nu\_2\]) can be thought of as binary regressions with sex as the covariate.
As regards $\mathcal E_{ij}$ of (\[eq:E\]), we first consider $a_0+a_1 E_i$ as a binary regression, and then write $$\mathcal E_{ij}=\exp\left(-\|(a_0+a_1E_i)-(a_0+a_1E_j)\|^2\right)=\exp\left[-a^2_1(E_i-E_j)^2\right],
\label{eq:E_binary}$$ with $b=a^2_1$ being the smoothness parameter. Observe that for the same sex, $\mathcal E_{ij}=1$ while for different sex, $\mathcal E_{ij}=\exp(-b)<1$.
[**Remarks on model implementation**]{} {#subsec:myo_implementation}
---------------------------------------
We first obtain the number of parameters to be updated by TMCMC in our case; other unknowns associated with the mixtures, to be updated using Gibbs steps in parallel. Note that in our case, the interaction matrix ${\boldsymbol{A}}$ is of order $32\times 32=1024$, and the associated Cholesky decomposition then consists of $33\times 16=528$ parameters. Also, ${\boldsymbol{\lambda}}$ is a $NJ=200\times 2=400$-dimensional random vector and ${\boldsymbol{\Sigma}}$ is of order $N\times N=200\times 200$, so that its Cholesky decomposition consists of $201\times 100=20100$ parameters. Furthermore, $\left\{(u_r,v_r):r=1,\ldots,L\right\}$, where $L=207$, consists of $2\times 207=414$ parameters, ${\boldsymbol{\mu}}$ and ${\boldsymbol{\beta}}$ consist of $64$ parameters each, and there are two more parameters $b$ and $\phi$. So, in all, there are $21572$ parameters to be updated simultaneously in a single block using TMCMC.
We implemented our parallel MCMC algorithm detailed in S-2 of the supplement on a VMware consisting of $50$ double-threaded, $64$-bit physical cores, each running at $2493.990$ MHz. In spite of the large number of parameters associated with the interaction part, our mixture of additive and additive-multiplicative TMCMC still ensured reasonable performance.
Our parallel MCMC algorithm takes about $11$ days to yield $100,000$ iterations in our aforementioned VMware machine. We discard the first $50,000$ iterations as burn-in. Informal convergence diagnostics such as trace plots exhibited adequate mixing properties of our parallel algorithm.
[**Results of the real data analysis**]{} {#subsec:realdata_results}
-----------------------------------------
### [**Effect of the sex variable**]{} {#subsubsec:sex_effect}
It turned out that $\varepsilon_{\phi}=1.043069$ and $P(\phi<\varepsilon_{\phi}|\mbox{Data})\approx 1$, so that $\phi$ is clearly insignificant, indicating no differential effect of sex on the genetic interactions. The posterior probabilities $P(|\beta_{1j1}-\beta_{1j0}|<\varepsilon|\mbox{Data})$ are shown in Figure \[fig:beta\_probs\]. As before, $\varepsilon$ is the $55$-th percentile of the posterior distribution of $|\beta_{1j1}-\beta_{1j0}|$ under the null model. Under the 0-1 loss function, the above posterior probability exceeding $0.5$ indicates significant environmental effect on the $j$th gene. From the figure it is interesting to note that there is significant differential effect due to sex on the marginal effects of several genes although sex does not affect the genetic interactions significantly.
![[**Index plots of posterior probabilities of no environmental effect with respect to $|\beta_{1j0}-\beta_{1j1}|<\varepsilon$, for $j=1,\ldots,32$.**]{}[]{data-label="fig:beta_probs"}](plots_realdata/plot_same_marginal_env_effect-crop.pdf){width="15cm" height="6cm"}
### [**Influence of genes and gene-gene interactions on MI based on our study**]{} {#subsubsec:influential_genes}
Our Bayesian hypotheses testing using the clustering metric yielded $P\left(d^*<\epsilon_1|\mbox{Data}\right)\approx 0.35202$ while that with the Euclidean distance we obtained $P\left(d^*_E<\epsilon_2|\mbox{Data}\right)\approx 0.51078$. In other words, it seems rather debatable whether or not the genes have significant overall effect on MI. This is in sharp contrast with the results obtained by BB where both clustering metric and Euclidean distance confirmed significant overall genetic influence on MI. However, both the posterior probabilities are substantially large, practically indicating that the genes are not very significant. As far as testing of significance of the individual genes are concerned, it turned out that under the clustering metric, except genes $SMARCA4$, $RBMS1$, $COL4A1$, $RP11-306G20.1$, $MRAS$, $SLC22A1$, $CDKAL1$, $PCSK9$, $ADAMTS9-AS2$, and $AP006216.5$, the rest turned out to be significant, while with respect to the Euclidean metric the only insignificant genes are $AP006216.10$, $CELSR2$, $MRAS$, $PCSK9$, $OR4A48P$ and $BUD13$. The posterior probabilities of the null hypotheses (of no significant genetic influence) are shown in Figure S-3 of the supplement. The figure reveals that the posterior probabilities of no significant genetic influence, although generally did not cross $0.5$, are not adequately small to reflect very strong evidence against the null hypotheses. This is consistent with the result on overall genetic significance that we obtained.
The actual gene-gene correlations based on medians of the posterior covariances, are shown in Figure S-4 of the supplement. The color intensities correspond to the absolute values of the correlations. Consistent with the figure, all the tests on interaction turned out to support the hypotheses of no interaction.
Thus, individual genes have impact on MI but not gene-gene interactions. Moreover, the relatively weak evidences against the null suggest that external factors, in our case sex, may be playing a bigger role in explaining case-control with respect to MI. As such, given our data set of size $200$ with $77$ cases, the empirical conditional probability of a male given case is $0.3766234$, while the empirical conditional probability of a male given control is $0.504065$, indicating that with respect to our data, females seem to be more at risk compared to males. Coherency of Bayesian models in general is instrumental in reflecting this information in our inference in the way of downplaying the genes, suggesting at the same time that the only external factor, namely, sex, must have more important effect.
A detailed investigation of the disease predisposing loci detected by our model and methods, and the role of SNP-SNP interactions behind such disease predisposing loci, is carried out in Section S-5 of the supplement, and a discussion on the posterior distribution of the number of distinct mixture components is provided in Section S-6 of the supplement.
[**Discussion of our Bayesian methods and GWAS in light of our findings**]{} {#subsec:discussion}
----------------------------------------------------------------------------
Our results of Bayesian analysis of the MI data set demonstrate that sex plays more significant role than the genes in triggering the disease, and in particular, do not support gene-gene interaction. In these regards, our results significantly differ from those obtained by BB, who do not consider the sex variable in their model. Since as per our inference sex seems to be far more influential compared to the genes with respect to MI, there is internal consistency of our more general gene-gene and gene-environment interaction model with the gene-gene interaction model of BB. It is important to note that [Lucas12]{} analyzed the same MI dataset using logistic regression and reached the same conclusion as ours that there is no significant gene-gene interaction. Since two completely different methods of analyses are in such strong agreement, it is pertinent to presume that the data contains enough information on the lack of gene-gene interaction. However, as we demonstrated, SNP-SNP correlations have important roles to play in determining the DPLs. These are responsible for suppression of the SNPs considered influential in the literature by implicit induction of negative correlations between Euclidean distances between cases and controls for the associated SNPs. Thus, even though the genes did not turn out to be as significant, it is clear that sophisticated nonparametric modeling of gene-gene and SNP-SNP interactions is of utmost importance.
[**Summary and conclusion**]{} {#sec:conclusion}
==============================
In this paper, we have extended the Bayesian semiparametric gene-gene interaction model of BB to realistically include the case of gene-environment interactions. Careful attention has been paid to the fact that in the absence of mutation, the environmental variable does not affect the marginal genotypic distributions, in spite of influencing gene-gene interaction. Needless to mention, our model considers dependence between SNPs as well to account for LD effects, in addition to gene-gene, gene-environment and dependencies between individuals. Besides, our model, via Dirichlet processes, facilitates learning about the number of genotypic sub-populations associated with the individuals and the genes, while accounting for the environmental effect at the same time. We extend the Bayesian hypotheses testing methods introduced in BB to enable test for significances of marginal genetic and environmental effects, gene-gene interactions, effect of environment on gene-gene interaction and mutational effect. The basis for our tests are extensions of the clustering metric based tests proposed by BB to account for the environmental variables, in conjunction with the tests based on Euclidean metric. We recommended careful application of our tests based on the clustering metric, followed by re-confirmation with respect to the Euclidean metric.
On the Bayesian computational side, we propose a powerful parallel processing algorithm that takes advantage of the conditional independence structures built within our model through the Dirichlet process based mixture framework for parallelisation, and is complemented by the efficiency of TTMCMC, which updates the interaction parameters within a single processor.
We validate our model and methodologies with applications to biologically realistic datasets generated from under $5$ different set-ups characterized by different combinations and structures associated with gene-gene and gene-environment interactions. Adequate performance of our model and methods are demonstrated in every situation. Additionally, our ideas correctly captured the true number of genetic sub-populations in each case, and attempted to capture the DPL adequately even in the face of highly complex dependence structures. We apply our model and methods to the MI Gen data set also studied by BB and because of inclusion of the sex variable, succeeded in obtaining results that are quite compatible with those reported in the literature. Although the gene-gene interactions turned out to be insignificant, the SNP-SNP correlations associated with case-control Euclidean distances facilitated understanding the mismatch of our DPL with those reported in the literature as having significant impact on MI. Interestingly, our Bayesian approach allowed us obtain insightful results even with a sample consisting of only $200$ individuals, showing the importance of building sophisticated models and prior structures, and efficient computational methods and technologies.
[^1]: Durba Bhattacharya is an Assistant Professor in St. Xavier’s College, Kolkata, pursuing PhD in Interdisciplinary Statistical Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108. Sourabh Bhattacharya is an Associate Professor in Interdisciplinary Statistical Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108. Corresponding e-mail: [email protected].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the orbital structure in a series of self-consistent $N$-body configurations simulating rotating barred galaxies with spiral and ring structures. We perform frequency analysis in order to measure the angular and the radial frequencies of the orbits at two different time snapshots during the evolution of each $N$-body system. The analysis is done separately for the regular and the chaotic orbits. We thereby identify the various types of orbits, determine the shape and percentages of the orbits supporting the bar and the ring/spiral structures, and study how the latter quantities change during the secular evolution of each system. Although the frequency maps of the chaotic orbits are scattered, we can still identify concentrations around resonances. We give the distributions of frequencies of the most important populations of orbits. We explore the phase space structure of each system using projections of the 4D surfaces of section. These are obtained via the numerical integration of the orbits of test particles, but also of the real $N$-body particles. We thus identify which domains of the phase space are preferred and which are avoided by the real particles. The chaotic orbits are found to play a major role in supporting the shape of the outer envelope of the bar as well as the rings and the spiral arms formed outside corotation.'
date: Released 2007 April 12
title: 'Orbital structure in $N$-Body models of barred-spiral galaxies'
---
\[firstpage\]
galaxies: structure, kinematics and dynamics, spiral.
Introduction
============
The study of the orbits in barred galaxies begun in the late 70’s [@b45; @b46] when the properties of the orbits in simple models of barred galaxies were derived theoretically by use of a ‘third integral’ of motion besides the Hamiltonian. @b47 made an extended study of the [*periodic orbits*]{} in weak and strong bars, establishing the standard nomenclature thenceforward (see @b103 and @b43 for a review of the various families of periodic orbits both inside and outside corotation). Many works have been devoted to the study of both periodic and non-periodic orbits in 2D [e.g. @b56; @b18; @b104; @b27; @b44] or 3D models [e.g. @b57; @b50; @b22; @b23; @b24; @b25; @b21]. In most studies an ‘ad hoc’ choice of model is made for the gravitational potential. However, some studies have addressed the question of the orbital structure in $N$-body systems of barred galaxies [e.g. @b32; @b33; @b34; @b35; @b36; @b16] in which, by definition, the orbits support the system self-consistently. Studies of the latter type are, in fact, necessary in order to identify which families of orbits are most relevant to the maintenance of self-consistency, i.e. to the production, by the orbits, of ‘response’ patterns matching those of the imposed potential/density field.
![The face-on and edge-on density profiles (left and right column respectively) for the experiment QR3 at $t=20T_{hmct}$ and at $t=300T_{hmct}$ (first and second row respectively).[]{data-label="fig:1"}](fig01.eps){width="8.5cm"}
![The rotation curves (a) and the velocity dispersions (b), for all the experiments at $t=20T_{hmct}$, along the major axis considering the line-of-sight along the middle axis. Lines from light gray to black correspond to the experiments from QR1 to QR4 respectively.[]{data-label="fig:2"}](fig02.eps){width="8.5cm"}
By constructing self-consistent models simulating real barred-spiral galaxies, @b49 pointed out the role of the [*chaotic orbits*]{} in supporting the inner parts of the spiral arms emanating beyond the bar. The orbits considered by Kaufmann & Contopoulos belong to the so-called ‘hot population’ (Sparke and Sellwood 1987), i.e. they wander stochastically, partly inside and partly outside corotation. On the other hand, the same authors found that the outer parts of the spiral arms are supported mainly by regular orbits. The study of the role of the chaotic orbits in the spiral structure was pursued by @b31, using self consistent $N$-body simulations of barred galaxies. These authors found long living spiral arms composed almost entirely by chaotic orbits. A new mechanism was proposed [@b10; @b100; @b39; @b101; @b140] according to which the invariant manifolds of the short period unstable periodic orbits around the stationary Lagrangian points $L_1$ and $L_2$ are responsible for the long-term support of the spiral structure. On the other hand @b40 argued that the innermost parts of the spiral arms, formed as a continuation of the bar, a little inside corotation, are due to chaotic orbits exhibiting, for long time intervals, a 4:1 resonance orbital behavior.
In @b1 (hereafter paper I) a detailed investigation was made of the orbital structure in one system belonging to the series of $N$-body experiments reported in @b31. This particular simulation represents a barred galaxy with nearly no spiral structure. Using i) the method of frequency analysis (Laskar 1990), and ii) 2D projections of the 4D surface of sections, obtained via the numerical integration both for test particles and of the real $N$-body particles, the relative percentages were found of a number of distinct populations which support the bar. The analysis was carried out separately for the regular and the chaotic orbits. A significant fraction of chaotic orbits were found to lie inside corotation, thus supporting the shape of the bar. Moreover it was shown that in a 2D approximation in which the system’s thickness is ignored, the chaotic orbits are limited by tori or cantori so that many orbits remain confined inside limited domains outside corotation for times comparable to the Hubble time. Arnold diffusion through the third dimension was found ineffective over such time periods, thus the results are applicable in the 3D case as well.
In the present paper we extend the above study in the whole series of $N$-body experiments reported in @b31, focusing in particular in those systems which simulate secularly evolving barred galaxies with significant spiral structure. The simulations represent self-consistent systems of a wide range of different pattern speeds of the bar. One main goal is to determine the dependence of the percentage of the different populations (particles in different types of orbits) on the value of the pattern speed. It should be stressed that the pattern speed of one system slows down as the system evolves secularly. To account for this effect, we study the orbital structure in each experiment at two distinct and well separated time snapshots, one close to the beginning of the simulation and one close to the final state (corresponding to one Hubble time evolution). This analysis yields the changes induced upon the different types of orbits, as well as upon the distributions of the $N$-body particles in the orbital space, by the secular evolution of each galaxy towards an equilibrium state.
In order to accomplish the orbital classification, we implement the method of frequency analysis both for the regular and for the chaotic orbits of each system. The method of frequency analysis was first introduced in the study of bar like potentials [@b150] and in the study of resonances between planetary orbits in the Solar system [@b2], and within the framework of galactic dynamics it was implemented in the orbital study of elliptical galaxies [@b60; @b59; @b3; @b58; @b4; @b5], or barred galaxies [@b14; @b17; @b1; @b16]. Here we extend the implementation of the method to the problem of spiral structure as well.
{width="12.0cm"}
Besides identifying various types of regular orbits, concentrated around well known resonances, we find that many chaotic orbits, especially those being confined inside corotation, also exhibit a large degree of concentration around specific resonances. The main difference is that the chaotic orbits yield more diffused frequency maps than the regular orbits. Most chaotic orbits inside corotation can be further characterized as ‘weakly chaotic’, i.e. they have low values of the Lyapunov exponents. These orbits support the outer region of the bar. On the other hand, the chaotic orbits outside corotation are in principle capable of escaping. In most cases, however, the escape time is much longer than one Hubble time. Thus, in practice many chaotic orbits outside corotation are also trapped by various resonances exhibiting ‘stickiness’ along the invariant manifolds of the unstable periodic orbits near the Lagrangian points $L_1, L_2,
L_4$ and $L_5$. We demonstrate that the different populations of such orbits are responsible for practically all observed significant morphological features beyond the bar, and in particular for the observed rings and/or spiral arms.
![The $c$-parameter as a function of the pattern speed of the bar for all the experiments and time snapshots.[]{data-label="fig:4"}](fig04.eps){width="6cm"}
Further examination of the phase space is performed using projections of the 4D surface of section of test particles. Using the same technique for the real $N$-body particles, an important information is recovered, namely the location of those domains of the phase space which are preferred and those which are avoided by the real particles.
The paper is organized as follows. In section 2 we describe briefly the $N$-body models and discuss the correlation between the pattern speed and the boxiness of the bar. In section 3.1 we present briefly the methods used for the classification of the orbits into regular and chaotic, we introduce the frequency analysis and we give the main results regarding the detailed orbital structure for all the models. In section 3.2 we examine the phase space structure through surfaces of sections using test particles as well as real $N$-body particles. Finally section 4 summarizes the conclusions of the present study.
Description of the models
=========================
The series of simulations referred to in the sequel are described in detail in @b31. The experiments were evolved using the **S**mooth **F**ield **C**ode (hereafter SFC) of @b37. The initial conditions were created by introducing rotation in the so called ‘Q-model’ of @b30 in the way described below.
A measure of the angular momentum $J$, for a gravitational system of total mass M and total binding energy E, can be given in terms of Peebles’ (1969) ‘spin parameter’ $\lambda$: $$\lambda=\frac{J|E|^{1/2}}{GM^{5/2}}$$
The Q-model of @b30 has initially an almost zero spin parameter, simulating an E7 elliptical galaxy [in contrast, the value of $\lambda$ for, say, the disc of our Galaxy is $\lambda_G=0.22$, see @b102]. In fact, a value of $\lambda$ close to the one observed in real barred-spiral galaxies cannot be obtained via a purely collisionless evolution of either a monolithically collapsing object, embedded in the tidal field of surrounding density perturbations, or a system formed by mergers between subclumps in a hierarchical clustering process. In order to produce systems with a $\lambda$ parameter appropriate for barred galaxies, while running a collisionless simulation, @b31 introduced a process called ‘re-orientation of the velocities’. Starting from an equilibrium triaxial system (e.g. the Q-system), all the particles’ velocity vector components on the plane defined by the intermediate and long axes of the triaxial equilibrium figure are reoriented as to render each particle’s velocity vector perpendicular to its position vector. Furthermore, the re-orientation is such that the new velocity vector defines a clockwise sense of the particle’s orbital revolution. This process yields the maximum possible rotation of a system, on the above referenced plane, with respect to the progenitor system which has the same density and equal distribution of the velocity moduli. The experiment obtained from the Q-system via ‘velocity re-orientation’ was called QR1 [see @b31]. This was left to run self-consistently, using an improved SFC version, for one Hubble time. In a similar way, the experiment QR2 was produced by introducing a new velocity re-orientation at a snapshot of the QR1 run corresponding to 20 half mass crossing times (hereafter $T_{hmct}$). The experiment QR3 was produced in the same way from QR2, and so on. We shall consider four $N$-body experiments of this sequence (QR1 to QR4). All these experiments have the same number of particles ($\approx
1.5\times10^5$) and the same binding energy but different amounts of total angular momentum, resulting therefore in different values of the bar’s pattern speed. Further details are provided in @b31. It must be pointed out here that in our self-consistent systems the particles cannot be identified a priori as belonging to a disc, halo or bulge component. The systems are flattened as a whole and when viewed edge-on they present a shape called ‘thick disc’ in @b31. The dynamical effects caused by the particles in the ‘thick disk’ were analyzed in details in @b101, in subsection 2.1, where it was shown that the particles with large vertical oscillations create a gravitational field which partly mimics the effect of a halo up to the code’s truncation radius. Moreover, it is possible to recognize even the bulge that is created self-consistently in barred galaxies, using a method proposed by observers, i.e. by plotting the density profile along and parallel to the major or minor axis of the bar. The bulge length is marked by the increased light distribution over the exponential disc, well above the bar (see Paper I for details).
In each experiment, the time unit is taken equal to the $T_{hmct}$. A Hubble time corresponds to $\approx300T_{hmct}$. The length unit is taken equal to the half mass radius (hereafter $r_{hm}$). Finally, the plane of rotation is the $y-z$ plane (intermediate-long axes) and the sense of rotation is clockwise.
In Fig.1 the density of the particles of the experiment QR3 is plotted face-on (Fig.1a,c) and edge-on (Fig.1b,d), for two different snapshots $t=20T_{hmct}$ (Fig.1a,b) and $t=300T_{hmct}$ (Fig.1c,d). Some other snapshots of these four experiments in the ordinary space can be seen in fig.1 of @b31.
Figure 2a shows the rotation curves obtained by calculating the mean values of the line-of-sight velocity profiles of the particles when the systems are viewed edge-on and the bar is side-on (with the line of sight along the middle axis), for all the experiments from QR1 (light gray line) to QR4 (black line) at time $t=20T_{hmct}$. In Fig.2b the corresponding velocity dispersion profiles are plotted.
The slow down of the pattern speed in our experiments (see fig.4a in Voglis et al. 2006a) is smaller compared with other simulations using ‘live halos’ (see for example Athanassoula 2003 and Martinez-Valpuesta et al. 2006). Moreover, the evolution towards an equilibrium state seems faster. One possible reason is that, in our simulations, a bar already exists from the beginning of the calculations $(t=0)$, where rotation has been inserted, via re-orientation of the velocities’ vectors, in all the particles of the system. Moreover, our systems present an important velocity dispersion especially in the central region (see Fig.2b) at time $t=20T_{hmct}$ which is close to the initial conditions, and this favors the faster slow down of the bar’s pattern speed according to the main conclusion of Athanassoula (2003).
Apart from the QR1 experiment, in which the spiral structure does not survive for more that $\approx15T_{hmct}$, in all the other experiments there are $m=2$ spiral modes surviving for 300 $T_{hmct}$, i.e. for one Hubble time. The amplitude of these modes undergoes oscillations with an overall tendency to decay. However, during this secular evolution the spiral modes are more prominent at particular time snapshots. In all three models QR2, QR3 and QR4 the spiral structures are relatively strong at t=20$T_{hmct}$.
Figure 3 shows the ellipticity $e_{yz}=1-b/a$ as a function of the major semi-axis $a$ of an isophote ($b$ is the isophote’s minor semi-axis) for the isophotes corresponding to the projected surface density on the $y-z$ plane of each system. Fig.3a corresponds to an early snapshot $t=20T_{hmct}$ far from equilibrium, while Fig.3b corresponds to $t=300T_{hmct}$, at which all systems have approached an equilibrium state. We see that, for $t=20T_{hmct}$, all the experiments present a nearly constant ellipticity along the bar, while, as the systems relax, the ellipticity profile acquires a declining slope, the isophotes becoming nearly round, i.e. the 3D matter distribution becoming nearly oblate beyond the end of the bar. The calculations for $t=20T_{hmct}$ (Fig.3a) are limited within a radius $R=1r_{hm}$ because all the experiments, except QR1, present conspicuous spiral structures beyond this radius at $t=20T_{hmct}$ so that the isophotes in this region cannot approximated by ellipses.
Boxiness of the isophotes is a well known feature in barred galaxies. This effect can be measured by the shape parameter $c$, which can be determined by the equation of the generalized ellipse (Athanassoula et al. 1990): $$\label{cparam}
\left(\frac{|y|}{b}\right)^c+\left(\frac{|a|}{b}\right)^c=1$$ where $a$ and $b$ are the major and minor semi-axes of an isophote and $c$ is the parameter describing the shape of a generalized ellipse best-fitting the isophote. For $c=2$ we obtain a standard ellipse, for $c>2$ the shape approaches a rectangular parallelogram and for $c<2$ we have a shape similar to lozenge.
\[tabt150\]
experiments QR1 QR1 QR2 QR2 QR3 QR3 QR4 QR4
------------------------------------ -------- --------- -------- --------- -------- --------- -------- ---------
snapshots $t=20$ $t=300$ $t=20$ $t=300$ $t=20$ $t=300$ $t=20$ $t=300$
$\Omega_p (rad/T_{hmct})$ 0.4 0.36 0.7 0.46 0.82 0.52 0.96 0.59
% chaotic motion 59.6 57.9 61.8 57.3 63.0 59.4 56.6 56.4
% chaotic motion inside corotation 28.5 27.8 12.6 16.0 9.8 14.9 7.4 11.1
% regular motion (2:1) 12.6 16.5 18.4 21.9 14.9 17.8 21.6 21.2
% regular motion (3:1) - - 18.7 - 19.8 - 18.5 -
% regular motion (A) 25.5 20.4 - 15.8 - 21.5 - 20.8
% regular motion (B) - 4.0 - 2.6 - 1.1 - 1.2
% chaotic motion (2:1) 1.8 8.7 1.7 2.7 1.3 2.0 1.1 1.7
% chaotic motion (3:1) 5.4 4.7 5.6 3.7 5.0 3.7 4.5 3.0
% chaotic motion (4:1) 4.2 6.5 4.3 4.4 4.2 1.5 3.1 2.7
% chaotic motion (A) 4.8 12.4 - 9.7 - 7.9 - 6.0
% chaotic motion (cor) 6.2 6.4 2.9 2.6 2.5 1.7 2.6 2.1
% chaotic motion (-2:1) 2.4 3.2 2.4 1.7 2.5 0.7 3.1 0.6
% chaotic motion (-1:1) - 2.0 - 2.3 4.4 1.9 4.4 2.1
% chaotic motion ($q<-1$) 14.8 8.7 30.0 23.1 35.2 33.4 26.1 35.7
Fig.4 shows the shape parameter $c$ as a function of the pattern speed of the bar at the time snapshots specified within the figure for each experiment. Each point’s ordinate corresponds to the mean value of the parameter $c$ of the generalized ellipses having major semi-axis between $0.4r_c$ and $0.7r_c$, where $r_c$ is the system’s corotation radius at the indicated snapshot. The abscissa of each point corresponds to the associated pattern speed of the bar. For values of $\Omega_p$ smaller that $\approx0.7 rad/T_{hmct}$ the corresponding $c$-parameter seems to have a flat behavior with $c>2$. However, for values greater than $\approx0.7 rad/T_{hmct}$ a declining relation is apparent indicating that the boxiness decreases as the pattern speed $\Omega_p$ increases. Similar study for elliptical galaxies has been presented in @b53 [@b52].
![The frequency map for the regular orbits at $20T_{hmct}$ (left column) and at $300T_{hmct}$ (right column) for the four experiments. The ‘disc’ resonances of the form $m_1\frac{\omega_\theta}{\omega_x}+m_2\frac{\kappa}{\omega_x}=0$ are shown by gray lines.[]{data-label="fig:5"}](fig05.eps){width="8.1cm"}
Study of the orbital structure
==============================
Frequency analysis of the orbits
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The distinction of the regular from the chaotic orbits is based on a combination of two methods, namely the [*Specific Finite Time Lyapunov Characteristic Number*]{} (SFTLCN) or simply $L_j$, introduced by @b31, and the [*Smaller ALignment Index*]{}, (SALI), or simply Alignment Index ($AI_j$) [@sk2001; @b30]. For more details on the implementation and efficiency of these combined methods see Paper I.
The percentage of chaotic orbits in all the experiments is close to $60\%$ (see Table 1), which is almost twice the corresponding percentage found in the non-rotating progenitor model, i.e. the Q-model [@b30]. The general conclusion in @b31 is that rotation enhances chaos, not only as regards the fraction of mass in chaotic motion, but also as regards the magnitude of the Lyapunov numbers that are seriously shifted towards higher values. The authors argued that rotation introduces instabilities and a sequence of important overlapping resonances between the pattern speed $\Omega_p$ and the frequencies of orbits. This fact is imprinted in the increase of the chaotic motion. The same conclusion has been reached in other studies dealing with orbits in bar like galaxies. For example, @b170, noted that an increase of the pattern speed of the bar model simulating the Milky way increases significantly the chaotic regions on the Poincaré map because of resonance overlap phenomena. Also, @b180, found that the combination of a triaxial halo with a fast-rotating bar leads to the development of chaos. Finally, @b160 found that the fraction of the chaotic orbits is enhanced in triaxial stellar systems when rotation is inserted. However, in all the experiments the fraction of mass that can develop effective chaotic diffusion within a Hubble time is less than $45\%$ [fig.12b in @b31] which means that a significant percentage of orbits are weakly chaotic.
In the experiment QR1 there appears a transient trailing spiral structure as a result of the transfer of angular momentum to the material outwards [@b11; @b12; @b13; @b14]. The transient spiral arms disappear quickly (after $\approx15 T_{hmct}$). As a result, the system resembles a bar-like galaxy without any spiral structure. The orbital analysis of this experiment at $t=300T_{hmct}$ was the subject of Paper I.
Here, we exploit the tool of frequency analysis [@b2; @b60; @b59] using the improved code of @b48. We apply this technique separately for the regular and the chaotic orbits in all four experiments and for two different snapshots corresponding to 20$T_{hmct}$ and 300$T_{hmct}$, respectively. In order to obtain the frequency analysis of the orbits at one snapshot, all the orbits are integrated forward in the ‘frozen’ potential corresponding to that snapshot, and for a time equal to 150 radial periods. The Jacobi constant $E_j$ of one orbit is given by the relation $$E_j=\frac{1}{2}(v_x^2+v_y^2+v_z^2)+V(x,y,z)-\frac{1}{2} \Omega_p^2
R_{yz}^2$$ where $V(x,y,z)$ is the full 3D ‘frozen’ potential, given by the SFC code as an expansion of a bi-orthogonal basis set, $v_x, v_y,
v_z$ are the velocities in the rotating frame of reference and $\Omega_p$ is the angular velocity of the bar at the studied snapshot (see Table 1 for the values of $\Omega_p$ for all the models in the two snapshots). The value $R_{yz}$ is the distance from the rotation axis.
The integrated orbits belong in a wide range of Jacobi integral values which implies a wide range of orbital periods (e.g. radial periods) of each orbit. This means that within a certain time (e.g a Hubble time) each orbit has a quite different dynamical evolution. In order to treat this integration process with the minimum computational cost, we have used a Runge-Kutta 7(8) order integrator with variable time step. This technique guaranties a relative error of the Jacobi constant less than $10^{-8}$ for all the integrations we provide below. Moreover, by requiring a relative error less than $10^{-9}$ we obtained very similar results, which implies that the presented results are robust.
By running the orbits in the ‘frozen’ potential, we then calculate the following frequencies for each orbit:
a\) the radial frequency $\kappa$ (frequency of epicyclic oscillations) which is derived from the time series of the radial velocity $\dot{r}(t)$
b\) the angular frequency in the rotating frame of reference, $\omega_{\theta}$, or the inertial frame, $\Omega$, which are derived from the time series of the corresponding polar angle
c\) the vertical frequency, $\omega_x$ which is derived from the time series of the coordinate $x(t)$.
![The distribution of the frequency ratio $q$ for the regular orbits at 20$T_{hmct}$ (left column) and at 300$T_{hmct}$ (right column). The main type of orbits supporting the bar is the 2:1 resonant orbit (or ‘x1’ type of orbits, using the nomenclature of Contopoulos & Papayannopoulos, 1980) for both snapshots. The second important type of orbits is either the 3:1 resonant orbit or the group ‘A’ depending on the system and on the time.[]{data-label="fig:6"}](fig06.eps){width="8.1cm"}
The main resonances can now be detected using frequency maps. Figure 5 shows the rotation numbers $(\frac{\kappa}{\omega_x},\frac{\omega_{\theta}}{\omega_x})$ of the ensemble of [*regular*]{} orbits for all the experiments at the two studied snapshots. Each row (column) of panels corresponds to a different system (snapshot) as indicated in the figure. In each panel, one point corresponds to one orbit of a real $N$-body particle that turns to be regular. In the ‘frozen potential’ approximation the position of this point on the $(\frac{\kappa}{\omega_x},\frac{\omega_{\theta}}{\omega_x})$ plane remains invariant in time, since the particle’s orbit stays confined on an invariant torus with the given values of the rotation numbers.
The most important resonances are the ‘disc’ resonances which are of the form $m_1\frac{\omega_\theta}{\omega_x}+m_2\frac{\kappa}{\omega_x}=0$, with $m_1,m_2$ integers. Such resonances are shown by gray straight lines in Fig.5. Orbits lying exactly on them belong to two groups:
![The three projections of real $N$-body orbits corresponding to the most important resonances. Regular orbits are plotted in the upper panel and chaotic ones in the lower panel. Note that the bar’s major axis is along the $z$-axis.[]{data-label="fig:7"}](fig07.eps){width="7cm"}
a1) if no second resonance relation of the form $m_1'\frac{\omega_\theta}{\omega_x}+
m_2'\frac{\kappa}{\omega_x}+m_3'=0$, with all three integers $m_i'\neq 0$, is satisfied, the orbit lies on a torus of dimension smaller than or equal to three. The orbit is quasi-periodic, and it is the product of a vertical oscillation and of a motion on the $y-z$ plane which is either exactly periodic or a ‘tube’ around a planar periodic orbit. The motion in the third dimension is essentially decoupled from the motion on the 2D plane. In practice, we find that the great majority of $N$-body particles in exactly resonant orbits populate precisely the above group. Furthermore, most of them exhibit vertical oscillations of a rather small amplitude, compared to the motion on the $y-z$ plane (this is expected since the system as a whole is substantially flattened in the $x$-axis). Thus a study of dynamics via a 2D approximation is sufficient to unravel all interesting properties of such orbits.
a2) if a second resonance relation of the form $m_1'\frac{\omega_\theta}{\omega_x}+
m_2'\frac{\kappa}{\omega_x}+m_3'=0$ is satisfied, the orbits are called ‘doubly resonant’. The orbits can be exactly 3D-periodic, corresponding e.g. to a vertical bifurcation from the x1 family (see Skokos et. al 2002 for a classification and nomenclature of such periodic orbits), or thin tubes around these periodic orbits. Particles on such orbits influence mainly the ‘edge-on’ profiles of our systems. The study of these orbits is outside the scope of the present paper and will be undertaken in a separate study. The connection between the ‘disc’ resonances and the orbital frequencies is explicitly defined, in Athanassoula (2003).
On the other hand, particles satisfying a ‘disc’ resonance relation only approximately, i.e. $$\label{apres}
m_1\frac{\omega_\theta}{\omega_x}+m_2\frac{\kappa}{\omega_x}\approx 0$$ can also be distinguished in two cases:
b1) if there is one exact resonance relation of the form $m_1'\frac{\omega_\theta}{\omega_x}+m_2'\frac{\kappa}{\omega_x}+m_3'=0$, $m_i'\neq 0$, the orbit lies on an invariant 2D torus. When projected on the $y-z$ plane, however, the orbit is [*not*]{} captured in the associated $m_1:-m_2$ disc resonance, i.e., it lies entirely outside the separatrix domain corresponding to that particular resonance under a 2D approximation of the dynamics. Such orbits exhibit a slow precession around the associated exact disc resonances and they are responsible for a number of interesting features of the ‘face on’ appearance of the galaxies, examined in detail below.
b2) if there is no exact resonance relation of the form $m_1'\frac{\omega_\theta}
{\omega_x}+m_2'\frac{\kappa}{\omega_x}+m_3'=0$ the orbit is quasi-periodic, lying on a 3D torus. When projected on the $y-z$ plane, these orbits share most features of the orbits of group (b1).
In Fig.5, we observe that the majority of particles in regular orbits are concentrated along and around specific resonance lines. A very clear concentration is seen along the resonance 2:1 (that corresponds to $m_1=2$ and $m_2=-1$ in Eq.4). These are quasi-periodic orbits forming thin tubes around the ‘x1’ type of periodic orbits [@b47] at values of the Jacobi constant corresponding to distances near the inner Lindblad resonance (hereafter ILR). Both the periodic and quasi-periodic orbits of this type have ellipsoidal forms in the 3D space, which are projected to elliptical figures in the $y-z$ plane with a long axis aligned to the long axis of the bar (Fig.7).
The 2:1 is the main resonance supporting the bar and therefore it is well populated in all four experiments both at 20$T_{hmct}$ and at 300$T_{hmct}$. On the other hand, depending on the system and snapshot considered, other significant groups of regular orbits supporting the bar are distinguishable. We note that the simple resonances of the form $n:1$, $n>0$, implying $n$ epicyclic per one azimuthal oscillation, are concentrated, as $n$ increases, to domains of the phase space (or values of the Jacobi constant) closer and closer to corotation. Since the accumulation of these resonances causes chaos by the mechanism of resonance overlap (see Contopoulos 2002 p.185), the regular orbits are expected to lie at resonances $n:1$ with a low value of $n$, or some nearby non-simple resonances $n':m' \simeq n:1$, while the chaotic orbits fill the remaining parts of the resonance web, i.e. the simple resonances $n:1$ with high values of $n$ or their nearby non-simple resonances $n':m'\simeq n:1$.
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Figure 6 gives the distribution of regular orbits along the quantity $q=\frac{\Omega-\Omega_p}{\kappa}$ calculated for all the systems at 20$T_{hmct}$ (left hand column) and at 300$T_{hmct}$ (right hand column). Note that the pattern speed $\Omega_p$ of the bar at any snapshot can be derived from fig.4a of @b31. It is well known that the quantity $q$ distinguishes very efficiently the various resonances and it has been used for the orbital classification in previous studies [e.g. @b14; @b16].
Each peak in Fig.6 corresponds to the indicated population. Besides the ILR (at $q=\frac{1}{2}$) we identify other important peaks in the $q$-value distribution of the $N$-Body particles. One prominent peak which appears at early snapshots (except in the experiment QR1) is at the 3:1 resonance (that corresponds to $m_1=3$ and $m_2=-1$ in Eq.(4) and to $q=1/3$ in Fig.6). This peak nearly disappears at late snapshots, being replaced, instead, by a different peak, called hereafter group ‘A’. This corresponds to particles with a $q$-value $q\simeq\frac{2}{5}$. This population has been found in other $N$-body simulations of bar-like galaxies [see for example @b17; @b16]. The transition from a distribution peaked at the 3:1 resonance to a distribution peaked at the group ‘A’ is related to a morphological transition in the bar, discussed in detail below.
A less populated group called group ‘B’, is also observed at snapshots near equilibrium (e.g. $t=300T_{hmct}$). This consists mainly of quasi-periodic retrograde orbits with a $y-z$ projection of the ‘x4’ type (Contopoulos & Papayannopoulos 1980), i.e. elongated perpendicularly to the bar and correspond to $m_1=2$ and $m_2=-1$ in Eq.(4). Remarkably, many orbits of this group exhibit also vertical oscillations (see Fig.7). The group ‘B’ orbits cannot exist at snapshots close to the beginning of the simulation, as for example $t=20T_{hmct}$, since the ‘velocity re-orientation’ process, by which the initial conditions are produced, implies that there be no retrograde orbits initially. Thus the group ‘B’ population develops in the course of the simulation, as the systems evolve towards the equilibrium. From the right panels of Fig.6 we can conclude that the number of particles in the group ‘B’ decreases with increasing pattern speed (e.g. from QR1 to QR4). This is further substantiated in section 3.2 in which we show that in all the self-consistent systems the phase space corresponding to the ‘x4’ family is almost empty.
Finally, in the experiment QR1, which does not have any spiral structure, there is a small concentration of particles in regular orbits along the resonance line 1:1 (see Fig.5). This resonance corresponds to the short period orbits PL4 and PL5 (using the nomenclature of Voglis et al. 2006b for the short period periodic orbits, bifurcating from the corresponding Lagrangian points), around the Lagrangian points $L_4$ and $L_5$ (see Fig.7) and exists for values of the Jacobi constant close to the one at corotation. In all the other experiments there are no regular orbits of this type.
The implications of the transition from a 3:1 peak at early snapshots to a group ‘A’ peak at late snapshots can be discussed with the help of Figs.7 and 8. Figure 7 shows the projections on various planes of some regular orbits trapped in major disc resonances. It is obvious that the group ‘A’ orbits yield a ‘boxy’ form, while the resonant 3:1 orbits are asymmetric with respect to the bar’s major axis (we can find a different orbit, symmetric to the 3:1 orbit of Fig.7, with respect to the axis of symmetry $y=0$). However, other orbits of the $q=\frac{1}{3}$ group of Fig.6, which are selected so that they are not exactly resonant (i.e. $q\simeq
\frac{1}{3}$), exhibit a precession of the apocentres which corresponds to a motion in the phase space close to, but outside the separatrices marking the 3:1 resonant domain. Such an orbit is shown in Fig.8b. Orbits such as in Fig.8b are considerably less boxy than the orbits of group ‘A’ (Fig.8c), although they are more boxy than the 2:1 resonant orbits (Fig.8a). We conclude that if the $q=\frac{1}{3}$ peak of the distribution is pronounced compared to other peaks, the bar isophotes appear more ‘disky’, i.e lower $c$ values (see Fig.4 for the early snapshots of the experiments QR2 to QR4). On the contrary, the enhancement of group ‘A’ orbits is responsible for the ‘boxy’ isophotes (high $c$ values in Fig.4) at the late snapshots of the same experiments. This implies also a correlation between the position of the peaks of the $q$-distribution and the value of the pattern speed of a particular galaxy. The 3:1 type of orbits appears mainly in systems with a high value of $\Omega_p$ (e.g. the systems QR2, QR3, QR4 at $t=20_T{hmct}$), while, as the systems slow down, the $q=1/3$ peak becomes gradually depleted and, instead, the $N$-body particles gradually populate the group ‘A’ type of orbits, which, hence, becomes a dominant type in systems with a low value of $\Omega_p$ (QR2, QR3, QR4 at $t=300T_{hmct}$). Note that the pattern speed of the QR1 experiment is quite small already at $t=20T_{hmct}$, and, in this particular experiment, the group ‘A’ peak is prominent also at this snapshot. This discussion suggests that $\Omega_p$ is indeed, the relevant parameter to which the position of the main peaks of the $q$-distribution should be correlated.
![Same as in Fig.5 but for the chaotic orbits. We see that chaotic orbits appear quite scattered. Nevertheless, many chaotic orbits are trapped near various resonance lines in some of which no regular orbits are observed.[]{data-label="fig:9"}](fig09.eps){width="8.1cm"}
Figure 9 shows the frequency maps of the [*chaotic*]{} orbits, for all the systems and for two different snapshots as indicated in the figure. It is obvious that the chaotic orbits appear quite scattered in these diagrams. Nevertheless, many chaotic orbits are trapped near various resonance lines. Such orbits are called ‘sticky’ and they behave similarly to regular orbits for long times. An interesting remark is that there are chaotic orbits near resonance lines where no regular orbits are observed. For example, in Fig.9 we can see points around the 3:1 and the 4:1 resonance as well as around the 1:1 resonance (see Fig.7 for their shapes), at snapshots when there are only a few regular orbits in these resonances (compare Figs. 5 and 9).
![The distribution of the frequency ratio $q$ for the chaotic orbits at $20T_{hmct}$ (left column) and at $300T_{hmct}$ (right column). Orbits inside corotation have positive $q$ values, while orbits extending outside corotation have negative $q$ values. The chaotic orbits are distributed in a greater variety of $q$ than the regular ones.[]{data-label="fig:10"}](fig10.eps){width="8.1cm"}
Figure 10 shows the distribution of the $q$ values for the chaotic orbits of all the systems. The first remark is that the chaotic orbits are distributed in a greater variety of $q$-values than the regular ones. The $q$-distributions present peaks at the same values of $q$ as for the regular orbits. Nevertheless, important peaks appear also at other resonances, as seen in the figure. Note that corotation, ILR and outer Lindblad resonance (hereafter OLR) correspond to $0,\frac{1}{2}$ and $-\frac{1}{2}$ values of $q$, respectively. In general orbits located inside (outside) corotation have positive (negative) $q$ values. The orbits corresponding to smaller $q$ values reach larger distances on the $y-z$ plane. Since at $t=300T_{hmct}$ all the systems have lower $\Omega_p$ values than at $t=20T_{hmct}$, the corotation radius is at larger distances. This explains the smaller percentage of orbits found outside corotation ($q \leq 0$) at $t=300T_{hmct}$.
The percentages of the most important populations of regular and chaotic orbits, derived from Figs.6 and 10, are given in Table 1.
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We now come to see the role of the various populations in supporting particular morphological features of the studied systems. In subsequent plots we focus on one example, the QR3 experiment, which exhibits all interesting phenomena. The ‘backbone’ of the system, i.e. its appearance on the $y-z$ plane, is plotted as a gray background in a number of subsequent plots.
Figure 11 shows the ‘backbone’ of the system QR3 at $t=20T_{hmct}$ together with the instantaneous positions of the particles belonging to the various identified populations of regular and chaotic orbits. The backbones of QR2, QR3 and QR4 are similar at $t=20T_{hmct}$. All four experiments present a bar, two spiral arms and a faint ring. We observe that the regular orbits contribute to the form of the bar. On the contrary, the chaotic orbits support the structures both inside and outside corotation. The chaotic orbits inside corotation ($q \ge 0$) create an envelope of the bar, the particles being located at the outer bar layers. Similar orbits were found in @b301. The particles in chaotic orbits near the 4:1 resonance ($q=\frac{1}{4}$) support segments of the spiral arms connected to the bar. The population corresponding to corotation $(q=0)$ contributes to the outer bar as well as to the ring. The chaotic orbits located near the OLR $\left(q=-\frac{1}{2}\right)$, as well as near the $-1:1$ resonance $\left(q=-1\right)$, contribute to the ring and to the spiral arms. The chaotic orbits below the $-1:1$ resonance $\left(q \le -1\right)$ contribute mainly to the spiral arms.
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In Fig.11 we only see which features of the system are supported by the instantaneous positions of the particles. Thus we have no information about the trend for morphological changes induced by the chaotic orbits undergoing slow chaotic diffusion. In order to see such effects, Fig.12 shows again the backbone of the system QR3 at $t=20T_{hmct}$ together with the isodensities of the high density areas (black contours) that we get from the superposition of the orbits of the same particles as in Fig.11, integrated for a time interval $400T_{hmct}$ (a little longer than a Hubble time). The isodensities are plotted separately for each population, while the numerical integration is carried under the fixed gravitational potential of the system at $t=20T_{hmct}$. We now see very clearly which features of the system are supported by each type of orbits in the long run. An important remark is that the chaotic orbits outside corotation $\left(q \le 0\right)$, which can in principle escape, continue in practice to support the basic features of the system after an integration time comparable to the Hubble time. In particular, the populations with $q=0$ and $q=-\frac{1}{2}$ support the ring around the bar, the population with $q=-1$ supports the spiral arm segments connected to the end of the bar and the population with $q \leq -1$, supports the outer parts of the spiral arms, as well. As shown in Section 3.2, these orbits have higher values of the Lyapunov exponents, implying a faster chaotic diffusion. The chaotic diffusion is a major factor driving the secular evolution of the systems within times comparable to the Hubble time.
![The evolution of a real chaotic orbit (of the experiment QR4 at $t=20T_{hmct}$), that presents ‘stickiness’ to several resonances before escaping from the system after more than a Hubble time. Different panels correspond to different intervals of time integration, of the same orbit. The orbit is located successively around frequency ratios: (a) $q=-1$ (b) $q=0$ (c) $q=\frac{1}{4}$ and (d) $q=\frac{1}{2}$.[]{data-label="fig:13"}](fig13.eps){width="8.5cm"}
It should be stressed here that the percentages of the various types of chaotic orbits change in time as a result of i) the chaotic diffusion, and ii) the gradual change of the pattern speed. Thus, orbits of a specific type can be converted to orbits of another type, or they can escape. At the equilibrium state this process must be in dynamical equilibrium (the number of orbits changing type should be equal to the number of orbits joining every same type). Thus, in equilibrium, such an exchange can only be due to the chaotic diffusion. The studied systems at $t=20T_{hmct}$ are far from equilibrium. Therefore, the diffusion of the orbits, especially those not trapped by some resonance, produces a strong secular evolution. An example of such a process is given in Fig.13 where the evolution of the chaotic orbit of a real particle, of the experiment QR4, is plotted before its final escape from the system. It is obvious that this chaotic orbit stays consecutively localized in a sequence of resonances, spending a considerable time near each resonance, before escaping finally from the system, after approximately three Hubble times. In particular, the orbit has a frequency ratio close to -1:1 ($q=-1$) up to about one Hubble time (Fig.13a). Then it stays located around the Lagrangian point ‘$L_4$’ corresponding to a frequency ratio $q=0$ (Fig.13b). Figure 13c presents the part of the orbit that gives a frequency ratio close to $q=\frac{1}{4}$ and finally Fig.13d corresponds the part of the orbit having frequency ratio close to $q=\frac{1}{2}$. The explanation of such behaviour of chaotic orbits is given using 2D surfaces of sections (see section 3.2). For certain Jacobi constants the area outside corotation can communicate with the area inside corotation and chaotic orbits present long time ‘stickiness’ in unstable asymptotic curves of several resonances, until they finally escape from the system following the path carved by these curves.
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Phase space structure
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Interesting remarks about the phase space structures can be revealed by plotting 2D projections of the 4D surfaces of sections (hereafter SOS) of all the self-consistent $N$-body models.
Using the ‘frozen’ potential and the instantaneous value of the pattern speed $\Omega_p$ at each studied snapshot (Table 1) we construct SOS of test particles for all the systems and thereby identify the main features of the phase space structure. On the other hand, plotting the real $N$-Body particles on the same SOS reveals which domains of the phase space are preferred and which are avoided by the real particles. A similar technique has been used in Paper I. A 4D SOS can be constructed by the intersection of the orbits with the plane $z=0$, $\dot{z}<0$. We then plot the projection of this particular SOS on the ($y,\dot{y}$) plane. Alternatively, we can use the 2D approximation of the gravitational potential on the disc plane and integrate the orbits of test particles on the same plane. This is hereafter called a ‘2D approximation’.
Figure 14 gives the phase space structure at various values of the Jacobi constant for the system QR3 at t=20$T_{hmct}$, via the 2D approximation. The central value of the potential (potential well) and the value of the Jacobi constant at the Lagrangian points $L_1$, $L_2$ are $E_0=-2.15\times10^6$ and $E_{L_1}=-1.265\times10^6$ respectively. The panels of the two upper rows of Fig.14 show the phase space structure for Jacobi constants below the value corresponding to $L_4, L_5$ $E_{L_4}=-1.225\times10^6$ (Note that the phase space areas corresponding to inside and outside corotation are divided). It is obvious that chaos becomes dominant for $E_j>-1.3\times10^6$. The areas inside corotation with $y<0$ correspond to retrograde orbits of the ‘x4’ type (and their quasi-periodic orbits), while the areas with $y>0$ are filled with ‘x1’ orbits and their bifurcations.
Figure 15 shows the projections of the real $N$-body particles with energies $E_j=-1.5\times10^6\pm 10^4$ on the SOS. The orbit of each real particle is integrated for 200 iterations. A distribution of the particles on layers corresponding to a foliation of invariant tori is discerned in this figure, despite the projection effects caused by the fact that the SOS is actually 4D. The central energy value is the same as in Fig.14c. At this value of the Jacobi constant there are almost no chaotic orbits inside corotation. Therefore, the orbits of Fig.15 are regular orbits close to the 2:1 and 3:1 resonances. Note that the area with $y>0$, corresponding to orbits around the ‘x1’ periodic orbit, is well populated, while the area with $y<0$ corresponding to the ‘x4’ retrograde type of orbits is nearly devoid of real particles.
Figure 16 shows the energy (Jacobi constant) distribution of the real $N$-body particles for the regular orbits (solid lines), and for the chaotic orbits (dashed lines) for the experiment QR3 at $t=20T_{hmct}$ (gray) and $t=300T_{hmct}$ (black). From Figs.14 and 16 we conclude that most chaotic orbits, with $E_j\lesssim
-1.3\times10^6$, are located outside corotation (the chaos is negligible inside corotation). On the other hand, for values greater than the above threshold there are chaotic orbits found both outside and inside corotation (mostly at the outer regions of the bar). In the 2D approximation, the chaotic orbits with energies $E_j\lesssim
E_{L_1}$ which are located inside corotation, (Fig.14) are surrounded by invariant curves and, consequently, they cannot communicate with the regions outside corotation. The important remark is that the above result remains essentially valid for the orbits in the full 3D potential. Namely, the majority of the real chaotic orbits of the same energy levels which are located inside corotation remain there at least up to the end of our numerical integration, i.e. for times much longer than a Hubble time, although in the full 3D potential these orbits are in principle able to escape via the phenomenon of ‘Arnold diffusion’ (Arnold 1964).
![ (a) The projection on the ($y, \dot{y}$) SOS of real particles for the region inside corotation for the QR3 experiment, for $Ej=-1.5\times10^6\pm10^4$ at $t=20T_{hmct}$. The area around ‘x1’ is well populated while the area around ‘x4’ corresponding to retrograde orbits is almost empty.[]{data-label="fig:15"}](fig15.eps){width="6.3cm"}
We have found that $\approx 26\%$ of the particles in chaotic orbits of this experiment are located inside corotation and close to the end of the bar, therefore contributing to the backbone of the bar. The percentages of the particles in chaotic orbits which remain inside corotation, are shown in Table 1, for all the experiments. From this we conclude that this ‘chaotic component’ represents an important percentage of particles in all the experiments. The value of the Specific Finite Time Lyapunov Characteristic Number $L_j$ of these orbits is in general smaller than the value of $L_j$ of the chaotic orbits outside corotation. This can be inferred from Fig.17, in which the $\log(L_j)$ values of the orbits are plotted as functions of their $q$ values. The chaotic orbits have $L_j\geq
10^{-2.8}$ (see Voglis et al., 2006a). The regular orbits are almost completely located inside corotation (around the resonances 2:1 and 3:1) and have the smallest values of $L_j$. The chaotic orbits, on the other hand, are spread in the whole range of resonances, but it is obvious that the most weakly chaotic orbits (having smaller $L_j$) are located inside corotation and therefore support the bar.
![The energy distribution for regular orbits (solid lines) and chaotic orbits (dashed lines) for the experiment QR3 at $t=20T_{hmct}$ (gray) and at $t=300T_{hmct}$ (black).[]{data-label="fig:16"}](fig16.eps){width="7.3cm"}
Figure 18a shows another SOS, obtained from the 2D approximation, for the experiment QR3 at $t=20T_{hmct}$ and for the Jacobi constant $E_j=-1.1\times10^6$. In this figure we see also the area outside corotation (which is located at $y\simeq 1.2$). A sticky region is apparent here between corotation and $y\simeq 2$. This stickiness phenomenon is related to the unstable manifolds emanating from unstable periodic points with fixed points in the same area. This phenomenon, called ‘stickiness in chaos’ has been studied thoroughly in simple dynamical systems [see @b15 and references therein]. The corresponding Jacobi constant is the maximum of the energy distribution of the chaotic particles having the same angular velocity as the pattern speed of the bar ($q=0$). The two stable periodic orbits located inside the sticky zone correspond to bifurcations of the main orbit around the point $L_5$ (named ‘PL5’) which has become unstable at this value of Jacobi constant. All the unstable asymptotic curves are necessarily parallel to the asymptotic curve of the simplest periodic orbit i.e. the short period orbit around $L_5$. Thus this orbit is called ‘PL5’. In Fig.18b we plot the projection on the same SOS of the real particles. We observe again the depopulation of the ‘x4’ area and the ‘sticky’ region just outside corotation around the unstable ‘PL5’ orbit. Such ‘stickiness’ is observed in many energy levels and it is responsible for supporting various features of the system (QR3 at $t=20T_{hmct}$). In Fig.18c we see the unstable periodic orbit (black), starting at the point ‘PL5’ of Fig.18a, superimposed to the backbone of the system QR3 at $t=20T_{hmct}$ (gray). The ‘sticky’ behaviour of some orbits around the unstable ‘PL5’ orbit supports the ring type shape.
![The logarithm of the Specific Finite Lyapunov Characteristic Number $(\log L_j)$ for the regular and for the chaotic orbits as a function of their $q$ value for the experiment QR3 at $t=20T_{hmct}$.[]{data-label="fig:17"}](fig17.eps){width="8.cm"}
Figure 19 shows the evolution of a real chaotic orbit which presents ‘stickiness’ around the unstable ‘PL5’ orbit of Fig.18a. The calculated frequency of this orbit is $q\approx 0.0$, i.e. it has the same angular velocity as the bar and it forms precessing ‘bananas’. Such orbits amplify the formation of the ring, which can be classified as R1, as it is perpendicular to the bar [see @b100].
{width="17cm"}
In Fig.20 we present the evolution of a real particle in chaotic orbit, of the experiment QR4 at $t=20T_{hmct}$, in the configuration space (Fig.20a) as well as in the phase space ($z,\dot{z}$) (Fig.20b). This orbit has $q<-1$ and $E_j\approx-1.2\times10^6$ (this Jacobi constant lies between $E_{L_1}$ and $E_{L_4}$). We see that the orbit on the configuration space is located outside corotation, filling the area all along the spiral arms (Fig.20a). In Fig.20b the SOS ($z,\dot{z}$) is presented for 50 iterations of test particles, for $E_j=-1.2\times10^6$, only for the area outside corotation. The integration time corresponds to 1.5-2.0 Hubble times. The ‘stickiness’ along the asymptotic curves of the periodic orbits is evident. In this value of the Jacobi constant the main unstable periodic orbit is the short period orbit ‘PL1’ or ‘PL2’, around the Lagrangian points $L_1$ and $L_2$. We note that the density of points drops abruptly outside the radius that corresponds approximately to the end of the spiral arms. As there are no islands of stability or obvious cantori, the only possible reason of stickiness could be the trapping along the asymptotic curves of the unstable orbits. In Fig.20b the projections of 350 iterations of the real 3D chaotic orbit (black dots), are superimposed. We notice that these sections stay located inside the radius of the spiral arms. In general, the stickiness of the chaotic orbits along the asymptotic curves of the unstable curves of the short period orbit ‘PL1’ or ‘PL2’ as well as all the other curves from unstable periodic orbits of greater multiplicity, is responsible for the support of features like rings and spiral arms. In particular, the invariant manifolds from all the unstable periodic orbits (for values of $E_j$ between $E_{L_1}$ and $E_{L_4}$) cannot intersect each other. Thus they are forced to follow nearly parallel paths in the phase space, and this enhances the structures supported by them, i.e. the rings and the spiral arms. A thorough study of the role of asymptotic curves in the maintenance of spiral structure is done in @b10 and @b101. They showed that the apocentres of all the chaotic orbits with initial conditions along the manifolds corresponding to the unstable periodic orbits with Jacobi constants $E_{L_1}\leq
E_j\leq E_{L_4}$ create an invariant locus supporting the spiral arms. In Fig.20c we plot only the apocentres of the orbit plotted in Fig.20a (black dots in Fig.20c), where we observe that they coincide very well with the outer parts of the ring and with the spiral arms (gray background).
{width="18cm"}
{width="17cm"}
CONCLUSIONS
===========
We present a detailed investigation of the orbital structure of four different $N$-body experiments simulating barred-spiral galaxies exhibiting significant secular evolution within one Hubble time. The study has been made for two different snapshots during the evolution of the systems, an early snapshot (t=20$T_{hmct}$) and a late one in which the systems are close to equilibrium (t=300$T_{hmct}$). The amplitude of the bar is large in all the experiments and the spin parameter has a value close to the one of our Galaxy.
The main conclusions of our study are the following:
\(i) The ellipticity profile of the bar’s isophotes evolves from flat, at t=20$T_{hmct}$, to declining at t=300$T_{hmct}$, when the systems are close to equilibrium.
\(ii) The boxiness of the bar’s isophotes and the pattern speed ($\Omega_p$) of the bar are correlated, i.e the boxiness decreases with increasing $\Omega_p$.
\(iii) Using frequency analysis we have found the main resonances around which the $N$-body particles in regular orbits are concentrated. The particles in the chaotic orbits are more scattered in their frequency maps, however we can still detect concentrations around a variety of resonances, larger than in the case of the particles in regular orbits.
\(iv) Almost the whole mass component in regular orbits are particles located inside corotation. These particles support the bar. The most populated type of resonant regular orbit is the 2:1 or ‘x1’ type of orbit. Another important group of regular orbits corresponds to the 3:1 resonance at t=20$T_{hmct}$ (except from the experiment QR1). This gradually transforms to a group ‘A’ type of orbit at t=300$T_{hmct}$. Group ‘A’ is responsible for the boxiness of the bar’s isophotes.
\(v) Group ‘B’ corresponds to retrograde ‘x4’ type of orbits and exhibits a very small component both in regular and chaotic orbits, which becomes almost negligible in experiments with great values of the bar’s pattern speed $\Omega_p$.
\(vi) Chaotic orbits exist only beyond a threshold of values of the Jacobi constant value $E_j$. An important percentage of them are located inside corotation near the resonances 2:1, 3:1, 4:1 and the group ‘A’ and support the outer regions of the bar. This component of the chaotic population has the smallest values of the Specific Finite Time Lyapunov Characteristic Number among all the chaotic orbits. Therefore, these orbits can be considered as weakly chaotic, resembling to regular orbits even when they are integrated for times much longer than a Hubble time.
\(vii) Comparing 2D surfaces of sections for test particles with projections on the same surfaces of sections for real $N$-body particles we conclude that the areas corresponding to the ‘x4’ type of orbits are depopulated. The chaotic domain in the surface of section, inside corotation, are limited by invariant curves in the 2D approximation of test particles. The same seems to be true for the projections on the surface of section of the real $N$-body chaotic orbits, despite the third dimension.
\(viii) There are chaotic domains in the surface of section, just outside corotation, that show ‘stickiness’ due to the existence of asymptotic manifolds of some unstable periodic orbits. The stickiness is observed in both the test particles and the real $N$-body particles. These domains correspond to chaotic orbits that present a very slow diffusion before escaping from their systems, and therefore they are responsible for the maintenance of the general features of the systems such as rings and spiral arms.
\(ix) The corresponding frequencies responsible for the formation of the ‘R1’ rings around the bar, in all our experiments have values around $q=0$ and $q=-\frac{1}{2}$, while the chaotic orbits that are responsible for the spiral arms have frequencies $q\leq-1$.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank professor G. Contopoulos and Dr. C. Efthymiopoulos for useful discussions and for a careful reading of the manuscript with many suggestions for improvement. Finally, we would like to thank the referee for his useful remarks that helped us to improve our paper.
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\[lastpage\]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We pursue applications of the light-front reduction of current matrix elements in the Bethe-Salpeter formalism. The normalization of the reduced wave function is derived from the covariant framework and related to non-valence probabilities using familiar Fock space projection operators. Using a simple model, we obtain expressions for generalized parton distributions that are continuous. The non-vanishing of these distributions at the crossover between kinematic regimes (where the plus component of the struck quark’s momentum is equal to the plus component of the momentum transfer) is tied to higher Fock components. Moreover continuity holds due to relations between Fock components at vanishing plus momentum. Lastly we apply the light-front reduction to time-like form factors and derive expressions for the generalized distribution amplitudes in this model.'
author:
- 'B. C. Tiburzi'
- 'G. A. Miller'
title: |
Current in the light-front Bethe-Salpeter formalism II:\
Applications
---
Introduction
============
More than a half century ago, Dirac’s paper on the forms of relativistic dynamics [@Dirac:1949cp] introduced the front-form Hamiltonian approach. Applications to quantum mechanics and field theory were overlooked at the time due to the appearance of covariant perturbation theory. The reemergence of front-form dynamics was largely motivated by simplicity as well as physicality. The light-front approach has the largest stability group [@Leutwyler:1977vy] of any Hamiltonian theory. Today the physical connection to light-front dynamics is transparent: hard scattering processes probe a light-cone correlation of the fields. Not surprisingly, then, many perturbative QCD applications can be treated on the light front, see e.g. [@Lepage:1980fj]. Outside this realm, physics on the light cone has been extensively developed for non-perturbative QCD [@Brodsky:1997de] as well as applied to nuclear physics [@Carbonell:1998rj; @Miller:2000kv].
Recently we investigated current matrix elements in the light-front Bethe-Salpeter formalism [@future], at task originally attempted in [@Tiburzi:2002mn]. As an example, we considered the ladder approximation for the covariant kernel in the weak-binding limit. We calculated $(1+1)$-dimensional electromagnetic form factors in this reduction scheme to demonstrate the replacement of non-wave function vertices (as coined in [@Bakker:2000rd]) with contributions from higher Fock states. These higher Fock states originate from the light-front energy pole-structure of the Bethe-Salpeter and photon vertices.
Below, we take the model into $(3+1)$ dimensions and make clear the connection to higher Fock components by explicitly constructing the three-body wave function from our expressions for form factors. Additionally we show how the normalization of the covariant Bethe-Salpeter equation turns into a familiar many-body normalization in the light-front reduction. The immediate application of our development for form factors (in a frame where the plus-component of the momentum transfer is non-zero) is to compute generalized parton distributions. Connection is again made to the Fock space representation and continuity of the distributions is put under scrutiny. The formalism is also employed to obtain time-like form factors. This latter application in interesting since no Fock space expansion in terms of bound states is possible.
The paper is organized as follows. First in section \[thenorm\] we review the normalization of the covariant Bethe-Salpeter wave function and then proceed to derive the light-front reduced version. This necessitates a review of the light-front reduction notation introduced in [@Sales:1999ec]. The normalization condition resembles a diagonal matrix element of a pseudo current and has contributions from higher Fock states. We calculate the explicit normalization condition for the ladder model at next-to-leading order in perturbation theory. The connection to the familiar many-body Fock state normalization is made in Appendix \[oftopt\]. Next in section \[gpds\], we use the reduction scheme to calculate generalized parton distributions. We do so by using the integrand of the electromagnetic form factor calculated in an arbitrary frame (these expressions in $(3+1)$ dimensions are collected in Appendix \[fff\] where the leading-order bound-state equation for the wave function also appears). We discuss the continuity of these distributions in terms of relations between Fock components at vanishing plus momentum. Connection is also made to the overlap representation of generalized parton distributions. Application of the reduction scheme to time-like form factors and their related generalized distribution amplitudes is presented in section \[gda\]. Finally we conclude with a brief summary (section \[summy\]).
Normalization {#thenorm}
=============
In [@Sales:1999ec] the relation between the four-dimensional Bethe-Salpeter wave function and the reduced light-front wave function was presented. The normalization of the reduced wave function, however, was not discussed in much detail and will be addressed below.
Let $G(R)$ denote the two-particle disconnected propagator, where $R$ labels the total momentum. Thus between effective single-particle states of momenta $p$ and $k$, we have $\langle p | G(R) | k \rangle = (2\pi)^4 \delta^4(p-k) G(k,R)$, where $G(k,R) = d(k) d(R-k)$ and the scalar single-particle propagator is $$d(k) = \frac{i}{(k^2 - m^2)[1 + (k^2 - m^2) f(k^2)] + \operatorname{i \epsilon}}.$$ Here the function $f(k^2)$ characterizes the renormalized, one-particle irreducible self-interactions and for simplicity shall be ignored below. The Bethe-Salpeter equation for the bound-state amplitude $|\Psi_R\rangle$ with mass $R^2 = M^2$ reads $$\label{BS}
|\Psi_R\rangle = G(R) V(R) |\Psi_R\rangle,$$ where $V(R)$ is the irreducible two-to-two scattering kernel (which we shall often call the potential). From the behavior of the reducible four-point function near the bound-state pole $R^2 = M^2$ one can deduce the covariant normalization condition [@Itzykson:rh] by application of l’Hôpital’s rule $$\label{normcov}
2 i R^\mu = \langle \Psi_{R} | \frac{\partial}{\partial R_{\mu}} \Big( G^{-1}(R) - V(R) \Big) |\Psi_{R} \rangle \; \Bigg|_{R^2 = M^2}.$$ The normalization takes the form of a diagonal matrix element of a pseudo current.
The light-front reduction is performed by integrating out the minus-momentum[^1] dependence with the help of an auxiliary Green’s function $\operatorname{\tilde{G}}(R)$. For simplicity, we denote the integration $\int \frac{d\operatorname{k^{--}}}{2\pi} \langle \operatorname{k^{--}}| \mathcal{O}(R) = \Big| \mathcal{O}(R)$. With this notation, we will always work in $(3+1)$-dimensional momentum space for which the only sensible matrix elements of $\Big| \mathcal{O}(R)$ are of the form $\langle \operatorname{k^{+}},\operatorname{\mathbf{k}^{\perp}}| \; \Big| \mathcal{O}(R) | \operatorname{p^{--}}, \operatorname{p^{+}},\operatorname{\mathbf{p}^{\perp}}\rangle$. The operator $\mathcal{O}(R) \Big|$ is defined similarly. To obtain light-front time-ordered perturbation theory one chooses $$\operatorname{\tilde{G}}(R) = G(R) \Big| g^{-1}(R) \Big| G(R),$$ where $g(R) = \Big| G(R) \Big|$. This form of $\operatorname{\tilde{G}}$ allows for a systematic approximation scheme for the light-front energy poles of the Bethe-Salpeter vertex [@future]. Lastly one defines an auxiliary kernel $W(R)$ by [@Woloshyn:wm] $$\label{W}
W(R) = V(R) + \Big(G(R) - \operatorname{\tilde{G}}(R)\Big) W(R).$$
In what follows we shall omit total four-momentum labels since they are all identically $R$. The normalization condition for the reduced wave function is then deduced by using the conversion [@Sales:1999ec] $$\label{324}
| \Psi_{R} \rangle = \Bigg( 1 + \Big(G - \operatorname{\tilde{G}}\Big) W \Bigg) G \Big| \; |\gamma_{R} \rangle,$$ and the definition of the reduced wave function $|\psi_R\rangle$, namely $$\label{psi}
|\psi_R\rangle \equiv \Big| \; | \Psi_R \rangle = g(R) |\gamma_R \rangle.$$ Hence taking the plus component of Eq. $$\begin{gathered}
\label{normlf}
2 i R^+ = \langle \gamma_{R} | \; \Big| G \Bigg( 1 + W ( G - \operatorname{\tilde{G}}) \Bigg) \\
\times \Bigg( \frac{\partial}{\partial R^-} \Big[ G^{-1} - V \Big] \Bigg) \Bigg( 1 + (G - \operatorname{\tilde{G}}) W \Bigg) G \Big| \; |\gamma_{R}\rangle.\end{gathered}$$ The complicated normalization condition is indicative of the effects of higher Fock space components. To see this explicitly, we work in the ladder model in perturbation theory for which $$\label{V}
V(k,p) = \frac{-g^2}{(k-p)^2 - \mu^2 + \operatorname{i \epsilon}}.$$ Notice $\partial V/\partial R^\mu = 0$.
Let us start with the contribution at leading order in $G - \operatorname{\tilde{G}}$ to the reduced wave function’s normalization. $$\label{normLO}
\frac{-i}{2 R^+}\langle \gamma_{R} | \; \Big| G
\Big( \frac{\partial}{\partial R^-} G^{-1} \Big) G \Big| \; | \gamma_{R} \rangle = 1.$$ To perform the integration, we note $$\frac{\partial}{\partial R^-} G^{-1}(k,R) = - 2 i R^+ d^{-1}(k) (1 - x),$$ where we have customarily chosen $x = \operatorname{k^{+}}/R^+$. Evaluation of the integral in equation is standard and yields $$\label{2to2}
N^{\text{LO}} \equiv \int \frac{dx d\operatorname{\mathbf{k}^{\perp}}}{2 (2\pi)^3 x (1-x)} \psi^*(x,\operatorname{\mathbf{k}^{\perp}}) \psi(x,\operatorname{\mathbf{k}^{\perp}}) = 1,$$ a simple overlap of the two-body wave function.
To analyze the normalization to first order in $G - \operatorname{\tilde{G}}$, we expand equation to first order $$N^{\text{LO}} + \delta N + \ldots = 1,$$ where $N^{\text{LO}}$ is the integral appearing in and the first-order correction arising from is $$\begin{gathered}
\label{deltaN}
\delta N = \frac{-i}{2 R^+} \langle \gamma_{R} | \; \Big| G \Big( \frac{\partial}{\partial R^-} G^{-1} \Big) \Big( G - \operatorname{\tilde{G}}\Big) V G \\
+ G V \Big( G - \operatorname{\tilde{G}}\Big) \Big( \frac{\partial}{\partial R^-} G^{-1} \Big) G \Big| \; | \gamma_{R} \rangle.\end{gathered}$$ The presence of $\operatorname{\tilde{G}}$ merely subtracts the leading-order result $N^{\text{LO}}$. Considering for the moment just the first term in the above equation (and omitting the subtraction $\operatorname{\tilde{G}}$), we have $$\begin{gathered}
- i \int \frac{d^4 k}{(2\pi)^4} \; \frac{d^4p}{(2\pi)^4} (1-x) \gamma^* (x,\operatorname{\mathbf{k}^{\perp}}|M^2) d(k) \\
\times d(R-k)^2 V(k,p) d(p) d(R-p) \gamma (y,\operatorname{\mathbf{p}^{\perp}}|M^2).\end{gathered}$$ The minus-momentum integrals above are similar to those considered in deriving the bound-state equation to leading order in Appendix \[fff\]. The only difference is the double pole due to the extra propagator $d(R-k)$. With $x = \operatorname{k^{+}}/R^+$ and $y = \operatorname{p^{+}}/R^+$, for $x>y$ we avoid picking up the residue at the double pole and the result is the same as in Eq. after using the bound-state equation. This term is then subtracted by the $\operatorname{\tilde{G}}$ term in equation . On the other hand, when $x<y$ we pick up the residue at the double pole. Part of the residue is subtracted by the $\operatorname{\tilde{G}}$ term; the other half depends on $\partial V(k,p)/\partial k^-$. The second term in Eq. is evaluated identically up to $\{k \leftrightarrow p\}$. Now combining the two terms and their relevant $\theta$ functions, we can rewrite the result using the explicit form of the one-boson exchange potential Eq. \[OBE\], namely
$$\label{nonN}
\delta N = \int \frac{dx d\operatorname{\mathbf{k}^{\perp}}}{2(2\pi)^3 x (1-x)} \; \frac{dy d\operatorname{\mathbf{p}^{\perp}}}{2 (2\pi)^3 y (1-y)} \psi^*(x,\operatorname{\mathbf{k}^{\perp}}) \Bigg( - \frac{\partial}{\partial M^2}
V(x, \operatorname{\mathbf{k}^{\perp}}; y, \operatorname{\mathbf{p}^{\perp}}|M^2) \Bigg) \psi(y, \operatorname{\mathbf{p}^{\perp}}).$$
Thus although the covariant derivative’s action on the potential vanishes, we can manipulate the correction to the normalization into the form of a derivative’s action on the light-front, time-ordered potential. With this form, we can compare to the familiar nonvalence probability discussed in Appendix \[oftopt\] (in the frame where $R^+ = R^- = M/\sqrt{2}$ with $M/\sqrt{2}$ as the eigenvalue of the light-front Hamiltonian, denoted $\operatorname{p^{--}}$ in Eq. ).
Here we have seen that the normalization of the light-cone wave function includes effects from higher Fock states. Since this normalization condition stemmed from a diagonal matrix element of a pseudo current, we should not be surprised that matrix elements of the electromagnetic current, when treated in this reduction scheme, pick up contributions from higher Fock states. These higher Fock contributions appear explicitly and are the subject of the next section.
Application to GPDs {#gpds}
===================
Having worked through matrix elements of the electromagnetic current in a frame where $\Delta^+ \neq 0$ [@future], we can now make the connection to generalized parton distributions (GPDs). These distributions, which in some sense are the natural interpolating functions between form factors and quark distribution functions, turn up in a variety of hard exclusive processes, e.g. deeply virtual Compton scattering, wide-angle Compton scattering and the electro-production of mesons [@Muller:1994fv]. The scattering amplitude for these processes factorizes into a convolution of a hard part (calculable from perturbative QCD) and a soft part which the GPDs encode. Since light-cone correlations are probed in these hard processes, the soft physics has a simple interpretation and expression in terms of light-front wave functions [@Brodsky:2001xy]. In this section, we cast our results for form factors [@future] in the language of GPDs and the light cone Fock space expansion. The $(3+1)$-dimensional expressions for form factors are presented in Appendix \[fff\]. Additionally one can obtain these results directly from time-ordered perturbation theory using two-body projection operators as explicated in Appendix \[oftopt\].
The GPD for our meson model is defined by a non-diagonal matrix element of bilocal field operators $$\label{bilocal}
F(x, \zeta, t) = \int \frac{dy^- }{4\pi} e^{i x P^+ y^-}
\langle \Psi_{P^\prime} | \; q(y^-) i \overset{\leftrightarrow}\partial{}^+ q(0) \; | \Psi_{P} \rangle,$$ where $q(x)$ denotes the quark field operator and $\overset{\leftrightarrow}\partial{}^\mu = \overset{\rightarrow}\partial{}^\mu -
\overset{\leftarrow}\partial{}^\mu$. Comparing to the current matrix element $J^\mu$ in Appendix \[fff\], the definition of the GPD leads immediately to the sum rule $$\label{sumrule}
\int \frac{dx}{1 - \zeta/2} F(x, \zeta, t) = F(t).$$ Hence one can calculate these distributions from the integrand of the form factor. In this way, the light-cone correlation defined in Eq. has a natural description in terms of light-front time-ordered perturbation theory, e.g. for $x>\zeta$ the relevant graphs contributing to the GPD are in Figure \[ftri\] and \[ftri2\], and those for $x<\zeta$ are in Figure \[fZZZ\].
Continuity
----------
Conversion of the contributions to the form factor into GPDs is straightforward using Eq. ; we merely remove $- 2 i P^+$ and $\int dx$ from Eqs. (\[FFLO\]-\[fs2\]). In order for the deeply virtual Compton scattering amplitude to factorize into hard and soft pieces (at leading twist), the GPDs $F(x,\zeta,t)$ must be continuous at $x = \zeta$. Maintaining continuity at the crossover is more pressing because experiments which measure the beam-spin asymmetry are limited to the crossover [@Diehl:1997bu]. The leading-order expressions are continuous. This is easy to see since the contribution for $x < \zeta$ is identically zero. The valence contribution for $x>\zeta$ is a convolution of wave functions one of which is $\psi^*(x^\prime,\ldots)$ which is probed at the end point since $x^\prime \equiv \frac{x - \zeta}{1-\zeta} \to 0$. From the bound-state equation Eq. , we see the two-body wave function vanishes quadratically at the end points. Taking into account the overall weight $x^{\prime -1}$, the valence piece vanishes linearly at the crossover. At leading order then, continuity is maintained at the crossover, while the derivative is discontinuous. Working only in the valence sector, valence quark models will never be of any use to beam-spin asymmetry measurements since the value at the crossover requires one wave function to be at an end point. In the three-body bound state problem (e.g. the nucleon), the valence GPD will vanish only if the three-body interaction is non-singular at the end points (which is physically reasonable and perturbatively true).
Let us now check the next-to-leading order contributions to the GPD for continuity. First we shall deal with the term stemming from iterating the Bethe-Salpeter equation for the initial state (see diagram $B$ of Figure \[ftri2\]). Since there is no Z-graph generated from iterating the initial state, we expect this contribution to vanish. Looking at the expression, we see again $\psi^*(x^\prime,\ldots)/x^\prime$ which vanishes linearly as $x \to \zeta$. Moreover there are the interaction terms: $D(y,\operatorname{\mathbf{p}^{\perp}};x,\operatorname{\mathbf{k}^{\perp}}|M^2) $ which is finite as $x \to \zeta$, and $$D(\operatorname{y^\prime},\operatorname{\mathbf{p}^{\prime\perp}};\operatorname{x^\prime},\operatorname{\mathbf{k}^{\prime\perp}}|M^2) \overset{x\to \zeta}{=} \frac{-\operatorname{x^\prime}}{(\operatorname{\mathbf{k}^{\perp}}+ \mathbf{\Delta}^\perp)^2 + m^2},$$ which vanishes at the crossover. Thus not only does the initial-state iteration term vanish at the crossover, its derivative does so as well.
Now we investigate the Born terms \[bt1\] (see diagram $A$ of Figure \[ftri2\]) and \[bt2\] (see diagram $D$ of Figure \[fZZZ\]) at the crossover. Approaching $\zeta$ from above , we have the finite contribution at the crossover
$$\label{bcross}
F(\zeta, \zeta, t)^{\text{Born}} = \int \frac{d\operatorname{\mathbf{k}^{\perp}}dy d\operatorname{\mathbf{p}^{\perp}}}{(16\pi^3)^2 y (1-y)\operatorname{y^\prime}}
\psi^*(y^\prime, \operatorname{\mathbf{p}^{\prime\perp}}) \frac{g^2 \theta(y - \zeta)/(y - \zeta) }{(\operatorname{\mathbf{k}^{\perp}}+ \mathbf{\Delta}^\perp)^2 + m^2}
D(y,\operatorname{\mathbf{p}^{\perp}};\zeta,\operatorname{\mathbf{k}^{\perp}}|M^2) \psi(y, \operatorname{\mathbf{p}^{\perp}}).$$
On the other hand, approaching the crossover from below we have to deal with singularities as $x^{\prime\prime} = x/ \zeta \to 1$. Writing out the propagator for the quark-antiquark pair heading off to annihilation, we see $$\label{qqbarprop}
\operatorname{D_{\text{W}}}(x^{\prime\prime}, \operatorname{\mathbf{k}^{\prime\prime\perp}}| t ) \to - \frac{1 - x^{\prime\prime}}{(\operatorname{\mathbf{k}^{\perp}}+ \mathbf{\Delta}^\perp)^2 + m^2}.$$ This linear vanishing cancels the weight $(1 - x^{\prime\prime})^{-1}$. Taking the limit $x \to \zeta$ then produces equation and thus the Born terms are continuous.
Lastly we must see how the final-state iteration terms match up at the crossover. Using Eq. to approach $\zeta$ from above (see diagram $C$ of Figure \[ftri2\]), we have the contribution $$\label{fcross}
F(\zeta, \zeta, t)^{\text{final}} = \int \frac{d\operatorname{\mathbf{k}^{\perp}}d\operatorname{y^\prime}d\operatorname{\mathbf{p}^{\prime\perp}}}{(16\pi^3)^2 (1-\zeta)\operatorname{y^\prime}(1-\operatorname{y^\prime}) }
\psi^*(y^\prime, \operatorname{\mathbf{p}^{\prime\perp}}) \frac{g^2/y }{(\operatorname{\mathbf{k}^{\perp}}+ \mathbf{\Delta}^\perp)^2 + m^2}
D(y,\operatorname{\mathbf{p}^{\perp}};\zeta,\operatorname{\mathbf{k}^{\perp}}|M^2) \psi(\zeta, \operatorname{\mathbf{k}^{\perp}}).$$ Approaching $\zeta$ from below (see diagram $E$ of Figure \[fZZZ\]), we utilize equation in taking the limit of . The result is and hence we have demonstrated continuity to first order, i.e. $$\label{contequ}
F(\zeta,\zeta,t) = F(\zeta,\zeta,t)^{\text{Born}} + F(\zeta,\zeta,t)^{\text{final}}$$ no matter how we approach $x = \zeta$.
Fock space representation
-------------------------
We now write the GPDs in terms of Fock component overlaps. In the diagonal overlap region $x > \zeta$ this will be a mere rewriting of our results, while there is a subtlety for the non-diagonal overlaps. To handle the zeroth-order term , we define the two-body Fock component as $$\label{twofock}
\psi_{2}(x_{1},\mathbf{k}^\perp_1, x_{2}, \mathbf{k}^\perp_2) = \frac{1}{\sqrt{ x_{1} x_{2} }} \psi(x_{1}, \mathbf{k}^\perp_{\text{rel}}),$$ noting that the relative transverse momentum can be defined as $\mathbf{k}^\perp_{\text{rel}} = x_{2} \mathbf{k}^\perp_{1} - x_{1} \mathbf{k}^\perp_{2}$. In terms of Eq. , the GPD appears as $$\label{222}
F(x, \zeta, t)^{\text{LO}} = \frac{\theta(x - \zeta)}{\sqrt{1-\zeta}} \int [dx]_{2} [d\operatorname{\mathbf{k}^{\perp}}]_{2} \sum_{j = 1,2} \delta(x - x_{j})
\psi_{2}^*(x^{\prime}_{i}, \mathbf{k}^\prime_{i}{}^\perp) \frac{2 x_j - \zeta}{\sqrt{ x^{\prime}_{j} x_j }} \psi_{2}(x_{i},\mathbf{k}_{i}^\perp),$$ where the primed variables are given by $$\label{primed}
\begin{cases}
x^\prime_{i} = \frac{x_{i}}{1-\zeta}\\
\mathbf{k}^\prime_{i}{}^\perp = \mathbf{k}_{i}^\perp - x^\prime_{i} \mathbf{\Delta}^\perp, \; \text{for} \; i \neq j
\end{cases}
\begin{cases}
x^\prime_{j} = \frac{x_{j} - \zeta}{1 - \zeta}\\
\mathbf{k}^\prime_{j}{}^\perp = \mathbf{k}_{j}^\perp + (1 - x^\prime_{j}) \mathbf{\Delta}^\perp
\end{cases}$$ and the integration measure is given by $$\begin{aligned}
[dx]_{N} & = \prod_{i = 1}^{N} dx_{i} \; \delta \Big( 1 - \sum_{i = 1}^{N} x_{i} \Big)\\
[d\operatorname{\mathbf{k}^{\perp}}]_{N} & = \frac{1}{[2(2\pi)^3]^{N-1}} \prod_{i=1}^N d\mathbf{k}^\perp_{i} \; \delta \Big( \sum_{i=1}^N \mathbf{k}^\perp_{i} \Big).\end{aligned}$$ Notice the sum over transverse momenta in the delta function is zero since our initial meson has $\mathbf{P}^\perp = 0$. The sum over $j$ in Eq. produces the overall factor of two for our case of equally massive (equally charged) constituents.
To cast the next-to-leading order expressions for $x > \zeta$ in terms of diagonal Fock space overlaps, we must write out the three-body Fock component. Looking at the diagrams in Figure \[ftri2\], it is constructed from the two-body wave function $$\begin{gathered}
\label{threefock}
\psi_{3} (x_{i},\mathbf{k}_{i}^\perp) = g \frac{2 (2\pi)^3}{\sqrt{x_{1} x_{2} x_{3}}}
\int [dy]_{2} [d\operatorname{\mathbf{p}^{\perp}}]_{2} \Bigg[ \theta(y_{1} - x_{1}) x_{3} \delta(y_{2} - x_{3})
\delta(\mathbf{p}^\perp_{2} - \mathbf{k}^\perp_{3}) D(y_1,\mathbf{p}^\perp_1;x_1,\mathbf{k}^\perp_1|M^2)
\\ + \theta(y_2 - x_3) x_1 \delta(y_{1} - x_{1}) \delta(\mathbf{p}^\perp_{1} - \mathbf{k}^\perp_{1})
D(y_2,\mathbf{p}^\perp_2;x_3,\mathbf{k}^\perp_3|M^2)
\Bigg] \frac{\psi_{2}(y_{j},\mathbf{p}_{j}^\perp)}{\sqrt{y_{1} y_{2}}},\end{gathered}$$ where $i$ runs from one to three and the label $j$, which stems from the integration measure, runs from one to two. We discuss how to obtain this three-body wave function directly from time-ordered perturbation theory in Appendix \[oftopt\]. Using $\psi_3$ in Eq. , the terms in the GPD at first order in the weak coupling can then be written compactly as $$\label{323}
F(x,\zeta,t)^{\text{NLO}} = \frac{\theta(x - \zeta)}{1 - \zeta} \int [dx]_3 [d\operatorname{\mathbf{k}^{\perp}}]_{3} \sum_{j = 1,3}
\delta( x - x_{j}) \psi_{3}^*(x_{i}^\prime,\mathbf{k}_{i}^\prime{}^\perp) \frac{2 x_j - \zeta}{\sqrt{ x^{\prime}_{j} x_j }}
\psi_{3}(x_{i},\mathbf{k}^\perp_i).$$
One can verify that the diagrams in Figure \[ftri2\] are generated by . Additionally there is a fourth diagram generated by Eq. which does not appear in the figure. This missing diagram is characterized by the spectator quark’s one-loop self interaction and is absent since we have ignored $f(k^2)$ and the scale dependence of light-cone wave functions.[^2] The absence of this diagram does not affect continuity at the crossover. The missing diagram vanishes at $x = \zeta$ since the final-state wave function is $\psi^*(x^\prime,\ldots)$. Lastly we note the above Fock component overlaps satisfy the positivity constraint for a composite scalar composed of scalar constituents [@Tiburzi:2002kr].
Now we must come to terms with the non-diagonal overlap region, $x < \zeta$. At first order, the diagrams of Figure \[fZZZ\] correspond to four-to-two Fock component overlaps. We have been cavalier about time ordering, however. The expressions Eqs. \[bt2\] and \[fs2\] do not correspond to time-ordered graphs. Both terms contain a product of time-ordered propagators: one for the two quarks leading to the final-state vertex and another for the quark-antiquark pair heading off to annihilation. But for an interpretation in terms of a four-body wave function, all four particles must propagate at the same time. This is a subtle issue as a graph containing the product of two independently time-ordered pieces (where one leads to a bound-state vertex) corresponds to a sum of infinitely many time-ordered graphs. It is easiest to write out the terms of concern in terms of the propagators’ poles. The quark, anti-quark heading to annihilation have propagators $d(k)$ and $d(k+\Delta)$ and poles we label $k^-_a$ and $k^-_c$, respectively. The remaining propagators of interest $d(p^\prime)$ and $d(P^\prime - p^\prime)$ have poles $p^-_a$ and $p^-_b$, respectively. We can then manipulate as follows $$\begin{gathered}
\frac{1}{k^-_{c} - k^-_{a}} \; \frac{1}{p^-_{b} - p^-_{a}} \\
= \frac{1}{p^-_{b} - p^-_{a} + k^-_{c} - k^-_{a}} \; \Bigg(
\frac{1}{p^-_{b} - p^-_{a}} + \frac{1}{k^-_{c} - k^-_{a}} \Bigg). \label{trickery}\end{gathered}$$ In this form, we have produced the correct energy denominator for the instant of light-front time where four particles are propagating. Multiplying this denominator by the three-body wave function yields the four-body wave function (up to constants). This is the part of the four-body wave function relevant for GPDs (there are additional pieces for two-quark, two-boson states, see Appendix \[oftopt\]). In the resulting sum , the first term will produce the two-body wave function for the final state and we will have a genuine four-to-two overlap. We do not write this out explicitly.
The second term in Eq. , however, contains again the propagator for the pair heading to annihilation. Using the light-front Bethe-Salpeter equation for the vertex (which contains infinitely many times) we can introduce a factor of the time-ordered interaction. The resulting product of independent time orderings can again be manipulated as in Eq. . The result produces another overall four-body denominator which contributes to the four-body Fock component of the initial state. Since we iterated the interaction, however, this new contribution is no longer at leading order and can be neglected. Thus the second term in does not contribute at this order.
Having manipulated the GPDs into non-diagonal overlaps for $x < \zeta$, we must wonder if continuity at the crossover is still maintained. In the limit $x \to \zeta$ the light-front energy of the struck quark goes to infinity. Consequently $k^-_{c}$, which contains this on shell energy, is infinite and dominates the four-body energy denominator. This is identical to the reasoning in Eqs. \[bt2\] and \[fs2\] where instead of the four-body denominator, we have $\operatorname{D_{\text{W}}}(x^{\prime\prime}, \operatorname{\mathbf{k}^{\perp}}+ x^{\prime\prime}
\mathbf{\Delta}^\perp |t)$ which is dominated by $k^-_{c}$ at the crossover. Either way, we arrive at the expressions found above for the crossover and .
Having cast our expressions for generalized parton distributions in terms of the Fock components, we can enlarge our understanding of the sum rule and continuity at the crossover. Both must deal with the relation between higher Fock components. The way $\zeta$-dependence disappears from mandates a relation between the diagonal and non-diagonal Fock component overlaps that make up the GPD. The relation between Fock components must follow from the field-theoretic equations of motion. Continuity itself is a special case of the relation between Fock components, specifically at the end points. Above we have seen our expressions are continuous (and non-vanishing) at the crossover and explicitly that the three- and four-body components match at the end point (where $x - \zeta = 0$). This weak binding model for behavior at the crossover is a simple example of the relations between Fock components at the end points (see also [@Antonuccio:1997tw]). More general relations must be permitted from the equations of motion to guarantee Lorentz covariance (e.g. in the structure of the Mellin moments of the GPDs, of which the sum rule is a special case). Here, of course, Lorentz symmetry is broken. Infinitely many light-cone time-ordered graphs are needed in the reduced kernel to reproduce the covariant one-boson exchange . Thus exactly satisfying polynomiality requires not only infinitely many exchanges in the kernel but contributions from infinitely many Fock components. It should be possible, however, to show how the sum rule and polynomiality are improved order by order.
Application to GDAs {#gda}
===================
Below we study time-like form factors to demonstrate the versatility of this approach and make the connection to the generalized distribution amplitudes (GDAs) for this model. Analogous to GPDs, GDAs encode the soft physics of two-meson production and can thus be thought of as crossed versions of the GPDs. The GDAs enter in convolutions for various two-meson production amplitudes [@Diehl:1998dk]. These distribution functions as well as time-like form factors are a theoretical challenge for light-front dynamics, since there is no direct decomposition in terms of meson Fock components alone. Furthermore, we shall see the leading-order expressions are non-valence contributions (which necessarily excludes a description in terms of most constituent quark models).
The time-like form factor $F(s)$ for our model meson is defined by (see Figure \[ftimetri\]) $$\langle \Psi_{p} \; \Psi_{p^\prime} | \; \Gamma^\mu \; | 0 \rangle = - i (p - p^\prime)^\mu F(s),$$ where $s = (p + \operatorname{p^{\prime}})^2$ is the center of mass energy squared. Now define $P^\mu = p^\mu + \operatorname{p^{\prime}}{}^\mu$ and $\zeta = \operatorname{p^{+}}/ \operatorname{P^{+}}$. We can work out the kinematics of this reaction in a frame where $\mathbf{P}^\perp = 0$ $$\begin{aligned}
P^- & = \frac{s}{2 P^+} \notag \\
p^- & = \frac{(1 - \zeta) s}{2 \operatorname{P^{+}}} \notag \\
\mathbf{p}^\perp{}^2 & = s (1 - \zeta) \zeta - M^2,\end{aligned}$$ where $M$ is the meson mass.
Similar to GPDs, the GDA for our model has a definition in terms of a non-diagonal matrix element of bilocal field operators $$\operatorname{\Phi(z, \zeta, s)}= \int \frac{dx^-}{2 \pi} e^{i z \operatorname{P^{+}}x^-} \langle \Psi_{p} \; \Psi_{\operatorname{p^{\prime}}} | \; q(x^-) i \
\overset{\leftrightarrow}\partial{}^+ q(0) \; | 0 \rangle.$$ Such a definition of the GDA leads directly to a sum rule for the time-like form factor $$\label{sumtime}
\int \frac{dz}{2 \zeta - 1} \operatorname{\Phi(z, \zeta, s)}= F(s),$$ and hence a means to calculate $\Phi$ from the integrand of the time-like form factor.
Taking the appropriate residues of the five-point function, we arrive at Fig. \[ftimetri\] for the time-like form factor. Keeping only the leading order piece of the electromagnetic vertex $\Gamma^\mu$, we have $$\begin{gathered}
\operatorname{\Phi(z, \zeta, s)}= i \operatorname{P^{+}}(2z -1) \int \frac{d\operatorname{k^{--}}d\operatorname{\mathbf{k}^{\perp}}}{(2\pi)^4}
\\ \times \gamma^*( \operatorname{z^{\prime\prime}}, \operatorname{\mathbf{k}^{\perp}}- \operatorname{z^{\prime\prime}}\operatorname{\mathbf{p}^{\perp}}| M^2 ) G(k,p) d^{-1}(k-p)
\\ \times G(P-k, \operatorname{p^{\prime}}) \gamma^*\big( \operatorname{z^\prime}, \operatorname{\mathbf{k}^{\perp}}- (1- \operatorname{z^\prime}) \operatorname{\mathbf{p}^{\perp}}\big| M^2 \big), \end{gathered}$$ where we have made use of $z = \operatorname{k^{+}}/ \operatorname{P^{+}}$, $\operatorname{z^\prime}= \frac{z - \zeta}{1 - \zeta}$ and $\operatorname{z^{\prime\prime}}= z / \zeta$. Recall $\gamma(x|M^2) \propto \theta[ x(1-x)]$. This translates to: $0< \operatorname{z^{\prime\prime}}< 1$ and $0< \operatorname{z^\prime}< 1$, and hence we do not pick up a contribution at zeroth order in the coupling.
To work at first order, we pick up three terms analogous to those in Appendix \[fff\]. We denote these as $\delta J^\mu_{\gamma}, \delta J^\mu_{p}$ and $\delta J^\mu_{\operatorname{p^{\prime}}}$. The Born term for the three-point electromagnetic vertex $\delta J^\mu_{\gamma}$ is quite simple. For the same reason as the zeroth-order result, the restriction of $\gamma(x|M^2) \propto \theta[x(1-x)]$ and momentum conservation force the contribution $\delta J^\mu_{\gamma}$ to vanish. This leaves us to consider only diagrams that arise from iteration of the Bethe-Salpeter equation of either final-state meson.
Considering first the term $\delta J^\mu_{p}$, we have the contribution to the GDA $$\begin{gathered}
\label{blah}
\Phi_{p}(z,\zeta,s) = i \operatorname{P^{+}}(2 z -1) \int \frac{d\operatorname{k^{--}}d\operatorname{\mathbf{k}^{\perp}}}{(2\pi)^4} \; \frac{d^4q}{(2\pi)^4}
\\ \times \gamma^*(\operatorname{y^{\prime\prime}}, \operatorname{\mathbf{q}^{\perp}}- \operatorname{y^{\prime\prime}}\operatorname{\mathbf{p}^{\perp}}) G(q,p) V(q,k) G(k,p) G(P - k, \operatorname{p^{\prime}})
\\ \times d^{-1}(k-p) \gamma^*(\operatorname{z^\prime}, \operatorname{\mathbf{k}^{\perp}}- (1-\operatorname{z^\prime}) \operatorname{\mathbf{p}^{\perp}}), \end{gathered}$$ where we have chosen to abbreviate $y = \operatorname{q^+}/ \operatorname{P^{+}}$ and hence the label $\operatorname{y^{\prime\prime}}= y / \zeta$. We have customarily omitted the subtracted term containing $\operatorname{\tilde{G}}$, which is zero because there is no leading-order term to subtract. Requiring wave function vertices mandates $0<\operatorname{y^{\prime\prime}}<1$ and $\zeta < z < 1$. Thus $\Phi_{p}$ produces one contribution to the GDA
$$\begin{gathered}
\label{jp}
\Phi_{p}(z,\zeta,s) = \frac{\theta(z - \zeta)}{(16\pi^3)^2 \zeta} \int \frac{d\operatorname{\mathbf{k}^{\perp}}dy d\operatorname{\mathbf{q}^{\perp}}(2 z - 1)}{z (1-z) \operatorname{z^\prime}\operatorname{y^{\prime\prime}}(1-\operatorname{y^{\prime\prime}})}
\operatorname{D_{\text{W}}}(z, \operatorname{\mathbf{k}^{\perp}}|s) \frac{g^2 \theta(z - y)}{z - y}
\\ \times D(z,\operatorname{\mathbf{k}^{\perp}};y,\operatorname{\mathbf{q}^{\perp}}|s) \psi^*(\operatorname{z^\prime}, \operatorname{\mathbf{k}^{\perp}}- (1-\operatorname{z^\prime}) \operatorname{\mathbf{p}^{\perp}}) \psi^*(\operatorname{y^{\prime\prime}}, \operatorname{\mathbf{q}^{\perp}}- \operatorname{y^{\prime\prime}}\operatorname{\mathbf{p}^{\perp}}). \end{gathered}$$
As a contribution to the time-like form factor, we can interpret Eq. as the time-ordered diagram $b$ of Figure \[fgda\].
At first order in the weak coupling, we have one term remaining to consider $\delta J^\mu_{\operatorname{p^{\prime}}}$. Again omitting the superfluous subtraction of $\operatorname{\tilde{G}}$, we have $$\begin{gathered}
\Phi_{p^\prime}(z,\zeta,s) = i \operatorname{P^{+}}(2 z -1) \int \frac{d\operatorname{k^{--}}d\operatorname{\mathbf{k}^{\perp}}}{(2\pi)^4} \; \frac{d^4 q}{(2 \pi)^4} \gamma^*(\operatorname{z^{\prime\prime}}, \operatorname{\mathbf{k}^{\perp}}- \operatorname{z^{\prime\prime}}\operatorname{\mathbf{p}^{\perp}}| M^2) G(k,p)
d^{-1}(p-k) \\ \times G(P - k , \operatorname{p^{\prime}}) V(P-k, P-q) G(P-q, \operatorname{p^{\prime}}) \gamma^*(\operatorname{y^\prime}, \operatorname{\mathbf{q}^{\perp}}- (1- \operatorname{y^\prime}) \operatorname{\mathbf{p}^{\perp}}|M^2), \end{gathered}$$ with $\operatorname{y^\prime}= \frac{y - \zeta}{1-\zeta}$. Both $\operatorname{y^\prime}$ and $z$ are restricted: $0 < \operatorname{y^\prime}< 1$ and $0 < z < \zeta$, and the final contribution to the GDA is $$\begin{gathered}
\label{jpp}
\Phi_{p^\prime}(z,\zeta,s) = \frac{\theta(\zeta - z)}{(16\pi^3)^2 \zeta} \int \frac{d\operatorname{\mathbf{k}^{\perp}}d\operatorname{y^\prime}d\operatorname{\mathbf{q}^{\perp}}(2 z - 1)}{(1-z) \operatorname{z^{\prime\prime}}(1-\operatorname{z^{\prime\prime}}) \operatorname{y^\prime}(1-\operatorname{y^\prime})}
\operatorname{D_{\text{W}}}(z, \operatorname{\mathbf{k}^{\perp}}| s) \frac{g^2 \theta(y - z)}{y-z}
\\ \times D(y,\operatorname{\mathbf{q}^{\perp}};z,\operatorname{\mathbf{k}^{\perp}}|s) \psi^*(\operatorname{z^{\prime\prime}}, \operatorname{\mathbf{k}^{\perp}}- \operatorname{z^{\prime\prime}}\operatorname{\mathbf{p}^{\perp}}) \psi^*(\operatorname{y^\prime}, \operatorname{\mathbf{q}^{\perp}}+ \operatorname{y^\prime}\operatorname{\mathbf{p}^{\perp}}), \end{gathered}$$
In this form we recognize this contribution as diagram $a$ of Figure \[fgda\].
Having found the leading non-vanishing contribution to the GDA namely $\Phi = \Phi_{p} + \Phi_{p^\prime}$, we observe that the higher Fock components derived in section \[gpds\] (as well as in Appendix \[oftopt\]) do not fit naturally into or . One needs a Fock space expansion for the photon wave function in order to have an expression for the GDA in terms of various Fock component overlaps. With the expressions derived for the GDA we can use Eq. to obtain the time-like form factor.
Summary {#summy}
=======
Above we have investigated various applications of the light-front reduction of current matrix elements. First we considered the normalization of the light-front wave function in the reduction formalism deriving Eq. from the covariant normalization Eq. . The complicated form of the reduced normalization was linked to effects of higher Fock components (which we illustrated by using the ladder model in perturbation theory). Using the explicit form of the leading-order kernel, we were able to derive Eq. , which is the familiar many-body normalization condition .
In Appendix \[fff\], we reviewed the derivation of the form factor at next-to-leading order in the $(3+1)$-dimensional ladder model. These expressions were then converted into the GPD for the model. Continuity of these distributions at the crossover (where the plus momentum of the struck quark is equal to the plus component of the momentum transfer) was explicitly demonstrated, *cf* Eq. . Connection was made to the overlap representation of GPDs by constructing the three-body wave function to leading order in perturbation theory. As a check on our results, we also reviewed the construction of higher Fock states from the valence sector in old-fashioned time-ordered perturbation theory (Appendix \[oftopt\]). The derived overlaps Eqs. and satisfy the relevant positivity constraint. The non-vanishing of the GPDs at the crossover could then be tied to higher Fock components, specifically at vanishing plus momentum, and are hence essential for any phenomenological modeling of these distributions. This rewriting allowed us to understand how continuity arises perturbatively from the small-$x$ behavior of Fock state wave functions. In perturbation theory, the diagonal valence overlap vanishes at the crossover, while the higher Fock component overlaps do not. In general the $n$-to-$n$ overlap matches up with the $(n+1)$-to-$(n-1)$ overlap at the crossover due to the dominance of the rebounding quark’s infinite energy. The same seems to be true perturbatively for a three-body bound state due to the nature of the kernel.
Unfortunately issues involving Lorentz invariance (such as the sum rule for the electromagnetic form factor and the polynomiality constraints) are left untouched. To maintain covariance one would need infinitely many time-ordered exchanges in the kernel as well as infinitely many Fock components. It should be possible, however, to understand perturbatively how the $\zeta$-dependence disappears from Eq. . This requires further relations between Fock components and these should be afforded by the field-theoretic equations of motion.
Lastly we considered application of the reduction formalism for currents to the time-like form factor. We did so by calculating the ladder model’s GDA Eqs. (\[jp\]-\[jpp\]), which is related to the time-like form factor via the sum rule in Eq. , systematically in perturbation theory. This is in contrast to the non-existent Fock space expansion for these types of processes.
With the formalism explored here, one could use phenomenological Lagrangian based models to explore both generalized parton distributions and generalized distributions amplitudes within the light-front framework. Such an investigation is interesting not only for testing phenomenological models, but also for anticipating problems for approximate non-perturbative solutions for the light-cone Fock states. Nonetheless more model studies are warranted before truly realistic calculations can be pursued.
We thank M. Diehl for enthusiasm, questions and critical comments. This work was funded by the U. S. Department of Energy, grant: DE-FG$03-97$ER$41014$.
Wave functions and form factors in $(3+1)$ dimensions {#fff}
=====================================================
In this Appendix we collect results relevant above for wave functions and form factors in the $(3+1)$-dimensional ladder model Eq. .
Wave functions
--------------
Using the Bethe-Salpeter equation and the definition of the light-cone wave function $|\psi_R\rangle$ we have the light-cone bound-state equation $$\label{bse}
|\psi_{R}\rangle = g(R) w(R) |\psi_R \rangle,$$ where $w(R)$ is the reduced auxiliary kernel $$w(R) = g^{-1}(R) \Big| G(R) W(R) G(R) \Big| g^{-1}(R),$$ with $W(R)$ defined in Eq. .
To leading order in $G - \operatorname{\tilde{G}}$, one calculates $w(R)$ for the ladder model to be: $$\begin{gathered}
\label{OBE}
V(x,\operatorname{\mathbf{k}^{\perp}};y,\operatorname{\mathbf{p}^{\perp}}|M^2) \equiv - \langle \; xR^+,\operatorname{\mathbf{k}^{\perp}}| \; w(R) \; | \; yR^+,\operatorname{\mathbf{p}^{\perp}}\rangle \\
= \frac{g^2}{x-y} \Big[ \theta(x-y) D(x,\operatorname{\mathbf{k}^{\perp}};y,\operatorname{\mathbf{p}^{\perp}}|M^2) \\
- \big\{(x,\operatorname{\mathbf{k}^{\perp}}) \longleftrightarrow (y, \operatorname{\mathbf{p}^{\perp}}) \big\} \Big] \theta[x(1-x)] \theta[y(1-y)],\end{gathered}$$ where we have defined $$\begin{gathered}
D^{-1}(x,\operatorname{\mathbf{k}^{\perp}};y,\operatorname{\mathbf{p}^{\perp}}|M^2) =
M^2 - \frac{\operatorname{\mathbf{p}^{\perp}}^2 + m^2}{y} \\ - \frac{(\operatorname{\mathbf{k}^{\perp}}- \operatorname{\mathbf{p}^{\perp}})^2 + \mu^2}{x - y} - \frac{\operatorname{\mathbf{k}^{\perp}}^2 + m^2}{1-x},\end{gathered}$$ and taken $\mathbf{R}^\perp = 0$. Graphically this one-boson exchange potential Eq. is depicted in Figure \[fOBEP\].
For reference, the bound-state equation appears as $$\begin{gathered}
\label{wavefunction}
\psi(x,\operatorname{\mathbf{k}^{\perp}}) = \operatorname{D_{\text{W}}}(x,\operatorname{\mathbf{k}^{\perp}}|M^2) \int \frac{dy d\operatorname{\mathbf{p}^{\perp}}}{2(2\pi)^3 y (1-y)} \\ \times V(x,\operatorname{\mathbf{k}^{\perp}};y,\operatorname{\mathbf{p}^{\perp}}|M^2) \psi(y,\operatorname{\mathbf{p}^{\perp}})\end{gathered}$$
Form factors
------------
To calculate form factors, we use the electromagnetic vertex $\Gamma^\mu$ constructed in [@future] up to the first Born approximation (notice that the ladder model’s gauged interaction $V^\mu = 0$) $$\Gamma^\mu = \Big( \overset{\leftrightarrow}\partial{}^\mu + V G \overset{\leftrightarrow}\partial{}^\mu \Big) d^{-1}_2,$$ where $\overset{\leftrightarrow}\partial{}^\mu$ denotes the electromagnetic coupling to scalars. Now using Eq. to first order in $G - \operatorname{\tilde{G}}$, the matrix element $J^\mu = \langle \Psi_{P^\prime}| \Gamma^\mu(-\Delta)
| \Psi_P \rangle$ then appears $$\begin{gathered}
J^\mu \approx \; \langle \gamma_{P^\prime} | \; \Big| G(P^\prime) \Big( 1 + V(P^\prime) ( G(P^\prime) - \operatorname{\tilde{G}}(P^\prime)) \Big)
\\ \times \Big( \overset{\leftrightarrow}\partial{}^\mu (-\Delta) d_{2}^{-1} + V(-\Delta) G(-\Delta) \overset{\leftrightarrow}\partial{}^\mu (-\Delta) d_{2}^{-1} \Big)
\\ \times \Big( 1 + (G(P) - \operatorname{\tilde{G}}(P)) V(P) \Big) G(P) \Big| \; | \gamma_{P} \rangle \\
= \Big( J^\mu_{\text{LO}} + \delta J^\mu_{i} + \delta J^\mu_{f} + \delta J^\mu_{\gamma} \Big) + \mathcal{O}[V^2],\end{gathered}$$ with the leading-order result $$\label{LO}
J^\mu_{\text{LO}} = \langle \gamma_{P^\prime} | \; \Big| G(P^\prime) \overset{\leftrightarrow}\partial{}^\mu (-\Delta) d_{2}^{-1} G(P) \Big| \; | \gamma_{P} \rangle.$$ The first-order terms are $$\begin{gathered}
\delta J^\mu_{i} = \langle \gamma_{P^\prime} | \; \Big| G(P^\prime) \overset{\leftrightarrow}\partial{}^\mu(-\Delta) d_{2}^{-1}
\\ \times \Big(G(P) - \operatorname{\tilde{G}}(P) \Big) V(P) G(P) \Big| \; |\gamma_{P} \rangle \notag\end{gathered}$$ $$\begin{gathered}
\delta J^\mu_{f} = \langle \gamma_{P^\prime} | \; \Big| G(P^\prime) V(P^\prime)
\\ \times \Big(G(P^\prime) - \operatorname{\tilde{G}}(P^\prime) \Big)
\overset{\leftrightarrow}\partial{}^\mu (-\Delta) d_{2}^{-1} G(P) \Big| \; | \gamma_{P} \rangle \notag\end{gathered}$$ $$\begin{gathered}
\delta J^\mu_{\gamma} = \langle \gamma_{P^\prime} | \; \Big| G(P^\prime) \Big(V(-\Delta) G(-\Delta)
\overset{\leftrightarrow}\partial{}^\mu (-\Delta) \Big) \\ \times d_{2}^{-1} G(P) \Big| \; | \gamma_{P} \rangle. \label{NLO}\end{gathered}$$
As outlined in [@future], Eqs. and can be evaluated by residues being careful to remove two-particle reducible contributions by utilizing . Here we state the results of these calculations in $(3+1)$ dimensions. We denote $\Delta^\mu$ as the momentum transfer and define $\Delta^+ = - \zeta P^+$ (see Figure \[ftri\]). The leading-order result appears
$$\label{FFLO}
J^+_{LO} = - 2 i P^+ \int \frac{\theta(x - \zeta) \; dx \; d\operatorname{\mathbf{k}^{\perp}}}{2(2\pi)^3 x(1-x) x^\prime} (2 x - \zeta)
\psi^*(x^\prime,\mathbf{k}^{\prime\perp}) \psi(x,\operatorname{\mathbf{k}^{\perp}}),$$
where $x^\prime = \frac{x - \zeta}{1-\zeta}$ and $\mathbf{k}^{\prime\perp} = \operatorname{\mathbf{k}^{\perp}}+ (1-x^\prime) \operatorname{\mathbf{\Delta}^\perp}$ denotes the momentum of the final state. Using $J^\mu = - i (P + P^\prime)^\mu F(t)$, Eq. reduces to the Drell-Yan formula [@Drell:1969km] for $\zeta = 0$.
The first of the leading order corrections is the Born term $\delta J^+_\gamma$. For $x>\zeta$ we have $$\begin{gathered}
\label{bt1}
\delta J^+_{\gamma \; \; (x>\zeta)} = \frac{+ 2iP^+}{(16\pi^3)^2} \int \frac{\theta(x - \zeta) dx d\operatorname{\mathbf{k}^{\perp}}dy d\operatorname{\mathbf{p}^{\perp}}(2x - \zeta)}{x x^\prime y (1-y) y^\prime}
\\ \times \psi^*(y^\prime,\mathbf{p}^{\prime\perp}) D(y^\prime,\mathbf{p}^{\prime\perp};x^\prime,\mathbf{k}^{\prime\perp} |M^2)
\frac{g^2 \theta(y-x)}{y - x} D(y,\operatorname{\mathbf{p}^{\perp}};x,\operatorname{\mathbf{k}^{\perp}}|M^2) \psi(y,\operatorname{\mathbf{p}^{\perp}}),\end{gathered}$$ where $y^\prime = \frac{y - \zeta}{1-\zeta}$ and $\operatorname{\mathbf{p}^{\prime\perp}}= \operatorname{\mathbf{p}^{\perp}}+ (1- y^\prime) \operatorname{\mathbf{\Delta}^\perp}$. This contribution corresponds to diagram $A$ in Figure \[ftri2\]. On the other hand, for $x<\zeta$ we have
$$\begin{gathered}
\label{bt2}
\delta J^+_{\gamma \; \; (x<\zeta)} = \frac{+2 i P^+}{(16\pi^3)^2} \int \frac{\theta(\zeta - x) dx d\operatorname{\mathbf{k}^{\perp}}dy d\operatorname{\mathbf{p}^{\perp}}(2 x - \zeta)/\zeta}{y(1-y) \operatorname{y^\prime}\operatorname{x^{\prime\prime}}(1-\operatorname{x^{\prime\prime}})} \\ \times
\psi^*(\operatorname{y^\prime},\operatorname{\mathbf{p}^{\prime\perp}})
\operatorname{D_{\text{W}}}(\operatorname{x^{\prime\prime}},\operatorname{\mathbf{k}^{\prime\prime\perp}}|t) \frac{g^2 \theta(y - x)}{y - x} D(y,\operatorname{\mathbf{p}^{\perp}};x,\operatorname{\mathbf{k}^{\perp}}|M^2) \psi(y,\operatorname{\mathbf{p}^{\perp}}),\end{gathered}$$
where $\operatorname{x^{\prime\prime}}= x / \zeta$ and $\operatorname{\mathbf{k}^{\prime\prime\perp}}= \operatorname{\mathbf{k}^{\perp}}+ \operatorname{x^{\prime\prime}}\operatorname{\mathbf{\Delta}^\perp}$ denotes the photon’s relative momentum. This expression is diagram $D$ in Figure \[fZZZ\].
The next leading-order term is the initial-state iteration $\delta J^+_i$. The only contribution is for $x>\zeta$, namely $$\begin{gathered}
\label{is1}
\delta J^+_{i} = \frac{+2iP^+}{(16\pi^3)^2} \int \frac{\theta(x - \zeta) dx d\operatorname{\mathbf{k}^{\perp}}dy d\operatorname{\mathbf{p}^{\perp}}(2x - \zeta)}{x x^\prime (1- x^\prime) y (1-y)} \\
\times
\psi^*(x^\prime,\mathbf{k}^{\prime\perp})
D(y^\prime,\mathbf{p}^{\prime\perp};x^\prime,\mathbf{k}^{\prime\perp} |M^2)
\frac{g^2 \theta(y-x)}{y - x} D(y,\operatorname{\mathbf{p}^{\perp}};x,\operatorname{\mathbf{k}^{\perp}}|M^2) \psi(y,\operatorname{\mathbf{p}^{\perp}}),\end{gathered}$$ which corresponds to diagram $B$ of Figure \[ftri2\].
Lastly there is the final-state iteration term $\delta J^+_f$. For $x>\zeta$ we have $$\begin{gathered}
\label{fs1}
\delta J^+_{f \; \; (x>\zeta)} = \frac{+2iP^+}{(16\pi^3)^2} \int \frac{\theta(x - \zeta) dx d\operatorname{\mathbf{k}^{\perp}}d\operatorname{y^\prime}d\operatorname{\mathbf{p}^{\prime\perp}}(2 x - \zeta)}{x(1-x)\operatorname{x^\prime}\operatorname{y^\prime}(1-\operatorname{y^\prime})}
\\ \times \psi^*(\operatorname{y^\prime},\operatorname{\mathbf{p}^{\prime\perp}}) D(\operatorname{y^\prime},\operatorname{\mathbf{p}^{\prime\perp}};\operatorname{x^\prime},\operatorname{\mathbf{k}^{\prime\perp}}|M^2) \frac{g^2 \theta(y - x)}{y - x} D(y,\operatorname{\mathbf{p}^{\perp}};x,\operatorname{\mathbf{k}^{\perp}}|M^2) \psi(x,\operatorname{\mathbf{k}^{\perp}}), \end{gathered}$$ where implicitly $y = \zeta + (1-\zeta) \operatorname{y^\prime}$ and $\operatorname{\mathbf{p}^{\perp}}= \operatorname{\mathbf{p}^{\prime\perp}}- (1- \operatorname{y^\prime}) \operatorname{\mathbf{\Delta}^\perp}$. This corresponds to diagram $C$ in Figure \[ftri\]. While for $x<\zeta$, the expression $$\begin{gathered}
\label{fs2}
\delta J^+_{f \; \; (x< \zeta)} = \frac{+2iP^+}{(16\pi^3)^2} \int \frac{\theta(\zeta - x) dx d\operatorname{\mathbf{k}^{\perp}}d\operatorname{y^\prime}d\operatorname{\mathbf{p}^{\prime\perp}}(2 x - \zeta)/\zeta}{(1-x) \operatorname{x^{\prime\prime}}(1-\operatorname{x^{\prime\prime}}) \operatorname{y^\prime}(1-\operatorname{y^\prime})}
\\ \times \psi^*(\operatorname{y^\prime},\operatorname{\mathbf{p}^{\prime\perp}}) \operatorname{D_{\text{W}}}(\operatorname{x^{\prime\prime}},\operatorname{\mathbf{k}^{\prime\prime\perp}}|t) \frac{g^2 \theta(y - x)}{y - x} D(y,\operatorname{\mathbf{p}^{\perp}};x,\operatorname{\mathbf{k}^{\perp}}|M^2) \psi(x,\operatorname{\mathbf{k}^{\perp}}),\end{gathered}$$ corresponds to diagram $E$ of Figure \[fZZZ\]. Eqs. (\[FFLO\]-\[fs2\]) are then the complete expressions for the form factor up to first order.
Old-fashioned time-ordered perturbation theory {#oftopt}
==============================================
The results of this paper can similarly be achieved directly from “old-fashioned” time-ordered perturbation theory in a form which utilizes projecting onto the two-body subspace of the full Fock space. For a nice, complete discussion of this formalism for the light-cone ladder model, see [@Cooke:2000ef]. In this Appendix, we show how to derive higher Fock space components in this formalism, thereby demonstrating the generation of higher components from the lowest sector we found indirectly for GPDs and form factors in section \[gpds\].
We write the light-cone Hamiltonian as a sum of a free piece and an interacting piece which carries an explicit power of the weak coupling $g$. In an obvious notation this is $$P^- = P^-_o + g P_{I}^-.$$ The free term $P^-_o$ is diagonal in the Fock state basis, while the interaction generally mixes components of different particle number (in the scalar model we consider above, the interaction is completely off-diagonal since there are no instantaneous terms). Let us suppose that in the full Fock basis, we have an eigenstate of the Hamiltonian, i.e. $$\Big( P^-_o + g P_{I}^- \Big) \operatorname{|\psi\rangle}= \operatorname{p^{--}}\operatorname{|\psi\rangle},$$ where the eigenvalue is labeled by $\operatorname{p^{--}}$. Since the coupling is presumed small, the mixing of Fock components with a large number of particles will be small. Thus one imagines our bound state will be dominated by the two-body Fock component.
To make this observation formal, we define projections operators on the Fock space $\operatorname{\mathcal{P}}$ and $\operatorname{\mathcal{Q}}$ in the usual sense. The operator $\operatorname{\mathcal{P}}$ projects out only the two-particle subspace of the full Fock space and hence $\operatorname{\mathcal{Q}}$ projects out the compliment. Let us define the action of these operators on our eigenstate $$\begin{aligned}
\operatorname{\mathcal{P}}\operatorname{|\psi\rangle}& = \operatorname{|\psi_{2}\rangle}\label{2ket}\\
\operatorname{\mathcal{Q}}\operatorname{|\psi\rangle}& = \operatorname{|\psi_{\operatorname{\mathcal{Q}}}\rangle}\label{qket}.\end{aligned}$$ As is well known, combination of Eqs. and leads to the following equation for the two-body Fock component $$P^-_{\text{eff}} \operatorname{|\psi_{2}\rangle}= \operatorname{p^{--}}\operatorname{|\psi_{2}\rangle},$$ where the effective two-body Hamiltonian is $$\begin{aligned}
\label{veff}
P^-_{\text{eff}} & \equiv P^-_{\operatorname{\mathcal{P}}\operatorname{\mathcal{P}}} + V_{\text{eff}} \\
& = P^-_{\operatorname{\mathcal{P}}\operatorname{\mathcal{P}}} + P^-_{\operatorname{\mathcal{P}}\operatorname{\mathcal{Q}}} \frac{1}{\operatorname{p^{--}}- P^-_{\operatorname{\mathcal{Q}}\operatorname{\mathcal{Q}}}} P^-_{\operatorname{\mathcal{Q}}\operatorname{\mathcal{P}}},\end{aligned}$$ and we have defined the following notation for any operators $A$ and $B$, $P^-_{A B} \equiv A P^- B$. The effective two-body interaction $V_{\text{eff}}$ defined in equation is dependent upon the energy eigenvalue $\operatorname{p^{--}}$ since we have suppressed the degrees of freedom of the $\operatorname{\mathcal{Q}}$ subspace. The relation between the $\operatorname{\mathcal{Q}}$-space probability (i.e. the non-valence contribution) and the effective interaction appears $$\label{quspace}
\langle \psi_{\operatorname{\mathcal{Q}}} \operatorname{|\psi_{\operatorname{\mathcal{Q}}}\rangle}= - \frac{\partial}{\partial \operatorname{p^{--}}} \operatorname{\langle\psi_{2}|}V_{\text{eff}} \operatorname{|\psi_{2}\rangle}.$$
In a weak-binding limit, we can series expand the effective interaction in powers of the coupling and thereby re-derive the light-front potential. Given that every boson emitted must be absorbed in the two-quark sector, we can have only an even number of interactions and hence $$V_{\text{eff}} = \operatorname{\mathcal{P}}g P^-_{I} \operatorname{\mathcal{Q}}\frac{1}{\operatorname{p^{--}}- P_{o}^-} \sum_{n=0}^{\infty} \Bigg( \frac{g P^-_{I}}{\operatorname{p^{--}}- P^-_{o}} \Bigg)^{2 n} \operatorname{\mathcal{Q}}g P^-_{I} \operatorname{\mathcal{P}}.$$ So, for example, at leading order we have all possible ways to propagate from the two-body sector and back with only two interactions in between. The diagrams in Figure \[fOBEP\] correspond to the two possibilities distinguished by the action of $\frac{1}{\operatorname{p^{--}}- P^-_{o}}$ between interactions. At the next order, we have all possible ways to propagate from two bodies to two bodies with four interactions in between, *etc*.
To generate higher Fock components from the two-body sector, we necessarily must look at the $\operatorname{\mathcal{Q}}$-space state arrived at from Eqs. and $$\operatorname{|\psi_{\operatorname{\mathcal{Q}}}\rangle}= \frac{1}{\operatorname{p^{--}}- P^-_{\operatorname{\mathcal{Q}}\operatorname{\mathcal{Q}}}} P^-_{\operatorname{\mathcal{Q}}\operatorname{\mathcal{P}}} \operatorname{|\psi_{2}\rangle}.$$ To generate an $n$-body Fock component from this state, we merely act with an $n$-body projection operator which we shall denote $\operatorname{\mathcal{Q}}_{n}$. Similar to the above, we expand in powers of the coupling to find $$| \psi_{n} \rangle = \operatorname{\mathcal{Q}}_{n} \frac{1}{\operatorname{p^{--}}- P^-_{o}} \sum_{n = 0}^{\infty} \Bigg( \frac{g P^-_{I}}{\operatorname{p^{--}}- P^-_{o}} \Bigg)^n \operatorname{\mathcal{Q}}g P^-_{I} \operatorname{|\psi_{2}\rangle}.$$ For example, the leading-order three-body state is obtained by attaching a boson to a quark line in the only two possible ways (and adding the light-front energy denominator at the end). With these three-body states, we can consider all possible three-to-three overlaps that would contribute to the form factor. These are depicted in Figure \[ftri2\] (with the exception of a quark self-interaction). The four-body sector is richer since there are two-boson, two-quark states as well as four-quark states. The two-to-four overlaps required for GPDs must have four quarks. At leading order, we generate the diagrams encountered above in Figure \[fZZZ\]. Not surprisingly directly applying time-ordered perturbation theory from a light-front Hamiltonian agrees with our derivation above from covariant perturbation theory in the Bethe-Salpeter formalism.
[99]{}
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[^1]: For any vector $a^\mu$, we define the light-cone variables $a^\pm = \frac{1}{\sqrt{2}} (a^0 \pm a^3)$.
[^2]: Recently the scale dependence of light-cone Fock components has been investigated [@Burkardt:2002uc].
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Data from the wide-angle, moderately deep ESO Imaging Survey have been used to produce target lists for the first year of the VLT. About 250 candidate clusters of galaxies have been identified from the $I-$band images covering $\sim$ 17 square degrees. In addition, using the multicolor data available over an area of 1.3 square degrees over 300 potentially interesting point-sources have been selected. The color-selected targets include low-mass stars/brown dwarfs, white-dwarfs and quasars. Images, object catalogs and derived target lists are available from the world-wide web (http://www.eso.org/eis).'
address: 'European Southern Observatory, Karl-Schwarzschild-Str 2, D-85748 Garching b. München, Germany '
author:
- Luiz Nicolaci da Costa
---
1[$h^{-1}$]{} \#1
Introduction
============
The ESO Imaging Survey (EIS) was conceived as a public survey to provide the ESO community with suitable data for producing target lists for the first year of VLT operation [@Renzini]. EIS has been carried out as a concerted effort of ESO and its community to optimize the one-year period available between the re-commissioning of the NTT in July 1997 and the deadline for proposals for the first period of scientific operation of the VLT. To maximize the return a new framework was established by ESO and the Observing Program Committee whereby a Working Group was created to define the goals and to oversee the execution of a public imaging survey. A Visitor Program was also created to allow tapping on the expertise of different European groups to develop the tools required for the efficient translation of raw images into potentially useful target lists. As a by-product the project has also been used to establish the infra-structure (pipeline processing, archive, object-oriented database) required to cope with the large increase in the data flow expected from new CCD-mosaic cameras under development for La Silla and Paranal, fully dedicated for wide-field imaging.
The first phase of EIS (EIS-wide) consisted of a moderately deep ($I\lsim 23.5$) wide-angle survey, covering four patches of sky spread over the right ascension range $22^h<\alpha<9^h$, thus providing targets nearly year-round. The observations of EIS-wide have already been completed and all the data in the form of photometrically and astrometrically calibrated pixel maps, object and derived catalogs are publicly available. These data can be retrieved from the web via an user interface built in collaboration with the ESO Science Archive group as a prototype for future distribution of public data. Table 1 summarizes the position of the EIS patches and the area covered in each of the passbands considered.
[**Table 1.**]{} EIS-Wide Sky Coverage
Patch $\alpha$ $\delta$ B V I
------- ---------- ----------- ----- ----- ----- --
A 22:42:54 -39:57:32 - 1.2 3.2
B 00:49:25 -29:35:34 1.7 1.7 1.8
C 05:38:24 -23:51:00 - - 6.0
D 09:51:36 -21:00:00 - - 6.0
- - 1.7 2.9 17
A full description of the EIS pipeline and the quality of the data for each patch can be found in papers that accompanied each data release [@Nonino] [@Prandoni] [@Benoist]. The computed star counts are, in general, consistent with model predictions, while the galaxy counts obtained for each patch are internally consistent and in good agreement with those of other authors. Moreover, we find that the $I-$band observations are sufficiently deep to search for distant clusters. Internal and external comparisons of the galaxy angular two-point correlation function are also in good agreement, indicating that the derived galaxy catalogs are uniform.
At the time of this writing, we have already started the second phase of EIS (EIS-deep) which consists of a deep, multicolor survey in five optical and three infrared passbands covering 75 arcmin$^2$ of the HST/Hubble Deep Field South (HDFS), including the WFPC2, STIS and NICMOS fields, and a region of 100 arcmin$^2$ selected for deep X-ray observations with AXAF. The data will be used to find U- and B-dropouts and to produce galaxy samples with photometric redshifts from which galaxies in the redshift range $ 1 < z < 2$ can be drawn for follow-up spectroscopic observations in the infrared.
Cluster Candidates
==================
Cluster candidates were identified using a cluster finding pipeline implemented at the back-end of the EIS pipeline [@Olsena]. The search algorithm is based on the matched-filter technique [@Postman] and it was chosen to facilitate the comparison with the results of one of the few systematic searches for optically-selected distant clusters [@Postman]. It should be emphasized that the primary goal of the EIS team has been to prepare a list of cluster [*candidates*]{} for follow-up observations and not to produce a well-defined sample for statistical analysis. Instead, our main concern has been to minimize the number of false detections, thereby increasing the yield in future follow-up work. For this purpose the analysis has been restricted to the most uniform surveyed areas and parameters were chosen conservatively, using an extensive set of simulations. We point out that the derived catalog is not unique, given the various underlying assumptions of the method and our particular choice of parameters. However, since the data are public, other groups may produce their own catalogs using different methods (Lobo, this conference). A comparison of various catalogs will be instructive in evaluating the strengths and weaknesses of different algorithms.
The total sample of EIS $I-$band cluster candidates consists of 252 objects in the redshift range $0.2 \leq z \leq 1.3$ [@Scodeggio]. The redshift distribution of the sample is shown in figure \[fig:zall\]. The median redshift of the distribution is $z\sim
0.4$. Note that the EIS redshift distribution differs somewhat from that observed by PDCS, also shown in the figure. The number of EIS candidates decreases monotonically with redshift up to $z \sim 0.6$ with an extended tail beyond, in contrast to the PDCS which shows a relatively flat distribution peaking at $z \sim 0.4$.
Another way of further testing the reality of the detections is to use data in different passbands. Using the $V$-band data available for $\sim$ 2.7 square degrees, clusters were identified and cross-correlated to the $I$ detections. The results can be summarized as follows [@Olsenb]: 1) About 90% of the cluster candidates with $z \le 0.5$ and about 25% with $z>0.5$, primarily rich clusters, are confirmed using the $V$ candidates; 2) Candidates at low-redshift show the red envelope in the C-M relation expected for ellipticals. The CM relation serves as an independent confirmation of the candidate clusters and an independent redshift estimate, by and large consistent with the estimates from the matched filter method.
Color-Selected Targets
======================
Preliminary lists of other potentially interesting targets were also extracted from the multicolor data obtained for a 1.7 square degree region near the South Galactic Pole. The region was observed in $B, V$ and $I$ and offers a unique combination of area and depth. These lists contain a total of 358 objects ($I \lsim 21.5$) over 1.27 square degrees, after eliminating regions observed under less than ideal conditions. Among the color selected targets are candidate very low mass stars/brown dwarfs (62), white-dwarfs (32), and quasars (264) [@Zaggia]. These objects are natural candidates for follow-up spectroscopic observations and illustrate the usefulness of the EIS data for a broad range of science. The selected objects can be found in the web and can be examined by displaying side-by-side image postage stamps in the three passbands. Improvements in the sample selection are certainly possible and we encourage interested groups to produce their own samples. From the present data the derived samples typically include 50 to 100 candidates each. However, much larger samples will be available from the Pilot Survey to be carried out with the new wide-field camera on the 2.2 m telescope at La Silla.
Summary
=======
One year after the first observations all the data for EIS-wide are available to the community. In addition, a basic pipeline for the processing of images is available which allows for the processing of individual frames, the coaddition of overlapping images, the extraction of objects and the preparation of color catalogs. The pipeline is currently being generalized to handle data from from different detectors and CCD mosaics, an essential step for its use with the wide-field imager [email protected] to be commissioned later this year.
The EIS candidate cluster catalog consists of about 250 candidates in the redshift range $0.2 < z < 1.3$, distributed over a wide range of right ascension, with candidates available year-round. This sample is currently the largest available in the southern hemisphere and confirmation work will start soon. EIS has also produced over 300 color-selected targets for different scientific goals. More importantly, these samples can grow in time if similar public surveys are carried out on the 2.2m telescope, with an efficiency at least 6 times higher than the original EIS at the NTT. It should be point out that a Pilot Survey on the 2.2m, which aims at completing EIS-wide, has already been scheduled for the beginning of 1999.
[99]{}[ Benoist , 1998, submitted to A&A (astro-ph/9807334) Nonino, M., et al. 1998, A&A, [*in press*]{}, (astro-ph/9803336) Olsen, L.F., et al. 1998a, A&A, [*in press*]{}, (astro-ph/9803338) Olsen, L.F., et al. 1998b, submitted to A&A (astro-ph/9807156) Postman, M., Lubin, L.M., Gunn, J.E., Oke, J.B., Hoessel, J.G., Schneider, D.P., Christensen, J.A. 1996, AJ, 111, 615 Prandoni, I., et al. 1998, submitted to A&A (astro-ph/9807153) Renzini, A. & da Costa, L. N. 1997, Messenger 87, 23 Scodeggio, M. , 1998, submitted to A&A (astro-ph/9807336) Zaggia, S. , 1998, submitted to A&A (astro-ph/9807152) ]{}
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{
"pile_set_name": "ArXiv"
}
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---
author:
- 'V.A.Berezin,'
- 'V.I.Dokuchaev'
- 'and Yu.N.Eroshenko'
title: 'Particle creation phenomenology, Dirac sea and the induced Weyl and Einstein–dilaton gravity'
---
Introduction
============
One of the most intriguing consequences of the quantum field theory is the phenomenon of particle creation. It is explained by the distortion of the vacuum state by the presence of some external fields. Everybody knows that the quantum fields must be renormalized in order to produce finite physically acceptable results. In the curved space-times the situation become much more subtle. First of all, there appear two new aspects in the renormalization procedure, local and global ones. The local aspect consists in that the counter–terms, needed to compensate the divergences in one–loop quantum calculations, contain the quadratic contributions of the Riemann curvature tensor and its convolutions, the Ricci tensor and scalar curvature, which are absent in the primordial Einstein–Hilbert action. This led A.D.Sakharov to the idea that the gravitational field is not fundamental but is just the manifestation of the vacuum fluctuations of all other fields [@Sakh], known nowadays as “the induced gravity”. The global aspect is that in the curved space-times there can exist the event horizons. They will change drastically their global geometrical structure and will influence the behavior of the quantum wave functions. The remarkable example is the black hole evaporation discovered by S.W.Hawking [@Hawking74; @Hawking75]. The event horizon may appear even in the locally flat space-times with conical singularity, accompanying by the analogous thermal effect what was demonstrated by W.G.Unruh [@Unruh]. This is connected to the non-inertial motion of the observers. In this paper we will be interested in the local aspects only
The cosmological particle creation in the framework of General Relativity were studied extensively in 70-s of the last century by many authors [@Parker69; @GribMam69; @Zeld70; @ZeldStar71; @ZeldPit71; @ParkerFull73; @HuFullPar73; @FullParHu74; @FullPar74; @LukashStar74; @ZS77; @BerKuzTk83]. Due to results of their works we know much about the structure of the counter-terms, the importance of the trace anomaly in the particle creation processes, the rate of particle production and so on.
All the above-mentioned investigations was confined to considering the quantum scalar field on the given background metrics, namely, cosmological homogeneous, but slightly anisotropic, space-times. What about the back reaction? The main obstacle in accounting for the back reaction is that the rigorous solution of the quantum problem requires the knowledge of the boundary conditions, while the latter can be imposed only after solving the (classical) Einstein equations. Thus, we have got the “vicious circle”.
Meanwhile, the back reaction seems very important, because not only the already created particles will change the geometry, but the very process of creation, being the pure quantum phenomenon of changing the vacuum energetic structure, should affect the classical gravitational field and may violate the well known energy dominance condition (see, e.g., [@Ber87a; @Ber14]). Therefore, the back reaction influence may appear crucial in constructing the global space-time geometry.
To avoid this difficulty, we propose to describe the particle creation process phenomenologically, on the classical level, what should be rather reasonable when the gravitational field is strong enough (e.g., in the early universe and inside black holes). We will use the fundamental result by Ya.B.Zel‘dovich and A.A.Starobinsky [@ZS77] that the rate of particle production is proportional the square of the Weyl tensor. It will also be shown that in our approach the conformal gravity action is actually incorporated into the formalism
The conformal gravity was invented by H.Weyl in 1918 [@Weyl]. His motivation was to construct the unified theory of two (known at the time) fundamental fields: electromagnetic and gravitational ones. Since the electromagnetic field (“identified with the Maxwell equations”) is invariant under the conformal transformations, H.Weyl proposed the conformal invariant Lagrangian for the gravitational field. Then, it was recognized that the Weyl’s gravity allows only massless particles to exist. On this ground the theory was rejected by H.Weyl himself and by A. Einstein. But, nowadays, this unpleasant feature can be “corrected” by Braut–Englert–Higgs mechanism for the spontaneous symmetry breaking [@tHooft14]. The vacuum space-time with very high symmetry is a good candidate for the creation of the universe from “nothing” [@Vil82]. It can be easily verified that all the homogeneous isotropic space-times have zero Weyl tensor. In other words, these space-times are the vacuum solutions of the conformal gravity. The idea that the initial state of the universe should be conformal invariant is advocated also by R.Penrose [@Penr10; @Penr14] and G.‘tHooft [@tHooft15].
This paper is devoted to the detailed description of our model for particle creation in the conformal gravity. We will use, in particular, the specific formalism of conformal gravity from our previous papers [@bde1; @bde2].
Throughout the paper we use the units $\hbar=c=1$ and the sign convention as in [@LL2], i.e., the signature of the metric tensor is $g_{\mu\nu}$ is $(+,-,-,-)$, the Riemann curvature tensor is defined as $$R_{\phantom{0}\nu\lambda\sigma}^{\mu} =
\frac{\partial\Gamma_{\nu\sigma}^{\mu}}{\partial x^\lambda}
- \frac{\partial\Gamma_{\nu\lambda}^{\mu}}{\partial x^\sigma}
+ \Gamma_{\varkappa\lambda}^{\mu}\Gamma_{\nu\sigma}^{\varkappa}
- \Gamma_{\varkappa\sigma}^{\mu}\Gamma_{\nu\lambda}^{\varkappa},
\label{Riemann}$$ while the Ricci tensor is the following convolution $$R_{\nu\sigma}= R_{\phantom{0}\nu\mu\sigma}^{\mu}.
\label{Ricci}$$ The scalar curvature $R=g^{\nu\sigma}R_{\nu\sigma}$, and $\Gamma_{\mu\nu}^{\lambda}$ are the metric connections, i.e., the covariant derivatives of the metric tensor are zero.
Phenomenology of particle creation
==================================
We start with construction of the hydrodynamical part of our model. In the “classical” hydrodynamics there exist two different sets of dynamical variables, the so called Lagrangian and Eulerian coordinates. The first of them are comoving, i.e., the observer is sitting on some world-line. So, using the least action principle, one has to vary the trajectory of the (quasi)-particles. Since in such a case we cannot take into account the very processes of both creation and annihilation of particles ( i.e., trajectories), it is not appropriate for our purposes. Therefore, we need to use the Eulerian description, when the dynamical variables are fields, namely, the particle number density $n(x0)$ and the four-velocities. The action integral in this case is [@Ray] (for details see also [@Ber87]): $$\begin{aligned}
S_{\rm hydro} &=& -\int\!\varepsilon(X,n)\sqrt{-g}\,dx+
\int\!\lambda_0(u^\mu u_\mu-1)\sqrt{-g}\,dx
+\int\!\lambda_1(nu^\mu)_{;\mu}\sqrt{-g}\,dx \nonumber \\
&& +\int\!\lambda_2X_{,\mu}u^\mu\sqrt{-g}\,dx,
\label{Shydro}\end{aligned}$$ where $\varepsilon(X,n)$ is the invariant energy density, $n(x)$ — invariant particle number density, $u^\mu(x)$ — four-velocity of the particle flow, $X(x)$ is the auxiliary dynamical variable introduced in order to avoid the identically zero vorticity of particle flow. It enters the action integral with the Lagrange multiplier $\lambda_2$, indicating the constraint $X_{,\mu}u^\mu=0$, i.e., $X(x)=const$ on the trajectories, thus enumerating them. The other two Lagrange multipliers, $\lambda_0(x)$ and $\lambda_1(x)$ are responsible, respectively, for the constraints $u^\mu u_\mu=1$ (natural normalization of the four-velocities) and $(nu^\mu)_{;\mu}=0$ — particle number conservation law. The semicolon “;” denotes a covariant derivative with respect to the metric $g_{\mu\nu}$.
Our aim is to incorporate into the formalism the particle “creation law” $$(nu^\mu)_{;\mu}=\Phi(inv)\neq0.
\label{Phi}$$ Evidently, the function $\Phi$ should depend on some invariants of the fields causing this particle creation. Here we would like to explore the fundamental result by Ya.B.Zel‘dovich and A.A.Starobinsky [@ZS77] obtain for the cosmological particle production $$(nu^\mu)_{;\mu}=\beta C^2.
\label{beta}$$ where $C^2$ is the square of the Weyl tensor $C^{\mu}_{\phantom{0}\nu\lambda\sigma}$ (its definition as well as some most important properties see e.g., in [@bde1; @bde2]) and the coefficient $\beta$ depends on the type of particles under consideration. We will consider this “creation law” as our first postulate. So, the hydrodynamical part of the action integral now becomes $$\begin{aligned}
S_{\rm hydro} &=& -\int\!\varepsilon(X,n)\sqrt{-g}\,dx
+\int\!\lambda_0(u^\mu u_\mu-1)\sqrt{-g}\,dx
+\int\!\lambda_1\left((nu^\mu)_{;\mu}-\beta C^2\right)\sqrt{-g}\,dx
\nonumber \\
&&+\int\!\lambda_2X_{,\mu}u^\mu\sqrt{-g}\,dx.
\label{Shydro2} \end{aligned}$$ Very important note. The Lagrange multiplier $\lambda_1$ is, actually, defined up to the additive constant. Indeed, let us replace $\lambda_1\rightarrow\lambda_1+\gamma_0$, $\gamma_0=const$, then $$\gamma_0\int\!\left((nu^\mu)_{;\mu}-\beta C^2\right)\sqrt{-g}\,dx \nonumber \\
=\gamma_0\int\!\left((n\sqrt{-g}u^\mu)_{,\mu}-\beta C^2\sqrt{-g}\right)\,dx
\label{gamma0}$$ Due to the identity $(nu^\mu)_{;\mu}\sqrt{-g}=(n\sqrt{-g}u^\mu)_{,\mu}$, the corresponding volume integral transforms into the surface integral with no effect on the dynamical equations. In result, we are left with the same “creation law” as before plus the Weyl gravitational action $$S_{\rm grav}^{\rm Weyl}=-\,\gamma_0\beta\int C^2\sqrt{-g}\,dx.
\label{Weulbeta}$$ Thus, the conformal gravity is intrinsically contained in our hydrodynamical part of the total action integral, prior to the introducing the gravitational action itself!
Scalar field and conformal invariance
=====================================
By the conformal transformation we will understand the space-time dependent scaling of the metric tensor $g_{\mu\nu}$, $$ds^2=g_{\mu\nu}(x)dx^\mu dx^\nu=\Omega^2\hat g_{\mu\nu}(x)dx^\mu dx^\nu
=\Omega^2(x)\hat ds^2\!.
\label{hat}$$ The conformal invariance means $$\frac{\delta S_{\rm tot}}{\delta\Omega}=0.
\label{inv}$$ Therefore, we can (and will) consider the conformal factor $\Omega$ as a dynamical variable and make variations independently in $\Omega$ and in $\hat g_{\mu\nu}$ [@tHooft15].
Let us $S_{\rm tot}=S_{\rm grav}+S_{\rm matter}$. By definition $$\delta S_{\rm matter}=\frac{1}{2}\int T_{\mu\nu}\sqrt{-g}\,\delta g^{\mu\nu}dx, \quad
\delta S_{\rm matter}=
\frac{1}{2}\int\hat T_{\mu\nu}\sqrt{-\hat g}\,\delta\hat g^{\mu\nu}dx,
\label{Shydro2}$$ where $T_{\mu\nu}(\hat T_{\mu\nu})$ is the matter energy-momentum tensor. Consider, first, the following transformation of the metric tensor $$\delta g^{\mu\nu}=-\,\frac{2}{\Omega^3}\hat g^{\mu\nu}\delta\Omega
=-\,\frac{2}{\Omega}g^{\mu\nu}\delta\Omega.
\label{metrictrans}$$ Suppose $$\frac{\delta S_{\rm grav}}{\delta\Omega}=0,
\label{grav}$$ then $$0=\delta S_{\rm matter}=
-\int T_{\mu\nu}g^{\mu\nu}\frac{\delta\Omega}{\Omega}\sqrt{-g}\,dx,
\label{deltamatter}$$ that is, the trace of the energy-momentum tensor should be zero: $${\rm Tr}\,(T_{\mu\nu})={\rm Tr}\,(\hat T_{\mu\nu})=0.
\label{trace}$$ If one considers the metric tensor transformation of the kind $$\delta g^{\mu\nu}=\Omega^2\delta\hat g^{\mu\nu},
\label{metrictrans2}$$ then, as can be easily seen, $$\hat T_{\mu\nu}=\Omega^2T_{\mu\nu}, \quad \hat T^\mu_\nu=\Omega^4T^\mu_\nu,
\quad \hat T^{\mu\nu}=\Omega^6T^{\mu\nu}.
\label{grav3}$$
Let us go further on. The question arises: quanta of what kind a field are creating? The most simple choice is the scalar field. And the simplest action integral is $$S_{\rm scalar} =\int\left(\frac{1}{2}\chi^\mu \chi_\mu - \frac{1}{2}m^2\chi^2\right)\sqrt{-g}\,dx.
\label{scalatact}$$ Here $\chi_\mu=\chi_{,\mu}$ (comma denotes the partial derivative), $\chi^\mu=g^{\mu\nu}\chi_\nu$ and $m$ is some constant with the dimension of mass. After the “standard” conformal transformations, namely $$g^{\mu\nu}=\Omega^2\hat g^{\mu\nu}, \quad \chi=\frac{1}{\Omega}\hat \chi,
\label{transf}$$ one gets $$S_{\rm scalar}=\int\!\left(\frac{1}{2}\hat \chi^\mu \hat \chi_\mu
-\frac{1}{\Omega}\hat \chi_\mu \Omega^\mu \frac{1}{2}\frac{\hat \chi^2}{\Omega^2}\Omega_\mu\Omega^\mu
-\frac{1}{2}m^2\Omega^2\hat\chi^2\!\right)\sqrt{-\hat g}\,dx.
\label{scalarS}$$ Now indices are raising and lowering with the metric $\hat g_{\mu\nu}(\hat g^{\mu\nu})$. How to make this action conformally covariant? The recipe is well known: one should add into the Lagrangian the term $(R/12)\chi^2$, where $R$ is the scalar curvature, constructing from the metric $g_{\mu\nu}$. The result is $$\begin{aligned}
S_{\rm scalar}&=&\int\!\left(\frac{1}{2}\chi^\mu \chi_\mu+\frac{R}{12}\chi^2- \frac{1}{2}m^2\chi^2\right)\sqrt{-g}\,dx \nonumber \\
&=&\int\!\left(\frac{1}{2}\hat \chi^\mu\hat \chi_\mu+\frac{\hat R}{12}\hat\chi^2- \frac{1}{2}m^2\Omega^2\hat \chi^2\!\right)\sqrt{-\hat g}\,dx
-\frac{1}{2}\int\!\left(\hat \chi^2\frac{\Omega^\lambda}{\Omega}\right)_{|\lambda}
\sqrt{-\hat g}\,dx.
\label{scalarScov} \end{aligned}$$ Here the vertical line “$|$” denotes the covariant derivative with respect to the metric $\hat g_{\mu\nu}$. The last term can be transformed to the surface integral, it does not effect the dynamics. Remarkably enough, that started with no gravitational action at all, we have got now both the conformal gravity (as a part of the “creation law”) and the Einstein-Hilbert-dilaton gravity (as a part of the conformally covariant scalar field Lagrangian).
It seems that if one puts $m=0$, everything else will be all right. But, it is not so easy, there exists a problem [@tHooft15]. This problem concerns the signs. With the “correct” sign for the kinetic term $(1/2)\chi^\mu\chi_\nu$, we have the “wrong” sign for the Einstein-Hilbert-dilaton part, $+(1/12)\hat R\hat\chi^2$ (with our sign convention there should be “-” instead of “+”), and vice-versa. Our choice is the “correct” sign for $\hat R$, i.e., $-(1/12)\hat R\hat\chi^2$, and the “wrong” sign for the kinetic term, i.e., $-(1/2)\chi^\mu\chi_\nu$. This requires some explanation. First of all, we do not care about the “correct” sign for the kinetic term, because our scalar $\chi$ is not the “genuine” (i.e., fundamental) one. Some part of it we have already “used” as the created particles. The residual part can be viewed as the vacuum fluctuations that consist of virtual particles, including the conformal anomaly, which is responsible for the creation process. Moreover, the “wrong” sign in the kinetic term means the absence of the lower bound for the energy and allows even infinite number of the created particles (let us remember the C-field in the “steady state” cosmological model by F.Hoyle and J.V.Narlikar [@HoyleNar]). Besides, we are not going to consider our field $\chi$ as an independent dynamical variable. One more thing. If the scalar field $\chi$ is an independent dynamical variable, then, why it “knows” about the conformal transformation $g_{\mu\nu}=\Omega^2\hat g_{\mu\nu}$ and adjusts itself properly, i.e., $\hat\chi=\Omega\chi$? Only, when this field is a part of it! Fortunately, in our case it is not so, and one can always choose the conformal factor, $\Omega=\varphi$, in such a way that $$\hat\chi=\frac{1}{\ell}\varphi,
\label{scalatact}$$ where $\ell$ is some factor having dimension of length (it is introduced in order to keep the action integral dimensionless). Then, the action integral for the scalar field takes the form $$S_{\rm scalar}=
-\,\frac{1}{\ell^2}\,\int\!\left(\frac{1}{2}\,\varphi^\mu\varphi_\mu
+\frac{\hat R}{12}\varphi^2
+\frac{1}{2}\,m^2\varphi^4\!\right)\!\sqrt{-\hat g}\,dx.
\label{scalarfin}$$ There appears the self-interaction term, $\varphi^4$. It must be noted that the power $4$ in this term is only in the case of the four-dimensional space-time (it depends on the space-time dimensions). Here, two comments are in order. First, the above action is covariant under the conformal transformation, $\varphi(new)=\hat\Omega\varphi(old)$, $g_{\mu\nu}(old)=\hat\Omega^2g_{\mu\nu}(new)$, $\sqrt{- g}(old)=\hat\Omega\sqrt{-g(new)}$, what can be easily checked. Second, it is now evident, that $3m^2=\Lambda$ plays the role of the (bare) cosmological term.
To finish this Section we write down the energy-momentum tensor $T_{\mu\nu}$ for our (new) scalar field $\varphi$, obtained by varying $S_{\rm scalar}$ in $\hat g_{\mu\nu}$: $$\begin{aligned}
\hat T_{\mu\nu}^{\rm scalar}&=&-\,\frac{1}{\ell^2}\varphi_\mu\varphi_\mu
+\frac{1}{2\ell^2}\varphi^\sigma\varphi_\sigma\hat g_{\mu\nu}
+\frac{1}{2\ell^2}m^2\varphi^4\hat g_{\mu\nu} \\
&&-\,\frac{1}{6\ell^2}\left(\varphi^2(\hat R_{\mu\nu}
-\frac{1}{2}\hat g_{\mu\nu}\hat R) -2\left((\varphi\varphi_\nu)_{|\mu}
-(\varphi\varphi^\sigma)_{|\sigma}\,\hat g_{\mu\nu}\right)\!\right)\!. \nonumber
\label{scalarT}\end{aligned}$$ Note the appearance of the second derivatives. The trace of this tensor equals $${\rm Tr}\,(\hat T_{\mu\nu}^{\rm scalar})=-\,\frac{1}{\ell^2}\left(\varphi\varphi^\sigma_{|\sigma}
-\frac{\hat R}{6}\varphi^2-2m^2\varphi^4\right)\!.
\label{scalarTr}$$
Hydrodynamics and conformal covariance
======================================
Since we consider now the conformal factor $\varphi$ and transformed metric tensor $\hat g_{\mu\nu}$ as the independent dynamical variables, the above-written hydrodynamical action integral should be properly “updated”. Let us start with analyzing the “creation law”, $$0=((nu^\mu)_{;\mu}-\beta C^2)\sqrt{-g}=((n\sqrt{-g}u^\mu)_{,\mu}-\beta C^2\sqrt{-g}).
\label{creation}$$ It is well known that in the four-dimensional space-time the combination $C^2\sqrt{-g}$ is invariant under conformal transformation, i.e., $$C^2\sqrt{-g}=\hat C^2\sqrt{-\hat g}.
\label{C2}$$ So should be the full derivative $(nu^\mu\sqrt{-g})_{,\mu}$. The square of the interval $ds^2$ transforms as $$ds^2=g_{\mu\nu}dx^\mu dx^\nu=\varphi^2\hat g_{\mu\nu}dx^\mu dx^\nu =\varphi^2d\hat s^2,
\label{ds2}$$ therefore, the four-velocity $u^\mu$ behaves as $$u^\mu=\frac{dx^\mu}{ds}=\frac{1}{\varphi}\frac{dx^\mu}{d\hat s}=\frac{1}{\varphi} \hat u^\mu,
\label{4u}$$ and, respectively, $$u_\mu=g_{\mu\nu}u^\nu=\varphi\hat g_{\mu\nu}\hat u^\nu= \varphi \hat u_\mu.
\label{4u}$$ Thus $$n\sqrt{-g}u^\mu=n\varphi^3\sqrt{-\hat g}\hat u^\mu=\hat n \hat u^\mu,
\label{nsqrt}$$ where we introduced the new notation $$\hat n =n\varphi^3\sqrt{-\hat g}.
\label{hatn}$$ It is clear that in the comoving coordinate system $\hat n$ is nothing but the particle number per unit spatial coordinate volume, and, thus, the conformally invariant quantity. So, the “creation law” does not contain the conformal factor $\varphi$ explicitly. Therefore, the hydrodynamical part of the total action integral becomes now $$\begin{aligned}
S_{\rm hydro} &=&
-\int\!\varepsilon\left(X,\frac{\hat n}{\varphi^3\sqrt{-\hat g}}\right)\varphi^4\sqrt{-\hat g}\,dx
+\int\!\lambda_0(\hat u^\mu\hat u_\mu-1)\varphi^4\sqrt{-\hat g}\,dx \nonumber \\
&&+\int\!\lambda_1\left((\hat n\hat u^\mu)_{,\mu}-\beta\hat C^2\sqrt{-\hat g}\right)\,dx
+\int\!\lambda_2X_{,\mu}\hat u^\mu\varphi^3\sqrt{-\hat g}\,dx\end{aligned}$$ and now the hydrodynamical variables are $\hat n$, $\hat u^\mu$ and $X$. Let us write down the corresponding equations of motion $$\begin{aligned}
\frac{\delta S_{\rm hydro}}{\delta\hat n}&=&
-\,\frac{\partial\epsilon}{\partial n}\frac{1}{\varphi^3\sqrt{-\hat g}}
-\lambda_{1,\sigma}\hat u^\sigma=0, \\
\frac{\delta S_{\rm hydro}}{\delta\hat u^\mu}&=&
2\lambda_0\hat u_\mu\varphi^4+\lambda_2\varphi^3X_{,\mu}
-\lambda_{1,\mu}\frac{\hat n}{\sqrt{-\hat g}}=0,
\\
\frac{\delta S_{\rm hydro}}{\delta X}&=&
-\,\frac{\partial\epsilon}{\partial X}\varphi^4
-\frac{(\lambda_2\varphi^3\sqrt{-\hat g}\hat u^\sigma)_{,\sigma}}{\sqrt{-\hat g}}=0.
\label{motion}\end{aligned}$$ To these we should add, of course, the constraints that follow from variation of the action integral in Lagrange multipliers $\lambda_0$, $\lambda_1$ and $\lambda_2$: $$\hat u^\sigma\hat u_\sigma=u^\sigma u_\sigma=1, \quad
X_\sigma\hat u^\sigma=X_\sigma u^\sigma=0, \quad
(\hat n\hat u^\mu)_{,\mu}=\beta \hat C^2\sqrt{-\hat g},
\label{constr}$$ the last of them being equivalent to $(nu^\mu)_{;\mu}=\beta C^2$. The above equations of motion can be also written in terms of the quantities without “hats”, namely $$\begin{aligned}
&&-\,\frac{\partial\epsilon}{\partial n}-\lambda_{1,\sigma}u^\sigma=0, \\
&& 2\lambda_0u_\mu+\lambda_2X_{,\mu}-n\lambda_{1,\mu}=0, \\
&& -\,\frac{\partial\epsilon}{\partial X}-(\lambda_2u^\sigma)_{,\sigma}=0.
\label{nohats}\end{aligned}$$ It is not difficult to extract the Lagrange multiplier $\lambda_0$ from these equations. Indeed, by making the convolution of the second of the equations with the four-velocity vector $u^\mu$ and using the constraints, we get, after comparing the results with the first of the equations, that $$2\lambda_0=-\,n\,\frac{\partial\epsilon}{\partial n}.
\label{lambda0}$$ Then, introducing the pressure $p$ in the usual way, $p=-\epsilon+n\frac{\partial\epsilon}{\partial n}$, one obtains $$2\lambda_0=-(\epsilon+p).
\label{lambda0b}$$ The next step is to compute the hydrodynamical part of the total energy-momentum tensor. Omitting the details, we present here the result: $$\begin{aligned}
\hat T_{\mu\nu}^{\rm hydro}&=& -\,\frac{\hat n}{\varphi^3\sqrt{-\hat g}}
\frac{\partial\epsilon}{\partial n}\hat g_{\mu\nu}\varphi^4+\epsilon\varphi^4\hat g_{\mu\nu}
-2\lambda_0\varphi^4\hat u^\mu\hat u^\nu
-\lambda_0(\hat u^\sigma\hat u_\sigma-1)\varphi^4\hat g_{\mu\nu}
-\lambda_2X_{,\sigma}\hat u^\sigma\varphi^3\hat g_{\mu\nu} \nonumber \\
&&-4\beta\left((\lambda_1\hat C_{\mu\sigma\nu\lambda})^{|\lambda|\sigma}
+\frac{1}{2}\lambda_1\hat C_{\mu\lambda\nu\sigma}\hat R^{\lambda\sigma}\right)\!.
\label{Thydro}\end{aligned}$$ or $$\hat T_{\mu\nu}^{\rm hydro}= (\varepsilon+p)\varphi^4\hat u_\mu\hat u_\nu
-p\varphi^4\hat g_{\mu\nu}
-4\beta\left((\lambda_1\hat C_{\mu\sigma\nu\lambda})^{|\lambda|\sigma}
+\frac{1}{2}\lambda_1\hat C_{\mu\lambda\nu\sigma}\hat R^{\lambda\sigma}\right)\!.
\label{Thydro2}$$ with the trace, equals to $${\rm Tr}\,(T_{\mu\nu}^{\rm hydro})=(\varepsilon-3p)\varphi^4.
\label{Trhydro}$$ The total trace equals $${\rm Tr}\,(T_{\mu\nu}^{\rm tot})=-\,\frac{1}{\ell^2}\left(\varphi\varphi^\sigma_{|\sigma}
-\frac{\hat R}{6}\varphi^2-2m^2\varphi^4\right)+(\varepsilon-3p)\varphi^4.
\label{Trtotal}$$ Finally, let us write the result of the variation of the total action integral in $\varphi$, which can be considered as one of the equations of motion as well as the consequence of the postulated conformal invariance. One gets $$\frac{1}{\ell^2}(\varphi^\sigma_{\phantom{0}|\sigma}-\frac{1}{6}\hat R\varphi-2m^2\varphi^3)+(\varepsilon-3p)\varphi^3=0,
\label{varphieq}$$ as it should be: ${\rm Tr}\,(T_{\mu\nu}^{\rm tot})=0$! The relation $\hat T_{\mu\nu}=\varphi T_{\mu\nu}$ can be also easily verified. This proves the self-consistency of our model. Note that in no way $\varphi$ can to be zero value, since this would lead to to the degeneracy of the whole space-time.
Dirac sea and Weyl gravity
==========================
Introducing new notations, $6l^2=8\pi G$ and $3m^2=\Lambda$, we are able to write our equations in a more familiar form (without “hats”!) $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R-\Lambda g_{\mu\nu}=8\pi G T_{\mu\nu}^{\rm hydro}.
\label{induced}$$ These equations look like the ordinary Einstein equations with a cosmological constant, but now the hydrodynamical energy-momentum tensor is modified by the presence of terms originated from the “creation law”, namely $$T_{\mu\nu}^{\rm hydro}=(\epsilon+p)u_\mu u_\nu-pg_{\mu\nu}-4\beta B_{\mu\nu}[\lambda_1],
\label{Thydro3}$$ where $$B_{\mu\nu}[\lambda_1]=(\lambda_1C_{\mu\sigma\nu\lambda})^{;\lambda;\sigma}
+\frac{1}{2}\lambda_1C_{\mu\lambda\nu\sigma}R^{\lambda\sigma},
\label{Thydro2}$$ for $\lambda_1=1$ it is just the Bach tensor $B_{\mu\nu}$. Note, that $\hat B_{\mu\nu}=\varphi^2B_{\mu\nu}$ and $\hat T_{\mu\nu}=\varphi^2T_{\mu\nu}$. It can be checked that derived equations are confomally covariant, i.e., if one makes the conformal transformation $\hat g_{\mu\nu}=\Omega^2\hat{\hat g}_{\mu\nu}\;\left(= g_{\mu\nu}=(\varphi^2\Omega^2)\hat{\hat g}_{\mu\nu}\right)$, then the equations written in terms of $\{(\varphi\Omega),\hat{\hat {\cal O}}\}$ look the same as for $\{\varphi,\hat {\cal O}\}$.
Let us consider the vacuum space-times. In the absence of real particles $(\epsilon=p=0)$ the energy-momentum tensor does not reveal the structure dictated by the presence of the trace anomaly. All these vacua are absolutely empty. The way out of such a situation we see in introducing yet another type of particles, but with the [*negative energies*]{}. This is something like the Dirac sea. For the vacuum solutions they must compensate each other. One should not be afraid of fluctuations having negative energies above the vacuum state. Due to the self-antigravitaion they will be gone away, while those with positive energies will undergo the usual gravitational instability and form the structures. Thus we need two (instead of one) parts of hydrodynamical action with, correspondingly, two sets of dynamical variables (labeled by “$\pm$”). Let us write down the corresponding equations of motion $$\begin{aligned}
\label{vect}
\left(\epsilon_{(\pm)}+p_{(\pm)}\right)u_\mu^{(\pm)}+\lambda_2^{(\pm)} X_{,\mu}^{(\pm)}
-n_{(\pm)}\lambda_{1,\mu}^{(\pm)}&=&0, \\
\frac{\partial \epsilon_{(\pm)}}{\partial X^{(\pm)}}
-\left(\lambda_2^{(\pm)} u^{{(\pm)}\sigma}\right)_{;\sigma}&=&0.
\label{scalar}\end{aligned}$$ In the vacuum, exactly as in the Dirac sea, from $\epsilon_+=-\epsilon_-$ and $n_+=n_-$, it follows that $p_+=-p_-$. Since there must be no energy or particle number flows in the vacuum, we get $$u^{(+)\mu}=u^{(-)\mu},
\label{upm}$$ i.e., the trajectories of these two types of “matter” are the same. For this reason, the auxiliary variables $X^{(\pm)}$ are also the same. Therefore, the second (scalar) equation of motion (\[scalar\]) gives us $$X^{(+)}=X^{(-)} \quad \Rightarrow \quad \lambda_2^{(+)}=-\lambda_2^{(-)},
\label{Xpm}$$ and from the first (vectorial) equation of motion (\[vect\]) it follows that $$\lambda^{(+)}_{1,\mu}=-\lambda^{(-)}_{1,\mu}.
\label{lambdap1m}$$ The Lagrange multiplier $\lambda^{(\pm)}_1$ will enter as a sum in our vacuum equation, so $$\lambda^{(+)}_{1}+\lambda^{(-)}_{1}=const.
\label{lambdap1sum}$$ Finally, we obtain the following equation for what can be called “the dynamical vacuum” $$4\alpha_0B_{\mu\nu}+\frac{1}{16\pi G}\,G_{\mu\nu}-\frac{\Lambda}{16\pi G}\,g_{\mu\nu}=0,
\label{dynvac}$$ where $$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}\,g_{\mu\nu}R
\label{dynvac}$$ is the Einstein tensor, and $$B_{\mu\nu}=C_{\mu\sigma\nu\lambda}^{\phantom{abcd};\lambda;\sigma}
+\frac{1}{2}\,C_{\mu\lambda\nu\sigma}R^{\lambda\sigma}.
\label{Thydro2}$$ is the Bach tensor.
Conclusions and Discussions
===========================
[*Conclusions*]{}
We have constructed the self-consistent conformally invariant phenomenological model for particle creation in the presence of strong gravitational fields. The word “phenomenological” means that we adopted classical description both for the created particles (hydrodynamics) and for the “creation law”.
This “creation law” enters the action integral with the corresponding Lagrange multiplier and substitute the particle number conservation law in the conventional hydrodynamics. The idea (and our hope) is that such an inclusion of the particle creation law straight into the least action formalism will cause the essential change in the structure of the energy-momentum tensor and will lead to the violation of the energy dominance condition and, thus, will take into account (to some extent) the quantum character of the particle creation process. This idea is not quite new, it was already explored by one of the authors. The new thing is combining of the method with the postulated conformal invariance of the whole theory. This allows to restrict the possible functional form of the “creation law” up to the square of the Weyl tensor.
It appeared, to our surprise, that the above mentioned Lagrange multiplier can be determined only uo to an arbitrary constant. This means that the Weyl gravitational action is, actually, already incorporated into the formalism and does not need to be introduced it artificially. The local conformal invariance, taken as the fundamental symmetry, has one more important consequence. In order to make it possible to create particles we need some fields which quanta are these very particles. The simplest is the scalar field. One needs it also because it is the scalar self-interacting field that gives the masses to particles through the Brout–Englert–Higgs mechanism (which also makes the conformal gravity meaningful). If one uses the simplest (again!) form for the scalar field Lagrangian, i.e., “the kinetic term $+$ the mass term”, then, in order to make it conformally covariant, it is necessary to introduce also the term proportional to the scalar curvature. Therefore, starting from hydrodynamics, needed for the description of the created particles and introducing the conformally invariant creation law plus the conformally covariant scalar field Lagrangian, we arrived at the conformal gravity theory with the Weyl Lagrangian plus the Einstein–Hilbert–dilaton gravity. This supports the idea, first discussed by A.D.Sakharov, about the induced gravity.
One more thing. In order to have the “correct” sign for the scalar curvature one has to choose the “wrong” sign for the kinetic term in the scalar field Lagrangian. But this causes no conceptual difficulties at all, since our scalar field is not the genuine (fundamental), it is simply the “vacuum residual” part of some entity, the “above-vacuum” part of which is already present in the form of the created particles, and its “conformal anomalous” part is already included into the “creation law”. Therefore, it is very “natural” to identify the “vacuum residual” part with the conformal factor of the metric tensor. The mass term now plays the twofold role, it produces the self-interaction and the cosmological term, both initially absent.
It is the “vacuum residual” part that will become the subject of our future investigations. We would like to study the possibility to have the spontaneous symmetry breaking allowing the particles to acquire the masses as well as the very appearing of the observers. The plausible result would be that the uniformly accelerated observer sees the thermal bath with the Unruh temperature as the vacuum state. Also, we would like to extend the form of our “creation law” by inclusion of other possible terms, like, say, the so called “Euler characteristic density”, etc. We are also intending to investigate whether the dark energy problem could be solved using our model, without introducing any other sophisticated fields, couplings, and so on.
1\. The model presented above, is very minimalistic. The matter is not only in that we did not include into consideration the electromagnetic (abelian) and other (nonabelian) gauge fields, causing creation of the particle–antiparticle pairs with opposite charges. Here we restricted ourselves by the specific form of the “creation law”, when the rate of particle production is proportional to the square of the Weyl tensor. The absence of other possible terms may be explained by the adopted conformal invariance principle. Indeed, $C^2\sqrt{-g}$ is conformally invariant. It is usually claimed that the latter is the only conformally invariant combination quadratic in Riemann curvature tensor in four dimensions. But, there exists yet another quadratic conformally invariant combination, namely, the so-called Hirzebruch–Pontryagin density $R^{\mu\nu\lambda\sigma}\,{^*}R_{\mu\nu\lambda\sigma}\sqrt{-g}$, where “star” means that ${^*}R_{\mu\nu\lambda\sigma}=\epsilon_{\mu\nu\alpha\beta}R^{\alpha\beta}_{\phantom{ab}\lambda\sigma}$. It is the total derivative and, therefore, when in the action integral, does not alter the equations of motion. In our model, however, it would enter together with the Lagrange multiplier ($\lambda_1$) and would have an influence on the whole situation. Of course, it is not a genuine scalar, but pseudoscalar. And may be , it is good, manifesting the $T$–violation in the irreversible particle creation processes.
2\. The most important problem is how to organize the Braut–Englert–Higgs mechanism for generating particle’s rest masses. The conventional line of reasoning is inapplicable here, because the usual (and the most convenient) solution $\varphi=0$ is impossible in our scheme, since it would mean the conformal factor vanishes and, thus, the very notion if the metrical space–time would become meaningless.
3\. Let us imagine that the above problem is already solved. Then, we are able to construct the observers equipped with the clocks and other measurement devices, engines for correcting trajectories and all that. In the self-consistent theory these observers cannot be arbitrary at all. For example, the so called vacuum observers (those, “sitting” outside the matter distribution) should see ($=$ measure) different things, depending on their trajectories: the uniformly accelerated ones must be surrounded by the thermal bath, having the Unruh temperature.
4\. All these problems are for the future investigations.
The reported study was partially supported by RFBR, research project No. 15-02-05038a. The authors would like to thank Alexey Smirnov for numerous discussions and comments. One of us (V.B) is also grateful to Artyom Starodubtsev, Michail Smolyakov and Igor Gulamov for helpful discussions.
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---
abstract: 'In this paper we define invariants for primitive Legendrian knots in lens spaces $L(p,q), q\neq 1$. The main invariant is a differential graded algebra $(\mathcal{A}, \partial)$ which is computed from a labeled Lagrangian projection of the pair $(L(p,q), K)$. This invariant is formally similar to a DGA defined by Sabloff which is an invariant for Legendrian knots in smooth $S^1$-bundles over Riemann surfaces. The second invariant defined for $K\subset L(p,q)$ takes the form of a DGA enhanced with a free cyclic group action and can be computed from the $p$-fold cover of the pair $(L(p,q), K)$.'
author:
- |
Joan E. Licata\
*Max-Planck-Institute für Mathematik/*\
*Stanford University*\
*[email protected]*
bibliography:
- 'contactbib.bib'
title: Invariants for Legendrian knots in lens spaces
---
Introduction
============
Endowing a three-manifold with a contact structure refines the associated knot theory by introducing new notions of equivalence among knots, and these in turn require invariants sensitive to the added geometry. In addition to more classical numerical invariants, invariants taking the form of differential graded algebras (DGAs) have seen success in distinguishing Legendrian non-isotopic knots in a variety of contact manifolds.
The first DGA invariants were developed for Legendrian knots in the standard contact $\mathbb{R}^3$. Chekanov constructed a combinatorial invariant, and an equivalent invariant was introduced independently in a geometric context by Eliahsberg [@C], [@E]. In the former case, the algebra is generated by the crossings in a Lagrangian projection, and the boundary map counts immersed discs in the diagram. In Eliashberg’s relative contact homology, the algebra is generated by Reeb chords and the differential counts rigid $\mathcal{J}$-holomorphic curves in the symplectization of ${\mathbb{R}}^3$. The two constructions were shown to produce the same DGA in [@ENS]. In [@S], Sabloff adapted further work of Eliashberg, Givental, and Hofer to construct a combinatorial DGA for Legendrian knots in a class of contact manifolds characterized by their distinctive Reeb dynamics [@EGH], [@S]. His algebra is again generated by Reeb chords, but he introduces additional technical machinery in order to handle periodic Reeb orbits. Sabloff’s invariant is defined for smooth $S^1$ bundles over Riemann surfaces, a class of manifolds which includes $S^3$ and $L(p,1)$, but does not admit other lens spaces.
In this paper we develop an invariant for Legendrian knots in the lens spaces $L(p,q)$ for $q \neq 1$ with the unique universally tight contact structure. For primitive $K \subset L(p,q)$, we define a *labeled diagram* to be the Lagrangian projection of the pair $(L(p,q), K)$ to $(S^2, \Gamma)$, together with some ancillary decoration which uniquely identifies the Legendrian knot. Numbering the crossings of $\Gamma$ from one to $n$, we consider the tensor algebra on $2n$ generators: $$\mathcal{A}=T(a_1, b_1, ...a_n, b_n).$$ We equip this algebra with a differential $\partial:\mathcal{A}\rightarrow \mathcal{A}$ counting certain immersed discs in $(S^2, \Gamma)$. The algebra is graded by a cyclic group, and the boundary map is graded with degree $-1$. The pair $(\mathcal{A}, \partial)$ is a semi-free DGA, and the natural equivalence on such pairs is that of stable tame isomorphism type.
Our main theorem is the following:
Up to equivalence, the semi-free DGA $(\mathcal{A}, \partial)$ is an invariant of the Legendrian type of $K\subset L(p,q)$.
The proof of Theorem \[thm:main\] applies Sabloff’s invariant to a freely periodic knot ${\widetilde{K}}\subset S^3$ which is a $p$-to-one cover of $K\subset L(p,q)$. The Legendrian type of ${\widetilde{K}}$ is an invariant of the Legendrian type of $K$, so Sabloff’s invariant for ${\widetilde{K}}$ is therefore also an invariant of $K$ (Proposition \[prop:easyinvt\]). In order to prove Theorem \[thm:main\], we endow Sabloff’s DGA with additional structure related to the covering transformations.
Given $K$ in $L(p,q)$ with $q\neq1$, let $(\widetilde{\mathcal{A}}, \widetilde{\partial})$ denote Sabloff’s low-energy DGA for the knot ${\widetilde{K}}$ in $S^3$. The algebra $(\widetilde{\mathcal{A}}, \widetilde{\partial})$ may be enhanced with a cyclic group action $\gamma:{\mathbb{Z}_p}\times \widetilde{\mathcal{A}} \rightarrow \widetilde{\mathcal{A}}$ which commutes with the boundary map. We define a notion of equivariant equivalence on DGAs with such actions in Section \[sect:triplepf\], and we associate to $K$ the equivariant DGA $(\widetilde{\mathcal{A}}, \gamma, \widetilde{\partial})$. Our second main theorem asserts that this is also an invariant of the Legendrian knot in the lens space.
The equivalence class of the equivariant DGA $(\widetilde{\mathcal{A}}, \gamma, \widetilde{\partial})$ is an invariant of the Legendrian type of $K$.
The major technical work of the paper lies in proving Theorem \[thm:triple\], and this occupies Section \[sect:triplepf\]. The proof of Theorem \[thm:main\] identifies $(\mathcal{A}, \partial)$ with a distinguished ${\mathbb{Z}_p}$-equivariant subalgebra of $(\widetilde{\mathcal{A}}, \widetilde{\partial})$ and follows as a consequence of Theorem \[thm:triple\]. The final section contains examples computed for knots in $L(3,2)$ and $L(5,2)$.
Finally, we note that although the arguments in this paper are developed for primitive knots in $L(p,q)$, they in fact construct invariants for any Legendrian knot in a lens space which is covered by a Legendrian knot in some $L(p,1)$. In this adaptation, $L(p,1)$ replaces $S^3$ as the contact manifold where Sabloff’s invariant is defined.
I would like to thank Josh Sabloff for helpful correspondence in the course of writing this paper. A portion of this work was conducted while visiting the Max Planck Institute for Mathematics in Bonn, Germany, and their hospitality and support are much appreciated.
Background
==========
This section contains a brief summary of the basic definitions from contact geometry and their realizations in three examples: the standard contact ${\mathbb{R}}^3$, $S^3$, and $L(p,q)$. A more thorough introduction to the topic is provided in [@Et] or [@Ge].
Basic definitions
-----------------
A *contact structure* $\xi$ on a three-manifold $M$ is an everywhere non-integrable $2$-plane field. A non-degenerate one-form $\alpha$ defines a contact structure by $\xi_{\alpha}= \ker \alpha$ at each point of $M$. Two contact manifolds $(M_1, \xi_1)$ and $(M_2, \xi_2)$ are *contactomorphic* if there is a diffeomorphism between the manifolds which takes contact planes to contact planes.
Given a contact form $\alpha$, the *Reeb vector field* is the unique vector field $X$ which satisfies $$\begin{aligned}
&\alpha(X)=1\notag \\
& d\alpha(X, \cdot)=0. \notag\end{aligned}$$ Integral curves of $X$ are known as *Reeb orbits*, and they inherit an orientation from $X$.
A knot $K$ in $(M, \xi)$ is *Legendrian* if its tangent lies in the contact plane at each point.
Two Legendrian knots are equivalent if they are isotopic through Legendrian knots. In general, two knots which are topologically equivalent may not be Legendrian equivalent; any topological isotopy class of knots will be represented by countably many Legendrian isotopy classes.
The *Lagrangian projection* of a contact manifold $(M, \xi_{\alpha})$ is the quotient space of $M$ which collapses each Reeb orbit of $\alpha$ to a point. If $K$ is a Legendrian knot in a contact manifold, the *Lagrangian projection* of $K$ is the image of the knot under Lagrangian projection of the manifold.
If $K$ is a Legendrian knot in $(M, \xi)$, a *Reeb chord* is a segment of a Reeb orbit with both endpoints on $K$. In the Lagrangian projection, a Reeb chord with distinct endpoints will map to a crossing in the knot projection.
First example: ${\mathbb{R}}^3$
-------------------------------
The standard contact structure $\xi_{std}$ on ${\mathbb{R}}^3$ is induced by the contact form $$\alpha_{std}=dz-ydx.$$ The Reeb vector field on $({\mathbb{R}}^3, \xi_{std})$ has trivial $dx$ and $dy$ coordinates at every point, so the Reeb orbits are vertical lines. Thus, the Lagrangian projection is simply projection to the $xy$-plane.
Second example: $S^3$ {#sect:s3}
---------------------
$S^3$ sits inside ${\mathbb{R}}^4$ as the unit sphere: $$S^3=\{ (r_1, \theta_1, r_2, \theta_2) | r_1^2+r_2^2=1\}.$$ The torus $r_1=\frac{1}{\sqrt{2}}=r_2$ separates $S^3$ into two solid tori, and it will be convenient to treat this torus as a Heegaard surface. The curves $r_1=0$ and $r_2=0$ are the core curves of the Heegaard tori, and the complement of the cores is foliated by tori of fixed $r_i$.
The standard tight contact structure on $S^3$ is $$\alpha_0=\frac{1}{2}(r_1^2d\theta_1+r_2^2 d\theta_2).$$ The punctured manifold $(S^3 - \{p\}, \xi_0)$ is contactomorphic to $({\mathbb{R}}^3, \xi_{std})$, but the Reeb dynamics are quite different. In particular, the Reeb orbits of $\alpha_0$ are $(1,1)$ curves on each torus of fixed $r_i$. This foliation of $S^3$ by circles gives the Hopf fibration of $S^3$, and Lagrangian projection in $(S^3, \xi_0)$ is projection to the $S^2$ base space of this fibration. Note that the core curves are each Reeb orbits, and their images under Lagrangian projection are the poles of the two-sphere. The contact form $\alpha$ also induces a *curvature* form $\Omega$ on the $S^2$ base space; for the standard contact structure, this is just the Euler class of the bundle, where $S^3$ is viewed as the unit sphere in $\mathbb{R}^4$. [@Ge].
Third example: Lens spaces {#sect:lens}
--------------------------
Define $F_{p,q}:S^3 \rightarrow S^3$ by $$\label{eq:fp}
F_{p,q}(r_1, \theta_1, r_2, \theta_2)=(r_1, \theta_1+ \frac{2 \pi}{p}, r_2,\theta_2 + \frac{2 q\pi}{p}).$$
The map $F_{p,q}$ generates a cyclic group of order $p$, and the quotient of $S^3$ by the action of this group is the lens space $L(p,q)$. Thus $\pi:S^3\rightarrow L(p,q)$ is a $p$-to-one covering map. Since $F_{p,q}$ preserves the contact structure on $S^3$, $\pi$ induces a contact structure on $L(p,q)$ [@BG]. The Reeb orbits of $(L(p,q), \xi_{p,q})$ again foliate the manifold by circles, and the Lagrangian projection of $(L(p,q), \xi_{p,q})$ is a two-sphere. As an $S^1$ bundle over $S^2$, $L(p,q)$ is smooth if and only if $q=1$.
\[def:fp\] A knot ${\widetilde{K}}$ in $S^3$ is *freely periodic* if it is preserved by a free periodic automorphism of $S^3$.
The map in Equation \[eq:fp\] has order $p$, so if ${\widetilde{K}}$ is freely periodic with respect to $F_{p,q}$, then $\pi({\widetilde{K}})$ is a knot in $L(p,q)$. Conversely, any $K$ in $L(p,q)$ which is primitive in $H_1(L(p,q))$ has a freely periodic lift ${\widetilde{K}}\subset S^3$. (Knots which are not primitive will lift to links in $S^3$.) This definition makes sense in both the topological and contact categories; with respect to the contact structures defined above, $K$ is Legendrian if and only if ${\widetilde{K}}$ is Legendrian. An explicit construction of a freely periodic lift is described in Section 6.2 of [@GRS], and we refer the reader to [@HLN] or [@R2] for a fuller treatment of freely periodic knots.
Throughout the paper, each topological manifold will be equipped with the contact structure associated to it in this section; we will write only $S^3$ and $L(p,q)$ for the contact manifolds $(S^3, \xi_{std})$, and $(L(p,q), \xi_{p,q})$. Furthermore, tildes will be used to distinguish objects in $S^3$ from their counterparts in $L(p,q)$; thus $\widetilde{\Gamma}$ will denote the Lagrangian projection of a knot ${\widetilde{K}}$ in $S^3$, whereas the Lagrangian projection of $K\subset L(p,q)$ will be denoted by $\Gamma$.
Differential graded algebra invariants for Legendrian knots {#DGA}
===========================================================
In this section we introduce Sabloff’s DGA invariant for Legendrian knots in smooth $S^1$ bundles over Riemann surfaces. We begin by defining differential graded algebras and the relevant notion of equivalence among them.
Equivalence of semi-free DGAs {#sect:eq}
-----------------------------
Let $V=\text{Span}_{\mathbb{Z}_2} \{x_1, x_2, ...x_n\}$. Define $$\mathcal{A}= T(x_1, x_2, ...x_n)=\bigoplus_{n=0}^{\infty} V^{\otimes n}$$ to be the tensor algebra on the elements $\{x_1, x_2, ...x_n\}$. If $V$ is graded by a cyclic group $G$ so that the $x_i$ are homogeneous, this induces a cyclic grading on $\mathcal{A}$ via the rule $|x_ix_j|=|x_i|+|x_j|$. When $\partial:\mathcal{A}\rightarrow \mathcal{A}$ is a degree $-1$ map satisfying $\partial^2=0$ and the Leibnitz rule $\partial (ab)=(\partial a)b+a(\partial b)$, then the pair $(\mathcal{A}, \partial)$ is a *semi-free differential graded algebra* (DGA). The modifier “semi-free" emphasizes that we keep track of the preferred generators $\{x_i\}_{i=1}^n$, which will be important in defining DGA equivalence.
An *elementary automorphism* of $\mathcal{A}$ is a map $g^i: \mathcal{A}\rightarrow \mathcal{A}$ such that $$g^i(x_j)= \begin{cases} x_i+ v_i, \text{ for } v_i \in T(a_1,....\hat{x}_i ...b_n) & \text{if }j=i \\
x_j & \text{if } j\neq i. \end{cases}$$ When $v_i$ is homogeneous in the same grading as $x_i$, we say that $g^i$ is a *graded elementary automorphism*. A *graded tame automorphism* is a composition of graded elementary automorphisms.
Given a DGA $(\mathcal{A}, \partial)=(T(x_1, ...x_n), \partial)$, let $\mathcal{E}=T(e_1, e_2)$ be a DGA which is graded by the same cyclic group and satisfies $\partial_{\mathcal{E}}e_1=e_2$ and $\partial_{\mathcal{E}}e_2=0$. A *stabilization* of $\mathcal{A}$ is the differential graded algebra $(T(x_1, ...x_n, e_1, e_2), \partial \coprod \partial_{\mathcal{E}})$.
\[def:equiv\] Two semi-free differential graded algebras $(\mathcal{A}_1, \partial_1)$ and $(\mathcal{A}_2, \partial_2)$ are *equivalent* if some stabilization of $(\mathcal{A}_1, \partial_1)$ is graded tame isomorphic to some stabilization of $(\mathcal{A}_2, \partial_2)$.
We will have reason to consider DGAs equipped with an action of a cyclic group ${\mathbb{Z}_p}$, so we extend the notion of equivalence to one respecting the group action. The cyclic group ${\mathbb{Z}_p}$ should not be confused with the cyclic group $G$ which grades the algebra.
An *equivariant DGA* $(\mathcal{A}, \gamma, \partial)$ is a semi-free DGA $(\mathcal{A}, \partial)$ together with an automorphism $\gamma: \mathcal{A} \rightarrow \mathcal{A}$ of order $p$ such that $\partial \circ \gamma=\gamma \circ \partial$ and $|\gamma x|=|x|$.
Suppose that $(\mathcal{A}, \gamma, \partial)$ is an equivariant DGA, where $\mathcal{A}=T(x_1, ...x_n)$. A *free* ${\mathbb{Z}_p}$ *stabilization* of $(\mathcal{A}, \gamma, \partial)$ is the equivariant DGA $(\mathcal{A}\coprod \mathcal{E}^p, \gamma, \partial \coprod \partial_{\mathcal{E}^p})$, where
- $\mathcal{A}\coprod \mathcal{E}^p = T(x_1, ...x_n, e_{1,1}, e_{1,2}, ...e_{1,p}, e_{2,1}, ...e_{2,p})$;
- $\gamma(e_{j,i})=e_{j, i+1}$;
- $\partial_{\mathcal{E}^p} e_{1,i}=e_{2,i}$;
- $\partial_{\mathcal{E}^p} e_{2,i}=0$.
A ${\mathbb{Z}_p}$ *elementary isomorphism* is a $\gamma$-equivariant map $f: (\mathcal{A}, \gamma, \partial)\rightarrow (\mathcal{A}, \gamma, \partial)$ which can be written as $f=g^1\circ g^2\circ...\circ g^p$, where each $g^i$ is an elementary isomorphism. If $f$ is graded, we say it is a *graded* ${\mathbb{Z}_p}$ *elementary isomorphism*. A composition of ${\mathbb{Z}_p}$ elementary isomorphisms is a ${\mathbb{Z}_p}$ *tame isomorphism*.
Two equivariant DGAs $(\mathcal{A}_1, \gamma_1, \partial_1)$ and $(\mathcal{A}_2, \gamma_2, \partial_2)$ are ${\mathbb{Z}_p}$ *equivalent* if they have free ${\mathbb{Z}_p}$ stabilizations which are graded ${\mathbb{Z}_p}$ tamely isomorphic.
Sabloff’s DGA for knots in $S^1$ bundles over Riemann surfaces {#sect:sab}
--------------------------------------------------------------
In [@S], Sabloff considers contact manifolds whose Reeb orbits are the fibers of a smooth $S^1$ bundle over a Riemann surface. For a Legendrian knot ${\widetilde{K}}$ in such a manifold, he defines an algebra generated by the Reeb chords with both endpoints on ${\widetilde{K}}$. Since each Reeb orbit is periodic, there are infinitely many such chords, and he also defines a finitely-generated *low-energy algebra* generated by chords which are strictly shorter than the fiber. The low-energy algebra sits inside the full invariant as a subalgebra, but the equivalence type of the low-energy algebra is also an invariant of ${\widetilde{K}}$. The following section introduces the the low-energy algebra $({\widetilde{\mathcal{A}}}, {\widetilde{K}})$ for Legendrian knots in $S^3$, and we refer the reader to [@S] for a description of the full invariant.
### Labeled Lagrangian diagram
Let ${\widetilde{K}}\subset S^3$ be a Legendrian knot, and denote the Lagrangian projection of ${\widetilde{K}}$ to $S^2$ by $\widetilde{\Gamma}$. Number the crossings of the diagram from $1$ to $n$, and associate two generators $a_i$ and $b_i$ to the $i^{th}$ crossing. These correspond to the complementary short chords in the fiber which intersects the crossing strands of ${\widetilde{K}}$. Because the Reeb orbit is oriented, each chord identifies the crossing strands locally as “sink" and “source". Select a preferred chord and indicate this choice with a plus sign in the two (opposite) quadrants where traveling sink-to-source orients the quadrant positively. Furthermore, assign each quadrant either $a_i^+$ and $b_i^-$ or $a_i^-$ and $b_i^+$ as indicated in Figure \[fig:crosslabels\]. Note that signs are used in two distinct ways; a “positive quadrant" will always mean one marked with a “+" to denote the preferred chord, and each generator $x$ labels every quadrant of the associated crossing as either $x^+$ or $x^-$.
If $\widetilde{\Gamma}$ is a labeled diagram with $n$ crossings, define ${\widetilde{\mathcal{A}}}(\widetilde{\Gamma})$ to be the tensor algebra generated by the associated Reeb chords: $${\widetilde{\mathcal{A}}}(\widetilde{\Gamma})=T(a_1, b_1, ...a_n, b_n).$$
The following definitions will prove useful in defining the defect and the boundary map:
If $x_i$ is a generator of ${\widetilde{\mathcal{A}}}(\widetilde{\Gamma})$, let $\mathit{l}(x_i)$ denote the length of the associated chord in $S^3$, where the length of an $S^1$ fiber is normalized to $1$.
We extend this to a length function $\mathit{l}'$ on words written in the signed generators $a_i^{\pm}$ and $b_i^{\pm}$. Let $\epsilon(a_i^+)=\epsilon(b_i^+)=1$ and $\epsilon(a_i^-)=\epsilon(b_i^-)=-1$. If $w$ is a word in the signed generators $x_i^{\pm}$, define $$\mathit{l}'(w)=\sum_{x_i^{\pm} \in w} \epsilon(x_i^{\pm}) \mathit{l}(x_i).$$
\[def:adm\] Let $(\Sigma, \partial \Sigma)$ be a disc with $m$ marked points on the boundary. An *admissible disc* is a map $f:(\Sigma, \partial \Sigma) \rightarrow (S^2, \widetilde{\Gamma})$ which satisfies the following:
1. each marked point maps to a crossing of $\widetilde{\Gamma}$;
2. $f$ is an immersion on the interior of $\Sigma$;
3. $f$ extends smoothly to $\partial \Sigma$ away from the marked points;
4. $f(\partial \Sigma)$ has a corner at each marked point, and $f(\Sigma)$ fills one quadrant there.
Two admissible discs $f$ and $g$ are *equivalent* if there is a smooth automorphism $\phi: \Sigma \rightarrow \Sigma$ such that $f=g\circ \phi$.
Let $R$ be a component of $S^2-\widetilde{\Gamma}$. To each corner of $R$, one may associate the signed generator corresponding to the preferred chord of the crossing, where the sign is dictated by the quadrant filled by $R$. Traveling counterclockwise around $\partial R$ and reading off these labels defines a cyclic word $w(R)$. (See Figure \[fig:word\] for an example.)
\[def:def\] Let $f$ be an admissible disc whose image is $R$. The *defect* of $R$ is given by: $$\label{eq:def}
n(f)=\frac{1}{2\pi}\int_{\Sigma} f^*\Omega + \mathit{l}'(w(R)).$$
Geometrically, the defect encodes the interaction between the knot and the fiber structure. Without this decoration, $\widetilde{\Gamma}$ does not specify even the topological type of the knot, as displacement in the Reeb direction is obscured by the projection. The curve $\partial R$ lifts to $S^3$ as a simple closed curve composed of alternating Legendrian and Reeb segments, and the defect measures the winding number of this lifted curve around the fiber with respect to an appropriate trivialization. Together with the signs at each crossing, the defects of components of $S^2-\widetilde{\Gamma}$ determine the Legendrian type of the knot.
The defect extends additively to unions of regions counted with multiplicity, so Equation \[eq:def\] holds for any admissible disc $f$. Since $S^2$ is simply connected, one may also define the defect of the knot $n({\widetilde{K}})$ to be the defect of any contracting disc bounded by the projection of ${\widetilde{K}}$.
### Gradings
A *capping path* for a generator $x_i$ is a path along ${\widetilde{K}}$ in $\widetilde{\Gamma}$ which begins and ends adjacent to the same $x_i^+$ quadrant. For each crossing, one of $a_i$ or $b_i$ will have two capping paths, and the other will have none. The *rotation number* of a capping path for $x_i$ is the number of counterclockwise rotations performed by the tangent vector, computed as a winding number in a trivialization over a contracting disc in $S^2$. Taking the edges at a crossing to be orthogonal, this value lies in ${{\mathbb{Z}}}-\frac{1}{4}$, and we denote it by $r(x_i)$.
Suppose that $f:\Sigma \rightarrow S^2$ is an admissible disc such that $f(\partial \Sigma )$ is a capping path for $x_i$ . Then the grading of $x_i$ is given by the following: $$\label{eq:grad1}
|x_i|= 2 r(x_i) -\frac{1}{2}+4n(f).$$ If $y_i$ is the other generator at the same crossing, $$\label{eq:grad2}
|y_i|= 3-|x_i|.$$
These gradings are well-defined modulo $2r(x_i) + 4n({\widetilde{K}})$.
### The algebra $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}), \widetilde{\partial})$
\[def:word\] Let $f:\Sigma\rightarrow S^2$ be an admissible disc with one corner filling a quadrant labeled $x_i^+$. The *boundary word* $w(f,x_i)$ is the concatenation of the $y_j^-$ generators associated to the other quadrants filled by $f(\Sigma)$, read counterclockwise around $\partial \Sigma$. $$w(f,x_i)=y_2 y_3...y_{m}.$$
See Figure \[fig:word\] for an example.
If $f:\Sigma\rightarrow S^2$ is an admissible disc, the *$x_i$ defect* $\tilde{n}_{x_i}(f)$ is given by $$\tilde{n}_{x_i}(f)=\frac{1}{2\pi}\int_{\Sigma} f^*\Omega + \mathit{l}(x_i)-\sum_{j=2}^{m} \mathit{l}(y_j).$$
Note that the $x_i$ defect of an admissible disc may differ from the defect of its image in the diagram, as the two are computed by associating (possibly) different words to the same disc. To compute $\tilde{n}_{x_i}(f)$, add one to $n(f)$ if $x_i^+$ occupies a non-positive quadrant, and subtract one from $n(f)$ for each $y_j^-$ in $w(f, x_i)$ which occupies a positive quadrant. Thus both types of defects may be computed from the labeled diagram without further data regarding the lengths of chords.
\[def:bndry\] The differential $\widetilde{\partial}:{\widetilde{\mathcal{A}}}(\widetilde{\Gamma})\rightarrow {\widetilde{\mathcal{A}}}(\widetilde{\Gamma})$ is defined by on the generator $x_i$ by $$\widetilde{\partial} x_i=\sum_{f: \tilde{n}_{x_i}(f)=0} w(f,x_i),$$ and $\widetilde{\partial}$ extends to other elements in ${\widetilde{\mathcal{A}}}(\widetilde{\Gamma})$ via the Leibnitz rule $\widetilde{\partial}(ab)=(\widetilde{\partial}a)b+a(\widetilde{\partial}b)$.
\[sabinvt\] The boundary map in Definition \[def:bndry\] satisfies $\widetilde{\partial}^2=0$, and the stable tame isomorphism type of $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}), \widetilde{\partial})$ is an invariant of the Legendrian knot type of ${\widetilde{K}}$ in $S^3$.
Invariants for Legendrian knots in lens spaces {#sect:invt}
==============================================
As noted above, Sabloff’s invariant is defined for contact manifolds which are smooth $S^1$ bundles, a class which excludes the lens spaces $L(p,q)$ for $q\neq1$. Although they do not induce smooth bundles, the Reeb orbits of these lens spaces nevertheless define an $S^1$ bundle structure, and this similarity is strong enough to permit a DGA invariant $(\mathcal{A},\partial)$ computable from the Lagrangian projection to $S^2$. The invariant is formally similar to Sabloff’s invariant $({\widetilde{\mathcal{A}}}, \widetilde{\Gamma})$ for knots in $S^3$, and in fact, the proof of invariance exploits the covering relationship between these manifolds.
Except if otherwise indicated, in the remainder of the paper every lens space $L(p,q)$ is assumed to have $q\neq1$.
The DGA $(A, \partial)$
-----------------------
Let $K$ be a knot in $L(p,q)$ which generates $H_1(L(p,q))$. Following Rasmussen, we call such knots *primitive* [@R2]. If $K$ is a primitive Legendrian knot, we begin by defining a labeled Lagrangian diagram. At the $i^{th}$ crossing of $\Gamma$, mark each quadrant with $a_i^+$ and $b_i^-$ or with $a_i^-$ and $b_i^+$ as in Figure \[fig:crosslabels\]. At each crossing, indicate a preferred choice of chord by decorating a pair of opposite quadrants with plus signs.
Recall that to $K$, we may associate its freely perioidic lift ${\widetilde{K}}\subset S^3$. The $p$-fold covering map $\pi:(S^3, {\widetilde{K}})\rightarrow (L(p,q), K)$ descends to a $p$-to-one branched cover of Lagrangian projections $\pi_*:(S^2, \widetilde{\Gamma})\rightarrow (S^2, \Gamma)$, where the branch points are the images of the core curves $r_i=0$ for $i=1,2$. Thus a choice of preferred chords in $\Gamma$ lifts to a choice of preferred chords in $\widetilde{\Gamma}$. Let $R$ be a region in $(S^2 - \Gamma)$, and let $f:\Sigma\rightarrow S^2$ be an admissible disc whose image is $\pi_*^{-1}(R)$. Define the *defect* of $R$ to be $n(R)=\frac{1}{p}n(f)$.
A *labeled diagram* for $K$ is a generic Lagrangian projection $\Gamma$ decorated with preferred chords and defects which are compatible with a labeled diagram for ${\widetilde{K}}$ as described above.
Let $\Gamma$ be a labeled diagram for a Legendrian knot $K\subset L(p,q)$. If $\Gamma$ has $n$ crossings, define $$\mathcal{A}(\Gamma)=T(a_1, b_1, ...a_n, b_n).$$
If $x_i$ is a generator of $\mathcal{A}(\Gamma)$, choose a lift $x_i^*\in \pi_*^{-1}(x_i)$ and define the grading of $x_i$ by $|x_i|=|x_i^*|$. This value is independent of the choice of lift, and $\mathcal{A}(\Gamma)$ is graded by the same cyclic group as ${\widetilde{\mathcal{A}}}(\widetilde{\Gamma})$.
**Remark** The grading can also be defined intrinsically. Given a labeled diagram, consider a capping path for $x_i$ which has winding number $p$ with respect to the poles. With only slight modification, the formulae in Equations \[eq:grad1\] and \[eq:grad2\] can be used to compute the grading directly from $\Gamma$.
Definitions \[def:adm\] and \[def:word\] may be applied verbatim in the context of labeled diagrams for knots in $L(p,q)$.
\[def:bndry2\] The differential $\partial:\mathcal{A}(\Gamma)\rightarrow \mathcal{A}(\Gamma)$ is defined on generators by $$\partial x_i=\sum_{f: \tilde{n}_{x_i}(f)=0} w(f,x_i),$$ where the sum is over admissible discs which satisfy the additional condition that $f(\partial \Sigma)$ has winding number $p$ with respect to the poles of $S^2$. Extend $\partial$ to other elements in $\mathcal{A}(\Gamma)$ via the Leibnitz rule.
\[thm:main\] Up to equivalence as a semi-free DGA, $(\mathcal{A}(\Gamma), \partial)$ is an invariant of the Legendrian knot type of $K\subset L(p,q)$.
In order to prove Theorem \[thm:main\], we will study the relationship between $(\mathcal{A}(\Gamma), \partial)$ and $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}), \widetilde{\partial})$.
It is clear that the Legendrian type of the freely periodic lift ${\widetilde{K}}\subset S^3$ is an invariant of the Legendrian type of $K$, so Sabloff’s construction has the following easy consequence:
\[prop:easyinvt\] The stable tame isomorphism type of $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}), \widetilde{\partial})$ is an invariant of the Legendrian isotopy class of $K$.
However, a stronger notion of equivalence yields a more interesting invariant, and the next section shows that we may associate an equivariant DGA to the freely-periodic lift of $K$.
The ${\mathbb{Z}_p}$ action on $(\mathcal{A}(\Gamma), \partial)$ {#sect:zpact}
----------------------------------------------------------------
Theorem 3.15 of [@S] states that the equivalence type of $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}), \widetilde{\partial})$ is independent of the choice of preferred chords, but we will restrict attention to diagrams where the signs at each crossing are preserved by $\frac{2 \pi}{p}$ rotation of $(S^2, \widetilde{\Gamma})$.
If $K$ is a Legendrian knot in $L(p,q)$, then there is a natural automorphism $\gamma: {\widetilde{\mathcal{A}}}(\widetilde{\Gamma}) \rightarrow {\widetilde{\mathcal{A}}}(\widetilde{\Gamma})$ with order $p$ such that $\widetilde{\partial}\circ \gamma=\gamma \circ \widetilde{\partial}$, and $|\gamma x|=|x|$.
Fix a representative of the isotopy class of $K$ and lift this to the freely periodic knot ${\widetilde{K}}$. The Lagrangian projection of $(S^3,{\widetilde{K}})$ is invariant under $\frac{2 \pi}{p}$ rotation about the axis through the points representing the fibers $r_i=0$ for $i=1,2$. If the projection of ${\widetilde{K}}$ is not generic, any local perturbation of $K$ will lift to $p$ local perturbations of ${\widetilde{K}}$, maintaining the contact covering relationship between $(S^3, {\widetilde{K}})$ and $(L(p,q),K)$ while removing singularities in the projection. In particular, $K$ (or equivalently, ${\widetilde{K}}$) may be assumed disjoint from the cores of the Heegaard tori.
If the Reeb chord $x$ is a generator of $\mathcal{A}$, then $\pi^{-1}(x)$ is a free ${\mathbb{Z}_p}$ orbit of generators of ${\widetilde{\mathcal{A}}}(\widetilde{\Gamma})$. This relationship descends to the Lagrangian diagrams, via the $p$-fold branched covering map $\pi_*:(S^2, \widetilde{\Gamma})\rightarrow(S^2, \Gamma)$. Since capping paths for crossings in a single orbit are permuted by the cyclic action, each member of the orbit has the same grading. Similarly, any disc which represents a term in the boundary is part of an orbit of $p$ discs. This proves that the ${\mathbb{Z}_p}$ action on ${\widetilde{\mathcal{A}}}(\widetilde{\Gamma})$ commutes with the differential.
**Remark** Recall that Sabloff’s invariant is defined for knots in lens spaces $L(p,1)$. The above discussion highlights another sense in which this case is exceptional. When $q=1$, the map $\pi_*:(S^2, \widetilde{\Gamma})\rightarrow(S^2, \Gamma)$ induced on Lagrangian projections is one-to-one. In this case, for any point $x$ on $\widetilde{\Gamma}$, the preimage $F^{-1}_{p,1}(x)$ consists of $p$ points on ${\widetilde{K}}$.
\[thm:triple\] If $K$ is a Legendrian knot in $L(p,q)$, then the equivariant DGA $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}), \gamma, \widetilde{\partial})$ is an invariant of $K$, up to ${\mathbb{Z}_p}$ equivalence.
The proof of Theorem \[thm:triple\] appears later, but we note a corollary of the statement here:
Let $\mathcal{A}^{\gamma}(K)$ be the subalgebra of $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}, \gamma, \widetilde{\partial})$ fixed by the ${\mathbb{Z}_p}$ action: $$\mathcal{A}^{\gamma}(K)=\{ a \in {\widetilde{\mathcal{A}}}(\widetilde{\Gamma}) | \gamma a=a \}.$$ The subalgebra $\mathcal{A}^{\gamma}(K)$ is a subcomplex and the homology of $\mathcal{A}^{\gamma}(K)$ is an invariant of the Legendrian type of $K$.
The statement that $\mathcal{A}^{\gamma}(K)$ is a subcomplex follows from the fact that $\gamma$ commutes with the differential. A ${\mathbb{Z}_p}$-equivariant isomorphism between equivariant DGAs induces an isomorphism on their homologies; the proof is similar to that in $\cite{C}$. The statement then follows from Theorem \[thm:triple\],
Proof of Theorem \[thm:main\]
-----------------------------
In this section we will show how Theorem \[thm:main\] follows from Theorem \[thm:triple\], postponing the proof of Theorem \[thm:triple\] until Section \[sect:triplepf\].
Setting $\phi: {\widetilde{\mathcal{A}}}(\widetilde{\Gamma}) \rightarrow {\widetilde{\mathcal{A}}}(\widetilde{\Gamma})$ to be the algebra homomorphism defined on the generators of ${\widetilde{\mathcal{A}}}(\Gamma)$ by $$\phi(x)=\bar{x}=\sum_{i=1}^{p} \gamma^i x,$$ define $\bar{\mathcal{A}}(\widetilde{\Gamma})$ to be the image of ${\widetilde{\mathcal{A}}}(\widetilde{\Gamma})$ under $\phi$.
$\bar{\mathcal{A}}(\widetilde{\Gamma})$ is a ${\mathbb{Z}_p}$-equivariant subalgebra of both ${\widetilde{\mathcal{A}}}$ and $\mathcal{A}^{\gamma}(K)$, and for every Legendrian knot, the following containments are proper: $$\bar{\mathcal{A}}(\widetilde{\Gamma}) \subset \ \mathcal{A}^{\gamma}(K) \subset {\widetilde{\mathcal{A}}}.$$
However, $\bar{\mathcal{A}}$ is also a semi-free DGA in its own right. The generators of ${\widetilde{\mathcal{A}}}$ are naturally grouped into ${\mathbb{Z}_p}$ orbits, and if $\{x_i\}_{i=1}^n$ is a set containing exactly one representative from each orbit, then $$\bar{\mathcal{A}}(\widetilde{\Gamma})=T(\bar{x}_1, \bar{x}_2, ...\bar{x}_n\}.$$
\[lem:chain\] $\phi$ is a chain map: $$\widetilde{\partial} \circ \phi =\phi \circ \widetilde{\partial}.$$
Using Lemma \[lem:chain\], we may define $\bar{\partial}: \bar{\mathcal{A}}(\widetilde{\Gamma})\rightarrow \bar{\mathcal{A}}(\widetilde{\Gamma})$ by $$\bar{\partial}\bar{w}=\bar{\partial}(\phi(w))= \phi(\widetilde{\partial} w).$$ This identifies the image of $\phi$ with the semi-free DGA $(\bar{\mathcal{A}}(\widetilde{\Gamma}), \bar{\partial})$.
\[lem:mainpf\] $(\mathcal{A}(\Gamma), \partial)$ and $(\bar{\mathcal{A}}(\widetilde{\Gamma}), \bar{\partial})$ are isomorphic as semi-free DGAs.
\[lem:equiv\] A ${\mathbb{Z}_p}$ stabilization of $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}), \gamma, \widetilde{\partial})$ induces an ordinary stabilization of $(\bar{\mathcal{A}}(\widetilde{\Gamma}), \bar{\partial})$, and a ${\mathbb{Z}_p}$ tame automorphism of $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}), \gamma, \widetilde{\partial})$ induces an ordinary tame automorphism of $(\bar{\mathcal{A}}(\widetilde{\Gamma}),\bar{\partial})$.
Theorem \[thm:main\] asserts that the equivalence type of $(\mathcal{A}(\Gamma), \partial)$ is an invariant of $K$. Lemma \[lem:mainpf\] replaces this with a statement about $(\bar{\mathcal{A}}(\widetilde{\Gamma}), \bar{\partial})$. Assuming Theorem \[thm:triple\] holds, Lemma \[lem:equiv\] then completes the proof.
Let $x$ and $y$ be two generators of ${\widetilde{\mathcal{A}}}(\widetilde{\Gamma})$.
$$\begin{aligned}
(\partial \circ \phi) (xy)&=\partial(\bar{x}\bar{y})\notag \\
&= \partial[ (\sum_{i=1}^p \gamma^i x)(\sum_{j=1}^p \gamma^j y)]\notag \\
&=\sum_{i=1}^p \sum_{j=1}^p \partial [(\gamma^i x)(\gamma^j y)]\notag \\
&= \sum_{i=1}^p \sum_{j=1}^p (\partial \gamma^i x)(\gamma^j y)+ (\gamma^ix)(\partial \gamma^j y)\notag\\
&= \sum_{i=1}^p \sum_{j=1}^p (\gamma^i \partial x)(\gamma^j y)+ (\gamma^ix)(\gamma^j \partial y)\notag\end{aligned}$$
On the other hand, $$\begin{aligned}
(\phi \circ \partial )(xy)&= \phi [(\partial x)y + x(\partial y)]\notag\\
&=\phi(\partial x) \phi (y) + \phi(x)\phi(\partial y))\notag\\
&= (\sum_{i=1}^p \gamma^i \partial x)(\sum_{j=1}^p \gamma^j y)+ (\sum_{i=1}^p \gamma^ix)(\sum_{j=1}^p \gamma^j \partial y)\notag\end{aligned}$$
Each crossing in $\Gamma$ lifts to a ${\mathbb{Z}_p}$ orbit of crossings in $\widetilde{\Gamma}$. If $x$ is a generator of ${\widetilde{\mathcal{A}}}(\widetilde{\Gamma})$, there is a one-to-one correspondence between generators $\bar{x}\in \bar{\mathcal{A}}(\widetilde{\Gamma})$ and generators $\pi (x) \in \mathcal{A}(\Gamma)$. The gradings of these generators agree, so $\bar{\mathcal{A}}(\widetilde{\Gamma})$ and $\mathcal{A}(\Gamma)$ are isomorphic as graded algebras.
The boundary map on $\mathcal{A}(\Gamma)$ counts discs whose boundary has winding number $p$ with respect to the poles, and any such curve lifts to a simple closed curve in $\widetilde{\Gamma}$. On the other hand, any disc in $(S^2, \widetilde{\Gamma})$ projects to a disc in $(S^2, \Gamma)$ whose boundary has winding number $p$ with respect to the poles of $S^2$.
By construction, the $x$ defect of any admissible disc mapping into $(S^2, \widetilde{\Gamma})$ will agree with the $\pi(x)$ defect of its image in $\pi_*(S^2, \widetilde{\Gamma})$. Thus a word $w(f,x)$ appears in $\partial \pi(x)$ if and only if $\phi(w(f, x))$ appears in the boundary of $\bar{x}$ in $\bar{\mathcal{A}}(\widetilde{\Gamma})$.
The proof is almost immediate from the definitions. A ${\mathbb{Z}_p}$ stabilization adds the generators $\bar{e}_1$ and $\bar{e}_2$ to $\bar{\mathcal{A}}(\widetilde{\Gamma})$, where $\bar{\partial} \bar{e}_1=\bar{e}_2$. Similarly, if $f$ is an elementary ${\mathbb{Z}_p}$ automorphism which sends $a_i$ to $a_i+w_i$, then the map $\bar{f}$ which sends $\bar{f}(\bar{a})$ to $\bar{a}+\bar{w}$ is a tame automorphism of $\bar{\mathcal{A}}(\widetilde{\Gamma})$ which intertwines $\phi$.
Proof of Theorem \[thm:triple\] {#sect:triplepf}
===============================
Theorem \[thm:triple\] asserts that the equivalence type of the equivariant DGA $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}), \gamma, \widetilde{\partial})$ is an invariant of the Legendrian type of the knot $K \subset L(p,q)$. This requires proving that Legendrian isotopy of $K$ changes $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}), \gamma, \widetilde{\partial})$ only by free ${\mathbb{Z}_p}$ stabilizations and ${\mathbb{Z}_p}$ tame isomorphisms. When an isotopy occurs in the complement of the core curves $r_1=0$ and $r_2=0$, the proof is similar to the proof of ordinary invariance for $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}), \widetilde{\partial})$. However, the core curves are Reeb orbits where the bundle fails to be smooth, and more care must be taken with isotopies which pass $K$ across these fibers.
Reidemeister moves and isotopy away from the cores
--------------------------------------------------
\[lem:sablem\] If $\widetilde{\Gamma}_1$ and $\widetilde{\Gamma}_2$ are Lagrangian projections for Legendrian isotopic knots in $S^3$, then they differ by a sequence of the Reidemeister moves shown in Figure \[fig:Reids\].
If $K_1$ and $K_2$ are Legendrian knots in $L(p,q)$ which are Legendrian isotopic in the complement of the core curves, then the Lagrangian projections $\widetilde{\Gamma}_1$ and $\widetilde{\Gamma}_2$ of their freely-periodic lifts differ by a sequence of $p$-tuples of the Reidemeister moves shown in Figure \[fig:Reids\].
Away from the poles, the pair $(S^2, \widetilde{\Gamma})$ is a $p$-fold cover of the pair $(S^2, \Gamma)$. Employing the argument in the proof of Lemma 6.3 of [@S], the Lagrangian image of the isotopy is a homotopy of immersions away from the Reidemeister moves shown, and each of these lifts to $p$ disjoint copies of the same move in $(S^2, \widetilde{\Gamma})$.
If $K_1$ and $K_2$ are Legendrian knots in $L(p,q)$ whose Lagrangian projections differ by a Reidemeister move, then the equivariant DGAs $(\widetilde{\mathcal{A}}(\widetilde{\Gamma}_1), \gamma, \widetilde{\partial}_1)$ and $(\widetilde{\mathcal{A}}(\widetilde{\Gamma}_2), \gamma, \widetilde{\partial}_2)$ are equivalent.
This proposition is proved in Section \[sect:Reids\].
Star moves and isotopy across the cores {#sect:stars}
---------------------------------------
We turn now to isotopies which pass $K$ through the core of one of the Heegaard tori. In a smooth bundle over the two-sphere, (e.g. $S^3$ or $L(p,1)$), such an isotopy would project to an isotopy of $\Gamma$ across one of the poles. Recall, however, that $L(p,q)$ is the quotient space of $S^3$ under a cyclic group action which maps each core curve onto itself and each non-core curve into an orbit of $p$ fibers. This implies the existence of pairs of fibers in $L(p,q)$ which are not homotopic in the Heegaard tori: any fiber on the boundary of an $\epsilon$ neighborhood of the core fiber has slope $\frac{x}{p}$ (where $x$ depends on the basis). Thus there is in an orbifold point at each pole of the Lagrangian projection. An isotopy passing $K$ across a core curve therefore passes the Lagrangian projection through a non-Reidemeister singularity, as shown in Figure \[fig:lpqstar\].
An isotopy passing $K$ across a core curve lifts to an isotopy passing ${\widetilde{K}}$ across a core curve in $S^3$ $p$ times. The projection of this isotopy simultaneously moves $p$ strands of $\widetilde{\Gamma}$ across the corresponding pole of $S^2$ as shown in Figure \[fig:sp\].
An isotopy passing $K$ across a core curve preserves the Legendrian type of the lift ${\widetilde{K}}$, so Lemma \[lem:sablem\] implies that $\widetilde{\Gamma}^-$ and $\widetilde{\Gamma}^+$ are related by a sequence of Reidemeister moves and the associated DGAs are stably tame isomorphic. However, it is not clear that such a sequence respects the ${\mathbb{Z}_p}$ action. This prompts the introduction of a new move relating generic Lagrangian diagrams.
If $K_-$ and $K_+$ are Legendrian knots in $L(p,q)$ which differ only by an isotopy passing one strand across a core curve, then we say that a *star move* relates the Lagrangian projections $\widetilde{\Gamma}^-$ and $\widetilde{\Gamma}^+$ of their freely periodic lifts.
\[prop:star\] If two labeled diagrams $\widetilde{\Gamma}^-$ and $\widetilde{\Gamma}^+$ differ only by a star move, the equivariant DGAs $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}^-), \gamma, \widetilde{\partial}^-)$ and $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}^+), \gamma, \widetilde{\partial}^+)$ are ${\mathbb{Z}_p}$ equivalent.
The proof of this proposition will occupy Section \[sect:star\]
**Remark** Readers familiar with grid diagrams may note that there is a set of Legendrian grid moves which relates the grid diagrams of Legendrian equivalent knots in both $L(p,q)$ and $S^3$ [@BG]. A single Legendrian grid move on a toroidal diagram in $L(p,q)$ corresponds to $p$ copies of a Legendrian grid move on a toroidal diagram in $S^3$. This proves Proposition \[prop:easyinvt\], but it is not strong enough to show Theorem \[thm:triple\], as a grid move does not clearly translate into a ${\mathbb{Z}_p}$ equivalence of equivariant DGAs.
Invariance under star moves {#sect:star}
---------------------------
This section is devoted to proving that a star move preserves the ${\mathbb{Z}_p}$ equivalence type of the equivariant DGA $({\widetilde{\mathcal{A}}}, \gamma, \widetilde{\partial})$. The proof is given for the case when $p$ is odd, but the proof for even $p$ is similar. The structure of this argument is based on Chekanov’s proof of Reidemeister II invariance in [@C].
### A closer look at $(\mathcal{A}^+,\partial^+)$
For simplicity, let $(\mathcal{A}^+,\partial^+)$ and $(\mathcal{A}^-, \partial^-)$ denote $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}^+), \gamma, \widetilde{\partial}^+)$ and $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}^-), \gamma, \widetilde{\partial}^-)$, respectively. Away from the star, assume all labels on the two diagrams agree.
Consider first the algebra associated to $\widetilde{\Gamma}^+$, focusing only on the star region where it differs from $\widetilde{\Gamma}^-$. The crossings in the star are grouped into $p-1$ orbits so that $\gamma(a_{j,i})=a_{j, i+1}$. (See Figure \[fig:splab\].) Furthermore, each crossing in $\widetilde{\Gamma}^+$ forms one corner of a bigon in the star, and the labels are assigned so that $a^+_{j,i}$ and $b^-_{p-j, i}$ appear at opposite ends of the same bigon.
Recall the length function $\mathit{l}$ defined on generators of $\mathcal{A}^+$ in Section \[sect:sab\], and note that it is constant on the generators in a fixed orbit. In fact, the lengths of the orbits associated to the star crossings in $\widetilde{\Gamma}^+$ are clustered around the values $\frac{m}{p}$, $m \in \mathbb{N}$. The next lemma makes this statement precise and shows that the only maps involving multiple generators within such a cluster are given by the obvious bigons connecting $a_{k,i}$/$b_{p-k,i}$ pairs.
\[lem:abndry\] If $b_{p-l,j}$ appears in a word $w \in \widetilde{\partial}a_{k,i}$ and $|\textit{l}(b_{p-l,j})-\mathit{l}(a_{k,i})| < \frac{1}{2p}$, then $w= b_{p-k, i}$.
The star results from passing $p$ segments of ${\widetilde{K}}$ across a core curve of $S^3$. This isotopy can be assumed to take place in $p$ arbitrarily small balls distributed evenly around the core, and crossings in the star correspond to chords connecting the curve segments in different balls. Thus the length associated to any generator is approximately $\frac{m}{p}$, $m\in \{ 1, 2, ...p-1 \}$, with the diameter of the ball providing an upper bound for the error.
Let $\epsilon'$ be the shortest length associated to a generator occurring outside the star, and set $\epsilon=\min(\epsilon', \frac{1}{4p})$. Choosing the isotopy balls small enough ensures that the lengths of the star generators lie in the intervals $(\frac{m}{p}-\epsilon, \frac{m}{p}+\epsilon)$.
Recall that an admissible disc $f:\Sigma\rightarrow S^2$ represents a term in the boundary of $a_{k,i}$ if and only if it satisfies $$0=\tilde{n}_{a_{k,i}}(f)=\frac{1}{2 \pi}\int_{\Sigma} f^*\Omega +\mathit{l}(a_{k,i}) - \sum_{y_j^- \in w(f,a_{k,i})} \mathit{l}(y_j).$$
We next show that $b_{p-k,i}$ always appears in $\partial a_{k,i}$. Let $f$ be an admissible disc whose image is a bigon in the star. The integral term in the defining equation for the defect can be made arbitrarily small by restricting the star to live in a sufficiently small neighborhood of the pole. Since the length of each generator lies strictly between zero and one and the defect is always an integer, $\tilde{n}_{a_{k,i}}(f)$ must be zero. Thus $b_{p-k,i}$ is as a summand in $\widetilde{\partial} a_{k,i}$.
Finally, suppose that for some $m'$, both $\mathit{l}(a_{k,i})$ and $\mathit{l}(b_{p-l,j})$ lie in $(\frac{m'}{p}-\epsilon, \frac{m'}{p}+\epsilon)$. The value $\mathit{l}(a_{k,i}) - \mathit{l}(b_{p-l, j})$ is smaller than the length of any generator, so any word $w(a_{k,i}, f)\in \widetilde{\partial}a_{k,i}$ containing $b_{p-l, j}$ cannot contain any other generators. This implies that the image of $f$ is a bigon, so $b_{p-l, j}=b_{p-k, i}$.
### Overview of proof
In order to prove that $(\mathcal{A}^+, \partial^+)$ and $(\mathcal{A}^-, \partial^-)$ are ${\mathbb{Z}_p}$ equivalent, we replace $\mathcal{A}^-$ with its free ${\mathbb{Z}_p}$ stabilization $\mathcal{A}'$: $$\mathcal{A}'=\mathcal{A}^- \coprod \mathcal{E}^p_1\coprod \mathcal{E}^p_2 ...\coprod \mathcal{E}^p_{p-1}.$$ The $2(p-1)$ new orbits are assigned gradings so that $(\mathcal{A}', \partial')$ is isomorphic to $(\mathcal{A}^+, \partial^+)$ as a graded algebra: $$|e^k_{1,i}|=|a_{k,i}|\text{ for }1\leq k \leq p-1,$$ and in fact they are tamely isomorphic equivariant DGAs. The proof begins with the construction of an explicit isomorphism $s:\mathcal{A}^+\rightarrow \mathcal{A}'$ (Section \[sect:s\]). Conjugating by $s$ gives a new boundary map $\hat{\partial}$ on $\mathcal{A}'$: $$\hat{\partial}=s\circ \partial^+ \circ s^{-1}.$$
$$\begin{diagram}
\node{\mathcal{A}^+}\arrow{e,t}{s}
\node{ \mathcal{A}'}
\arrow{s,r}{g}
\node{\mathcal{A}^-}
\arrow{w,t}{{\mathbb{Z}_p}\text{ stab.}}\\
\node{}
\arrow{e,t,!}{}
\node{\mathcal{A}'}
\end{diagram}$$
In Section \[sect:g\] we define a graded ${\mathbb{Z}_p}$ tame isomorphism $g:\mathcal{A}'\rightarrow\mathcal{A}'$. The map $g$ is constructed so that $\partial'=g\circ\hat{\partial} \circ g^{-1}$, which implies the ${\mathbb{Z}_p}$ tame isomorphism of $\mathcal{A}^+$ and $\mathcal{A}'$.
### The map $s:\mathcal{A}^+\rightarrow \mathcal{A}'$ {#sect:s}
As a first step towards defining the isomorphism $s:\mathcal{A}^+\rightarrow \mathcal{A}'$, we extend the length function $\mathit{l}$ to generators of $\mathcal{A}'$. The generators of $\mathcal{A}'$ are of two types: generators coming from crossings in $\widetilde{\Gamma}^-$ and generators in the $\mathcal{E}_k^p$. To a generator of the first type, assign the length of the corresponding generator in $\mathcal{A}^-$. Similarly, the star generators of $\mathcal{A}^+$ can be used to assign lengths to the generators of $\mathcal{A}'$ coming from the stabilizing orbits. Set $$\begin{aligned}
\mathit{l}(e_{1,i}^k)&=\mathit{l}(a_{k,i})\notag \\
\mathit{l}(e_{2,i}^k)&=\mathit{l}(b_{p-k,i}).\notag\end{aligned}$$
Note that this implies $\mathit{l}(e_{j,i}^k)$ lies in $(\frac{m}{p}-\epsilon, \frac{m}{p}+\epsilon)$ for some $m\in \{1,2, ...p-1\}$. Perturbing $K$ slightly and letting $\epsilon\rightarrow 0$, we may assume that no other generators’ lengths lie in these intervals.
Let $\mathcal{O}_{[0]}$ denote the set of generators of $\mathcal{A}'$ whose length is less than or equal to $\frac{1}{p}+\epsilon$. Order the remaining orbits by increasing length, and label them as $\mathcal{O}_{[j]}, j=1, 2, 3...$ so that if $x_j \in \mathcal{O}_{[j]},$ then $\mathit{l}'(x_{1})<\mathit{l}'(x_2)<\mathit{l}'(x_3)...$. Let $\mathcal{A}_{[j]}$ be the subalgebra generated by the elements $ \mathcal{O}_{[i]}$ for $0\leq i \leq j$. $$A_{[j]} = T(\mathcal{O}_{[0]}, \mathcal{O}_{[1]}, ....\mathcal{O}_{[j]}).$$
\[lem:length\] If $x_k$ is in $\mathcal{O}_{[j]}$ for $j\geq1$, then $\partial'(x_k) \subset \mathcal{A}_{[j-1]}$.
If $w(f,x_l)$ appears in $\partial'x_k$, then $\tilde{n}_{x_k}(f)=0$: $$0=\frac{1}{2\pi}\int_{\Sigma} f^*\Omega +\mathit{l}(x_k) - \sum_{x_l^- \in w(f,x_k)} \mathit{l}(x_l).$$ The integral term is negative, and each $\mathit{l}(x_l)$ is positive, which proves the lemma.
For each $a_{k,i}$, write $\partial^+a_{k,i}=b_{p-k,i}+v_{k,i}+w_{k,i}$, where words in $v_{k,i}$ involve only generators from crossings outside the star, and each word in $w_{k,i}$ contains at least one generators coming from a star crossing. Let $\mathcal{O}_M = \{ x\in \mathcal{A}^+ | \mathit{l}'(x)\leq \frac{M}{p}+\epsilon\}$. If $a_{k,i} \in \mathcal{O}_M \backslash \mathcal{O}_{M-1}$, then Lemma \[lem:abndry\] shows that $w_{k,i} \in \mathcal{O}_{M-1}$. We define maps $s_M:\mathcal{O}_M \rightarrow \mathcal{A}'$ inductively: $$s_1(x) = \begin{cases}
x& \text{if }x \text{ comes from a crossing in } \widetilde{\Gamma}^-\\
e_{1,i}^k & \text{if} \ x=a_{k,i} \\
e_{2,i}^k + v_{k,i} & \text{if}\ x=b_{p-k,i}. \end{cases}$$ For the inductive step, suppose that $s_i$ is defined for $i\in \{ 1,2, ...M-1\}$. $$s_M(x) = \begin{cases}
x& \text{if }x \text{ comes from a crossing in } \widetilde{\Gamma}^-\\
e_{1,i}^k & \text{if} \ x=a_{k,i} \\
e_{2,i}^k + v_{k,i} + s_{M-1}(w_{k,i}) & \text{if}\ x=b_{p-k,i}. \end{cases}$$
Define $s:\mathcal{A}^+ \rightarrow \mathcal{A}'$ by $$s(x) = \begin{cases}
x&\text{if }x \text{ comes from a crossing in } \widetilde{\Gamma}^-\\
e_{1,i}^k & \text{if} \ x=a_{k,i} \\
e_{2,i}^k + v_{k,i} +s_{p-1}(w_{k,i})& \text{if}\ x=b_{p-k,i}. \end{cases}$$
By construction, $s$ preserves gradings and intertwines the ${\mathbb{Z}_p}$ actions on $\mathcal{A}^+$ and $\mathcal{A}'$.
### The projection map $\tau:\mathcal{A}'\rightarrow \mathcal{A}^-$ {#sect:tau}
As a vector space, $\mathcal{A}'$ decomposes as $\mathcal{A}^-\oplus \mathcal{I}_{\mathcal{E}}$, where $\mathcal{I}_{\mathcal{E}}$ is the two-sided ideal generated by elements in the $\mathcal{E}^p_k$. Define $F: \mathcal{A}'\rightarrow \mathcal{A}'$ by $$F(x) = \begin{cases} ye_{1,i}^kz & \text{if} \ x=ye_{2,i}^kz \text{ and }y\in \mathcal{A}^-\\
0 & otherwise.\end{cases}$$ Note that $F$ is graded with degree $1$. Let $\tau: \mathcal{A}'\rightarrow \mathcal{A}'$ be projection to $\mathcal{A}^-$.
$\tau$ satisfies $$\label{lem:tau}
\tau +id_{\mathcal{A}'}=F \circ \partial ' + \partial' \circ F.$$
The proof is a straightforward computation.
\[lem:taucomp\] $\tau \circ \partial'=\tau \circ \hat{\partial}.$
We first show that on the $\mathcal{E}_k^p$, the stronger statement $\partial'=\hat{\partial}$ holds. This fact will be used of this in Section \[sect:g\].
First recall that $\partial'(e_{1,i}^k)=e_{2,i}^k$ and $\partial'(e_{2,i}^k)=0$. Compare this to $\hat{\partial}(e_{j,k}^k)$: $$\begin{aligned}
\hat{\partial}(e_{1,i}^k)&=s\circ \partial^+ \circ s^{-1}(e_{1,i}^k)\notag\\
&=s\circ \partial^+(a_{k,i})\notag\\
&= s (b_{p-k,i}+v_{k,i}+w_{k,i})\notag\\
&= (e_{2,i}^k+v_{k,i}+s_{p-1}(w_{k,i}))+s(v_{k,i})+s(w_{k,i})\notag\\
&=e_{2,i}^k\notag\\
\newline\notag\\
\hat{\partial}(e_{2,i}^k)&=s\circ \partial^+ \circ s^{-1}(e_{2,i}^k)\notag\\
&=s\circ \partial^+(b_{p-k,i}+ v_{k,i}+w_{k,i})\notag\\
&=s\circ \partial^+(\partial^+a_{k,i})\notag\\
&=0\notag\end{aligned}$$
Now consider some generator $x$ which is associated to a crossing in $\widetilde{\Gamma}^-$. To prove the lemma, we compare the words appearing in $\tau \circ \partial'(x)$ and in $\tau \circ \hat{\partial}(x)$; it suffices to show that any word with no generators in the $\mathcal{E}^p_k$ appears in both $\hat{\partial}(x)$ and $\partial'(x)$. This argument is similar to the proof of Step 5 in [@S].
Terms in $\partial'x$ come from one of the following types of discs (Figure \[fig:neck\]):
1. Discs which have the same multiplicity throughout the star region;
2. Discs which flow through the star region in $\widetilde{\Gamma}^-$.
Clearly, $\tau\circ \partial'(x)=\partial'(x)$.
Now expand $\hat{\partial}$ as $s\circ \partial^+ \circ s^{-1}$. Since $s^{-1}(x)=x$, terms in $\hat{\partial}x$ are the $s$-images of terms in $\partial^+$. These come in three flavors:
1. \[1a\] Words involving none of the generators associated to crossings in the star;
2. \[2\] Words involving some $a_{k,i}$ but no $b_{j,l}$;
3. \[3\] Words involving $b_{j,i}$.
Boundary terms of the first type in both lists agree. In the second list, note that $s$ sends words involving only $a_{k,i}$ terms to $\mathcal{I}_{\mathcal{E}}$, so these will vanish under $\tau$.
To see that the remaining terms agree, recall that $s(b_{p-k,i})=e_{2,i}^k+v_{k,i}+s_{p-1}(w_{k,i})$. Any word containing $e_{2,i}^k$ vanishes under $\tau$. However, the terms coming from $v_{k,i}+s_{p-1}(w_{k,i})$ which are not killed by $\tau$ represent boundary discs which start at $a_{k,i}^+$. These can be “glued" to boundary discs for $x$ with a corner at $b_{p-k,i}^-$ to produce boundary discs for $x$ in $\widetilde{\Gamma}^-$. See Figure \[fig:glue\]. However, these are exactly the discs in $\partial'x$ which flow through the star. The proof that this gluing operation is smooth comes from the argument in [@S].
### Constructing $g:\mathcal{A}'\rightarrow \mathcal{A}'$ {#sect:g}
Following Chekanov, we construct $g:\mathcal{A}'\rightarrow \mathcal{A}'$ as a composition of maps $g_j$. Each $g_j$ is a graded ${\mathbb{Z}_p}$ elementary isomorphism which is the identity away from the orbit $\mathcal{O}_j$. Furthermore, each $g_j$ inductively defines a new boundary map $\partial_{[j]}$ on $\mathcal{A}'$ by conjugation: $$\partial_{[j]}=g_{j}\partial_{[j-1]}g_{j}^{-1}.$$
Setting $g=g_n \circ g_{n-1}\circ ...\circ g_2 \circ g_1$, we will prove that $$\partial'=g\circ \hat{\partial} \circ g^{-1}=g\circ s\circ \partial^+\circ s^{-1}\circ g^{-1}.$$ This establishes a ${\mathbb{Z}_p}$ tame isomorphism between $(\mathcal{A}^+,\partial^+)$ and $(\mathcal{A}', \partial')$, and thus the ${\mathbb{Z}_p}$ equivalence of $(\mathcal{A}^+, \partial^+)$ and $(\mathcal{A}^-, \partial^-)$.
The $g_j$ should satisfy $\partial_{[j]}|_{\mathcal{A}_{[j]}}=\partial'|_{\mathcal{A}_{[j]}}$. We begin by setting $\partial_{[0]}=\hat{\partial}$. As noted in the proof of Lemma \[lem:taucomp\], $\hat{\partial}=\partial'$ on the generators in the stabilizing orbits, and Lemma \[lem:length\] implies that they also agree on generators with length less that $\frac{1}{p}-\epsilon$. This establishes the base case $\partial_{[0]}|_{\mathcal{A}_{[0]}}=\partial'|_{\mathcal{A}_{[0]}}$.
For the inductive step, suppose that for $1\leq k \leq j-1$, the maps $g_k$ satisfy $\partial_{[k]}|_{A_{[k]}}=\partial'|_{A_{[k]}}$. Define $g_j: \mathcal{A}'\rightarrow\mathcal{A}'$ by $$g_j(x) = \begin{cases} x+F(\partial'(x)+\partial_{[j-1]}(x)) & \text{if}\ x \in \mathcal{O}_j\\
x & \text{otherwise}. \end{cases}$$
It follows from Lemma \[lem:length\], the proof of Lemma \[lem:taucomp\], and the definition of $F$ that if $x\in \mathcal{O}_{[j]}$, then $F(\partial'(x)+\partial_{[j-1]}(x)) \in A_{[j-1]}$. By the inductive hypothesis, the restriction of $\partial_{[j-1]}$ to $A_{[j-1]}$ agrees with $\partial'$. We have the following for $x \in \mathcal{O}_{[j]}$: $$\begin{aligned}
\partial_{[j]}(x)& =g_j \circ \partial_{[j-1]} \circ g_j^{-1}(x) \notag \\
&=g_j \circ \partial_{[j-1]}(x+F(\partial'(x)+\partial_{[j-1]}(x))) \notag \\
&= g_j \circ \partial_{[j-1]}(x) +g_j\circ \partial_{[j-1]}(F(\partial'(x)+\partial_{[j-1]}(x)))\notag \\
&= \partial_{[j-1]}(x) +g_j \circ \partial'(F(\partial'(x)+\partial_{[j-1]}(x))) \notag \\
&= \partial_{[j-1]}(x) +(\partial'\circ F)(\partial'(x)+\partial_{[j-1]}(x))\notag \end{aligned}$$
Again taking $x\in \mathcal{O}_{[j]}$, we apply Lemma \[lem:tau\] to the last line: $$\begin{aligned}
\partial_{[j]}(x) &= \partial_{[j-1]}(x)+(\tau+Id_{\mathcal{A}'}+F \circ \partial') (\partial'(x)+\partial_{[j-1]}(x))\notag \\
&= \partial_{[j-1]}(x)+ \tau\circ \partial'(x)+\tau\circ \partial_{[j-1]}(x) + \partial'(x)+\partial_{[j-1]}(x)+ F\partial'\partial'(x)+F\partial'\partial_{[j-1]}(x).\notag \\
&= \tau\circ \partial'(x)+\tau\circ \partial_{[j-1]}(x) ) + \partial'(x) \notag
\end{aligned}$$
Expand the middle term in the last line: $$\tau\circ \partial_{[j-1]}(x)=\tau\circ g_{j-1}\circ g_{j-2}\circ...\circ g_1\circ \hat{\partial}(x)= \tau\circ \hat{\partial}(x).$$
Thus $$\tau\partial'(x)+\tau\partial_{[j-1]}(x) =(\tau\circ \partial'+\tau\circ \hat{\partial})(x)=0$$ by Lemma \[lem:taucomp\] and $$\partial_{[j]}(x)= \partial'(x)$$ as desired. This proves that $\partial'$ and $\partial_{[j]}$ agree on $\mathcal{A}_{[j]}$. Inductively, this implies $g\circ \hat{\partial} \circ g^{-1} =\partial'$, which completes the proof of Proposition \[prop:star\].
Reidemeister invariance {#sect:Reids}
-----------------------
To complete the proof of Theorem \[thm:triple\], we show that the equivalence type of the equivariant DGA $({\widetilde{\mathcal{A}}}(\widetilde{\Gamma}), \gamma, \widetilde{\partial})$ is preserved by a Reidemeister move on $\Gamma$. Recall that each Reidemeister move in $(S^2,\Gamma)$ lifts to a $p$-tuple of Reidemeister moves in $(S^2, \widetilde{\Gamma})$.
A Reidemeister II move on $\Gamma$ adds $2p$ new crossings to $\widetilde{\Gamma}$. The proof that the “before" and “after" diagrams yield equivalent equivariant DGAs is similar to the proof of Proposition \[prop:star\], so we turn to Reidemeister III. This argument is based on the proof in [@S]. Up to rotation and switching $a$- and $b$-type generators, there are two versions of this move in $(S^2, \Gamma)$. (See Figure \[fig:r3\]). Each of these lifts to a $p$-tuple of identical local moves in $(S^2, \widetilde{\Gamma})$.
Suppose that each of the Reidemeister triangles in $\widetilde{\Gamma}_1$ (respectively, $\widetilde{\Gamma}_2$) look locally like the left (right) diagram shown in the top row of Figure \[fig:r3\]. (This is the case left to the reader in [@S].) Set $n=0$. Since ${\widetilde{\mathcal{A}}}(\widetilde{\Gamma}_1)$ and ${\widetilde{\mathcal{A}}}(\widetilde{\Gamma}_2)$ have the same number of generators with the same labels, we identify the algebras and show that the boundary maps corresponding to the two diagrams give isomorphic equivariant DGAs.
Define the the tame isomorphisms $f_i: ({\widetilde{\mathcal{A}}}, \widetilde{\partial}_1) \rightarrow ({\widetilde{\mathcal{A}}}, \widetilde{\partial}_2)$: $$f_i(x) = \begin{cases} a_{2,i}+a_{1,i}a_{3,i}& \text{if} \ x=a_{2,i} \\
b_{1,i}+a_{3,i}b_{2,i} &\text{if} \ x=b_{1,i}\\
b_{3,i}+b_{2,i}a_{1,i} &\text{if}\ x=b_{3,i} \\
x &\text{otherwise.} \end{cases}$$
Let $f=f_p\circ f_{p-1}\circ ...\circ f_1$.
The map $f$ is a graded tame automorphism which satisfies $f \circ \widetilde{\partial}_1=\widetilde{\partial}_2 \circ f$.
That $f$ is tame follows from the definition. To see that $f$ is graded, consider a generator $x$ associated to a crossing away from the Reidemeister triangle, and suppose that some disc representing a term in $\widetilde{partial}_2x$ which crosses the triangle and has a corner at $a_2^-$. Since $n=0$, truncating the disc at the $a_1a_3$ edge gives a new disc which has the same defect. This amounts to replacing $a_2^-$ by $a_1^-a_3^-$ and getting a new boundary word. Since the boundary map is graded, this implies that $|a_1a_3|=|a_2|$. The arguments for the other generators are similar.
To prove $f \circ \widetilde{\partial}_1=\widetilde{\partial}_2 \circ f$, apply the two compositions to an arbitrary generator and compare the resulting terms. We demonstrate this comparison for $x=b_{1,i}$.
Figure \[fig:bndry1\] shows that $\partial(b_{1,i})=Ub_{2,i} + a_{3,i}V + W + X$, where each capital letter represents a sum of words not involving any of the other local generators. The map $f_i$ fixes $b_{2,i}$ and $a_{3,i}$, so $$(f _i \circ \widetilde{\partial}_1 )(b_{1,i})=Ub_{2,i} + a_{3,i}V + W + X.$$
On the other hand, $f_i (b_{1,i})=b_{1,i}+a_{3,i}b_{2,i}$ and Figure \[fig:bndry2\] shows the following: $$\begin{aligned}
( \widetilde{\partial}_2 \circ f _i )(b_{1,i})& = \widetilde{\partial}_2b_{1,i}+ (\widetilde{\partial}_2 a_{3,i})b_{2,i}+a_{3,i}(\partial b_{2,i})\notag \\
&=W + a_{3,i}Z+ X +Yb_{2,i}+(U+Y)b_{2,i}+a_{3,i}(V+Z).\notag\end{aligned}$$
Working modulo two, this shows $f \circ \widetilde{\partial}_1(b_{1,i})=\widetilde{\partial}_2 \circ f(b_{1,i})$. The arguments are similar for generators associated to other crossings.
It remains to show that the map $f$ commutes with the ${\mathbb{Z}_p}$ action $\gamma$. First note that $f\circ \gamma=f$ because the $p$ Reidemeister triangles are permuted by $\gamma$.
Although $f$ is defined as $f_p\circ f_{p-1}\circ ...\circ f_1$, in fact it is independent of the order of the composition. This allows us to write $f=\sum_{i=1}^p f_i$. Since $ \gamma \circ f_i =f_{i+1}$, we have $$\gamma \circ f=\gamma\circ (f_1 +f_2 + ... + f_p)=f_ 2 ...+ f_p + f_1=f.$$
This completes the proof for the chosen case. When $n=1$, the map which sends each generator of ${\widetilde{\mathcal{A}}}(\widetilde{\Gamma}_1)$ to the generator of ${\widetilde{\mathcal{A}}}(\widetilde{\Gamma}_2)$ with the same label intertwines the $\widetilde{\partial}_i$. The proof for the other Reidemeister III move is given explicitly in [@S], and the argument is similar to the one provided here.
Examples
========
We conclude with two examples.
A pair of knots which are not Legendrian isotopic
-------------------------------------------------
For the first example, we consider two knots in the lens space $L(3,2)$. (See Figure \[fig:ex2\].) The two knots differ by a Legendrian stabilization, a topological isotopy which does not preserve the Legendrian type of the knot.
The boundary map $\partial_1$ is the zero map, so the homology of $(\mathcal{A}_1, \partial_1)$ is just a free group on two generators.
In the case of the stabilized knot, the images of the generators are as follow: $$\begin{aligned}
\partial_2 a_1&=1+b_2+a_2a_2\notag\\
\partial_2 b_1&= a_1a_2a_2 + a_2a_2a_1\notag\\
\partial_2 a_2&=1\notag\\
\partial_2 b_2&= a_2a_1a_1+a_1a_1a_2.\end{aligned}$$
In general, distinguishing equivalence classes of DGAs can be difficult, and a variety of algebraic tools have been developed to make this problem more tractable. We refer the reader to [@C] and [@Ng] for a discussion of Chekanov polynomials, augmentations, linearized homology, and the characteristic algebra, but we note that the following suffices to distinguish $K_1$ and $K_2$:
An *augmentation* of a DGA is an algebra homomorphism $\epsilon:\mathcal{A}\rightarrow \mathbb{Z}_2$ such that $\epsilon(1)=1$, $\epsilon \circ \partial=0$ and $\epsilon(x)=0$ if $|x|\neq 0$. The existence of augmentations is an invariant of the equivalence type of the algebra.
The identity map is an augmentation of $(\mathcal{A}_1, \partial_1)$, whereas $(\mathcal{A}_2, \partial_2)$ has no augmentations. Thus the two DGAs are not equivalent, and $K_1$ is not Legendrian isotopic to $K_2$. The knots in this example can also be distinguished by the classical invariants of their lifts to $S^3$; we would be interested in studying pairs of Legendrian non-isotopic knots in $L(p,q)$ which are not distinguished by classical invariants.
An example in $L(5,2)$
----------------------
For the second example, we compute $(\mathcal{A}, \partial)$ for a knot in $L(5,2)$. The Lagrangian projection is shown as a rectangle; to recover $S^2$, collapse each of the top and bottom edges to a point and identify the vertical edges as indicated. The knot shown here is $K(5,2,2)$ in the notation of [@R2].
The cyclic group grading $\mathcal{A}$ is $\mathbb{Z}_2$, and we have $|a_i|=1$ and $|b_i|=0$ for $i=1,2$. $$\begin{aligned}
\partial a_1&=a_2a_2\notag \\
\partial b_1&=0\notag \\
\partial a_2&=1\notag \\
\partial b_2&=a_2 b_1+b_1a_2\notag \\\end{aligned}$$
In addition, both capping paths for $a_1$ bound admissible discs with $\tilde{n}_{a_2}(f)=0$; the corresponding terms cancel modulo two, and these discs are not shown. We note that although $\partial a_2=1$, $K$ is not a stabilization of any other Legendrian knot in $L(5,2)$. The proof of this fact relies on the classification of Legendrian unknots in $S^3$ due to Eliashberg and Fraser [@EF].
Lemma \[lem:mainpf\] implies that we could also compute the differential from the $\phi$-image of $\widetilde{\partial}$ in ${\widetilde{\mathcal{A}}}(\widetilde{\Gamma})$. In this computation there are additional discs representing non-canceling terms in $\widetilde{\partial}$ which nevertheless cancel in the image of $\phi$. Representatives of these discs are shown in Figure \[fig:ex\]. In $({\widetilde{\mathcal{A}}}(\widetilde{\partial}), \widetilde{\partial})$, the terms in $\widetilde{\partial}(x_{i,0})$ are as follow: $$\begin{aligned}
\widetilde{\partial} a_{1,0}&=b_{1,1}+a_{2,0}a_{2,1}+b_{1,-1}\notag\\
\widetilde{\partial} b_{1,0}&=a_{2,2}+a_{2,-1}\notag\\
\widetilde{\partial} a_{2,0}&=1\notag\\
\widetilde{\partial} b_{2,0}&=a_{1,-2}+a_{1,1}+a_{2,1}b_{1,0}+b_{1,-1}a_{2,-1}\notag\end{aligned}$$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
In $\mathrm{ZFC}$, the class $\mathrm{Ord}$ of ordinals is easily seen to satisfy the definable version of strong inaccessibility. Here we explore deeper $\mathrm{ZFC}$-verifiable combinatorial properties of $\mathrm{Ord}$, as indicated in Theorems A & B below. Note that Theorem A shows the unexpected result that $\mathrm{Ord}$ is never definably weakly compact in any model of $\mathrm{ZFC}$.
**Theorem A. ***Let* $\mathcal{M}$* be any model of* .
**(1)** *The definable tree property fails in* $%
\mathcal{M}$: *There is an* $\mathcal{M}$*-definable* $\mathrm{%
Ord}$-*tree with no* $\mathcal{M}$-*definable cofinal branch*.**
**(2)** *The definable partition property fails in* $%
\mathcal{M}$: *There is an* $\mathcal{M}$-*definable* $2$*-coloring* $f:[X]^{2}\rightarrow 2$* for some* $\mathcal{M}$*-definable proper class* $X$* such that no* $\mathcal{M}$*-definable proper classs is monochromatic for* $f$.**
**(3)** *The definable compactness property for* $%
\mathcal{L}_{\infty \mathrm{,\omega }}$ *fails in* $\mathcal{M}$:* There is a definable theory* $\Gamma $* in the logic* $%
\mathcal{L}_{\infty \mathrm{,\omega }}$ (*in the sense of* $\mathcal{M%
}$) *of size* $\mathrm{Ord}$ *such that every set-sized subtheory of* $\Gamma $ *is satisfiable in* $\mathcal{M}$*, but there is no* $\mathcal{M}$*-definable model of* $\Gamma $.
**Theorem B. ***The definable* $\Diamond _{\mathrm{Ord}%
}$ *principle holds in a model* $\mathcal{M}$* of* * iff* $\mathcal{M}$* carries an* $\mathcal{M}$*-definable* *global well-ordering*.
Theorems A and B above can be recast as theorem schemes in $\mathrm{ZFC}$, or as asserting that a single statement in the language of class theory holds in all ‘spartan’ models of $\mathrm{GB}$ (Gödel-Bernays class theory); where a spartan model of $\mathrm{GB}$ is any structure of the form $(\mathcal{M},D_{\mathcal{M}})$*,* where $\mathcal{M}\models
\mathrm{ZF}$* *and $D_{\mathcal{M}}$ is the family of* *$%
\mathcal{M}$-definable classes. Theorem C gauges the complexity of the collection $\mathrm{GB}_{\mathrm{spa}}$ of (Gödel-numbers of) sentences that hold in all spartan models of $\mathrm{GB.}$**
**Theorem C. **$\mathrm{GB}_{\mathrm{spa}}$ *is* $\Pi
_{1}^{1}$*-complete.*
author:
- 'Ali Enayat & Joel David Hamkins'
title: ZFC proves that the class of ordinals is not weakly compact for definable classes
---
**1. Introduction & Preliminaries**
In $\mathrm{ZFC}$, the class $\mathrm{Ord}$ of ordinals satisfies the definable version of strong inaccessibility since the power set axiom and the axiom of choice together make it evident that $\mathrm{Ord}$ is closed under cardinal exponentiation; and the scheme of replacement ensures the definable regularity of $\mathrm{Ord}$** **in the sense that for each cardinal $\kappa <\mathrm{Ord}$, the range of every definable ordinal-valued map $f$ with domain $\kappa $ is bounded in $\mathrm{Ord}$. In this paper we investigate more subtle definable combinatorial properties of $\mathrm{Ord}$** **in the context of $\mathrm{ZFC}$ to obtain results, each of which takes the form of a *theorem scheme* within $\mathrm{ZFC}$. In Section 2 we establish a number of results that culminate in Theorem 2.6, which states that the tree property fails for definable classes across all models of $\mathrm{ZFC}$; this result is then used in Section 3 to show the failure of the partition property for definable classes, and the failure of weak compactness of $\mathrm{Ord}$ for definable classes in all models of $%
\mathrm{ZFC.}$ *Thus, the results in Sections 2 and 3 together demonstrate the unexpected* $\mathrm{ZFC}$*-provable failure of the definable version of a large cardinal property for* $\mathrm{Ord}$. In Section 4 we establish the equivalence of the combinatorial principle $%
\Diamond _{\mathrm{Ord}}$ and the existence of a definable global choice function across all models of $\mathrm{ZFC.}$
The results in Sections 2 through 4 can be viewed as stating that certain sentences in the language of class theory hold in all ‘spartan’ models of (Gödel-Bernays class theory), i.e., in all models of $%
\mathrm{GB}$ of the form $(\mathcal{M},\mathcal{D}_{\mathcal{M}})$, where $%
\mathcal{M}$ is a model of $\mathrm{ZF}$ and $\mathcal{D}_{\mathcal{M}}$ is the collection of $\mathcal{M}$-definable subsets of $M$. For example Theorem 2.6 is equivalent to the veracity of the statement if the axiom of choice for sets holds, then there is an $\mathrm{Ord}$-Aronszajn tree in every spartan model of In Section 5 we show that the *theory* of all spartan models of $\mathrm{%
GB}$, when viewed as a subset of $\omega $ via Gödel-numbering, is $\Pi
_{1}^{1}$-complete; and a fortiori, it is not computably axiomatizable.
We now turn to reviewing pertinent preliminaries concerning models of set theory. Our meta-theory is $\mathrm{ZFC}$.
**1.1. Definition. **Suppose $\mathcal{M}=(M,\in ^{\mathcal{M}%
})$ and $\mathcal{N}=(N,\in ^{\mathcal{N}})$ are models of set theory. Note that we are not assuming that either $\mathcal{M}$ or $\mathcal{N}$ is well-founded.****
**(a)** For $m\in M,$ let $m_{\mathcal{M}}:=\{x\in M:x\in ^{%
\mathcal{M}}m\}.$ If $\mathcal{M}$ $\subseteq \mathcal{N}$ (i.e., $\mathcal{M%
}$ is a submodel of $\mathcal{N}$) and $m\in M$, then $\mathcal{N}$ *fixes* $m$ if $m_{\mathcal{M}}=m_{\mathcal{N}}.$ $\mathcal{N}$ *end extends* $\mathcal{M}$, written $\mathcal{M}\subseteq _{e}\mathcal{N}$, iff $%
\mathcal{N}$ fixes every $m\in M.$ Equivalently: $\mathcal{M}\subseteq _{e}%
\mathcal{N}$ iff $\mathcal{M}$ is a transitive submodel of $\mathcal{N}$ in the sense that if $x\in ^{\mathcal{N}}y$ for some $x\in N$ and some $y\in M,$ then $x\in ^{\mathcal{M}}y$. ****
**(b)** Given $n\in \omega $, $\mathcal{N}$ is a *proper* $\Sigma _{n}$*-e.e.e.*** **of $\mathcal{M}$ (e.e.e. stands for elementary end extension), iff $\mathcal{M\subsetneq }_{e}%
\mathcal{N}$, and $\mathcal{M}\prec _{\Sigma _{n}}\mathcal{N}$ (i.e., $%
\Sigma _{n}$-statements with parameters from $M$ are absolute in the passage between $\mathcal{N}$ and $\mathcal{M)}$. It is well-known that if $\mathcal{%
M}\prec _{\Sigma _{2}}\mathcal{N}$ and $\mathcal{M}\models \mathrm{ZF}$, then $\mathcal{N}$ is a *rank extension* of $\mathcal{M}$, i.e., whenever $a\in M$ and $b\in N\backslash M$, then $\mathcal{N}\models \rho
(a)\in \rho (b),$ where $\rho $ is the usual ordinal-valued rank function on sets.****
**(c)** Given $\alpha \in \mathrm{Ord}^{\mathcal{M}}$, $%
\mathcal{M}_{\alpha }$ denotes the structure $(\mathrm{V}_{\alpha },\in )^{%
\mathcal{M}},$ and $M_{\alpha }=\mathrm{V}_{\alpha }^{M}$.****
**(d)** For $X\subseteq M^{n}$ (where $n\in \omega ),$ we say that $X$ is $\mathcal{M}$-*definable* iff $X$ is parametrically definable in $\mathcal{M}$.****
**(e)** $\mathcal{N}$ is a* conservative* extension of $\mathcal{M}$, written $\mathcal{M}\subseteq _{\mathrm{cons}}\mathcal{N}$, iff the intersection of any $\mathcal{N}$-definable subset of $N$ with $M$* *is $\mathcal{M}$-definable.
For models of $\mathrm{ZF}$, the set-theoretical sentence $\exists
p\left( \mathrm{V}=\mathrm{HOD}(p)\right) $ expresses: there is some $p$ such that every set is first order definable in some structure of the form $(\mathrm{V}_{\alpha },\in ,p)$ with $p\in \mathrm{V}%
_{\alpha }$. The following theorem is well-known; the equivalence of (a) and (b) will be revisited in Theorem 4.2.
**1.2. Theorem. ***The following statements are equivalent for* $\mathcal{M}\models \mathrm{ZF}$****
**(a)** $\mathcal{M}\models \exists p\left( \mathrm{V}=%
\mathrm{HOD}(p)\right) .$****
**(b)** *For some* $p\in M$ *and some set-theoretic formula* $\varphi (x,y,\overline{p})$ (*where* $%
\overline{p}$ *is a name for* $p$) $\mathcal{M}$ *satisfies* $\varphi $ *well-orders the universe*.****
**(c)** *For some* $p\in M$ *and some* $\Sigma
_{2}$-*formula* $\varphi (x,y,\overline{p})$ $\mathcal{M}$ *satisfies* $\varphi $ *well-orders the universe*.****
(**d)** $\mathcal{M}\models \forall x(x\neq \varnothing
\rightarrow f(x)\in x)$ *for some* $\mathcal{M}$-*definable* $%
f:M\rightarrow M.$
Next we use definable classes to lift certain combinatorial properties of cardinals to the class of ordinals.
**1.3. Definitions. **Suppose* *$\mathcal{M}\models
\mathrm{ZFC.}$
**(a)** Suppose $\tau =\left( T,\ <_{T}\right) $ is a tree ordering, where both $T$ and $<_{T}$ are $\mathcal{M}$-definable. $\tau $ is an $\mathrm{Ord}$-*tree in* $\mathcal{M}$ iff $\mathcal{M}$ satisfies $\tau $ is a well-founded tree of height $\mathrm{Ord}$** **and for all $\alpha \in \mathrm{Ord}$, the collection $T_{\alpha
} $ of elements of $T$ at level $\alpha $ of $\tau $ form a set. Such a tree $\tau $ is said to be a *definably* $\mathrm{Ord}$-*Aronszajn* tree in $\mathcal{M}$ iff no cofinal branch of $\tau $ is $\mathcal{M}$-definable.
**(b)** *The definable tree property for* $\mathrm{Ord}
$ *fails* *in* $\mathcal{M}$* *iff there exists a definably $\mathrm{Ord}$-*Aronszajn* tree in $\mathcal{M}$.[^1]
**(c)** *The definable proper class partition property fails in* $\mathcal{M}$ iff there is an $\mathcal{M}$-definable proper class $X$ of $M$ with an $\mathcal{M}$-definable $2$-coloring $f:[X]^{2}%
\rightarrow 2$ such that there is no $\mathcal{M}$-definable monochromatic proper class for $f$. We also say that $\mathrm{Ord}\rightarrow \left(
\mathrm{Ord}\right) _{2}^{2}$ *fails in* $\mathcal{M}$ iff there is an $\mathcal{M}$-definable $2$-coloring $f:[\mathrm{Ord}]^{2}\rightarrow 2$ such that there is no $\mathcal{M}$-definable monochromatic proper class for $f$.
**(d)** *The definable* *compactness property for* $\mathcal{L}_{\mathrm{\infty },\mathrm{\omega }}$ *fails in* $%
\mathcal{M}$ iff there is an* *$\mathcal{M}$-definable theory $%
\Gamma $ formulated in the logic $\mathcal{L}_{\mathrm{\infty },\mathrm{%
\omega }}$ (in the sense of $\mathcal{M}$) such that every set-sized subtheory of $\Gamma $ is satisfiable in $\mathcal{M}$, but there is no* *$\mathcal{M}$*-*definable model* *of $T$. Here $%
\mathcal{L}_{\mathrm{\infty },\mathrm{\omega }}$ is the extension of first order logic that allows conjunctions and disjunctions applied to *sets* of formulae (of any cardinality) with only a finite number of free variables, as in [@Barwise Ch.III].
**(e)** An $\mathcal{M}$-definable subset $E$ of $\mathrm{Ord}%
^{\mathcal{M}}$ is said to be *definably* $\mathcal{M}$-*stationary* iff $E\cap C\neq \varnothing $ for every $\mathcal{M}$-definable subset $C$ of $\mathcal{M}$ such that $C$ is closed and unbounded in $%
\mathrm{Ord}^{\mathcal{M}}$.
**(f)** *The definable*** **$\Diamond _{\mathrm{%
Ord}}$ *holds in* $\mathcal{M}$ iff there is some $\mathcal{M}$-definable $\vec{A}=\left\langle A_{\alpha }:\alpha \in \mathrm{Ord}^{%
\mathcal{M}}\right\rangle $ such that $\mathcal{M}$ satisfies $A_{\alpha }\subseteq \alpha $ for all $\alpha \in \mathrm{%
Ord}$, and for all $\mathcal{M}$-definable $A\subseteq
\mathrm{Ord}^{\mathcal{M}}$ there is $E\subseteq \mathrm{Ord}^{\mathcal{M}}$ such that $E$ is definably $\mathcal{M}$-stationary and $A_{\alpha }=A\cap
\alpha $ for all $\alpha \in E$. Here $\vec{A}$ is said to be $\mathcal{M}$-definable if there is an $\mathcal{M}$-definable $A$ such that $A_{\alpha
}=\{m:\left\langle m,\alpha \right\rangle \in A\}$ for each $\alpha \in
\mathrm{Ord}^{\mathcal{M}}.$
**2. The failure of the definable tree property for the class of ordinals**
The proof of the main result of this section (Theorem 2.6) is based on a number of preliminary model-theoretic results which are of interest in their own right. We should point out that a proof of a special case of Theorem 2.6 was sketched in [@Ali; @Power-like Remark 3.5] for models of set theory with built-in global choice functions, using a more technical argument than the one presented here.
We begin with the following theorem which refines a result of Kaufmann [@Matt; @Topless; @and; @blunt Theorem 4.6]. The proof uses an adaptation of Kaufmann’s proof based on a strategy introduced in [@Ali; @TAMS Theorem 1.5(a)]**.**
**2.1. Theorem. ***No model of* $\mathrm{ZFC}$ *has a proper conservative* $\Sigma _{3}$-*e.e.e*.* *
**Proof.** Suppose to the contrary that $\mathcal{M}\models
\mathrm{ZF}$ and $\mathcal{M}\prec _{\Sigma _{3},\mathrm{e},\mathrm{cons}}%
\mathcal{N}$ for some $\mathcal{N}$. Let $\varphi $ be the statement that expresses the following instance of the reflection theorem:
$\forall \lambda \in \mathrm{Ord}\mathbf{\ }\exists \beta \in \mathrm{Ord}%
\left( \lambda \in \beta \wedge \ (\mathrm{V}_{\beta },\in )\prec _{\Sigma
_{1}}\left( \mathrm{V},\in \right) \right) .$
Using the fact that the satisfaction predicate for $\Sigma _{1}$-formulae is $\Sigma _{1}$-definable it is easy to see that $\varphi $ is a $%
\Pi _{3}$-statement, and thus $\varphi $ also holds in $\mathcal{N}$ since $%
\varphi $ holds in $\mathcal{M}$ by the reflection theorem.[^2] So we can fix some $\lambda \in \mathrm{Ord}^{\mathcal{N}}\backslash \mathrm{Ord}^{\mathcal{M}}$ and some $\mathcal{N}$-ordinal $\beta >\lambda $ of $\mathcal{N}$ such that:
$\mathcal{N}_{\beta }\prec _{\Sigma _{1}}\mathcal{N}$.
Note that this implies that $\mathcal{N}_{\beta }$ can meaningfully define the satisfaction predicate for every set-structure ‘living in’ $\mathcal{N}_{\beta }$ since that $\mathcal{N}_{\beta }$ is a model of a substantial fragment of $\mathrm{ZF}$, including $\mathrm{KP}$ (Kripke-Platek set theory), and already $\mathrm{KP}$ is sufficient for this purpose [@Barwise III.2]. Also, since the statement every set can be well-ordered is a $\Pi
_{2}$-statement which holds in $\mathcal{M}$ by assumption, it also holds in $\mathcal{N}$, and therefore we can fix a binary relation $w$ in $\mathcal{N}
$ such that, as viewed in $\mathcal{N}$, $w$ is a well-ordering of $\mathrm{V%
}_{\beta }.$ Hence for any $\alpha \in \mathrm{Ord}^{\mathcal{M}}$ with $%
\alpha <\beta $, *within* $\mathcal{N}$ one can define the submodel $%
\mathcal{K}_{\alpha }$ of $\mathcal{N}_{\beta }$ whose universe $K_{\alpha }$ is defined via:
$K_{\alpha }:=\{a\in \mathrm{V}_{\beta }:a$ is first order definable in $(%
\mathcal{N}_{\beta },w,\lambda ,m)_{m\in \mathrm{V}_{\alpha }}$}.
Clearly $M_{\alpha }\cup \{\lambda \}\subsetneq \mathcal{K}%
_{\alpha }\prec \mathcal{N}_{\beta }$, and of course $\mathcal{K}_{\alpha }$ is a member of $\mathcal{N}$. Next let:
$\mathcal{K}:=\bigcup\limits_{\alpha \in \mathrm{Ord}^{\mathcal{M}}}\mathcal{%
K}_{\alpha }.$
Note that we have:
$\mathcal{M}\subsetneq _{\text{e}}\mathcal{K}\preceq \mathcal{N}_{\beta
}\prec _{1}\mathcal{N}.$
We now make a crucial case distinction: either (a) $\mathrm{Ord}^{%
\mathcal{K}}\backslash \mathrm{Ord}^{\mathcal{M}}$ has minimum element, or (b) it does not. The proof will be complete once we verify that both cases lead to a contradiction.
**Case (a)**. Let $\eta =\min (\mathrm{Ord}^{\mathcal{K}}\backslash
\mathrm{Ord}^{\mathcal{M}})$. We claim that $\mathcal{M\prec N}_{\eta }$. To see this, we use Tarski’s test for elementarity: suppose $\mathcal{N}_{\eta
}\models \exists x\varphi (x,\overline{m})$ for some $m\in M$ and some formula $\varphi (x,y),$ and let $\theta _{0}$ be defined in $\mathcal{N}%
_{\beta }$ as the least ordinal $\theta $ such that $x\in \mathrm{V}_{\theta
}$ and $\mathrm{V}_{\eta }\models \exists x$ $\varphi (x,\overline{m})$. Then $\theta _{0}\in K$ and clearly $\theta _{0}<\eta $, which shows that $%
\theta _{0}\in \mathrm{Ord}^{\mathcal{M}}$. Hence $\mathcal{N}_{\eta
}\models \varphi (\overline{m_{0}},\overline{m})$ for some $m_{0}\in M,$ thus completing the proof of $\mathcal{M\prec N}_{\eta }.$ But if $\mathcal{%
M\prec N}_{\eta }$, then we can choose $S$ in $\mathcal{N}$ such that:
$\mathcal{N}\models S=\{\ulcorner \varphi (\overline{m})\urcorner \in
\mathrm{V}_{\eta }:\mathcal{N}\models $ $(\mathrm{V}_{\eta
},\in )\models \varphi (\overline{m})$$\}.$
Based on the assumption that $\mathcal{N}$ is a conservative extension of $\mathcal{M}$, $S\cap M$ should be an $\mathcal{M}$-definable satisfaction predicate for $\mathcal{M}$, which contradicts (a version of) Tarski’s undefinability of truth theorem.
**Case (b)**. This is the more difficult case, where $\mathrm{Ord}^{%
\mathcal{K}}\backslash \mathrm{Ord}^{\mathcal{M}}$ has no least element. Let $\Phi :=\bigcup\limits_{\alpha \in \mathrm{Ord}^{\mathcal{M}}}\Phi _{\alpha
} $, where
$\Phi _{\alpha }:=\{\ulcorner \varphi (c,\overline{m})\urcorner \in M:%
\mathcal{N}\models $ $(\mathrm{V}_{\beta },\in ,w,\lambda
,m)_{m\in \mathrm{V}_{\alpha }}\models \varphi (c,\overline{m})$$\}.$
In the above definition of $\Phi _{\alpha }$, the constant $c$ is interpreted as $\lambda $ and $\varphi (c,\overline{m})$ ranges over first order formulae in the sense of $\mathcal{M}$ (or equivalently: in the sense of $\mathcal{N}$) in the language
$\mathcal{L}_{\alpha }=\{\in ,\vartriangleleft ,c\}\cup \{\overline{m}:m\in
\mathrm{V}_{\alpha }\}$,
where $c$ is a new constant symbol and $\vartriangleleft $ is a binary relation symbol interpreted by $w$. Thus $\Phi $ can be thought of as the type of $\lambda $ in $\mathcal{N}_{\beta }$ over $M$. Since $\mathcal{N}
$ is assumed to be a conservative extension of $\mathcal{M}$, $\Phi $ is $%
\mathcal{M}$-definable via some unary formula $\phi $. Hence $\Gamma $ below is also $\mathcal{M}$-definable via some unary formula $\gamma :$
$\overset{\Gamma }{\overbrace{\left\{ \ulcorner t(c,\overline{m})\urcorner
\in M:\phi \left( \ulcorner t(c,\overline{m})\in \mathrm{Ord}\mathbf{%
\urcorner }\right) \ \mathrm{and}\ \forall \theta \in \mathrm{Ord}(\phi
\left( \ulcorner t(c,\overline{m})>\overline{\theta }\mathbf{\urcorner }%
\right) \right\} }},$
where $t$ is a definable term in the language $\mathcal{L}$, i.e., $t(c,\overline{m})$ is an $\mathcal{L}$-definition $\varphi (c,\overline{m}%
,x)$ of some element $x$. So, officially speaking, $\Gamma $ consists of $%
\ulcorner \varphi (c,\overline{m},x)\urcorner \in M$ that satisfy the following three conditions:
\(1) $\phi \left( \ulcorner \exists !x\varphi (c,\overline{m},x%
\mathbf{\urcorner }\right) .$
\(2) $\phi \left( \ulcorner \forall x\left( \varphi (c,\overline{m}%
,x)\rightarrow x\in \mathrm{Ord}\right) \mathbf{\urcorner }\right) .$
\(3) $\forall \theta \in \mathrm{Ord}\ \phi \left( \ulcorner
\forall x\left( \varphi (c,\overline{m},x)\rightarrow x>\overline{\theta }%
\right) \mathbf{\urcorner }\right) .$
Since $\mathrm{Ord}^{\mathcal{K}}\backslash \mathrm{Ord}^{\mathcal{%
M}}$ has no minimum element (recall: we are analysing case (b)), $\mathcal{M}%
\models \psi $, where:
$\psi :=\forall t\left( \gamma (t)\rightarrow \exists t^{\prime }(\gamma
(t^{\prime })\wedge \phi (\ulcorner t^{\prime }\in t\mathbf{\urcorner }%
)\right) .$
Choose $k$ such that $\psi $ is a $\Sigma _{k}$-statement, and use the reflection theorem in $\mathcal{M}$ to pick $\mu \in \mathrm{Ord}^{%
\mathcal{M}}$ such that $\mathcal{M}_{\mu }\prec _{\Sigma _{k}}\mathcal{M}.$ Then $\psi $ holds in $\mathcal{M}_{\mu }$, so by DC (dependent choice, which holds in $\mathcal{M}$ since AC holds in $\mathcal{M}$), there is some function $f_{c}$ in $\mathcal{M}$ such that:
$\mathcal{M}\models \forall n\in \omega \ \phi \left( \ulcorner
f_{c}(n+1)\in f_{c}(n)\mathbf{\urcorner }\right) .$
Let $\alpha \in \mathrm{Ord}^{\mathcal{M}}$ be large enough so that $M_{\alpha }$ contains all constants $\overline{m}$ that occur in any of the terms in the range of $f$; let $f_{\lambda }(n)$ be defined in $%
\mathcal{N}$ as the result of replacing all occurrences of the constant $c$ with $\overline{\lambda }$ in $f_{c}(n)$; and let $g(n)$ be defined in $%
\mathcal{N}$ as the interpretation of $f_{\lambda }(n)$ in $(\mathrm{V}%
_{\beta },\in ,w,\lambda ,m)_{m\in \mathrm{V}_{\alpha }}.$ Then $\mathcal{N}$ satisfies:
$\forall n\in \omega \ \left( g(n)\in g(n+1)\right) $,
which contradicts the foundation axiom in $\mathcal{N}$. The proof is now complete.$\square $
**2.2. Definition.**
**(a)** Given ordinals $\alpha <\beta ,$ $\mathcal{V}_{\beta
,\alpha }$ denotes the structure $(\mathrm{V}_{\beta },\in ,a)_{a\in \mathrm{%
V}_{\alpha }}$, and for a model $\mathcal{M}\models \mathrm{ZF},$
$\mathcal{M}_{\beta ,\alpha }:=\left( \mathcal{V}_{\beta ,\alpha }\right) ^{%
\mathcal{M}}.$
**(b)** Given a meta-theoretic natural number $n$, $\tau _{n}$ denotes the definable tree whose nodes at level $\alpha $ consist of first order theories of the form $\mathrm{Th}(\mathcal{V}_{\beta ,\alpha },s)$, where $s\in \mathrm{V}_{\beta }\backslash \mathrm{V}_{\alpha },$ and $\beta $ is $n$-correct[^3]. The language of $\mathrm{Th}(\mathcal{V}_{\beta ,\alpha },s)$ consists of $\{\in \}$ plus constants $\overline{m}$ for each $m\in \mathrm{V}_{\alpha },$ and a new constant $c$ whose denotation is $s$. The ordering of the tree is by set-inclusion.
**2.3. Lemma. ***For each meta-theoretic natural number* $n$, $\mathrm{ZFC}$ *proves* $\tau _{n}$ *is an* $\mathrm{Ord}$-*tree*.
**Proof. **Thanks to the Montague-Vaught reflection theorem, there are plenty of nodes at any ordinal level $\alpha $. On the other hand, since each $\mathrm{Th}(\mathcal{V}_{\beta ,\alpha },s)$ can be canonically coded as a subset of $\mathrm{V}_{\alpha },$ and $\left\vert \mathrm{V}%
_{\omega +\alpha }\right\vert =\beth _{\alpha },$ there are at most $\beth
_{\alpha }$-many nodes at level $\alpha $ $\square $
**2.4**.** Remark. **One may ‘prune’ every $\mathrm{Ord}$-tree $\tau $ to obtain a definable subtree $\tau ^{\ast }$ which has nodes of arbitrarily high level in $\mathrm{Ord}$ by simply throwing away the nodes whose set of successors have bounded height and then using the replacement scheme to verify that the subtree $\tau ^{\ast }$ thus obtained has height $\mathrm{Ord}$. See [@Kunen; @Text Lemma 3.11] for a similar construction for $\kappa $-trees (where $\kappa $ is a regular cardinal).
**2.5. Lemma. ***Suppose* $\mathcal{M}$ *is a model of* $\mathrm{ZFC}$ *that carries an* $\mathcal{M}$*-definable global well-ordering*. *Furthermore, suppose that* $n\geq
3 $ *and the tree* $\tau _{n}^{\mathcal{M}}$ *has a branch* $B$.* **Then:***
**(a)*** There is a model* $\mathcal{N}$* and a proper embedding* $j:\mathcal{M}\rightarrow \mathcal{N}$ *such that* $j(\mathcal{M})\prec _{\mathrm{e},n}\mathcal{N}$*.*****
**(b)** *Both* $\mathcal{N}$* and* $j$ *are* $\mathcal{M}$-*definable if* $B$* is* $\mathcal{M}$*-definable.*****
**(c)** $\mathcal{N}$* is a conservative extension of* $j(\mathcal{M})$* if* $B$* is* $\mathcal{M}$*-definable.*
**Proof. **We will only prove (a) since the proof of (b) will be clear by an inspection of the proof of (a), and (c) is an immediate consequence of (b). Let $B$ be a branch of $\tau _{n}^{\mathcal{M}}.$ Each node in $B$ is a first order theory in the sense of $\mathcal{M}$ and is of the form $\left( \mathrm{Th}(\mathcal{\mathrm{V}}_{\beta ,\alpha },s)\right)
^{\mathcal{M}}$. Note that $\left( \mathrm{Th}(\mathcal{V}_{\beta ,\alpha
},s)\right) ^{\mathcal{M}}$ is not the necessarily the same as $\mathrm{Th}(%
\mathcal{M}_{\beta ,\alpha },s)$, since the latter is the collection of *standard* sentences in $\left( \mathrm{Th}(\mathcal{V}_{\beta
,\alpha },s)\right) ^{\mathcal{M}}$. In particular, if $\mathcal{M}$ not $%
\omega $-standard, then:
$\mathrm{Th}(\mathcal{M}_{\beta ,\alpha },s)\subsetneq \left( \mathrm{Th}(%
\mathcal{V}_{\beta ,\alpha },s)\right) ^{\mathcal{M}}.$
For each $\alpha \in \mathrm{Ord}^{\mathcal{M}}$, let $b_{\alpha }$ be the node of $B$ at level $\alpha $. We may choose some $\beta _{\alpha
}\in \mathrm{Ord}^{\mathcal{M}}$ and some $s_{\alpha }\in \left( \mathrm{V}%
_{\beta _{\alpha }}\backslash \mathrm{V}_{\alpha }\right) ^{\mathcal{M}}$ such that:
$b_{\alpha }=\left( \mathrm{Th}(\mathcal{V}_{\beta _{\alpha },\alpha
},s_{\alpha })\right) ^{\mathcal{M}}.$
The above choices of $\beta _{\alpha }$ and $s_{\alpha }$ are performed at the meta-theoretic level (where $\mathrm{ZFC}$ is assumed); however if $B$ is $\mathcal{M}$-definable, then so is the map $\alpha
\mapsto b_{\alpha }$, which in turn shows that the maps $\alpha \mapsto
\beta _{\alpha }$ and $\alpha \mapsto s_{\alpha }$ can also be arranged to be $\mathcal{M}$-definable since $\mathcal{M}$ is assumed to carry an $%
\mathcal{M}$-definable global well-ordering (the definability of these two maps plays a key role in verifying that an inspection of the proof of (a) yields a proof of (b)).
We now explain how to use $B$ to construct the desired structure $\mathcal{N}
$. In order to do so, we need some definitions:
$(i)$ Let $\mathcal{L}$ be the language consisting of the usual language $\{\in \}$ of set theory, augmented with a binary relation symbol $%
\vartriangleleft $, constants $\overline{m}$ for each $m\in M,$ and a new constant $c.$
$(ii)$ For each $\alpha \in \mathrm{Ord}^{\mathcal{M}}$ let $%
\mathcal{N}_{\alpha }$ be the submodel of $\mathcal{M}_{\beta _{\alpha }}$ whose universe $N_{\alpha }$ consists of elements of $M_{\beta _{\alpha }}$ that are first order definable in the structure $\left( \mathcal{V}_{\beta
_{\alpha },\alpha },s_{\alpha }\right) $, *as viewed from* $\mathcal{M%
}$ (so the available parameters for the definitions come from $M_{\alpha
}\cup \{s_{\alpha }\}$ and consequently $M_{\alpha }\cup \{s_{\alpha
}\}\subseteq N_{\alpha }$)$.$ By Theorem 1.2 we may assume that for some formula $W(x,y,\overline{m})$ the sentence $W$ is a global well-ordering" is equivalent to a $\Pi _{3}$-statement in* *$%
\mathcal{M}$. Therefore, since $n\geq 3$, the statement there is a well-ordering of $\mathrm{V}_{\beta _{\alpha }}$ that is definable in $(\mathrm{V}_{\beta _{\alpha }},\in )$ holds in $\mathcal{M}$, which immediately shows (by Tarski’s elementarity test) that the statement expressing $\mathcal{N}_{\alpha }\prec \mathcal{V}_{\beta
_{\alpha }}$ holds in $\mathcal{M}.$[^4] It is important to have in mind that, as viewed from $\mathcal{M}$, each member of $\mathcal{N}_{\alpha }$ can be written as the denotation $\delta ^{\mathcal{N%
}_{\alpha }}$ of a definable term $\delta =$ $\delta (\overline{m_{\delta }}%
,c)$ for some $m\in M$ in the language $\mathcal{L}$ described above (where $%
c$ is interpreted by $s_{\alpha })$ so $\delta $ might be of nonstandard length if $\mathcal{M}$ is not $\omega $-standard (here we are taking advantage of the definability of a sequence-coding function in $\mathcal{M}%
_{\beta _{\alpha }}$ to reduce the number of parameters of a definable term that come from $M_{\alpha }$ to one).
$(iii)$ Given ordinals $\alpha _{1},\alpha _{2}\in \mathrm{Ord}^{%
\mathcal{M}}$ with $\alpha _{1}<\alpha _{2},$ in $\mathcal{M}$ consider:
$j_{\alpha _{1},\alpha _{2}}:\mathcal{N}_{\alpha _{1}}\rightarrow \mathcal{N}%
_{\alpha _{2}}$, where $j_{\alpha _{1},\alpha _{2}}(\delta ^{\mathcal{N}%
\alpha _{1}}):=\delta ^{\mathcal{N}_{\alpha _{2}}}.$
It is not hard to see that $j_{\alpha _{1},\alpha _{2}}$ is an *elementary* embedding as viewed from $\mathcal{M}$. This follows from the following key facts:
- $\left( \mathrm{Th}(\mathcal{V}_{\beta _{\alpha _{1}},\alpha
_{1}},s_{\alpha _{1}})\right) ^{\mathcal{M}}=$ $\left( \mathrm{Th}(\mathcal{V%
}_{\beta _{\alpha _{2}},\alpha _{1}},s_{\alpha _{2}})\right) ^{\mathcal{M}}$, whenever $\alpha _{1},\alpha _{2}\in \mathrm{Ord}^{\mathcal{M}}$ with $%
\alpha _{1}<\alpha _{2};$ and
- $\mathcal{M}\models \mathcal{N}_{\alpha }\prec \mathcal{V}_{\beta
_{\alpha }}$ for each $\alpha \in \mathrm{Ord}^{\mathcal{M}}$.
$(iv)$ Hence $\left\langle j_{\alpha _{1},\alpha _{2}}:\alpha
_{1}<\alpha _{2}\in \mathrm{Ord}^{\mathcal{M}}\right\rangle $ is a*directed system of elementary embeddings*. The desired $\mathcal{N}$ is the direct limit of this system. Thus, the elements of $\mathcal{N}$ are equivalence classes $[f]$ of strings $f$ of the form:
$f:\{\alpha \in \mathrm{Ord}^{\mathcal{M}}:\alpha \geq \alpha
_{0}\}\rightarrow \bigcup\limits_{\alpha \in \mathrm{Ord}^{\mathcal{M}%
}}N_{\alpha },$
where $\alpha _{0}\in \mathrm{Ord}^{\mathcal{M}}$ and there is some $\mathcal{L}$-term $\delta $ such that $m_{\delta }\in M_{\alpha _{0}}$ and $f(\alpha )=\delta ^{\mathcal{N}_{\alpha }}\in N_{\alpha }$ (two strings are identified iff they agree on a tail of $\mathrm{Ord}^{\mathcal{M}})$. In particular, for each $\alpha \in \mathrm{Ord}^{\mathcal{M}}$ there is an embedding:
$j_{\alpha ,\infty }:\mathcal{N}_{\alpha }\rightarrow \mathcal{N}$, where $%
j_{\alpha ,\infty }(\delta ^{\mathcal{N}_{\alpha }}):=[h],$ and
$h(\alpha ):=\delta ^{\mathcal{N}_{\alpha }}$ for all $\alpha $ such that $%
m_{\delta }\in M_{\alpha }.$
A routine variant of Tarski’s elementary chains theorem guarantees that $j_{\alpha ,\infty }$ is an *elementary embedding* for all $%
\alpha \in \mathrm{Ord}^{\mathcal{M}}.\medskip $
$(v)$ For $m\in M,$ let $f_{m}(\alpha ):=m=\overline{m}^{^{%
\mathcal{N}_{\alpha }}}$ for all $\alpha \in \mathrm{Ord}^{\mathcal{M}}$ such that $m\in M_{\alpha },$ and consider the embedding
$j:\mathcal{M}\rightarrow \mathcal{N}$, where $j(m):=[f_{m}(\alpha )]$.
By identifying $m$ with $[f_{m}]$ we can, without loss of generality, construe $\mathcal{M}$ as a *submodel of* $\mathcal{N}$.$%
\medskip $
A distinguished element of $\mathcal{N}$ is $[g]$, where $g(\alpha
)=s_{\alpha }$ for $\alpha \in \mathrm{Ord}^{\mathcal{M}}.$ $[g]\neq \lbrack
f_{m}]$ for all $m\in M$ since $s_{\alpha }\notin \mathrm{V}_{\alpha }$ for all $\alpha $ and therefore $g$ and $f_{m}$ differ on a tail of $\alpha \in
\mathrm{Ord}^{\mathcal{M}}.$ This shows that $\mathcal{M}$ is a *proper submodel* of $\mathcal{N}$. To see that $\mathcal{N}$ *end* extends $\mathcal{M}$, suppose $m\in M$ and for some $\mathcal{L}$-definable term $\delta $, $\delta ^{\mathcal{N}_{\alpha }}\in \overline{m}$ holds in $%
\mathcal{N}_{\alpha }$ for sufficiently large $\alpha $, i.e., for any $%
\alpha $ such that $\{m,m_{\delta }\}\subseteq M_{\alpha }.$ Therefore there is some $m_{0}\in \mathrm{V}_{\alpha }^{\mathcal{M}}$ such that $\delta ^{%
\mathcal{N}_{\alpha }}=\overline{m_{0}}$ holds in $\mathcal{N}_{\alpha }$ for sufficiently large $\alpha ,$ and therefore also in $\mathcal{N}$, hence $\mathcal{N}$ end extends $\mathcal{M}$.
Finally, let’s verify that $\mathcal{M}\prec _{\Sigma _{n}}\mathcal{N}$. Suppose $\mathcal{M}\models \varphi (\overline{m})$, where $\varphi $ is $%
\Sigma _{n}$ and $m\in M.$ Then $\varphi (\overline{m})$ holds for all sufficiently large $\mathcal{N}_{\alpha }$, since by design we have:
$\mathcal{N}_{\alpha }\prec \mathcal{M}_{\beta _{\alpha }}\prec _{\Sigma
_{n}}\mathcal{M}$.
This shows that $\mathcal{N}\models \varphi (\overline{m})$ since, as observed earlier, each $\mathcal{N}_{\alpha }$ is elementarily embeddable in $\mathcal{N}$ via $j_{\alpha ,\infty }$.$\square $
We are now ready to verify that the tree property for $\mathrm{Ord}$ fails in the sense of $\mathcal{M}$ for all $\mathcal{M}\models \mathrm{ZFC}$.
**2.6. Theorem. ***Every model* $\mathcal{M}$ *of* $\mathrm{ZFC}$ *carries an* $\mathcal{M}$-*definable* $%
\mathrm{Ord}^{\mathcal{M}}$-*tree no cofinal branch of* *which* *is* $\mathcal{M}$-*definable*.
**Proof.** The proof splits into two cases, depending on whether $\mathcal{M}$ satisfies $\exists p\left( \mathrm{V}=\mathrm{HOD}%
(p)\right) $ or not.[^5]
**Case 1**. Suppose that $\exists p\left( \mathrm{V}=\mathrm{%
HOD}(p)\right) $ fails in $\mathcal{M}$. Within $\mathrm{ZFC}$ we can define the tree $\tau _{\mathrm{Choice}}$ whose nodes at level $\alpha $ are choice functions $f$ for $\mathrm{V}_{\alpha }$, i.e., $f:\mathrm{V}_{\alpha
}\rightarrow \mathrm{V}_{\alpha }$, where $f(x)\in x$ for all nonempty $x\in
\mathrm{V}_{\alpha }$, and the tree ordering is set inclusion. Clearly $%
\mathrm{ZFC}$ can verify that $\tau $ is an $\mathrm{Ord}$-tree. It is also clear that every $\mathcal{M}$-definable branch of $\tau ^{\mathcal{M}}$ (if any) is an $\mathcal{M}$-definable global choice function. By Theorem 1.2 this shows that no branch of $\tau _{\mathrm{Choice}}$ is $\mathcal{M}$-definable.
**Case 2**. Now suppose $\exists p\left( \mathrm{V}=\mathrm{%
HOD}(p)\right) $ holds in $\mathcal{M}$. Then by Theorem 1.2 there is some $%
\Sigma _{2}$-formula $W(x,y)$ that defines a global well-ordering of $%
\mathcal{M}$. Note that $W$ is a global well-ordering" is $%
\Pi _{3}$*-*expressible in $\mathcal{M}$. We claim that for any fixed $n\geq 3$, no branch of $\tau _{n}^{\mathcal{M}}$ is $\mathcal{M}$-definable. If not, then by Lemma 2.4 there is an $\mathcal{M}$-definable structure $\mathcal{N}$, and an $\mathcal{M}$-definable embedding $j:%
\mathcal{M}\rightarrow \mathcal{N}$ such that $\mathcal{N}$ is a proper is a $\Sigma _{n}$-e.e.e. of $j(\mathcal{M)}$, which contradicts Theorem 2.1.$\square $
**3. Consequences of the failure of the definable tree property for the class of ordinals**
In this section we use Theorem 2.6 to establish further results about definable combinatorial properties of proper classes within $\mathrm{%
ZFC}$. Our first result improves Theorem 2.6 by combining its proof with appropriate combinatorial and coding techniques so as to obtain the description of a single subtree of $^{<\mathrm{Ord}}2$ that is -Aronszajn across all models of $\mathrm{ZFC}$; here
$^{<\mathrm{Ord}}2=\bigcup\limits_{\alpha \in \mathrm{Ord}}{}^{\alpha }2,$
where${}\ ^{\alpha }2$ is the set of binary sequences of length $%
\alpha .$ The ordering on $2^{<\mathrm{Ord}}$ is ‘end extension’, denoted $%
\sqsubseteq $. Given a tree $\tau $ we say $\tau $ is a *subtree* of $%
\left( ^{<\mathrm{Ord}}2,\ \sqsubseteq \right) $ if each node of $\tau $ is an element of $^{<\mathrm{Ord}}2$, and the nodes of $\tau $ are ordered by $%
\sqsubseteq $ **3.1. Theorem. ***There is a definable class* $\sigma
$ *that satisfies the following three properties*:
**(a)** $\mathrm{ZFC}\vdash \sigma $ *is a subtree of* $\left( ^{<\mathrm{Ord}}2,\ \sqsubseteq \right) .$
**(b)** $\mathrm{ZFC}\vdash $ $\sigma $ *is an* $%
\mathrm{Ord}$-*tree*.
**(c)** *For all formulae* $\beta (x,y)$ *of set theory,* $\mathrm{ZFC}\vdash $ $\{x:\beta (x,y)\}$ *is not a branch of* $\sigma $ *for any parameter* $y$.
**Proof. **The proof has two stages. In the first stage we construct an $\mathrm{Ord}$-tree that satisfies properties (b) and (c); and then in the second stage we construct an appropriate variant of the tree constructed in the first stage which satisfies properties (a), (b) and (c).
**Stage 1. **Given $\mathrm{Ord}$-trees $\sigma _{1}=(S_{1},<_{1})$ and $\sigma _{2}=(S_{2},<_{2})$, let $\sigma _{1}\otimes \sigma _{2}$ be the tree whose set of nodes is:
$S_{1}\otimes S_{2}:=\{(p,q)\in S_{1}\times S_{2}:h_{1}(p)=h_{2}(q)\},$
where $h_{i}(x)$ is the height (level) of $x$, i.e., the ordinal that measures the order-type of the set of predecessors of $x$ in $\tau _{i}$. The ordering on $\sigma _{1}\otimes \sigma _{2}$ is given by:
$(p,q)\vartriangleleft (p^{\prime },q^{\prime })$ iff $p<_{1}p^{\prime }$ and $q<_{2}q^{\prime }.$
Routine considerations show that the following two assertions are verifiable in $\mathrm{ZFC}$:
$(i)$ $\sigma _{1}\otimes \sigma _{2}$ is an $\mathrm{Ord}$-tree.
$(ii)$ Every branch $B$ of $\sigma _{1}\otimes \sigma _{2}$ is of the form:
$\{(p,q)\in S_{1}\otimes S_{2}:p\in B_{1}$ and $q\in B_{2}\}$,
where $B_{i}$ is the branch of $\tau _{i}$ obtained by projecting $%
B$ on its $i$-th coordinate. In particular, for any model $\mathcal{M}%
\models \mathrm{ZFC}$ we have:
$(iii)$ If $\left( \sigma _{1}\otimes \sigma _{2}\right) ^{%
\mathcal{M}}$ has an $\mathcal{M}$-definable branch, so do $\sigma _{1}^{%
\mathcal{M}}$ and $\sigma _{2}^{\mathcal{M}}.$
Let $\sigma _{0}:=\tau _{\mathrm{Choice}}\otimes \tau _{3};$ where $\tau _{%
\mathrm{Choice}}$ and $\tau _{3}$ are as in the proof of Theorem 2.6. It is easy to see that $\sigma _{0}$ is an -tree (provably in ). The proof of Theorem 2.6, coupled with (*iii*) above shows that no branch of $\sigma _{0}^{\mathcal{M}}$ is $\mathcal{M}$-definable for any $\mathcal{M}\models \mathrm{ZFC}$.
**Stage 2. **The tools of this stage of the construction are Lemmas 3.1.1 and 3.1.2. Recall that the ordering on both trees $\tau _{\mathrm{%
Choice}}$ and $\tau _{3}$ is set-inclusion $\subseteq .$
**Lemma 3.1.1. ***Given* $\mathcal{M}\models \mathrm{%
ZFC}$ *and* $\mathrm{Ord}$*-trees* $\sigma _{1}$ *and* $%
\sigma _{2}$ *in* $\mathcal{M}$ *whose ordering* (*as viewed in* $\mathcal{M})$ *are set-inclusion, there is an* $\mathcal{M%
}$*-definable* $\mathrm{Ord}$*-tree* $\sigma _{1}\oplus \sigma
_{2}$ *whose ordering is also set-inclusion such that* $\sigma
_{1}\otimes \sigma _{2}$* is isomorphic to* $\sigma _{1}\oplus
\sigma _{2}$ *via an* $\mathcal{M}$-*definable isomorphism.*
**Proof.** Let $S_{i}$ be the collection of nodes of $\sigma
_{i},$ and consider the tree $\sigma _{1}\oplus \sigma _{2}$ whose sets of nodes, $S_{1}\oplus S_{2}$, is defined as:
$\left\{ \left( p\times \{0\}\right) \cup \left( q\times \{1\}\right)
:\left( p,q\right) \in S_{1}\otimes S_{2}\right\} ,$
and whose ordering is set inclusion. It is easy to see the desired isomorphism between $\sigma _{1}\otimes \sigma _{2}$ and $\sigma _{1}\oplus
\sigma _{2}$ is described by:
$(p,q)\mapsto \left( p\times \{0\}\right) \cup \left( q\times \{1\}\right) .$
$\square $ (Lemma 3.1.1)
**Lemma 3.1.2. ***Given* $\mathcal{M}\models \mathrm{%
ZFC}$ *and any* $\mathrm{Ord}$*-tree* $\tau $ *in* $%
\mathcal{M}$ *whose ordering* (*as viewed in* $\mathcal{M})$ *is set-inclusion, there is an* $\mathrm{Ord}$*-tree* $%
\widetilde{\tau }$ *of* $\mathcal{M}$ *satisfying the following properties*:
**(a)** $\widetilde{\tau }=(\widetilde{T},\sqsubseteq ),$* for some* $\widetilde{T}\subseteq \ ^{<\mathrm{Ord}}2$ .
**(b)** *If* $\widetilde{\tau }$ *has an* $%
\mathcal{M}$-*definable branch*, *then* $\tau $ *has an* $\mathcal{M}$*-definable branch*.$\medskip $
**Proof.** We will first describe a $\mathrm{ZFC}$-construction that should be understood to be carried out within $\mathcal{M}
$. Given a set $s$, let $\overline{s}$ be the transitive closure of $\left\{
s\right\} $, and let $\kappa _{s}:=\left\vert \overline{s}\right\vert .$ It is well-known that given a bijection $g:\overline{s}\rightarrow \left\vert
\overline{s}\right\vert $, $s$ can be canonically coded by some binary sequence $v_{g}(s)\in \ ^{\left\vert \overline{s}\right\vert }2$. More specifically, the $\in $ relation on $\overline{s}$ can be readily copied over $\kappa _{s}$ with the help of $g$ so as to obtain a binary relation $%
R_{g}(s)$ such that $\left( \overline{s},\in \right) \cong \left( \kappa
_{s},R_{g}(s)\right) .$ Since, $R_{g}(s)$ is an extensional well-founded relation, $s$ can thus be recovered from $R_{g}(s)$ as the top element of the transitive collapse of $R_{g}(s)$$.$ On the other hand, $R_{g}(s)$ can be coded-up as $X_{g}(s)\subseteq \kappa
_{s}$ with the help of a canonical pairing function $p:\mathrm{Ord}%
^{2}\rightarrow \mathrm{Ord}$. Thus, if $v_{g}(s):$ $\kappa _{s}\rightarrow
\{0,1\}$ is defined as the characteristic function of $X_{g}(s)$, then $%
s=F(v_{g}(s))$, where $F(x)$ is the parameter-free definable class function given by:
If $x\in \ ^{<\mathrm{Ord}}2$, and $\overset{x^{\circ }}{\overbrace{%
\{p^{-1}(t):x(t)=1\}}}$ is well-founded, extensional, and has a top element, then $F(x)$ is the top element of the transitive collapse of $x^{\circ };$ otherwise $F(x)=0.\medskip $
Given an $\mathrm{Ord}$-tree $\tau =(T,\subseteq )$, let $%
T_{\alpha }$ be the set of elements of $T$ of height $\alpha \in \mathrm{Ord}%
,$ and for $s\in T_{\alpha }$, and $\beta \leq \alpha ,$ let $s_{\beta }$ be the unique element in $T_{\beta }$ that is a subset of $s.$ Let
$h_{g}(s):=\bigoplus\limits_{\beta \leq \alpha }v_{g}(s_{\beta }),$
where $g:\overline{s}\rightarrow \left\vert \overline{s}%
\right\vert $ is a bijection and the operation $\oplus $ is defined as follows: given a transfinite sequence $\left\langle m_{\beta }:\beta \leq
\alpha \right\rangle $ of binary sequences, $\bigoplus\limits_{\beta \leq
\alpha }m_{\beta }$ is the *ternary* sequence obtained by concatenating the sequence of sequences $\left\langle m_{\beta }\ast
\left\langle 2\right\rangle :\beta \leq \alpha \right\rangle $, where $%
m_{\beta }\ast \left\langle 2\right\rangle $ is the concatenation of the sequence $m_{\beta }$ and the sequence $\left\langle 2\right\rangle .$ Thus the ‘maximal binary blocks’ of $\bigoplus\limits_{\beta \leq \alpha
}m_{\beta }$ are precisely sequences of the form $m_{\beta }$ for some $%
\beta \leq \alpha $. This makes it clear that $s$ can be readily ‘read off’ $%
h_{g}(s)$ as the result of applying $F$ to last binary block of $h_{g}(s)$.
Let $\widetilde{T}_{0}:=\{h_{g}(s):s\in T$, and $g$ is a bijection between $%
\overline{s}$ and $\left\vert \overline{s}\right\vert \}.$ We are now ready to define the desired $\widetilde{T}.$ Fix a canonical embedding $G$ of $^{<%
\mathrm{Ord}}3$ into $^{<\mathrm{Ord}}2$, and let:
$\widetilde{T}:=\{G(v):v\in \widetilde{T}_{0}\}.$
It is easy to see, using the assumption that $(T,\subseteq )$ is an $\mathrm{Ord}$-tree, that $\widetilde{\tau }:=(\widetilde{T},\sqsubseteq
) $ is an $\mathrm{Ord}$-tree. Since $\widetilde{T}\subseteq \ ^{<\mathrm{Ord%
}}2$, it remains to show that if $\widetilde{\tau }$ has an $\mathcal{M}$-definable branch, then $\tau $ also has an $\mathcal{M}$-definable branch. Suppose $\widetilde{B}=\{\widetilde{b}_{\alpha }:\alpha \in \mathrm{Ord}^{%
\mathcal{M}}\}$ is a branch of $\widetilde{\tau }.$ Let
$\widetilde{B}_{0}:=\{G^{-1}(\widetilde{b}_{\alpha }):\alpha \in \mathrm{Ord}%
^{\mathcal{M}}\}$
Note that $\widetilde{B}_{0}$ is a cofinal branch of the tree $%
\widetilde{\tau }_{0}$; and the maximal binary blocks of $\widetilde{B}_{0}$ form a proper class, and are linearly ordered by set-inclusion (in the sense of $\mathcal{M}$) by design. Let $B$ be the collection of elements $b\in T$ that are of the form $F(m)$, where $m$ is the last binary block of $G^{-1}(%
\widetilde{b}_{\alpha }).$ Then $B$ is a cofinal branch of $\tau $ and is definable from $\widetilde{B}.$$\square $ (Lemma 3.1.2)
Let $\delta :=\left( \tau _{\mathrm{Choice}}\oplus \tau _{3}\right) $, and $%
\tau :=\widetilde{\delta }$. Theorem 2.6 together with Lemmas 3.1.1 and 3.1.2 make it clear that in every model $\mathcal{M}$ of $\mathrm{ZFC}$, $%
\tau ^{\mathcal{M}}$ is a definably $\mathrm{Ord}$-Aronszajn subtree of $%
\left( ^{<\mathrm{Ord}}2\right) ^{\mathcal{M}}$; so by the completeness theorem of first order logic, the proof is complete. $\square $ (Theorem 3.1)$\medskip $
Theorem 3.1 has the following immediate consequence for spartan models of $%
\mathrm{GB+AC}$, where $\mathrm{AC}$ is the axiom of choice for sets:
**3.2. Corollary. ***There is a definable class* $%
\sigma $ *in the language of class theory satisfying the following properties:*
**(a)** $\mathrm{GB+AC}\vdash \sigma $ *is a subtree of* $^{^{<\mathrm{Ord}}}2$ *and* $\sigma $ *is a proper class.*
**(b)** *The statement* $\sigma $ *is an* $\mathrm{Ord}$-*Aronszajn treeholds in every spartan model of* $\mathrm{GB+AC}$.
**3.3. Remark.** It is known [@Ali; @NFUA Corollary 2.2.1] that the set-theoretical consequences of $\mathrm{GB+AC}$ + $\mathrm{Ord}$ has the tree property is precisely $\mathrm{ZFC}+\Phi $, where $\Phi $ is the scheme whose instances are of the form there is an $n$-Mahlo cardinal $\kappa $ such that $\kappa $ is $n$-correct, and $n$ ranges over meta-theoretic natural numbers. Also note that one can derive global choice from local choice in $\mathrm{GB+AC}$ + $\mathrm{Ord}$ is weakly compact (using $\tau _{\mathrm{Choice}}$ of the proof of Theorem 2.6). Moreover, by an unpublished result of the first-named-author, there are (non $\omega $-) models $(\mathcal{M},\mathcal{%
S})$ of $\mathrm{GB+AC}$ + $\mathrm{Ord}$ has the tree property in which the partition property $\mathrm{Ord}%
\rightarrow \left( \mathrm{Ord}\right) _{2}^{k}$ fails for some nonstandard $%
k\in \omega ^{\mathcal{M}}$, which implies that for models of $\mathrm{GB+AC}
$, the condition $\forall k\in \omega $ $\mathrm{Ord}%
\rightarrow \left( \mathrm{Ord}\right) _{2}^{k}$ is strictly stronger than $\mathrm{Ord}$ has the tree property.[^6] But of course in the Kelley-Morse theory of classes these two statements are equivalent.
**3.4. Theorem.** *The definable proper class partition property fails in every model of* $\mathit{\mathrm{ZFC}}$.*That is, there is a definable* $2$-*coloring of pairs of sets having no definable monochromatic proper class.* $\medskip $
**Proof**. Let $\tau =(T,\sqsubseteq )$ be as in Theorem 3.1 and $\mathcal{M}\models \mathrm{ZFC}$. We argue in $\mathcal{M}$. For $p,q$ in $T$, we will say that $p$ is *to the right* of $q$, written $%
p\vartriangleright q$, if $p>_{T}q$, or at the point of first difference, the bit of $p$ is larger than $q$ at that coordinate. Also, as in the proof of Theorem 3.1, we use $h(p)$ for the height of $p$ in $\tau $. Define a coloring $f:[T]^{2}\rightarrow \{0,1\}$ by:
$f(\{p,q\})=\begin{cases}
0,\ \mathrm{if\ }h(p)>h(q),\ \mathrm{and}\ p\vartriangleright q\mathrm{;}\\
1,\ \mathrm{otherwise}.\\
\end{cases}$
Suppose that $H$ is a definable proper subclass of $T$ that is $f$-monochromatic. Next, color pairs from $H$ with color blue if they are of the same height, and red otherwise. Since the collection of elements of $%
\tau $ of a given height are sets, there cannot be a proper subclass colored blue, and so we can find a subclass of $H$ with all elements on different levels. So without loss of generality, all elements on $H$ are on different levels. If the monochromatic value of pairs from $H$ is $0$, then as one goes up the tree, the nodes in $H$ are always to the right. Let $B$ consist of the nodes in $\tau $ that are eventually below the nodes of $H$, that is, $p\in B$ just in case there is some ordinal $\alpha $ such that all nodes in $H$ above $\alpha $ are above $p$. It is clear that $B$ is downward closed. We claim that $B$ is a branch through $\tau $. $B$ is linearly ordered, since there can be no first point of nonlinearity: if eventually the nodes of $H$ are above $p\ast 1$, then they cannot be eventually above $p\ast 0$ (where $\ast $ is the concatenation operation on sequences). Finally, $B$ is closed under limits, since if $p$ has length $\delta $ and $p|\alpha $ is in $B$ for all $\alpha <\delta $, then take the supremum of the levels witnessing that, so you find a single level such that all nodes in $H$ above that level are above every $p|\alpha $, and so they are above $p$. Thus, $B$ is a branch through $\tau $. But $\tau $ has no definable branches, and so there cannot be such a monochromatic set $H$. Finally, if the monochromatic value of $H$ is $1$, then as one goes up, the nodes go to the left, and a similar argument works. $\square \medskip $
**3.5. Corollary.** $\mathrm{Ord}\rightarrow \left( \mathrm{%
Ord}\right) _{2}^{2}$ *fails for definable classes in* *every model of* $\mathrm{ZFC}+\exists p\left( \mathrm{V}=\mathrm{HOD}(p)\right) .$ *Indeed,* $\mathrm{Ord}\rightarrow \left( \mathrm{Ord}\right)
_{2}^{2} $ *fails for definable classes in* *every model of* $%
\mathrm{ZFC}$ *in which there is a definable well-ordering of* $^{^{<%
\mathrm{Ord}}}2.\footnote{%
The existence of a global definable well-ordering of $^{^{<\mathrm{Ord}}}2$
is equivalent over $\mathrm{ZF}$ to the so-called Leibniz-Mycielski
principle (LM), explored in \cite{Ali LM}, which includes a result of
Solovay that shows that if $\mathrm{ZF}$ is consistent, then there is a
model of $\mathrm{ZF}+\mathrm{LM}$\ in which \textrm{AC} fails (such a
model, a fortiori, does not carry a parametrically definable global
well-ordering). The conjecture that there is a model of $\mathrm{ZFC}+%
\mathrm{LM}$\ which does not carry a parametrically definable global
well-ordering remains open.}\medskip $
**3.6. Remark.** We do not know whether $\mathrm{Ord}%
\rightarrow \left( \mathrm{Ord}\right) _{2}^{2}$ fails for definable classes in every model of $\mathrm{ZFC.}$ Some of the usual proofs of the infinite Ramsey theorem use König’s lemma, which is exactly what is going wrong with our definably $\mathrm{Ord}$-Aronszajn tree; this suggest that perhaps there is a definable coloring of pairs of ordinals for which there is no definable monochromatic proper class of ordinals.$\medskip $
**3.7. Theorem.** *The definable compactness property fails for* $\mathcal{L}_{\mathrm{\infty },\mathrm{\omega }}$ *in every model* $\mathcal{M}$* of* $\mathrm{ZFC}$.$\medskip $
**Proof.** Fix a definable $\mathrm{Ord}$-Aronszajn tree $%
\tau =(T,<_{T})$ of $\mathcal{M}$, and let $\mathcal{L}$ be the language having a constant $\overline{p}$ for every element $p\in T$ and a binary relation $<$ for the order of $\tau $, together with a new constant $c$. Let $\Gamma $ be the theory in $\mathcal{M}$ consisting of the atomic diagram of $\tau $, together with the assertion that $<$ is a tree order and the assertions of the form:
$\varphi _{\alpha }:=\bigvee\limits_{p\in T_{\alpha }}(\overline{p}<c)$.
That is, $\varphi _{\alpha }$ asserts that the new constant $b$ lies above one of the elements on the $\alpha $-th level $T_{\alpha \text{ }%
} $of $\tau $. In $\mathrm{ZFC}$, having ‘size $\mathrm{Ord}$’ is a stronger property than ‘proper class’, if global choice fails. Nevertheless, we can organize $\Gamma $ into an equivalent theory of size $\mathrm{Ord}$ as follows. Instead of taking the whole atomic diagram as separate statements, which may not be well-orderable, since we can’t seem to well-order the nodes of $\tau $, we instead for each ordinal $\alpha $ let $\sigma _{\alpha }$ be the conjunction of the set of atomic assertions that hold in the tree up to level $\alpha $. Recall that the logic $\mathcal{L}_{\mathrm{Ord,\ \omega }}$ allows the formation of conjunctions of any set of assertions, without needing to put them into any order. Hence $\Gamma $ is defined in $\mathcal{M%
}$ as $\left\{ \sigma _{\alpha }\wedge \varphi _{\alpha }:\alpha \in \mathrm{%
Ord}\right\} $ plus the sentence that expresses that $<$ is a tree order$.$
Every set-sized subtheory of $\Gamma $ mentions only bounded many sentences of the form $\sigma _{\alpha }\wedge \varphi _{\alpha },$ so we can find a model in $\mathcal{M}$ of the subtheory by interpreting $c$ as any element of the tree $\tau $ on a sufficiently high level. But if there is an $%
\mathcal{M}$-definable model of $\Gamma $, then from that model we can extract the predecessors of the interpretation of the element $c$, and this will give an $\mathcal{M}$-definable branch through $\tau $, contradicting that $\tau $ is definably $\mathrm{Ord}$-Aronszajn in $\mathcal{M}$. $%
\square \medskip $
We close this section with a conjecture. In what follows $\mathcal{D}_{%
\mathcal{M}}$ is the collection of $\mathcal{M}$-definable subsets of $M$, and $\tau $ is a definably $\mathrm{Ord}$-Suslin tree in $%
\mathcal{M}$ means that $(\mathcal{M},\mathcal{D}_{%
\mathcal{M}})$ satisfies $\tau $ is an* *$\mathrm{%
Ord}$-Aronszajn tree and every anti-chain of $\tau $ has cardinality less than $\mathrm{Ord}$$.$
**3.8. Conjecture. ***Suppose*** **$\mathcal{M}$ *is a model of* $\mathrm{ZFC}+\mathrm{V}=\mathrm{L}$. *Then there is some* $\tau _{S}\in $ $\mathcal{D}_{\mathcal{M}}$ *such that* $\tau _{S}$ *is a definably* $\mathrm{Ord}$*-Suslin tree in* $%
\mathcal{M}$.* *
Let us motivate the above conjecture. By a theorem of Jensen [@Devlin Theorem VII.1.3] , if $\mathrm{V}=\mathrm{L}$ holds, then every cardinal $\kappa $ that is not weakly compact carries a $\kappa $-Suslin tree. The relevant case for us of Jensen’s proof is when $\kappa $ is a strongly inaccessible cardinal. Jensen’s proof takes advantage of (1) the existence of a $\kappa $-Aronszajn tree, and (2) the combinatorial principle for some stationary subset set $E$ of $\kappa $, $\square
_{\kappa }(E)$ holds. We know, by Theorem 2.6, that the definable version of (1) can be arranged for $\mathrm{Ord}$. On the other hand, by adapting Jensen’s proof to the definable context, the analogue of (2) might also be true (using the $\mathrm{V}=\mathrm{L}$ assumption) in $(%
\mathcal{M},\mathcal{D}_{\mathcal{M}}).$ The result in the next section suggests that perhaps the definable version of (2) holds with the assumption $\mathrm{V}=\mathrm{L}$ weakened to $\exists p\left( \mathrm{V}=\mathrm{HOD}%
(p)\right) .$ This motivates a stronger form of Conjecture 3.8 in which the assumption that $\mathrm{V}=\mathrm{L}$ holds in $\mathcal{M}$ is weakened to the $\mathcal{M}$-definability of a global well-ordering of the universe.
**4. The definable version of** $\Diamond _{\mathrm{Ord}}$ **and global definable well-orderings**
In this section we show that the definable version of $\Diamond _{\mathrm{Ord%
}}$ holds in a model $\mathcal{M}$ of $\mathrm{ZFC}$ iff $\mathcal{M}$ carries a definable well-ordering of the universe. In light of Theorem 1.2 it follows as a consequence that the definable $\Diamond _{\mathrm{Ord}}$, although seeming to be fundamentally scheme-theoretic, is actually expressible in the first-order language of set theory as $\exists p\left(
\mathrm{V}=\mathrm{HOD}(p)\right) $.
In set theory, the diamond principle asserts the existence of a sequence of objects, of growing size, such that any large object at the end is very often anticipated by these approximations. In the case of diamond on the ordinals, what we will have is a definable sequence of $A_{\alpha }\subseteq
\alpha $, such that for any definable class of ordinals $A$ and any definable class club set $C$, there are ordinals $\theta \in C$ with $A\cap
\theta =A_{\theta }$. This kind of principle typically allows one to undertake long constructions that will diagonalize against all the large objects, by considering and reacting to their approximations $A_{\alpha }$. Since every large object $A$ is often correctly approximated that way, this enables many such constructions to succeed.
**4.1. Theorem. ***For any model* $\mathcal{M}$ *of* $\mathrm{ZFC}$,* if there is an* $\mathcal{M}$*-definable well-ordering of the universe, then the definable* $\Diamond _{%
\mathrm{Ord}}$ *holds in* $\mathcal{M}$*.*
**Proof.** We argue in $\mathcal{M}$ to establish the theorem as a theorem scheme; namely, we shall provide a specific definition within $%
\mathcal{M}$ for the sequence $\vec{A}=\left\langle A_{\alpha }:\theta <%
\mathrm{Ord}\right\rangle $, using the same parameter $p$ as the definition of the global well-order and with a definition of closely related syntactic complexity, and then prove as a scheme, a separate statement for each $%
\mathcal{M}$-definable class $A\subseteq \mathrm{Ord}$ and class club $%
C\subseteq \mathrm{Ord}$, that there is some $\theta \in C$ with $A\cap
\theta =A_{\alpha }.$ The definitions of the classes $A$ and $C$ may involve parameters and have arbitrary complexity.
Let $\vartriangleleft $ be the definable well-ordering of the universe, definable by a specific formula using some parameter $p$. We define the $%
\Diamond _{\mathrm{Ord}}$-sequence $\vec{A}=\left\langle A_{\alpha }:\theta <%
\mathrm{Ord}\right\rangle $ by transfinite recursion. Suppose that $\vec{A}%
\upharpoonright \theta $ has been defined. We shall let $A_{\theta
}=\varnothing $ unless $\theta $ is a $\beth $-fixed point above the rank of $p$ and there is a set $A\subseteq \theta $ and a closed unbounded set $%
C\subseteq \theta $, with both $A$ and $C$ definable in the structure $%
\left( \mathrm{V}_{\theta },\in \right) $ (allowing parameters), such that $%
A\cap \theta \neq A_{\alpha }$ for every $\alpha \in C$. In this case, we choose the least such pair $(A,C)$, minimizing first on the maximum of the logical complexities of the definitions of $A$ and of $C$, and then minimizing on the total length of the defining formulas of $A$ and $C$, and then minimizing on the Gödel codes of those formulas, and finally on the parameters used in the definitions, using the well-order $\lhd \
\upharpoonright \mathrm{V}_{\theta }$. For this minimal pair, let $A_{\theta
}=A$. This completes the definition of the sequence $\vec{A}=\left\langle
A_{\alpha }:\theta <\mathrm{Ord}\right\rangle $.
Let us remark on a subtle point, since the meta-mathematical issues loom large here. The definition of $\vec{A}$ is internal to the model $\mathcal{M}
$, and at stage $\theta $ we ask about subsets of $\theta $ definable in $%
\left( \mathrm{V}_{\theta },\in \right) $, using the truth predicate for this structure. If we were to run this definition inside an $\omega $-nonstandard model $\mathcal{M}$, it could happen that the minimal formula we get is nonstandard, and in this case, the set $A$ would not actually be definable by a standard formula. Also, even when $A$ is definable by a standard formula, it might be paired (with some constants), with a club set $%
C$ that is defined only by a nonstandard formula (and this is why we minimize on the maximum of the complexities of the definitions of $A$ and $C$ together). So one must give care in the main argument keeping straight the distinction between the meta-theoretic natural numbers and the internal natural numbers of the object theory $\mathrm{ZFC}$.
Let us now prove that the sequence $\vec{A}$ is indeed a $\Diamond _{\mathrm{%
Ord}}$-sequence for $\mathcal{M}$-definable classes. The argument follows in spirit the classical proof of $\Diamond $ in the constructible universe $%
\mathrm{L}$, subject to the metamathematical issues we mentioned. If the sequence $\vec{A}$ does not witness the veracity of the definable $\Diamond
_{\mathrm{Ord}}$ in $\mathcal{M}$, then there is some $\mathcal{M}$-definable class $A\subseteq \mathrm{Ord}$, defined in $\mathcal{M}$ by a specific formula $\varphi $ and parameter $z$, and definable club $%
C\subseteq \mathrm{Ord}$, defined by some $\psi $ and parameter $y$, with $%
A\cap \alpha \neq A_{\alpha }$ for every $\alpha \in C$. We may assume without loss of generality that these formulas are chosen so as to be minimal in the sense of the construction, so that the maximum of the complexities of $\varphi $ and $\psi $ are as small as possible, and the lengths of the formulas, and the Gödel codes and finally the parameters $%
z,y$ are $\vartriangleleft $-minimal, respectively, successively. Let $m$ be a sufficiently large natural number, larger than the complexity of the definitions of $\vartriangleleft ,$ $A$, $C$, and large enough so that the minimality condition we just discussed is expressible by a $\Sigma _{m}$ formula. Let $\theta $ be any $\Sigma _{m}$-correct ordinal above the ranks of the parameters used in the definitions. It follows that the restrictions $%
\vartriangleleft \ \upharpoonright \mathrm{V}_{\theta }$ and also $A\cap
\theta $ and $C\cap \theta $ are definable in $\left( \mathrm{V}_{\theta
},\in \right) $ by the same definitions and parameters as their counterparts in $\mathrm{V}$, that $C\cap \theta $ is club in $\theta $, and $A\cap
\theta $ and $C\cap \theta $ form a form a minimal pair using those definitions $A\cap \alpha \neq \alpha $ for any $\alpha \in C\cap \theta .$ Thus, by the definition of $\vec{A}$, it follows that $A_{\theta }=A\cap
\theta .$ Since $C\cap \theta $ is unbounded in $\theta $ and $C$ is closed, it follows that $\theta \in C$ , and so $A_{\theta }=A\cap \theta $ contradicts our assumption about $A$ and $C$. So there are no such counterexample classes, and thus $\vec{A}$ is a $\Diamond _{\mathrm{Ord}}$-sequence with respect to $\mathcal{M}$-definable classes, as claimed.$\square $
**4.2. Theorem. ***The following are equivalent for* $%
\mathcal{M}\models \mathrm{ZFC}$.
$\mathbf{(a)}$ $\mathcal{M}$ *carries an* $\mathcal{M}$*-definable global well-ordering.*
$\mathbf{(b)}$ $\exists p\left( \mathrm{V}=\mathrm{HOD}(p)\right) $ *holds in* $\mathcal{M}$.
$\mathbf{(c)}$ *The definable* $\Diamond _{\mathrm{Ord}}$ *holds in* $\mathcal{M}$.
**Proof. **We will first give the argument, and then in Remark 4.3 discuss some issues about the formalization, which involves some subtle issues.
$\mathbf{(a)}\Rightarrow \mathbf{(b)}.$ Suppose that $%
\vartriangleleft $ is a global well-ordering that is definable in $\mathcal{M%
}$ from a parameter $p$. In particular in $\mathcal{M}$ every set has a $%
\vartriangleleft $-minimal element. Let us refine this order by defining $%
x\vartriangleleft ^{\prime }y$, just in case $\rho (x)<\rho (y)$ or $\rho
(x)=\rho (y)$ and $x\vartriangleleft y$ (where $\rho $ is the usual ordinal-valued rank function). The new order is also a well-order, which now respects rank. In particular, the order $\vartriangleleft ^{\prime }$ is set-like, and so every object $x$ is the $\theta $-th element with respect to the $\vartriangleleft ^{\prime }$-order, for some ordinal $\theta $. Thus, every object is definable in $\mathcal{M}$ from $p$ and an ordinal, and so $\mathrm{V}=\mathrm{HOD}(p)$ holds in $\mathcal{M}$, as desired.
$\mathbf{(b)}\Rightarrow \mathbf{(a)}.$ If $\mathcal{M}$ satisfies $\exists p\ \mathrm{V}=\mathrm{HOD(}p\mathrm{)}$, then we have the canonical well-order of $\mathrm{HOD}$ using parameter $p$, similar to how one shows that the axiom of choice holds in $\mathrm{HOD}$. Namely, define $%
x\vartriangleleft y$ if and only if $\rho (x)<\rho (y)$, or the ranks are the same, but $x$ is definable from $p$ and ordinal parameters in some $%
\mathrm{V}_{\theta }$ with a smaller $\theta $ than $y$ is, or the ranks are the same and the $\theta $ is the same, but $x$ is definable in that $%
\mathrm{V}_{\theta }$ by a formula with a smaller Gödel code, or with the same formula but smaller ordinal parameters. It is easy to see that this is an $\mathcal{M}$-definable well-ordering of the universe.
$\mathbf{(a)}\Rightarrow \mathbf{(c)}.$ This is the content of the Theorem 4.1.
$\mathbf{(c)}\Rightarrow \mathbf{(a)}.$ If $\vec{A}$ is an $%
\mathcal{M}$-definable $\Diamond _{\mathrm{Ord}}$-sequence for $\mathcal{M}$-definable classes, then it is easy to see that if $A$ is a set of ordinals in the sense of $\mathcal{M}$, then $A$ must arise as $A_{\theta }$ for unboundedly many $\theta \in \mathrm{Ord}^{\mathcal{M}}$. As recalled in the proof of Lemma 3.1.2, in $\mathrm{ZFC}$ every set is coded by a set of ordinals. So let us define that $x\vartriangleleft y$, just in case $x$ is coded by a set of ordinals that appears earlier on $\vec{A}$ than any set of ordinals coding $y$. This is clearly a well-ordering, since the map sending $%
x$ to the ordinal $\theta $ for which codes $x$ is an $\mathrm{Ord}$-ranking of $\vartriangleleft $. So there is an $\mathcal{M}$-definable well-ordering of the universe.$\square $
**4.3. Remark. **An observant reader will notice some meta-mathematical issues concerning Theorem 4.2. The issue is that statements (a) and (b) are known to be expressible by statements in the first-order language of set theory, as single statements, but for statement (c) we have previously expressed it only as a scheme of first-order statements. So how can they be equivalent? The answer is that the full scheme-theoretic content of statement (3) follows already from instances in which the complexity of the definitions of $A$ and $C$ are bounded. Basically, once one gets the global well-order, then one can construct a $%
\Diamond _{\mathrm{Ord}}$-sequence that works for all definable classes. In this sense, we may regard the diamond principle $\Diamond _{\mathrm{Ord}}$ for definable classes as not really a scheme of statements, but rather equivalent to a single first-order assertion.
Lastly, let us consider the content of Theorem 4.2 in Gödel-Bernays set theory or Kelley-Morse set theory. Of course, we know that there can be models of these theories that do not have $\Diamond _{\mathrm{Ord}}$ in the full second-order sense. For example, it is relatively consistent with $%
\mathrm{ZFC}$ that an inaccessible cardinal $\kappa $ does not have $%
\Diamond _{\kappa }$, and in this case, the structure $\left( \mathrm{V}%
_{\kappa +1},\mathrm{V}_{\kappa },\in \right) $ will satisfy $\mathrm{GBC}$ and even $\mathrm{KMC}$, but it will not satisfy $\Diamond _{\mathrm{Ord}}$ with respect to all classes, even though it has a well-ordering of the universe (since there is such a well-ordering in $\mathrm{V}_{\kappa +1}$). But meanwhile, there will be a $\Diamond _{\mathrm{Ord}}$-sequence that works with respect to classes that are definable from that well-ordering and parameters, simply by following the construction given in Theorem 4.2.
**4.4.** A minor adaptation of the proof of Theorem 4.1 shows that if $\mathcal{M}$ is a model of $\mathrm{ZFC}$ that carries an $%
\mathcal{M}$-definable global well-ordering, then the definable version of $%
\Diamond _{\mathrm{Ord}}(E)$ holds in* *$\mathcal{M}$ for any definably $\mathcal{M}$-stationary $E\subseteq \mathrm{Ord}^{\mathcal{M}}$: use the same argument, but only define $A_{\alpha }$ for $\alpha \in E;$ and in the reflection step of the argument use $\theta \in E\cap C.$ Theorem 4.2 can be also accordingly strengthened.
**5. The theory of spartan models of GB**
Recall from Section 1 that $\mathrm{GB}_{\mathrm{spa}}$ is the collection of all sentences that hold in all spartan models of $\mathrm{GB}$. As mentioned earlier, each theorem scheme of Sections 2 through 4 can be readily reformulated as demonstrating that a certain sentence belongs to $\mathrm{GB}%
_{\mathrm{spa}}$. Note that the purely set-theoretical consequences of $%
\mathrm{GB}_{\mathrm{spa}}$ coincides with the deductive closure of $\mathrm{%
ZF}$; this is an immediate consequence of coupling the completeness theorem for first order logic with the fact that $(\mathcal{M},\mathcal{D}_{\mathcal{%
M}})$ is a model of $\mathrm{GB}$ whenever $\mathcal{M}$ is a model of $%
\mathrm{ZF}$. A natural question is whether $\mathrm{GB}_{\mathrm{spa}}$ is computably axiomatizable. The following result provides a strong negative answer to this question.
**5.1. Theorem. **$\mathrm{GB}_{\mathrm{spa}}$ *is* $%
\Pi _{1}^{1}$-*complete.*
**Proof. **We need to use both the meta-theoretic natural numbers, which we will denote by $\omega $, and the object-theoretic natural numbers, which we denote by $\mathbb{N}$. It is not hard to see that $%
\mathrm{GB}_{\mathrm{spa}}$ has* *a* *$\Pi _{1}^{1}$-description. To see this, consider the following predicates, where $%
r,s\subseteq \omega :\medskip $
\(1) $\mathrm{Sat}_{\mathrm{ZF}}(r)$ expresses the structure canonically coded by $r$ is a model of $\mathrm{ZF}$.
\(2) $s=\mathrm{Def}(r)$ expresses $\mathrm{Sat}_{%
\mathrm{ZF}}(r)$ and $s$ codes the collection of $r$-definable subsets of the domain of discourse of the structure (coded by) $r$.
\(3) $\mathrm{Sat}((s,r),\varphi )$ expresses $s=%
\mathrm{Def}(r),$ $\varphi $ is a sentence of $\mathcal{L}_{\mathrm{GB}}$, and the $\mathrm{GB}$-model coded by $(r,s)$ satisfies $\varphi $$.\medskip $
Usual arguments show that each of the above three predicates is $%
\Delta _{1}^{1}$ in the Baire space. In light of the fact that $\Delta
_{1}^{1}$-predicates are closed under Boolean operations, this makes it clear that $\mathrm{GB}_{\mathrm{spa}}$ is $\Pi _{1}^{1}$, since by the Löwenheim-Skolem theorem, we have:
$\varphi \in \mathrm{GB}_{\mathrm{spa}}$ iff $\forall r\subseteq \omega \
\forall s\subseteq \omega \ \left( \left( \mathrm{Sat}_{\mathrm{ZF}%
}(r)\wedge s=\mathrm{Def}(r)\right) \rightarrow \mathrm{Sat}((s,r),\varphi
)\right) $
We next show that $\mathrm{GB}_{\mathrm{spa}}$ is $\Pi _{1}^{1}$-complete. The revelatory idea here is that within $\mathrm{GB}$ one can define – via an existential quantification over classes – a nonempty ‘cut’ $\mathrm{I}$ of ambient natural numbers $\mathbb{N}$ (i.e., a nonempty initial segment $%
\mathrm{I}$ of $\mathbb{N}$ that contains 0 and is closed under successors) such that:
$(\ast )$ If $(\mathcal{M},D_{\mathcal{M}})$ is a spartan model of , then $\mathrm{I}^{(\mathcal{M},D_{\mathcal{M}})}\cong \omega $; i.e., $\mathrm{I}^{(\mathcal{M},D_{\mathcal{M}})}$ has no nonstandard elements.
The cut $\mathrm{I}$ has a simple definition within $\mathrm{GB}$. In the definition below $F_{n}$ is the collection of set theoretical formulae of complexity at most $n,$ where ‘complexity’ can be taken as the number of occurrences of logical symbols (i.e. the Boolean connectives and the quantifiers)[^7]
$\mathrm{I}:=\{n\in \mathbb{N}:$ there is a proper class $C$ such that $C$ is the satisfaction-predicate for $F_{n}\}$,
*The relevant insight is that in spartan models of* $%
\mathrm{GB}$*, the only members of* $\mathrm{I}$* are thestandard natural numbers* $\mathbb{\omega }$*, thanks to Tarski’s undefinability of truth theorem,* which explains the veracity of $(\ast )$.
Using $(\ast )$, and the fact that every real can be included in the standard system of a model of $\mathrm{ZF}$, we will show that every $\Pi
_{1}^{1}$-subset of $\omega $ is many-one reducible to $\mathrm{GB}_{\mathrm{%
spa}}.$ Suppose $P$ is a $\Pi _{1}^{1}$-subset of $\omega $, and let $\omega
^{\omega }$ be the Baire space. Then by Kleene normal form for $\Pi _{1}^{1}$-sets [@Hartley], there is some recursive predicate $R(x,y)$ such that:
$\forall n\left( n\in P\leftrightarrow \forall F\in \omega ^{\omega }\
\exists m\in \omega \ R(F\upharpoonright m,n)\right) ,$
where $F\upharpoonright m$ is the canonical code for the finite set of ordered pairs of the form $\left\langle i,F(i)\right\rangle $ with $%
i<m$. Let $\mathsf{R}$ be the formula that numeralwise represents $R$ in $%
\mathrm{GB}$, and given $n\in \omega $, consider the sentence $\varphi _{n}$ in the language of $\mathrm{GB}$ that expresses:
$\forall s\left( s\in \mathbb{N}\backslash \mathrm{I}\rightarrow \exists
m\in I\ \mathsf{R}^{I}(\mathrm{F}_{s}\upharpoonright m,n)\right) $,
where $\mathsf{R}^{\mathrm{I}}$ is the result of restricting all of the quantifiers of the $\mathsf{R}$ to $\mathrm{I}$, and $\mathrm{F}_{s}$ is the function defined in with domain $\mathbb{N}$ such that:
$\mathrm{GB}\vdash \ $$\mathrm{F}_{s}(x)$ is the $x$-th digit of the binary expansion of $s$.
It is evident that $n\mapsto \ulcorner \varphi _{n}\urcorner $ is a computable function. We claim:
$(\ast \ast )$ $\forall n(n\in P\leftrightarrow \varphi _{n}\in
\mathrm{GB}_{\mathrm{spa}}).$
The left-to-right direction of $(\ast \ast )$ should be clear. The right-to-left direction is also easy to see, using the fact (proved by a simple compactness argument) that for every $F\in \omega ^{\omega }$ there is a non $\omega $-standard model $\mathcal{M}\models \mathrm{ZF}$ and some nonstandard $s\in \mathbb{N}^{\mathcal{M}}$ such that the ‘standard part’ of the $\mathcal{M}$-finite function coded by $s$ agrees with $F$, i.e., $%
\forall m\in \omega ~\mathcal{M}\models \left( \mathrm{F}_{s}\upharpoonright
m=F\upharpoonright m\right) .$$\square $
**5.2. Remark. **The above proof strategy can be used to show that the following theories are also $\Pi _{1}^{1}$-complete:
**(a)** The theory $\left( \mathrm{ACA}_{0}\right) _{\mathrm{%
spa}}$ of all spartan[^8] models of $%
\mathrm{ACA}_{0}.$
**(b)** The theory of all models of the form $(\mathcal{M}%
,\omega )$, where $\mathcal{M}$ is a model of $\mathrm{ZF}$ or $\mathrm{PA,}$ and $(\mathcal{M},\omega )$ is the expansion of $\mathcal{M}$ by a new predicate $\omega $ consisting of all *standard* natural numbers in $%
\mathcal{M}$.
**(c)** The theory of all models the form $(\mathcal{M},%
\mathrm{Sat}_{\mathcal{M}})$, where $\mathcal{M}$ is a model of $\mathrm{ZF}$ or $\mathrm{PA}$, and $\mathrm{Sat}_{\mathcal{M}}$ is the satisfaction predicate for $\mathcal{M}$.
[En-1]{} J. Barwise, **Admissible Sets and Structures**, Springer-Verlag, Berlin, 1975.
K. D. Devlin, **Constructibility**, Springer-Verlag, 1984.
W. Easton, Doctoral Dissertation, Princeton University, 1964.
A. Enayat, *On certain elementary extensions of models of set theory*, **Trans. Amer. Math**.** Soc.,** vol. 283 (1984), pp.705-715.
\_\_\_\_\_\_\_\_\_\_\_, *Power-like models of set theory*, **J. Sym. Log.** ** **vol. 66, (2001), pp.1766-1782
\_\_\_\_\_\_\_\_\_\_\_, *Automorphisms, Mahlo cardinals, and NFU,* in **Nonstandard Models of Arithmetic and Set Theory** (A. Enayat and R. Kossak eds.), Contemporary Mathematics Series, American Mathematical Socity (2004), pp. 37-59.
\_\_\_\_\_\_\_\_\_\_\_, *The Leibniz-Mycielski axiom in set theory*, **Fund. Math.**, vol. 181 (2004), pp.215-231.
U. Felgner, *Choice functions on sets and classes*, in **Sets and Classes** (on the work by Paul Bernays), Studies in Logic and the Foundations of Math., 84, North-Holland, Amsterdam, 1976, pp.217–255.
P. Hájek and P. Pudlák, **Metamathematics of First-order Arithmetic**, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998.
J. D. Hamkins, *Does ZFC prove the universe is linearly orderable*? MathOverflow answer, 2012 URL:http://mathoverflow.net/q/110823
C. Jockusch, *Ramsey’s theorem and recursion theory*, **J. Sym. Logic,** vol. 37 (1972), pp.268-280.
M. Kaufmann, *Blunt and topless end extensions of models of set theory*, **J. Sym. Log. **vol. 48 (1983), pp.1053-1073.
K. Kunen, **Set theory**, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1983.
A. Leshem, *On the consistency of the definable tree property on* $\aleph _{1}$, **J. Sym. Log.** vol. 65 (2000), pp.1204–1214.
A. Mostowski, *Some impredicatve definitions in the axiomatic set-theory,* **Fund. Math. **vol. 37 (1950), pp.111-124.
H. Rogers, **Theory of Recursive Functions and Effective Computability**. McGraw-Hill, 1967.
H. Wang, **Popular Lectures on Mathematical Logic**, Dover Publications, Mineola (1993).
-------------------------------------------------------------------------------- -------------------------------------
[Ali Enayat]{} [Joel David Hamkins]{}
[Department of Philosophy, Linguistics, & Theory of Science]{} [The Graduate Center]{}
[University of Gothenburg]{} [The City University of New York]{}
[Box 200, SE Sweden]{} [365 Fifth Ave]{}
[E-mail: [email protected]]{} [New York, NY 10016, USA]{}
[URL: http://flov.gu.se/english/about/staff?languageId=100001&userId=xenaal]{} [and]{}
[College of Staten Island]{}
[The City University of New York]{}
[2800 Victory Boulevard]{}
[Staten Island, NY 10314, USA]{}
[E-mail: [email protected]]{}
[URL: http://jdh.hamkins.org]{}
-------------------------------------------------------------------------------- -------------------------------------
[^1]: This notion should not be confused with the *definable tree property of a cardinal* $\kappa $, first introduced and studied by Leshem $\cite%
{Leshem},$ which stipulates that every $\kappa $-tree that is first order definable (parameters allowed) in the structure $\left( H(\kappa ),\in
\right) $ has a cofinal branch $B$ (where $H(\kappa )$ is the collection of sets that are hereditarily of cardinality less than $\kappa $). Note that in this definition $B$ is not required to be first order definable in $\left(
H(\kappa ),\in \right) ;$ so every weakly compact cardinal has the definable tree property.
[^2]: Recall that, provably in , the ordinals $\beta $ such that $(%
\mathrm{V}_{\beta },\in )\prec _{\Sigma _{1}}\left( \mathrm{V},\in \right) $ are precisely the fixed points of the $\beth $-function.
[^3]: An ordinal $\beta $ is $n$-correct when $\left( \mathrm{V}_{\beta },\in
\right) \prec _{\Sigma _{n}}\left( \mathrm{V},\in \right) .$
[^4]: This is the only part of the proof that takes advantage of the assumption that $\mathcal{M}$ carries a definable global well-ordering.
[^5]: Easton proved (in his unpublished dissertation [@Easton; @Thesis]) that assuming Con($\mathrm{ZF}$) there is a model $\mathcal{M}$ of $\mathrm{ZFC}$ which carries no $\mathcal{M}$-definable global choice function for the class of pairs in $\mathcal{M}$; and in particular $\exists p\left( \mathrm{V%
}=\mathrm{HOD}(p)\right) $ fails in $\mathcal{M}$. Easton’s theorem was exposited by Felgner [@Felgner p.231]; for a more recent and streamlined account, see Hamkins’ MathOverflow answer [@Joel-failure; @of; @class; @choice; @for; @pairs].
[^6]: A similar phenomena occurs in the arithmetic setting in relation to Ramsey’s Theorem: even though the predicative extension $\mathrm{ACA}_{\mathrm{0}}$ of $\mathrm{PA}$ can prove every instance of Ramsey’s Theorem of the form $%
\mathbf{\omega }\rightarrow \left( \mathbf{\omega }\right) _{2}^{n}$, where $%
n$ is any meta-theoretic natural number (by a routine arithmetization of any of the usual proofs of Ramsey’s theorem), $\mathrm{ACA}_{\mathrm{0}}$ cannot prove the stronger statement $\forall k\in \omega $ $\mathbf{\omega }%
\rightarrow \left( \mathbf{\omega }\right) _{2}^{k}$. This natural incompleteness phenomena follows from a subtle recursion-theoretic theorem of Jockusch [@Jockusch], which states that for each natural number $%
n\geq 2$ there is a recursive partition $P_{n}$ of $[\omega ]^{n}$ into two parts such that $P_{n}$ has no infinite $\Sigma _{n}^{0}$-homogeneous subset. For more detail, see Wang’s exposition [@Wang p.25]; note that Wang refers to $\mathrm{ACA}_{\mathrm{0}}$ as $\mathrm{PPA}$.
[^7]: The idea of defining the cut $\mathrm{I}$ goes back to** **Mostowski [@Mostowski], who used it to show that the scheme of induction over $%
\mathbb{N}$ is not provable in .
[^8]: Spartan models of $\mathrm{ACA}_{0}$ are of the form $\left( \mathcal{M},D_{%
\mathcal{M}}\right) $, where $\mathcal{M}\models \mathrm{PA}.$
|
{
"pile_set_name": "ArXiv"
}
|
**The Second Subconstituent of some Strongly Regular Graphs**
Norman Biggs
Department of Mathematics
London School of Economics
Houghton Street
London WC2A 2AE
U.K.
[email protected]
February 2010
**Abstract**
This is a report on a failed attempt to construct new graphs $X^h$ that are strongly regular with parameters $((h^4 + 3h^2 +4)/2, \, h^2+1,\, 0, 2)$. The approach is based on the assumption that the second subconstituent of $X^h$ has an equitable partition with four parts. For infinitely many odd prime power values of $h$ we construct a graph $G^h$ that is a plausible candidate for the second subconstituent. Unfortunately we also show that the corresponding $X^h$ is strongly regular only when $h=3$, in which case the graph is already known.
[**1. Introduction**]{}
We should like to be able to construct graphs $X$ that have the following properties:
$\bullet$ $X$ is regular with degree $k$;
$\bullet$ $X$ is triangle-free;
$\bullet$ any two non-adjacent vertices have just two common neighbours.
Standard calculations with eigenvalues and multiplicities show that $k-1$ must be a square $h^2$, with $h$ not congruent to $0$ modulo $4$. In the standard terminology \[5,11\], $X$ is a strongly regular graph with parameters $$((h^4 + 3h^2 +4)/2, \, h^2+1,\, 0, 2).$$ Although there are infinitely many possibilities, only a few graphs are known, even when the number of common neighbours is allowed to be an arbitrary constant $c \neq 2$ \[10\]. The topic is particularly interesting because the known graphs are associated with remarkable groups.
For each vertex $v$ of $X$ we denote by $X_1(v)$, $X_2(v)$ the sets of vertices at distance $1,2$ respectively from $v$. We call the graph induced by $X_2(v)$ the [*second subconstituent*]{} of $X$. We shall usually write it as $X_2$, although there is no reason why it should be independent of $v$. The results in \[2,11\] establish that $X_2$ is a connected graph of degree $k-2$ with diameter 2 or 3. Furthermore, the only numbers that can be eigenvalues of $X_2$ are: $k-2$, $-2$ and the eigenvalues $\lambda_1$, $\lambda_2$ of $X$.
There are three known examples: $h=1,2,3$ corresponding to $k=2,5,10$. When $k=2$ we have the $4$-cycle. When $k=5$ we have a graph with 16 vertices known as the Clebsch graph, a name suggested by Coxeter \[6\] because the graph represents a geometrical configuration discussed by Clebsch. For this graph $X_2$ is the Petersen graph. When $k=10$ we have the Gewirtz graph with $56$ vertices. It can be represented by taking the vertices to be a set of $56$ ovals in PG$(2,4)$, and making two vertices adjacent when the corresponding ovals are disjoint. (An historical note about this graph is appended to this paper.) The algebraic properties of the Gewirtz graph have been studied in detail by Brouwer and Haemers \[4\], and a list of the $56$ ovals may be found at \[12\]. Using this list, it can be verified that the 45 vertices of $X_2$ are partitioned with respect to their distance from a given vertex $w \in X_2$ as $\{w\} \cup P \cup S$, where $|P| = 8$ and $|S| = 36$. The 36 vertices are of two types. One set $Q$ of 16 vertices has the property that each is adjacent to 1 vertex in $P$, while the complementary set $R$ of 20 vertices is such that each is adjacent to 2 vertices in $P$. Further analysis shows that the partition $\{w\}\cup P \cup Q \cup R$ is equitable \[11, p.195\], with the intersection numbers given by the matrix
$$\pmatrix{ 0 &1 &0 &0 \cr
8 &0 &1 &2 \cr
0 &2 &2 &4 \cr
0 &5 &5 &2 \cr }.$$
The eigenvalues of this intersection matrix are $8, 2, -2, -4$. The numbers $2$ and $-4$ are the eigenvalues of the Gewirtz graph, while $8$ $(=k-2)$ and $-2$ are the only other eigenvalues permitted by the general theorem mentioned above. On this basis, it seems worthwhile to investigate possible generalizations.
[**2. Properties of the second subconstituent**]{}
We begin by constructing a suitable intersection matrix for the second subconstituent, for a general value of $k$.
Let $G = (V,E)$ be a graph with vertex-set $V = K^{(2)}$, the set of unordered pairs of elements of a set $K$, where $|K| = k$, and suppose the edge-set $E$ is defined so that the following conditions hold.
[**C1**]{} [*For each vertex $ab \in V$ there is a partition of $V$ with four parts,*]{} $$V = \{ab\} \cup P_{ab} \cup Q_{ab} \cup R_{ab}$$ [*such that $P_{ab} = \{cd \mid \{ab,cd\} \in E\}$ and $Q_{ab} = \{cd \mid |ab \cap cd| = 1\}$.*]{}
[**C2**]{} [*This partition is equitable with intersection matrix*]{} $$M = \pmatrix { 0 &1 &0 &0 \cr
k-2 &0 &1 &2 \cr
0 &2 &2 &4 \cr
0 &k-5 &k-5 &k-8 \cr }.$$
The fact that $P_{ab}$ and $Q_{ab}$ are disjoint implies that if $\{ab,cd\}$ is an edge, then $a,b,c,d$ are distinct. It follows from the definition of $Q_{ab}$ that $|Q_{ab}| = 2(k-2)$. The other parameters then imply that $$|P_{ab}| = k-2, \quad |R_{ab}| = \frac{1}{2}(k-2)(k-5).$$ It is easy to see that a graph $G$ with the given properties would be triangle-free, regular with degree $k-2$, and have diameter 2.
The conditions are clearly meaningful only when $k \ge 8$. Since we know that the corresponding strongly regular graphs can exist only when $k=h^2 + 1$, and that they do exist when $k=2$ and $k=5$, we shall assume that $k\ge 10$ in what follows.
[**Theorem 1**]{} Let $G$ be a graph satisfying conditions [**C1**]{} and [**C2**]{}, and let $C$ be the bipartite graph (claw) with vertex-set $\{*\} \cup K$. Denote by $G \oplus C$ the graph formed from the disjoint union of $G$ and $C$ by adding edges joining each vertex $ab$ in $G$ to the vertices $a$ and $b$ in $C$. Then $G \oplus C$ is a strongly regular graph with degree $k$, it is triangle-free, and each pair of non-adjacent vertices has just 2 common neighbours.
[*Proof*]{} Since $G$ is regular with degree $k-2$, $G \oplus C$ is regular with degree $k$. Since $C$ is a claw, there are no triangles containing the vertex $*$. A triangle containing the vertex $a \in K$ would have to contain vertices $ab$ and $ab'$ in $G$, but these vertices are not adjacent. Finally, a triangle containing the vertex $ab$ would lie wholly in $G$, but $G$ is triangle-free.
It remains to check that any two non-adjacent vertices in $G \oplus C$ have exactly two common neighbours. If the two vertices are $*$ and $ab$, the neighbours are $a$ and $b$, and if the two vertices are $a$, $b$, the common neighbours are $*$ and $ab$. If the two vertices are of the form $ab$ and $ac$, then $ac \in Q_{ab}$, and the neighbours are $a$ and the unique vertex in $P_{ab}$ that is adjacent to $ac$. If the two vertices are of the form $ab$ and $cd$, where $cd \in R_{ab}$, then the neighbours are the two vertices in $P_{ab}$ that are adjacent to $cd$.
[**3. Construction of triangle-free graphs $G^q$**]{}
We attempt to construct graphs satisfying conditions [**C1**]{} and [**C2**]{}. We know that $k-1$ must be a square, say $h^2$. Let $h= q$, where $q$ is a prime power, so that $k= q^2 + 1$. Take $K$ to be the set of points on the projective line $PG(1, q^2)$, that is
$$K= {\mathbb F}_{q^2} \cup \{\infty\} = \langle t \rangle \cup \{0, \infty \}.$$
Here ${\mathbb F}_{q^2}$ is the finite field of order $q^2$, $\infty$ is the conventional ‘point at infinity’, and $t$ is a primitive element of the field, so that $\langle t \rangle$ is a cyclic group of order $q^2 -1 = k-2$. The group PGL$(2, q^2)$ of projective linear transformations acts 3-transitively on $K$, and hence transitively on the unordered pairs $ab$ in $K^{(2)}$, and the stabilizer of the pair $0\infty$ is generated by $x \mapsto tx$ and $x \mapsto x^{-1}$. When $q$ is an odd prime power (so that $k$ is even) its orbits on $K^{(2)}$ are as follows:
$$\{0 \infty\}, \quad O_0 = \{0x \mid x \in \langle t \rangle\} \cup
\{ \infty x \mid x \in \langle t \rangle\},$$ and $\frac{1}{2} (k-2)$ orbits of the form
$$O_v = \{ vx\; x \mid x \in \langle t \rangle \} \quad (v = t, t^2, \ldots, t^{(k-2)/2}).$$
The orbit $O_0$ has size $2(k-2)$, the orbits $O_v$ ($v \neq -1$) have size $k-2$, and the orbit $O_{-1}$ has size $(k-2)/2$. Note that when $q$ is a power of $2$ the orbit-partition takes a slightly different form: this is consistent with the fact that no construction can work when $k-1= h^2$ with $h \equiv 0$ (mod $4$), by the feasibility conditions. (The exceptional case $q=2$ has already been covered.)
The idea of the following construction is to define a graph with vertex-set $V = K^{(2)}$ such that for a suitable value of $u$, the partition postulated in condition [**C1**]{} (taking the vertex $ab$ to be $0\infty$) is given by
$$P= O_u, \qquad Q = O_0, \qquad R = \bigcup_{v \neq u, 0} O_v.$$
The construction depends on the [*cross-ratio*]{}, which is defined for any points $a,b,c,d \in K$ by the rule
$$(ab|cd) = \frac{(a-c)(b-d)}{(a-d)(b-c)},$$
with the usual conventions about $\infty$. The cross-ratio is $1$ if and only if $a=b$ or $c=d$ or both, and so this value does not occur when $ab$ and $cd$ are in $V = K^{(2)}$. Given the unordered pair of unordered pairs $ab$ and $cd$, the cross-ratio $(ab|cd)$ takes only two values $\rho$ and $\rho^{-1}$, which it is convenient to write in the form $(ab|cd) = \rho^{\pm}$.
Let $V = K^{(2)}$, and given $u \in \langle t \rangle, u \neq \pm 1$ define $E_u$ to be the set of pairs $\{ab, cd\}$ such that $(ab|cd) = u^{\pm}$. Since $(0 \infty \mid ux\, x) = u$, it follows that in the graph $G_u = (V, E_u)$ the set of vertices adjacent to $0\infty$ is the orbit $O_u$, as defined above.
We consider the possibility that, for a suitable value of $u$, $G_u$ is a graph in which the partition given above is equitable, with the intersection matrix $M$ as in condition [**C2**]{}. The first step is to ensure that $M_{PP} = 0$, which means that $G_u$ is triangle-free.
[**Lemma**]{} Let
$$\Omega \; = \; \{v \in K \mid v= (x+x^{-1} -1)^{\pm} \; {\rm for\; some} \;
x \in \langle t \rangle, x \neq 1\}.$$
Then the graph $G_u$ is triangle-free if and only if $u$ is not in $\Omega$.
[*Proof*]{} The group PGL$(2, q^2)$ acts as a group of automorphisms of $G_u$ since it preserves cross-ratios, and so $G_u$ is vertex-transitive. Hence we need only consider the possibility of triangles containing a given vertex, say $0\infty$. The stabilizer of $0 \infty$ contains $x \mapsto tx$ and $x \mapsto 1/x$, and so we can assume that two edges of the triangle are $\{ 0 \infty, u \;1\}$ and $\{ 0 \infty, ux \;x \}$, with $x \neq 1$. Now, the vertices $u\;1$ and $ux\;x$ are adjacent if and only if
$$\frac{ (ux-u)(u-1)}{ (ux-1)(x-u) } = u^{\pm}.$$
After some rearrangement, this reduces to
$$u = (x + x^{-1} -1)^{\pm}.$$
In other words, there is a triangle if and only if $u$ is in $\Omega$.
Since $x + x^{-1} -1$ is symmetrical with respect to inverting $x$, the set $\Omega$ can be found by calculating at the values of $x + x^{-1} -1$ for $x= t^j$, $j = 1, 2, 3, \ldots, (k-2)/2$. For example, when the field is ${\mathbb F}_9$ with the primitive element $t$ satisfying $t^2 + t + 2 =0$, we have the table
$$\matrix{ j\quad &x &x+ x^{-1} - 1 &(x+x^{-1}-1)^{-1} \cr
& & & \cr
1\quad &t &2t &2+2t \cr
2\quad &1+2t &2 &2 \cr
3\quad &2+2t &1+t &t \cr
4\quad &2 &0 &\infty \cr }.$$
Since $1+2t$ and its inverse $2+t$ are not in $\Omega$, we conclude that the graph $G_{1+2t}$ is triangle-free.
Generally, the special orbit $O_u$ could be any one of the $O_v$ except $O_0$ and $O_{-1}$, thus $\{u, u^{-1}\}$ could be any one of the $(k-4)/2$ pairs $\{t^j, t^{-j} \}$, $ j= 1, 2, \ldots,(k-4)/2$. As a working definition let us say that $u$ is [*admissible*]{} if (1) $u = \neq 0, \infty, -1, 1$, and (2) $\{u, u^{-1}\} \notin \Omega$. At first sight it appears that as many as $(k-2)/2$ pairs are not admissible, because they are in $\Omega$, but fortunately things are not so bad.
[**Theorem 2**]{} Suppose that $q$ is an odd prime power and there is an element $\zeta$ in ${\mathbb F}_{q^2}$ such that $\zeta^2 = 3$. Then there is at least one $u$ of the form $t^j$ with $j \in \{1,2, \ldots, (k-4)/2\}$ such that $u$ is not in $\Omega$, and hence the graph $G_u$ is triangle-free.
[*Proof*]{} When $q$ is odd $q^2$ is congruent to $1$ mod $4$, and so there is an element $\iota$ such that $\iota^2 = -1$. Then
$$\iota + \iota^{-1} - 1 = (\iota + \iota^{-1} -1)^{-1} = -1,$$ which means that the ‘pair’ $\{-1, -1\}$ occurs in $\Omega$. But $u = -1$ is not admissible anyway, and so the number of non-admissible pairs in $\Omega$ is effectively reduced to at most $(k-4)/2$.
Similarly, if we can find an $x$ such that $x + x^{-1} -1 = 0$ then the pair $\{0, \infty\}$ will occur in $\Omega$, and since this pair is also not admissible, the number of non-admissible pairs in $\Omega$ will be reduced to at most $(k-6)/2$. It is easy to see that this happens if there is an element $\zeta \in {\mathbb F}_{q^2}$ such that $\zeta^2 = 3$. In that case, let
$$\theta = 2^{-1}(1+\iota \zeta), \quad {\rm so\;that} \quad
\theta^{-1} = 2^{-1}(1- \iota\zeta) \quad {\rm and} \quad \theta + \theta^{-1} -1 = 0.$$
Hence an admissible $u$ must exist.
For example, in the field ${\mathbb F}_{25}$ with the primitive element $t$ satisfying $t^2 + t + 2 =0$, we have $\iota =2$, $\zeta= 4+3t$, $\theta = 2+3t$. Hence the pair $\{2+3t, 4+2t\}$ is admissible. A complete check shows that there are two other admissible pairs $\{1+2t, 2+4t\}$, and $\{3+t, 4+3t\}$.
It is easy to see that there are infinitely many fields ${\mathbb F}_{q^2}$ which contain an element with $\zeta^2 = 3$, for example by applying the law of quadratic reciprocity. We do not pursue this matter, since the final step is to rule out all fields except ${\mathbb F}_9$, where explicit calculation shows that the construction works.
[**4. Failure of the construction in general**]{}
We now know that in many cases a graph $G_u$ can be constructed satisfying the condition $M_{PP} = 0$. But it remains to check that the other entries of $M$ are correct. In fact, several of them can be verified, but it turns out that the condition $M_{PR} =2$ cannot be satisfied in general.
[**Theorem 3**]{} Let $G_u$ be defined for an odd prime power $q$ as in Section 3, and let the partition $\{0 \infty\}\cup P \cup Q \cup R$ of the vertices of $G_u$ be as stated there. Then this partition is equitable with an intersection matrix of the form required by condition [**C2**]{} only when $q =3$.
[*Proof*]{} We shall show that the condition $M_{PR}=2$ cannot hold, except when $q=3$.
A typical vertex in $R$ is $vw\, w$ where $v \neq u, 0, \infty, 1$ and $w \in \langle t \rangle$ . This vertex is adjacent to the vertices $ux\, x$ in $P$ for which $(ux\, x \mid vw\, w) = u$ or $u^{-1}$. These two equations can be written as quadratics in $x$:
$$ux^2 - (1+u) vwx + vw^2 = 0, \qquad ux^2 -(1+u)wx + vw^2 =0,$$
and their discriminants are
$$\Delta^+ = w^2((1+u)^2v^2 - 4uv), \qquad \Delta^- = w^2((1+u)^2 - 4uv).$$
If there are just two solutions for $x$ then either (1) exactly one of $\Delta^+$, $\Delta^-$ is a square in ${\mathbb F}_{q^2}$, or (2) $\Delta^+$ and $\Delta^-$ are both zero.
Consider first the case $v = -1$. Here $\Delta^+ = \Delta^-$ and so their common value must be zero. That is, $$(1 + u^2) +4u =0.$$
Then for all $v \neq -1$ $$\Delta^+ = w^2 (-4uv^2 -4uv) = v w^2 (-4u - 4uv) = v \Delta^-.$$
Hence in order that exactly one of $\Delta^+$, $\Delta^-$ is a square, $v$ must be a non-square, and this must hold for all the orbits $O_v \subseteq R$ except $O_{-1}$. In the case $q=3$ we chose $u = 1+t = t^2$, so $R = O_t \cup O_{t^3} \cup O_{-1}$, and the condition is satisfied. But for $q \ge 5$ there must be at least one square among the relevant values of $v$ and the condition cannot be satisfied.
[**Historical note on the Gewirtz graph**]{}
Gewirtz discussed his graph in two papers published in 1969 \[8,9\]. Brouwer \[3\] says that the graph was discovered by Sims, and calls it the Sims-Gewirtz graph.
My own interest in strongly regular graphs dates from the late 1960s, when I was told by John McKay about the exciting discoveries of new simple groups. The topic (but not the Gewirtz graph) is mentioned in a paper I gave at the 1969 Oxford Conference \[1\]. I do not wish to claim any originality for myself, but I am fairly sure that I initially derived my knowledge of the Gewirtz graph from a 1965 paper of W.L. Edge ‘On some implications of the geometry of the 21-point plane’ \[7\]. In that paper the three sets of 56 ovals in PG$(2,4)$ are clearly described, with the critical property that any one of the sets of 56 has the property that two of them intersect in 0 or 2 points.
[**References**]{}
1. N.L. Biggs. Intersection matrices for linear graphs. In: [*Combinatorial Mathematics and its Applications*]{} (ed. D.J.A. Welsh), Academic Press, 1971, 15-23.
2. N.L. Biggs. Strongly regular graphs with no triangles. [*arXiv*]{} 0911.2160v1, September 2009. Families of Parameters for SRNT Graphs. [*arXiv*]{} 0911.2455v1, October 2009.
3. A.E. Brouwer. Sims-Gewirtz graph. www.win.tue.nl/aeb/graphs/Sims-Gewirtz.html (accessed 10/11/09).
4. A.E. Brouwer, W. Haemers. The Gewirtz Graph: an exercise in the theory of graph spectra. [*Europ. J. Combinatorics*]{} 14 (1993) 397-407.
5. P.J. Cameron, J. van Lint. [*Designs, Graphs, Codes and their Links*]{}. Cambridge University Press, 1991.
6. H.S.M. Coxeter. Self-dual configurations and regular graphs. [*Bull. Amer. Math. Soc.*]{} 56 (1950) 413-458.
7. W.L. Edge. Some implications of the geometry of the 21-point plane. [*Math. Zeitschr.*]{} 87 (1965) 348-362.
8. A. Gewirtz. The uniqueness of $g(2,2,10,56)$. [*Trans. New York Acad. Sci.*]{} 31 (1969) 658-675.
9. A. Gewirtz. Graphs with maximal even girth. [*Canad. J. Math*]{} 21 (1969) 915-934.
10. C.D. Godsil. Problems in algebraic combinatorics. [*Elect. J. Combinatorics*]{} 2 (1995) F1.
11. C.D. Godsil, G.F. Royle. [*Algebraic Graph Theory*]{}. Springer, 2001.
12. Wolfram MathWorld. mathworld.wolfram.com/GewirtzGraph.html (accessed 10/11/09).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We introduce a new notion of viscosity solutions for the level set formulation of the motion by crystalline mean curvature in three dimensions. The solutions satisfy the comparison principle, stability with respect to an approximation by regularized problems, and we also show the uniqueness and existence of a level set flow for bounded crystals.'
address:
- 'Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914, Japan.'
- 'Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kakuma town, Kanazawa, Ishikawa 920-1192, Japan.'
author:
- Yoshikazu Giga
- Norbert Požár
title: A level set crystalline mean curvature flow of surfaces
---
Introduction
============
A crystalline mean curvature flow is a typical example of an anisotropic mean curvature flow, which can be regarded as a mean curvature flow under a Minkowski or Finsler metric [@BP96]. A crystalline mean curvature flow was proposed by S. B. Angenent and M. E. Gurtin [@AG89] and independently by J. Taylor [@T91] to describe the motion of an anisotropic antiphase boundary in materials science. There is a large amount of literature devoted to the study of the motion by crystalline mean curvature. However, even local-in-time unique solvability of its initial value problem has been a long-standing open problem except in the case of planar motion or convex initial data. The main reason is that the surface energy density is not smooth and hence the speed of evolution is determined by a nonlocal quantity.
Our goal in this paper is to solve this long-standing open problem for purely crystalline mean curvature flow in ${\ensuremath{\mathbb{R}}}^3$. In fact, we shall introduce a new notion of solutions which corresponds to a generalization of a level set flow for the mean curvature flow equation and establish its unique existence.
To motivate the problem, let us explain an example of anisotropic mean curvature flow equation and its level set formulation; see e.g. [@CGG; @GG92; @G06]. Let $\gamma:S^2 \to (0,\infty)$ be a given interfacial energy density on the unit sphere $S^2$. For a given closed surface $\Gamma$ we define the interfacial energy $$I_\gamma(\Gamma) = \int_\Gamma \gamma(\mathbf{n})\;d\mathcal{H}^2,$$ and call $I_\gamma$ the interfacial energy of $\Gamma$ with density $\gamma$. Here $\mathbf{n}$ denotes the unit exterior normal of $\Gamma$ and $d\mathcal{H}^2$ denotes the area element. The anisotropic mean curvature $\kappa_\gamma$ is the first variation of $I_\gamma$ with respect to change of volume enclosed by $\Gamma$. Its explicit form is $$\kappa_\gamma = -\operatorname{div}_\Gamma \left(\nabla_p \gamma(\mathbf{n})\right)$$ where $\gamma$ is $1$-homogeneously extended as $\gamma(p)=|p|\gamma\left(p/|p|\right)$ for $p\in {\ensuremath{\mathbb{R}}}^3\backslash\{0\}$ and $\gamma(0)=0$; $\operatorname{div}_\Gamma$ denotes the surface divergence [@Si83; @G06]. If $\gamma(p)=|p|$, $I_\gamma$ is the surface area and $\kappa_\gamma=-\operatorname{div}_\Gamma
\mathbf{n}$, which is nothing but (two times) the classical mean curvature. When the interfacial energy density $\gamma$ is not a constant function on $S^2$, we say $\kappa_\gamma$ is an anisotropic mean curvature. Let $\{\Gamma_t\}_{t>0}$ be a smooth family of closed surfaces in ${\ensuremath{\mathbb{R}}}^3$ and let $V$ be its normal velocity in the direction of $\mathbf{n}$. The equation for $\{\Gamma_t\}$ of the form $$V = \kappa_\gamma \quad \text{on} \quad \Gamma_t$$ is a simple example of an anisotropic mean curvature flow equation. Of course, if $\gamma(p)=|p|$, then this equation is nothing but the standard mean curvature flow equation $V=\kappa$. A typical feature of this equation is that even if one starts with a smooth surface $\Gamma_0$, the solution $\Gamma_t$ may pinch in finite time, for example a dumbbell with thin neck [@Gr89]. So a weak formulation is necessary to track the evolution after the formation of singularities. There are two standard approaches for the (isotropic) mean curvature flow equation. One is a variational way like a varifold solution initiated by K. Brakke [@B78] and developed further by T. Ilmanen [@Il93] and K. Takasao and Y. Tonegawa [@TT]. Another approach is a level set method based on a comparison principle introduced by [@CGG; @ES]. As already noted in [@CGG] the level set method is very flexible and it applies to anisotropic curvature flow equation [@GG92] while a varifold solution is still limited to the isotropic mean curvature flow equation.
Let us explain the idea of the level set formulation. We introduce an auxiliary function $u:{\ensuremath{\mathbb{R}}}^3\times [0,\infty)\to{\ensuremath{\mathbb{R}}}$ so that its zero level set agrees with $\Gamma_t$. To fix the idea we assume that $u>0$ in a region $D_t$ enclosed by $\Gamma_t$ and $u<0$ outside of $D_t \cup \Gamma_t$. Then the equation $V=\kappa_\gamma$ is represented as $$\frac{u_t}{|\nabla u|}=-\operatorname{div}\left(\nabla_p \gamma\left(-\frac{\nabla u}{|\nabla u|}\right)\right) \quad \text{on} \quad \Gamma_t$$ since $V=u_t/|\nabla u|$, $\mathbf{n}=-\nabla u/|\nabla u|$. The idea of the level set method is to consider this equation not only on $\Gamma_t$ but also in ${\ensuremath{\mathbb{R}}}^3$, i.e. each level set of $u$ is required to move by $V=\kappa_\gamma$. In other words, we consider $$\label{level set mcf}
u_t-|\nabla u|\left(-\operatorname{div}\left(\nabla_p \gamma \left(-\nabla u/|\nabla u|\right)\right)\right) = 0
\quad \text{in} \quad {\ensuremath{\mathbb{R}}}^3 \times (0,\infty)$$ with initial condition $$\label{level set mcf initial}
u(x,0)=u_0(x), \ x \in {\ensuremath{\mathbb{R}}}^3.$$ Here $u_0$ is taken so that $\Gamma_0$ is its zero level set. In the case $\gamma(p)=|p|$, is nothing but the famous level set mean curvature flow equation $$u_t - |\nabla u|\operatorname{div} \left(\nabla u/|\nabla u|\right)=0.$$ The level set equation is degenerate even if $\gamma$ is convex. It is unexpected that the problem can be solved even locally-in-time in classical sense even if $u_0$ is smooth.
Fortunately, if $\gamma$ is $C^2$ on ${\ensuremath{\mathbb{R}}}^3 \setminus {{\left\{0\right\}}}$ and convex, the notion of viscosity solutions [@CIL] is adjustable to solve – uniquely and globally-in-time for any uniformly continuous initial data [@CGG; @G06]. One shall notice that there is a large freedom to choose $u_0$ for given $\Gamma_0$. However, it is known [@CGG; @G06] that the zero level set is uniquely determined by $\Gamma_0$ (independently of the choice of $u_0$). Although the zero level set of $u$ may fatten, it is often called a level set flow (solution) of $V=\kappa_\gamma$ with initial data $\Gamma_0$. The theory is based on a comparison principle for viscosity solutions and it applies when $\gamma$ is not necessarily $C^2$ but the singularity is weak. For example, in planar motion even if the second derivative of $\gamma\in C^1\left({\ensuremath{\mathbb{R}}}^2 \backslash\{0\}\right)$ is allowed to jump at finitely many point in $S^1$, the result of [@CGG] is extendable [@OS93; @GSS]; see [@I96] for higher dimensional problem. However, if the singularity of $\gamma$ is strong, such that the first derivative of $\gamma$ may have jumps, then the situation is completely different. The equation becomes very singular in the sense that the speed becomes a nonlocal quantity and establishing the level set method becomes totally non-trivial even if only a planar motion is considered, although it has been established in [@GG01]. However, it has been a long-standing open problem for surface evolution even if $\gamma$ is (purely) crystalline, i.e. $\gamma$ is piecewise linear and convex in ${\ensuremath{\mathbb{R}}}^3$. Such functions are often in convex analysis referred to as *polyhedral* [@Rockafellar].
Our purpose is to establish a level set method for a crystalline mean curvature flow, whose typical example includes $V=\kappa_\gamma$ for crystalline $\gamma$. Our theory can apply to more general equations such as $V=\kappa_\gamma+1$. We shall introduce a new notion of viscosity solutions so that the following well-posedness result holds.
\[th:unique existence\] Let $\gamma$ be crystalline in ${\ensuremath{\mathbb{R}}}^3$. Assume that $f=f(m,\lambda)$ is continuous on $S^2\times{\ensuremath{\mathbb{R}}}^3$ and $\lambda \mapsto f(m,\lambda)$ is non-decreasing. Assume that $\left|f(m,\lambda)\right|/\left(|\lambda|+1\right)$ is bounded in $S^2\times{\ensuremath{\mathbb{R}}}$. Let $D_0$ be a bounded open set in ${\ensuremath{\mathbb{R}}}^3$ with the boundary $\Gamma_0=\partial D_0$. Then there exists a global unique level set flow $\{\Gamma_t\}_{t\geq 0}$ with $$\label{general mcf}
V=f(\mathbf{n},\kappa_\gamma) \quad \text{on} \quad \Gamma_t$$ and initial data $\Gamma_0$.
The assumption of the linear growth for $f$ in $\lambda$ is just for simplicity. One can remove it by introducing a special class of test functions [@IS; @G06] or by a flattening argument [@Go].
To prove the uniqueness part a key step is to establish a comparison principle for the level set equation of which is of the form $$\label{P}
u_t + F \left( \nabla u, \operatorname{div}\partial W(\nabla u) \right) = 0,$$ where $$\begin{aligned}
\label{geometric F}
F(p,\Lambda) = -|p|f \left(-p/|p|,\Lambda \right),\ W(p)=\gamma(-p).\end{aligned}$$ Here we rather use the subdifferential notion $\partial W$ instead of $\nabla W$ since $W$ is piecewise linear and so not everywhere differentiable. To prove the existence part, one cannot unfortunately apply Perron’s method since the nonlocal quantity “$\operatorname{div}\partial W(\nabla u)$” is not constant in a flat part of the solution (which is different from planar case.) We thus construct a solution by smoothing $W$. Here we need to establish a stability of our viscosity solutions. The basic idea of proofs is an elaboration on the idea for establishing uniqueness based on the comparison principle and stability for the total variation flow of non-divergence type [@GGP13JMPA; @GGP13AMSA]. We shall establish comparison principle for a more general nonlinearity $F$ than , see Remark \[rem:general F\] below.
The bibliography of [@GGP13AMSA] includes many references on unique solvability. We take this opportunity to mention related results for evolution of closed surfaces by crystalline or more general singular interface energy. In three dimensions and higher, the crystalline mean curvature $\kappa_\gamma$ is not only a nonlocal quantity as mentioned above, but it might be non-constant on facets of the crystal [@BNP99]. In fact, it might be discontinuous, and in general it is known to be only a function of bounded variation [@BNP01a; @BNP01b]. Therefore facet breaking and bending might occur and we cannot restrict the solutions only to surfaces with facets parallel to those of the Wulff shape corresponding to the crystalline energy density $\gamma$. A more general notion of solutions is necessary. The variational approach have led to a significant progress by understanding the properties of $\kappa_\gamma$. A notion of solutions via an approximation by reaction-diffusion equations for $V
= \gamma \kappa_\gamma$ was established in [@BGN00; @BN00]. An approximation via minimizing movements was used in [@CasellesChambolle06; @BCCN06; @BCCN09]. However, all these results only provide existence for *convex* initial data.
We also establish a convergence result which is useful to discuss approximation by an Allen-Cahn type equation.
\[th:convergence\] Under the assumption of Theorem \[th:unique existence\], let $u$ be a viscosity solution of with initial data $u_0 \in C({\ensuremath{\mathbb{R}}}^3)$ such that $u_0(x)=-c$ for $|x|\geq R$ with some $R$ and $c>0$. Assume that $\gamma_\varepsilon$ is smooth in ${\ensuremath{\mathbb{R}}}^3\setminus{{\left\{0\right\}}}$, convex and $1$-homogeneous and $\gamma_\varepsilon\to\gamma$ uniformly on $S^2$. Let $u^\varepsilon$ be a viscosity solution of with $W=W_\varepsilon(p)=\gamma_\varepsilon(-p)$, with initial data $u_0^\varepsilon$ such that $u_0^\varepsilon(x)=-c$ for $|x|\geq R$. Assume that $u_0^\varepsilon\to u_0$ uniformly. Then $u^\varepsilon$ converges locally uniformly to $u$ in ${\ensuremath{\mathbb{R}}}^3\times [0,\infty)$.
This gives a convergence of diffuse interface model to the sharp interface model even if $\gamma$ is crystalline; see [@GOS; @TC98].
After this work had been completed, the authors learned of a recent work by A. Chambolle, M. Morini and M. Ponsiglione [@CMP], where they established a unique global solvability (up to fattening) for $V=\gamma \kappa_\gamma$ for any convex $\gamma$ by introducing a new notion of a solution related to the anisotropic distance function. Their approach applies to all dimension and all initial data not necessarily bounded. However, their approach requires a special form of the equation so that the mobility is proportional to the interfacial energy density $\gamma$ and it does not apply to $V=\kappa_\gamma$ or $V=\kappa_\gamma +1$. Our approach applies to all $V=f(\mathbf{n},\kappa_\gamma)$ including these equations but the dimension $n$ is limited as $n \leq 3$ and $\gamma$ is limited to crystalline. It is not yet clear whether or not our solution agrees with theirs in the case when both approaches are available although it is very likely.
\[rem:general F\] In full generality, we will assume that $F \in C({{\ensuremath{{\mathbb{R}^{{n}}}}}}\times {\ensuremath{\mathbb{R}}})$, $n \geq 1$, and that it is nonincreasing in the second variable, that is, $$\begin{aligned}
\label{F ellipticity}
F(p, \xi) \leq F(p, \eta) \qquad \text{for all } p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}},\ \xi \geq \eta.\end{aligned}$$ For simplicity, we shall also assume that $$\begin{aligned}
F(0,0) = 0.\end{aligned}$$ In particular, constants are solutions of .
Viscosity solutions and the contribution of this paper {#viscosity-solutions-and-the-contribution-of-this-paper .unnumbered}
------------------------------------------------------
We extend the notion of viscosity solutions to the problem with crystalline $W$. The main strength of the viscosity solution approach is that it can handle general problems that are not of divergence form by exploiting their comparison principle structure [@CIL; @G06].
The main difficulty in defining a solution of is the singular, nonlocal operator ${\operatorname{div}}\partial W(\nabla \cdot)$. We interpret this operator as the minimal section (also known as the canonical restriction) of the subdifferential of the anisotropic total variation energy in the Hilbert space $L^2(\Omega)$, $$\begin{aligned}
E(\psi) :=
\begin{cases}
\int_\Omega W(D \psi) {\;dx}, & \psi \in L^2(\Omega) \cap BV(\Omega),\\
+\infty, & \text{otherwise},
\end{cases}\end{aligned}$$ where $\Omega$ is the flat torus ${\ensuremath{\mathbb{R}}}^n / L{\ensuremath{\mathbb{Z}}}^n$ for some $L > 0$, $n \geq 1$, and $BV(\Omega)$ is the space of functions of bounded variation. That is, we only consider this energy for periodic functions $\psi$ to avoid issues with handling the boundary of $\Omega$. Since $D\psi$ is in general only a Radon measure, the functional $E$ is understood as the lower semi-continuous envelope (closure) of the functional defined for Sobolev functions $W^{1,1}(\Omega)$.
It is well-known that the subdifferential of $E$ defined above is the set of divergences of certain vector fields, often called *Cahn-Hoffman vector fields* [@Moll]. More precisely, if $\psi$ is a Lipschitz function on $\Omega$, then $$\begin{aligned}
\partial E(\psi) = {{\left\{-{\operatorname{div}}z: z(x) \in \partial W(\nabla \psi(x)) \text{ for a.e. $x$, } {\operatorname{div}}z \in L^2(\Omega)\right\}}}.\end{aligned}$$ The subdifferential $\partial E(\psi)$ is a closed convex, possibly empty subset of the Hilbert space $L^2(\Omega)$. If it is nonempty, we say that $\psi \in {\operatorname{\mathcal{D}}}(\partial E)$ and the unique element of the subdifferential with the minimal $L^2$-norm is called the minimal section of $\partial E(\psi)$ and is denoted as $\partial^0 E(\psi)$. In such a case we will interpret ${\operatorname{div}}\partial W(\nabla \psi)$ as $- \partial^0 E(\psi)$.
This interpretation is consistent with the classical theory of monotone operators for the solvability of problems of the form $$\begin{aligned}
u'(t) \in - \partial E(u(t)).\end{aligned}$$ Indeed, it is known that a solution is right-differentiable and the right derivative $d^+u/dt (t) = -\partial^0
E(u(t))$. As we noted above, the mean curvature flow can be viewed as the gradient flow of the surface energy functional.
The viscosity solutions are defined via a comparison with a suitable class of test functions. It is therefore necessary to identify a sufficiently large class of functions for which we can define ${\operatorname{div}}\partial W(\nabla \cdot)$ so that they can serve as test functions in the definition of viscosity solutions. In particular, it must be possible to prove both uniqueness (via a comparison principle) and existence (via a stability property of solutions).
Since the energy density $W$ is crystalline, that is, piecewise linear, the domain of the subdifferential of $E$ can be understood as functions that have flat parts with gradients that fall into the set where $W$ is not differentiable. These flat parts then correspond to the features of the crystal—facets and edges—depending on the dimension of the subdifferential $\partial
W(\nabla \psi)$ on the given flat part of $\psi$. This then leads to an idea of energy stratification with respect to the subdifferential dimension. It turns out that the value of ${\operatorname{div}}\partial W(\nabla \psi)$ at a point $x$ depends only on the shape of $\psi$ in the directions parallel to $\partial W(\nabla \psi(x))$, and it is basically independent of the shape in the orthogonal direction.
Because of the simple structure of $W$, the local behavior of $W$ (and $\partial W$) in a neighborhood of a given gradient $p$ can be completely captured by a one-homogeneous function that is linear in directions orthogonal to the subspace spanned by the directions in $\partial W(p)$, Proposition \[pr:direction-decomposition\]. We therefore for a given slope $p$ define a sliced energy ${E^{\rm sl}}_p$ to capture the interesting behavior, and reduce the analysis to a space ${\ensuremath{\mathbb{R}}}^k$, where $k$ is the dimension of $\partial W(p)$. Then we consider *stratified faceted functions* by separating the variables into the directions parallel to $\partial W(p)$, in which we assume that the function has a “nice” facet, and the orthogonal directions where the function can be of any form (as long as it is differentiable), Definition \[def:strat-faceted-test-function\].
It can be easily seen that ${\operatorname{div}}\partial W(\nabla \psi)(x) = 0$ whenever $\psi$ is twice continuously differentiable in a neighborhood of $x$ and $W$ is differentiable at $\nabla \psi(x)$. We therefore have to identify the value of this operator at points where $\partial W(\nabla \psi)$ is not a singleton, that is, on the flat parts of the stratified faceted functions. These flat parts can be thought of as $k$-dimensional facets, and they can be described by a pair of open sets $(A_-, A_+)$, which specify where the function is below ($A_-$) or above $(A_+)$ the flat part. It turns out that ${\operatorname{div}}\partial W(\nabla \psi)$ is independent of the particular choice of $\psi$, Corollary \[co:lambda support func indep\], but only depends on the sets $(A_-, A_+)$ and the slope $p = \nabla \psi$ of the flat part. We call this value $\Lambda_p(\psi)$ to emphasize this dependence on $p$, and connect this to the previous results [@GGP13JMPA; @GGP13AMSA], see Section \[sec:crystalline curvature\]. While $\Lambda_p(\psi)$ might be discontinuous on the flat parts, it satisfies a comparison principle property with respect to a natural ordering of the $k$-dimensional facets.
We use the stratified faceted functions as the test functions for the definition of viscosity solutions. Heuristically speaking, a continuous function $u$ is a viscosity solution of if it satisfies a comparison principle with all stratified faceted functions that are local solutions of .
To show that this definition of viscosity solutions is reasonable, we have to establish a general comparison principle and stability of solutions (with respect to approximation by regularized problems). For the comparison principle, we need a sufficiently large class of stratified faceted test functions. In particular, for any given gradient $p$ such that $\partial W(p)$ is not a singleton and a pair of smooth disjoint open sets $(A_-, A_+)$ in ${\ensuremath{\mathbb{R}}}^k$, $k = \dim \partial W(p)$, we need to be able to construct a $k$-dimensional facet arbitrarily close to the facet given by $(A_-, A_+)$ such that there exists a stratified faceted function with this facet, and for which $\Lambda_p(\psi)$ is well-defined. See Corollary \[co:approximate pair sliced\] for details. This unfortunately seems to be quite nontrivial, and we currently know how to do this construction in one and two dimensions. This allows us to prove the comparison principle for in three dimensions. However, if this approximated admissible facet construction in Corollary \[co:approximate pair sliced\] can be extended to higher dimensions, our results Theorem \[th:unique existence\] and Theorem \[th:convergence\] will automatically apply to the higher dimensions as well.
The proof of the comparison principle Theorem \[th:comparison principle\] follows the standard doubling-of-variables argument with an additional parameter as in [@GGP13JMPA; @GGP13AMSA]. This is substantially extended to handle the stratified energy and the stratified faceted test functions. We consider two solutions $u$, $v$ of that are ordered as $u \leq v$ at $t = 0$ and consider the function $$\begin{aligned}
\Phi_{\zeta,{\ensuremath{\varepsilon}}}(x,t,y,s) := u(x,t) - v(y,s) - \frac{{\left|x-y-\zeta\right|}^2}{2{\ensuremath{\varepsilon}}}-
S_{\ensuremath{\varepsilon}}(t,s),\end{aligned}$$ on $(x, t, y, s) \in {\ensuremath{\mathbb{R}}}^n \times (0, T) \times {\ensuremath{\mathbb{R}}}^n \times (0, T)$, where $S_{\ensuremath{\varepsilon}}$ is defined in , and $T, {\ensuremath{\varepsilon}}> 0$ are fixed. We then analyze the maxima of $\Phi_{\zeta, {\ensuremath{\varepsilon}}}$ for $\zeta
\in {\ensuremath{\mathbb{R}}}^n$ small. This extra parameter $\zeta$ allows us to recover additional information about the behavior of $u$ and $v$ near the maximum of $\Phi_{\zeta, {\ensuremath{\varepsilon}}}$. We then argue by contradiction: if $u > v$ at some point, we can construct stratified faceted test functions for $u$ and $v$ near the maximum of $\Phi_{\zeta, {\ensuremath{\varepsilon}}}$. These test functions have ordered facets, which then together with the comparison principle for $\Lambda_p$ yields a contradiction.
The stability of solutions with respect to approximation of by regularized problems then follows from an extension of the argument developed in [@GGP13JMPA]. We have to again overcome the discrepancy between the test functions of the regularized problem, which are only smooth functions, and the stratified faceted functions for the limit problem . This is related to the fact that we are approximating a singular, nonlocal operator by local operators. The idea is to perturb the test function by solving the resolvent problem for the energy $E$ and the regularized (elliptic) energy $E_m$ with a small parameter $a > 0$: $$\begin{aligned}
\psi_a = (I + a \partial E)^{-1} \psi, \qquad \psi_{a,m} = (I + a \partial E_m)^{-1} \psi,\end{aligned}$$ which amounts to solving one step of the implicit Euler discretization of the gradient flow of those energies. This transfers the nonlocal information onto the perturbed test function and allows passing in the limit, Theorem \[th:stability quadratic\]. The main extension in this paper is the handling of the sliced energy. An elaboration on this argument yields also stability with respect to an approximation by one-homogeneous energies $E_m$, Theorem \[th:linear growth stability\].
Combining the above results we obtain the existence of a unique solution of . Since the level set of the solution does not depend on the choice of the initial level set function, we have uniqueness of the level set flow.
Outline {#outline .unnumbered}
-------
We open with a review of the theory for convex functionals with linear growth in Section \[sec:convex func lin growth\]. This will allow us to introduce the idea of energy stratification and the slicing of the energy density $W$ according to its features, Section \[sec:energy-stratification\]. We then define the crystalline mean curvature $\Lambda$ on various features of the evolving surface such as edges and facets, Section \[sec:crystalline curvature\], and establish its properties, including a comparison principle. At this point we introduce the notion of viscosity solutions, Section \[sec:viscosity solutions\], and construct faceted test functions in Section \[sec:faceted functions\]. The comparison principle for viscosity solutions is established in Section \[sec:comparison principle\], followed by the stability results, Section \[se:stability\]. Finally, the main result on the well-posedness of is presented in Section \[sec:well-posedness\].
Convex functionals with linear growth {#sec:convex func lin growth}
=====================================
There are a considerable number of publications on the topic of convex functionals with linear growth, see [@ACM] for a list of references. In this section we review the rather standard notation and results that we will use throughout the paper, and prove two important lemmas that will allow us to better understand the crystalline mean curvature later.
Suppose that $W: {\ensuremath{\mathbb{R}}}^d \to {\ensuremath{\mathbb{R}}}$, $d \geq 1$, is a convex function that satisfies the growth condition $$\begin{aligned}
\label{growth-condition}
{\left|W(p)\right|} \leq M (1 + {\left|p\right|}), \qquad p \in {\ensuremath{\mathbb{R}}}^d,\end{aligned}$$ for some $M > 0$. Note that it is usually also assumed that $W(p) \geq c{\left|p\right|}$ for some $c > 0$, or that $W(p) = W(-p)$, but we make no such assumption since they are unnecessary for our purposes, and in fact we need the generality.
Let $\Omega$ be either an bounded open subset of ${\ensuremath{\mathbb{R}}}^d$ or the $d$-dimensional flat torus ${\ensuremath{\mathbb{R}}}^d / L {\ensuremath{\mathbb{Z}}}^d$. We are interested in the functional $E_W(\cdot; \Omega): L^2(\Omega) \to {\ensuremath{\mathbb{R}}}$ defined as $$\begin{aligned}
\label{EW}
E_W(\psi; \Omega) &=
\begin{cases}
\int_\Omega W(D\psi) & \psi \in L^2(\Omega) \cap BV(\Omega),\\
+\infty & \text{otherwise}
\end{cases}\end{aligned}$$ that is understood as the *relaxation* (also the *closure* or the *lower semi-continuous envelope*) of the functional $$\begin{aligned}
\label{functional-w11}
\psi \mapsto
\begin{cases}
\int_\Omega W(\nabla\psi) & \psi \in L^2(\Omega) \cap W^{1,1}(\Omega),\\
+\infty & \text{otherwise}.
\end{cases}\end{aligned}$$
The relaxed functional $E_W$ can be expressed more explicitly following [@GiaquintaModicaSoucek; @BouchitteDalMaso]. Indeed, we introduce the recession function of $W$, $$\begin{aligned}
W0^+(p) = \lim_{\lambda\to 0+} \lambda W(\lambda^{-1} p),\end{aligned}$$ which is a positively one-homogeneous convex function on ${\ensuremath{\mathbb{R}}}^d$ due to the growth condition . If $W$ is one-homogeneous itself, we have $W0^+ = W$.
For $\psi \in BV(\Omega)$, $\nabla\psi$ will denote the Radon-Nikodým derivative of the absolutely continuous part of $D\psi$ with respect to the Lebesgue measure $L^d \lfloor \Omega$ and $D^s \psi$ will be the singular part. Then we have $$\begin{aligned}
D \psi = \nabla \psi L^d \lfloor \Omega + D^s \psi,\end{aligned}$$ and we can write $E_W$ as $$\begin{aligned}
\label{ew-decomp}
E_W(\psi; \Omega) =
\int_\Omega W(\nabla \psi) {\;dx}+ \int_\Omega W0^+{\left(\frac{D^s \psi}{{\left|D^s \psi\right|}}\right)} \;d{\left|D^s\psi\right|},\end{aligned}$$ where $\frac{D^s \psi}{{\left|D^s \psi\right|}}$ is the Radon-Nikodým derivative of $D^s\psi$ with respect to ${\left|D^s\psi\right|}$. We note that if $\psi \in L^2(\Omega) \cap W^{1,1}(\Omega)$, or even $\psi \in {{\rm Lip}}(\Omega)$, then this formula simplifies to since $D^s \psi = 0$.
Subdifferentials
----------------
Since $E_W(\cdot; \Omega)$ is a proper closed (that is, lower semi-continuous) convex functional on $L^2(\Omega)$, its subdifferential $$\begin{aligned}
\partial E_W(\psi; \Omega) = {{\left\{
v \in L^2(\Omega):
E_W(\psi + h; \Omega) - E_W(\psi; \Omega) \geq (h, v)
\text{ for all $h \in L^2(\Omega)$}\right\}}}\end{aligned}$$ is a closed convex, possibly empty subset of the Hilbert space $L^2(\Omega)$ equipped with the inner product $(h, v) := \int_\Omega h v {\;dx}$. If $\partial E_W(\psi; \Omega)$ is nonempty, we say that $\psi \in {\operatorname{\mathcal{D}}}(\partial E_W(\cdot; \Omega))$, the *domain* of the subdifferential, and we define the *minimal section* (also known as the *canonical restriction*) $\partial^0 E_W(\psi; \Omega)$ of the subdifferential as the unique element of $\partial E_W(\psi; \Omega)$ with the minimal norm in $L^2(\Omega)$.
The characterization of the subdifferential of $E_W$ is well-known when $W$ is a positively one-homogeneous function, that is, when $$\begin{aligned}
W(tp) = t W(p) \qquad t \geq 0.\end{aligned}$$ We will need this characterization for Lipschitz functions only, and we therefore present it in this simplified settings. Let $\Omega$ be an open subset of ${\ensuremath{\mathbb{R}}}^d$ or a $d$-dimensional torus ${\ensuremath{\mathbb{R}}}^d / L {\ensuremath{\mathbb{Z}}}^d$ for some $L > 0$. Following [@Anzellotti], let us introduce the space of vector fields with $L^2$ divergence, $$\begin{aligned}
X_2(\Omega) = {{\left\{z \in L^\infty(\Omega; {\ensuremath{\mathbb{R}}}^d): {\operatorname{div}}z \in L^2(\Omega)\right\}}}.\end{aligned}$$ For given $\psi \in {{\rm Lip}}(\Omega)$, we define the set of *Cahn-Hoffman vector fields* on $\psi$ as $$\begin{aligned}
\label{cahn-hoffman}
{{CH}}_W(\psi; \Omega) := {{\left\{z \in X_2(\Omega): z(x) \in \partial W(\nabla \psi(x)) \text{ a.e. $x \in \Omega$}\right\}}}.\end{aligned}$$ Note that the set $$\begin{aligned}
\label{div-cahn-hoffman}
{\operatorname{div}}{{CH}}_W(\psi; \Omega) := {{\left\{{\operatorname{div}}z: z \in {{CH}}_W(\psi; \Omega)\right\}}}\end{aligned}$$ is a closed convex, possibly empty subset of $L^2(\Omega)$. We have the well-known characterization of the subdifferential of $E_W$ in the periodic case, see [@ACM Section 1.3] or [@Moll].
\[pr:subdiff-char-periodic\] Let $\Omega = {\ensuremath{\mathbb{R}}}^d / L {\ensuremath{\mathbb{Z}}}^d$ for some $d \in {\ensuremath{\mathbb{N}}}$ and $L > 0$, and assume that $W$ is a positively one-homogeneous convex function on ${\ensuremath{\mathbb{R}}}^d$. If $\psi \in {{\rm Lip}}(\Omega)$ then $$\begin{aligned}
\partial E_W(\psi; \Omega) = {{\left\{-{\operatorname{div}}z: z \in {{CH}}_W(\psi; \Omega)\right\}}} =
-{\operatorname{div}}{{CH}}_W(\psi; \Omega).\end{aligned}$$
If $\Omega$ is a bounded open subset of ${\ensuremath{\mathbb{R}}}^d$ with a Lipschitz boundary, then the subdifferential is given by the vector fields $z \in {{CH}}_W(\psi; \Omega)$ such that $[z \cdot \nu] = 0$ on $\partial\Omega$; see [@ACM] for details. We will work on periodic domains to not have to deal with this technicality. We will see later (Lemma \[le:cahn-hoffman-patch\] and Properties \[pr:lambda-well-defined\]) that this does not change the value of the crystalline curvature on the facet.
Let us also mention one trivial result concerning the subdifferential of one-homogeneous convex functions on ${\ensuremath{\mathbb{R}}}^d$.
\[le:one-homogeneous-subdiff\] Suppose that $W$ is positively one-homogeneous convex function on ${\ensuremath{\mathbb{R}}}^d$. Then $\partial W(p) \subset \partial W(0)$ for any $p \in {\ensuremath{\mathbb{R}}}^d$. We also have $(x - y) \perp p$ for any $x, y \in \partial W(p)$ and any $p \in {\ensuremath{\mathbb{R}}}^d$.
The resolvent problem and the approximation by regularized functionals {#sec:resolvent-approximation}
----------------------------------------------------------------------
Let $W$ be a convex function satisfying the growth condition . For some flat torus $\Gamma = {\ensuremath{\mathbb{R}}}^d / L {\ensuremath{\mathbb{Z}}}^d$, $d \geq 1$, we want to approximate $E_W(\cdot; \Gamma)$ defined in by certain regularized functionals.
Suppose therefore that ${{\left\{W_m\right\}}}_{m\in {\ensuremath{\mathbb{N}}}}$ is a sequence of convex functions on ${\ensuremath{\mathbb{R}}}^d$ that satisfies the following:
1. ${{\left\{W_m\right\}}}_{m\in{\ensuremath{\mathbb{N}}}}$ is a decreasing sequence,
2. $W_m \in C^2({\ensuremath{\mathbb{R}}}^d)$,
3. $W_m \searrow W$ as $m\to\infty$ locally uniformly on ${\ensuremath{\mathbb{R}}}^d$,
4. there exist positive numbers $a_m$ such that $a_m^{-1} I \leq \nabla_p^2 W_m(p) \leq a_m I$ for all $p \in {\ensuremath{\mathbb{R}}}^d$, $m\in{\ensuremath{\mathbb{N}}}$, where $I$ is the $d\times d$ identity matrix.
We introduce the regularized functionals $$\begin{aligned}
E_m(\psi; \Gamma) := \begin{cases}
\int_{{\ensuremath{{\mathbb{R}^{{n}}}}}}W_m(\nabla \psi) {\;dx}& \psi \in H^1(\Gamma),\\
+\infty & \psi \in L^2(\Gamma) \setminus H^1(\Gamma),
\end{cases}\end{aligned}$$ where $H^k(\Gamma) := W^{k,2}(\Gamma)$ is the standard Sobolev space of $L{\ensuremath{\mathbb{Z}}}^d$-periodic functions.
Let us give an example of a regularized $W_m$ first.
\[ex:wm-example\] Let $\eta_m$ be the standard mollifier with support of radius $1/m$. Define the smoothing $$\begin{aligned}
W_m(p) = (W * \eta_m)(p) + \frac 1{2m} {\left|p\right|}^2 \qquad p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}.\end{aligned}$$ By convexity we have $W_m \geq W$ and $W_m$ convex, $W_m \in C^\infty({\ensuremath{\mathbb{R}}}^d)$, $\nabla^2 W_m \geq \frac 1m I$ and $W_m \searrow W$ as $m\to0$ locally uniformly. The uniform upper bound on $\nabla^2 W_m$ follows immediately from $\partial_{p_i p_j}(W * \eta_m)
= \partial_{p_i} W * \partial_{p_j} \eta_m$, and the right-hand side is bounded since $\nabla
W$ is bounded.
We need the following result similar to [@GGP13JMPA Proposition 5.1].
\[pr:energy-convergence\]
1. $E_m(\cdot; \Gamma)$ form a decreasing sequence of proper closed convex functionals on $L^2(\Gamma)$.
2. The subdifferential $\partial E_m$ is a singleton for all $$\begin{aligned}
\psi \in {\operatorname{\mathcal{D}}}(\partial E_m) = H^2(\Gamma)\end{aligned}$$ containing the unique element $$\begin{aligned}
-{\operatorname{tr}}{\left[{\left(\nabla_p^2 W_m\right)} {\left(\nabla \psi\right)} \nabla^2 \psi\right]} \qquad \text{a.e.}\end{aligned}$$
3. $(\inf_m E_m(\cdot; \Gamma))_* = E_W(\cdot; \Gamma)$, the lower semi-continuous envelope of $\inf_m E_m$ in $L^2(\Gamma)$.
For (a) and (b) see [@Evans Section 9.6.3].
(c): ${{\left\{E_m(\cdot; \Gamma)\right\}}}$ is decreasing since ${{\left\{W_m\right\}}}$ is decreasing. Therefore $E_m(\psi; \Gamma) \to E_W(\psi; \Gamma)$ for any $\psi \in H^1(\Gamma)$ by the Dominated convergence theorem, since $E_W$ is of the form in this case. If $\psi \notin H^1(\Gamma)$, $E_m(\psi; \Gamma) = \infty$ by definition and therefore $E_W(\cdot; \Gamma) \leq \inf_m E_m(\cdot; \Gamma)$, with equality on $H^1(\Gamma)$. Let us now denote $F(\psi) = \inf_m E_m(\psi; \Gamma)$. By a standard approximation result, for any $\psi \in BV(\Gamma)$ there exists a sequence ${{\left\{\psi_k\right\}}} \subset C^\infty(\Gamma) \cap BV(\Gamma) \subset H^1(\Gamma)$ such that $\psi_k \to \psi$ in $L^2(\Omega)$ and $\int_\Gamma {\left|D\psi_k\right|} \to \int_\Gamma {\left|D\psi\right|}$, which yields $E_W(\psi_k; \Gamma) \to E_W(\psi; \Gamma)$ due to [@Resetnjak]; see [@GiaquintaModicaSoucek]. In particular, $$\begin{aligned}
F_*(\psi) \leq \liminf_{k\to\infty} F(\psi_k) = \liminf_{k\to\infty} E_W(\psi_k; \Gamma)
= E_W(\psi; \Gamma).\end{aligned}$$ Hence $F_* = E_W$ by the lower semi-continuity of $E_W$.
We will need the following approximation and convergence result for the resolvent problems.
\[pr:resolvent-problems\] For $\psi \in {{\rm Lip}}(\Gamma)$, and $m \in {\ensuremath{\mathbb{N}}}$, $a > 0$, the resolvent problems $$\begin{aligned}
\psi_a + a \partial E_W(\psi_a; \Gamma) \ni \psi,\\
\psi_{a,m} + a \partial E_m(\psi_{a,m}; \Gamma) \ni \psi,\end{aligned}$$ admit unique solutions $\psi_a$ and $\psi_{a,m}$ in $L^2(\Gamma)$, respectively. Moreover, $\psi_a$ and $\psi_{a,m}$ are Lipschitz continuous and $$\begin{aligned}
{\left\|\nabla \psi_a\right\|}_\infty, {\left\|\nabla \psi_{a,m}\right\|}_\infty \leq
{\left\|\nabla \psi\right\|}_\infty.\end{aligned}$$ Finally, $\psi_{a,m} \in C^{2,{\ensuremath{\alpha}}}(\Gamma)$ for some ${\ensuremath{\alpha}}= \alpha_m > 0$.
We also introduce the functions $$\begin{aligned}
h_a := \frac{\psi_a - \psi}{a}, &&&
h_{a,m} := \frac{\psi_{a,m} - \psi}{a} = -{\operatorname{tr}}{\left[(\nabla_p^2 W_m)(\nabla \psi_{a,m}) \nabla^2 \psi_{a,m}\right]}.\end{aligned}$$ Then, for fixed $a>0$, $$\begin{aligned}
\psi_{a,m} &\rightrightarrows \psi_a && \text{uniformly as $m \to \infty$, and},\\
h_{a,m} &\rightrightarrows h_a && \text{uniformly as $m \to \infty$}.
\intertext{Moreover,}
\psi_a &\rightrightarrows \psi && \text{uniformly as $a \to 0$.}
\intertext{If furthermore $\psi \in {\operatorname{\mathcal{D}}}{\left(\partial E_W(\cdot; \Gamma)\right)}$ then
also}
h_a &\to -\partial^0 E(\psi; \Gamma)&& \text{in $L^2(\Gamma)$ as $a \to 0$.}\end{aligned}$$
We follow the proof of [@GGP13JMPA Proposition 5.3]. Due to Proposition \[pr:energy-convergence\](a), [@Attouch Theorem 3.20] implies the *Mosco convergence* of $E_m$ to $E$. This yields the resolvent convergence [@Attouch Theorem 3.26], namely, for fixed $a > 0$ we have $$\begin{aligned}
\label{res-l2-conv}
\psi_{a,m} \to \psi_a \quad \text{in $L^2(\Gamma)$.}\end{aligned}$$
The $C^{2,\alpha}$ regularity of $\psi_{a,m}$ is standard from the elliptic theory, as $I + a \partial^0 E_m(\cdot; \Gamma)$ is a quasilinear uniformly elliptic operator as noted in Proposition \[pr:energy-convergence\].
Since the $E_m$-resolvent problem is translation invariant and has a maximum principle, we find that $\psi_{a,m}$ is Lipschitz since $\psi$ is Lipschitz, and $$\begin{aligned}
{\left\|\nabla \psi_{a,m}\right\|}_\infty \leq {\left\|\nabla \psi\right\|}_\infty.\end{aligned}$$ Therefore the Arzelá-Ascoli theorem and yield the uniform convergence of $\psi_{a,m} \to \psi_a$ and $h_{a,m} \to h_a$ as $m\to\infty$ for fixed $a > 0$, and hence also the Lipschitz bound ${\left\|\nabla \psi_a\right\|}_\infty \leq {\left\|\nabla \psi\right\|}_\infty$. Moreover, since the $E_m$-resolvent problem has a maximum principle, the $E_W$-resolvent problem has a maximum principle as well.
Finally, a standard result implies that $\psi_a \to \psi$ in $L_2(\Gamma)$ as $a\to0$ [@Attouch Theorem 3.24], therefore with Arzelá-Ascoli and the uniform Lipschitz bound we conclude that $\psi_a \to \psi$ uniformly. If furthermore $\psi \in {\operatorname{\mathcal{D}}}(\partial E_W(\cdot; \Gamma))$, also $h_a \to -\partial^0 E_W(\psi; \Gamma)$ [@Attouch Proposition 3.56].
We give a lemma on the Mosco convergence of functionals with linear growth.
\[le:lingrowthapproximation\] Suppose that $W_m$ are convex positively one-homogeneous functions such that $W_m \rightrightarrows
W$ uniformly on the unit ball. Then $E_m(\psi) = \int_\Gamma W_m(\nabla \psi)$ Mosco-converges to $E(\psi) = \int_\Gamma W(D\psi)$ as $m \to \infty$.
By [@Attouch Proposition 3.19], we need to show that for every $\psi$, $\psi_m
\stackrel{w}{\to} \psi$ weakly in $L^2(\Gamma)$ we have $E(\psi) \leq \liminf_m E_m(\psi_m)$ and that for every $\psi \in L^2(\Gamma)$ there exists a sequence $\psi_m \to
\psi$ strongly in $L^2(\Gamma)$ such that $E(\psi) = \lim_m E_m(\psi_m)$.
If $\psi_m \stackrel{w}\to \psi$ weakly in $L^2(\Gamma)$, we can deduce $E(\psi) \leq \liminf_m
E_m(\psi_m)$ from the formula [@AB] $$\begin{aligned}
E(\psi) := \sup \Big\{\int_\Gamma \psi {\operatorname{div}}\varphi: &\varphi \in
C^1(\Gamma), {\left\|\varphi\right\|}_\infty \leq 1,\\ &\varphi(x) \cdot p \leq 1 \text{ whenever } W(p) \leq
1, x \in \Gamma\Big\}.\end{aligned}$$
By a standard approximation result, for any $\psi \in BV(\Gamma)$ there exists a sequence ${{\left\{\psi_k\right\}}} \subset C^\infty(\Gamma) \cap BV(\Gamma) \subset W^{1, 1}(\Gamma)$ such that $\psi_k \to \psi$ in $L^2(\Omega)$ and $\int_\Gamma {\left|D\psi_k\right|} \to \int_\Gamma {\left|D\psi\right|}$, which yields $E(\psi_m) \to E(\psi)$ by the theorem of Rešetnjak [@Resetnjak]. On the other hand, by the uniform convergence of $W_m$ to $W$ on the unit ball we have for any $\xi \in W^{1,1}(\Gamma)$ $$\begin{aligned}
{\left|\int_\Gamma W_m(\nabla\xi) - W(\nabla \xi) {\;dx}\right|} \leq &\int_\Gamma |\nabla \xi|
{\left|W_m{\left(\frac{\nabla\xi}{|\nabla \xi|}\right)} - W{\left(\frac{\nabla\xi}{|\nabla \xi|}\right)}\right|} {\;dx}\\
&\leq \int_\Gamma |\nabla \xi| {\;dx}{\left\|W_m - W\right\|}_{L^\infty(B_1(0))}.\end{aligned}$$ Therefore $E(\psi) = \lim_m E_m(\psi_m)$.
Cahn-Hoffman vector field patching
----------------------------------
We shall use the minimal section $\partial^0 E_W(\psi; \Omega)$ of the subdifferential of $E_W$ to define the crystalline curvature for a given Lipschitz function $\psi$ on $\Omega$. However, the minimal section is a solution of a variational problem and therefore its value might depend strongly on the set $\Omega$, and nonlocally on the values of $\psi$. Fortunately, the situation is not as dire as it might appear at first, and in fact, the minimal section is nonlocal only on flat parts (facets) of $\psi$. This restriction of nonlocality is expressed by the following lemma. Intuitively, we can patch the Cahn-Hoffman vector fields as much as we please as long as we do it across the level sets of $\psi$.
\[le:cahn-hoffman-patch\] Let $W: {\ensuremath{\mathbb{R}}}^d \to {\ensuremath{\mathbb{R}}}$ be a positively one-homogeneous convex function, $d \geq 1$. Suppose that $\psi_1 \in {{\rm Lip}}(\Omega_1)$ and $\psi_2 \in {{\rm Lip}}(\Omega_2)$ are two Lipschitz functions on two open subsets $\Omega_1, \Omega_2$ of ${\ensuremath{\mathbb{R}}}^d$. Let $G = {{\left\{x \in \Omega_1: a < \psi_1(x) < b\right\}}}$ for some $a < b$ such that ${\overline{G}} \subset \Omega_1 \cap \Omega_2$ and $\psi_1 = \psi_2$ on $G$. If $z_i \in {{CH}}_W(\psi_i; \Omega_i)$ are two Cahn-Hoffman vector fields, then $$\begin{aligned}
\label{z-patch}
z(x) =
\begin{cases}
z_2(x) & x \in G,\\
z_1(x) & x \in \Omega_1 \setminus G,
\end{cases}\end{aligned}$$ is also a Cahn-Hoffman vector field $z \in {{CH}}_W(\psi_1; \Omega_1)$, and $$\begin{aligned}
\label{divz-patch}
{\operatorname{div}}z(x) =
\begin{cases}
{\operatorname{div}}z_2(x) & \text{a.e. } x \in G,\\
{\operatorname{div}}z_1(x) & \text{a.e. } x \in \Omega_1 \setminus G.
\end{cases}\end{aligned}$$
Since adding the same constant to both $\psi_1$ and $\psi_2$ does not change anything, we can assume that $a = -\delta$ and $b = \delta$ for some $\delta > 0$. For given ${\ensuremath{\varepsilon}}\in (0, \delta)$ we introduce the Lipschitz function $$\begin{aligned}
\zeta_{\ensuremath{\varepsilon}}(x) = 1 + \max {\left( -1, \min {\left(0, \frac{{\left|\psi_1(x)\right|} - \delta}{\ensuremath{\varepsilon}}\right)}\right)}.\end{aligned}$$ Note that $\zeta_{\ensuremath{\varepsilon}}= 0$ on ${{\left\{{\left|\psi_1\right|} \leq \delta - {\ensuremath{\varepsilon}}\right\}}}$ and $\zeta_{\ensuremath{\varepsilon}}= 1$ on ${{\left\{{\left|\psi_1\right|} \geq \delta\right\}}}$. Furthermore, $$\begin{aligned}
\label{zeta-deriv}
\nabla \zeta_{\ensuremath{\varepsilon}}(x) =
\begin{cases}
{\operatorname{sign}}\psi_1(x)\frac{\nabla \psi_1(x)}{\ensuremath{\varepsilon}}& \delta - {\ensuremath{\varepsilon}}< \psi_1(x) < \delta,\\
0 & \text{otherwise}
\end{cases}\end{aligned}$$ for a.e. $x$. Finally, $\zeta_{\ensuremath{\varepsilon}}\searrow \chi_{\Omega_1 \setminus G}$ monotonically pointwise as ${\ensuremath{\varepsilon}}\to 0$.
Now for $\rho > 0$ we define $z_i^\rho = z_i * \eta_\rho$, where $\eta_\rho$ is the standard mollifier with radius $\rho$, and we extend $z_i$ as $0$ to $\Omega_i^c$. We have $z_i^\rho \to z_i$ in $L^\infty(\Omega_i)$-weak$^*$ and strongly in $L^p_{\rm loc}(\Omega_i)$ for any $1 \leq p < \infty$ as well as ${\operatorname{div}}z_i^\rho \to {\operatorname{div}}z_i$ strongly in $L^2_{{\rm loc}}(\Omega_i)$ as $\rho \to 0$, $i = 1,2$. Define $$\begin{aligned}
z_{\ensuremath{\varepsilon}}^\rho = z_1^\rho \zeta_{\ensuremath{\varepsilon}}+ z_2^\rho (1 - \zeta_{\ensuremath{\varepsilon}}).\end{aligned}$$ This function is clearly Lipschitz.
On $G$ we have $z_i(x) \in \partial W(\nabla \psi_1(x)) = \partial W(\nabla \psi_2(x))$ for a.e. $x$. Therefore $(z_1(x) - z_2(x)) \cdot \nabla \psi_1(x) = 0$ for a.e. $x \in G$ by Lemma \[le:one-homogeneous-subdiff\], which together with implies $$\begin{aligned}
\label{zetae-ortho}
\nabla \zeta_{\ensuremath{\varepsilon}}\cdot (z_1 - z_2) = 0 \qquad \text{a.e.}\end{aligned}$$ Thus we have for any $\varphi \in C^\infty_c(\Omega_1)$ $$\begin{aligned}
\int z_{\ensuremath{\varepsilon}}^\rho \cdot \nabla \varphi
&= -\int \varphi {\operatorname{div}}z^\rho\\
&= -\int \varphi {\left[\zeta_{\ensuremath{\varepsilon}}{\operatorname{div}}z_1^\rho + (1- \zeta_{\ensuremath{\varepsilon}}) {\operatorname{div}}z_2^\rho + \nabla \zeta_{\ensuremath{\varepsilon}}\cdot (z_1^\rho - z_2^\rho)\right]}.\end{aligned}$$ Now we send $\rho \to 0$ and obtain $$\begin{aligned}
\int z_{\ensuremath{\varepsilon}}\cdot \nabla \varphi &=
-\int \varphi {\left[\zeta_{\ensuremath{\varepsilon}}{\operatorname{div}}z_1 + (1- \zeta_{\ensuremath{\varepsilon}}) {\operatorname{div}}z_2 + \nabla \zeta_{\ensuremath{\varepsilon}}\cdot (z_1 - z_2)\right]}\\
&= -\int \varphi {\left[\zeta_{\ensuremath{\varepsilon}}{\operatorname{div}}z_1 + (1- \zeta_{\ensuremath{\varepsilon}}) {\operatorname{div}}z_2\right]},\end{aligned}$$ where we used . Finally we send ${\ensuremath{\varepsilon}}\to 0$ and use the Dominated convergence theorem to conclude that $$\begin{aligned}
\int z \cdot \nabla \varphi = - \int \varphi {\left[\chi_{\Omega_1 \setminus G} {\operatorname{div}}z_1 + \chi_G {\operatorname{div}}z_2\right]}.\end{aligned}$$ Since this holds for any test function, we see that ${\operatorname{div}}z \in L^2(\Omega_1)$ and it can be expressed as in .
\[arb-convex-patch\] We can take an arbitrary convex combination of $z_1$ and $z_2$ on $G$ in . Indeed, take $z$ as in $\eqref{z-patch}$. Then $\lambda z_1 + (1-\lambda) z = (\lambda z_1 + (1 - \lambda) z_2) \chi_G + z_1 \chi_{G_1 \setminus G} \in {{CH}}_W(\psi_1; G_1)$ by convexity.
\[re:patching-on-facet-boundaries\] In the proof of [@GGP13JMPA Proposition 2.10] in the case of $W$ with a smooth $1$-level set we used the fact that Cahn-Hoffman vector fields can be patched across the boundary of a facet arbitrarily, as a consequence of [@GGP13JMPA Proposition 2.8]. This is stronger than Lemma \[le:cahn-hoffman-patch\] above where we can patch the Cahn-Hoffman vector field only if the support functions coincide on a neighborhood of the facet. We believe that this requirement can be removed as in [@GGP13JMPA], but we do not pursue this matter further in the current paper.
Finally, let us briefly consider the characterization of the subdifferential of $E_W$ in the case when $W$ is not positively one-homogeneous. Proposition \[pr:subdiff-char-periodic\] does not apply in such a case. However, if $W$ is equal to a positively one-homogeneous function $W'$ in the neighborhood of the origin, the subdifferentials of $E_W$ and $E_{W'}$ coincide at least for functions with small Lipschitz constant.
\[le:subdiff-homog-relation\] Suppose that $W$ is a convex function and $W'$ is a positively one-homogeneous convex function on ${\ensuremath{\mathbb{R}}}^d$, $d \geq 1$, and there exists ${\ensuremath{\varepsilon}}> 0$ such that $W(p) = W'(p)$ for ${\left|p\right|} < {\ensuremath{\varepsilon}}$. Suppose that $\Omega$ is a bounded open subset of ${\ensuremath{{\mathbb{R}^{{n}}}}}$ or the torus ${\ensuremath{\mathbb{R}}}^d / L {\ensuremath{\mathbb{Z}}}^d$ for some $L > 0$. If $\psi \in {{\rm Lip}}(\Omega)$ and ${\left\|\nabla \psi\right\|}_\infty < {\ensuremath{\varepsilon}}$, then $$\begin{aligned}
\partial E_W(\psi; \Omega) = \partial E_{W'}(\psi; \Omega).\end{aligned}$$
We shall denote the functionals as $E$ and $E'$ for short. Fix $\psi \in {{\rm Lip}}(\Omega)$ with ${\left\|\nabla \psi\right\|}_\infty < {\ensuremath{\varepsilon}}$. By definition of the functionals and our assumption on the equality of $W$ and $W'$, we have $$\begin{aligned}
\label{en-equality}
E(\psi + h) = E'(\psi + h) \qquad h \in {{\rm Lip}}(\Omega),\ {\left\|\nabla h\right\|}_\infty < \delta = {\ensuremath{\varepsilon}}- {\left\|\nabla \psi\right\|}_\infty.\end{aligned}$$
The convexity of $W$, $W'$, and one-homogeneity of $W'$ imply for $p \in {\ensuremath{\mathbb{R}}}^d$ and $\lambda \in (0,1)$ such that $\lambda {\left\|p\right\|} < {\ensuremath{\varepsilon}}$ $$\begin{aligned}
\lambda W(p) \geq W(\lambda p) - (1 -\lambda) W(0) = W'(\lambda p) = \lambda W'(p).\end{aligned}$$ In particular, $W(p) \geq W'(p)$ on ${\ensuremath{\mathbb{R}}}^d$. Therefore $E(\psi + h) - E(\psi) \geq E'(\psi + h) - E'(\psi)$ for all $h \in L^2(\Omega)$ since $E(\psi) = E'(\psi)$. We conclude that $\partial E'(\psi) \subset \partial E(\psi)$.
To prove the opposite inclusion, take $v \in \partial E(\psi)$, if such an element exists. We want to prove $$\begin{aligned}
\label{subdiff-def}
E'(\psi + h) - E'(\psi) \geq (h, v) \qquad \text{for all $h \in L^2(\Omega)$.}\end{aligned}$$ If $h \notin BV(\Omega)$, $E'(\psi + h) = \infty$ by definition. Thus we can assume that $h \in BV(\Omega)$. By a standard approximation result, there exists a sequence ${{\left\{h_m\right\}}} \subset C^\infty(\Omega) \cap BV(\Omega)$ such that $h_m \to h$ in $L^2(\Omega)$ and $Dh_m \to Dh$ weakly$^*$ as measures, which yields $E'(\psi + h_m) \to E'(\psi + h)$ due to [@Resetnjak]; see [@GiaquintaModicaSoucek]. But we can choose $\lambda_m \in (0,1)$ such that $\lambda_m {\left\|\nabla h_m\right\|}_\infty < \delta$.
Then implies $$\begin{aligned}
E'(\psi + \lambda_m h_m) - E'(\psi) =
E(\psi + \lambda_m h_m) - E(\psi) \geq \lambda_m (h_m, v).\end{aligned}$$ By convexity, we have $$\begin{aligned}
E'(\psi + h_m) - E'(\psi) \geq (h_m, v).\end{aligned}$$ Indeed, $$\begin{aligned}
\lambda_m E'(\psi + h_m) + (1-\lambda_m) E'(\psi)
&\geq E'(\lambda_m(\psi + h_m) + (1-\lambda_m)\psi)\\
&=E'(\psi +\lambda_m h_m) \geq E'(\psi) +\lambda_m(h_m, v).\end{aligned}$$ Sending $m \to \infty$ yields .
Energy stratification {#sec:energy-stratification}
=====================
In this section we shall assume that $W$ is a convex **polyhedral** function on ${{\ensuremath{{\mathbb{R}^{{n}}}}}}$. Since $W$ is polyhedral, it can be locally viewed as a positively one-homogeneous convex function; we will give a detailed explanation in this section. The features of $W$ correspond to the dual features of the crystal such as facets, edges and vertices, depending on the dimension. For each gradient, we will decompose the space into orthogonal subspaces of interesting directions, corresponding to the given feature of the crystal, and the directions in which $W$ is linear and therefore its behavior simple.
Slicing of $W$ {#sec:slicing of W}
--------------
To perform the decomposition, we need a few standard concepts from convex analysis (see for example [@Rockafellar]). For a given convex set $C$ let ${\operatorname{aff}}C$ denote the affine hull of $C$, that is, the smallest affine space containing $C$. The dimension of the convex set is defined as the dimension of its affine hull, $\dim C := \dim {\operatorname{aff}}C$. Let ${\operatorname{ri}}C$ be the relative interior of $C$ with respect to ${\operatorname{aff}}C$. A convex set is said to be relatively open if $C = {\operatorname{ri}}C$. We know that ${\operatorname{ri}}C \neq \emptyset$ if $C \neq \emptyset$ ([@Rockafellar Theorem 6.2]). We say that ${\operatorname{aff}}C$ is parallel to a subspace $V \subset {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ if ${\operatorname{aff}}C = p + V$ for some $p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$.
We can decompose ${{\ensuremath{{\mathbb{R}^{{n}}}}}}$ based on the features of the crystal, which correspond to the value of $\partial W$.
\[pr:feature-decomposition\] For given $W$ polyhedral with $W < \infty$ on ${{\ensuremath{{\mathbb{R}^{{n}}}}}}$ there exist a finite number of mutually disjoint maximal sets $\Xi_i$, $i \in \mathcal N$, such that ${{\ensuremath{{\mathbb{R}^{{n}}}}}}= \bigcup_{i \in \mathcal N} \Xi_i$ and $\partial W$ is constant on each $\Xi_i$. Furthermore, each $\Xi_i$ is a relatively open convex set and ${\operatorname{aff}}\Xi_i \perp {\operatorname{aff}}\partial W(p)$ for $p \in \Xi_i$ in the sense that whenever $p,
q \in \Xi_i$ and $\xi, \zeta \in \partial W(p)$ then $p - q \perp \xi - \zeta$.
We use the projections of relative interiors of the non-empty faces of the epigraph ${\operatorname{epi}}W :=
{{\left\{(p, \lambda): \lambda \geq W(p), p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}\right\}}}$, other than ${\operatorname{epi}}W$ itself, onto ${{\ensuremath{{\mathbb{R}^{{n}}}}}}$. For the definition of a face of a convex set see [@Rockafellar Section 18]. By [@Rockafellar Corollary 18.1.3], all faces of ${\operatorname{epi}}W$ other than ${\operatorname{epi}}W$ itself must lie in the relative boundary of ${\operatorname{epi}}W$. The relative boundary of ${\operatorname{epi}}W$, the set ${\operatorname{epi}}W \setminus
{\operatorname{ri}}{\operatorname{epi}}W$, is just the regular boundary and therefore it is the graph of $W$, ${\operatorname{graph}}W := {{\left\{(p, W(p)): p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}\right\}}} \subset {\ensuremath{\mathbb{R}}}^{n+1}$. By [@Rockafellar Theorem 18.2], the relative interiors $\hat \Xi_i$ of the faces of ${\operatorname{epi}}W$ other than ${\operatorname{epi}}W$ itself form a partition of ${\operatorname{graph}}W$. By projecting these relative interiors $\hat \Xi_i$ onto ${{\ensuremath{{\mathbb{R}^{{n}}}}}}$ we obtain sets $\Xi_i$, which form a partition of ${{\ensuremath{{\mathbb{R}^{{n}}}}}}$ and are again relatively open by [@Rockafellar Theorem 6.6].
Let us now prove that $\partial W$ is constant on $\Xi_i$. Fix two points $p, q \in \Xi_i$. Since $\hat \Xi_i$ are relatively open, there exists $\mu > 1$ such that $W(\mu p + (1 - \mu) q) = \mu
W(p) + (1- \mu) W(q)$. Let $\xi \in \partial W(p)$. By definition of the subdifferential, we have $W(\mu p + (1 - \mu) q) \geq W(p) + (\mu - 1) \xi \cdot (p - q)$ and $W(q) \geq W(p) + \xi \cdot (q
- p)$. Using the equality in the first inequality and dividing by $\mu - 1$ we obtain $W(q) \leq
W(p) + \xi \cdot (q - p)$. Therefore $W(q) - W(p) = \xi \cdot (q - p)$ and we deduce that $\xi \in
\partial W(q)$. Finally, if $\zeta \in \partial W(p)$ as well, we have $(\zeta - \xi)
\cdot (q - p) = 0$. Maximality, that is, that $\partial W(p) \neq \partial W(q)$ for $p \in \Xi_i$, $q \in \Xi_j$, $i
\neq j$, follows from the definition of convex faces.
\[le:aff Xi origin\] Suppose that $\Xi_i$ are as in Proposition \[pr:feature-decomposition\] and suppose that $W$ is also positively one-homogeneous. Then $0 \in {\operatorname{aff}}\Xi_i$ for every $i$.
This follows immediately from one-homogeneity since $\partial W(p) = \partial W(tp)$ for any $p \in
{\ensuremath{\mathbb{R}}}^n$, $t > 0$.
Since $W$ is finite everywhere, $\partial W(p)$ is a nonempty closed convex set for any $p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$. For given $p_0 \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ we introduce the one-sided directional derivative of $W$ at $p_0$ with respect to a vector $p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ as ([@Rockafellar Section 23]) $$\begin{aligned}
W_{p_0}'(p) := \lim_{\lambda \to 0+} \frac{W(p_0 + \lambda p) - W(p_0)}{\lambda}.\end{aligned}$$ Then $W'(p_0; \cdot)$ is a positively one-homogeneous convex function, and ([@Rockafellar Theorem 23.4]) $$\begin{aligned}
\label{W'-convex-conjugate}
W_{p_0}'(p) \equiv \delta^*(p \mid \partial W(p_0)) := \sup {{\left\{p \cdot \xi: \xi \in \partial W(p_0)\right\}}}.\end{aligned}$$ In particular, $W_{p_0}'$ is the convex conjugate of the indicator function of $\partial W(p_0)$. Therefore by [@Rockafellar Theorem 13.4] the lineality space of $W_{p_0}'$ (the subspace of directions in which $W_{p_0}'$ is affine) is the orthogonal complement of the subspace parallel to ${\operatorname{aff}}\partial W(p_0)$. This provides the orthogonal decomposition of ${{\ensuremath{{\mathbb{R}^{{n}}}}}}$ for a given gradient.
\[pr:direction-decomposition\] Let $W$ be a polyhedral convex function on ${{\ensuremath{{\mathbb{R}^{{n}}}}}}$ finite everywhere and let $p_0 \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$. Let $V$ be the subspace of ${{\ensuremath{{\mathbb{R}^{{n}}}}}}$ parallel to ${\operatorname{aff}}\partial W(p_0)$ and set $U = V^\perp$. Then $W_{p_0}'$ is linear on $U$ and $$\begin{aligned}
\label{W'-linear}
W_{p_0}'(p) = W_{p_0}'(P_V p) + \xi \cdot P_U p \qquad \text{for any $p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$, $\xi \in {\operatorname{aff}}\partial W(p_0)$},\end{aligned}$$ where $P_U$ and $P_V$ are the orthogonal projections onto $U$ and $V$, respectively.
Moreover, there exists $\delta > 0$ such that $$\begin{aligned}
W(p) = W_{p_0}'(p - p_0) + W(p_0) \qquad \text{for all ${\left|p - p_0\right|} < \delta$.}\end{aligned}$$
follows from and from the orthogonality of $U$ and $V$.
The existence of $\delta > 0$ can be proved by contradiction: suppose that there exists a sequence ${{\left\{p_k\right\}}}$, $p_k \to p_0$ such that $W(p_k) - W(p_0) > W_{p_0}'(p_k - p_0)$ (it is clear that $W_{p_0}'(p - p_0) \leq W(p) - W(p_0)$ by convexity). Since $W$ is polyhedral, it is given as the maximum of a finite number of affine functions, and therefore by taking a subsequence we can assume that $W(p_k) = \xi \cdot p_k + c$ for some fixed $\xi \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$, $c \in {\ensuremath{\mathbb{R}}}$. By continuity we have $W(p_k) - W(p_0) = \xi \cdot (p_k - p_0)$. Therefore $\xi \in \partial W(p_0)$. But this yields a contradiction since then implies $W_{p_0}'(p_k - p_0) \geq \xi \cdot (p_k - p_0)$.
The previous proposition tells us that the behavior of $W$ is interesting only in the directions parallel to ${\operatorname{aff}}\partial W$. That motivates the following notation. For given $W: {{\ensuremath{{\mathbb{R}^{{n}}}}}}\to {\ensuremath{\mathbb{R}}}$ convex polyhedral and $p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ let $V$ be the subspace of ${{\ensuremath{{\mathbb{R}^{{n}}}}}}$ parallel to ${\operatorname{aff}}\partial W(p)$, $U = V^\perp$, $k = \dim V$, and we fix an arbitrary rotation $$\begin{aligned}
\label{rotation}
{{\mathcal T}}: {{\ensuremath{{\mathbb{R}^{{n}}}}}}\to {{\ensuremath{{\mathbb{R}^{{n}}}}}}\end{aligned}$$ that maps ${\ensuremath{\mathbb{R}}}^k \times {{\left\{0\right\}}}$ onto $V$ and ${{\left\{0\right\}}} \times {\ensuremath{\mathbb{R}}}^{n-k}$ onto $U$. For given $x \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$, we define the unique $x' \in {\ensuremath{\mathbb{R}}}^k$ and $x'' \in {\ensuremath{\mathbb{R}}}^{n-k}$ such that $$\begin{aligned}
\label{rotation'}
{{\mathcal T}}(x', x'') = x.\end{aligned}$$ We set ${{\mathcal T}}_V: {\ensuremath{\mathbb{R}}}^k \to V$ and ${{\mathcal T}}_U: {\ensuremath{\mathbb{R}}}^{n-k} \to U$ by $$\begin{aligned}
\label{rotationUV}
{{\mathcal T}}_V x' = {{\mathcal T}}(x', 0), \qquad {{\mathcal T}}_U x'' = {{\mathcal T}}(0, x'').\end{aligned}$$ In the above we also allow for $k = 0$ and $k = n$, in which case terms containing $x'$ respectively $x''$ simply do not appear in the formulas, and ${{\mathcal T}}_V$ respectively ${{\mathcal T}}_U$ are trivial maps. Note that $$\begin{aligned}
\left( {{\mathcal T}}_V z \right)' = z, \quad \left( {{\mathcal T}}_U w \right)'' = w, \qquad z \in {\ensuremath{\mathbb{R}}}^k, w \in
{\ensuremath{\mathbb{R}}}^{n-k},\end{aligned}$$ and $$\begin{aligned}
{{\mathcal T}}_V x' = P_V x, \quad {{\mathcal T}}_U x'' = P_U x, \qquad x \in {\ensuremath{\mathbb{R}}}^n,\end{aligned}$$ where $P_V$ and $P_U$ are respectively the orthogonal projections on $V$ and $U$. Since ${{\mathcal T}}$ is a linear isometry, it preserves the inner product $$\begin{aligned}
z_1 \cdot z_2 = {{\mathcal T}}_V z_1 \cdot {{\mathcal T}}_V z_2, \qquad z_1, z_2 \in {\ensuremath{\mathbb{R}}}^k,\end{aligned}$$ and similarly for ${{\mathcal T}}_U$.
We are free to choose any such ${{\mathcal T}}$, as long as we keep this choice consistent throughout the paper for given $W$ and $p$. We can in fact choose the same ${{\mathcal T}}$ for all $p \in \Xi_i$ from Proposition \[pr:feature-decomposition\].
We will introduce the sliced energy density ${W^{\rm sl}}$ that locally captures behavior of $W$ in the directions $V$.
\[def:sliced-W\] We define the *sliced density* ${W^{\rm sl}}_p: {\ensuremath{\mathbb{R}}}^k \to {\ensuremath{\mathbb{R}}}$ as $$\begin{aligned}
\label{W-red}
\begin{aligned}
{W^{\rm sl}}_p &:= W_p' \circ {{\mathcal T}}_V.
\end{aligned}\end{aligned}$$
\[le:decomposition-subdiff-W\] For any fixed $p_0 \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ we have $$\begin{aligned}
\partial W_{p_0}'(p) =
{{\left\{{{\mathcal T}}(\zeta', \xi''): \zeta' \in \partial {W^{\rm sl}}_{p_0}(p')\right\}}}
\qquad \text{for all $p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$, $\xi \in \partial W_{p_0}'(0)$}.\end{aligned}$$
The following lemma states that the behavior of $W$ in the neighborhood of some $p$ is completely captured by the sliced density ${W^{\rm sl}}_p$.
\[le:W-decomposition\] For every $p_0 \in {\ensuremath{\mathbb{R}}}^n$ there exists ${\ensuremath{\varepsilon}}> 0$ such that $$\begin{aligned}
\label{W-decomposition}
W(p) = {W^{\rm sl}}_{p_0}(p'-p_0')
+ P_U \xi \cdot (p - p_0) + W(p_0)\end{aligned}$$ for any $p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$, ${\left|p - p_0\right|} <{\ensuremath{\varepsilon}}$, $\xi \in \partial W(p)$. Since ${{\mathcal T}}$ is an isometry, we have $P_U \xi \cdot (p - p_0) = \xi'' \cdot (p'' - p_0'')$.
The claim follows from Definition \[def:sliced-W\] and Proposition \[pr:direction-decomposition\].
\[le:linear growth SW\] Suppose that $p_0 \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ and $\xi_0 \in {\operatorname{ri}}\partial W(p_0)$. Then there exists $\delta > 0$ such that $$\begin{aligned}
{W^{\rm sl}}_{p_0}(z) - \xi_0' \cdot z \geq \delta |z|, \qquad z \in {\ensuremath{\mathbb{R}}}^k,\end{aligned}$$ where $k = \dim {\operatorname{aff}}\partial W(p_0)$.
Let again $V$ be the subspace parallel to ${\operatorname{aff}}\partial W(p_0)$. Then ${\operatorname{aff}}\partial W(p_0)
= \xi_0 + V$. Since $\xi_0 \in {\operatorname{ri}}\partial W(p_0)$, there exists $\delta > 0$ with $\xi \in
\partial W(p_0)$ for all $|\xi - \xi_0| \leq \delta$, $\xi \in \xi_0 + V$. Take $z \in {\ensuremath{\mathbb{R}}}^k$ and set $\zeta = \xi_0 + \delta \frac{{{\mathcal T}}_V z}{|z|} \in \partial W(p_0)$. From the definition of ${W^{\rm sl}}_{p_0}$ we have from and $$\begin{aligned}
{W^{\rm sl}}_{p_0}(z) = W_{p_0}'({{\mathcal T}}_V z) =\sup {{\left\{{{\mathcal T}}_V z \cdot \xi : \xi \in \partial W(p_0)\right\}}} \geq {{\mathcal T}}_V
z \cdot \zeta = \xi_0'\cdot z + \delta |z|.\end{aligned}$$ This yields the lower bound.
Sliced energy
-------------
Suppose now that $p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ such that $k = \dim \partial W(p) > 0$ and recall the definition of ${{\mathcal T}}$ in . We shall consider the rotated flat torus $\Gamma = {\ensuremath{\mathbb{R}}}^n / L {{\mathcal T}}{\ensuremath{\mathbb{Z}}}^n$ for some $L > 0$. We can write $\Gamma = {{\mathcal T}}(\Gamma' \times \Gamma'')$, where $\Gamma' = {\ensuremath{\mathbb{R}}}^k / L {\ensuremath{\mathbb{Z}}}^k$ and $\Gamma'' = {\ensuremath{\mathbb{R}}}^{n - k} / L{\ensuremath{\mathbb{Z}}}^{n-k}$, and $x \in \Gamma$ is given as $x = {{\mathcal T}}(x', x'')$ for $x' \in \Gamma'$, $x'' \in \Gamma''$.
We define the functionals $$\begin{aligned}
E_p(\psi) &:= E_{W(\cdot + p) - W(p)}(\psi; \Gamma), && \psi \in L^2(\Gamma),\\
E'_p(\psi) &:= E_{W_p'}(\psi; \Gamma), && \psi \in L^2(\Gamma),\\
{E^{\rm sl}}_p(\psi) &:= E_{{W^{\rm sl}}_p}(\psi; \Gamma'), && \psi \in L^2(\Gamma').\end{aligned}$$ All three functionals are proper closed convex functions on $L^2(\Gamma)$ resp. $L^2(\Gamma')$.
Since $W_p'$ and ${W^{\rm sl}}_p$ are positively one-homogeneous, the characterization of the subdifferential in Proposition \[pr:subdiff-char-periodic\] applies.
The function $q \mapsto W(q + p) - W(p)$ is not one-homogeneous in general, however, and therefore the same characterization does not apply for the subdifferential of $E_p$. Nevertheless, it coincides with the subdifferential of $E_p'$ at $\psi \in {{\rm Lip}}(\Gamma)$ when ${\left\|\nabla \psi\right\|}_\infty$ is small by Lemma \[le:subdiff-homog-relation\]. This observation allows us to use the simpler, positively one-homogeneous energy $E_p'$ when defining the crystalline curvature of a facet.
What follows is the main justification of the energy stratification. We show that since $W_p'$ is linear on the subspace $U$, we need to only consider the directions in $V = U^\perp$ when computing the crystalline curvature of a **stratified** function.
\[le:subdiff-slicing\] Let $p$ be as above. Suppose that $\bar\psi \in {{\rm Lip}}(\Gamma')$ and $f \in C^1(\Gamma'')$ are given functions and let $\psi(x) = \bar\psi(x') + f(x'')$. Let $\psi_a$ and $\bar \psi_a$ be the unique solutions of the resolvent problems $$\begin{aligned}
\psi_a + a \partial E_p'(\psi_a) &\ni \psi,\\
\bar\psi_a + a \partial {E^{\rm sl}}_p(\bar\psi_a) &\ni \bar\psi,\end{aligned}$$ for given $a > 0$. Then $$\begin{aligned}
\psi_a(x) = \bar \psi_a(x') + f(x''), \qquad \text{$x = {{\mathcal T}}(x', x'') \in \Gamma$}.\end{aligned}$$ or, equivalently, $$\begin{aligned}
(I +a\partial E_p')^{-1}(\psi)(x) = (I +a \partial {E^{\rm sl}}_p)^{-1}(\bar\psi)(x') + f(x'').\end{aligned}$$ If moreover $\bar\psi \in {\operatorname{\mathcal{D}}}(\partial {E^{\rm sl}}_p)$, then $\psi \in {\operatorname{\mathcal{D}}}(\partial E_p')$, $\partial^0 E_p'(\psi)$ is independent of $x''$ and $$\begin{aligned}
\label{minsectionslicing}
\partial^0 E_p'(\psi)(x) = \partial^0 {E^{\rm sl}}_p(\bar \psi)(x') \qquad \text{a.e. $x = {{\mathcal T}}(x', x'') \in \Gamma$}.\end{aligned}$$
Suppose that $\psi(x) = \bar \psi(x') + f(x'')$ for some $\bar \psi \in {{\rm Lip}}(\Gamma')$ and $f \in C^1(\Gamma'')$. By the characterization of the subdifferentials in Proposition \[pr:subdiff-char-periodic\], we have $$\begin{aligned}
\partial E'_p(\psi) = - {\operatorname{div}}{{CH}}_{W_p'}(\psi; \Gamma), \qquad
\partial {E^{\rm sl}}_p(\bar \psi) = - {\operatorname{div}}{{CH}}_{{W^{\rm sl}}_p}(\bar\psi; \Gamma').\end{aligned}$$ The decomposition lemma \[le:decomposition-subdiff-W\] implies $$\begin{aligned}
\label{subdiffslicing}
\partial W'_p(\nabla \psi(x)) = {{\left\{{{\mathcal T}}(\xi', \xi''):
\xi' \in \partial {W^{\rm sl}}_p(\nabla \bar\psi(x'))\right\}}}\end{aligned}$$ for some fixed $\xi'' \in {\ensuremath{\mathbb{R}}}^{n-k}$ since $$\begin{aligned}
\nabla \psi(x) = {{\mathcal T}}(\nabla \bar \psi(x'), \nabla f(x'')).\end{aligned}$$
By Proposition \[pr:resolvent-problems\], both $\psi_a$ and $\bar\psi_a$ are Lipschitz. As $\bar\psi_a$ is the unique solution of the resolvent problem, the characterization of the subdifferential of ${E^{\rm sl}}_p$ above yields that there exists $\bar z_a \in {{CH}}_{{W^{\rm sl}}_p}(\bar\psi_a; \Gamma')$ such that $\bar\psi_a - \bar\psi = a {\operatorname{div}}\bar z_a$. Set $z_a(x) = {{\mathcal T}}(\bar z_a(x'), \xi'')$ for some fixed $\xi''$ as above and $\zeta_a(x) = \bar\psi_a(x') + f(x'')$. Note that $z_a \in {{CH}}_{W_p'}(\zeta_a; \Gamma)$ by . Moreover ${\operatorname{div}}_x z_a(x) = {\operatorname{div}}_{x'} \bar z_a(x')$. Therefore $$\begin{aligned}
\label{hgyfhbs}
\begin{aligned}
\zeta_a(x) - \psi(x) &= \bar\psi_a(x') +f(x'') - \bar\psi(x') + f(x'') =
\bar \psi_a(x') - \bar\psi(x')\\
&= a {\operatorname{div}}_{x'} \bar z_a(x') =
a {\operatorname{div}}_x z_a(x).
\end{aligned}\end{aligned}$$ The characterization of the subdifferential of $E_p'$ above implies that $\zeta_a - \psi \in - \partial E_p'(\zeta_a; \Gamma)$ and therefore $\zeta_a$ is a solution of the resolvent problem. However, the solution is unique and therefore $\psi_a = \zeta_a$ almost everywhere.
Now we suppose that $\bar \psi \in {\operatorname{\mathcal{D}}}(\partial {E^{\rm sl}}_p)$, that is, that $\partial {E^{\rm sl}}_p(\bar\psi)$ is nonempty. And so there exists $\bar z\in {{CH}}_{{W^{\rm sl}}_p}(\bar\psi; \Gamma')$. But then $z(x) = {{\mathcal T}}(\bar z(\bar x), \xi'') \in {{CH}}_{W_p'}(\psi; \Gamma)$ as we just observed. In particular, $\psi \in {\operatorname{\mathcal{D}}}(\partial E_p')$.
Let us set $h_a = (\psi_a - \psi)/a$ and $\bar h_a = (\bar \psi_a - \bar\psi)/a$ as in Proposition \[pr:resolvent-problems\]. Observe that due to $$\begin{aligned}
h_a(x) = \bar h_a(x').\end{aligned}$$ Since $-h_a \to \partial^0 E_p'(\psi; \Gamma)$ in $L^2(\Gamma)$ and $-\bar h_a \to \partial^0 {E^{\rm sl}}_p(\bar \psi; \Gamma')$ in $L^2(\Gamma')$ as $a \to 0$, we conclude .
Crystalline curvature {#sec:crystalline curvature}
=====================
We introduce an operator $\Lambda_p$ that assigns the crystalline curvature to a facet with slope $p$ given by a faceted function, as long as the faceted function is admissible in a certain sense.
Facets
------
To describe facets, let us recall the notation for pairs that was introduced in [@GGP13AMSA]. Since we need to construct facets of various dimensions, depending on the dimension of $\partial W(p)$, $\mathcal P^k$ will denote the set of pairs on ${\ensuremath{\mathbb{R}}}^k$:
For any $k \in {\ensuremath{\mathbb{N}}}$ we will denote by $\mathcal P^k$ the set of all ordered pairs $(A_-, A_+)$ of disjoint sets $A_\pm \subset {\ensuremath{\mathbb{R}}}^k$, $A_- \cap A_+ = \emptyset$.
We will introduce a partial ordering $(\mathcal P^k, \preceq)$ by $$\begin{aligned}
(A_-, A_+) \preceq (B_-, B_+)
\qquad \Leftrightarrow \qquad
A_+ \subset B_+ \text{ and } B_- \subset A_-\end{aligned}$$ for $(A_-, A_+), (B_-, B_+) \in \mathcal P^k$, as well as the *reversal* $$\begin{aligned}
-(A_-, A_+) := (A_+, A_-).\end{aligned}$$ Clearly, if $(A_-, A_+) \preceq (B_-, B_+)$ then $-(B_-, B_+) \preceq -(A_-, A_+)$.
A pair $(A_-, A_+) \in \mathcal P^k$ is said to be *open* if both $A_-$ and $A_+$ are open.
A *smooth* pair is then an open pair $(A_-, A_+) \in \mathcal P^k$ for which we also have
(i) ${\operatorname{dist}}(A_-, A_+) > 0$, where we use the convention ${\operatorname{dist}}(\emptyset, E) = + \infty$ for any $E$, and
(ii) $\partial A_- \in C^\infty$ and $\partial A_+ \in C^\infty$.
We will refer to the set $$\begin{aligned}
{\ensuremath{\mathbb{R}}}^k \setminus {\left(A_- \cup A_+\right)} = A_-^c \cap A_+^c\end{aligned}$$ as the *facet* of the pair $(A_-, A_+) \in \mathcal P^k$.
We will drop $k$ if the dimension is understood from the context or is irrelevant.
We will add the notion of a bounded pair.
We say that a pair $(A_-, A_+) \in \mathcal P^k$ is *bounded* if either $A_-^c$ or $A_+^c$ is bounded.
Note that if $(A_-, A_+)$ is a bounded pair, then the facet $A_-^c \cap A_+^c$ is bounded. If $(A_-, A_+)$ is an open pair, the reverse implication also applies.
Let us also recall the useful notion of a support function.
A Lipschitz function $\psi \in {{\rm Lip}}({\ensuremath{\mathbb{R}}}^k)$ is called a *support function* of an open pair $(A_-, A_+) \in \mathcal P^k$ if $$\begin{aligned}
\psi(x)
\begin{cases}
> 0 & x \in A_+,\\
= 0 & x \in A_-^c \cap A_+^c,\\
< 0 & x \in A_-.
\end{cases}\end{aligned}$$
On the other hand, for any function $\psi$ on ${\ensuremath{\mathbb{R}}}^k$ we define the pair $$\begin{aligned}
{\operatorname{Pair}}(\psi) := {\left({{\left\{x \in {\ensuremath{\mathbb{R}}}^k : \psi(x) <0\right\}}},
{{\left\{x \in {\ensuremath{\mathbb{R}}}^k: \psi(x) >0\right\}}}\right)}.\end{aligned}$$
\[ex:trivial-support-function\] For any open pair $(A_-, A_+) \in \mathcal P^k$ the function $$\begin{aligned}
\psi(x) := {\operatorname{dist}}(x, A_+^c) - {\operatorname{dist}}(x, A_-^c)\end{aligned}$$ is a support function of the pair $(A_-, A_+)$.
Finally, let us recall the notion of a generalized neighborhood of a subset of ${\ensuremath{\mathbb{R}}}^k$.
For any set $E \subset {\ensuremath{\mathbb{R}}}^k$ and $\rho \in {\ensuremath{\mathbb{R}}}$ the *generalized neighborhood* is defined as $$\begin{aligned}
{{\mathcal U}^{\rho}}(E) :=
\begin{cases}
E + {\overline{B}}_\rho(0) & \rho > 0,\\
E & \rho = 0,\\
{{\left\{x \in E : {\overline{B}}_{{\left|\rho\right|}}(x) \subset E\right\}}} & \rho < 0.
\end{cases}\end{aligned}$$
For a pair $(A_-, A_+) \in \mathcal P^k$ we introduce the generalized neighborhood $$\begin{aligned}
{{\mathcal U}^{\rho}}(A_-,A_+) := {\left({{\mathcal U}^{-\rho}}(A_-),{{\mathcal U}^{\rho}}(A_+)\right)}.\end{aligned}$$
A part of the following proposition was stated in [@GGP13AMSA] for the $n$-dimensional torus, but it can be easily restated for ${{\ensuremath{{\mathbb{R}^{{n}}}}}}$. The proof is straighforward.
\[pr:nbd-properties\]
1. $\mathcal U^{-\rho}(A) \subset A \subset \mathcal U^\rho(A)$ for $\rho > 0$.
2. (complement) $$\begin{aligned}
\label{compl-nbd}
{\left({{\mathcal U}^{\rho}}(A)\right)}^c = {{\mathcal U}^{-\rho}}(A^c)
\qquad \text{for any set $A \subset {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ and $\rho \in {\ensuremath{\mathbb{R}}}$}\end{aligned}$$
3. (monotonicity) $$\begin{aligned}
{{\mathcal U}^{\rho}}(A_1) \subset {{\mathcal U}^{\rho}}(A_2)\qquad
\text{for $A_1 \subset A_2 \subset {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ and $\rho \in {\ensuremath{\mathbb{R}}}$.}\end{aligned}$$
4. ${{\mathcal U}^{\rho}}(A_1 \cap A_2) \subset {{\mathcal U}^{\rho}}(A_1) \cap {{\mathcal U}^{\rho}}(A_2)$ for all $\rho \in {\ensuremath{\mathbb{R}}}$, with equality for $\rho \leq 0$.
5. ${{\mathcal U}^{r}}({{\mathcal U}^{\rho}}(A)) \subset {{\mathcal U}^{r+\rho}}(A)$ for $r \geq 0$ and $\rho \in {\ensuremath{\mathbb{R}}}$; equality holds if $\rho \geq 0$.
6. For any $\rho \in {\ensuremath{\mathbb{R}}}$, we have ${{\mathcal U}^{\rho}}(A_1) \subset A_2$ if and only if $A_1 \subset {{\mathcal U}^{-\rho}} (A_2)$.
7. (interior and closure) $$\begin{aligned}
\bigcup_{\rho > 0} {{\mathcal U}^{-\rho}}(A) = {\operatorname{int}}A \subset A \subset {\overline{A}} = \bigcap_{\rho>0} {{\mathcal U}^{\rho}}(A) \qquad \text{for any set $A \subset {{\ensuremath{{\mathbb{R}^{{n}}}}}}$.}\end{aligned}$$
8. (distance) $$\begin{aligned}
{\operatorname{dist}}(A_1, A_2) = \sup {{\left\{\rho \geq 0: {{\mathcal U}^{\rho}}(A_1) \subset A_2^c\right\}}} \qquad \text{for all $A_1, A_2 \subset {{\ensuremath{{\mathbb{R}^{{n}}}}}}$.}\end{aligned}$$
Definition of crystalline curvature
-----------------------------------
We assume for the rest of the paper that $W$ is a convex polyhedral function on ${{\ensuremath{{\mathbb{R}^{{n}}}}}}$. Let $p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ such that $k = \dim \partial W(p) > 0$. Let $\psi$ be a support function of a bounded open pair $(A_-, A_+) \in \mathcal P^k$. We say that $\psi$ is an *$p$-admissible support function* if there exists an open set $G \supset A_-^c \cap A_+^c$ such that the set of Cahn-Hoffman vector fields $$\begin{aligned}
{CH^{\rm sl}}_p(\psi; G) := {{CH}}_{{W^{\rm sl}}_p}(\psi; G)\end{aligned}$$ is nonempty. We denote this for short as $\psi \in {\operatorname{\mathcal{D}}}(\Lambda_p)$. If for a given bounded open pair $(A_-, A_+)$ there exists at least one $p$-admissible support function, we say that $(A_-, A_+)$ is a *$p$-admissible pair*. If $p$ is understood from the context, we refer to them as an admissible support function and an admissible pair.
Let $\psi \in {\operatorname{\mathcal{D}}}(\Lambda_p)$ be an admissible support function of an admissible pair $(A_-, A_+)$. We define the function $\Lambda_p[\psi] \in L^2(A_-^c \cap A_+^c)$ on the facet as $$\begin{aligned}
\label{crystalline-curvature}
\Lambda_p[\psi](x) = {\operatorname{div}}z_{\rm min}(x), \qquad x \in A_-^c \cap A_+^c,\end{aligned}$$ where $z_{\rm min}$ is an element of ${CH^{\rm sl}}_p(\psi; G)$ that minimizes ${\left\|{\operatorname{div}}z\right\|}_{L^2(G)}$. We call $\Lambda_p$ the *crystalline curvature*.
As we shall see later in Corollary \[co:lambda support func indep\] at the end of this section, the crystalline curvature satisfies a comparison principle and therefore its value on a facet of a given admissible pair is independent of the choice of an admissible support function of this pair.
We first prove that the crystalline curvature is well-defined $\Lambda_p$.
\[pr:lambda-well-defined\] The quantity $\Lambda_p[\psi]$ is well-defined in the sense that the value is unique a.e. and it does not depend on $G$ nor on the value of $\psi$ away from the facet. More precisely, if $\psi_1$ and $\psi_2$ are two support functions of a bounded open pair $(A_-, A_+) \in \mathcal P^k$ with $\psi_i \in {\operatorname{\mathcal{D}}}(\Lambda_p)$ for some $p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ with $k = \dim \partial W(p) > 0$ such that $\psi_1 = \psi_2$ on a neighborhood of the facet $A_-^c \cap A_+^c$, then $\Lambda_p[\psi_1] = \Lambda_p[\psi_2]$ a.e. on $A_-^c \cap A_+^c$.
Let $\psi_i \in {\operatorname{\mathcal{D}}}(\Lambda_p)$, $i = 1,2$, be two support functions that satisfy the hypothesis. Then there are open sets $G_i \supset A_-^c \cap A_+^c$ and associated Cahn-Hoffman vector fields $z_i \in {CH^{\rm sl}}_p(\psi_i; G_i)$ that minimize ${\left\|{\operatorname{div}}z_i\right\|}_{L^2(G_i)}$ over ${CH^{\rm sl}}_p(\psi_i; G_i)$. Since the facet $A_-^c \cap A_+^c$ is assumed to be bounded, we can find a bounded open set $H \supset A_-^c \cap A_+^c$ with $\psi_1 = \psi_2$ on $H$ and $H \subset G_1 \cap G_2$. Let us take $0 < \delta < \min_{\partial H} {\left|\psi_1\right|}$ and set $G = {{\left\{x \in H: {\left|\psi_1\right|} < \delta\right\}}} \subset\subset H$.
Set $z = z_1 \chi_{G_1 \setminus G} + z_2 \chi_G$. By Lemma \[le:cahn-hoffman-patch\] we have that $z \in {CH^{\rm sl}}_p(\psi_1; G_1)$ and therefore ${\left\|{\operatorname{div}}z\right\|}_{L^2(G_1)} \geq {\left\|{\operatorname{div}}z_1\right\|}_{L^2(G_1)}$, which with implies $$\begin{aligned}
{\left\|{\operatorname{div}}z_2\right\|}_{L^2(G)} \geq {\left\|{\operatorname{div}}z_1\right\|}_{L^2(G)}.\end{aligned}$$ Reversing the roles of $\psi_1$ and $\psi_2$, and $G_1$ and $G_2$, we get the opposite inequality.
Therefore the strict convexity of the $L^2$-norm implies that ${\operatorname{div}}z_1 = {\operatorname{div}}z_2$ a.e. on $L$. Indeed, if they are not equal, we can decrease the norm by taking the vector field $z = \frac 12 {\left(z_1 + z_2\right)}$ on $G$ which is still admissible due to Remark \[arb-convex-patch\].
The following crucial result will allow us to express the crystalline curvature as the minimal section of the subdifferential of the sliced energy on a periodic domain.
\[pr:curvature-as-min-section\] Let $p \in {\ensuremath{\mathbb{R}}}^k$ be such that $k = \dim \partial W(p) > 0$. Suppose that $\psi \in {\operatorname{\mathcal{D}}}(\Lambda_p)$, that is, $\psi$ is an admissible support function of a bounded open pair $(A_-, A_+)$. Let $L > 0$ be such that $A_-^c \cap A_+^c \subset B_{L/4}(0)$. Denote $\Gamma = {\ensuremath{\mathbb{R}}}^k / L {\ensuremath{\mathbb{Z}}}^k$.
There exists an $L$-periodic Lipschitz function $\psi_2 \in {{\rm Lip}}(\Gamma)$ such that $\psi_2$ is a support function of the open pair $(A_- + L {\ensuremath{\mathbb{Z}}}^k, A_+ + L {\ensuremath{\mathbb{Z}}}^k)$ and ${CH^{\rm sl}}_p(\psi_2; \Gamma)$ is nonempty, and for some open set $H$, $A_-^c \cap A_+^c \subset H \subset B_{L/4}(0)$ we have $\psi_1 = \psi_2$ on $H$. Moreover, $$\begin{aligned}
\label{lambda-per-expr}
\Lambda[\psi_1](x) = -\partial^0 {E^{\rm sl}}_p(\psi_2; \Gamma)(x) \qquad \text{a.e. $x \in A_-^c \cap A_+^c$.}\end{aligned}$$
Let us first show if we have function $\psi_2$ with the properties stated in the proposition. We use the characterization of the differential in Proposition \[pr:subdiff-char-periodic\]. Let therefore $z_2 \in {CH^{\rm sl}}_p(\psi_2; \Gamma)$ be a Cahn-Hoffman vector field that minimizes ${\left\|{\operatorname{div}}z_2\right\|}_2$ in this set. Note that we have $\partial^0 {E^{\rm sl}}_p(\psi_2; \Gamma)(x) = - {\operatorname{div}}z_2$ by the characterization of the subdifferential in Proposition \[pr:subdiff-char-periodic\].
Now we can proceed as in the proof of Proposition \[pr:lambda-well-defined\]. Let $z_1$ minimize ${\left\|{\operatorname{div}}z_1\right\|}_2$ in ${CH^{\rm sl}}_p(\psi_1; G_1)$ for some open set $G_1 \supset A_-^c \cap A_+^c$. We can assume that $G_1 \subset H$. We proceed as follows: given that $G \subset\subset H$ with $G$ as defined in that proof, Lemma \[le:cahn-hoffman-patch\] can be applied to $\psi_1$ on $G_1$ and $\psi_2$ on $G_2 = H$, and since we are only modifying the vector fields away from the boundary of $H$, replacing $z_2$ by $z_1$ on the set $G$ yields again a vector field in ${CH^{\rm sl}}_p(\psi_2; \Gamma)$. We again deduce that ${\operatorname{div}}z_1 = {\operatorname{div}}z_2$ on $A_-^c \cap A_+^c$,
We shall now construct $\psi_2$. Since $\psi_1$ is admissible there are an open set $G \supset A_- \cap A_+$, $G \subset B_{L/4}(0)$, and a vector field $z \in {CH^{\rm sl}}_p(\psi_1; G)$. Let us choose a positive $\delta$ such that $\delta < \min_{\partial G} {\left|\psi_1\right|}$. This is possible since $\psi_1$ is continuous and $\partial G \subset A_- \cup A_+ = {{\left\{\psi_1 \neq 0\right\}}}$. We set $$\begin{aligned}
\xi(x) =
\begin{cases}
-\delta & x \in A_- \setminus G,\\
\max(-\delta, \min(\delta, \psi_1(x))) & x \in G\\
\delta & x \in A_+ \setminus G.
\end{cases}\end{aligned}$$ Note that $\xi$ is Lipschitz on ${\ensuremath{\mathbb{R}}}^k$, $\nabla \xi(x) = \nabla \psi_1$ whenever ${\left|\xi(x)\right|} < \delta$ and $\nabla \xi(x) = 0$ if ${\left|\xi(x)\right|} = \delta$, almost everywhere. Moreover, we see that $$\begin{aligned}
\label{xi-H-eq}
\xi = \psi_1 \qquad \text{on } H := {{\left\{x \in G: {\left|\psi_1\right|} < \delta\right\}}}.\end{aligned}$$ Since the complement of $B_{L/4}(0)$ is connected and $A_-$, $A_+$ are open disjoint sets, we must have either $A_- \subset B_{L/4}(0)$ or $A_+ \subset B_{L/4}(0)$. In any case, $\xi$ is constant outside of $B_{L/4}(0)$.
Let $\phi \in C^\infty({\ensuremath{\mathbb{R}}}^k)$ be such that $0 \leq \phi \leq 1$, ${\operatorname{supp}}\phi \subset G$ and $\phi = 1$ on ${{\left\{x \in G: {\left|\psi\right|} \leq \delta\right\}}}$. Define $$\begin{aligned}
w(x) =
\begin{cases}
z(x) \phi(x) & x \in G,\\
0 & \text{otherwise}.
\end{cases}\end{aligned}$$ Clearly $w \in L^\infty({\ensuremath{\mathbb{R}}}^k; {\ensuremath{\mathbb{R}}}^k)$, ${\operatorname{div}}w \in L^2({\ensuremath{\mathbb{R}}}^k)$, ${\operatorname{supp}}w \subset G$. Moreover, since $\partial {W^{\rm sl}}_p(q) \subset \partial {W^{\rm sl}}_p(0) \ni 0$ for any $q \in {\ensuremath{\mathbb{R}}}^k$ by Lemma \[le:one-homogeneous-subdiff\] and $\partial {W^{\rm sl}}_p(0)$ is convex, we have $w(x) \in \partial {W^{\rm sl}}_p(\nabla \xi(x))$ for a.e. $x \in {\ensuremath{\mathbb{R}}}^k$. Therefore $\xi$ is an admissible support function of the pair $(A_-, A_+)$.
Since $\xi = \psi$ in a neighborhood of $A_-^c \cap A_+^c$, we conclude that $\Lambda_p[\xi] = \Lambda_p[\psi_1]$ a.e. on $A_-^c \cap A_+^c$ due to Proposition \[pr:lambda-well-defined\].
Now we $L$-periodically extend $\xi$ and $w$ from $[-L/2, L/2)^k$ to ${\ensuremath{\mathbb{R}}}^k$ and call them $\psi_2$ and $z_2$, respectively. This gives a support function of an open pair $(A_- + L {\ensuremath{\mathbb{Z}}}^k, A_+ + L {\ensuremath{\mathbb{Z}}}^k)$ and clearly ${CH^{\rm sl}}_p(\psi_2; \Gamma) \ni z_2$.
By construction, $\psi_2 = \psi_1$ on $H$ due to .
Comparison principle for the crystalline curvature
--------------------------------------------------
We can prove the following comparison theorem for the crystalline curvature of ordered facets, as in [@GGP13JMPA]. This will imply that $\Lambda_p[\psi]$ on a given admissible pair is in fact independent of the choice of an admissible support function $\psi$, Corollary \[co:lambda support func indep\] below.
\[pr:comparison Lambda\] Let $p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ such that $k = \dim \partial W(p) > 0$. Suppose that $(A_{1,-}, A_{1,+})$ and $(A_{2,-}, A_{2,+})$ are two $p$-admissible pairs in $\mathcal P^k$. If the pairs are ordered in the sense of $$\begin{aligned}
(A_{1,-}, A_{1,+}) \prec (A_{2,-}, A_{2,+}),\end{aligned}$$ then for any two $p$-admissible support functions $\psi_1$ and $\psi_2$ of the respective pairs we have $$\begin{aligned}
\label{curv-ordered-facet}
\Lambda_p[\psi_1](x) \leq \Lambda_p[\psi_2](x) \quad \text{a.e. $x \in A_{1,-}^c \cap A_{1,+}^c \cap
A_{2,-}^c \cap A_{2,+}^c$.}\end{aligned}$$
Before proceeding with the proof, we first give a technical lemma, which is a variant of [@G06 Lemma 4.2.9]; such a result goes back to [@CGG; @ES] to establish a uniqueness of a level set flow.
\[le:Lipschitz ordering\] Suppose that $\psi$ and $\varphi$ are two nonnegative periodic Lipschitz functions on ${\ensuremath{\mathbb{R}}}^d$, $d \geq 1$, such that ${{\left\{\psi = 0\right\}}} \subset {{\left\{\varphi = 0\right\}}}$. Then there exists a Lipschitz continuous function $\theta: [0, \infty) \to [0, \infty)$ such that $\theta(0) = 0$, $\theta(s) > 0$ for $s > 0$ and $\theta'(s) > 0$ for almost every $s > 0$ and we have $$\begin{aligned}
\theta \circ \varphi \leq \psi \qquad \text{on ${\ensuremath{\mathbb{R}}}^d$.}\end{aligned}$$
We may assume that ${{\left\{\psi = 0\right\}}} \neq \emptyset$, otherwise the statement is trivial. We define $$\begin{aligned}
\eta(s) := \inf {{\left\{\psi(x): \varphi(x) \geq s\right\}}}.\end{aligned}$$ Clearly by compactness $\eta(0) = 0$, $\eta(s) > 0$ for $s > 0$. Furthermore, $\eta$ is nondecreasing since $s \mapsto {{\left\{\varphi \geq s\right\}}}$ is nonincreasing. Finally, $\eta \circ \varphi \leq \psi$ as $$\begin{aligned}
\eta(\varphi(x)) = \inf {{\left\{\psi(y): \varphi(y) \geq \varphi(x)\right\}}} \leq \psi(x).\end{aligned}$$
As $\eta$ can have jumps or be infinite, we now consider $$\begin{aligned}
\sigma(s) := \inf {{\left\{\eta(t) + |s - t|: 0 \leq t \leq s\right\}}}.\end{aligned}$$ We immediately obtain $0\leq \sigma(s) \leq \eta(s)$ and $\sigma(0) = 0$. On the other hand, $\eta(t) + |s - t| \geq \min {{\left\{\frac s2, \eta(\frac s2)\right\}}} > 0$ for $s > 0$, $t \in [0, s]$, and so $\sigma(s) > 0$ for $s > 0$. As for monotonicity, a simple estimate for $s \geq u \geq 0$ yields $$\begin{aligned}
\sigma(s) &= \min {{\left\{\inf {{\left\{\eta(t) + |s - t|: 0 \leq t \leq u\right\}}}, \inf {{\left\{\eta(t) + |s - t|: u
\leq t \leq s\right\}}}\right\}}}\\
&\geq \min {{\left\{\sigma(u) + |s - u|, \eta(u)\right\}}}\\
&\geq \sigma(u).\end{aligned}$$ We also show that $\sigma$ is Lipschitz. Take $0 \leq u \leq s$ and $\delta > 0$ and find $t \in
[0,u]$ such that $\sigma(u) > \eta(t) + |u - t| - \delta$. Then we have $$\begin{aligned}
\sigma(s) \leq \eta(t) + |s - t| = \eta(t) + |u - t| + |s - u| < \sigma(u) + |s - u| + \delta.\end{aligned}$$ Since $\delta$ was arbitrary, $\sigma$ is Lipschitz.
Finally, set $$\begin{aligned}
\theta(s) := (1 - e^{-s}) \sigma(s).\end{aligned}$$ Clearly $\theta(0) = 0$, $\theta(s) > 0$ for $s > 0$. The product rule yields $\theta'(s) > 0$ for almost every $s > 0$. By construction, $$\begin{aligned}
\theta \circ \varphi \leq \sigma \circ \varphi \leq \eta \circ \varphi \leq \psi.\end{aligned}$$
Now we complete the proof of the comparison principle for the crystalline curvature $\Lambda_p$.
By Proposition \[pr:curvature-as-min-section\], we can for a sufficiently large $L > 0$ find $L$-periodic functions, called $\tilde\psi_1$ and $\tilde\psi_2$, such that ${CH^{\rm sl}}_p(\tilde\psi_i; \Gamma)$ is nonempty, $\Gamma = {\ensuremath{\mathbb{R}}}^k / L {\ensuremath{\mathbb{Z}}}^k$, and $\tilde\psi_i$ coincides with the original $\psi_i$ on the neighborhood of the facet $A_{i,-}^c \cap A_{i,+}^c$, $i = 1, 2$, and that $$\begin{aligned}
\label{l-is-cononrest}
\Lambda_p[\psi_i] = -\partial^0 {E^{\rm sl}}_p(\tilde\psi_i; \Gamma)
\qquad \text{a.e. on $A_{i,-}^c \cap A_{i,+}^c$,
$i =1,2$.}\end{aligned}$$
Since the pairs are ordered, if we consider the sets $A_{i, \pm}$ as subsets of $\Gamma$ we have $$\begin{aligned}
\{\tilde\psi_{2,+} = 0\} = A_{2,+}^c \subset A_{1,+}^c = \{\tilde\psi_{1,+} = 0\},\\
\{\tilde\psi_{1,-} = 0\} = A_{1,-}^c \subset A_{2,-}^c = \{\tilde\psi_{2,-} = 0\},\end{aligned}$$ where $\tilde\psi_{i,\pm} := \max (\pm\tilde\psi_i, 0)$ denote the positive and negative parts. By Lemma \[le:Lipschitz ordering\], there exist Lipschitz functions $\theta^-$ and $\theta^+$ on $[0, \infty)$ such that $\theta^\pm(0) = 0$, $\theta^\pm(s) > 0$ for $s > 0$, and $(\theta^\pm)'(s) > 0$ for almost all $s > 0$, such that $\theta^+ \circ \tilde\psi_{1,+} \leq
\tilde\psi_{2,+}$ and $\theta^- \circ \tilde\psi_{2,-} \leq \tilde\psi_{1,-}$. We introduce $$\begin{aligned}
\theta_1(s) :=
\begin{cases}
s, & s <0,\\
\theta^+(s), & s \geq 0,
\end{cases}
\qquad
\theta_2(s) :=
\begin{cases}
-\theta^-(-s), & s <0,\\
s, & s \geq 0.
\end{cases}\end{aligned}$$ and $$\begin{aligned}
\xi_1 := \theta_1 \circ \tilde\psi_1, \qquad \xi_2 := \theta_2 \circ \tilde\psi_2.\end{aligned}$$ By construction we have that $\xi_i$ are Lipschitz on $\Gamma$, $$\begin{aligned}
\xi_1 \leq \xi_2,\end{aligned}$$ and the chain rule for Lipschitz functions yields $$\begin{aligned}
\nabla \xi_i(x) = \theta_i'(\xi_i(x)) \nabla \tilde\psi_i(x), \qquad \text{for almost every $x$},\end{aligned}$$ if we interpret the right-hand side to be equal to zero if $\nabla \tilde\psi_i(x)$ is zero, no matter if $\theta_i'$ is differentiable at $\xi_i(x)$ or not. Since $\theta_i'(s) > 0$ for almost every $s \in {\ensuremath{\mathbb{R}}}$, we have by the positive one-homogeneity of ${W^{\rm sl}}_p$ $$\begin{aligned}
\partial {W^{\rm sl}}_p(\nabla \xi_i(x)) = \partial {W^{\rm sl}}_p(\nabla \tilde\psi_i(x)) \qquad \text{for almost
every $x$},\end{aligned}$$ and therefore $$\begin{aligned}
\label{subd-bent-same}
{CH^{\rm sl}}_p(\xi_i; \Gamma) = {CH^{\rm sl}}_p(\tilde\psi_i; \Gamma) \neq \emptyset.\end{aligned}$$
The functional ${E^{\rm sl}}_p(\cdot; \Gamma)$ is proper closed convex and therefore the resolvent problems $$\begin{aligned}
\zeta_i + \lambda \partial {E^{\rm sl}}_p(\zeta_i; \Gamma) \ni \xi_i\end{aligned}$$ have unique solutions $\zeta_i \in L^2(\Gamma)$.
By approximation by smooth problems that have a comparison principle, as in Proposition \[pr:resolvent-problems\] and its proof, we can deduce that $\zeta_i$ are Lipschitz since $\xi_i$ are Lipschitz, and $$\begin{aligned}
\zeta_1 \leq \zeta_2.\end{aligned}$$ On the intersection of the facets $K = A_{1,-}^c \cap A_{1,+}^c \cap A_{2,-}^c \cap A_{2,+}$ we have $\xi_1 = \xi_2 = 0$ and therefore $$\begin{aligned}
\label{resol-ordering}
\frac{\zeta_1 - \xi_1}\lambda \leq \frac{\zeta_2 - \xi_2}\lambda \qquad \text{on $K$.}\end{aligned}$$ By and the characterization of the subdifferential Proposition \[pr:subdiff-char-periodic\], we know that $\xi_i \in {\operatorname{\mathcal{D}}}(\partial {E^{\rm sl}}_p(\cdot; \Gamma))$ and therefore the standard result [@Attouch Proposition 3.56] yields $$\begin{aligned}
\frac{\zeta_i - \xi_i}\lambda \to - \partial^0 {E^{\rm sl}}_p(\xi_i; \Gamma) \qquad \text{in $L^2(\Gamma)$
as $\lambda\to0$.}\end{aligned}$$ We can send $\lambda\to0$, and then use , and the ordering to conclude that $$\begin{aligned}
\Lambda_p[\psi_1] = -\partial^0 {E^{\rm sl}}_p(\xi_1; \Gamma)
\leq -\partial^0 {E^{\rm sl}}_p(\xi_2; \Gamma) = \Lambda_p[\psi_2] \qquad \text{a.e. on $K$.}\end{aligned}$$ This is the comparison principle for the $\Lambda_p$.
The following result is an immediate consequence of Proposition \[pr:comparison Lambda\].
\[co:lambda support func indep\] Suppose that $p \in {\ensuremath{\mathbb{R}}}^n$ with $k := \dim \partial W(p) > 0$ and let $(A_-, A_+) \in \mathcal
P^k$ be a $p$-admissible pair. Then the value of $\Lambda_p$ on the facet $A_-^c \cap A_+^c$ is independent of the choice of a $p$-admissible support function, that is, for any two $p$-admissible support functions $\psi, \xi$ of pair $(A_-, A_+)$ we have $$\begin{aligned}
\Lambda_p[\psi] = \Lambda_p[\xi] \qquad \text{a.e. on $A_-^c \cap A_+^c$.}\end{aligned}$$
Viscosity solutions {#sec:viscosity solutions}
===================
In this section we introduce viscosity solutions of problem . For the definition of viscosity solutions we shall use *stratified faceted functions* that rely on the concept of energy stratification that we have developed in Section \[sec:energy-stratification\]. Recall that for every $\hat p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ we have introduced the coordinate system $x = {{\mathcal T}}(x', x'')$ using the rotation ${{\mathcal T}}= {{\mathcal T}}_{\hat p}$ from .
\[def:strat-faceted-test-function\] Let $(\hat x, \hat t) \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}\times {\ensuremath{\mathbb{R}}}$ and $\hat p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$, $V \subset {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ be the subspace parallel to ${\operatorname{aff}}\partial W(\hat p)$, $U = V^\perp$, $k = \dim V$. We say that a function ${\ensuremath{\varphi}}(x,t)$ is a *stratified faceted test function at $(\hat x, \hat t)$ with gradient $\hat p$* if $$\begin{aligned}
{\ensuremath{\varphi}}(x,t) = \bar \psi{\left(x' - \hat x'\right)}
+ f{\left(x'' - \hat x''\right)}
+ \hat p \cdot x + g(t),\end{aligned}$$ where
- $\bar \psi: {\ensuremath{\mathbb{R}}}^k \to {\ensuremath{\mathbb{R}}}$ is a support function of a bounded facet $(A_-, A_+) \in \mathcal P^k$ with $0 \in {\operatorname{int}}(A_-^c \cap A_+^c)$ and $\bar \psi \in {\operatorname{\mathcal{D}}}(\Lambda_{\hat p})$,
- $f \in C^2({\ensuremath{\mathbb{R}}}^{n - k})$, $f(0) = 0$ and $\nabla f(0) = 0$,
- $g \in C^1({\ensuremath{\mathbb{R}}})$.
With this notion of test functions, we define viscosity solutions.
\[def:visc-solution\] An upper semi-continuous function $u: {\overline{Q}} \to {\ensuremath{\mathbb{R}}}$ is a *viscosity subsolution* of if the following hold:
(i) *(faceted test)* Let ${\ensuremath{\varphi}}$ be a stratified faceted test function at $(\hat x, \hat t) \in Q$ with gradient $\hat p \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ and pair $(A_-, A_+)$. Then if there is $\rho > 0$ such that $$\begin{aligned}
\label{general-position}
u(x + w,t) - {\ensuremath{\varphi}}(x,t) \leq u(\hat x, \hat t) - {\ensuremath{\varphi}}(\hat x, \hat t)\end{aligned}$$ for all $$\begin{aligned}
{\left|w'\right|} \leq \rho,\ w'' = 0, \quad \text{and }
x' - \hat x' \in {{\mathcal U}^{\rho}}({A_-^c \cap A_+^c}),\ {\left|x'' - \hat x''\right|} \leq \rho,\ {\left|t - \hat t\right|} \leq \rho,\end{aligned}$$ then there exists ${\ensuremath{\delta}}> 0$ such that $B_\delta(\hat x') \subset {\operatorname{int}}({A_-^c \cap A_+^c})$ and $$\begin{aligned}
{\ensuremath{\varphi}}_t(\hat x, \hat t) + F(\hat p, {\operatorname*{ess\,inf}}_{B_{\ensuremath{\delta}}(0)} \Lambda_{\hat p}[\bar \psi]) \leq 0.\end{aligned}$$
(ii) *(off-facet test)* Let ${\ensuremath{\varphi}}\in C^1(\mathcal U)$ where $\mathcal U$ is a neighborhood of some point $(\hat x, \hat t) \in Q$ and suppose that $\dim \partial W(\nabla {\ensuremath{\varphi}}(\hat x, \hat t)) = 0$. If $u - {\ensuremath{\varphi}}$ has a local maximum at $(\hat x, \hat t)$ then $$\begin{aligned}
{\ensuremath{\varphi}}_t(\hat x, \hat t) + F(\nabla {\ensuremath{\varphi}}(\hat x, \hat t), 0) \leq 0.\end{aligned}$$
Supersolutions are defined analogously.
If for some $p$ the value of $F(p, \xi)$ does not depend on $\xi$ in the sense below, we can replace the faceted test by a simpler test that does not need an admissible faceted function.
\[def:level-set-type\] We say that $F$ is of *curvature-free type* at $p_0 \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ if we have for any constant $C > 0$ $$\begin{aligned}
\lim_{p \to p_0} \sup_{{\left|\zeta\right|} \leq C} F(p, \zeta) = F(p_0, 0) =
\lim_{p \to p_0} \inf_{{\left|\zeta\right|} \leq C} F(p, \zeta).\end{aligned}$$
The function $F$ defined in is of curvature-free type at $p_0 = 0$.
\[def:level-set-test\] If $F$ is of curvature-free type at $p_0 = 0$, we replace the faceted test (i) in Definition \[def:visc-solution\] at $\hat{p} = p_0 = 0$ by the following test:
1. Let $g \in C^1({\ensuremath{\mathbb{R}}})$, ${\ensuremath{\varphi}}(x, t) := g(t)$ and suppose that $u - {\ensuremath{\varphi}}$ has a local maximum at $(\hat x, \hat t)$. Then $$\begin{aligned}
g'(\hat t) + F(0, 0) = g'(\hat t) \leq 0.\end{aligned}$$
Construction of faceted functions {#sec:faceted functions}
=================================
To prove the uniqueness of viscosity solutions of , we need to be able to construct a sufficiently wide class of test functions, the *faceted functions*. In this section we will assume that $W$ is convex, positively one-homogeneous and crystalline. We shall also assume that there exists $\delta > 0$ such that $W(p) \geq \delta {\left|p\right|}$. The important case for us is ${W^{\rm sl}}_p$ from Definition \[def:sliced-W\].
The polar function $W^\circ$ of $W$ is defined as $$\begin{aligned}
\label{polar}
W^\circ(x) = \sup {{\left\{x \cdot p: W(p) \leq 1\right\}}}.\end{aligned}$$ Clearly $$\begin{aligned}
(W^\circ)^\circ = W.\end{aligned}$$ We define the Wulff shape corresponding to $W$ as $$\begin{aligned}
{\operatorname{Wulff}}_W := {{\left\{x \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}: W^\circ(x) \leq 1\right\}}}.\end{aligned}$$ Note that the Wulff shape of a one-homogeneous crystalline (polyhedral) $W$ with linear growth is a bounded polyhedron containing the origin in its interior.
We want to establish a proposition similar to [@GGP13AMSA Proposition 2.12], but for a crystalline energy:
\[pr:approximate-pair\] Let $k = 1$ or $2$, $(A_-, A_+) \in \mathcal P^k$ be a *bounded* pair and let $0 \leq \rho_1 < \rho_2$. Suppose that $W: {\ensuremath{\mathbb{R}}}^k \to {\ensuremath{\mathbb{R}}}$ is a convex, positively one-homogeneous polyhedral function such that there exists $\delta > 0$ with $W(p) \geq \delta |p|$ for $p \in {\ensuremath{\mathbb{R}}}^k$. Then there exists an *admissible pair* $(G_-, G_+) \in \mathcal P^k$ such that $$\begin{aligned}
\label{admiss pair approx}
{{\mathcal U}^{\rho_1}}(A_-, A_+) \preceq (G_-, G_+) \preceq {{\mathcal U}^{\rho_2}}(A_-, A_+),\end{aligned}$$ that is, there exists a support function $\psi$ of pair $(G_-, G_+)$ such that ${{CH}}_W(\psi; {\ensuremath{\mathbb{R}}}^k)$ is nonempty.
We shall use this result in the following form:
\[co:approximate pair sliced\] Let $W: {\ensuremath{\mathbb{R}}}^n \to {\ensuremath{\mathbb{R}}}$ be a polyhedral convex function finite everywhere. Suppose that $p_0 \in {\ensuremath{\mathbb{R}}}^n$ such that $\dim \partial W(p_0) = k$ for $k = 1$ or $2$. Then for any bounded pair $(A_-, A_+) \in
\mathcal P^k$ and any $0 \leq \rho_1 < \rho_2$ there exists a $p_0$-*admissible pair* $(G_-,
G_+)$ satisfying .
Let us take $\xi_0 \in {\operatorname{ri}}\partial W(p_0)$. The function $\hat W(p) := {W^{\rm sl}}_{p_0}(p) - \xi_0' \cdot p$ satisfies the assumptions of Proposition \[pr:approximate-pair\] by Lemma \[le:linear growth SW\]. Therefore there exists a pair $(G_-, G_+) \in \mathcal P^k$, its support function $\psi$ and a Cahn-Hoffman vector field $z \in {{CH}}_{\hat W}(\psi; {\ensuremath{\mathbb{R}}}^k)$. It is easy to see that $z +
\xi_0 \in {CH^{\rm sl}}_{p_0}(\psi; {\ensuremath{\mathbb{R}}}^k)$, and therefore $(G_-, G_+)$ is $p_0$-admissible.
As of now we only know how to construct such admissible facets for dimensions $k = 1$ and $k = 2$.
For the construction of an admissible function we will basically use a signed-distance-like function induced by $W$, and then define a possible Cahn-Hoffman vector field for this function. For a given set $V \subset {\ensuremath{\mathbb{R}}}^k$ the signed-distance-like function $d_V$ is defined as $$\begin{aligned}
\label{d_V-def}
d_V(x) := \inf_{y\in V} W^\circ(x - y) -\inf_{y\in V^c} W^\circ(y - x), \quad x \in {\ensuremath{\mathbb{R}}}^k,\end{aligned}$$ where $W^\circ$ is the polar of $W$ given as .
One-dimensional admissible facets
---------------------------------
We will give an explicit construction as a proof of Proposition \[pr:approximate-pair\] in the one-dimensional case to illustrate the process and hopefully prepare the reader for the construction in the two-dimensional case.
The situation is depicted in Figure \[fig:addissible-pair-1d\].
Let $(A_-, A_+) \subset \mathcal P^1$ be a bounded pair in ${\ensuremath{\mathbb{R}}}$. By making $\rho_1$ larger if necessary, we can assume that $0 < \rho_1 < \rho_2$. Let us set ${\ensuremath{\varepsilon}}:= \frac{\rho_2 - \rho_1}3$.
We define the open sets $$\begin{aligned}
G_- := {\operatorname{int}}{\overline{{{\mathcal U}^{{\ensuremath{\varepsilon}}}}{\left({{\mathcal U}^{-\rho_2}}(A_-)\right)}}} \qquad \text{and} \qquad
G_+ := {\operatorname{int}}{\overline{{{\mathcal U}^{\rho_1 + {\ensuremath{\varepsilon}}}}(A_+)}}.\end{aligned}$$ Due to the properties of the set neighborhood, we have for all $\eta > 0$ $$\begin{aligned}
\label{G-pm-eta}
{{\mathcal U}^{-\rho_2}} (A_-) \subset G_- \subset {{\mathcal U}^{-\rho_2 + {\ensuremath{\varepsilon}}+ \eta}}(A_-), \qquad {{\mathcal U}^{\rho_1}}(A_+) \subset G_+ \subset {{\mathcal U}^{\rho_1 + {\ensuremath{\varepsilon}}+ \eta}}(A_+).\end{aligned}$$ In particular, we take the interior of the closure in the definition of $G_\pm$ to regularize the boundary so that $G^c_\pm$ has no isolated points.
By definition $A_-\subset A_+^c$, and therefore Proposition \[pr:nbd-properties\] together with imply that for any $\eta \in (0, 2{\ensuremath{\varepsilon}})$ $$\begin{aligned}
G_- &\subset {{\mathcal U}^{{\ensuremath{\varepsilon}}+\eta}}{\left({{\mathcal U}^{-\rho_2}}(A_-)\right)} \subset {{\mathcal U}^{-\rho_2 + {\ensuremath{\varepsilon}}+ \eta}}(A_-)\\
&\subset {{\mathcal U}^{-\rho_2 + {\ensuremath{\varepsilon}}+ \eta}}(A_+^c) \subset {{\mathcal U}^{-{\ensuremath{\varepsilon}}+ 2\eta}}{\left({{\mathcal U}^{-\rho_1 - {\ensuremath{\varepsilon}}- \eta}}(A_+^c)\right)}\\
&= {{\mathcal U}^{-{\ensuremath{\varepsilon}}+ 2\eta}}{\left({{\mathcal U}^{\rho_1 + {\ensuremath{\varepsilon}}+ \eta}}(A_+)^c\right)}\\
&= {{\mathcal U}^{{\ensuremath{\varepsilon}}- 2\eta}}{\left({{\mathcal U}^{\rho_1 + {\ensuremath{\varepsilon}}+ \eta}}(A_+)\right)}^c \subset {{\mathcal U}^{{\ensuremath{\varepsilon}}- 2\eta}}{\left(G_+\right)}^c\end{aligned}$$ We conclude that $$\begin{aligned}
{\operatorname{dist}}(G_-, G_+) = {\ensuremath{\varepsilon}}> 0.\end{aligned}$$ Therefore $(G_-, G_+)$ is an open pair, and due to $$\begin{aligned}
{{\mathcal U}^{\rho_1}}(A_-, A_+) \preceq (G_-, G_+) \preceq {{\mathcal U}^{\rho_2}}(A_-, A_+),\end{aligned}$$ To prove that the pair $(G_-, G_+)$ is bounded, we recall that $(A_-, A_+)$ is a bounded pair therefore there exists $R > 0$ such that $B_R^c(0) \subset A_-$ or $B_R^c(0) \subset A_+$. From we have that ${{\mathcal U}^{-\rho_2}}(B_R^c(0)) \subset G_-$ or ${{\mathcal U}^{\rho_1}}(B_R^c(0)) \subset A_+$. Therefore $B_{\tilde R}^c(0) \subset G_-$ or $B_{\tilde R}^c(0) \subset G_+$ for $\tilde R = R + \rho_2$, which implies that $(G_-, G_+)$ is bounded.
Since $G_\pm$ are open, we can write the union $G_- \cup G_+$ as at most a countable union of disjoint open intervals. Since the facet $G_-^c \cap G_+^c$ is bounded, and the sets $G_\pm$ have the interior ball property with radius ${\ensuremath{\varepsilon}}$ by construction, the length of the intervals must be greater than or equal to $2{\ensuremath{\varepsilon}}$. In particular, there must only be finitely many of them. Since moreover ${\operatorname{dist}}(G_-, G_+) > 0$, we can find $m \in {\ensuremath{\mathbb{N}}}$ and ${{\left\{a_i\right\}}}_{i=0}^m$, ${{\left\{b_i\right\}}}_{i=0}^m$ such that $$\begin{aligned}
-\infty = a_0 < b_0 < a_1 < b_1 < \cdots < a_m < b_m = \infty\end{aligned}$$ and $$\begin{aligned}
G_- \cup G_+ = \bigcup_{i=0}^m (a_i, b_i).\end{aligned}$$ Finally, by construction, $$\begin{aligned}
\label{def-of-delta}
\delta := \frac13 \min {{\left\{\min_{0 \leq i \leq m} b_i - a_i, \min_{1 \leq i \leq m} a_i - b_{i-1}\right\}}} > 0.\end{aligned}$$ The facet $G_-^c \cap G_+^c$ is closed and $$\begin{aligned}
G_-^c \cap G_+^c = \bigcup_{i=1}^m [b_{i-1}, a_i].\end{aligned}$$
Let us now introduce the sign function $$\begin{aligned}
\sigma(x) :=
\begin{cases}
1 & x \in {\overline{G_+}},\\
-1 & x \in {\overline{G_-}},\\
0 & \text{otherwise}.
\end{cases}\end{aligned}$$ This allows us to define the function $$\begin{aligned}
\psi(x) := \min{{\left\{\delta, {\operatorname{dist}}(x, G_+^c)\right\}}} - \min {{\left\{\delta, {\operatorname{dist}}(x, G_-^c)\right\}}},\end{aligned}$$ as a clipped version of the function in Example \[ex:trivial-support-function\], which is again clearly a support function of the pair $(G_-, G_+)$. Moreover, $$\begin{aligned}
\psi(x) =
\begin{cases}
\delta \sigma(x) & x \in [a_i + \delta, b_i - \delta] \text{ for some $i$},\\
0 & x \in [b_{i-1}, a_i] \text{ for some $i$},\\
(x-a_i)\sigma(x) & x \in (a_i, a_i + \delta) \text{ for some $i$},\\
(b_i -x)\sigma(x) & x \in (b_i - \delta, b_i) \text{ for some $i$}.
\end{cases}\end{aligned}$$ Therefore the function $\psi$ is differentiable everywhere except at the points $a_i, b_i, a_i + \delta, b_i - \delta$ for $0 \leq i \leq m$. We can evaluate the derivative at the other points as $$\begin{aligned}
\psi'(x) =
\begin{cases}
0 & x \in (a_i + \delta, b_i -\delta) \cup (b_{i-1}, a_i) \text{ for some $i$},\\
\sigma(x) & x \in (a_i, a_i + \delta) \text{ for some $i$},\\
-\sigma(x) & x \in (b_i - \delta, b_i) \text{ for some $i$}.
\end{cases}\end{aligned}$$
In one dimension, the subdifferential of one-homogeneous $W$ can be expressed as $$\begin{aligned}
\partial W(p) =
\begin{cases}
{{\left\{w_-\right\}}} & p < 0,\\
[w_-, w_+] & p = 0,\\
{{\left\{w_+\right\}}} & p < 0,
\end{cases}\end{aligned}$$ for $w_\pm = W'(\pm 1)$, $w_- < 0 < w_+$.
Let us define the continuous Cahn-Hoffman vector field as $$\begin{aligned}
z(x) :=
\begin{cases}
W'(\sigma(x)) & x \in (a_i, a_i + \delta) \text{ for some $i$},\\
W'(-\sigma(x)) & x \in (b_i - \delta, b_i) \text{ for some $i$},\\
W'(\sigma(b_0)) & x \leq b_0 - \delta,\\
W'(\sigma(a_m)) & x \geq a_m + \delta,\\
\text{linear} & \text{otherwise},
\end{cases}\end{aligned}$$ One can easily see that the function $z$ is Lipschitz continuous on ${\ensuremath{\mathbb{R}}}$ and ${\left\|\nabla z\right\|}_\infty \leq \frac{w_+ - w_-}\delta \leq \infty$ by the definition of $\delta$ in . Therefore $\psi \in {\operatorname{\mathcal{D}}}(\partial E)$ and the facet $(G_-, G_+)$ is admissible, which finishes the proof of Proposition \[pr:approximate-pair\] in the case of $k = 1$.
Two-dimensional admissible facets
---------------------------------
In this section we give a proof of Proposition \[pr:approximate-pair\] in the two-dimensional case. We can without loss suppose that $\rho_1 = 0$ and $\rho_2 = \rho > 0$. Let us stress again that we do not assume that the Wulff shape of $W$ is symmetric with respect to the origin.
The proof of Proposition \[pr:approximate-pair\] for $k = 2$ uses a rather simple idea of an explicit construction that is unfortunately quite technical. It will be split in several steps:
1. Approximate a general bounded facet by a smooth facet.
2. Rotate the smooth facet by a small angle so that the boundary has nonzero curvature at the points where the normal is pointing in the direction of a corner of $W$.
3. Flatten the boundary locally at these points.
4. Use the Fenchel distance-like function induced by $W$ to construct a support function and a Cahn-Hoffman vector field in the neighborhood of the boundary.
We define the set of critical directions, $$\begin{aligned}
\mathcal N := {{\left\{p \in S^1: \partial W(p) \text{ is not a singleton}\right\}}} = {{\left\{p \in S^1: W \text{
is not differentiable at $p$}\right\}}},\end{aligned}$$ where $S^1 := {{\left\{p \in {\ensuremath{\mathbb{R}}}^2: |p| = 1\right\}}}$ is the unit circle. Since $W$ is polyhedral, $\mathcal N$ is finite.
\[le:\] $\partial W: p \to 2^{{\ensuremath{\mathbb{R}}}^2}$ is constant on every connected component of $S^1 \setminus \mathcal
N$. Moreover, $\partial W(p)$ is a singleton for every such $p$.
This follows from the fact that $W$ is polyhedral.
We will also use some basic results of the convex analysis. In particular, recall the definition of the polar $W^\circ$ in . We will for short denote the associate Wulff shape as $$\begin{aligned}
\mathcal W := {{\left\{x: W^\circ(x) \leq 1\right\}}}.\end{aligned}$$ This is a polygon in two dimensions, with a finite number of vertices, corresponding to the number of critical directions $\mathcal N$. We have the following basic result:
\[le:wulff-frank-rel\] If $p \neq 0$ and $x \in \partial W(p)$ then $W^\circ(x) = 1$ and $x \cdot p = W(p)$. Similarly, if $x \neq 0$ and $p \in
\partial W^\circ(x)$ then $W(p) = 1$ and $x \cdot p = W^\circ(x)$. Suppose now that $x \neq 0$ and $p \neq 0$. Then $$\begin{aligned}
\frac x{W^\circ(x)} \in \partial W(p) \quad \Leftrightarrow \quad \frac p{W(p)} \in \partial W^\circ(x).\end{aligned}$$
### Smooth pair approximation
By the smooth approximation lemma, [@GGP13AMSA Lemma 2.11], we can find smooth disjoint open sets $H_-, H_+$ such that $$\begin{aligned}
\label{A-Hausdorff-approx}
\begin{aligned}
{{\mathcal U}^{\rho/2}}(A_-, A_+) \preceq (H_-, H_+) \preceq {{\mathcal U}^{3\rho/4}}(A_-, A_+).
\end{aligned}\end{aligned}$$ We note that $(H_-, H_+)$ is an smooth bounded pair.
We claim that we can choose $H_-, H_+$ in such a way that $$\begin{aligned}
\label{nonzero curvature}
\begin{aligned}
\text{the curvature of $\partial H_-$ and
$\partial H_+$ at $x$ is nonzero}\\
\text{whenever $\nu_{\partial H_-}(x) \in \mathcal N$ or $-\nu_{\partial
H_+}(x) \in \mathcal N$, respectively.}
\end{aligned}\end{aligned}$$ Indeed, let $V$ be $H_-$ or ${\operatorname{int}}H_+^c$. Since $\partial V$ is smooth and bounded, it is a union of finitely many disjoint closed curves. Each of these curves is a one-dimensional manifold without boundary and the unit outer normal vector map $\nu: \partial V
\to S^1$ is smooth. By Sard’s theorem we have $\mathcal H^1{\left(\nu{\left({{\left\{x \in \partial V: d\nu(x) \text{
has rank } < 1\right\}}}\right)}\right)} = 0$. Note that the curvature $\kappa(x)$ of $\partial V$ at $x \in \partial V$ is zero if and only if the rank of $d\nu(x)$ is zero. Since the set of critical directions $\mathcal N \subset S^1$ is finite, we can find a rotation $R$ of ${\ensuremath{\mathbb{R}}}^2$ by an arbitrary small angle such that $R(\mathcal N) \cap \nu({{\left\{x \in \partial V : \kappa(x) = 0\right\}}}) = \emptyset$. We therefore rotate the set $V$ by $R^{-1}$ with a sufficiently small such angle so that the rotated set still approximates the original one. Therefore whenever $x \in R^{-1}(V)$ such that $\kappa_{R^{-1}(\partial V)}(x) = 0$, we have $\nu_{R^{-1}(\partial V)}(x) \notin \mathcal N$. We can therefore replace $H_-$ and $H_+$ with the rotated ones by a sufficiently small angle if necessary and then $H_\pm$ satisfy .
### Flattening of $\partial H_\pm$ in the critical directions
Let $V$ denote either $H_-$ or ${\operatorname{int}}H_+^c$ in what follows and let $\nu(x) = \nu_{\partial V}(x)$ be the unit outer normal to $\partial V$ at $x \in \partial V$. We will modify $V$ in the neighborhood of the critical points of its boundary $x \in \partial V$ with $\nu(x)
\in \mathcal N$ so that the boundary of the modified set has a flat part of nonzero length with the same normal. Let us denote the set of these critical points by $S$, $$\begin{aligned}
S := {{\left\{x \in \partial V: \nu(x) \in \mathcal N\right\}}}.\end{aligned}$$ Note that $S$ is compact since $\nu$ is smooth and $\partial V$ is bounded.
We claim that $S$ is finite. Indeed, suppose that $S$ is infinite. Since $S$ is compact, there is $\hat x \in S$ such that $B_{\ensuremath{\varepsilon}}(\hat x) \cap S$ is infinite for every ${\ensuremath{\varepsilon}}> 0$. Since $\mathcal N$ is discrete and $\nu$ is continuous, there exists ${\ensuremath{\varepsilon}}_0 > 0$ such that $\nu(x) \equiv \nu(\hat x)$ for all $x \in B_{{\ensuremath{\varepsilon}}_0}(\hat x) \cap S$. But that is a contradiction with $d \nu(\hat x) \neq 0$ from .
Let us choose $\eta > 0$ such that $$\begin{aligned}
\eta < \min {{\left\{\frac 1{40}{\operatorname{dist}}(\partial H_-,\partial H_+), \frac \rho8,
\min_{\substack{x, y \in S\\x \neq y}} |x - y|\right\}}}.\end{aligned}$$ Since for any $\hat x \in S$ we have $\kappa(\hat x) \neq 0$, by making $\eta$ smaller if necessary, we may also assume that $\partial V \cap B_{20\eta}(\hat x)$ is a graph of a convex or a concave function $g = g_{\hat x}$ in the sense that $$\begin{aligned}
V \cap B_{20\eta}(\hat x) = {{\left\{y + \hat x \in B_{20\eta}(\hat x) : y \cdot \nu(\hat x) < g(y
\cdot \tau(\hat x))\right\}}},\end{aligned}$$ where $\tau(\hat x) \perp \nu(\hat x)$, $|\tau(\hat x)| = 1$. Note that $g(0) = g'(0) = 0$. Since $\kappa(\hat x) \neq 0$, we have $g''(\hat x) \neq 0$ and by Taylor expansion we may also assume that $$\begin{aligned}
\frac 14 |g''(0)|s^2 \leq |g(s)| \leq |g''(0)| s^2, \quad |s| < 20 \eta.\end{aligned}$$
With this set-up, we can for every $\hat x \in S$ find $L_{\hat x} > 0$ such that ${{\left\{s:
|g_{\hat x}(s)| < L_{\hat x}\right\}}} \times [-L_{\hat x}, L_{\hat x}] \subset B_\eta(0)$. We then define $\hat V$, the set with flattened boundary in the critical directions, as $$\begin{aligned}
\hat V := &{\left(V \setminus \bigcup_{\hat x \in S} B_{10 \eta}(\hat x)\right)}\\
&\bigcup_{\substack{\hat x \in S\\g_{\hat x}''(0) > 0}} {{\left\{y + \hat x \in B_{10\eta}(\hat x) :
y \cdot \nu(\hat x) < \max(L_{\hat x}, g_{\hat x}(y \cdot \tau(\hat x)))\right\}}}\\
&\bigcup_{\substack{\hat x \in S\\g_{\hat x}''(0) < 0}} {{\left\{y + \hat x \in B_{10\eta}(\hat x) :
y \cdot \nu(\hat x) < \min(-L_{\hat x}, g_{\hat x}(y \cdot \tau(\hat x)))\right\}}}.\end{aligned}$$ Note that $\partial V \subset {{\mathcal U}^{\eta}}(\partial \hat V)$ and $\partial \hat V \subset
{{\mathcal U}^{\eta}}(\partial V)$.
We finish our construction of the admissible pair by defining $G_- = \hat V$ when starting with $V = H_-$, and $G_+ = {\operatorname{int}}\hat V^c$ when starting with $V = {\operatorname{int}}H_+^c$.
### Construction of the support function and the Cahn-Hoffman vector field
In this part we shall finally define a candidate for the admissible function with an appropriate Cahn-Hoffman vector field in a small neighborhood of the flattened boundary $\partial \hat V$, where $\hat V = G_-$ or $\hat V = {\operatorname{int}}G_+^c$.
Let $\mathcal V$ denote the set of vertices of the Wulff shape $\mathcal W$. We define $\mathcal C_0$ to be the family of connected components of $\partial \hat V \setminus
\partial V$, and $\mathcal C_r$ to be the family of connected components of $\partial \hat V \cap
\partial V$. We also define $\mathcal C = \mathcal C_0 \cup \mathcal C_r$. Every $\Gamma_0 \in
\mathcal C_0$ is the flattened part of the boundary $\partial \hat V$, the line segment with a normal vector $\nu_0 \in \mathcal N$. Similarly, every $\Gamma \in \mathcal C_r$ is a connected piece of the original smooth boundary $\partial V$, and by construction there exists a unique vertex $v \in \mathcal V$ of the Wulff shape such that ${{\left\{v\right\}}} = \partial W(\nu(x))$ for $x
\in \Gamma$. We set $\mathcal V(\Gamma) = {{\left\{v\right\}}}$.
Given $\Gamma \in \mathcal C_0$ with normal $\nu_0$, there exists exactly two distinct vertices $v, w \in \mathcal V$ such that ${{\left\{v, w\right\}}} \subset \partial W(\nu_0)$. In this case we set $\mathcal V(\Gamma) =
{{\left\{v, w\right\}}}$. There exist exactly two sets $\Gamma', \Gamma'' \in \mathcal C_r$ such that $\mathcal V(\Gamma') =
{{\left\{v\right\}}}$, $\mathcal V(\Gamma'') = {{\left\{w\right\}}}$, and ${\overline{\Gamma}} \cap \Gamma' = {{\left\{x_v\right\}}}$, ${\overline{\Gamma}}
\cap \Gamma'' = {{\left\{x_w\right\}}}$ for some points $x_v, x_w$; see Figure \[fig:local-admissible\].
Since $v, w$ are linearly independent, we have a unique point $c^\Gamma$ at the intersection of $L_v(x_v)$ and $L_w(x_w)$. We have $c^\Gamma + t v = x_v$ and $c^\Gamma + sw = x_w$ for some $t, s \in {\ensuremath{\mathbb{R}}}\setminus {{\left\{0\right\}}}$. However, since $(x_w - x_v) \cdot \nu_0 = 0$, we must have $c^\Gamma \cdot \nu_0 + s w \cdot \nu_0
= c^\Gamma \cdot \nu_0 + t v \cdot \nu_0$. As $v \cdot \nu_0 = w \cdot \nu_0 = W(\nu_0)$, it follows that $t = s$ and we set $\alpha^\Gamma := t$. This induces a coordinate system on ${\ensuremath{\mathbb{R}}}^2$ with coordinates $x = \xi_v^\Gamma(x) v + \xi_w^\Gamma(x) w + c^\Gamma$ for every $x \in {\ensuremath{\mathbb{R}}}^2$. We note that $$\begin{aligned}
\label{Gamma alpha level}
\Gamma = {{\left\{x: \xi_v^\Gamma(x) + \xi_w^\Gamma(x) = \alpha^\Gamma,\ \xi_v^\Gamma(x) \xi_w^\Gamma(x)
> 0\right\}}}.\end{aligned}$$ Clearly $\xi_v^\Gamma(x_v) = \xi_w^\Gamma(x_w) = \alpha^\Gamma$ and $\xi^\Gamma_v(x_w) =
\xi^\Gamma_w(x_v) = 0$.
We define the line through a point $x$ in the direction $v$ as $$\begin{aligned}
L_v(x) := {{\left\{x + tv: t \in {\ensuremath{\mathbb{R}}}\right\}}},\end{aligned}$$ and the cylinder through set $\Gamma$ $$\begin{aligned}
L_v(\Gamma) := {{\left\{x + tv: x \in \Gamma,\ t \in {\ensuremath{\mathbb{R}}}\right\}}}.\end{aligned}$$ The thickness of a cylinder is denoted by $$\begin{aligned}
\theta(L_v(\Gamma)) := \sup_{x, y \in \Gamma} {\operatorname{dist}}{\left(L_v(x), L_v(y)\right)}.\end{aligned}$$
We collect a few basic properties of the relationship between the components $\Gamma \in \mathcal
C$ and the associated cylinders. These results follow from the construction of $\hat V$ in the previous section.
\[le:one intersect\] Suppose that $\Gamma \in \mathcal C$ and $x, y \in \Gamma$. Let $v \in \mathcal V(\Gamma)$. Then there exists $p \in \partial
W^\circ(v)$ with $v \cdot p = 1$ such that $(x-y) \cdot p = 0$. In particular, if $x \in L_v(\Gamma)$ then $L_v(x) \cap \Gamma = {{\left\{y\right\}}}$ for some $y$, that is, there exists a unique $t \in {\ensuremath{\mathbb{R}}}$ such that $x - tv \in \Gamma$.
Since $\Gamma$ is a smooth curve, by the mean value theorem there exists $\xi \in \Gamma$ such that $(x-y) \cdot \nu(\xi) = 0$. But $p:= \frac{\nu(\xi)}{W(\nu(\xi))} \in \partial W^\circ(v)$ by construction. Then $v \cdot p = 1$ follows from the characterization of the subdifferential of $W^\circ$ in Lemma \[le:wulff-frank-rel\].
Now let $x \in L_v(\Gamma)$. By definition, there exists $t \in {\ensuremath{\mathbb{R}}}$ such that $x - tv \in \Gamma$. Suppose that $x - sv \in \Gamma$ for $s \in {\ensuremath{\mathbb{R}}}$. Then from the above there exists $p$ such that $v \cdot p = 1$ and $0 = (x - tv - x + sv) \cdot p = (s - t) v \cdot p = s - t$. We have $s = t$.
\[le:neighbor cylinders\] Let $\Gamma \in \mathcal C_0$ and $\Gamma' \in \mathcal C_r$ such that ${\operatorname{dist}}(\Gamma, \Gamma') =
0$. Then there exist $v$ such that ${{\left\{v\right\}}} = \mathcal V(\Gamma) \cap \mathcal V(\Gamma')$, and $\xi$ such that ${\overline{\Gamma}} \cap \Gamma' = {{\left\{\xi\right\}}}$. Moreover, $L_v(\Gamma) \cap L_v(\Gamma') = \emptyset$ and ${\overline{L_v(\Gamma)}} \cap L_v(\Gamma') = L_v(\xi)$.
If ${\operatorname{dist}}(\Gamma, \Gamma') = 0$, then $\Gamma$ must be the flattened part and $\Gamma'$ must be the adjacent smooth part of $\partial \hat V$. By construction, $\mathcal V(\Gamma) \cap \mathcal V(\Gamma') = {{\left\{v\right\}}}$ for some $v \in \mathcal
V$, and ${\overline{\Gamma}} \cap \Gamma' = {{\left\{\xi\right\}}}$ for some $\xi$. In particular, ${\overline{L_v(\Gamma)}} \cap
L_v(\Gamma') \subset L_v(\xi)$. Now suppose that there exist distinct points $x \in {\overline{\Gamma}}$, $y \in \Gamma'$ such that $L_v(x) = L_v(y)$. Then by connectedness of ${\overline{\Gamma}}$ and $\Gamma'$, we can find such points arbitrarily close to $\xi$. But this is a contradiction with the fact that $L_v(x)$ can intersect both $\Gamma$ and $\Gamma'$ at most once by Lemma \[le:one intersect\], and the line $L_v(x)$ in the direction of $v$ travels from $\hat V$ to $\hat V^c$ at two consecutive points $x, y$, with no transition from $\hat
V^c$ to $\hat V$ in between.
\[co:on intersect neighbor\] Suppose that $\Gamma \in \mathcal C_r$, $\Gamma', \Gamma'' \in \mathcal C_0$ are the adjacent flat parts, ${\operatorname{dist}}(\Gamma', \Gamma) = {\operatorname{dist}}(\Gamma'', \Gamma) = 0$, and $x, y \in \Gamma \cup \Gamma'
\cup \Gamma''$. Let $v \in \mathcal V(\Gamma)$. Then there exists $p \in \partial
W^\circ(v)$ with $v \cdot p = 1$ such that $(x-y) \cdot p = 0$. In particular, if $x \in L_v(\Gamma \cup \Gamma' \cup \Gamma'')$ then $L_v(x) \cap (\Gamma \cup \Gamma' \cup \Gamma'') = {{\left\{y\right\}}}$ for some $y$, that is, there exists a unique $t \in {\ensuremath{\mathbb{R}}}$ such that $x - tv \in \Gamma \cup \Gamma' \cup \Gamma''$.
This follows by combining Lemma \[le:neighbor cylinders\] and Lemma \[le:one intersect\] for the neighboring components $\Gamma,
\Gamma', \Gamma''$, since the flat ones have normals $\nu', \nu'' \in \partial W^\circ(v)$.
Given $\mu > 0$, we define the sets $U_\Gamma$ for $\Gamma \in \mathcal C$ by $$\begin{aligned}
U_\Gamma :=
\begin{cases}
{{\left\{x + tv: x \in \Gamma, \ |t| \leq \mu,\ v \in \mathcal V(\Gamma)\right\}}} & \text{if $\Gamma \in \mathcal
C_r$,}\\
{{\left\{x: |\xi_v^\Gamma(x) + \xi_w^\Gamma(x) - \alpha^\Gamma| \leq \mu, \ \xi_v^\Gamma(x) \xi_w^\Gamma(x) > 0\right\}}} &
\text{if $\Gamma \in \mathcal C_0$.}
\end{cases}\end{aligned}$$ We shall show below in that ${{\left\{U_\Gamma\right\}}}_{\Gamma \in \mathcal C}$ cover a neighborhood of $\partial
\hat V$. Note that if we take $\mu \leq |\alpha^\Gamma|/2$ we must have ${\operatorname{sign}}\xi_v^\Gamma(x) =
{\operatorname{sign}}\xi_w^\Gamma(x) = {\operatorname{sign}}\alpha^\Gamma$ on $U_\Gamma$ for $\Gamma \in \mathcal C_0$.
If we choose $\mu >0$ small enough, the sets $U_\Gamma$ are pair-wise disjoint.
\[le:UG disjoint\] Suppose that $0 < \mu < \min {{\left\{\mu_1, \mu_2\right\}}}$, where $$\begin{aligned}
\mu_1 := \frac{1}{3\max_{W^\circ(v)} |v|}
\min_{\substack{\Gamma, \Gamma' \in \mathcal C\\{\operatorname{dist}}(\Gamma, \Gamma')> 0
}}{\operatorname{dist}}(\Gamma, \Gamma'),\qquad
\mu_2 := \min_{\Gamma \in \mathcal C_0}
\frac{|\alpha^\Gamma|}2.\end{aligned}$$ Then $U_\Gamma \cap U_{\Gamma'} = \emptyset$ for all $\Gamma, \Gamma' \in \mathcal C$, $\Gamma \neq \Gamma'$.
Suppose that ${\operatorname{dist}}(\Gamma, \Gamma') > 0$. Then $U_\Gamma \subset {{\mathcal U}^{t}}(\Gamma)$ and $U_{\Gamma'} \subset {{\mathcal U}^{t}}(\Gamma')$ with $t = \mu \max_{W^\circ(v) \leq 1} |v|$. Hence $U_\Gamma \cap U_{\Gamma'} = \emptyset$ by $\mu < \mu_1$.
On the other hand, if ${\operatorname{dist}}(\Gamma, \Gamma') = 0$, then one of the sets, say $\Gamma$, belongs to $\mathcal C_0$, and the other belongs to $\mathcal C_r$. Suppose that $y \in U_\Gamma \cap U_{\Gamma'}$. We will show that this leads to a contradiction. Indeed, set $v \in \mathcal V(\Gamma')$ and note that $U_{\Gamma'} \subset L_v(\Gamma')$. We have $c^\Gamma \in L_v(\Gamma')$. Therefore $y(\lambda) := \lambda y + (1-\lambda) c^\Gamma \in L_v(\Gamma')$ for every $\lambda \in [0,1]$. By Lemma \[le:neighbor cylinders\], we have $\Gamma \cap
L_v(y(\lambda)) =\emptyset$ for all $\lambda \in [0,1]$.
Let $t:= \xi_v^\Gamma(y) + \xi_w^\Gamma(y) - \alpha^\Gamma$. Since $y \in U_\Gamma$, we have $|t|
\leq \mu < \mu_2 \leq |\alpha^\Gamma|/2$ and $\xi_v^\Gamma(y) \xi_w^\Gamma(y) > 0$. If $t \alpha^\Gamma \leq 0$, we have $y - t
v \in \Gamma$ by , and this is a contradiction with $\Gamma \cap L_v(y) = \emptyset$. If $t \alpha^\Gamma > 0$, we set $\lambda := \frac{\alpha}{\alpha + t} \in (0,1)$. A simple computation using shows that $y(\lambda) \in \Gamma$, which is a contradiction with $\Gamma \cap
L_v(y(\lambda)) = \emptyset$. The conclusion $U_\Gamma \cap U_{\Gamma'} = \emptyset$ follows.
We choose $\mu$ satisfying the assumption in Lemma \[le:UG disjoint\]. Then on the pair-wise disjoint collection of sets ${{\left\{U_\Gamma: \Gamma \in \mathcal C\right\}}}$, we define functions $\psi$ and $z$ by $$\begin{aligned}
\psi(x) :=
\begin{cases}
t \text{ such that $x - tv \in \Gamma$, $v \in \mathcal V(\Gamma)$},& x \in U_\Gamma,\ \Gamma \in
\mathcal C_r,\\
\xi_v^\Gamma(x) + \xi_w^\Gamma(x) - \alpha^\Gamma, & x \in U_\Gamma,\ \Gamma \in \mathcal C_0,
\end{cases}\end{aligned}$$ and $$\begin{aligned}
z(x) :=
\begin{cases}
v \text{, where $v \in \mathcal V(\Gamma)$}& x \in U_\Gamma,\ \Gamma \in \mathcal C_r,\\
\frac{\xi_v^\Gamma(x) v + \xi_w^\Gamma(x) w}{\xi_v^\Gamma(x) + \xi_w^\Gamma(x)}, \text{ where $v, w
\in \mathcal V(\Gamma)$}, & x \in U_\Gamma,\ \Gamma \in \mathcal C_0.
\end{cases}\end{aligned}$$
Both $\psi$ and $z$ are well-defined by Lemma \[le:one intersect\]. Note that $|\psi| \leq
\mu$ on $U_\Gamma$. We can easily see that $\psi$ is differentiable in the interior of $U_\Gamma$ for all $\Gamma \in
\mathcal C$ by the inverse function theorem. Moreover, the level set ${{\left\{x: \psi(x) =
\psi(y)\right\}}}$ in a neighborhood of $y \in {\operatorname{int}}U_\Gamma$ is just a translation of $\Gamma$. Therefore $\nabla \psi(y) = s \nu'$, where $\nu' = \nu^\Gamma$ for $\Gamma \in \mathcal C_0$, or $\nu'
= \nu(y - \psi(y) v)$ for $\Gamma \in \mathcal C_r$, with $s = v \cdot \nu' > 0$. In particular, $$\begin{aligned}
z(y) \in \partial W(\nabla \psi(y)) \qquad \text{for $z \in {\operatorname{int}}U_\Gamma$, $\Gamma \in
\mathcal C$.}\end{aligned}$$
We now conclude this part by showing that for small $\delta > 0$, the functions $\psi$ and $z$ are well-defined, Lipschitz continuous functions on ${{\mathcal U}^{\delta}}(\partial \hat V)$. We shall use the following two lemmas that we prove first. We set $$\begin{aligned}
K := \max_{W(p)\leq 1} |p| \qquad \text{and} \qquad
\delta_\theta = \min_{\Gamma \in \mathcal C} \min_{v \in \mathcal V(\Gamma)} \theta(L_v(\Gamma)),
\qquad \text{and} \qquad \delta_\mu := \frac \mu K.\end{aligned}$$ Finally, we find $\delta_s > 0$ such that for every $\Gamma \in \mathcal C_0$, $\Gamma' \in
\mathcal C_r$ the adjacent component to $\Gamma$, ${\operatorname{dist}}(\Gamma', \Gamma) = 0$, $v \in \mathcal
V(\Gamma')$, $v \neq w \in \mathcal V(\Gamma)$, we have $$\begin{aligned}
\label{delta sign}
{\operatorname{dist}}(\Gamma' \cap {{\mathcal U}^{\delta_s}}(L_v(\Gamma)), L_w(c^\Gamma)) > \delta_s.\end{aligned}$$ This is possible since $\Gamma' \cap {\overline{L_v(\Gamma)}} = {{\left\{x_v\right\}}}$, $\Gamma'$ is smooth (in fact detaching from $L_v(\Gamma)$ linearly), with $\xi_v^\Gamma(x_v) =
\alpha^\Gamma \neq 0$, and $L_w(c^\Gamma) = {{\left\{x: \xi_v^\Gamma(x) = 0\right\}}}$.
\[le:Cr case\] Let $x \in \Gamma \in \mathcal C_r$, $v \in \mathcal V(\Gamma)$, and let $\Gamma', \Gamma'' \in
\mathcal C_0$ be the neighboring components with ${\operatorname{dist}}(\Gamma', \Gamma) = {\operatorname{dist}}(\Gamma'', \Gamma) = 0$. Then for every $y$, $|y - x| \leq \min{{\left\{\delta_\theta, \delta_\mu, \delta_s\right\}}}$ there exists a unique $t(y)$ such that $y - t(y) v \in \Gamma
\cup \Gamma' \cup \Gamma''$. Moreover, $|t(y)| \leq \mu$. Finally, ${\operatorname{sign}}\xi_v^{\Gamma'}(y) = {\operatorname{sign}}\alpha^{\Gamma'}$ and ${\operatorname{sign}}\xi_v^{\Gamma''}(y) = {\operatorname{sign}}\alpha^{\Gamma''}$.
Using Lemma \[le:neighbor cylinders\] and $|x - y| \leq \delta_\theta$, that is, that the distance between $x$ and $y$ is smaller than the width of the cylinders $L_v(\Gamma')$ and $L_v(\Gamma'')$, we are guaranteed that $y \in L_v(\Gamma \cup \Gamma' \cup \Gamma'')$. Therefore there exists a unique $t \in {\ensuremath{\mathbb{R}}}$ with $y - tv \in \Gamma \cup \Gamma' \cup \Gamma''$. By Corollary \[co:on intersect neighbor\], there exists $p \in \partial W^\circ(v)$ such that $$\begin{aligned}
0 = (x - y + tv) \cdot p = (x - y) \cdot p + t.\end{aligned}$$ By Cauchy-Schwarz $|t| \leq K |x - y| \leq \mu$ and the conclusion follows. The sign of $\xi_v^{\Gamma'}(y)$ and $\xi_v^{\Gamma''}(y)$ must match the sign at $\Gamma \cap
{\overline{\Gamma}}'$, $\Gamma \cap \Gamma''$, which matches that of $\alpha^{\Gamma'}$, $\alpha^{\Gamma''}$, respectively, since $\delta \leq \delta_s$ and $\delta_s$ satisfies .
\[le:C0 case\] Let $x \in \Gamma \in \mathcal C_0$, $v \in \mathcal V(\Gamma)$, and let $\Gamma', \Gamma'' \in
\mathcal C_r$ be the neighboring components with ${\operatorname{dist}}(\Gamma', \Gamma) = {\operatorname{dist}}(\Gamma'', \Gamma) = 0$. Let $v \in
\mathcal V(\Gamma')$ and $w \in \mathcal V(\Gamma'')$. Then for every $y$, $|y - x| \leq \min{{\left\{\delta_\theta, \delta_\mu\right\}}}$ where $\delta_\mu = \mu/K$. Then exactly one of the following holds:
1. $\xi_v^\Gamma(y) \xi_w^\Gamma(y) > 0$, $|\xi_v^\Gamma(y) + \xi_w^\Gamma(y) - \alpha| \leq \mu$, or
2. $y \in L_v(\Gamma')$, there exists $t$ such that $y - t v \in \Gamma'$, and $|t| \leq \mu
$, or
3. $y \in L_w(\Gamma'')$, there exists $t$ such that $y - t w \in \Gamma''$, and $|t| \leq \mu$.
Let us set $t = \xi_v^\Gamma(y) + \xi_w^\Gamma(y) - \alpha$. Then $\xi_v^\Gamma(y - tv) +
\xi_w^\Gamma(y - tv) - \alpha^\Gamma = 0$ and therefore $(x - y + t v) \cdot \nu_0 =0$, where $\nu_0$ is the normal of $\Gamma$. In particular, $t = (x - y) \cdot \frac{\nu_0}{W(\nu_0)}$ and hence $|t| \leq K |x - y| \leq \mu$, which implies the estimate in (a).
Since $|y - x| \leq \delta_\mu$, we have $|t| \leq \mu < \mu_2 \leq |\alpha^\Gamma|/2$ and therefore $\xi_v^\Gamma(y) + \xi_w^\Gamma(y)$ has the same sign as $\alpha^\Gamma$. We conclude that at least one of $\xi_v^\Gamma(y)$, $\xi_v^\Gamma(y)$ has the same sign as $\alpha^\Gamma$. Due to Lemma \[le:neighbor cylinders\], $L_v(\Gamma') \subset {{\left\{\xi_w^\Gamma \alpha^\Gamma \leq 0\right\}}}$ and $L_w(\Gamma'') \subset {{\left\{\xi_v^\Gamma \alpha^\Gamma \leq 0\right\}}}$. Therefore $y \notin L_v(\Gamma') \cap L_w(\Gamma'')$.
If $\xi_v^\Gamma(y) \xi_w^\Gamma(y) > 0$ then we are at case (a). Otherwise since $|y - x| \leq \delta_\theta$, $y$ must be in exactly one of the cylinders $L_v(\Gamma')$ or $L_w(\Gamma'')$ due to the discussion above.
Suppose therefore $y \in
L_v(\Gamma')$. Then there exists a unique $t$ such that $y - t v \in \Gamma'$, and Corollary \[co:on intersect neighbor\] implies the estimate $|t| \leq K |y -
x| \leq \mu$ as in Lemma \[le:Cr case\]. The case $y \in L_w(\Gamma'')$ can be handled similarly.
We therefore take $$\begin{aligned}
0 < \delta < \min{{\left\{\delta_\theta, \delta_\mu, \delta_s\right\}}}.\end{aligned}$$ With this choice, $$\begin{aligned}
\label{partialVcover}
{{\mathcal U}^{\delta}}(\partial \hat V) \subset \bigcup_{\Gamma \in \mathcal C} U_\Gamma.\end{aligned}$$ Indeed, let us fix $y \in {{\mathcal U}^{\delta}}(\partial \hat V)$. Then there exists $x \in \partial \hat V$ with $|x - y| \leq \delta$.
In the case that $x \in \Gamma \in \mathcal C_r$, we apply Lemma \[le:Cr case\] to conclude that there is a unique $t$, $|t| \leq K |y-x| \leq K\delta \leq \mu$, such that $y - t v \in \Gamma \cup \Gamma'
\cup \Gamma''$ where $v \in \mathcal V(\Gamma)$. If $y - tv \in \Gamma$, then clearly $y \in U_\Gamma$. On the other hand, if $y - tv \in \Gamma'$, we have $$\begin{aligned}
0 = \xi_v^{\Gamma '}(y - tv) + \xi_w^{\Gamma '}(y - tv) - \alpha^{\Gamma'} =
\xi_v^{\Gamma '}(y) + \xi_w^{\Gamma '}(y) - \alpha^{\Gamma'} - t.\end{aligned}$$ Since also ${\operatorname{sign}}\xi_v^{\Gamma'}(y) = {\operatorname{sign}}\alpha^{\Gamma'}$, we conclude that $y \in
U_{\Gamma'}$. An analogous argument works if $y - tv \in \Gamma''$.
Now if $x \in \Gamma \in \mathcal C_0$, we apply Lemma \[le:C0 case\], and we argue as above to conclude that $y \in U_{\Gamma}$, $U_{\Gamma'}$ or $U_{\Gamma''}$. Therefore we recover .
Now we finally show that $\psi$ and $z$ are Lipschitz on ${{\mathcal U}^{\delta}}(\partial \hat V)$. Since $\psi$ and $z$ are smooth in the interior of $U_\Gamma$, we only need to address the continuity across the transition between $U_\Gamma$, $U_{\Gamma'}$, $\Gamma \in \mathcal C_0$, $\Gamma' \in
\mathcal C_r$, with ${\operatorname{dist}}(\Gamma, \Gamma') = 0$. The function $z$ is clearly Lipschitz across this boundary, since we can alternatively define $z$ in the neighborhood of this boundary using $$\begin{aligned}
\zeta(x):=
\begin{cases}
\xi_w^\Gamma(x), & \xi_w^\Gamma(x) \alpha^\Gamma > 0,\\
0, & \text{otherwise}.
\end{cases}\end{aligned}$$ Then we have in the neighborhood of the boundary between $U_\Gamma$ and $U_{\Gamma'}$ that $$\begin{aligned}
z(x) = \frac{\xi_v^\Gamma(x) v + \zeta(x) w}{\xi_v^\Gamma(x) + \zeta(x)},\end{aligned}$$ which is clearly a Lipschitz function when ${\left|\xi_v^\Gamma(x)\right|} > {\ensuremath{\varepsilon}}> 0$, as is the case near the boundary.
Similarly, we can alternatively define $\psi$ in the neighborhood of the boundary between $U_\Gamma$ and $U_{\Gamma'}$ as $$\begin{aligned}
\psi(x) = t \quad \text{where $t$ is such that $x - t v \in \Gamma \cup \Gamma'$.}\end{aligned}$$ This function is Lipschitz continuous by Corollary \[co:on intersect neighbor\].
### Completion of the proof of Proposition \[pr:approximate-pair\]
We now have two Lipschitz functions $\psi^-, \psi^+$ and Lipschitz continuous vector fields $z^-,
z^+$ defined in ${{\mathcal U}^{\delta}}(\partial G_-)$ and ${{\mathcal U}^{\delta}}(\partial G_+)$, respectively, such that $z^\pm(x) \in \partial W(\nabla \psi^\pm)$. Furthermore, $\partial G_\pm = {{\left\{\psi^\pm =
0\right\}}}$ almost everywhere. We now have to connect them to produce an admissible support function of the pair $(G_-,
G_+)$. We define the constant $\eta = \min (\eta^-, \eta^+) > 0$ by $$\begin{aligned}
\eta^\pm := \frac 12 \min {{\left\{|\psi_\pm(x)|: \delta/2 \leq {\operatorname{dist}}(x, \partial G_\pm) \leq \delta\right\}}}.\end{aligned}$$ We find smooth cutoff functions $\varphi^\pm \in C^\infty_c$ such that $$\begin{aligned}
\text{
$0 \leq \varphi^\pm \leq 1$,
${\operatorname{supp}}\varphi^\pm \subset {{\mathcal U}^{3\delta/4}}(\partial G_\pm)$, $\varphi_\pm = 1$ on
${{\mathcal U}^{\delta/2}}(\partial G_\pm)$.
}\end{aligned}$$ We define the support function of the pair $(G_-, G_+)$ as $$\begin{aligned}
\psi(x) :=
\begin{cases}
\eta & x\in G_+ \setminus {{\mathcal U}^{\delta}}(\partial G_+),\\
\min(\eta, \max (\psi^+, 0)) & x\in {{\mathcal U}^{\delta}}(\partial G_+),\\
0 & x \in G_-^c \cap G_+^c \setminus {{\mathcal U}^{\delta}}(\partial G_- \cup \partial
G_+),\\
\max(-\eta, \min (\psi^-, 0)) & x\in {{\mathcal U}^{\delta}}(\partial G_-),\\
-\eta & x\in G_- \setminus {{\mathcal U}^{\delta}}(\partial G_-).
\end{cases}\end{aligned}$$ It is easy to check that $\psi$ is a Lipschitz support function of $(G_-, G_+)$. Moreover, it is admissible with the Lipschitz Cahn-Hoffman vector field $$\begin{aligned}
z(x) := z^-(x) \varphi^-(x) + z^+(x) \varphi^+(x).\end{aligned}$$ by Lemma \[le:one-homogeneous-subdiff\].
Comparison principle {#sec:comparison principle}
====================
In this section we prove the comparison principle on a spacetime cylinder $Q := {{\ensuremath{{\mathbb{R}^{{n}}}}}}\times (0,T)$ for some $T > 0$.
\[th:comparison principle\] Let $W: {\ensuremath{\mathbb{R}}}^n \to {\ensuremath{\mathbb{R}}}$ be a positively one-homogeneous convex polyhedral function such that the conclusion of Corollary \[co:approximate pair sliced\] holds for $1 \leq k \leq n-1$, and let $F$ be of curvature-free type at $p_0 = 0$. Suppose that $u$ and $v$ are a subsolution and a supersolution of on ${{\ensuremath{{\mathbb{R}^{{n}}}}}}\times [0, T]$ for some $T > 0$, respectively. Moreover, suppose that there exist a compact set $K\subset {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ and constants $c_u \leq c_v$ such that $u \equiv c_u$, $v \equiv c_v$ on ${\left({{\ensuremath{{\mathbb{R}^{{n}}}}}}\setminus K\right)} \times [0,T]$. Then $u(\cdot, 0) \leq v(\cdot, 0)$ on ${{\ensuremath{{\mathbb{R}^{{n}}}}}}$ implies $u \leq v$ on ${{\ensuremath{{\mathbb{R}^{{n}}}}}}\times [0, T]$.
We will use the standard doubling-of-variables technique with an additional parameter to enforce a certain facet-like behavior of the functions at a contact point, which will allow us to construct faceted test functions there.
Let us suppose that the comparison theorem does not hold for a given subsolution $u$ and supersolution $v$, that is, suppose that $$\begin{aligned}
\label{m_0}
m_0 := \sup_Q [u - v] > 0.\end{aligned}$$
For arbitrary $\zeta \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$, ${\ensuremath{\varepsilon}}> 0$ we define $$\begin{aligned}
\Phi_{\zeta,{\ensuremath{\varepsilon}}}(x,t,y,s) := u(x,t) - v(y,s) - \frac{{\left|x-y-\zeta\right|}^2}{2{\ensuremath{\varepsilon}}}-
S_{\ensuremath{\varepsilon}}(t,s),\end{aligned}$$ where $$\begin{aligned}
\label{Se}
S_{\ensuremath{\varepsilon}}(t,s) := \frac{{\left|t-s\right|}^2}{2{\ensuremath{\varepsilon}}} + \frac{{\ensuremath{\varepsilon}}}{T - t}
+\frac{\ensuremath{\varepsilon}}{T-s}.\end{aligned}$$
As in [@GG98ARMA], we define the maximum of $\Phi_{\zeta,{\ensuremath{\varepsilon}}}$ as $$\begin{aligned}
\ell(\zeta, {\ensuremath{\varepsilon}}) = \max_{{\overline{Q}} \times {\overline{Q}}} \Phi_{\zeta,{\ensuremath{\varepsilon}}}\end{aligned}$$ and the set of maxima of $\Phi_{\zeta,{\ensuremath{\varepsilon}}}$ over ${\overline{Q}} \times {\overline{Q}}$ $$\begin{aligned}
\mathcal A(\zeta,{\ensuremath{\varepsilon}}) := {\operatorname*{arg\,max}}_{{\overline{Q}} \times {\overline{Q}}} \Phi_{\zeta,{\ensuremath{\varepsilon}}}
:= {{\left\{(x,t,y,s) \in {\overline{Q}} \times {\overline{Q}} :
\Phi_{\zeta,{\ensuremath{\varepsilon}}}(x,t,y,s) = \ell(\zeta,{\ensuremath{\varepsilon}})\right\}}}.\end{aligned}$$ Moreover, we define the set of gradients $$\begin{aligned}
\mathcal B(\zeta,{\ensuremath{\varepsilon}}) :=
{{\left\{\frac{x - y -\zeta}{{\ensuremath{\varepsilon}}} : (x,t,y,s) \in \mathcal A(\zeta,{\ensuremath{\varepsilon}})\right\}}}.\end{aligned}$$
\[pr:maxima-interior\] There exists ${\ensuremath{\varepsilon}}_0 > 0$ such that for all ${\ensuremath{\varepsilon}}\in (0, {\ensuremath{\varepsilon}}_0)$ we have $$\begin{aligned}
\mathcal A(\zeta,{\ensuremath{\varepsilon}})\subset Q \times Q
\qquad \text{for all ${\left|\zeta\right|} \leq \kappa({\ensuremath{\varepsilon}})$},\end{aligned}$$ where $\kappa({\ensuremath{\varepsilon}}) := \frac 18 (m_0 {\ensuremath{\varepsilon}})^{\frac 12}$.
From now on, we **fix** ${\ensuremath{\varepsilon}}\in (0, {\ensuremath{\varepsilon}}_0)$ so that Proposition \[pr:maxima-interior\] holds and we write $\kappa = \kappa({\ensuremath{\varepsilon}})$ for simplicity, and drop ${\ensuremath{\varepsilon}}$ from our notation.
We have the following properties of $\mathcal A(\zeta)$ and $\mathcal B(\zeta)$.
\[pr:max-graph\] The graphs of $\mathcal A(\zeta)$ and $\mathcal B(\zeta)$ over $\zeta \in {\overline{B}}_\kappa(0)$ are compact.
See [@GG98ARMA Proposition 7.3]. Since $\Phi_\zeta - \ell(\zeta) \leq 0$ by definition of $\ell$, we observe that $$\begin{aligned}
{\operatorname{graph}}\mathcal A(\zeta) &:=
\{(\zeta, x, t, y, s) \subset {\overline{B}}_\kappa(0) \times {\overline{Q}}
\times {\overline{Q}}:\\
&\qquad\qquad\Phi_\zeta(x,t,y,s) - \ell(\zeta) \geq 0\},\end{aligned}$$ which is closed since $\Phi_\zeta$ is an upper semi-continuous function of $(\zeta, x, t, y, s)$ and $\ell(\zeta)$ is a lower semi-continuous function. ${\operatorname{graph}}\mathcal B(\zeta)$ is a continuous image of ${\operatorname{graph}}\mathcal A(\zeta)$ and therefore also compact.
\[pr:ball-of-gradients\] With $\kappa = \kappa({\ensuremath{\varepsilon}})$ fixed above, there exists a maximal relatively open convex set $\Xi
\subset {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ on which $\partial W$ is constant, $\zeta_0 \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ and $\lambda > 0$ such that ${\left|\zeta_0\right|} + 2\lambda < \kappa$ and $$\begin{aligned}
\mathcal B(\zeta) \cap \Xi \neq \emptyset \qquad \text{for all } \zeta \in B_{2\lambda}(\zeta_0).\end{aligned}$$ Moreover, ${\operatorname{aff}}\Xi \perp {\operatorname{aff}}\partial W(p)$ for all $p \in \Xi$.
In other words, for every $\zeta \in B_{2\lambda}(\zeta_0)$ there exists a point of maximum $(\hat x, \hat t, \hat y, \hat s)
\in \mathcal A(\zeta)$ of $\Phi_\zeta$ such that $$\begin{aligned}
\label{grad-in-Xi}
\frac{\hat x - \hat y - \zeta}{{\ensuremath{\varepsilon}}} \in \Xi.\end{aligned}$$
Recall the decomposition of ${{\ensuremath{{\mathbb{R}^{{n}}}}}}$ from Proposition \[pr:feature-decomposition\] into relatively open convex sets $\Xi_i$, $i \in \mathcal N$. Moreover, since $\Xi_i$ is relatively open we can find an increasing sequence of compact sets $K_{i,j} \subset \Xi_i$ such that $$\begin{aligned}
\Xi_i = \bigcup_{j\in{\ensuremath{\mathbb{N}}}} K_{i,j}.\end{aligned}$$ Let us now define the sets $$\begin{aligned}
Z_{i,j} := {{\left\{\zeta \in {\overline{B}}_\kappa(0) : K_{i,j} \cap \mathcal B(\zeta)
\neq \emptyset\right\}}}.\end{aligned}$$ We observe that $Z_{i,j}$ are compact due to Proposition \[pr:max-graph\]. Since $$\begin{aligned}
{\overline{B}}_\kappa(0) = \bigcup_{i\in\mathcal N} \bigcup_{j\in{\ensuremath{\mathbb{N}}}} Z_{i,j},\end{aligned}$$ the Baire category theorem implies that there exists $i_0 \in \mathcal N$, $j_0 \in {\ensuremath{\mathbb{N}}}$ such that ${\operatorname{int}}Z_{i_0, j_0} \neq \emptyset$. In particular, we can find $\zeta_0$ and $\lambda > 0$ with $B_{2\lambda}(\zeta_0) \subset Z_{i_0,j_0}$, and we set $\Xi = \Xi_{i_0}$. Note that $\Xi$ is maximal by Proposition \[pr:feature-decomposition\].
Flatness at a contact point
---------------------------
We will now use the information about the behavior of $u$ and $v$ at the point of maximum to show that there is enough space to construct faceted test functions. We shall use the Constancy lemma from [@GG98ARMA].
\[le:constancy\] Let $1\leq k < N$, $K \subset {\ensuremath{\mathbb{R}}}^N$ be compact and $G \subset {\ensuremath{\mathbb{R}}}^k$ be a bounded domain. Denote $P: {\ensuremath{\mathbb{R}}}^N \to {\ensuremath{\mathbb{R}}}^k$ the natural projection $w \mapsto (w_1, \ldots, w_k)$. Assume that $h$ is an upper semi-continuous function and $\phi\in C^2({\ensuremath{\mathbb{R}}}^k)$, and define for $w \in K$ and $z \in G$ $$\begin{aligned}
h_z(w) &:= h(w) - \phi(Pw - z),\\
H(z) &:= \max_K h_z.\end{aligned}$$ If for all $z\in G$ there exists $w \in K$ such that $h_z(w) = H(z)$ and $(\nabla \phi)(P w - z) = 0$ then $H(z)$ is constant on $G$.
In what follows, we will decompose ${\ensuremath{\mathbb{R}}}^n$ into two orthogonal subspaces $V$ and $U$, as in Section \[sec:slicing of W\], of dimensions $k = \dim V$ and $n - k = \dim U$. Therefore we will use ${{\mathcal T}}$, ${{\mathcal T}}_U$, ${{\mathcal T}}_V$, and the decomposition $x = {{\mathcal T}}(x', x'')$ as introduced in , and .
\[le:max-constancy\] Suppose that there exist $p_0, \zeta_0 \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$, a subspace $U \subset {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ and $\lambda > 0$ such that ${\left|\zeta_0\right|} + 2\lambda < \kappa$ and for all $\zeta \in B_{2\lambda}(\zeta_0)$ we have $$\begin{aligned}
\mathcal B(\zeta) \cap (p_0 + U) \neq \emptyset.\end{aligned}$$ Then $$\begin{aligned}
\ell(\zeta) - p_0 \cdot \zeta = const
\quad \text{for } \zeta \in (\zeta_0 + V) \cap B_{2\lambda}(\zeta_0),\end{aligned}$$ where $V := U^\perp$.
We apply the constancy lemma, Lemma \[le:constancy\]. Set $k = \dim V$ and $N = 2 (n+1)$. We will denote $w = (\xi', \xi'', t, y, s)$ for $\xi, y \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$, $t, s \in {\ensuremath{\mathbb{R}}}$, so that the natural projection $P:{\ensuremath{\mathbb{R}}}^N \to {\ensuremath{\mathbb{R}}}^k$ is given as $Pw = \xi'$ (for the notation $'$ and $''$ see ). Additionally, let us set $$\begin{aligned}
K = {{\left\{(\xi', \xi'', t, y, s) : (\xi + y,t) \in {\overline{Q}},
\ (y,s) \in {\overline{Q}}\right\}}}
\quad \text{and} \quad G = B^d_{2\lambda}(0).\end{aligned}$$ And, finally, let us define the functions $$\begin{aligned}
h(w) &= u(\xi + y) - v(y) - p_0' \cdot \xi'
- \frac{{\left|\rho'' - \zeta_0''\right|}^2}{2{\ensuremath{\varepsilon}}} - S(t,s), &&w \in {\ensuremath{\mathbb{R}}}^N,\\
\phi(\eta) &= \frac{{\left|\eta - \zeta_0'\right|}^2}{2{\ensuremath{\varepsilon}}} - p_0' \cdot \eta, \qquad &&\eta \in {\ensuremath{\mathbb{R}}}^k.\end{aligned}$$ The Pythagorean theorem $|\xi' - \zeta_0' -z|^2 + |\xi'' - \zeta_0''|^2 = |\xi - \zeta_0 - {{\mathcal T}}_V
z|^2$, due to the fact that ${{\mathcal T}}$ is a rotation, yields $$\begin{aligned}
h_z(w) := h(w) - \phi(\xi' - z) = \Phi_{\zeta_0 + {{\mathcal T}}_V z} (\xi + y, y) - p_0' \cdot z.\end{aligned}$$ We know, by assumption, that for every $\zeta = \zeta_0 + {{\mathcal T}}_V z$, $z \in G$, there exists a point of maximum $(x,t,y,s) \in \mathcal A(\zeta)$ of $\Phi_{\zeta}$ such that $\frac{x - y -\zeta}{{\ensuremath{\varepsilon}}} \in p_0 + U$. This yields $\frac{x' - y' - z -\zeta_0'}{{\ensuremath{\varepsilon}}} = p_0'$. In particular, $$\begin{aligned}
(\nabla \phi)(x' - y' - z) = 0.\end{aligned}$$ Thus, by Lemma \[le:constancy\], we infer that $$\begin{aligned}
H(z) = \max_K h_z = \ell(\zeta_0 + {{\mathcal T}}_V z) - p_0' \cdot z\end{aligned}$$ is constant for $z \in G$, which is what we wanted to prove since $p_0 \cdot \zeta = p_0' \cdot z + p_0 \cdot \zeta_0$.
The previous lemma has the following important corollary.
\[co:contact-ordering\] Suppose that we have $p_0$, $\zeta_0$, $\lambda$, $U$ and $V$ as in Lemma \[le:max-constancy\]. Define $$\begin{aligned}
{\ensuremath{\theta}}(x,t,y,s) := u(x,t) - v(y,s) - \frac{{\left|x'' - y''- \zeta_0''\right|}^2}{2{\ensuremath{\varepsilon}}}
- p_0' \cdot (x' - y'-\zeta_0') - S(t,s).\end{aligned}$$ Then for any $(\hat x,\hat t,\hat y,\hat s) \in \mathcal A(\zeta_0)$ such that $\frac{\hat x' - \hat y' - \zeta_0'}{{\ensuremath{\varepsilon}}} = p_0'$ we have $$\begin{aligned}
{\ensuremath{\theta}}(x,t,y,s) \leq {\ensuremath{\theta}}(\hat x, \hat t, \hat y,\hat s) \quad \text{for } (x,t),(y,s) \in {\overline{Q}},
\ {\left|x' - y' - (\hat x' - \hat y')\right|} \leq \lambda.\end{aligned}$$
For the sake of clarity, we will drop $t,s, \hat t$ and $\hat s$ from the following formulas. Let us fix $x, y, \hat x, \hat y$ that satisfy the assumptions and set $$\begin{aligned}
\label{choice-of-zeta}
\zeta = \zeta_0 + {{\mathcal T}}_V(x' - y' - (\hat x' - \hat y')).\end{aligned}$$ Since $|\zeta - \zeta_0| \leq \lambda$ and $\zeta \in \zeta_0 + V$, Lemma \[le:max-constancy\] implies $\ell(\zeta) - p_0 \cdot \zeta = \ell(\zeta_0) - p_0 \cdot \zeta_0$ and we infer from the definition of $\ell$ $$\begin{aligned}
\label{Phi-estim}
\Phi_\zeta(x,y) \leq
\ell(\zeta) =
\ell(\zeta_0) + p_0 \cdot (\zeta - \zeta_0) = \Phi_{\zeta_0} (\hat x, \hat y) + p_0 \cdot (\zeta - \zeta_0).\end{aligned}$$ Note also that $p_0 \cdot (\zeta - \zeta_0) = p_0' \cdot (\zeta' - \zeta_0')$. We express ${\ensuremath{\theta}}$ in terms of $\Phi_\zeta$ and use to obtain $$\begin{aligned}
{\ensuremath{\theta}}(x,y) &= \Phi_\zeta(x,y) + \frac{{\left|x'-y'-\zeta'\right|}^2}{2{\ensuremath{\varepsilon}}}
- p_0' \cdot (x' - y' - \zeta_0')\\
&\leq \Phi_{\zeta_0}(\hat x,\hat y) + \frac{{\left|x'-y'-\zeta')\right|}^2}{2{\ensuremath{\varepsilon}}}
+ p_0' \cdot (\zeta' - \zeta_0' - (x' - y' - \zeta_0'))\\
&= {\ensuremath{\theta}}(\hat x, \hat y) + \frac{{\left|x'-y'-\zeta'\right|}^2}{2{\ensuremath{\varepsilon}}}
-\frac{{\left|\hat x'-\hat y'-\zeta_0'\right|}^2}{2{\ensuremath{\varepsilon}}}\\
&\quad + p_0' \cdot (-(x' - y' - \zeta') + (\hat x' - \hat y' - \zeta_0'))\\
&= {\ensuremath{\theta}}(\hat x, \hat y) + \frac{{\left|w\right|}^2}{2{\ensuremath{\varepsilon}}} - \frac{{\left|z\right|}^2}{2{\ensuremath{\varepsilon}}}
+ p_0' \cdot (-w + z),\end{aligned}$$ where we set $w = x'-y'-\zeta'$ and $z = \hat x' - \hat y' -\zeta_0'$. We now just have to show that the extra terms cancel out. First, we see that $w - z = 0$ by . Furthermore, by the choice of $\hat x,\hat y$ we have $z/{\ensuremath{\varepsilon}}= p_0'$. Therefore we have, using ${\left|w -z\right|}^2 = {\left|w\right|}^2 + {\left|z\right|}^2 - 2w \cdot z$, $$\begin{aligned}
\frac{{\left|w\right|}^2}{2{\ensuremath{\varepsilon}}} - \frac{{\left|z\right|}^2}{2{\ensuremath{\varepsilon}}} =
\frac{{\left|w - z\right|}^2}{2{\ensuremath{\varepsilon}}} - \frac{{\left|z\right|}^2}{{\ensuremath{\varepsilon}}} + \frac{w \cdot z}{{\ensuremath{\varepsilon}}}
= 0 -p_0' \cdot z + p_0' \cdot w.\end{aligned}$$ Therefore ${\ensuremath{\theta}}(x,t,y,s) \leq {\ensuremath{\theta}}(\hat x, \hat t, \hat y, \hat s)$, which is what we wanted to prove.
Construction of faceted test functions
--------------------------------------
We shall use Corollary \[co:contact-ordering\] to construct test functions for $u$ and $v$ following the idea from [@GGP13AMSA].
Let us fix $\Xi, \zeta_0 \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ and $\lambda>0$ to be a triplet provided by Proposition \[pr:ball-of-gradients\]. Then we fix a point of maximum $(\hat x, \hat t, \hat y,
\hat s) \in \mathcal A(\zeta_0)$ that satisfies with $\zeta = \zeta_0$. We set $p_0 := \frac{\hat x - \hat y - \zeta_0}{\ensuremath{\varepsilon}}\in \Xi$, $U := {\operatorname{span}}(\Xi - p_0)$ and $V \subset {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ be the subspace parallel to ${\operatorname{aff}}\partial W(p_0)$. We have $U = V^\perp$ by Proposition \[pr:feature-decomposition\]. It is easy to check that $p_0$, $\zeta_0$, $\lambda$, $U$, $V$ and $(\hat x, \hat t, \hat y, \hat s)$ satisfy the hypothesis of Corollary \[co:contact-ordering\]. Let us also set $k = \dim V$ as usual.
Depending on the value $k$ and $p_0$, we split the situation into three cases:
- : $k = 0$;
- : $k = n$, $p_0 = 0$ and $F$ is of curvature-free type at $p_0 = 0$;
- : none of the above.
We will deal with each case individually in the following three subsections and show that they all lead to a contradiction. Therefore cannot occur and the comparison principle holds.
### Case I
We have $k = 0$. In this case we use the off-facet test in Definition \[def:visc-solution\](ii). Corollary \[co:contact-ordering\] in this case reduces to $$\begin{aligned}
\label{off-facet max order}
u(x, t) - v(y, s) - \frac{|x - y - \zeta_0|^2}{2{\ensuremath{\varepsilon}}} - S(t,s)
\leq
u(\hat x, \hat t) - v(\hat y, \hat s) - \frac{|\hat x - \hat y - \zeta_0|^2}{2{\ensuremath{\varepsilon}}} - S(\hat t,\hat s)\end{aligned}$$ for all $(x,t), (y,s) \in {\overline{Q}}$. We define the test functions $$\begin{aligned}
\varphi_u(x,t) &:= \frac{|x - \hat y - \zeta_0|^2}{2{\ensuremath{\varepsilon}}} + S(t,\hat s),\\
\varphi_v(x,t) &:= - \frac{|\hat x - x - \zeta_0|^2}{2{\ensuremath{\varepsilon}}} - S(\hat t, t).\end{aligned}$$ From we deduce that $u - \varphi_u$ has a global maximum at $(\hat x,
\hat t)$ and $v - \varphi_v$ has a global minimum at $(\hat y, \hat s)$. Therefore we must have from the definition of viscosity solutions $$\begin{aligned}
(\varphi_u)_t(\hat x, \hat t) + F(\nabla \varphi_u(\hat x, \hat t), 0) &\leq 0,\\
(\varphi_v)_t(\hat y, \hat s) + F(\nabla \varphi_v(\hat y, \hat s), 0) &\geq 0.\\\end{aligned}$$ Since $\nabla \varphi_u(\hat x, \hat t) = \nabla \varphi_v(\hat y, \hat s)$, subtracting the second inequality from the first and evaluating the time derivatives yields $$\begin{aligned}
0 \geq (\varphi_u)_t(\hat x, \hat t) - (\varphi_v)_t(\hat y, \hat s) = \frac{{\ensuremath{\varepsilon}}}{(T - \hat t)^2} + \frac{{\ensuremath{\varepsilon}}}{(T - \hat s)^2} > 0,\end{aligned}$$ a contradiction.
### Case II
Now $k = n$, or, in other words, $V = {\operatorname{aff}}\partial W(p_0) = {{\ensuremath{{\mathbb{R}^{{n}}}}}}$. Since we now assume that $p_0 =
0$ and that $F$ is of curvature-free type at $p_0 = 0$, we use Definition \[def:level-set-test\]. Then this case is just a minor modification of Case I. Indeed, Corollary \[co:contact-ordering\] now reads $$\begin{aligned}
\label{curvature-free order}
u(x, t) - v(y, s) - S(t,s)
\leq
u(\hat x, \hat t) - v(\hat y, \hat s) - S(\hat t,\hat s)\end{aligned}$$ for all $(x, t), (y, t) \in {\overline{Q}}$, $|x - y - (\hat x - \hat y)| \leq \lambda$. Thus if we define the test functions $$\begin{aligned}
\varphi_u(x,t) := S(t, \hat s), \quad \text{and} \quad \varphi_v(x,t):= - S(\hat t, t),\end{aligned}$$ we see from that $u - \varphi_u$ has a local maximum at $(\hat x, \hat
t)$, and $v - \varphi_v$ has a local minimum at $(\hat y, \hat s)$. The definition of viscosity solution for the curvature-free type case yields $$\begin{aligned}
(\varphi_u)_t(\hat x, \hat t) + F(0, 0) &\leq 0,\\
(\varphi_v)_t(\hat y, \hat s) + F(0, 0) &\geq 0.\\\end{aligned}$$ The contradiction then follows as in Case I.
### Case III
This is the most involved situation. Since $W$ is positively one-homogeneous, we have $p_0 \perp V$ by Lemma \[le:aff Xi origin\] and the orthogonality from Proposition \[pr:feature-decomposition\], and therefore $p_0' = 0$ in what follows. Nevertheless, we keep the terms with $p_0'$ below for completeness, they are necessary when handling a case of general polyhedral $W$. We first reduce the problem to the subspace $V$ by introducing the functions $$\begin{aligned}
\begin{aligned}
\hat u(w) &:= u({{\mathcal T}}_V w + \hat x, \hat t) - p_0' \cdot w - u(\hat x, \hat t),\\
\hat v(w) &:= v({{\mathcal T}}_V w + \hat y, \hat s) - p_0' \cdot w - v(\hat y, \hat s),
\end{aligned}
&&& w \in {\ensuremath{\mathbb{R}}}^k.\end{aligned}$$ Then we build facets on ${\ensuremath{\mathbb{R}}}^k$ using the closed sets $$\begin{aligned}
\hat U := {{\left\{w \in {\ensuremath{\mathbb{R}}}^k : \hat u(w) \geq 0\right\}}}, &&&
\hat V := {{\left\{w \in {\ensuremath{\mathbb{R}}}^k : \hat v(w) \leq 0\right\}}}.\end{aligned}$$ as in [@GGP13AMSA]; see Figure \[fig:facet-construction\]. Note that these sets were denoted there as $U$ and $V$. This allows us to create test functions for both subsolution and supersolution and arrive at a contradiction as before.
Let us review the construction. For convenience we set $$\begin{aligned}
\xi_u(x'', t) &:= \frac{|x'' - \hat y'' - \zeta_0''|^2}{2{\ensuremath{\varepsilon}}}
- \frac{|\hat x'' - \hat y'' - \zeta_0''|^2}{2{\ensuremath{\varepsilon}}} + S(t, \hat s) - S(\hat t, \hat s),\\
\xi_v(y'', s) &:= \frac{|\hat x'' - \hat y'' - \zeta_0''|^2}{2{\ensuremath{\varepsilon}}}
- \frac{|\hat x'' - y'' - \zeta_0''|^2}{2{\ensuremath{\varepsilon}}} + S(\hat t, \hat s) - S(\hat t, s).\end{aligned}$$ Then $$\begin{aligned}
u(x,t) - u(\hat x, \hat t) - p_0' \cdot (x' - \hat x') - \xi_u(x'', t) &\leq 0,
&&\text{for } (x,t) \in {\overline{Q}}, x' - \hat x \in {{\mathcal U}^{\lambda}}(\hat V),\\
v(y,s) - v(\hat y, \hat s) - p_0' \cdot (y' - \hat y') - \xi_v(y'', s) &\geq 0,
&&\text{for } (y,s) \in {\overline{Q}}, y' - \hat y \in {{\mathcal U}^{\lambda}}(\hat U).\end{aligned}$$
We set $r := \lambda/10$ and introduce the closed sets $$\begin{aligned}
X := {\overline{({{\mathcal U}^{r}}(\hat U))^c}}, \qquad Y:= {\overline{({{\mathcal U}^{r}}(\hat V))^c}}.\end{aligned}$$
Since ${\operatorname{dist}}(\hat U, X) = {\operatorname{dist}}(\hat V, Y) = r$, the semi-continuity of $u$ and $v$ imply that there exists $\delta > 0$ such that $$\begin{aligned}
u(x,t) - u(\hat x, \hat t) - p_0' \cdot (x' - \hat x') - \xi_u(x'', t) &< 0,
&&x' - \hat x' \in X, |x'' - \hat x''| \leq \delta, |t - \hat t| \leq \delta,\\
v(y,s) - v(\hat y, \hat s) - p_0' \cdot (y' - \hat y') - \xi_v(y'', s) &> 0,
&&y' - \hat y' \in Y, |y'' - \hat y''| \leq \delta, |s - \hat s| \leq \delta.\end{aligned}$$ Note that if $X$ is unbounded, then $u(x, t) = c_u < u(\hat x, \hat t)$ for all $x \notin K$ and therefore we only need to use semi-continuity of $u$ on a compact subset of $X$ to get the $\delta$ above. We can similarly handle the case of unbounded $Y$.
Therefore as in [@GGP13AMSA], we define the pairs $$\begin{aligned}
S_u := (\hat U^c, \hat U \setminus {{\mathcal U}^{\lambda-3r}}(\hat V)), \qquad
S_v := (\hat V^c, \hat V \setminus {{\mathcal U}^{\lambda-3r}}(\hat U)).\end{aligned}$$ We note that both $S_u$ and $S_v$ are bounded pairs. Indeed, $S_u$ bounded if $\hat U$ is bounded or $\hat U^c \cup \hat V$ is bounded. Since $u(\hat x, \hat t) - v(\hat y, \hat t) \geq m_0$, we deduce that $u(\hat x, \hat t) > v(\hat
y, \hat t)$. Then if $\hat U$ is unbounded, we have $u(\hat x, \hat t) \leq c_u$ and therefore $v(\hat y, \hat s) < u(\hat x, \hat t) \leq c_u \leq c_v$, and so we conclude that $\hat U^c \cup \hat V$ are both bounded. We can argue similarly for $S_v$.
Since both $S_u$ and $S_v$ are bounded pairs, Corollary \[co:approximate pair sliced\] (currently only for $k = 1, 2$) implies that there exist $p_0$-admissible pairs $(U_-, U_+)$ and $(V_-, V_+)$ such that $$\begin{aligned}
{{\mathcal U}^{2r}}(S_u) &\preceq (U_-, U_+) \preceq {{\mathcal U}^{3r}}(S_u),\\
{{\mathcal U}^{2r}}(S_v) &\preceq (V_-, V_+) \preceq {{\mathcal U}^{3r}}(S_v).\end{aligned}$$
We have the following lemma.
\[le:pair properties\] The pair $(U_-, U_+)$ and the pair $(V_-, V_+)$ have the following properties:
1. The pairs are strictly ordered in the sense $$\begin{aligned}
\label{pair order}
{{\mathcal U}^{r}}(U_-, U_+) \preceq (V_+, V_-) = - (V_-, V_+).\end{aligned}$$
2. The origin $0$ lies in the interior of the intersection of the facets, that is, $$\begin{aligned}
{\overline{B}}_r(0) \subset U_-^c \cap U_+^c \cap V_-^c \cap V_+^c.\end{aligned}$$
3. The pairs are in general position with respect to $R_u$ and $R_v$, that is, $$\begin{aligned}
{{\mathcal U}^{r}}(R_u) \preceq (U_-, U_+), \qquad {{\mathcal U}^{r}}(R_v) \preceq (V_-, V_+),\end{aligned}$$ where $$\begin{aligned}
R_u &:= (X, X^c \setminus {{\mathcal U}^{\lambda}}(\hat V)),\\
R_v &:= (Y, Y^c \setminus {{\mathcal U}^{\lambda}}(\hat U)).\end{aligned}$$
See [@GGP13AMSA Lemma 4.6].
Now we have all that we need to reach a contradiction. Let us define $$\begin{aligned}
\tilde u(x' - \hat x') &:=\sup_{|x'' - \hat x''| \leq \delta}\sup_{|t - \hat t| \leq \delta} {\left[ u(x,t) - u(\hat
x, \hat t) - p_0' \cdot (x' - \hat x') - \xi_u(x'', t)\right]},\\
\tilde v(y' - \hat y') &:= \inf_{|y'' - \hat y''| \leq \delta}\inf_{|s - \hat s| \leq \delta} {\left[v(y,s) - v(\hat
y, \hat s) - p_0' \cdot (y' - \hat y') - \xi_v(y'', s)\right]}.\end{aligned}$$ By the construction above, we have $\tilde u < 0$ on $X$ and $\tilde u \leq 0$ on $X \cup
{{\mathcal U}^{\lambda}}(\hat V)$. Similarly, we have $\tilde v > 0$ on $Y$ and $\tilde v \geq 0$ on $Y \cup
{{\mathcal U}^{\lambda}}(\hat U)$. Lemma \[le:pair properties\](c) implies that $X \supset {{\mathcal U}^{r}}(U_-)$ and $X \cup {{\mathcal U}^{\lambda}}(
\hat V) \supset {{\mathcal U}^{r}}(U_+^c)$. Therefore for any support function $\psi$ of pair $(U_-, U_+)$ we can by upper semi-continuity of $\tilde u$ find two constants $\alpha,
\beta > 0$ so that $\alpha \psi_+ - \beta \psi_- \geq \tilde u(\cdot - w)$ for all $|w| \leq r/2$ in a neighborhood of the facet $U_-^c \cap U_+^c$. An analogous reasoning applies to $\tilde v$.
Since the pairs $(U_-, U_+)$ and $(V_-,
V_+)$ are $p_0$-admissible, there exist faceted functions $\psi_u, \psi_v \in {\operatorname{\mathcal{D}}}(\Lambda_{p_0})$ that are the support functions of the respective pairs. By applying the observation in the previous paragraph, we can assume that $\tilde u(\cdot - w) \leq \psi_u$ and $\psi_v \leq \tilde v(\cdot -
w)$ for all $|w| \leq r/2$ in a neighborhood of the respective facets $U_-^c \cap U_+^c$ and $V_-^c
\cap V_+^c$. Therefore the functions $\varphi_u(x,t) := \psi_u(x') + p_0' \cdot (x' - \hat x') + \xi_u(x'',
t)$, $\varphi_v := \psi_v(x') + p_0' \cdot (y' - \hat y')
+ \xi_v(x'', t)$ are test functions for $u$ and $v$, respectively, in the sense of Definition \[def:visc-solution\](i). Due to Lemma \[le:pair properties\](a–b), the comparison principle for the crystalline curvature Proposition \[pr:comparison Lambda\] yields $$\begin{aligned}
\label{essinf sup order}
{\operatorname*{ess\,inf}}_{B_r(0)} \left[ \Lambda_{p_0}[\psi_u] \right] \leq {\operatorname*{ess\,sup}}_{B_r(0)} \left[
\Lambda_{p_0}[\psi_v] \right].\end{aligned}$$
From the definition of viscosity solutions, namely Definition \[def:visc-solution\](i), we infer $$\begin{aligned}
(\xi_u)_t(\hat t) + F{\left(p_0, {\operatorname*{ess\,inf}}_{B_r(0)} \left[ \Lambda_{p_0}[\psi_u] \right]\right)} &\leq 0,\\
(\xi_v)_t(\hat s) + F{\left(p_0, {\operatorname*{ess\,sup}}_{B_r(0)} \left[ \Lambda_{p_0}[\psi_v] \right]\right)} &\geq 0.\end{aligned}$$ Using and the ellipticity of $F$, we get after subtracting the above two inequalities $$\begin{aligned}
0 < \frac {\ensuremath{\varepsilon}}{(T - \hat t)^2} + \frac {\ensuremath{\varepsilon}}{(T - \hat s)^2} + F{\left(p_0, {\operatorname*{ess\,inf}}_{B_r(0)} \left[
\Lambda_{p_0}[\psi_u] \right]\right)} - F{\left(p_0, {\operatorname*{ess\,sup}}_{B_r(0)} \left[ \Lambda_{p_0}[\psi_v] \right]\right)}
\leq 0,\end{aligned}$$ a contradiction.
This finished the proof of the comparison principle Theorem \[th:comparison principle\] since we have shown that always yields a contradiction.
Stability {#se:stability}
=========
We will show the stability of under the approximation by parabolic problems $$\begin{aligned}
\label{regularized-problem}
\begin{cases}
u_t + F(\nabla u, {\operatorname{tr}}{\left[(\nabla_p^2 W_m)(\nabla u) \nabla^2 u\right]}) = 0,\\
{{{\left.{u}\right|}_{t=0}}} = u_0,
\end{cases}\end{aligned}$$ where $W_m$ approximate $W$ as in Section \[sec:resolvent-approximation\]. An example of such sequence ${{\left\{W_m\right\}}}$ is given in Example \[ex:wm-example\].
The main result of this section is the following stability theorem. We recall the definition of *half-relaxed limits* (semi-continuous limits) $$\begin{aligned}
{\operatorname*{\star-limsup}}_{m\to\infty} u_m(x,t) &:= \lim_{k \to \infty} \sup_{m > k} \sup_{|y - x| < \frac 1k} \sup_{|t-s| <
\frac 1k} u_m(y,s),\\
{\operatorname*{\star-liminf}}_{m\to\infty} u_m(x,t) &:= -{\operatorname*{\star-limsup}}_{m\to\infty} \big(-u_m(x,t)\big).\end{aligned}$$
\[th:stability quadratic\] Let $u_m$ be a locally bounded sequence of viscosity solutions of (without the initial condition). Then ${\operatorname*{\star-limsup}}_{m\to\infty} u_m$ is a viscosity subsolution of and ${\operatorname*{\star-liminf}}_{m\to\infty} u_m$ is a viscosity supersolution of .
*Proof of stability* We will only show the subsolution part, the proof of the supersolution part is analogous. Let $u = {\operatorname*{\star-limsup}}_m u_m$. Clearly $u$ is upper semi-continuous. We want to show that $u$ is a subsolution of .
We have to verify (i)–(ii) of Definition \[def:visc-solution\] and (i-cf) of Definition \[def:level-set-test\] for curvature-free type $F$.
Case (i) {#se:stability case (i)}
--------
Suppose that ${\ensuremath{\varphi}}$ is a stratified faceted test function at $(\hat x, \hat t)$ with gradient $\hat p$ and with $\bar \psi$, $f$ and $g$ as in Definition \[def:strat-faceted-test-function\], and suppose that this test function is a test function for $u$ in the sense of Definition \[def:visc-solution\](i), i.e., it satisfies with some $\rho > 0$. Let $(A_-, A_+)$ be the pair supported by $\bar \psi$. We will set $V \subset {{\ensuremath{{\mathbb{R}^{{n}}}}}}$ to be the subspace parallel to ${\operatorname{aff}}\partial W(\hat p)$, $U=V^\perp$ and $k = \dim V$. We recall that we have the rotated coordinate system $x = {{\mathcal T}}(x', x'')$ with ${{\mathcal T}}= {{\mathcal T}}_{\hat p}$ introduced in .
Let us define the function $\bar u: {\ensuremath{\mathbb{R}}}^k \to {\ensuremath{\mathbb{R}}}$ $$\begin{aligned}
\bar u(x') := \sup_{\substack{{\left|x''\right|} \leq \rho\\{\left|t - \hat t\right|} \leq \rho}} u(\hat x + x, t)
&- u(\hat x, \hat t) - f(x'') - g(t) + g(\hat t) - \hat p \cdot x\end{aligned}$$ and the closed subsets of ${\ensuremath{\mathbb{R}}}^k$ $$\begin{aligned}
Y &:= {{\left\{x' \in {\ensuremath{\mathbb{R}}}^k : \bar u(x') \geq 0\right\}}}, \qquad\\
Z &:= {{\left\{x'\in O: \bar \psi(x') \leq 0\right\}}} = A_+^c \cap O,\end{aligned}$$ where $O = {{\mathcal U}^{\rho}}({A_-^c \cap A_+^c})$, see Figure \[fig:stability-geometry\]. Note that with this definition of $\bar u$, the condition is equivalent to $$\begin{aligned}
\label{gp-bar}
\bar u(y') \leq \bar\psi(x') \qquad \text{for all $x' \in O$, ${\left|y' - x'\right|} \leq \rho$.}\end{aligned}$$
We immediately have the following “geometrical” lemma. Intuitively, since $\bar u$ and $\bar\psi$ are ordered even when shifted by a small distance, we must have that $\bar\psi$ is nonnegative in a neighborhood of the set $Y$ where $\bar u$ is nonnegative, and, analogously, $\bar u$ is nonpositive in a neighborhood of the set $Z$ where $\bar \psi$ is nonpositive.
\[le:u-psi-shift\] Suppose that $u$ and ${\ensuremath{\varphi}}$ satisfy for some $\rho > 0$, $(\hat x, \hat t) \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$. Then $$\begin{aligned}
\bar u(x') \leq 0 \qquad \text{for all $x' \in {{\mathcal U}^{\rho}}(Z)$,}\end{aligned}$$ or, more explicitly, $$\begin{aligned}
u(x,t) \leq f{\left(x''- \hat x''\right)} + g(t) - g(\hat t) + u(\hat x,\hat t)
+ \hat p \cdot (x - \hat x)\end{aligned}$$ for $x' - \hat x \in {{\mathcal U}^{\rho}}(Z)$, ${\left|x'' - \hat x''\right|} \leq \rho$, ${\left|t - \hat t\right|} \leq \rho$. Furthermore, we have $$\begin{aligned}
\bar \psi(x') \geq 0 \qquad \text{for } x' \in {{\mathcal U}^{\rho}}(Y) \cap O.\end{aligned}$$
Let us prove the first statement. If $x' \in {{\mathcal U}^{\rho}}(Z)$ then there exists $z' \in Z \subset O$ such that ${\left|x' - z'\right|} \leq \rho$. Thus and the definition of $Z$ imply $$\begin{aligned}
\bar u(x') \leq \bar\psi(z') \leq 0,\end{aligned}$$ and that is what we wanted to prove.
Similarly, if we suppose that $x' \in {{\mathcal U}^{\rho}}(Y) \cap O$, there exists $y' \in Y$ with ${\left|x' - y'\right|} \leq \rho$. Then and the definition of $Y$ imply $$\begin{aligned}
\bar\psi(x') \geq \bar u(y') \geq 0.\end{aligned}$$ The lemma is proved.
We obtain the following corollary.
Suppose that holds with $\rho > 0$. Then there exists $\delta$, $0 < \delta \leq \rho/5$, such that ${{\mathcal U}^{4\delta}}(N) \subset O$ where $$\begin{aligned}
N := {{\mathcal U}^{\delta}}(Z) \cap Y \cap O,\end{aligned}$$ and moreover $$\begin{aligned}
{\overline{B}}^k_\delta(0) \subset A_-^c \cap A_+^c\end{aligned}$$ and $$\begin{aligned}
\label{u-psi-scaled-comp}
\bar u(x') \leq \alpha \bar\psi(x' + z') \qquad \text{for all $\alpha > 0$, $x' \in {{\mathcal U}^{3\delta}}(N)$, ${\left|z'\right|} \leq \delta$,}\end{aligned}$$ with strict inequality for $x' \notin N$.
By definition, $Z \subset A_+^c$. Moreover, the second result in Lemma \[le:u-psi-shift\] is equivalent to $$\begin{aligned}
{{\mathcal U}^{\rho}}(Y) \cap O \subset A_-^c.\end{aligned}$$ We can therefore estimate $$\begin{aligned}
\label{N-bound}
N = {{\mathcal U}^{\delta}}(Z) \cap Y \cap O \subset {{\mathcal U}^{\delta}}(A_+^c) \cap A_-^c
\subset {{\mathcal U}^{\delta}}(\partial A_+) \cup (A_+^c \cap A_-^c).\end{aligned}$$ Since $A_+$ is open, we have $\partial A_+ \subset A_+^c$. But since $A_- \cap A_+ = \emptyset$ and $A_-$ is also open, we must also have $\partial A_+ \subset A_-^c$. Therefore $\partial A_+$ is in the facet, and by assumption on $O$ we have $$\begin{aligned}
\label{boundary-in-facet}
\partial A_+ \subset A_-^c \cap A_+^c \subset O.\end{aligned}$$ Since $O$ is open and $A_-^c \cap A_+^c$ is compact, and $0\in {\operatorname{int}}A_-^c \cap A_+^c$, for $\delta > 0$ small enough we will have $$\begin{aligned}
{{\mathcal U}^{5\delta}}(A_-^c \cap A_+^c) \subset O \qquad \text{and} \qquad
{\overline{B}}_\delta^k(0) \subset A_-^c \cap A_+^c.\end{aligned}$$ Using in , we obtain $$\begin{aligned}
\label{N-4delta}
{{\mathcal U}^{4\delta}}(N) \subset {{\mathcal U}^{5\delta}}(A_-^c \cap A_+^c) \subset O.\end{aligned}$$ Let us now fix $\alpha > 0$ and ${\left|z'\right|} \leq \delta$. Using the definition and , we can estimate $$\begin{aligned}
{{\mathcal U}^{3\delta}}(N) \subset {{\mathcal U}^{4\delta}}(Z) \cap {{\mathcal U}^{3\delta}}(Y) \cap
{{\mathcal U}^{-\delta}}(O).\end{aligned}$$ In particular, if $x' \in {{\mathcal U}^{3\delta}}(N)$ then $x' + z' \in {{\mathcal U}^{\rho}}(Y) \cap O$ and $x' \in {{\mathcal U}^{\rho}}(Z)$. Hence Lemma \[le:u-psi-shift\] applies, yielding $$\begin{aligned}
\bar u(x') \leq 0 \leq \bar\psi(x' + z'),\end{aligned}$$ and therefore follows. If $x' \in {{\mathcal U}^{3\delta}}(N) \setminus N$, then we must have $x' + z' \in O$ and at least one of the following:
- $x' \notin {{\mathcal U}^{\delta}}(Z)$: Thus $x' + z' \in O \setminus Z$ and therefore $\psi(x'+z') > 0$.
- $x' \notin Y$: Thus $u(x') < 0$.
We deduce the strict ordering in for $x \notin N$.
The previous corollary has the following important direct consequence.
\[le:strict\_order\] Suppose that is satisfied for some $\rho > 0$. By adding the term ${\left|y\right|}^2$ to $f(y)$ and ${\left|t - \hat t\right|}^2$ to $g(t)$ if necessary, there exists $0 < \delta < \rho/5$ such that for all ${\left|z'\right|} \leq \delta$ and $\alpha > 0$ $$\begin{aligned}
u(x,t) - \alpha \bar \psi(x' + z' - \hat x')
- f(x'' - \hat x'') - g(t) - \hat p \cdot (x - \hat x)
\leq u(\hat x, \hat t) - g(\hat t)\end{aligned}$$ whenever $$\begin{aligned}
x' - \hat x' \in {{\mathcal U}^{3\delta}}(N),\ {\left|x'' - \hat x''\right|} \leq \rho,\ {\left|t - \hat t\right|} \leq \rho,\end{aligned}$$ with a *strict* inequality outside of ${{\left\{(x,t): x' - \hat x' \in N,\ x'' = \hat x'',\ t = \hat t\right\}}}$.
We shall now proceed with the proof of stability. By Proposition \[pr:curvature-as-min-section\], for $L > 0$ sufficiently large and $\Gamma' = {\ensuremath{\mathbb{R}}}^k / L {\ensuremath{\mathbb{Z}}}^k$, we can find a function $\xi \in {{\rm Lip}}(\Gamma')$ such that $\xi(x') = \bar\psi(x')$ on a neighborhood of the facet ${A_-^c \cap A_+^c}$ such that $\xi \in {\operatorname{\mathcal{D}}}(\partial {E^{\rm sl}}_{\hat p}(\cdot; \Gamma'))$ and $\Lambda_{\hat p}(\bar \psi) = -\partial^0 {E^{\rm sl}}_{\hat p}(\xi; \Gamma')$ a.e. on ${A_-^c \cap A_+^c}$. By making the set $O$ smaller if necessary, we can assume that $\xi = \bar\psi$ on $O$. Let $\delta > 0$ be from Lemma \[le:strict\_order\].
Fix $\alpha > 0$. Since $\nabla f(0) = 0$, we can find $\theta_\alpha > 0$ and $f_\alpha \in {{\rm Lip}}(\Gamma'')$, $\Gamma'' = {\ensuremath{\mathbb{R}}}^{n-k}/L {\ensuremath{\mathbb{Z}}}^{n-k}$, such that $f_\alpha(x'') = f(x'')$ for ${\left|x''\right|} \leq 2\theta$ with ${\left\|\nabla f_\alpha\right\|}_\infty \leq \alpha {\left\|\nabla\xi\right\|}_\infty$. Let us define the function $$\begin{aligned}
\label{def-psi}
\psi(x) = \psi_\alpha(x) = \alpha \xi{\left(x'\right)} + f_\alpha{\left(x''\right)},
\qquad x \in \Gamma = {{\mathcal T}}(\Gamma' \times \Gamma'').\end{aligned}$$ We see that $\psi \in {{\rm Lip}}(\Gamma)$ and therefore by Lemma \[le:subdiff-slicing\] $\psi \in {\operatorname{\mathcal{D}}}(\partial E_{\hat p}'(\cdot; \Gamma))$. We can estimate $$\begin{aligned}
\label{lip-bound-psi-alpha}
{\left\|\nabla \psi_\alpha\right\|}_\infty \leq 2\alpha {\left\|\nabla \xi\right\|}_\infty.\end{aligned}$$ In particular, if $\alpha$ is sufficiently small, $\partial E_{\hat p}(\psi_\alpha) = \partial E_{\hat p}'(\psi_\alpha)$ by Lemma \[le:subdiff-homog-relation\].
From now on we fix one such $\alpha$ and we write $\psi = \psi_\alpha$. For given $a >0$ let $\psi_a$ and $\psi_{a,m}$ be the solutions of the resolvent problems in Proposition \[pr:resolvent-problems\] for energies $E_W = E_{\hat p}$ and $E_m = E_{W_m(\cdot - \hat p) - W(\hat p)}$, respectively, on $\Gamma$. Note that these energies satisfy all the assumptions of Proposition \[pr:resolvent-problems\].
For given $a > 0$ and ${\left|z'\right|} \leq \delta$, we define the set of maxima $$\begin{aligned}
A_{a, z'} := {\operatorname*{arg\,max}}_{(x,t) \in M_2} {\left[u(x + \hat x, t + \hat t) -
\psi_a(x + {{\mathcal T}}_V z') - \hat p \cdot x - g(t + \hat t)\right]}\end{aligned}$$ where $M_s := {{\left\{(x,t): x' \in {{\mathcal U}^{s\delta}}(N),\ {\left|x''\right|} \leq s \theta,\ {\left|t\right|} \leq s\delta\right\}}}$. Note that $\psi(x + {{\mathcal T}}_V z') = \alpha \bar \psi(x' + z') + f(x'')$ for $(x,t) \in M_2$, ${\left|z'\right|} \leq \delta$. Due to the uniform convergence $\psi_a \rightrightarrows \psi$ on $\Gamma$ from Proposition \[pr:resolvent-problems\], and the strict ordering of Lemma \[le:strict\_order\], we have that there exists $a_0 > 0$, independent of $z'$, such that $$\begin{aligned}
\label{Az-in-M1}
\emptyset \neq A_{a,z'} \subset M_1 \qquad \text{for all ${\left|z'\right|} \leq \delta$, $a < a_0$.}\end{aligned}$$
We now fix one such $a < a_0$ and find ${\left|z'\right|} \leq \delta$ such that $$\begin{aligned}
\label{choice-of-z'}
\psi_a({{\mathcal T}}_V z') - \alpha\bar\psi(z')
= \min_{{\left|w'\right|} \leq \delta} {\left[\psi_a({{\mathcal T}}_V w') - \alpha\bar\psi(w')\right]}.\end{aligned}$$ As in [@GGP13JMPA], $z'$ is chosen in such a way that Lemma \[le:resolvent-order\] below holds.
Due to the uniform convergence $\psi_{a,m} \rightrightarrows \psi_a$ as $m\to\infty$ there exists $(x_a, t_a) \in A_{a,z'}$ and a sequence $(x_{a,m}, t_{a,m})$ (for a subsequence of $m$) of local maxima of $$\begin{aligned}
(x,t) \mapsto u_m(x + \hat x,t + \hat t) - \psi_{a,m}(x + {{\mathcal T}}_V z') - \hat p \cdot x - g(t + \hat t)\end{aligned}$$ such that $(x_{a,m}, t_{a,m}) \to (x_a, t_a)$ as $m\to\infty$ (along a subsequence).
Recall the definitions of $h_a$ and $h_{a,m}$ from Proposition \[pr:resolvent-problems\]. Since $\psi_{a,m} \in C^{2,{\ensuremath{\alpha}}}(\Gamma)$ and $u_m$ is a viscosity subsolution of , we must have $$\begin{aligned}
&g'(t_{a,m} + \hat t) + F(\nabla \psi_{a,m}(x_{a,m} + {{\mathcal T}}_V z') + \hat p, h_{a,m}(x_{a,m} + {{\mathcal T}}_V z'))\\
&
\begin{aligned}
= g'(t_{a,m} + \hat t) + F\big(&\nabla \psi_{a,m}(x_{a,m} + {{\mathcal T}}_V z') + \hat p,\\
&{\operatorname{tr}}{\left[(\nabla_p^2
W_m)(\nabla \psi_{a,m} + \hat p) \nabla^2 \psi_{a,m}\right]}(x_{a,m} + {{\mathcal T}}_V z')\big) \leq 0.
\end{aligned}\end{aligned}$$
By the uniform Lipschitz bound ${\left\|\nabla \psi_{a,m}\right\|}_\infty \leq {\left\|\nabla \psi\right\|}_\infty \leq C\alpha$ from , and $h_{a,m} \rightrightarrows h_a$ as $m \to\infty$, we can find a point $p_a \in {{\ensuremath{{\mathbb{R}^{{n}}}}}}$, ${\left|p_a - \hat p\right|} \leq C\alpha$, and send $m\to\infty$ along a *subsequence* to recover $$\begin{aligned}
\label{subsol ha}
g'(t_a + \hat t) &+ F(p_a, h_a(x_a + {{\mathcal T}}_V z')) \leq 0.\end{aligned}$$ To estimate $h_a(x_a + {{\mathcal T}}_V z')$, we prove the following lemma.
\[le:resolvent-order\] We have $$\begin{aligned}
h_a(x_a + {{\mathcal T}}_V z') \leq h_a({{\mathcal T}}_V z') = \min_{{\left|w'\right|}\leq \delta} h_a({{\mathcal T}}_V w').\end{aligned}$$
We chose $z'$ so that holds and therefore the equality above holds as well. Therefore we only need to show the inequality. Recalling the definition of $h_a$, we have to show that $$\begin{aligned}
\label{main-ineq-order}
\psi_a(x_a + {{\mathcal T}}_Vz') - \psi(x_a + {{\mathcal T}}_V z') \leq
\psi_a({{\mathcal T}}_Vz') - \psi({{\mathcal T}}_V z').\end{aligned}$$ We begin by expressing the second term on the left-hand side using , which yields $$\begin{aligned}
- \psi(x_a + {{\mathcal T}}_V z') = -\alpha \bar\psi{\left(x_a'+ z'\right)} - f{\left(x_a''\right)}.\end{aligned}$$ Since $(x_a, t_a) \in A_{a,z'} \in M_1$ by , clearly $$\begin{aligned}
\label{point-in-neighb}
x_a' + z' \in {{\mathcal U}^{3\delta}}(Z) \cap
{{\mathcal U}^{2\delta}}(Y) \cap O\end{aligned}$$ and therefore $\bar\psi{\left(x_a' + z'\right)} \geq 0$ by Lemma \[le:u-psi-shift\]. This implies $$\begin{aligned}
\label{main-ineq-1}
- \psi(x_a + {{\mathcal T}}_V z') \leq -f{\left(x_a''\right)}.\end{aligned}$$ For the first term in , we use the fact that $(x_a, t_a)$ is a point of maximum and therefore $$\begin{aligned}
u(x_a + \hat x, t_a + \hat t) - \psi_a(x_a + {{\mathcal T}}_V z') - \hat p \cdot x_a -g(t_a + \hat t) \geq u(\hat x, \hat t) - \psi_a({{\mathcal T}}_V z') - g(\hat t).\end{aligned}$$ After rearranging the terms, we obtain $$\begin{aligned}
\psi_a(x_a + {{\mathcal T}}_V z') \leq {\left[u(x_a + \hat x, t_a + \hat t) - u(\hat x, \hat t) - \hat p \cdot x_a - g(t_a + \hat t) + g(\hat t)\right]} + \psi_a({{\mathcal T}}_V z').\end{aligned}$$ We use again and therefore the first inequality of Lemma \[le:u-psi-shift\] allows us estimate the term in the bracket from above by $f(x_a'')$, yielding $$\begin{aligned}
\label{main-ineq-2}
\psi_a(x_a + {{\mathcal T}}_V z') \leq \psi_a({{\mathcal T}}_V z')
+ f{\left(x_a''\right)}.\end{aligned}$$ Finally, by the choice of ${\ensuremath{\delta}}$ we have $z' \in A_-^c \cap A_+^c$ and therefore $$\begin{aligned}
\psi({{\mathcal T}}_V z') = 0.\end{aligned}$$ Hence using this observation, and taking the sum of and we arrive at and the proof of the lemma is finished.
Then, by the ellipticity of $F$ in , $$\begin{aligned}
g'(t_a + \hat t) + F(p_a + \hat p, \min_{{\left|w'\right|} \leq \delta} h_a({{\mathcal T}}_V w')) \leq g'(t_a + \hat t) + F(p_a, h_a(x_a + {{\mathcal T}}_V z')) \leq 0.\end{aligned}$$
We send $a \to 0$ along a subsequence $a_l$ such that $\min h_{a_l} \to \liminf_{a\to 0} \min h_a$ and $p_a \to p_0$ as $l \to\infty$, for some $p_0 \in {\ensuremath{\mathbb{R}}}^n$, ${\left|p_0 - \hat p\right|} \leq C\alpha$, to obtain $$\begin{aligned}
g'(\hat t) + F(p_0, \liminf_{a\to 0} \min_{{\left|w'\right|} \leq \delta} h_a(T_V w')) \leq 0.\end{aligned}$$ Now we use Lemma \[le:subdiff-slicing\], in particular the fact that $h_a(x) = \bar h_a(x')$ for some $\bar h_a = (\bar \psi_a - \bar \psi)/a \in {{\rm Lip}}(\Gamma')$ and that $\bar h_a \to -\partial^0 {E^{\rm sl}}_{\hat p}(\bar\psi; \Gamma')$ in $L^2(\Gamma')$. Thus, recalling Proposition \[pr:curvature-as-min-section\], $$\begin{aligned}
\liminf_{a\to 0} \min_{{\left|w'\right|} \leq \delta} h_a({{\mathcal T}}_V w')
=\liminf_{a\to 0} \min_{{\left|w'\right|} \leq \delta} \bar h_a(w')
\leq {\operatorname*{ess\,inf}}_{B_\delta(0)} -\partial^0 {E^{\rm sl}}_{\hat p}(\xi; \Gamma')
= {\operatorname*{ess\,inf}}_{B_\delta(0)} \Lambda_{\hat p}[\bar\psi],\end{aligned}$$ and ellipticity yields $$\begin{aligned}
g'(\hat t) + F(p_0, {\operatorname*{ess\,inf}}_{B_\delta(0)} \Lambda_{\hat p}[\bar\psi]) \leq 0.\end{aligned}$$ Since this holds for any $\alpha > 0$ small, and therefore continuity of $F(p,\xi)$ in $p$ and the estimate ${\left|p_0 - \hat p\right|} \leq C \alpha$ yields $$\begin{aligned}
g'(\hat t) + F(\hat p, {\operatorname*{ess\,inf}}_{B_\delta(0)} \Lambda_{\hat p}[\bar\psi]) \leq 0,\end{aligned}$$ which we needed to prove.
Case (ii)
---------
In this case the test function is also a test function and therefore the stability follows the standard viscosity solution argument.
Case (curvature-free type)
--------------------------
In this part we will assume that $F$ is of curvature-free type at $p = 0$ in the sense of Definition \[def:level-set-type\]. We need to verify Definition \[def:level-set-test\](i-cf).
Suppose therefore that $\phi(x,t) = g(t)$ on a neighborhood $U$ of a point $(\hat x, \hat t)$ and $u - \phi$ has a local maximum 0 at $(\hat x, \hat t)$. We want to show that $g_t (\hat t) + F(0, 0) \leq 0$.
This can be accomplished by perturbing the test function $\phi(x,t)$ and considering the function $$\begin{aligned}
\phi_{m,q}(x,t) = W^\star_{m; A, q}(x - \hat x) + g(t) +{\left|t - \hat t\right|}^2,\end{aligned}$$ with $W^\star_{m;A,q}$ given by [@GGP13AMSA Lemma 5.8], and with suitable parameters $A, q > 0$.
Let us recall that $W^\star_{m; A,q}$ is the Legendre-Fenchel transform of $$\begin{aligned}
W_{m;A,q}(p) := A{\left(W_m(p) + q \psi{\left(\frac pq\right)} - W_m(0)\right)}.\end{aligned}$$ Here $\psi : {{\ensuremath{{\mathbb{R}^{{n}}}}}}\to [0, \infty]$ is a lower semi-continuous nonnegative convex function such that $\psi \in C^\infty(B_1(0))$, $\psi(0) = 0$ and $\psi(p) = \infty$ for ${\left|p\right|} \geq 1$. The semi-continuity then implies $\psi(p) \to \infty$ as $ {\left|p\right|} \to 1^-$.
The following lemma was proved in [@GGP13AMSA].
\[le:Wstar-props\] For any $m, A, q$ positive, $W^\star_{m;A,q}$ is a strictly convex, nonnegative, $C^2$ function on ${{\ensuremath{{\mathbb{R}^{{n}}}}}}$ and $$\begin{aligned}
{\left|\nabla W^\star_{m;A,q}(x)\right|} \leq q, \qquad
0 \leq \mathcal L_m(W^\star_{m;A,q})(x) \leq A^{-1} n,
\qquad x \in {{\ensuremath{{\mathbb{R}^{{n}}}}}},\end{aligned}$$ where $\mathcal L_m(u)(x) := {\operatorname{tr}}{\left[(\nabla^2 W_m) (\nabla u(x)) \nabla^2 u(x)\right]}$ for $u \in C^2({{\ensuremath{{\mathbb{R}^{{n}}}}}})$.
We will add the following modification of [@GGP13AMSA Lemma 5.8].
\[le:Wstar-growth\] For every $\delta > 0$ there exists $A > 0$ such that for every $q > 0$ there exist ${\ensuremath{\varepsilon}}> 0$ and $m_0 > 0$ for which $$\begin{aligned}
W^\star_{m; A, q}(x) > {\ensuremath{\varepsilon}}, \qquad \text{for all } x, {\left|x\right|} \geq \delta,
\text{ and } m \geq m_0.\end{aligned}$$
Let us define $$\begin{aligned}
\label{def-mu}
\mu := \sup_{{\left|p\right|} = 1/2} {\left[W(p) + \psi(p)\right]} \in (0, \infty)\end{aligned}$$ and set for given ${\ensuremath{\delta}}> 0$ $$\begin{aligned}
A := \frac{\delta}{8\mu}.\end{aligned}$$ Now we fix $q > 0$ and set $$\begin{aligned}
{\ensuremath{\varepsilon}}:= \frac{q\delta}{8}.\end{aligned}$$ By the locally uniform convergence of $W_m \to W$, we can find $m_0 > 0$ such that $$\begin{aligned}
\label{def-m0}
\sup_{{\left|p\right|} = q/2} {\left|W_m(p) - W_m(0) - W(p)\right|} \leq q \mu
\qquad m \geq m_0.\end{aligned}$$ Now whenever ${\left|x\right|} \geq \delta$ and $m \geq m_0$, we can take $p = \frac q2 \frac{x}{{\left|x\right|}}$ and estimate, using , one-homogeneity of $W$, and , $$\begin{aligned}
W^\star_{m; A, q} &\geq x \cdot p - W_{m; A, q} (p)\\
&= \frac{q}{2} {\left|x\right|} - A {\left(W_m(p) + q \psi{\left(\frac pq\right)}
- W_m(0)\right)}\\
&\geq \frac{q}{2} {\left|x\right|} - A {\left(W(p) + q \psi{\left(\frac pq\right)} + q\mu\right)}\\
&= \frac{q}{2} {\left|x\right|} - A {\left(qW{\left(\frac pq\right)} + q \psi{\left(\frac pq\right)} + q\mu\right)}\\
&\geq \frac{q}{2} {\left|x\right|} - 2 Aq \mu \geq \frac{q\delta}{4} > {\ensuremath{\varepsilon}}.\end{aligned}$$
\[le:Wstar-zero\] For any $A, q$ positive $$\begin{aligned}
W^\star_{m; A, q}(0) \to 0 \qquad \text{as } m \to \infty.\end{aligned}$$
Since $W_m$ is a decreasing sequence converging to $W$ locally uniformly, we have $W_m \geq \min W = 0$ and $W_m(0) \to W(0) = 0$. As also $\psi \geq 0$, it follows that $$\begin{aligned}
0 \leq W^\star_{m; A, q}(0) \leq A W_m(0) \to 0.\end{aligned}$$
Let us now choose ${\ensuremath{\delta}}> 0$ small enough so that $Q := {\overline{B}}_\delta(\hat x) \times [\hat t - \delta, \hat t + \delta] \subset U$. We have $u - \phi \leq 0$ on $Q$ with equality at $(\hat x, \hat t)$. For this $\delta$ we fix $A > 0$ from Lemma \[le:Wstar-growth\].
Now due to the same lemma for any $q > 0$ we also have ${\ensuremath{\varepsilon}}, m_0 > 0$ such that $$\begin{aligned}
u - \phi_{m, q} < -{\ensuremath{\varepsilon}}\qquad \text{on } {\left(\partial B_\delta(\hat x)\right)}
\times [\hat t - \delta, \hat t + \delta],
\text{ for } m \geq m_0.\end{aligned}$$ Because $W^\star_{m;A,q} \geq 0$ by Lemma \[le:Wstar-props\], we also have $$\begin{aligned}
u - \phi_{m,q} \leq - \delta^2 \qquad \text{on } x \in B_\delta(\hat t),
\ t = \hat t \pm \delta, \text{ for all } m.\end{aligned}$$ Since $\phi_{m,q}(0) \to 0$ as $m \to \infty$ by Lemma \[le:Wstar-zero\] and since $\phi_{m,q}$ is uniformly Lipschitz in $m$ by Lemma \[le:Wstar-props\], we conclude that there must exist a subsequence $m_j$ and a sequence of points $(x_j, t_j)$ such that $u_{m_j} - \phi_{m_j, q}$ has a local maximum at $(x_j, t_j)$, $x_j \in B_\delta(\hat x)$, and, moreover, $t_j \to \hat t$.
Let us now choose $q_k = 1/k$. By the standard diagonalization argument we can find a subsequence $m_k$ such that $u_{m_k} - \phi_{m_k, q_k}$ has a local maximum at a point $(x_k, t_k)$, $x_k \in B_\delta(\hat x)$, and ${\left|t_k - \hat t\right|} \leq 1/k$. Thus we introduce $$\begin{aligned}
p_k &:= \nabla \phi_{m_k, q_k}(x_k, t_k) = \nabla W^\star_{m_k; A, q_k}(x_k - \hat x), \text{ and}\\
\xi_k &:= \mathcal L_{m_k}{\left(\phi_{m_k, q_k}(\cdot, t_k)\right)}(x_k) =
\mathcal L_{m_k}{\left(W^\star_{m_k; A, q_k}\right)}(x_k - \hat x).\end{aligned}$$ By the assumption that $u_{m_k}$ is a subsolution of , we have $$\begin{aligned}
g'(t_k) + 2 (t_k - \hat t) + F(p_k, \xi_k) \leq 0.\end{aligned}$$ Furthermore, from Lemma \[le:Wstar-props\] and the choice of $q_k$ we have the bounds $$\begin{aligned}
{\left|p_k\right|} \leq 1/k, \qquad {\left|\xi_k\right|} \leq A^{-1} n \qquad \text{for all } k,\end{aligned}$$ where $A$ is independent of $k$.
Since $F$ is of curvature-free type at $p = 0$, Definition \[def:level-set-type\], we finally obtain $$\begin{aligned}
g'(\hat t) + F(0, 0) &=
g'(\hat t) + \liminf_{p \to 0} \inf_{{\left|\xi\right|} \leq A^{-1} n} F(p, \xi)
\\
&\leq \liminf_{k \to \infty} {\left[g'(t_k) + 2(t_k - \hat t) + F(p_k, \xi_k)\right]} \leq 0.\end{aligned}$$
The supersolution case can be handled similarly with a test function $$\begin{aligned}
\phi_{m,q}(x,t) = -W^\star_{m; A, q}(-x + \hat x) + g(t) +{\left|t - \hat t\right|}^2.\end{aligned}$$ This finishes the proof of stability for the curvature-free test function case.
The proof of Theorem \[th:stability quadratic\] is complete.
Approximation by linear growth functionals
------------------------------------------
In this section we prove the following approximation result:
\[th:linear growth stability\] Suppose that $F$ is of curvature-free type at $p_0 = 0$ and that ${{\left\{W_m\right\}}}_{n\in {\ensuremath{\mathbb{N}}}} \subset C({\ensuremath{\mathbb{R}}}^n) \cap
C^2({\ensuremath{\mathbb{R}}}^n\setminus{{\left\{0\right\}}})$ are positively one-homogeneous functions with bounded, strictly convex sub-level sets ${{\left\{W_m
\leq 1\right\}}}$ such that $W_m \rightrightarrows W$ uniformly on ${\overline{B}}_1(0)$. Let $u_m$ be the unique viscosity solutions of $$\begin{aligned}
\left\{
\begin{aligned}
u_t + F(\nabla u, {\operatorname{div}}\nabla_p W_m(\nabla u)) &= 0, && \text{in ${\ensuremath{\mathbb{R}}}^n \times (0,\infty)$,}\\
u(\cdot, 0) &= u_{0, m}, && \text{in ${\ensuremath{\mathbb{R}}}^n$},
\end{aligned}
\right.\end{aligned}$$ where $u_{0, m} \in C({\ensuremath{\mathbb{R}}}^n)$ are uniformly bounded. Then $$\begin{aligned}
\overline{u} &:= {\operatorname*{\star-limsup}}_{m\to\infty} u_m, &
\underline{u} &:= {\operatorname*{\star-liminf}}_{m\to\infty} u_m\end{aligned}$$ are a viscosity subsolution and a viscosity supersolution of .
We will follow the proof of Theorem \[th:stability quadratic\] with an additional approximation because the solutions $\psi_{a,m}$ of the resolvent problem for the linear growth energy $E_m$ might not be smooth. Let us set for $\delta > 0$ $$\begin{aligned}
W_m^\delta(p) := (W_m * \eta_\delta)(p) + \delta |p|^2,\end{aligned}$$ where $\eta_\delta$ is the standard mollifier with radius $\delta$, and let $u_m^\delta$ be the unique viscosity solution of $$\begin{aligned}
\label{quad approximation}
\left\{
\begin{aligned}
u_t + F(\nabla u, {\operatorname{div}}\nabla_p W_m^\delta(\nabla u_m)) &= 0, && \text{in ${\ensuremath{\mathbb{R}}}^n \times (0,\infty)$,}\\
u_m(\cdot, 0) &= u_{0, m}, && \text{in ${\ensuremath{\mathbb{R}}}^n$}.
\end{aligned}
\right.\end{aligned}$$ From the standard theory we have that $u_m^\delta \rightrightarrows u_m$ as $\delta \to 0$ locally uniformly on ${\ensuremath{\mathbb{R}}}^n
\times [0, \infty)$.
Suppose now that $\varphi$ is a stratified test function at $(\hat x, \hat t)$ with gradient $\hat
p$, as in the proof in Section \[se:stability case (i)\], Case (i) above, for a subsolution. We proceed as in that proof, but we use an additional perturbation of the test function by solving the resolvent problem for the energy $E_m^\delta := E_{W_m^\delta(\cdot - \hat p) - W(\hat
p)}$: we define the unique solution $\psi_{a,m}^\delta \in L^2(\Gamma)$ of $$\begin{aligned}
\psi_{a,m}^\delta + a \partial E_m^\delta(\psi_{a,m}^\delta) \ni \psi,\end{aligned}$$ where $\psi$ and $\Gamma$ were given in . Recall that $\psi_{a,m}^\delta \in
C^{2,\gamma}(\Gamma)$ by the elliptic regularity.
We can apply Proposition \[pr:resolvent-problems\] to $E_m^\delta$ and $E_m$ for fixed $m$ in the limit $\delta \to 0$. We in particular have $\psi_{a,m}^\delta \rightrightarrows \psi_{a,m}$ and $h_{a,m}^\delta \rightrightarrows h_{a,m}$ as $\delta \to 0$ for fixed $a, m$.
Due to the Mosco convergence of $E_m$ to $E_{\hat p}$ in Lemma \[le:lingrowthapproximation\], we also can apply Proposition \[pr:resolvent-problems\] to $E_m$ and $E_{\hat p}$ in the limit $m \to \infty$.
We now fix $a$ and $z'$ as in . Due to the uniform convergence $\psi_{a,m}^\delta \rightrightarrows \psi_{a,m}$ as $\delta \to 0$ and $\psi_{a,m} \rightrightarrows \psi_a$ as $m\to\infty$, there exists $(x_a, t_a) \in A_{a,z'}$ and a sequence $(x_{a,m}, t_{a,m})$ (for a subsequence of $m$) of local maxima of $$\begin{aligned}
(x,t) \mapsto u_m(x + \hat x,t + \hat t) - \psi_{a,m}(x + {{\mathcal T}}_V z') - \hat p \cdot x - g(t + \hat t)\end{aligned}$$ such that $(x_{a,m}, t_{a,m}) \to (x_a, t_a)$ as $m\to\infty$ (along a subsequence), and for each $m$ in this subsequence there exist a sequence $(x_{a,m}^\delta, t_{a,m}^\delta)$ (for a subsequence of $\delta$ as $\delta \to 0$) of local maxima of $$\begin{aligned}
(x,t) \mapsto u_m^\delta(x + \hat x,t + \hat t) - \psi^\delta_{a,m}(x + {{\mathcal T}}_V z') - \hat p \cdot x
- g(t + \hat t),\end{aligned}$$ such that $(x_{a,m}^\delta, t_{a,m}^\delta) \to (x_{a,m}, t_{a,m})$ as $\delta \to 0$ (along a subsequence).
Since $u_m^\delta$ is a viscosity solution of , we have $$\begin{aligned}
&g'(t^\delta_{a,m} + \hat t) + F(\nabla \psi^\delta_{a,m}(x^\delta_{a,m} + {{\mathcal T}}_V z') + \hat p,
h^\delta_{a,m}(x^\delta_{a,m} + {{\mathcal T}}_V z'))\\
&
\begin{aligned}
= g'(t^\delta_{a,m} + \hat t) + F\Big(&\nabla \psi^\delta_{a,m}(x^\delta_{a,m} + {{\mathcal T}}_V z') + \hat p,\\
&{\operatorname{tr}}{\left[(\nabla_p^2
W^\delta_m)(\nabla \psi^\delta_{a,m} + \hat p) \nabla^2 \psi^\delta_{a,m}\right]}(x^\delta_{a,m} + {{\mathcal T}}_V
z')\Big) \leq 0.
\end{aligned}\end{aligned}$$ Sending $\delta \to 0$ along a *subsequence* and using the uniform convergence of $h^\delta_{a,m}
\rightrightarrows h_{a,m}$, we can find $p_{a,m}$ with $|p_{a,m} - \hat p| \leq C \alpha$ such that $$\begin{aligned}
\label{ham subsol}
&g'(t_{a,m} + \hat t) + F(p_{a,m},
h_{a,m}(x_{a,m} + {{\mathcal T}}_V z')) \leq 0.\end{aligned}$$ Sending $m \to \infty$ along a *subsequence*, we obtain $p_a$ and . Then we finish the proof as in the proof of Theorem \[th:stability quadratic\] for Case(i).
Case (ii) as well as the curvature-free case are both straightforward.
Well-posedness {#sec:well-posedness}
==============
Once the stability with respect to the approximation of the energy density $W$ is established, we get existence of solutions as in [@GGP13JMPA].
\[th:well-posedness\] Let $W: {\ensuremath{\mathbb{R}}}^n \to {\ensuremath{\mathbb{R}}}$ be a positively one-homogeneous convex polyhedral function such that the conclusion of Corollary \[co:approximate pair sliced\] holds for $1 \leq k \leq n-1$, and let $F$ be of curvature-free type at $p_0 = 0$. Then for given $u_0 \in C({\ensuremath{\mathbb{R}}}^n)$ such that $u \equiv
c$ on ${\ensuremath{\mathbb{R}}}^n \setminus K$ for some compact $K \subset {\ensuremath{\mathbb{R}}}^n$ and $c \in {\ensuremath{\mathbb{R}}}$ there exists a unique viscosity solution of $$\begin{aligned}
\label{limit problem}
\left\{
\begin{aligned}
u_t + F(\nabla u, {\operatorname{div}}\partial W(\nabla u)) &= 0, && \text{in ${\ensuremath{\mathbb{R}}}^n \times (0, \infty)$},\\
u(\cdot, 0) &= u_0, && \text{in ${\ensuremath{\mathbb{R}}}^n$.}
\end{aligned}
\right.\end{aligned}$$ Moreover, if $u_0$ is Lipschitz, then $$\begin{aligned}
{\left\|\nabla u(\cdot, t)\right\|}_\infty \leq {\left\|\nabla u_0\right\|}_\infty, \qquad t \geq 0.\end{aligned}$$
We follow a standard approximation argument using the stability result from Section \[se:stability\]. Let $W_m \in C({\ensuremath{\mathbb{R}}}^n) \cap C^2({\ensuremath{\mathbb{R}}}^n \setminus {{\left\{0\right\}}})$ be a sequence of convex positively one-homogeneous functions with ${{\left\{W_m \leq 1\right\}}}$ strictly convex, such that $W_m
\rightrightarrows W$ on ${\overline{B}}_1(0)$. We can find the unique viscosity solutions $u_m$ of the problem $$\begin{aligned}
\left\{
\begin{aligned}
u_t + F(\nabla u, {\operatorname{div}}\nabla_p W_m(\nabla u)) &= 0, && \text{in ${\ensuremath{\mathbb{R}}}^n \times (0, \infty)$},\\
u(\cdot, 0) &= u_0, && \text{in ${\ensuremath{\mathbb{R}}}^n$.}
\end{aligned}
\right.\end{aligned}$$ We define the limits $$\begin{aligned}
\overline{u} &:= {\operatorname*{\star-limsup}}_{m\to\infty} u_m, &
\underline{u} &:= {\operatorname*{\star-liminf}}_{m\to\infty} u_m.\end{aligned}$$ These limits are well-defined since $u_m$ are uniformly bounded. By the stability result Theorem \[th:linear growth stability\], we see that $\overline u$ is a viscosity subsolution and $\underline u$ is a viscosity supersolution of .
We need to prove that $\overline u$ and $\underline u$ have the correct initial data. We can compare $u_m$ with translations of barriers $$\begin{aligned}
\psi^+_{m; a, b} &:= a (W_m^\circ(x - x_0) - bt)_+, &
\psi^-_{m; a, b} &:= -a (-W_m^\circ(-x + x_0) + bt)_-,\end{aligned}$$ where $W_m^\circ$ is the polar of $W_m$. The comparison with such barriers shows that $\overline u(\cdot, 0) = \underline u(\cdot, 0) =
u_0$, and for every $T > 0$ there exists a compact set $K_T \subset {\ensuremath{\mathbb{R}}}^n$ such that $\overline u =
\underline u = c$ on $({\ensuremath{\mathbb{R}}}^n \setminus K_T) \times [0, T]$.
Then the comparison principle Theorem \[th:comparison principle\] yields $\overline u \leq
\underline u$ and thus $u := \overline u = \underline u$ is the unique solution of .
The Lipschitz continuity follows from the comparison principle.
We now present the proofs of Theorems \[th:unique existence\] and \[th:convergence\].
Find $R > 0$ such that $D_0 \subset B_{R/2}(0)$. Let $F$ and $W$ be as in . Let $u_0$ be a continuous function with $D_0 = {{\left\{u_0 > 0\right\}}}$ such that $u_0 = -c$ for some $c > 0$ for ${\left|x\right|} \geq R$. For instance, take a cutoff of the signed distance function to $\Gamma_0$, $u_0(x) :=
-\min({\operatorname{dist}}(x, D_0), 1) + {\operatorname{dist}}(x, D_0^c)$. Then there is a unique solution $u$ of with initial data $u_0$ by Theorem \[th:well-posedness\]. This establishes the existence of a level set flow ${{\left\{\Gamma_t\right\}}}_{t\geq0}$ as $\Gamma_t := {{\left\{x: u(x, t) = 0\right\}}}$.
We therefore only need to show that the zero level set of $u$ does not depend on $u_0$. For this we simply argue as in [@G06 Section 4.1.1] to show that $\theta \circ u := \theta(u)$ is also a viscosity solution of for any continuous, nondecreasing $\theta$. Then for any given two continuous level set functions $u_0$, $\tilde u_0$ of $\Gamma_0$ we can find $\theta_1$, $\theta_2
\in C({\ensuremath{\mathbb{R}}})$, strictly increasing, such that $\theta_1 \circ u_0 \leq \tilde u_0$ and $\theta_2 \circ
\tilde u_0 \leq u_0$. Let $u$, $\tilde u$ be the two unique viscosity solutions of with initial data $u_0$, $\tilde u_0$, respectively. By the comparison principle Theorem \[th:comparison principle\] we get $\theta_1 \circ u \leq
\tilde u$ and $\theta_2 \circ \tilde u \leq u$. Since $\theta_1 \circ u$ and $\theta_2 \circ \tilde
u$ have the same zero level sets as $u$ and $\tilde u$, respectively, we conclude that the level set flow ${{\left\{\Gamma_t\right\}}}_{t \geq 0}$ is unique.
The stability result of Theorem \[th:convergence\] follows from Theorem \[th:linear growth stability\] and the comparison principle Theorem \[th:comparison principle\].
Acknowledgments {#acknowledgments .unnumbered}
---------------
Y. G. is partially supported by Grants-in-Aid for Scientific Research No. 26220702 (Kiban S), No. 23244015 (Kiban A) and No. 25610025 (Houga) of Japan Society for the Promotion of Science (JSPS). N. P. is partially supported by JSPS KAKENHI Grant Number 26800068 (Wakate B).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We designed and built a new Four-Stokes-Parameter spectral line Polarimeter (FSPPol) for the Caltech Submillimeter Observatory (CSO). The simple design of FSPPol does not include any mirrors or optical components to redirect or re-image the radiation beam and simply transmits the beam to the receiver through its retarder plates. FSPPol is currently optimized for observation in the $200-260$ GHz range and measures all four Stokes parameters, $I$, $Q$, $U,$ and $V$. The very low level of instrument polarization makes it possible to obtain reliable measurements of the Goldreich-Kylafis effect in molecular spectral lines. Accordingly, we measured a polarization fraction of a few percent in the spectral line wings of $^{12}\mathrm{CO}$ $(J=2\rightarrow1)$ in Orion KL/IRc2, which is consistent with previous observations. We also used FSPPol to study the Zeeman effect in the $N=2 \rightarrow 1$ transition of CN in DR21(OH) for the first time. At this point we cannot report a Zeeman detection, but more observations are ongoing.'
author:
- 'Talayeh Hezareh, Martin Houde'
title: 'A Four-Stokes-Parameter Spectral Line Polarimeter at the Caltech Submillimeter Observatory'
---
Introduction
============
Astronomical polarimetry is a powerful tool for studying the characteristics of the interstellar medium, from the large scale galactic magnetic fields to the gravitational collapse of molecular cloud cores. The understanding of the physical phenomena responsible for polarized interstellar emission provides valuable information about the underlying astrophysical processes, an important example being the formation and evolution of stars. Since the 1990s, polarimetry techniques and instruments have been developed for a wide range of wavelengths, from optical (e.g., @Magalhaes [@Wiktor]) to the submillimeter and radio regimes (e.g., @Platt [@Hildebrand; @1997; @hiroko; @greaves; @Li; @thum; @Hildebrand; @2009; @houde; @2009; @heiles1; @heiles2; @heiles3]).
One of the important applications of astronomical polarimetry lies in the study of the effect of magnetic fields on the early stages of star formation. Hence, it is important to characterize the interstellar magnetic fields around star-forming regions. Indeed, it is possible to obtain the orientation and strength of the magnetic fields observationally, as they leave their signature on the emission of the interstellar dust grains and molecules. For example, the dust continuum radiation becomes linearly polarized in a magnetic field [@Hildebrand; @1999]. At submillimeter wavelengths this polarization is perpendicular to the field lines and a polarimetry map of dust continuum emission will therefore reveal the plane-of-the-sky orientation of the magnetic field. Furthermore, in the presence of anisotropic radiation, magnetic fields cause the emission from gas molecules to also be linearly polarized by a few percent [@goldreich]. It is also possible to determine whether this polarization is aligned parallel or perpendicular to the plane-of-the-sky component of the ambient magnetic field [@deguchi]. Finally, the only way to directly measure the strength of interstellar magnetic fields is through Zeeman line-broadening measurements. More precisely, the line-of-sight component of the field is obtained by measuring the circular polarization in the emission of the Zeeman components of molecular spectral lines (@Crutcher [@93; @crutcher; @99]).
We describe the Four-Stokes-Parameter spectral line Polarimeter (FSPPol) we recently designed and successfully commissioned at the Caltech Submillimeter Observatory (CSO) in November 2008 in $\S 2$. We present linear polarization measurements of $^{12}\mathrm{CO}$ $(J=2\rightarrow1)$ in Orion KL/IRc2 in $\S 3$ and preliminary Zeeman observations of the $N=2 \rightarrow 1$ transition of CN in DR21(OH) in $\S 4$. We end with a summary in $\S 5$.
Instrument Description
======================
The design of FSPPol is based on single beam polarimetry, as the current heterodyne receivers at the CSO can measure only one linear polarization state at one position at a time. The simple construction of the polarimeter enabled us to mount the instrument inside the elevation tube that traces much of the optical path from the tertiary mirror behind the telescope dish to the Nasmyth focus where the $200-300$ GHz receiver is located. A schematic diagram of FSPPol installed in the elevation tube is shown in Figure \[fig:polarimeter\]. The polarimeter is mounted on a bracket attached to the walls of the tube and its location with respect to the tertiary mirror and the receiver is depicted with the schematic diagram of the CSO telescope in Figure \[fig:cso\]. The third focus of the $230$ GHz telescope beam is virtual and located $1.07$ m behind the tertiary mirror and has a waist of $35.9$ mm. We calculated the beam waist upon incidence on FSPPol to be $39.5$ mm. As illustrated in Figure \[fig:polarimeter\], FSPPol is comprised of a half-wave plate (HWP) and a quarter-wave plate (QWP), each being $100$ mm in diameter and optimized for observations at 226 GHz. The wave plates were manufactured by Meller Optics, Inc.[^1] and subsequently anti-reflection coated by QMC Instruments Ltd.[^2]. More detailed specifications of the wave plates are listed in Table \[ta:hwp-qwp\]. These plates are installed in rotating rings mounted side by side on an aluminum translational stage that moves across the elevation tube, placing either wave plate in the path of the signal reflected from the tertiary mirror. The anti-reflection coating on the plates is efficient in eliminating standing waves along the optical path. Nevertheless, the mounting of the translational stage was adjusted in such a manner that the telescope beam impinged on the HWP or QWP at an incidence that is slightly off from normal (i.e., at most a few degrees) to avoid potential standing waves. The rotational and translational motions of these wave plates are precisely monitored and controlled by the instrument controlling software from the observatory’s control room. The system temperature increases by about $8\%$ when the HWP is placed in the path of the beam, and by about $6\%$ with the QWP in the beam.
We aligned the optical axes of the wave plates in relation with the polarization axis of the receiver by measuring the intensity from a cold load, located at the entrance of FSPPol, while rotating the wave plates over a wide range of angles. We previously placed a wire grid with its polarizing state parallel to the receiver’s axis between the wave plates and the cold load to maintain a specific incoming polarization state for testing purposes; this grid was removed for astronomical observations. It is straightforward to show that the location of a minimum in measured intensity with this set-up corresponds to an angle where either the fast or slow axis of a plate (HWP or QWP) is aligned with the receiver axis. This allows for a precise orientation of the HWP and QWP with the receiver axis (also see $\S 3$). It should also be noted that the slow axis is marked by the manufacturer on each plate, which we used as a further consistency check. A small circular cold load was also used to position the polarimeter within the elevation tube such that the telescope beam was centered on the surface of the HWP and QWP.
To test the instrument’s accuracy, we observed the linear polarization in the $J=2 \rightarrow 1$ transition of $^{12}$CO towards the Orion KL/IRc2 high mass star forming region. After successful linear polarization tests, we proceeded to perform Zeeman observations on the $N=2 \rightarrow 1$ transition of CN in DR21(OH) for the first time. The details of these observations are explained in the following sections.
Linear Polarization Measurements
================================
We observed $^{12}\mathrm{CO}\,\,\left(J=2\rightarrow1\right)$ at $230.54$ GHz in three different locations in Orion KL/IRc2 coincident with previous polarization observations by @girart, in order to assess the performance of FSPPol. These locations, defined in the equatorial coordinate system, were offset $(20\arcsec,20\arcsec)$, $(20\arcsec,-20\arcsec)$ and $(-20\arcsec,-20\arcsec)$ from the center of IRc2 ($\alpha=05^{\mathrm{h}}35^{\mathrm{m}}14.50^{\mathrm{s}}$ and $\delta=-05^{\circ}22^{'}30.4^{''}$, J2000.0). The observations were performed between September $27^{\mathrm{th}}$ and October $6^{\mathrm{th}}$ 2009 using the $200-300$ GHz receiver at the CSO, and the FFTS spectrometer with a bandwidth of 500 MHz and a channel width of 61 kHz that corresponds to a velocity resolution of about 0.08 km s$^{-1}$. The telescope efficiency was determined with scans on Jupiter and calculated to be $\simeq65$ % for a beam width of $\simeq32\arcsec$(FWHM). During these observations the typical optical depth at 225 GHz, as obtained with the CSO radiometer, was $\tau_{225}\thickapprox0.06$ and the typical system temperature with FSPPol in use was $T_{sys}\thickapprox370$ K.
Although the refractive indices for the ordinary and extraordinary rays vary significantly with frequency, the birefringence of crystal Quartz appears to be relatively unchanged over a wide band [@afsar] even though there is some uncertainty in the literature over its value (0.048 for the frequencies we are concerned with; [@marrone]). However, since the HWP was operated at approximately 230.5 GHz, i.e., more than 4 GHz away from its design frequency, we characterized its performance at the aforementioned frequency by measuring the power from a cold load at different orientations of the HWP slow axis with respect to the polarization axis of the facility receiver at the CSO. Figure \[fig:hwp\] shows the plot of the power at the receiver against different angles between the HWP and receiver axis. The power is measured in arbitrary units and the angles are displayed in degrees. A polarizing grid, with its polarization state parallel to the receiver axis, was placed between the cold load and the HWP to maintain a specific incoming polarization state. As seen in the figure the HWP response is very good at that frequency, although the power at successive minima is observed to slightly vary [@savini]. Our model for the fit to the data in Figure \[fig:hwp\] is a cosine function with the form
$$I(\theta)=\frac{(\mathrm{CL}+\mathrm{HL})}{2}+\frac{(\mathrm{CL}-\mathrm{HL})}{2}\mathrm{cos}(4(\theta-\delta)),\label{eq:hwp_fit}$$
where $I(\theta)$ is the power measured by the receiver at angle $\theta$ and $\delta$ determines the offset of the HWP slow axis from the orientation defining the polarization state of the receiver. CL is the power from the cold load and HL is the power due to emission from the receiver that reflects back on the wires of the polarizing grid; The HL signal incident on the HWP is therefore polarized perpendicular to the CL signal. It should also be noted that there is an equal contribution to CL and HL from receiver noise, although we cannot quantify it with these measurements alone. The fit values for CL, HL and $\delta$ are $722.2\pm2.1$, $1373.0\pm2.9$ and $1.45^{\circ}\pm0.11^{\circ}$, respectively.
The polarization efficiency of FSPPol for linear polarization measurements was previously determined in November 2008 by taking scans on Saturn. We placed a wire grid at the entrance of FSPPol with its polarization state parallel to the receiver’s axis to force a precise incoming polarization state, and observed Saturn for HWP angles $\theta=0^{\circ}$, $90^{\circ}$, $45^{\circ}$ and $135^{\circ}$, where $\theta$ is defined as the angle between the HWP slow axis and the receiver polarization axis. We performed a one-minute (on-source) integration for each measurement, with the system temperature being calibrated before every scan and the signal from Saturn being integrated over the whole available spectral bandwidth. Assuming that the wire grid is perfectly polarizing and that Saturn is unpolarized, we obtained a polarization efficiency of $\simeq99\%$. These measurements also allowed us to estimate the instrument polarization to be on the order of $\simeq0.3\%$. In view of this high polarization efficiency, we did not correct our Orion KL/IRc2 data for the efficiency and instrumental effects.
The polarimeter and the receiver are mounted in such a way that they rotate in elevation with the telescope and the polarization axis of the receiver on the sky, which is precisely oriented east-west (parallel to the horizon) when FSPPol is not used, is preserved regardless of the pointing of the telescope since the CSO telescope has an alt-azimuth mount. Our aim for linear polarization measurements was to obtain the Stokes $Q$ and $U$ parameters in the reference frame of the sky, i.e., in the equatorial coordinate system, as well as the Stokes $I$. Therefore, the rotation of the object’s orientation on the sky, or in other words the changes in its parallactic angle with time had to be considered. For this purpose, we obtained one-minute (on-source) long intensity measurements for different orientations of the HWP slow axis relative to the receiver polarization axis $\theta=(\gamma+\theta^{'}+90^{\circ})/2$, where $\gamma$ is the parallactic angle defining the orientation of the object on the sky and $\theta^{'}$ is the angle at which we seek to measure the linear polarization state in the reference frame of the source (i.e., the frame that is rotated by $\gamma$ in the equatorial system; see Figure \[fig:coordinates\]). These measurements are denoted by $I_{\theta^{'}}$. The instrument controlling software continuously obtained the updated value of the parallactic angle from the observatory’s antenna computer during the on-source integration, and compared it to the initial value at the beginning of the integration. Once the change in $\gamma$ exceeded a predetermined threshold (i.e., $1^{\circ}$), the software commanded the antenna computer to stop the integration, rotated the HWP by the updated angle using the new value for $\gamma$, and the integration resumed. Our observations were performed in cycles of four measurements for $\theta'=0^{\circ}$, $90^{\circ}$, $45^{\circ}$ and $135^{\circ}$. Similar to the observations on Saturn, we took one minute long scans for each measurement, and calibrated the system temperature before every scan. We adopted this conservative observing plan to minimize the effect of pointing or calibration errors in the polarization data. The telescope pointing was verified hourly with typical five points integrations on CRL865 ($\alpha=06^{\mathrm{h}}{03}^{\mathrm{m}}{59.8}^{\mathrm{s}}$ and $\delta=07^{\circ}25^{'}51.4^{''}$, J2000.0) as our reference star.
With the aforementioned definition for $I_{\theta^{'}}$ we have
$$\begin{aligned}
I & = & \frac{\left(I_{0^{\circ}}+I_{90^{\circ}}+I_{45^{\circ}}+I_{135^{\circ}}\right)}{2}\nonumber \\
Q & = & I_{0^{\circ}}-I_{90^{\circ}}\label{eq:stokes}\\
U & = & I_{135^{\circ}}-I_{45^{\circ}},\nonumber \end{aligned}$$
and
$$\begin{aligned}
p & = & \frac{\sqrt{Q^{2}+U^{2}}}{I}\label{eq:p-chi}\\
\mathrm{PA} & = & \frac{1}{2}\arctan\left(\frac{U}{Q}\right)\nonumber \end{aligned}$$
for the polarization fraction and angle (measured from north increasing eastwards), respectively. The results from our measurements are shown in Figure \[fig:1\], where the top graphs show the Stokes $I$ spectra (corrected for the beam efficiency), while the middle and bottom graphs are for the corresponding polarization fractions and angles, respectively. For $p$ and $\mathrm{PA}$ the data were binned using six adjacent velocity channels and only values for which $p\ge3\sigma_{p}$ are plotted, where $\sigma_{p}$ is the uncertainty in the polarization fraction.
Table \[ta:co\] displays the average values for $p$ and $\mathrm{PA}$ in the blue and red wings of the CO spectra, as well as the small polarization fraction detected at the center of the lines. These values were obtained after calculating the average of the Stokes $I$, $Q$ and $U$ across the stated velocity ranges. The small polarization level in the center of the lines is probably the contribution from the instrumental polarization, as it is expected that the linear polarization due to the Goldreich-Kylafis effect will be greatly reduced where the optical depth is high [@goldreich]. The values of $\mathrm{PA}$ are generally uniform across the spectral lines except for the center of the lines, where the relative contribution from the instrumental polarization is significant.
One potential problem in the linear polarization measurements is the presence of polarized sidelobes that introduce false polarization signals in the data [@frobrich]. This is more likely to happen when the telescope is pointed towards the edge of an extended source for the purpose of polarimetry measurements in low intensity regions. This way, it is possible that strong emission from the core of the source falls on these polarized sidelobes and contaminates the data. We have yet to determine the potential contribution of polarized sidelobes to data obtained with FSPPol at the CSO, we cannot therefore comment on their significance for our results on Orion KL. We hope to do so during a future observing run.
Our polarization results are in general consistent with the findings of @girart (see their Figure 1). The polarization fraction that we calculate in the spectral line wings of $^{12}$CO is in good agreement with their results, and the difference between their values for polarization angles and ours (i.e., $10^{\circ}-20^{\circ}$) could solely be the result of different telescope beam sizes. This could also explain the slight differences in line profiles and intensities between their Stokes $I$ spectra and ours. For example the secondary peak at $\approx12$ km $\mathrm{s}^{-1}$ in the offset positions obtained by @girart is not resolved in our spectral data.
Zeeman Measurements of CN ($N=2 \rightarrow 1$)
===============================================
As mentioned earlier, measurement of the Zeeman effect in interstellar molecular spectral lines is the only direct way to obtain the strength of magnetic fields in molecular clouds. The Zeeman line-broadening in an interstellar spectral line profile is directly proportional to the strength of the ambient magnetic field, and can be studied by observing the signature of circular polarization in the line profile. To this date, Zeeman detections have been reported in the spectral lines of a few species, namely , OH, $\mathrm{H_{2}O}$, $\mathrm{CH_{3}\mathrm{OH}}$ and CN [@sarma09; @troland; @falgarone; @sarma01; @crutcher; @99; @plante]. The high critical density of CN makes this molecule suitable for studying magnetic fields in dense regions. Additionally, the rotational transition lines of CN contain several hyperfine components that have different Zeeman splitting coefficients. This makes it possible to distinguish between the true Zeeman effect and instrumental effects that produce artificial circular polarization in the data [@crutcher; @96]. Although the Zeeman effect in the $N=1 \rightarrow 0$ transition of CN has previously been detected [@falgarone; @crutcher; @99], to the best of our knowledge it has never been attempted at $N=2 \rightarrow 1$. The higher critical density of the latter will make it possible to probe denser regions in molecular clouds and better establish observationally how the magnetic field strength scales with density.
There are nine strong hyperfine components in the $N=2 \rightarrow 1$ transition of CN. The frequencies, Zeeman coefficients and relative intensities of these lines are displayed in Table \[ta:hf\_z\]. Similar to the $N=1 \rightarrow 0$ transition previously observed in DR21(OH), the CN ($N=2 \rightarrow 1$) lines are double peaked, suggesting the existence of two velocity components [@crutcher; @99]. These two velocity components may be arising from different regions, and may exhibit different field strengths. It is possible to observe the nine hyperfine components simultaneously and fit the observed circular polarization data to the following expression for both velocity components [@crutcher; @99]
$$V_{i}=a(I_{i1}+I_{i2})+b_{1}(\frac{dI_{i1}}{d\nu})+b_{2}(\frac{dI_{i2}}{d\nu})+c_{1}(Z_{i}\frac{dI_{i1}}{d\nu})+c_{2}(Z_{i}\frac{dI_{i2}}{d\nu}),\label{eq:vfit}$$
where $V_{i}$ is the total circular polarization for hyperfine line $i$. In the above equation, the different sources of polarization are expressed in separate terms. The first term represents the polarization contribution from the error in intensity calibrations in the two polarization modes, $a$, that adds a small image of the intensity spectrum, $I_{i1}+I_{i2}$, to the $V_{i}$ spectra. The second and third terms are polarization contributions from the beam squint effect represented by $b_{1}$ and $b_{2}$, that is caused when the right and left handed polarization measurements do not probe exactly the same region, for which a non-zero velocity gradient exists in its vicinity [@Crutcher; @93]. This effect is brought about by pointing errors of the telescope, and also in some cases by mechanical deformations in the antenna. The last two terms are the true Zeeman signals for each velocity component of the spectrum with $c_{1}$ and $c_{2}$ representing the strength of the line-of-sight component of the magnetic fields $B_{\mathrm{los}1}/2$ and $B_{\mathrm{los}2}/2$, respectively, for each velocity component, and $Z_{i}$ being the Zeeman coefficient for hyperfine line $i$. This way, the real Zeeman effect is separated from instrumental effects by determining the above fitting parameters.
Observations
------------
We observed CN ($N=2 \rightarrow 1$) at 226.8 GHz in DR21(OH) ($\alpha=20^{\mathrm{h}}39^{\mathrm{m}}01^{\mathrm{s}}$ and $\delta=42^{\circ}22^{'}37.7^{''}$, J2000.0) during July $8^{\mathrm{th}}$ to $13^{\mathrm{th}}$ and September $27^{\mathrm{th}}$ to October $6^{\mathrm{th}}$ 2009, using the FFTS spectrometer with bandwidth of 500 MHz and channel resolution of $61$ kHz. The telescope efficiency was determined with scans on Saturn and Jupiter and calculated to be $\simeq65$ % for a beam width of $\simeq32\arcsec$(FWHM). We verified the telescope pointing on an approximately hourly rate by performing $^{12}$CO ($J=2 \rightarrow 1$) scans on $\chi$ Cygni ($\alpha=19^{\mathrm{h}}50^{\mathrm{m}}33.8^{\mathrm{s}}$ and $\delta=32^{\circ}{54}^{'}53.2^{''}$, J2000.0) as our reference star. Our first observing session suffered from issues with an unstable receiver and mediocre skies. The average system temperature at that time was about $500$ K and the typical $\tau_{225}$ was $\approx0.12$. The circular polarization observations were performed by taking scans of $\mathrm{CN}$ ($N=2 \rightarrow 1$) in DR21(OH) with the QWP slow axis rotated by $+45^{\circ}$ and $-45^{\circ}$ with respect to the receiver’s polarization axis. With our instrument set-up, our one minute on-source integrations at $+45^{\circ}$ project the right-handed circular polarization ($I_{RCP}$) emission (on the sky) on the receiver’s axis, while similar integrations at $-45^{\circ}$ probe the corresponding left-handed circular polarization ($I_{LCP}$) emission. The Stokes $I$ and $V$ are obtained from $I=I_{LCP}+I_{RCP}$ and $V=I_{LCP}-I_{RCP}$. As before, this observing strategy was chosen in order to mitigate any potential calibration or pointing errors. The system temperature was calibrated before every scan. We have so far obtained a total on-source integration time of 553 minutes, with an average system temperature of 450 K.
Figure \[fig:spectra\] shows the Stokes $I$ spectrum for the CN ($N=2 \rightarrow 1$) hyperfine lines that are labeled according to the order given in Table \[ta:hf\_z\] with the line temperatures corrected for the telescope beam efficiency. There are two further hyperfine lines at $\simeq-45$ km s$^{-1}$ and $\simeq200$ km s$^{-1}$ that are weak relative to the labeled lines and therefore not included in the study. Observing the $N=2 \rightarrow 1$ transition of CN brings more complications compared to the $N=1 \rightarrow 0$ transition. For example, there are three lines in Table \[ta:hf\_z\] that are labeled 5 because they are heavily blended and appear as a single broad line in the observed spectrum, as seen in Figure \[fig:spectra\]. Furthermore, due to the double peaked feature of the CN lines, the other hyperfine lines with small frequency separations (i.e., lines 2 and 3 and lines 6 and 7) are blended together as well, although to a lesser level.
An inspection of the ratio of line temperatures reveals that our CN lines are not consistent with the LTE assumption. The spacing between the two velocity components of every hyperfine line varies with the line strength, and the rotational diagram for the $N=2 \rightarrow 1$ lines produced a negative excitation temperature, suggesting that the lines are affected by self absorption. A more complete spectral line analysis will be presented in a forth-coming paper.
Preliminary Analysis & Discussion
---------------------------------
The following is only a preliminary analysis on the data we have gathered so far, as more data will be obtained in upcoming observing runs, and a more thorough discussion will be presented then.
Since the CN hyperfine lines in DR21(OH) are double peaked, we fitted two to three Gaussian profiles to each line depending on its shape. In the fitting procedure, the line widths of the Gaussian profiles within each velocity component of a hyperfine line profile were kept the same. For the three blended hyperfine components of line 5 (in Figure \[fig:spectra\]), we fitted one Gaussian profile to each component and fixed their relative frequencies according to the information given in Table \[ta:hf\_z\]. Each hyperfine component of the other pairs of blended lines (lines 2 and 3 and lines 6 and 7) were treated as single lines, such that Gaussian profiles incorporating both lines were fitted to the blended line profiles. Once the fitting parameters for all the Gaussian profiles were obtained, the blended lines could thus be separated by subtracting the Gaussian fit of one line from the spectrum of the pair. The Gaussian fits for the Stokes $I_{i}=I_{i1}+I_{i2}$ for both velocity components of every hyperfine line $i$, together with their derivatives with respect to frequency and corresponding Zeeman coefficients were simultaneously fitted to the Stokes $V$ spectrum using Equation (\[eq:vfit\]). The resulting fitting parameters were $a=-0.0015\pm0.0007$, $b_{1}=1.24\pm1.13$ kHz, $b_{2}=1.28\pm1.47$ kHz, $c_{1}=-0.5\pm1.3$ mG and $c_{2}=-2.5\pm2.0$ mG. Although we have not obtained a Zeeman detection, the instrumental polarization contribution is seen to be very small for $a$ and kept to reasonable levels for $b_{1}$ and $b_{2}$ (i.e., of the same order as the expected values for $c_{1}$ and $c_{2}$).
Since it is not easy to distinguish the Zeeman signal in the $V$ spectra of individual lines from the noise level, @crutcher [@99] produced an averaged sum of the $N=1 \rightarrow 0$ hyperfine lines with strong Zeeman coefficients to display the Zeeman fit for their detection. In the case of the $N=2 \rightarrow 1$ transition, not all hyperfine lines have the same sign for their Zeeman coefficients and therefore it is only possible to average together the lines with same coefficient signs. The top panel of Figure \[fig:spec\_neg\] shows the weighted average Stokes $\overline{I}$ for lines 1, 2 and 4 that have negative Zeeman coefficients, with the weights being the relative sensitivity to the magnetic field, i.e., $|\mathrm{Z_{\mathit{i}}\times R.I_{\mathit{i}}}|$ given in Table \[ta:hf\_z\]. All the hyperfine lines are centered on the $V_{LSR}$ of the source, i.e., -3 km s$^{-1}$ and in order to remove the contamination of line 3 from line 2, the Gaussian fit for line 3 was subtracted from the blended spectrum. A similar procedure was performed for the averaged $V$ spectrum of the aforementioned lines, which is displayed in the lower panel of Figure \[fig:spec\_neg\] with the instrumental effects removed. The bold line is the Zeeman fit to the average $V$ data, expressed by $\overline{V}=\overline{Z}(C_{1}d\overline{I_{1}}/d\nu)+\overline{Z}(C_{2}d\overline{I_{2}}/d\nu)$, where $\overline{Z}$ is the weighted average of the negative Zeeman coefficients, with the weights being $|\mathrm{R.I_{\mathit{i}}}|$. The fits for $C_{1}$ and $C_{2}$ are $C_{1}=-0.15\pm2.25$ mG and $C_{2}=-3.57\pm3.22$ mG, which are consistent with the values obtained for $c_{1}$ and $c_{2}$ from the simultaneous Zeeman fit to individual $V$ spectra mentioned above.
The Zeeman Stokes $V$ signal due to a field of a few hundred $\mu$Gauss is small (i.e., a line broadening of a few hundred Hz), and the noise level in the spectra needs to be sufficiently low in order to obtain at least a $2\sigma$ detection. We developed simulations for the observed CN lines with different noise levels and magnetic field strengths, to estimate the remaining integration time required to obtain a detection. @crutcher [@99] obtained a value of $B_{\mathrm{los}}\simeq0.75$ mG for one of the velocity components of DR21(OH) by observing the $N=1 \rightarrow 0$ transition of CN, which is associated with a critical density $n_{c}\approx10^{5}\mathrm{cm^{-3}}$. Assuming that the magnetic field strength varies with $n^{1/2}$, where $n$ is the the average gas density of the observed region, we should expect that the magnetic field we probe with our observations will be somewhat stronger. Since $n_{c}\approx10^{6}\mathrm{cm^{-3}}$ at the $N=2 \rightarrow 1$ transition, $B_{\mathrm{los}}$ should be at most a few times stronger than the value obtained with the $N=1 \rightarrow 0$ transition. Our simulations show that for a 2 mG field, we need to have our noise level down to $\simeq5$ mK to be able to get at least a $2\sigma$ detection. The RMS noise in our data is currently $\simeq16$ mK, implying that we need more observations to obtain a credible detection. Assuming an average system temperature of 400 K, with a bandwidth of 61 kHz, we will require about 48 more hours of on-source integration time. This time estimate is comparable with the observing time that @crutcher [@99] spent on OMC1n and DR21OH to obtain a Zeeman detection in the CN ($N=1 \rightarrow 0$) transition.
Summary
=======
We recently designed and successfully commissioned a Four-Stokes-Parameter spectral line Polarimeter (FSPPol) at the CSO in November 2008. The simple design of FSPPol does not contain any mirrors or grids to redirect or split the radiation beam, and the instrument is conveniently mounted in the elevation tube between the tertiary mirror behind the telescope dish and the Nasmyth focus where the heterodyne receiver is located. FSPPol transmits the beam to the receiver through half-wave and quarter-wave plates that are presently optimized for observations at 226 GHz.
We used FSPPol for linear and circular polarization measurements in the spectral lines of interstellar molecules during the months of July, September and October 2009. We measured a linear polarization level of $\simeq1\%$ to $2.5\%$ due to the Goldreich-Kylafis effect in the spectral line wings of $^{12}\mathrm{CO}$ $(J=2\rightarrow1)$ in Orion KL/IRc2, and our results are consistent with previous observations [@girart]. We also started Zeeman observations on the $N=2 \rightarrow 1$ transition of CN in DR21(OH) for the first time. At this point we have obtained about 10 hours of on-source integration time, and our preliminary data analysis shows that although we have not detected a Zeeman signal in the CN ($N=2 \rightarrow 1$) lines, the overall contribution from the instrumental effects in the Stokes $V$ spectrum is low. Further observations of CN ($N=2 \rightarrow 1$) are ongoing for this source and other star forming regions.
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![Linear polarization measurements of $^{12}$CO ($J=2 \rightarrow 1$) in Orion KL/IRc2; ($top$) Stokes $I$, corrected for the telescope beam efficiency; ($middle$) The polarization level $p$ and ($bottom$) angles PA across the spectral line.[]{data-label="fig:1"}](f5a.eps "fig:"){height="2.9in"} ![Linear polarization measurements of $^{12}$CO ($J=2 \rightarrow 1$) in Orion KL/IRc2; ($top$) Stokes $I$, corrected for the telescope beam efficiency; ($middle$) The polarization level $p$ and ($bottom$) angles PA across the spectral line.[]{data-label="fig:1"}](f5b.eps "fig:"){height="2.9in"} ![Linear polarization measurements of $^{12}$CO ($J=2 \rightarrow 1$) in Orion KL/IRc2; ($top$) Stokes $I$, corrected for the telescope beam efficiency; ($middle$) The polarization level $p$ and ($bottom$) angles PA across the spectral line.[]{data-label="fig:1"}](f5c.eps "fig:"){height="2.9in"}
[ll]{}
Material & single crystal Quartz\
Indices of refraction: &\
Ordinary & $2.106 \pm 0.006 \tablenotemark{a}$\
Extraordinary & $2.154 \pm 0.007$\
Thickness & 13.82 mm (HWP), 6.90 mm (QWP)\
Anti-reflection coating: &\
Material & High density Polypropylene (HDPP)\
[cccccrr]{}
(20$\arcsec$, 20$\arcsec$) & 2.1 $\pm$ 0.3 & 0.35 $\pm$ 0.04 &
1.8 $\pm$ 0.2 & & 77.1 $\pm$ 4.2 & 81.2 $\pm$ 3.4\
(20$\arcsec$, -20$\arcsec$) & 1.6 $\pm$ 0.2 & 0.29 $\pm$ 0.04 &
2.4 $\pm$ 0.2 & & 97.8 $\pm$ 3.3 & 88.1 $\pm$ 2.9\
(-20$\arcsec$, -20$\arcsec$) & 1.1 $\pm$ 0.3 & 0.29 $\pm$ 0.07 &
1.5 $\pm$ 0.4 & & 154.8 $\pm$ 7.1 & 155.6 $\pm$ 8.1\
[cccccc]{}
1 & (2,3/2,3/2)$\rightarrow$(1,1/2,3/2) & 226632.19 & -0.72241 & 0.59259 & 0.42809\
2 & (2,3/2,5/2)$\rightarrow$(1,1/2,3/2) & 226659.58 & -0.70995 & 2.0 & 1.41991\
3 & (2,3/2,1/2)$\rightarrow$(1,-1/2,1/2) & 226663.70 & 0.62277 & 0.59259 & 0.36905\
4 & (2,3/2,3/2)$\rightarrow$(1,1/2,1/2) & 226679.38 & -1.18326 & 0.74074 & 0.87649\
5 & (2,5/2,5/2)$\rightarrow$(1,3/2,3/2) & 226874.17 & 0.70995 & 2.016 & 1.43127\
5 & (2,5/2,7/2)$\rightarrow$(1,3/2,5/2) & 226874.75 & 0.40035 & 3.200 & 1.28112\
5 & (2,5/2,3/2)$\rightarrow$(1,3/2,1/2) & 226875.90 & 1.18326 & 1.200 & 1.41991\
6 & (2,5/2,3/2)$\rightarrow$(1,3/2,3/2) & 226887.35 & 1.46973 & 0.3840 & 0.56438\
7 & (2,5/2,5/2)$\rightarrow$(1,3/2,5/2) & 226892.12 & 1.05692 & 0.3840 & 0.40586\
[^1]: 120 Corliss Street, Providence, RI 02904 USA
[^2]: Cardiff University, School of Physics and Astronomy, The Parade, Cardiff CF24 3AA UK
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Owing to BERT’s phenomenal success on various NLP tasks and benchmark datasets, industry practitioners have started to experiment with incorporating BERT to build applications to solve industry use cases. Industrial NLP applications are known to deal with much more noisy data as compared to benchmark datasets. In this work we systematically show that when the text data is noisy, there is a significant degradation in the performance of BERT. While this work is motivated from our business use case of building NLP applications for user generated text data which is known to be very noisy, our findings are applicable across various use cases in the industry. Specifically, we show that BERT’s performance on fundamental tasks like sentiment analysis and textual similarity drops significantly as we introduce noise in data in the form of spelling mistakes and typos. For our experiments we use three well known datasets - IMDB movie reviews, SST-2 and STS-B to measure the performance. Further, we identify the shortcomings in the BERT pipeline that are responsible for this drop in performance.'
author:
- |
Ankit Kumar\
Piyush Makhija\
Anuj Gupta Vahan Inc\
[email protected]\
bibliography:
- 'bertugc.bib'
title: ' User Generated Data: Achilles’ heel of BERT'
---
Introduction
============
In recent times, pre-trained contextual language models have led to significant improvement in the performance for many NLP tasks. Among the family of these models, the most popular one is BERT [@devlin2018bert], which is also the focus of this work. The strength of the BERT model \[fig:figure2\] stems from its transformer[@vaswani2017attention] based encoder architecture\[fig:figure\]. While it is still not very clear as to why BERT along with its embedding works so well for downstream tasks when it is fine tuned, there has been some work in this direction that that gives some important clues[@reif2019visualizing; @hewitt2019structural].
![BERT architecture [@devlin2018bert][]{data-label="fig:figure"}](bert.jpg)
![The Transformer model architecture [@vaswani2017attention][]{data-label="fig:figure2"}](transformer.png){width="9cm" height="7cm"}
At a high level, BERT’s pipelines looks as follows: given a input sentence, BERT tokenizes it using wordPiece tokenizer[@wu2016google]. The tokens are then fed as input to the BERT model and it learns contextualized embeddings for each of those tokens. It does so via pre-training on two tasks - Masked Language Model (MLM)[@devlin2018bert] and Next Sentence Prediction (NSP)[@devlin2018bert].
The focus of this work is to understand the issues that a practitioner can run into while trying to use BERT for building NLP applications in industrial settings. It is a well known fact that NLP applications in industrial settings often have to deal with the noisy data. There are different kinds of possible noise namely non-canonical text such as spelling mistakes, typographic errors, colloquialisms, abbreviations, slang, internet jargon, emojis, embedded metadata (such as hashtags, URLs, mentions), non standard syntactic constructions and spelling variations, grammatically incorrect text, mixture of two or more languages to name a few. Such noisy data is a hallmark of user generated text content and commonly found on social media, chats, online reviews, web forums to name a few. Owing to this noise a common issue that NLP models have to deal with is Out Of Vocabulary (OOV) words. These are words that are found in test and production data but not part of training data. In this work we highlight how BERT fails to handle Out Of Vocabulary(OOV) words, given its limited vocabulary. We show that this negatively impacts the performance of BERT when working with user generated text data and evaluate the same.
This evaluation is motivated from the business use case we are solving where we are building a dialogue system to screen candidates for blue collar jobs. Our candidate user base, coming from underprivileged backgrounds, are often high school graduates. This coupled with ‘fat finger’ problem[^1] over a mobile keypad leads to a lot of typos and spelling mistakes in the responses sent to the dialogue system. Hence, for this work we focus on spelling mistakes as the noise in the data. While this work is motivated from our business use case, our findings are applicable across various use cases in industry - be it be sentiment classification on twitter data or topic detection of a web forum.
To simulate noise in the data, we begin with a clean dataset and introduce spelling errors in a fraction of words present in it. These words are chosen randomly. We will explain this process in detail later. Spelling mistakes introduced mimic the typographical errors in the text introduced by our users. We then use the BERT model for tasks using both clean and noisy datasets and compare the results. We show that the introduction of noise leads to a significant drop in performance of the BERT model for the task at hand as compared to clean dataset. We further show that as we increase the amount of noise in the data, the performance degrades sharply.
Related Work
============
In recent years pre-trained language models ((e.g. ELMo[@peters2018deep], BERT[@devlin2018bert]) have made breakthroughs in several natural language tasks. These models are trained over large corpora that are not human annotated and are easily available. Chief among these models is BERT[@devlin2018bert]. The popularity of BERT stems from its ability to be fine-tuned for a variety of downstream NLP tasks such as text classification, regression, named-entity recognition, question answering[@devlin2018bert], machine translation[@lample2019cross] etc. BERT has been able to establish State-of-the-art (SOTA) results for many of these tasks. People have been able to show how one can leverage BERT to improve search[@patel2019tinysearch].
Owing to its success, researchers have started to focus on uncovering drawbacks in BERT, if any. [@jin2019bert] introduce TEXTFOOLER, a system to generate adversarial text. They apply it to NLP tasks of text classification and textual entailment to attack the BERT model. [@aspillaga2020stress] evaluate three models - RoBERTa, XLNet, and BERT in Natural Language Inference (NLI) and Question Answering (QA) tasks for robustness. They show that while RoBERTa, XLNet and BERT are more robust than recurrent neural network models to stress tests for both NLI and QA tasks; these models are still very fragile and show many unexpected behaviors. [@pal2020transfer] discuss length-based and sentence-based misclassification attacks for the Fake News Detection task trained using a context-aware BERT model and they show 78% and 39% attack accuracy respectively.
Our contribution in this paper is to answer that can we use large language models like BERT directly over user generated data.
Experiment
==========
For our experiments, we use pre-trained BERT implementation as given by huggingface [^2] transformer library. We use the BERTBase uncased model. We work with three datasets namely - IMDB movie reviews[@maas2011learning], Stanford Sentiment Treebank (SST-2) [@socher2013recursive] and Semantic Textual Similarity (STS-B) [@cer2017semeval].
IMDB dataset is a popular dataset for sentiment analysis tasks, which is a binary classification problem with equal number of positive and negative examples. Both STS-B and SST-2 datasets are a part of GLUE benchmark\[2\] tasks . In STS-B too, we predict positive and negative sentiments. In SST-2 we predict textual semantic similarity between two sentences. It is a regression problem where the similarity score varies between 0 to 5. To evaluate the performance of BERT we use standard metrics of F1-score for imdb and STS-B, and Pearson-Spearman correlation for SST-2.
In Table \[tab:table1\], we give the statistics for each of the datasets.
Dataset Training utterances Validation utterances
------------------- --------------------- -----------------------
IMDB Movie Review 25000 25000
SST-2 67349 872
STS-B 5749 1500
: Number of utterances in each datasets[]{data-label="tab:table1"}
We take the original datasets and add varying degrees of noise (i.e. spelling errors to word utterances) to create datasets for our experiments. From each dataset, we create 4 additional datasets each with varying percentage levels of noise in them. For example from IMDB, we create 4 variants, each having 5%, 10%, 15% and 20% noise in them. Here, the number denotes the percentage of words in the original dataset that have spelling mistakes. Thus, we have one dataset with no noise and 4 variants datasets with increasing levels of noise. Likewise, we do the same for SST-2 and STS-B.
All the parameters of the BERTBase model remain the same for all 5 experiments on the IMDB dataset and its 4 variants. This also remains the same across other 2 datasets and their variants. For all the experiments, the learning rate is set to 4e-5, for optimization we use Adam optimizer with epsilon value 1e-8. We ran each of the experiments for 10 and 50 epochs.
Results
=======
Let us discuss the results from the above mentioned experiments. We show the plots of accuracy vs noise for each of the tasks. For IMDB, we fine tune the model for the sentiment analysis task. We plot F1 score vs % of error, as shown in Figure \[imdb\]. Figure \[imdb\] shows the performance after fine tuning for 10 epochs, while Figure \[imdb\] shows the performance after fine tuning for 50 epochs.
Similarly, Figure \[sst\] and Figure \[sst\]) shows F1 score vs % of error for Sentiment analysis on SST-2 dataset after fine tuning for 10 and 50 epochs respectively.
Figure \[sts\] and \[sts\] shows Pearson-Spearman correlation vs % of error for textual semantic similarity on STS-B dataset after fine tuning for 10 and 50 epochs respectively.
Key Findings
------------
It is clear from the above plots that as we increase the percentage of error, for each of the three tasks, we see a significant drop in BERT’s performance. Also, from the plots it is evident that the reason for this drop in performance is introduction of noise (spelling mistakes). After all we get very good numbers, for each of the three tasks, when there is no error (0.0 % error). To understand the reason behind the drop in performance, first we need to understand how BERT processes input text data. BERT uses WordPiece tokenizer to tokenize the text. WordPiece tokenizer utterances based on the longest prefix matching algorithm to generate tokens [^3]. The tokens thus obtained are fed as input of the BERT model.
When it comes to tokenizing noisy data, we see a very interesting behaviour from WordPiece tokenizer. Owing to the spelling mistakes, these words are not directly found in BERT’s dictionary. Hence WordPiece tokenizer tokenizes noisy words into subwords. However, it ends up breaking them into subwords whose meaning can be very different from the meaning of the original word. Often, this changes the meaning of the sentence completely, therefore leading to substantial dip in the performance.
To understand this better, let us look into two examples, one each from the IMDB and STS-B datasets respectively, as shown below. Here, (a) is the sentence as it appears in the dataset ( before adding noise) while (b) is the corresponding sentence after adding noise. The mistakes are highlighted with italics. The sentences are followed by the corresponding output of the WordPiece tokenizer on these sentences: In the output ‘\#\#’ is WordPiece tokenizer’s way of distinguishing subwords from words. ‘\#\#’ signifies subwords as opposed to words.
**Example 1 (imdb example):**
1. *“that loves its characters and communicates something rather beautiful about human nature” (0% error)*
2. *“that loves 8ts characters abd communicates something rathee beautiful about human natuee” (5% error)*
**Output of wordPiece tokenizer:**
1. *\[’that’, ’loves’, ’its’, ’characters’, ’and’, ’communicate’, ’\#\#s’, ’something’, ’rather’, ’beautiful’, ’about’, ’human’,’nature’\] (0% error IMDB example)*
2. *\[’that’, ’loves’, ’8’, **’\#\#ts’**, ’characters’, ’abd’, ’communicate’,’\#\#s’, ’something’,’rat’, **’\#\#hee’**, ’beautiful’, ’about’, ’human’,**’nat’, ’\#\#ue’, ’\#\#e’**\] (5% error IMDB example)*
**Example 2(STS example):**
1. *“poor ben bratt could n’t find stardom if mapquest emailed himpoint-to-point driving directions.” (0% error)*
2. *“poor ben bratt could n’t find stardom if mapquest emailed him point-to-point drivibg dirsctioge.” (5% error)*
**Output of wordPiece tokenizer:**
1. *\[’poor’, ’ben’, ’brat’, ’\#\#t’, ’could’, ’n’, “’”, ’t’, ’find’,’star’, ’\#\#dom’, ’if’, ’map’, ’\#\#quest’, ’email’, ’\#\#ed’, ’him’,’point’, ’-’, ’to’, ’-’, ’point’, ’driving’, ’directions’, ’.’\] (0% error STS example)*
2. *\[’poor’, ’ben’, ’brat’, ’\#\#t’, ’could’, ’n’, “’”, ’t’, ’find’,’star’, ’\#\#dom’, ’if’, ’map’, ’\#\#quest’, ’email’, ’\#\#ed’, ’him’, ’point’, ’-’, ’to’, ’-’, ’point’, ’dr’, **’\#\#iv’, ’\#\#ib’,’\#\#g’**,’dir’,**’\#\#sc’, ’\#\#ti’, ’\#\#oge’,** ’.’\] (5% error STS example)*
In example 1, the tokenizer splits **communicates** into **\[‘communicate’, ‘\#\#s’\]** based on longest prefix matching because there is no exact match for **“communicates”** in BERT vocabulary. The longest prefix in this case is **“communicate”** and left over is **“s”** both of which are present in the vocabulary of BERT. We have contextual embeddings for both **“communicate” and “\#\#s”**. By using these two embeddings, one can get an approximate embedding for **“communicates”**. However, this approach goes for a complete toss when the word is misspelled. In example 1(b) the word **natuee** (‘nature’ is misspelled) is split into **\[’nat’, ’\#\#ue’, ’\#\#e’\]** based on the longest prefix match. Combining the three embeddings one cannot approximate the embedding of nature. This is because the word **nat** has a very different meaning (it means ‘a person who advocates political independence for a particular country’). This misrepresentation in turn impacts the performance of downstream subcomponents of BERT bringing down the overall performance of BERT model. Hence, as we systematically introduce more errors, the quality of output of the tokenizer degrades further, resulting in the overall performance drop.
Our results and analysis shows that one cannot apply BERT blindly to solve NLP problems especially in industrial settings. If the application you are developing gets data from channels that are known to introduce noise in the text, then BERT will perform badly. Examples of such scenarios are applications working with twitter data, mobile based chat system, user comments on platforms like youtube, reddit to name a few. The reason for the introduction of noise could vary - while for twitter, reddit it’s often deliberate because that is how users prefer to write, while for mobile based chat it often suffers from ‘fat finger’ typing error problem. Depending on the amount of noise in the data, BERT can perform well below expectations.
We further conducted experiments with different tokenizers other than WordPiece tokenizer. For this we used stanfordNLP WhiteSpace [@manning2014stanford] and Character N-gram [@mcnamee2004character] tokenizers. WhiteSpace tokenizer splits text into tokens based on white space. Character N-gram tokenizer splits words that have more than n characters in them. Thus, each token has at most n characters in them. The resultant tokens from the respective tokenizer are fed to BERT as inputs. For our case, we work with n = 6.
Results of these experiments are presented in Table \[tokens\]. Even though wordPiece tokenizer has the issues stated earlier, it is still performing better than whitespace and character n-gram tokenizer. This is primarily because of the vocabulary overlap between STS-B dataset and BERT vocabulary.
Error WordPiece WhiteSpce N-gram(n=6)
------- ----------- ----------- -------------
20% 0.35 0.22 0,25
: Comparative results on STS-B dataset with different tokenizers[]{data-label="tokens"}
Conclusion and Future Work
==========================
In this work we systematically studied the effect of noise (spelling mistakes) in user generated text data on the performance of BERT. We demonstrated that as the noise increases, BERT’s performance drops drastically. We further investigated the BERT system to understand the reason for this drop in performance. We show that the problem lies with how misspelt words are tokenized to create a representation of the original word.
There are 2 ways to address the problem - either (i) preprocess the data to correct spelling mistakes or (ii) incorporate ways in BERT architecture to make it robust to noise. The problem with (i) is that in most industrial settings this becomes a separate project in itself. We leave (ii) as a future work to fix the issues.
[^1]: https://en.wikipedia.org/wiki/Fat-finger\_error
[^2]: https://github.com/huggingface/transformers
[^3]: https://github.com/google-research/bert/blob/master/tokenization.py
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We compute the compactly supported cohomology of the standard realization of any locally finite building.
**AMS classification numbers**. Primary: 20F65\
Secondary: 20E42, 20F55, 20J06, 57M07.
**Keywords**: Building, cohomology of groups, Coxeter group.
author:
- 'M.W. Davis[^1]'
- 'J. Dymara[^2]'
- 'T. Januszkiewicz[^3]'
- 'J. Meier[^4]'
- 'B. Okun'
title: Compactly supported cohomology of buildings
---
Introduction {#introduction .unnumbered}
============
A building consists of a set $\Phi$ (the elements of which are called “chambers”) together with a family of equivalence relations (“adjacency relations”) on $\Phi$ indexed by a set $S$ and a “$W$-valued distance function,” $\Phi\times \Phi \to W$, where $W$ is a Coxeter group with fundamental set of generators $S$. So, associated to any building there is a Coxeter system $(W,S)$, its *type*.
There is a construction which associates a topological space to $\Phi$. This construction admits some freedom of choice. The idea is to choose a space $X$ as a “model chamber” and then glue together copies of it, one for each element of $\Phi$. To do this, it is first necessary to choose a family of closed subspaces $\{X_s\}_{s\in S}$ so that copies of $X$ corresponding to $s$-adjacent chambers are glued together along $X_s$. (We call such a family, $\{X_s\}_{s\in S}$, a “mirror structure” on $X$.) Let ${\mathcal {U}}(\Phi,X)$ denote the topological realization of $\Phi$ where each chamber is realized by a copy of the model chamber $X$. (Details are given in Section \[s:geom\].)
Classically, interest has centered on buildings of spherical or affine type, meaning that $W$ is a spherical or Euclidean reflection group, respectively. For example, each algebraic group over a local field has a corresponding affine building. However, we are mainly interested in buildings which are not classical in that their associated Coxeter systems are neither spherical nor affine. This is a large class of spaces, many of which have a great deal of symmetry. For example, such buildings arise in the theory of Kac-Moody groups (e.g., see [@cr; @remy; @rr]). Also, nonclassical buildings associated to arbitrary right-angled Coxeter groups have been a subject of recent interest in geometric group theory (e.g., see [@dbuild; @js; @thomas]).
Two choices for a model chamber $X$ stand out. The first is $X={\Delta}$, a simplex of dimension ${\operatorname{Card}}(S)-1$, with its codimension one faces indexed by $S$. This was Tits’ original choice (cf. [@ab]). We call ${\mathcal {U}}(\Phi, {\Delta})$ the “classical realization” of $\Phi$. The other choice for $X$ is the “Davis chamber” $K$, defined as the geometric realization of the poset ${\mathcal {S}}$ of spherical subsets of $S$ (see [@dbook Chapters 7, 18]). ${\mathcal {U}}(\Phi, K)$ is the “standard realization” of $\Phi$. Both realizations are contractible. The standard realization is important in geometric group theory, the reason being that in this field one is interested in discrete group actions which are both proper and cocompact and these conditions are more likely to hold for the action of a group of automorphisms on the standard realization than on the classical realization. (If $\Phi$ has finite thickness, ${\mathcal {U}}(\Phi,K)$ is locally finite, while ${\mathcal {U}}(\Phi,{\Delta})$ need not be.)
If a discrete group ${\Gamma}$ acts properly and cocompactly on a locally finite, contractible CW complex $Y$, then the compactly supported cohomology of $Y$ is isomorphic to the cohomology of ${\Gamma}$ with ${{\mathbf Z}}{\Gamma}$ coefficients. In particular, it determines the virtual cohomological dimension of ${\Gamma}$, as well as, the number of ends of ${\Gamma}$, and it determines if ${\Gamma}$ is a duality group. (For more information, see [@g Part IV].) However, as we will explain, even if one is only interested in cohomological computations in the case of the standard realization of $\Phi$, it is necessary to carry out similar computations for various other realizations, in particular, for the classical realization.
In the classical case of an (irreducible) affine building, the two notions of model chamber agree: ${\Delta}=K$. So, in the affine case the study of the cohomology of cocompact lattices in ${\operatorname{Aut}}(\Phi)$ is closely tied to the study of the cohomological properties of ${\mathcal {U}}(\Phi,{\Delta})$. For example, in [@bs] Borel and Serre calculated the compactly supported cohomology, $H^*_c({\mathcal {U}}(\Phi, {\Delta}))$, for any (irreducible) affine building and then used this calculation to derive information about the cohomology of “$S$-arithmetic” subgroups. The calculation of [@bs] was that $H^*_c({\mathcal {U}}(\Phi, {\Delta}))$ is concentrated in the top degree ($=\dim{\Delta}$) and is free abelian in that degree.
Our main result, Corollary \[c:main\], is a calculation of $H^*_c({\mathcal {U}}(\Phi, K))$, generalizing the theorem of Borel-Serre. In the case where $\Phi=W$, this was done in [@d98; @ddjo2; @dm]. For a general (thick) building, in the case where $(W,S)$ is right-angled, it was done in [@ddjo2 Theorem 6.6]. It was claimed in full generality in [@dm]; however, there is a mistake in the proof (see [@dm-erratum]).
In order to write the formula, we need more notation. Let $A$ be the free abelian group of finitely supported, ${{\mathbf Z}}$-valued functions on $\Phi$. For each subset $T\subseteq S$, let $A^T$ denote the subgroup of all functions $f\in A$ which are constant on each residue of type $T$. (A “residue” of type $T$ is a certain kind of subset of $\Phi$; in the case of the building $W$, a residue of type $T$ is a left coset of $W_T$, the subgroup of $W$ generated by $T$.) N.B. The empty set, $\emptyset$, is a spherical subset and a residue of type $\emptyset$ is just a single chamber; hence, $A^\emptyset=A$. If $U\supset T$, then $A^U\subset A^T$. Let $A^{{>T}}\subset A^T$ denote the ${{\mathbf Z}}$-submodule, $\sum_{U\supset T} A^U$. (Throughout this paper we will use the convention that $\subseteq$ denotes containment and $\subset$ will be reserved for strict containment. Also, we will use the symbol $\sum$ for an internal sum of modules and $\bigoplus$ to mean either an external direct sum or an internal sum which we have proved is direct.)
We shall show in Section \[s:decomp\] that $A^{{>T}}$ is a direct summand of $A^T$. Let ${{\hat{A}}}^T$ be a complementary summand. As in [@ddjo2] the main computation is a consequence the following Decomposition Theorem (proved as Theorem \[t:decomp\] in Section \[s:decomp\]).
$$A=\bigoplus_{U\in {\mathcal {S}}} {{\hat{A}}}^U\quad\text{and, in fact, for any $T\in{\mathcal {S}}$,}\quad A^T=\bigoplus_{U\supseteq T} {{\hat{A}}}^U.$$
The point is that this theorem provides a decomposition of a coefficient system which can be used to calculate the compactly supported cohomology of any of the various realizations of $\Phi$. The calculation in which we are most interested is the following (proved as Corollary \[c:main\] in Section \[s:cohomology\]).
Suppose $\Phi$ is a building of finite thickness and type $(W,S)$. Let $K$ be the geometric realization of the poset ${\mathcal {S}}$ of spherical subsets of $S$. Then $$H^*_c({\mathcal {U}}(\Phi,K))\cong \bigoplus_{T\in {\mathcal {S}}} H^*(K,K^{S-T})\otimes {{\hat{A}}}^T.$$
(For any subset $U$ of $S$, $K^U$ denotes the union of the $K_s$, with $s\in U$.)
The above theorem applies to all buildings. A general building $\Phi$ will not be highly symmetric, in that its automorphism group, ${\operatorname{Aut}}(\Phi)$, can have infinitely many orbits of chambers. However, if $\Phi/{\operatorname{Aut}}(\Phi)$ is finite and if ${\Gamma}$ is a torsion-free cocompact lattice in ${\operatorname{Aut}}(\Phi)$, then the Main Theorem implies that the cohomological dimension of ${\Gamma}$ is equal to the virtual cohomological dimension of the corresponding Coxeter group. Moreover, this dimension is $${\operatorname{cd}}({\Gamma})={\operatorname{vcd}}(W)=\max\{k\mid H^k(K,K^{S-T})\neq 0, \text{for some $T\in {\mathcal {S}}$}\}$$ (cf. Corollary \[c:dim\]). As another example, such a torsion-free cocompact lattice is an $n$-dimensional duality group if and only if for each $T\in {\mathcal {S}}$, $H^*(K,K^{S-T})$ is free abelian and concentrated in degree $n$ (cf. Corollary \[c:duality\]).
The central objective of [@ddjo2] was to calculate $H^*_c({\mathcal {U}}(W,K))$ as a $W$-module. In that paper we showed there is a filtration of $H^*_c({\mathcal {U}}(W,K))$ by $W$-submodules so that the associated graded terms look like the terms on the right hand side of the formula in the Main Theorem. Similarly, one can ask about the $G$-module structure of $H^*_c({\mathcal {U}}(\Phi,K))$ for any subgroup $G\subseteq {\operatorname{Aut}}(\Phi)$. The methods of [@ddjo2] are well adapted to the present paper. In particular, for each $T\in {\mathcal {S}}$, the free abelian group $A^T$ is a $G$-module, as is its quotient $D^T:=A^T/A^{{>T}}$. So, as in [@ddjo2], there is a filtration of $H^*_c({\mathcal {U}}(\Phi,K))$ by $G$-submodules and we get the following (proved as Theorem \[t:Gmodule\]).
Suppose $G$ is a group of automorphisms of $\Phi$. There is a filtration of $H^*_c({\mathcal {U}}(\Phi,K))$ by right $G$-submodules with associated graded term in filtration degree $p$: $$\bigoplus _{\substack{T\in {\mathcal {S}}\\|T|=p}} H^*(K,K^{S-T})\otimes D^T.$$
As we mentioned earlier, although the Main Theorem is the result of importance in geometric group theory, it is no harder to do similar calculations when the chambers are modeled on an arbitrary $X$ (or at least on $X^f$, the complement of the faces of $X$ which have infinite stabilizers in $W$). In fact, as we explain below, the proof of the Decomposition Theorem depends on first establishing a version of the Main Theorem for ${\mathcal {U}}(\Phi,{\Delta}^f)$. So, the Main Theorem ultimately depends on first proving a version of it in the case of the classical realization. This version (proved as Theorem \[t:SI\]) is the following.
When $W$ infinite, $H^*_c({\mathcal {U}}(\Phi, {\Delta}^f))$ is free abelian and is concentrated in the top degree $n$ ($=\dim {\Delta}$).
This result is obvious when $\Phi=W$, for then ${\mathcal {U}}(W,{\Delta}^f)$ is homeomorphic to Euclidean space ${{\mathbf R}}^n$ (see Section \[s:geom\]). We prove it for a general $\Phi$ by showing the following (Theorem \[t:cat\]).
${\mathcal {U}}(\Phi,{\Delta}^f)$ admits a ${\operatorname{CAT}}(0)$ metric (extending Moussong’s ${\operatorname{CAT}}(0)$ metric on the standard realization).
The existence of this ${\operatorname{CAT}}(0)$ metric on ${\mathcal {U}}(\Phi,{\Delta}^f)$ is of independent interest. (See [@moussong; @dbuild] or [@dbook] for a description of Moussong’s metric on ${\mathcal {U}}(\Phi, K)$.)
To finish the calculation of $H^*_c({\mathcal {U}}(\Phi, {\Delta}^f))$ we invoke a result of [@bbm] which asserts that the compactly supported cohomology of such a ${\operatorname{CAT}}(0)$ space is concentrated in the top degree provided the cohomology of each “punctured link” vanishes except in the top degree. These links are spherical buildings and the vanishing of the cohomology groups, in degrees below the top, of their punctured versions is a result of [@dymo] (and independently, [@someone]).
As we have said, special cases of the Decomposition Theorem were proved in [@ddjo2]. The method of [@ddjo2] was simply to find a basis for $A$ adapted to its decomposition into the ${{\hat{A}}}^T$. In the general case, finding an explicit description of such a basis seems problematic. We use instead an idea coming from an analogy with the argument of [@ddjo]. In that paper we proved $L^2$ versions of the Decomposition Theorem and of the Main Theorem. The proof of the $L^2$ version of the Decomposition Theorem was homological: the key step was to show that certain “weighted $L^2$-homology” groups of certain auxiliary spaces associated to $W$ vanished except in the bottom degree. We then applied a certain duality (not applicable here) to deduce the Decomposition Theorem in the cases of actual interest. Analogously, in this paper we prove the Decomposition Theorem by establishing the vanishing, except in the top degree, of the cohomology of certain auxiliary spaces. The most important of these auxiliary spaces is ${\mathcal {U}}(\Phi,{\Delta}^f)$ and we indicated in the previous paragraphs how we prove the result in that case.
Coxeter groups and buildings {#s:buildings}
============================
A *chamber system over a set $S$* is a set $\Phi$ of *chambers* together with a family of equivalence relations on $\Phi$ indexed by $S$. Two chambers are *$s$-equivalent* if they are related via the equivalence relation with index $s$; they are *$s$-adjacent* if they are $s$-equivalent and not equal. A *gallery* in $\Phi$ is a finite sequence of chambers $({\varphi}_0, \dots ,{\varphi}_k)$ such that ${\varphi}_{j-1}$ is adjacent to ${\varphi}_j, 1 \le j \le k$. The *type* of this gallery is the word ${{\mathbf s}}=(s_1, \dots, s_k)$ where ${\varphi}_{j-1}$ is $s_j$-adjacent to ${\varphi}_j$. If each $s_j$ belongs to a given subset $T$ of $S$, then the gallery is a *$T$-gallery*. A chamber system is *connected* (resp., *$T$-connected*) if any two chambers can be joined by a gallery (resp., a $T$-gallery). The $T$-connected components of a chamber system $\Phi$ are its *residues* of *type $T$*. A *Coxeter matrix over a set* $S$ is an $S\times S$ symmetric matrix $M=(m_{st})$ with each diagonal entry $=1
$ and each off-diagonal entry an integer $\ge 2$ or the symbol $\infty$. The matrix $M$ defines a presentation of a group $W$ as follows: the set of generators is $S$ and the relations have the form $(st)^{m_{st}}$ where $(s,t)$ ranges over all pairs in $S\times S$ such that $m_{st}\neq \infty$. The pair $(W,S)$ is a *Coxeter system* (cf. [@bourbaki; @dbook]). Given $T\subseteq S$, $W_T$ denotes the subgroup generated by $T$; it is called a *special subgroup*. $(W_T,T)$ is itself a Coxeter system. The subset $T$ is *spherical* if $W_T$ is finite.
The poset of spherical subsets of $S$ (partially ordered by inclusion) is denoted ${\mathcal {S}}$.
Also, ${\mathcal {P}}$ ($={\mathcal {P}}(S)$) will denote the poset of all proper subsets of $S$. (In Section \[s:cat0\] the poset ${\mathcal {P}}-{\mathcal {S}}$ plays a role.)
Suppose $(W,S)$ is a Coxeter system and $M=(m_{st})$ is its Coxeter matrix. Following [@ronan] (or [@ddjo2]), a *building of type $(W,S)$* (or of *type $M$*) is a chamber system $\Phi$ over $S$ such that
for all $s\in S$, each $s$-equivalence class contains at least two chambers, and
there exists a *$W$-valued distance function* ${\delta}: \Phi \times
\Phi \to W$. (This means that given a reduced word ${{\mathbf s}}$ for an element $w\in W$, chambers ${\varphi}$ and ${\varphi}'$ can be joined by a gallery of type ${{\mathbf s}}$ from ${\varphi}$ to ${\varphi}'$ if and only if ${\delta}({\varphi}, {\varphi}') = w$.)
\[ex:thin\] The group $W$ itself has the structure of a building: the $s$-equivalence classes are the left cosets of $W_{\{s\}}$ and the $W$-valued distance, ${\delta}:W\times W\to W$, is defined by ${\delta}(v,w)=v^{{-1}}w$.
A residue of type $T$ is a building; its type is $(W_T,T)$. A building of type $(W,S)$ is *spherical* if $W$ is finite. A building has *finite thickness* if each $s$-equivalence class is finite, for each $s\in S$. (This implies all spherical residues are finite.) *Henceforth, all buildings will be assumed to have finite thickness*.
Geometric realizations of Coxeter groups and buildings {#s:geom}
======================================================
A *mirror structure over a set* $S$ on a space $X$ is a family of subspaces $(X_s)_{s\in S}$ indexed by $S$. Given a mirror structure on $X$, a subspace $Y\subseteq X$ inherits a mirror structure by $Y_s:=Y\cap X_s$. If $X$ is a CW complex and each $X_s$ is a subcomplex, then $X$ is a *mirrored CW complex*. For each nonempty subset $T\subseteq S$, define subspaces $X_T$ and $X^T$ by $$\label{e:T}
X_T:=\bigcap_{s\in T} X_s\quad\text{and}\quad X^T:=\bigcup_{s\in T} X_s.$$ Put $X_\emptyset:=X$ and $X^\emptyset:=\emptyset$. Given a cell $c$ of (a CW complex) $X$ or a point $x \in X$, put $$\begin{aligned}
S(c)&:=\{s\in S\mid c\subseteq X_s\},\\
S(x)&:=\{s\in S\mid x\in X_s\}.\end{aligned}$$
Suppose now that $S$ is the set of generators for a Coxeter system $(W,S)$. Let ${\Upsilon}(X)$ denote the union of the nonspherical faces of $X$ and $X^f$ its complement in $X$, i.e., $$\label{e:xf}
{\Upsilon}(X):=\bigcup_{T\notin {\mathcal {S}}} X_T\quad \text{and}\quad X^f:=X-{\Upsilon}(X).$$ The mirror structure is *$W$-finite* if ${\Upsilon}(X)=\emptyset$.
Given a building $\Phi$ of type $(W,S)$ and a mirrored space $X$ over $S$, define an equivalence relation $\sim$ on $\Phi \times X$ by $({\varphi}, x) \sim ({\varphi}', x')$ if and only if $x = x'$ and ${\delta}({\varphi}, {\varphi}') \in W_{S(x)}$ (i.e., ${\varphi}$ and ${\varphi}'$ belong to the same $S(x)$-residue). The $X$-*realization* of $\Phi$, denoted ${\mathcal {U}}(\Phi,X)$, is defined by $$\label{e: defUPhi}
{\mathcal {U}}(\Phi,X):=(\Phi \times X)/\sim.$$ ($\Phi$ has the discrete topology.) Suppose $X$ is a mirrored CW complex and that we are given a cell $c$ of $X$ and a chamber ${\varphi}\in \Phi$. Then ${\varphi}\cdot c$ denotes the corresponding cell in ${\mathcal {U}}(\Phi,X)$. Let ${\mathcal {U}}^{(i)}$ denote the set of $i$-cells in ${\mathcal {U}}(\Phi,X)$. Each such cell has the form ${\varphi}\cdot c$ for some ${\varphi}\in \Phi$ and $i$-cell $c$ of $X$.
**The classical realization**. ${\Delta}$ denotes the simplex of dimension $|S|-1$, with its codimension one faces indexed by $S$. In other words, the mirror ${\Delta}_s$ is a codimension one face and ${\Delta}_T$ (the intersection of the ${\Delta}_s$ over all $s\in T$) is a face of codimension $|T|$.
The simplicial complex ${\mathcal {U}}(W,{\Delta})$ is the *Coxeter complex* of $(W,S)$ while ${\mathcal {U}}(\Phi, {\Delta})$ is the *classical realization* of the building $\Phi$.
Tits constructed a representation of $W$ on ${{\mathbf R}}^S$ called “the contragredient of the canonical representation” in [@bourbaki] and the “geometric representation” in [@dbook]. The elements of $S$ are represented by reflections across the codimension one faces of a simplicial cone $C$. The union of translates of $C$ is denoted by $WC$. It is a convex cone and $W$ acts properly on its interior ${\mathcal {I}}$. If $W$ is infinite, $WC$ is a proper cone. If $C^f$ denotes the complement of the nonspherical faces of $C$, then ${\mathcal {I}}$ is equivariantly homeomorphic to ${\mathcal {U}}(W,C^f)$. Assume $W$ is infinite. Then the image of $C-0$ in projective space can be identified with the simplex ${\Delta}$ obtained by intersecting $C$ with some affine hyperplane; moreover, ${\Delta}^f$ is identified with the intersection of $C^f$ and this hyperplane. The image of ${\mathcal {I}}$ in projective space is then identified with ${\mathcal {U}}(W,{\Delta}^f)$. Since this image is the interior of a topological disk (of dimension $|S|-1$), it follows that ${\mathcal {U}}(W,{\Delta}^f)$ is homeomorphic to a Euclidean space of that dimension. In particular, it is contractible.
**Geometric realizations of posets**. Given a poset ${\mathcal {T}}$, ${\operatorname{Flag}}({\mathcal {T}})$ denotes the set of finite chains in ${\mathcal {T}}$, partially ordered by inclusion, i.e., an element of ${\operatorname{Flag}}({\mathcal {T}})$ is a finite, nonempty, totally ordered subset of ${\mathcal {T}}$. If ${\alpha}=\{t_0,\dots,t_k\}\in {\operatorname{Flag}}({\mathcal {T}})$ where $t_0<\cdots<t_k$, then we will write ${\alpha}:=\{t_0<\cdots<t_k\}$ and $\min {\alpha}:=t_0$. ${\operatorname{Flag}}({\mathcal {T}})$ is an abstract simplicial complex with vertex set ${\mathcal {T}}$ and with $k$-simplices the elements of ${\operatorname{Flag}}({\mathcal {T}})$ of cardinality $k+1$. The corresponding topological simplicial complex is the *geometric realization* of the poset ${\mathcal {T}}$ and is denoted by $|{\mathcal {T}}|$. For example, if ${\mathcal {P}}$ is the poset of proper subsets of $S$, partially ordered by inclusion, then its opposite poset, ${\mathcal {P}}^{op}$, is the poset of nonempty faces of the simplex ${\Delta}$. ${\operatorname{Flag}}({\mathcal {P}})$ is the poset of simplices in its barycentric subdivision, $b{\Delta}$, and $|{\mathcal {P}}|=b{\Delta}$.
**The standard realization, $\boldsymbol{{\mathcal {U}}(\Phi,K)}$**. As before, ${\mathcal {S}}$ denotes the poset of spherical subsets of $S$. Put $K:=|{\mathcal {S}}|$. It is a subcomplex of $b{\Delta}$ (provided $W$ is infinite). The mirror structure on ${\Delta}$ induces one on $K$. More specifically, for each $s\in S$, put $K_s:=|{\mathcal {S}}_{\ge \{s\}}|$ and for each $T\in {\mathcal {S}}$, $K_T=|{\mathcal {S}}_{\ge T}|$. ($K$ is the “compact core” of ${\Delta}^f$.) $K$ is sometimes called the *Davis chamber* of $(W,S)$ and ${\mathcal {U}}(W,K)$, the *Davis complex*. Alternatively, ${\mathcal {U}}(\Phi,K)$ is the geometric realization of the poset of spherical residues of $\Phi$ (see [@dbuild]).
By construction ${\mathcal {U}}(\Phi, K)$ is locally finite (since $\Phi$ is assumed to have finite thickness). It is proved in [@dbuild] that ${\mathcal {U}}(\Phi,K)$ is contractible.
**The realization $\boldsymbol{{\mathcal {U}}(\Phi,{\Delta}^f)}$**. ${\Delta}^f$ and $K$ have the same poset of faces (indexed by ${\mathcal {S}}$) and there is a face-preserving deformation retraction ${\Delta}^f\to K$. That is to say, ${\Delta}^f$ is a “thickened version” of $K$. Similarly, ${\mathcal {U}}(\Phi,{\Delta}^f)$ is a thickened version of ${\mathcal {U}}(\Phi,K)$. Like ${\mathcal {U}}(\Phi, K)$, the space ${\mathcal {U}}(\Phi, {\Delta}^f)$ has the advantage of being locally finite; however, the chamber ${\Delta}^f$ is not compact whenever ${\Delta}^f\neq {\Delta}$.
A $\boldsymbol{{\operatorname{CAT}}(0)}$ metric on $\boldsymbol{{\mathcal {U}}(\Phi,{\Delta}^f)}$ {#s:cat0}
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Our goal in this section is to prove the following.
\[t:cat\] Let $\Phi$ be a building of type $(W,S)$ with $W$ infinite. Then there is a piecewise Euclidean, ${\operatorname{CAT}}(0)$ metric on ${\mathcal {U}}(\Phi, {\Delta}^f)$.
**Review of the Moussong metric**. Suppose $T$ is a spherical subset of $S$. $W_T$ acts on ${{\mathbf R}}^T$ via the canonical representation. The *Coxeter cell* of *type $T$*, denoted $P_T$, is defined to be the convex hull of the $W_T$-orbit of a point $x_0$ in the interior of the fundamental simplicial cone. As examples, if $W_T$ is a product of $n$ copies of the cyclic group of order $2$, then $P_T$ is an $n$-cube; if $W_T$ is the symmetric group on $n+1$ letters, then $P_T$ is an $n$-dimensional permutohedron. Its boundary complex, $\partial P_T$, is the dual of the Coxeter complex of $W_T$ (the Coxeter complex is a triangulation of the unit sphere in ${{\mathbf R}}^T$). The fact that $\partial P_T$ is dual to a simplicial complex means that $P_T$ is a “simple polytope”. The isometry type of $P_T$ is determined once we choose the distance from $x_0$ to each of the bounding hyperplanes of the simplicial cone. (We assume, without further comment, that such a choice of distance has been made for each $s\in S$.) The intersection of $P_T$ with the fundamental simplicial cone is denoted $B_T$ and called the *Coxeter block* of type $T$. It is a convex cell combinatorially isomorphic to a cube of dimension $|T|$ (because $P_T$ is simple). One can identify $B_T$ with the subcomplex $|{\mathcal {S}}_{\le T}|$ of $K$ in such a way that $x_0$ is identified with the vertex corresponding to $\emptyset$. To be more precise, $B_T$ is the union of simplices of ${\operatorname{Flag}}({\mathcal {S}})$ whose maximum vertex is $\le T$, i.e., $\vert {\mathcal {S}}_{\le T} \vert$ is a subdivision of $B_T$.
The convex polytope $B_T$ has two types of faces. First, there are the faces which contain the vertex $x_0$. Each such face is a Coxeter block of the form $B_V$ for some $V\subseteq T$. The other type of face is the intersection of $B_T$ with the face of the fundamental simplicial cone fixed by $W_V$ for some nonempty $V\subseteq T$. We denote such a face by $B_{T,V}$ and call it a *reflecting face* of $B_T$. For the purpose of unifying different cases, we shall sometimes write $B_{T,\emptyset}$ instead of $B_T$.
The *Moussong metric* on $K$ is the piecewise Euclidean metric on $K$ in which each Coxeter block $B_T$ is given its Euclidean metric as a convex cell in ${{\mathbf R}}^T$ (cf. [@moussong] or [@dbook Section 12.1]). This induces a piecewise Euclidean metric on ${\mathcal {U}}(W,K)$ as well as one on ${\mathcal {U}}(\Phi,K)$. The link of the central vertex corresponding to $\emptyset$ can be identified with a certain simplicial complex $L$ ($=L(W,S)$) called the *nerve* of the Coxeter system. The vertex set of $L$ is $S$ and a subset $T\subseteq S$ spans a simplex if and only if it is spherical. Thus, the poset of simplices in $L$ (including the empty simplex) is ${\mathcal {S}}$. The piecewise Euclidean metric on $K$ induces a piecewise spherical metric on $L$ such that whenever $m_{st}< \infty$, the length of the edge corresponding to $\{s,t\}$ is $\pi-\pi/m_{st}$. Moussong proved that this piecewise spherical metric on $L$ is ${\operatorname{CAT}}(1)$ and from this he deduced that the piecewise Euclidean metric on ${\mathcal {U}}(W,K)$ is ${\operatorname{CAT}}(0)$ (cf. [@moussong] and [@dbook Section 12.3]). Using this, it is proved in [@dbuild] that for any building $\Phi$, the Moussong metric on the standard realization, ${\mathcal {U}}(\Phi,K)$, is ${\operatorname{CAT}}(0)$.
**A piecewise Euclidean metric on $\boldsymbol{{\Delta}^f}$**. We will define a cell structure on ${\Delta}^f$ so that each cell will have the form $B_{T,V} \times [0,\infty)^m$ for some $T\in {\mathcal {S}}$, $V\subseteq T$ and nonnegative integer $m$. When $m>0$ such a cell will be noncompact. There are two types of such cells. First there are the compact cells $B_{T,V}$ where $T\in {\mathcal {S}}$ and $V\subseteq T$. The remaining cells are in bijective correspondence with triples $(T,V,{\alpha})$ where $T\in {\mathcal {S}}$, $V\subseteq T$ and ${\alpha}\in {\operatorname{Flag}}({\mathcal {P}}-{\mathcal {S}})$ is such that $T< \min {\alpha}$. Let ${\mathcal {F}}$ be the set consisting of pairs $(T,V)$ and triples $(T,V,{\alpha})$, where $T\in {\mathcal {S}}$, $V\subseteq T$ and ${\alpha}\in {\operatorname{Flag}}({\mathcal {P}}-{\mathcal {S}})$ is such that $T< \min {\alpha}$. ${\mathcal {F}}$ is partially ordered as follows:
- $(T', V')\le (T,V)$ if and only if $T'\subseteq T$ and $V'\subseteq V$,
- $(T',V',{\alpha}')\le (T,V,{\alpha})$ if and only if $(T',V')\le (T,V)$ and ${\alpha}'\subseteq {\alpha}$,
- $(T',V')\le (T,V,{\alpha})$ if and only if $(T',V')\le (T,V)$.
The cell $c(T,V,{\alpha})$ which corresponds to $(T,V,{\alpha})$ is defined to be $B_{T,V}\times [0,\infty)^{\alpha}$, where $[0,\infty)^{\alpha}$ means the set of all functions from the finite set ${\alpha}$ to $[0,\infty)$. For the most part, it will suffice to deal with the case $V=\emptyset$ since the cells of the form $B_T\times[0,\infty)^{\alpha}$ cover ${\Delta}^f$. The piecewise Euclidean structure on ${\Delta}^f$ will be defined by declaring each $B_T\times [0,\infty)^{\alpha}$ to have the product metric. Thus, the piecewise Euclidean metric on ${\Delta}^f$ will extend the one on $K$.
\[l:cell\] ${\Delta}^f$ has a decomposition into the cells, $\{B_{T,V}\}\cup \{ c(T,V,{\alpha})\}$, defined above, where $(T,V)$ and $(T,V,{\alpha})$ range over ${\mathcal {F}}$.
To prove this, we need to set up a standard identification of the open cone of radius 1 on a $k$-simplex ${\sigma}$ with the standard simplicial cone $[0,\infty)^{k+1}\subset {{\mathbf R}}^{k+1}$. Let $\{v_b\}_{b\in {\mathcal {B}}}$ be the vertex set of ${\sigma}$ for some finite index set ${\mathcal {B}}$ and let $(x_b)_{b\in{\mathcal {B}}}$ be barycentric coordinates on ${\sigma}$. Let ${{\mathbf R}}^{\mathcal {B}}$ be the Euclidean space of all functions ${\mathcal {B}}\to {{\mathbf R}}$. Let ${{\mathbb S}}_+({{\mathbf R}}^{\mathcal {B}})$ denote the intersection of the standard simplicial cone $[0,\infty)^{\mathcal {B}}$ with the unit sphere. Thus, ${{\mathbb S}}_+({{\mathbf R}}^{\mathcal {B}})$ is an “all right” spherical simplex (i.e., all edge lengths and all dihedral angles are $\pi/2$). Let $\{e_b\}_{b\in {\mathcal {B}}}$ be the standard basis for ${{\mathbf R}}^{\mathcal {B}}$. Define a homeomorphism $\theta_{\mathcal {B}}:{\sigma}\to {{\mathbb S}}_+({{\mathbf R}}^{\mathcal {B}})$, taking $v_b$ to $e_b$, by $$\label{e:theta}
\sum x_b v_b \to \sum x_be_b\ \bigg{/}\ (\sum x_b^2)^{1/2}.$$ Let $r:[0,1)\to [0,\infty)$ be some fixed homeomorphism. The open cone on ${\sigma}$ can be regarded as the points $[t,x]$ in the join, $v*{\sigma}$, of ${\sigma}$ with a point $v$ such that the join coordinate $t$ is $\neq 1$. The homeomorphism from the open cone to $[0,\infty)^{\mathcal {B}}$ is defined by $[t,x]\to r(t)\theta_{\mathcal {B}}(x)$.
Suppose ${\alpha}\in {\operatorname{Flag}}({\mathcal {P}})$. Then ${\alpha}=\{T_0<\cdots <T_{k+l}\}$, where $T_k\in {\mathcal {S}}$ and $T_{k+1}\notin {\mathcal {S}}$. Put ${\alpha}':=\{T_0<\cdots <T_{k}\}$ and ${\alpha}'':=\{T_{k+1}<\cdots <T_{k+l}\}$. The simplex ${\sigma}_{{\alpha}'}$ lies in $B_{T_k}$ while the simplex ${\sigma}_{{\alpha}''}$ is in the nonspherical face ${\Delta}_{T_{k+1}}$. We have ${\sigma}_{\alpha}={\sigma}_{{\alpha}'}*{\sigma}_{{\alpha}''}$ and a point in ${\sigma}_{\alpha}$ has coordinates $[t,x,y]$ where $t\in [0,1]$, $x\in {\sigma}_{{\alpha}'}$ and $y\in {\sigma}_{{\alpha}''}$. The points in ${\sigma}_{{\alpha}}-{\sigma}_{{\alpha}''}$ are those where the join coordinate $t$ is $\neq 1$. The identification ${\sigma}_{{\alpha}}-{\sigma}_{{\alpha}''} \to {\sigma}_{{\alpha}'} \times [0,\infty)^{{\alpha}''}$ is given by $[t,x,y]\to (x,r(t)\theta_{{\alpha}''}(y))$.
Next we want to consider the link of the central vertex $v_\emptyset$ (corresponding to $\emptyset$) in this cell structure. Let ${\Delta}^{op}$ denote the simplex on $S$. The nerve $L$ is a subcomplex of ${\Delta^{op}}$. If $W$ is spherical, then $L={\Delta^{op}}$, while if $W$ is infinite, then $L$ is a subcomplex of $\partial {\Delta^{op}}$. Moreover, $\partial{\Delta^{op}}$ is a triangulation of the $(n-1)$-sphere, for $n=|S|$. Assume $W$ is infinite. The link of $v_\emptyset$ in ${\Delta}^f$ is a certain subdivision $L'$ of $\partial{\Delta^{op}}$, which we shall now describe. The vertex set of $L'$ is the disjoint union $S\cup ({\mathcal {P}}-{\mathcal {S}})$. There are three types of simplices in $L'$:
simplices ${\sigma}_T$ in $L$ corresponding to spherical subsets $T\in {\mathcal {S}}_{>\emptyset}$,
simplices ${\sigma}_{\alpha}$ in $b\partial{\Delta}$ ($=b\partial{\Delta^{op}}$) corresponding to flags ${\alpha}\in {\operatorname{Flag}}({\mathcal {P}}-{\mathcal {S}})$,
joins ${\sigma}_T*{\sigma}_{\alpha}$, with $T\in {\mathcal {S}}_{>\emptyset}$, ${\alpha}\in {\operatorname{Flag}}({\mathcal {P}}-{\mathcal {S}})$ and $T<\min {\alpha}$.
\[l:subdivision\] $L'$ is a subdivision of $\partial {\Delta^{op}}$ and $L\subseteq L'$ is a full subcomplex.
Suppose $U$ is a minimal element of ${\mathcal {P}}-{\mathcal {S}}$. Let ${\sigma}_U$ denote the corresponding simplex in $\partial{\Delta^{op}}$. Introduce a “barycenter” $v_U\in {\partial\Delta^{op}}$ and then subdivide ${\sigma}_U$ to a new simplicial complex $({\sigma}_U)'$ by coning off the simplices in $\partial {\sigma}_U$. Each new simplex will have the form $v_U*{\sigma}_T$ for some $T\subset U$.
Next, let $U$ be an arbitrary element of ${\mathcal {P}}-{\mathcal {S}}$ and suppose by induction that we have defined the subdivision of $({\sigma}_{U'})$ for each $U'\subset U$ and hence, a subdivision $(\partial {\sigma}_U)'$ of $\partial {\sigma}_U$. Introduce a barycenter $v_U$ of ${\sigma}_U$ and subdivide by coning off $(\partial {\sigma}_U)'$. Each new simplex will have the form $v_U*{\sigma}_{U'}$ for some $U'\subset U$. In other words, each new simplex will be either of type 2 or type 3 above. It is clear that the subcomplex $L$ is full.
The following lemma is also clear.
\[l:link\] $L'$ is the link of the central vertex in the piecewise Euclidean cell structure on ${\Delta}^f$.
The piecewise Euclidean metric induces a piecewise spherical metric on $L'$ extending the given metric on $L$. Since the link of the origin in $[0,\infty)^{k+1}$ is the all right spherical $k$-simplex we get the following description of the metric on $L'$.
\[l:ps\] The simplices in $L'$ have spherical metrics of the following types.
For $T\in {\mathcal {S}}_{>\emptyset}$, the simplex ${\sigma}_T$ in $L$ is the dual to the fundamental simplex for $W_T$ on the unit sphere in ${{\mathbf R}}^T$. In other words, for distinct elements $s$, $t$ in $T$, the edge corresponding to $\{s,t\}$ has length $\pi-\pi/m_{st}$.
The simplex ${\sigma}_{\alpha}$ corresponding to ${\alpha}\in {\operatorname{Flag}}({\mathcal {P}}-{\mathcal {S}})$ has its all right structure. In other words, each edge of ${\sigma}_{\alpha}$ has length $\pi/2$.
The simplex ${\sigma}_T*{\sigma}_{\alpha}$ has the structure of a spherical join. In other words, the length of an edge connecting a vertex in ${\sigma}_T$ to one in ${\sigma}_{\alpha}$ is $\pi/2$.
\[r:mfld\] The simplicial complex $L'$ is the nerve of a Coxeter group $(W',S')$ which contains $W$ as a special subgroup. Namely, $S'$ is the disjoint union, $S \cup ({\mathcal {P}}-{\mathcal {S}})$. Two generators of $S$ are related as before. If $U$, $V\in {\mathcal {P}}-{\mathcal {S}}$, then put $m(U,V):=2$ whenever $U\subset V$ or $V\subset U$ and $m(U,V):=\infty$, otherwise. Similarly, if $s\in S$ and $U\in{\mathcal {P}}-{\mathcal {S}}$, then $m(s,U)=m(U,s)=2$ when $s\in U$ and it is $=\infty$, otherwise. Since $L'$ is a triangulation of $S^{n-1}$, with $n=|S|$, this shows that any $n$ generator Coxeter group is a special subgroup of a Coxeter group which acts cocompactly on a contractible $n$-manifold (such a Coxeter group is said to be *type* $HM^n$ in [@dbook]).
**$\boldsymbol{{\operatorname{CAT}}(0)}$ and $\boldsymbol{{\operatorname{CAT}}(1)}$ metrics**. Gromov [@gromov] proved that a piecewise Euclidean metric on a polyhedron $Y$ is locally ${\operatorname{CAT}}(0)$ if and only if the link in $Y$ of each cell is ${\operatorname{CAT}}(1)$. In his proof that $L(W,S)$ was ${\operatorname{CAT}}(1)$, Moussong [@moussong] gave a criteria for certain piecewise spherical structures on simplicial complexes to be ${\operatorname{CAT}}(1)$. We recall his criteria below. A spherical simplex *has size $\ge \pi/2$* if each of its edges has length $\ge \pi/2$. A spherical simplex ${\sigma}\subset {{\mathbb S}}^k\subset {{\mathbf R}}^{k+1}$ with vertex set $\{v_0,\dots,v_{k+1}\}$ and edge lengths $l_{ij}:=\cos^{{-1}}(v_i\cdot v_j)$ is determined up to isometry by the $(l_{ij})$. Conversely, a symmetric $(k+1)\times (k+1)$ matrix $(l_{ij})$ of real numbers in \[$0,\pi)$ can be realized as the set of edge lengths of a spherical simplex if and only if the matrix $(\cos l_{ij})$ is positive definite, cf. [@dbook Lemma I.5.1, p. 513]. Suppose $N$ is a simplicial complex with a piecewise spherical structure (i.e., each simplex has the structure of a spherical simplex). $N$ *has size $\ge \pi/2$* if each of its simplices does. $N$ is a *metric flag complex* if it satisfies the following condition: given any collection of vertices $\{v_0,\dots, v_k\}$ which are pairwise connected by edges, then $\{v_0,\dots, v_k\}$ is the vertex set of a simplex in $N$ if and only if the matrix of edge lengths $(l_{ij})$ can be realized as the matrix of edge lengths of an actual spherical simplex. (In other words, if and only if $(\cos l_{ij})$ is positive definite.) Moussong’s Lemma is the following.
\[l:mlemma\] . Suppose a piecewise spherical simplicial complex $N$ has size $\ge \pi/2$. Then $N$ is ${\operatorname{CAT}}(1)$ if and only if it is a metric flag complex.
The next result follows immediately from our previous description of the piecewise spherical complex $L'$.
\[l:L’\] $L'$ has size $\ge \pi/2$ and is a metric flag complex.
\[c:L’\] $L'$ is ${\operatorname{CAT}}(1)$. Moreover, $L$ is a totally geodesic subcomplex.
The last sentence follows from the fact that $L$ is a full subcomplex.
Since the link of any simplex in a metric flag complex of size $\ge\pi/2$ has the same properties (cf. [@moussong Lemma 8.3] or [@dbook Lemma I.5.11]), the link of any simplex in $L'$ is also ${\operatorname{CAT}}(1)$. Since the link in ${\Delta}^f\cap {\Delta}_V$ of any cell of the form $B_{T,V}\times [0,\infty)^{\alpha}$ can be identified with the link of the corresponding simplex ${\sigma}_T*{\sigma}_{\alpha}$ in $L'$, it follows that the link of each cell in ${\Delta}^f$ is ${\operatorname{CAT}}(1)$.
\[c:cat0\] The piecewise Euclidean metric on ${\mathcal {U}}(W,{\Delta}^f)$ is ${\operatorname{CAT}}(0)$.
The union of $W_T$-translates of a Coxeter block $B_T$ in ${\mathcal {U}}(W,{\Delta}^f)$ is a Coxeter cell $P_T$ and the complete inverse image of $B_T$ in ${\mathcal {U}}(W,{\Delta}^f)$ is a disjoint union of copies of $P_T$. Hence, ${\mathcal {U}}(W,{\Delta}^f)$ has a cell structure in which the cells are either translates of Coxeter cells of the form $P_T$ or translates of cells of the form $P_T\times [0,\infty)^{\alpha}$ for some ${\alpha}\in {\operatorname{Flag}}(({\mathcal {P}}-{\mathcal {S}})_{>T})$. In either case the link of such a cell in ${\mathcal {U}}(W,{\Delta}^f)$ is identified with the link of the corresponding cell in ${\Delta}^f$. By Corollary \[c:L’\], ${\mathcal {U}}(W,{\Delta}^f)$ is locally ${\operatorname{CAT}}(0)$. A space is ${\operatorname{CAT}}(0)$ if and only if it is locally ${\operatorname{CAT}}(0)$ and simply connected (cf. [@gromov p. 119] or [@bh Ch. II.4]). ${\mathcal {U}}(W,{\Delta}^f)$ is contractible (hence, simply connected), since it is homotopy equivalent to ${\mathcal {U}}(W,K)$. Therefore, it is ${\operatorname{CAT}}(0)$.
We also want to consider links of cells of the form $B_{T,V}$ or $B_{T,V} \times [0,\infty)^{\alpha}$ in various cell complexes. (Here $B_{T,V}$ is a reflecting face of $B_T$.)
\[l:links\] Suppose $c$ is a cell in ${\Delta}^f$ of the form $c=B_{T,V}$ or $c=B_{T,V}\times [0,\infty)^{\alpha}$ for some $V\subseteq T\in {\mathcal {S}}_{>\emptyset}$ and ${\alpha}\in{\operatorname{Flag}}(({\mathcal {P}}-{\mathcal {S}})_{>T})$. Let $d=B_T$ or $B_T\times [0,\infty)^{\alpha}$ be the corresponding larger cell in ${\Delta}^f$ and let ${\sigma}_d$ be the corresponding simplex in $L'$. Then $$\begin{aligned}
{\operatorname{Lk}}(c,{\Delta}^f)&={\operatorname{Lk}}({\sigma}_d,L')*{\tau}(V)\label{e:L1}\\
{\operatorname{Lk}}(c, {\mathcal {U}}(W,{\Delta}^f)&={\operatorname{Lk}}({\sigma}_d,L')* {{\mathbb S}}^V\label{e:L2}\end{aligned}$$ Here ${{\mathbb S}}^V$ is the unit sphere in the canonical representation of $W_V$ on ${{\mathbf R}}^V$ and ${\tau}(V)\subset {{\mathbb S}}^V$ is the fundamental simplex.
Moreover, suppose that $\Phi$ is a building of type $(W,S)$, that $R$ is the spherical residue of type $V$ containing the base chamber and that ${{\mathbb S}}(R):={\mathcal {U}}(R,{\tau}(V))$ is the spherical realization of $R$. Then $$\label{e:L3}
{\operatorname{Lk}}(c, {\mathcal {U}}(\Phi,{\Delta}^f)={\operatorname{Lk}}({\sigma}_d,L')* {{\mathbb S}}(R).$$
Let ${\Delta}^f_V$ denote the face of ${\Delta}^f$ fixed by $W_V$ (cf. . Then $$\begin{aligned}
{\operatorname{Lk}}(c,{\Delta}^f)&= {\operatorname{Lk}}(c,{\Delta}^f_V)* {\operatorname{Lk}}({\Delta}^f_V,{\Delta}^f)\\
&={\operatorname{Lk}}(d,{\Delta}^f)* {\tau}(V)\\
&={\operatorname{Lk}}({\sigma}_d,L')*{\tau}(V)\end{aligned}$$ and similarly for formulas and .
We can now prove the main result of this section.
Any spherical building, such as ${{\mathbb S}}(R)$, is ${\operatorname{CAT}}(1)$ (e.g., see [@dbuild]) and the spherical join of two ${\operatorname{CAT}}(1)$-spaces is ${\operatorname{CAT}}(1)$ (e.g., see [@bh]). So, the theorem follows from . Alternatively, it can be proved from Corollary \[c:cat0\] by using the argument in [@dbuild §11].
**A variation**. In Section \[s:morse\] we will need the following modification of the previous construction. Given a subset $U\subseteq S$, we will define a new piecewise Euclidean metric on ${\Delta}^f-{\Delta}^U$ and then show that it induces ${\operatorname{CAT}}(0)$ metrics on ${\mathcal {U}}(W_{S-U}, {\Delta}^f-{\Delta}^U)$ and ${\mathcal {U}}(R, {\Delta}^f-{\Delta}^U)$ for any $(S-U)$-residue $R$ of $\Phi$.
For each spherical subset $T$, let $C^*_T\subset {{\mathbf R}}^T$ be the simplicial cone determined by the bounding hyperplanes of $B_T$ passing through the vertex $x_0$. (In other words, $C^*_T$ is the dual cone to the fundamental simplicial cone.) Let ${\mathcal {S}}_{S-U}:={\mathcal {S}}(W_{S-U}, S-U)$ be the poset of spherical subsets of $S-U$ and let $L_{S-U}:=L(W_{S-U},S-U)$ be the nerve of $W_{S-U}$. ${\Delta}^f-{\Delta}^U$ has a cell structure with cells of the following two types:
$B_{T,V}$, where $T\in {\mathcal {S}}_{S-U}$ and $V\subseteq T$,
$B_T\times [0,\infty)^{\alpha}$, where ${\alpha}\in {\mathcal {P}}-{\mathcal {S}}_{S-U}$ and $T,V$ are as above.
As before, each such cell is given the natural product metric. In effect we are putting ${\Upsilon}({\Delta}) \cup {\Delta}^U$ at infinity.
The piecewise Euclidean metric on ${\Delta}^f-{\Delta}^U$ induces one ${\mathcal {U}}(R,{\Delta}^f-{\Delta}^U)$. Let us describe the link, $L'_{S-U}$, of the central vertex in the new metric on ${\Delta}^f-{\Delta}^U$. The vertex set of $L'_{S-U}$ is $(S-U)\cup ({\mathcal {P}}-{\mathcal {S}}_{S-U})$. As before, there are three types of simplices:
simplices ${\sigma}_T$ in $L_{S-U}$ corresponding to spherical subsets $T\in {\mathcal {S}}_{S-U}$,
simplices ${\sigma}_{\alpha}$ in $b\partial{\Delta}$ ($=b\partial{\Delta^{op}}$) corresponding to flags ${\alpha}\in {\operatorname{Flag}}({\mathcal {P}}-{\mathcal {S}}_{S-U})$,
joins ${\sigma}_T*{\sigma}_{\alpha}$, with $T\in {\mathcal {S}}_{S-U}$, ${\alpha}\in {\operatorname{Flag}}({\mathcal {P}}-{\mathcal {S}}_{S-U})$ and $T<\min {\alpha}$.
and $L'_{S-U}$ can be identified with a subdivision of ${\partial\Delta^{op}}$. Moreover, just as before, $L'_{S-U}$ has size $\ge\pi/2$ and is a metric flag complex; hence, it is ${\operatorname{CAT}}(1)$. This proves the following.
\[t:cat2\] Suppose $U\subseteq S$ and $R$ is any $(S-U)$-residue in $\Phi$. Then the piecewise Euclidean metric on ${\mathcal {U}}(R,{\Delta}^f-{\Delta}^U)$, defined above, is ${\operatorname{CAT}}(0)$.
Metric spheres in $\boldsymbol{{\mathcal {U}}(\Phi, {\Delta}^f)}$ {#s:morse}
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An $n$-dimensional cell complex $X$ is $CM$ (for “Cohen-Macaulay”) if ${\widetilde}{H}^*(X)$ is concentrated in degree $n$ and is a free abelian group in that degree. Similarly, an $n$-dimensional, noncompact, contractible space $X$ is $SI$ (for “spherical at infinity”) if $H^*_c(X)$ is concentrated in degree $n$ and is a free abelian group in that degree. We want to prove that ${\mathcal {U}}(\Phi, {\Delta}^f)$ is $SI$. This is a consequence of Theorems \[t:bbm\] and \[t:pi2\] below.
Suppose $N$ is a ${\operatorname{CAT}}(1)$, piecewise spherical polyhedron and that $p\in N$. Let $B(p,\pi/2)\subseteq N$ denote the open ball of radius $\pi/2$ centered at $p$. Define a space $PN_p$, called $N$ *punctured at* $p$, by $PN_p:=N-B(p,\pi/2)$. We are interested in this concept when $N={\operatorname{Lk}}(c)$, the link of a cell $c$ in some ${\operatorname{CAT}}(0)$ complex $X$. In this case we will write ${\operatorname{PLk}}_p(c)$ for $PN_p$ and call it the *punctured link* of $c$ at $p$.
\[t:bbm\] . Let $X$ be a ${\operatorname{CAT}}(0)$, piecewise Euclidean cell complex (with finitely many shapes of cells). If for each cell $c$ in $X$ and for each $p\in {\operatorname{Lk}}(c)$, the spaces ${\operatorname{Lk}}(c)$ and ${\operatorname{PLk}}_p (c)$ are $(n-\dim c)$-acyclic, then $X$ is $n$-acyclic at infinity. In particular, if $X$ satisfies this condition and is $n$-dimensional, then it is $SI$.
In [@bbm] the hypothesis of the above theorem is that $X$ is the universal cover of a finite, nonpositively curved complex; however, the proof clearly works with a weaker hypothesis such as finitely many shapes of cells (which holds in our case). The proof of the theorem uses Morse theory for polyhedral complexes. Roughly, it goes as follows. Let ${\rho}:X\to {{\mathbf R}}$ be the distance from some base point $x_0$, i.e., ${\rho}(x):=d(x,x_0)$. The spheres $S(r)$ of radius $r$ centered at $x_0\in X$ are the level sets of ${\rho}$. Call a point $x$ a *critical point* (of ${\rho}$) if it is the closest point to $x_0$ in some closed cell $c$. It follows from the ${\operatorname{CAT}}(0)$ hypothesis that the critical points are isolated. If, for sufficiently small ${\varepsilon}$, there is no critical point in the annular region between $S(r+{\varepsilon})$ and $S(r)$, then $S(r+{\varepsilon})$ and $S(r)$ are homeomorphic. On the other hand, the effect of crossing a critical point $x\in S(r)$ is to remove a contractible neighborhood of $x$ in $S(r)$ and replace it by the punctured link ${\operatorname{PLk}}(x)_p$ where $p$ is the direction at $x$ of the geodesic from $x$ to $x_0$. (If $x$ lies in the relative interior of a $k$-dimensional cell $c$, then ${\operatorname{Lk}}(x)\cong {{\mathbb S}}^{k-1} * {\operatorname{Lk}}(c)$.) So, the effect on the homotopy type of the level sets is to replace a contractible neighborhood by a copy of a suspension of a punctured link. It follows that, under the hypotheses of Theorem \[t:bbm\], each metric sphere is $CM$. Since $X$ is ${\operatorname{CAT}}(0)$, metric balls are contractible and since $H^*_c(X)=\varinjlim H^*(B(r),S(r))$ is concentrated in the top degree, $X$ is $SI$.
\[t:pi2\] . Suppose $R$ is a spherical building of type $(W_T,T)$ and that ${{\mathbb S}}(R)$ ($:={\mathcal {U}}(R, {\tau}(T))$) is its spherical realization. Then for any $p\in {{\mathbb S}}(R)$, the space $P{{\mathbb S}}(R)_p$ is $CM$.
An immediate corollary to Theorems \[t:bbm\] and \[t:pi2\] is the following.
\[t:SI\] With notation as in Section \[s:cat0\] and above, given any building $\Phi$, its realization ${\mathcal {U}}(\Phi,{\Delta}^f)$ is $SI$.
As a corollary to Theorem \[t:cat2\] we get the following relative version of Theorem \[t:bbm\].
\[t:SI2\] Suppose $U\subseteq S$ and $R$ is any $(S-U)$-residue in $\Phi$. Then ${\mathcal {U}}(R,{\Delta}^f-{\Delta}^U)$ is $SI$.
\[c:SI3\] Suppose $U\subseteq S$. Then ${\mathcal {U}}(\Phi,{\Delta}^f-{\Delta}^U)$ is $SI$.
${\mathcal {U}}(\Phi,{\Delta}^f-{\Delta}^U)$ is the disjoint union of the spaces ${\mathcal {U}}(R,{\Delta}^f-{\Delta}^U)$ where $R$ ranges over the $(S-U)$-residues. By the previous theorem, the compactly supported cohomology of each such component is free abelian and concentrated in the top degree.
Cohomology with finite support and compact support {#s:support}
==================================================
Given a CW complex $Y$, $C^*_{{{\mathrm {fin}}}}(Y)$ denotes the complex of finitely supported cellular cochains on $Y$ and $H^*_{{\mathrm {fin}}}(Y)$ its cohomology. When $Y$ is only required to be a topological space, $H^*_c(Y)$ denotes its compactly supported singular cohomology, i.e., $$H^*_c(Y):=\varinjlim H^*(Y, Y-C),$$ where the direct limit is over all compact subsets $C\subseteq Y$. If $Y$ is a locally finite CW complex, $H^*_{{\mathrm {fin}}}(Y)\cong H^*_c(Y)$.
Suppose $Z$, $Z'$ are mirrored spaces over $S$. A map $F:Z\to Z'$ is *mirrored* if $F(Z_T)\subseteq Z'_T$ for all $T\subseteq S$; it is a *mirrored homotopy equivalence* if $F\vert_{Z_{T}}:Z_T\to Z'_T$ is a homotopy equivalence for all $T$ (including $T=\emptyset$).
Given a mirrored space $X$, let us say that ${\Upsilon}(X)$ is *collared in $X$* if there is an increasing family ${\mathcal {N}}=\{N_{\varepsilon}\}_{{\varepsilon}\in (0,a]}$ of open neighborhoods of ${\Upsilon}(X)$ (“increasing means that $N_{\varepsilon}\subseteq N_{{\varepsilon}'}$ whenever ${\varepsilon}<{\varepsilon}'$) such that the following two properties hold:
$
\bigcap_{{\varepsilon}\in (0,a]} N_{\varepsilon}= {\Upsilon}(X)
$ and
For each ${\varepsilon}>0$, the inclusion ${\Upsilon}(X)\hookrightarrow {\overline}{N}_{\varepsilon}$ is a mirrored homotopy equivalence.
For example, if $X$ is the simplex ${\Delta}$, then ${\Upsilon}({\Delta})$ is collared in ${\Delta}$. (Proof: ${\Upsilon}({\Delta})$ is a union of closed faces and we can take $N_{\varepsilon}$ to be its ${\varepsilon}$-neighborhood in ${\Delta}$.) More generally, if $X$ is any finite CW complex, then ${\Upsilon}(X)$ is collared in $X$.
Property (ii) implies that ${\mathcal {U}}(\Phi, {\Upsilon}(X))\hookrightarrow {\mathcal {U}}(\Phi, N_{\varepsilon})$ is a homotopy equivalence. To simplify notation, in what follows we often write ${\mathcal {U}}_X$ instead of ${\mathcal {U}}(\Phi,X)$.
\[l:cfin\] Suppose $X$ is a finite, mirrored CW complex . Then $$H^*_c({\mathcal {U}}_{X^f})=H^*_{{\mathrm {fin}}}({\mathcal {U}}_X,{\mathcal {U}}_{{\Upsilon}(X)}).$$
Since $X$ is a finite complex, $
H^*_{{\mathrm {fin}}}({\mathcal {U}}_X,{\mathcal {U}}_{{\Upsilon}(X)})=H^*_c({\mathcal {U}}_X,{\mathcal {U}}_{{\Upsilon}(X)}).
$ Since ${\Upsilon}(X)$ is collared in $X$, we have a family ${\mathcal {N}}=\{N_{\varepsilon}\}$ of open neighborhoods. For a given $N= N_{\varepsilon}\in {\mathcal {N}}$, let $\partial N$ denote the boundary of ${\overline}{N}$. Since $N$ is mirrored homotopy equivalent to ${\Upsilon}(X)$, $$H^*_{{\mathrm {fin}}}({\mathcal {U}}_X,{\mathcal {U}}_{{\Upsilon}(X)})\cong H^*_c({\mathcal {U}}_X,{\mathcal {U}}_{{\overline}{N}})\cong
H^*_c({\mathcal {U}}_{(X-N)},{\mathcal {U}}_{\partial N}),$$ where the second isomorphism is an excision. As ${\varepsilon}\to 0$, $X-N_{\varepsilon}\to X^f$, so any compact subset $C$ of ${\mathcal {U}}_{X^f}$ lies within some ${\mathcal {U}}_{X-N_{\varepsilon}}$. Hence, $$H^*_c({\mathcal {U}}_{X^f})=\varinjlim H^*({\mathcal {U}}_{X^f},{\mathcal {U}}_{X^f}-C)=H^*_c({\mathcal {U}}_{X-N},{\mathcal {U}}_{\partial N}).$$ Combining these equations, we get the result.
${\Delta}^f$ is a thickened version of $K$ and ${\mathcal {U}}(\Phi,{\Delta}^f)$ is a thickened version of ${\mathcal {U}}(\Phi,K)$. Hence, ${\mathcal {U}}(\Phi,{\Delta}^f)$ and ${\mathcal {U}}(\Phi,K)$ are homotopy equivalent. On the other hand, their compactly supported cohomology groups can be completely different. For example, suppose $W$ is the free product of three copies of ${{\mathbf Z}}/2$. Then ${\Delta}^f$ is a triangle with its vertices deleted while $K$ is a tripod. ${\mathcal {U}}(W,{\Delta}^f)$ can be identified with the hyperbolic plane while ${\mathcal {U}}(W,K)$ is the regular trivalent tree. (This is the familiar picture of the congruence $2$ subgroup of $PSL(2,{{\mathbf Z}})$ acting on the hyperbolic plane.) The compactly supported cohomology of ${\mathcal {U}}(W,{\Delta}^f)$ is that of the plane (i.e., it is concentrated in degree $2$ and is isomorphic to ${{\mathbf Z}}$ in that degree), while the compactly supported cohomology of ${\mathcal {U}}(W,K)$ is that of a tree (i.e., it is concentrated in degree $1$ and is a countably generated, free abelian group in that degree).
Cohomology with coefficients in ${\mathcal {I}}(A)$ {#s:A}
===================================================
Let $A$ ($=A(\Phi)$) be the free abelian group of finitely supported, ${{\mathbf Z}}$-valued functions on $\Phi$. For each subset $T$ of $S$, define a ${{\mathbf Z}}$-submodule of $A$: $$A^T:=\{f\in A\mid f \text{ is constant on each residue of type $T$}\}.$$ Note that $A^\emptyset =A$. Also note that $A^T=0$ whenever $T$ is not spherical.
We have $A^U\subset A^T$ when $T\subset U$. Let $A^{{>T}}$ be the ${{\mathbf Z}}$-submodule of $A^T$ spanned by the $A^U$ with $T$ a proper subset of $U$. Put $$D^T:=A^T/A^{{>T}}.$$
When $\Phi=W$, $A$ is the group ring ${{\mathbf Z}}W$ and $A^T$ consists of elements in ${{\mathbf Z}}W$ which are constant on each left coset $wW_T$. Let us assume $T$ is spherical (otherwise $A^T=0$). Let ${\operatorname{In}}(w):=\{s\in S\mid l(ws)<l(w)\}$ be the set of letters with which a reduced expression for $w$ can end. Since $T$ is spherical, each left coset of $W_T$ has a unique representative $w$ which has a reduced expression ending in the longest element of $W_T$; hence, this representative has $T\subseteq {\operatorname{In}}(w)$. As a basis for $D^T$, we can take images of the elements in $A^T$ corresponding to cosets which have longest representatives with $T={\operatorname{In}}(w)$.
\[d:cochains\] The abelian group $A$ and the family of subgroups $\{A^T\}_{T\subseteq S}$ define a “coefficient system” ${\mathcal {I}}(A)$ on $X$ so that the *$i$-cochains with coefficients in ${\mathcal {I}}(A)$* are given by $${\mathcal {C}}^i(X; {\mathcal {I}}(A)):= \prod_{c\in X^{(i)}} A^{S(c)}$$ where $X^{(i)}$ denotes the set of $i$-cells in $X$.
We continue to write ${\mathcal {U}}_X$ for ${\mathcal {U}}(\Phi, X)$. Let ${\mathcal {U}}^{(i)}$ denote the set of $i$-cells in ${\mathcal {U}}_X$. Given a chamber ${\varphi}\in \Phi$ and an $i$-cell $c$ in $X$, let ${\varphi}\cdot c$ denote the corresponding $i$-cell in ${\mathcal {U}}_X$. Given a finitely supported function ${\alpha}:{\mathcal {U}}^{(i)}\to {{\mathbf Z}}$, and an $i$-cell $c\in X^{(i)}$ with $S(c)$ spherical, we get an element $f\in A^{S(c)}$ defined by $
f({\varphi}):={\alpha}({\varphi}\cdot c).
$ Of course, when $S(c)$ is not spherical, $A^{S(c)}=0$. So, for any finite, mirrored CW complex $X$, this establishes an isomorphism $$\label{e:cochainiso}
{\mathcal {C}}^i(X;{\mathcal {I}}(A))\cong C^i_{{\mathrm {fin}}}({\mathcal {U}}_X, {\mathcal {U}}_{{\Upsilon}(X)}).$$ In other words, the isomorphism is given by identifying a finitely supported function on the inverse image in ${\mathcal {U}}^{(i)}$ of a cell $c\in X^{(i)}$ with a function on $\Phi$ (i.e., with an element of $A$) which is constant on $S(c)$-residues containing the cells ${\varphi}\cdot c$.
The coboundary maps in ${\mathcal {C}}^*(X;{\mathcal {I}}(A))$ are defined by using these isomorphisms to transport the coboundary maps on finitely supported cochains to ${\mathcal {C}}^*(X;{\mathcal {I}}(A))$. This means that the coboundary maps in ${\mathcal {C}}^*(X;{\mathcal {I}}(A))$ are defined by combining the usual coboundary maps in $C^*(X)$ with the inclusions $A^U\hookrightarrow A^T$ for $U\supset T$. So, we have the following.
\[l:finite\] For $X$ a finite, mirrored CW complex, $${\mathcal {H}}^*(X;{\mathcal {I}}(A))=H^*_{{\mathrm {fin}}}({\mathcal {U}}_X,{\mathcal {U}}_{{\Upsilon}(X)})=H^*_c({\mathcal {U}}_{X^f}).$$
The second equation is from Lemma \[l:cfin\].
If $\Phi =W$, then $A={{\mathbf Z}}W$. ${\mathcal {C}}^*(X;{\mathcal {I}}({{\mathbf Z}}W))$ can be interpreted as the equivariant cochains on ${\mathcal {U}}(W,X)$ with coefficients in ${{\mathbf Z}}W$. When $X$ is a finite complex, Lemma \[l:finite\] asserts $${\mathcal {C}}^*(X;{\mathcal {I}}({{\mathbf Z}}W )=C^*_{{\mathrm {fin}}}({\mathcal {U}}(W,X),{\mathcal {U}}(W,{\Upsilon}(X))).$$ The corresponding cohomology groups were studied in [@ddjo2; @d98]. In particular, when $X=K$, these cohomology groups are isomorphic to $H^*(W;{{\mathbf Z}}W)$.
In what follows the coefficient system is usually ${\mathcal {I}}(A)$ and we shall generally omit it from our notation, writing ${\mathcal {H}}(\ )$ and ${\mathcal {C}}(\ )$ instead of ${\mathcal {H}}(\ ;{\mathcal {I}}(A))$ and ${\mathcal {C}}(\ ;{\mathcal {I}}(A))$. (In other words, the coefficients ${\mathcal {I}}(A)$ are implicit when we use the calligraphic ${\mathcal {C}}$ or ${\mathcal {H}}$ notation.) As usual, ${\Delta}$ is the simplex of dimension $n=|S|-1$ with its codimension one faces indexed by $S$.
\[t:gd\] ${\mathcal {H}}^*({\Delta})$ is concentrated in degree $n$ and is free abelian; moreover, ${\mathcal {H}}^n({\Delta})=D^\emptyset$. More generally, for any subset $U$ of $S$, ${\mathcal {H}}^*({\Delta}, {\Delta}^U)$ is concentrated in degree $n$ and is free abelian and $${\mathcal {H}}^n({\Delta},{\Delta}^U)=A\big/\sum_{s\in S-U} A^s$$
By Lemma \[l:finite\], ${\mathcal {H}}^*({\Delta})=H^*_c({\mathcal {U}}_{{\Delta}^f})$ and by Theorem \[t:SI\], the right hand side is concentrated in degree $n$ and is free abelian. The cochain complex looks like $$\cdots \to {\mathcal {C}}^{n-1}({\Delta}) \to {\mathcal {C}}^n({\Delta})\to 0,$$ where ${\mathcal {C}}^n({\Delta})=A$ and ${\mathcal {C}}^{n-1}({\Delta})=\bigoplus A^s$. It follows that cohomology in degree $n$ is the quotient $$A\big/\sum_{s\in S} A^s = D^\emptyset.$$ Similarly, ${\mathcal {H}}^*({\Delta},{\Delta}^U)=H^*_c({\mathcal {U}}_{{\Delta}^f-{\Delta}^U})$, and by Corollary \[c:SI3\] the right hand side is concentrated in degree $n$ and is free abelian. Since ${\mathcal {C}}^n({\Delta},{\Delta}^U)=A$ and ${\mathcal {C}}^{n-1}({\Delta}, {\Delta}^U)=\bigoplus_{s\notin U} A^s$, we get the final formula in the theorem.
\[c:Dfree\] $D^\emptyset$ is free abelian.
\[r:duality\] We saw in Section \[s:geom\] that whenever $W$ is infinite, ${\mathcal {U}}(W, {\Delta}^f)$ is homeomorphic to Euclidean space ${{\mathbf R}}^n$. Hence, the compactly supported cohomology of ${\mathcal {U}}(W, {\Delta}^f)$ is that of $H^*_c({{\mathbf R}}^n)$. Similarly, ${\mathcal {U}}(R,{\Delta}^f-({\Delta}^f)^U)$ is homeomorphic to ${{\mathbf R}}^n$, for each $(S-U)$-residue $R$.
\[r:pd\] ${\mathcal {U}}(W,{\Delta}^f)$ is a thickened version of ${\mathcal {U}}(W,K)$. In the proof of Corollary \[c:cat0\] we explained the cellulation of ${\mathcal {U}}(W,K)$ by “Coxeter cells.” The corresponding cellular chain complex, $C_*({\mathcal {U}}(W,K))$, has the form $${{\mathbf Z}}W\ \longleftarrow \bigoplus_{s\in S}H^s\ \longleftarrow
\bigoplus_{T\in {\mathcal {S}}^{(2)}}H^T \longleftarrow\cdots$$ where $H^T$ is the representation induced from the sign representation of $W_T$ and where ${\mathcal {S}}^{(k)}$ is the set of spherical subsets with $k$ elements (cf. [@ddjo §8]). The cochain complex ${\mathcal {C}}^*({\Delta})$ for $n-2\le *\le n$, looks like $$\cdots{\smash{\mathop{\longrightarrow}\limits^{}}}\ \bigoplus_{T\in {\mathcal {S}}^{(2)}}A^T{\smash{\mathop{\longrightarrow}\limits^{}}}\ \bigoplus_{s\in S}A^s{\smash{\mathop{\longrightarrow}\limits^{}}}\ {{\mathbf Z}}W .$$ So, $C_*({\mathcal {U}}(W,K))$ and $C^{n-*}_c({\mathcal {U}}(W,{\Delta}^f))$ are Poincaré dual.
The Decomposition Theorem {#s:decomp}
=========================
We want to prove a version of Theorem \[t:gd\] for lower dimensional spherical faces of ${\Delta}$. Suppose $T$ is a spherical subset of $S$. Put ${\sigma}={\Delta}_T$ and $m=n-|T|$. Let $U$ be an arbitrary subset of $S-T$. We continue the policy of omitting the coefficient system ${\mathcal {I}}(A)$ from our notation.
\[p:sigma\] Each of the following cohomology groups is concentrated in the top degree and is a free abelian group in that degree: $${\mathcal {H}}^*({\sigma},{\sigma}^U),\ {\mathcal {H}}^*({\sigma}^U,\partial ({\sigma}^U)), \text{ and }\ {\mathcal {H}}^*({\sigma}^U)$$ (The top degrees are $m$, $m-1$, and $m-1$, respectively.) Moreover, $$\begin{aligned}
{\mathcal {H}}^m({\sigma},{\sigma}^U)&=A^T\big/\sum_{s\in (S-T)-U} A^{T\cup\{s\}},\label{e:1} \\
{\mathcal {H}}^{m-1}({\sigma}^U,\partial ({\sigma}^U))&=\sum_{s\in U} A^{T\cup\{s\}},\label{e:2}\\
{\mathcal {H}}^{m-1}({\sigma}^U)&=\sum_{s\in U}A^{T\cup\{s\}} \big/ \sum_{\substack{s\in U\\t\in (S-T)-U}} A^{T\cup
\{s,t\}},\label{e:3}
$$
We first prove ${\mathcal {H}}^*({\sigma},{\sigma}^{U})$ is concentrated in degree $m$ and is free abelian. The proof is by induction on the number of elements in $T$. It holds for $|T|=0$ by Theorem \[t:gd\]. Suppose $T=T'\cup \{s\}$, $U'=U\cup \{s\}$ and ${\tau}={\Delta}_{T'}$. The exact sequence of the triple $({\tau},{\tau}^{U'},{\tau}^{U})$ gives $$\cdots \to{\mathcal {H}}^{*-1}({\sigma},{\sigma}^U) \to {\mathcal {H}}^*({\tau}, {\tau}^{U'})\to {\mathcal {H}}^*({\tau},{\tau}^{U}) \to\cdots$$ (This uses the excision, ${\mathcal {H}}^{*-1}({\tau}^{U'},{\tau}^{U})\cong {\mathcal {H}}^{*-1}({\sigma},{\sigma}^U)$.) By inductive hypothesis, the last two terms are free abelian and concentrated in degree $m+1$. Hence, ${\mathcal {H}}^*({\sigma},{\sigma}^{U})$ is concentrated in degree $m$. It is free abelian since it injects into a free abelian group.
That the other cohomology groups are free abelian and are concentrated in the top degree follows from various exact sequences. For example, for ${\mathcal {H}}^*({\sigma}^U,\partial ({\sigma}^U))$, consider the sequence of the triple $({\sigma},\partial {\sigma}, {\sigma}^{S-U})$: $$\label{e:triple1}
\to{\mathcal {H}}^{*-1}({\sigma}^U,\partial({\sigma}^U))\to {\mathcal {H}}^*({\sigma},\partial{\sigma})\to {\mathcal {H}}^*({\sigma},{\sigma}^{S-U})\to$$ where we have used the excision ${\mathcal {H}}^{*-1}(\partial{\sigma}, {\sigma}^{S-U})\cong {\mathcal {H}}^{*-1}({\sigma}^U,\partial({\sigma}^U))$ to identify the first term. The second and third terms are free abelian and concentrated in degree $m$; hence, the first term is free abelian and concentrated in degree $m-1$. For ${\mathcal {H}}^*({\sigma}^U)$, we have the exact sequence of the pair $({\sigma},{\sigma}^U)$: $$\label{e:pair1}
\to {\mathcal {H}}^{*-1}({\sigma}^U)\to {\mathcal {H}}^*({\sigma},{\sigma}^U)\to {\mathcal {H}}^*({\sigma})\to$$ where the second and third terms are free abelian and concentrated in degree $m$. Hence, ${\mathcal {H}}^*({\sigma}^U)$ is free abelian and is concentrated in degree $m-1$.
It remains to verify formulas , and . We have ${\mathcal {C}}^m({\sigma},{\sigma}^U)=A^T$ and ${\mathcal {C}}^{m-1}({\sigma},{\sigma}^U)=\bigoplus_{s\notin U} A^{T\cup\{s\}}$, so $${\mathcal {H}}^m({\sigma},{\sigma}^U)=A^T \big{/} \sum_{s\notin U} A^{T\cup\{s\}},$$ proving . (In particular, ${\mathcal {H}}^m({\sigma})=D^T$ and ${\mathcal {H}}^n({\Delta})=D^\emptyset$.) In the exact sequence , we have ${\mathcal {H}}^m({\sigma},\partial {\sigma})=A^T$ and, by , ${\mathcal {H}}^m({\sigma},{\sigma}^{S-U})=A^T/\sum_{s\in U} A^{T\cup\{s\}}$; hence, . Using to calculate the second and third terms of , we get $${\mathcal {H}}^{m-1}({\sigma}^U)=\sum_{s\in S-T}A^{T\cup\{s\}} \big/ \sum_{s\in (S-T)-U} A^{T\cup\{s\}}$$ and this can be rewritten as .
By Proposition \[p:sigma\], $D^T$ is the free abelian group ${\mathcal {H}}^m({\Delta}_T)$. So, for each $T\in {\mathcal {S}}$, we can choose a splitting ${\iota}_T:D^T\to A^T$ of the projection map $A^T\to D^T$.
\[d:ha\] Let ${{\hat{A}}}^T:={\iota}_T(D^T)$. (It is a direct summand of the free abelian group $A^T$.)
\[p:sums\] $$\begin{aligned}
{\mathcal {H}}^m({\sigma},{\sigma}^U)&=\bigoplus_{\substack{V\supseteq T\\V-T\subseteq U}} {{\hat{A}}}^V \label{e:sum1}\\
{\mathcal {H}}^{m-1}({\sigma}^U, \partial({\sigma}^U))&=\bigoplus_{\substack{V\supset T\\(V-T)\cap U\neq\emptyset}} {{\hat{A}}}^V\label{e:sum2}\\
{\mathcal {H}}^{m-1}({\sigma}^U)&=\bigoplus_{\substack{V\supset T\\V-T\subseteq U}} {{\hat{A}}}^V \label{e:sum3}
$$
Assume by induction that through hold for $\dim {\sigma}=m-1$ and assume as well that they hold when $\dim {\sigma}=m$ and $U$ is replaced by $U'$ with $|U'|<|U|$. Write $U=\{s\}\cup U'$, for some $s\in U$. Consider the exact sequence of the triple $({\sigma}^U,{\sigma}_s\cup \partial ({\sigma}^U),\partial ({\sigma}^U))$: $$\label{e:triple2}
0\to {\mathcal {H}}^{m-1}({\sigma}^{U'},\partial({\sigma}^{U'}))\to {\mathcal {H}}^{m-1}({\sigma}^U,\partial ({\sigma}^U))\to {\mathcal {H}}^{m-1}({\sigma}_s, ({\sigma}_s)^{S-U'})\to 0,$$ where we have used the excisions ${\mathcal {H}}^{*}({\sigma}^U,{\sigma}_s\cup \partial ({\sigma}^U))={\mathcal {H}}^{*}({\sigma}^{U'},\partial({\sigma}^{U'}))$ and ${\mathcal {H}}^{*}({\sigma}_s\cup \partial ({\sigma}^U),\partial ({\sigma}^U))={\mathcal {H}}^{*}({\sigma}_s,({\sigma}_s)^{S-U'})$ to rewrite the first and third terms. By induction, $$\begin{aligned}
{\mathcal {H}}^{m-1}({\sigma}^{U'},\partial({\sigma}^{U'}))
&=\bigoplus_{\substack{V\supset T\\V\cap U'\neq\emptyset}} {{\hat{A}}}^V ,\\
{\mathcal {H}}^{m-1}({\sigma}_s,({\sigma}_s)^{S-U'}) &=
\bigoplus_{\substack{V\supseteq T\cup\{s\}\\V\subseteq T\cup\{s\}\cup U'}} {{\hat{A}}}^V \end{aligned}$$ Substituting these into the last two terms of , we get $${\mathcal {H}}^{m-1}({\sigma}^U,\partial({\sigma}^U))=\bigoplus_{\substack{V\supset T\\V\cap U\neq \emptyset}} {{\hat{A}}}^V ,$$ which is . Next consider the Mayer-Vietoris sequence of ${\sigma}^U={\sigma}^{U'}\cup {\sigma}_s$: $$\label{e:MV1}
0\to {\mathcal {H}}^{m-2}(({\sigma}_s)^{U'})\to {\mathcal {H}}^{m-1}({\sigma}^U)\to {\mathcal {H}}^{m-1}({\sigma}^{U'})\oplus {\mathcal {H}}^{m-1}({\sigma}_s)\to 0.$$ By induction, $${\mathcal {H}}^{m-2}(({\sigma}_s)^{U'})=\bigoplus_{\substack{V\supset T\cup\{s\}\\V-(T\cup\{s\})\subseteq U'}} {{\hat{A}}}^V
\quad\text{and}\quad
{\mathcal {H}}^{m-1}({\sigma}^{U'})=\bigoplus_{\substack{V\supset T\\V-T\subseteq U'}} {{\hat{A}}}^V$$ and ${\mathcal {H}}^{m-1}({\sigma}_s))={{\hat{A}}}^{T\cup\{s\}}$. Substituting these into we get $${\mathcal {H}}^{m-1}({\sigma}^U)=\bigoplus_{\substack{V\supset T\\V-T\subseteq U}} {{\hat{A}}}^V,$$ which is . Sequence is $$0\to {\mathcal {H}}^{m-1}({\sigma}^U)\to {\mathcal {H}}^m({\sigma}, {\sigma}^U)\to {\mathcal {H}}^m({\sigma})\to 0.$$ Substituting for the first term and ${{\hat{A}}}^T$ for the third, we get formula for the middle term.
We have ${\mathcal {H}}^m({\sigma},\partial {\sigma})=A^T$. Hence, in the special case $U=S-T$ formulas and give the following theorem (cf. [@s], [@ddjo Thm. 9.11], [@ddjo2 Cor. 3.3]).
\[t:decomp\] For each subset $T$ of $S$ $$A^T=\bigoplus_{V\supseteq T} {{\hat{A}}}^V.$$
For any ${{\mathbf Z}}$-submodule $B\subset A$, put $B^T:=A^T\cap B$. Suppose we have a direct sum decomposition of ${{\mathbf Z}}$-modules, $A=B\oplus C$, so that for each $T\subseteq S$, $A^T=B^T\oplus C^T$. As explained in [@ddjo2 §2] this leads to a decomposition of coefficient systems ${\mathcal {I}}(A)={\mathcal {I}}(B)\oplus {\mathcal {I}}(C)$ so that for any mirrored CW complex $X$ we have a decomposition of cochain complexes ${\mathcal {C}}^*(X;{\mathcal {I}}(A))={\mathcal {C}}^*(X;{\mathcal {I}}(B))\oplus {\mathcal {C}}^*(X;{\mathcal {I}}(C))$. Since $$({{\hat{A}}}^V)^T=
\begin{cases}
{{\hat{A}}}^V &\text{if $V\supseteq T$,}\\
0 &\text{otherwise,}
\end{cases}$$ the formula in the Decomposition Theorem satisfies $(A^\emptyset)^T=\bigoplus_{V\supseteq T}({{\hat{A}}}^V)^T$, for all $T\subseteq S$. So, we get a decomposition of coefficient systems ${\mathcal {I}}(A)=\bigoplus {{\hat{A}}}^V$ and a corresponding decomposition of cochain complexes: $$\label{e:decomp}
{\mathcal {C}}^*(X;{\mathcal {I}}(A))=\bigoplus_{V} {\mathcal {C}}^*(X;{\mathcal {I}}({{\hat{A}}}^V))$$
Cohomology of buildings {#s:cohomology}
=======================
Just as in [@ddjo2 Thm. 3.5], the Decomposition Theorem (Theorem \[t:decomp\]) implies the following.
\[t:main\] Suppose $X$ is a finite, mirrored CW complex. Then $$H^*_c({\mathcal {U}}(\Phi,X^f))\cong \bigoplus_{T\in {\mathcal {S}}} H^*(X,X^{S-T})\otimes {{\widehat A}}^T.$$
By Lemma \[l:finite\], $H^*_c({\mathcal {U}}(\Phi,X^f))$ is the cohomology of ${\mathcal {C}}^*(X;{\mathcal {I}}(A))$. Formula gives a decomposition of cochain complexes: $${\mathcal {C}}^*(X;{\mathcal {I}}(A))=\bigoplus_{T\in {\mathcal {S}}} {\mathcal {C}}^*(X;{\mathcal {I}}({{\hat{A}}}^T))$$ We have $${\mathcal {C}}^k(X;{\mathcal {I}}({{\hat{A}}}^T))=\prod_{c\in X^{(k)}}{{\hat{A}}}^T\cap A^{S(c)}
=\prod_{\substack{c\in X^{(k)}\\ c\not\subseteq X^{S-T}}} {{\hat{A}}}^T.$$ So, an element of ${\mathcal {C}}^k(X;{\mathcal {I}}({{\hat{A}}}^T))$ is just an ordinary ${{\hat{A}}}^T$-valued cochain on $X$ which vanishes on $X^{S-T}$, i.e., $${\mathcal {C}}^*(X;{\mathcal {I}}({{\hat{A}}}^T))=C^*(X,X^{S-T})\otimes {{\hat{A}}}^T; \quad \text{so,}$$ $${\mathcal {C}}^*(X;{\mathcal {I}}(A))=\bigoplus_{T\in {\mathcal {S}}} C^*(X,X^{S-T})\otimes {{\hat{A}}}^T.$$ Taking cohomology, we get the result.
The most important special case of the previous theorem is the following.
\[c:main\] $$H^*_c({\mathcal {U}}(\Phi, K))\cong \bigoplus_{T\in {\mathcal {S}}} H^*(K,K^{S-T})\otimes {{\hat{A}}}^T.$$
The $G$-module structure on cohomology
======================================
Assume $X$ has a $W$-finite mirror structure (i.e., $X=X^f$). Suppose $G$ is a group of automorphisms of $\Phi$. Then $A$, $A^T$, $A^{{>T}}$ and $D^T$ are naturally right $G$-modules and so is the cochain complex $C^*_c({\mathcal {U}}(\Phi,X))$ as is its cohomology. The discussion in Section \[s:cohomology\] is well adapted to studying the $G$-module structure of $H^*_c({\mathcal {U}}(\Phi,X))$. As in [@ddjo2], we should not expect a direct sum splitting of $G$-modules analogous nonequivariant splitting of Theorem \[t:main\]; rather there should be a filtration of cohomology by $G$-submodules with associated graded terms similar to those in the direct sum. We show below that this is the case.
For each nonnegative integer $p$ define a $G$-submodule $F_p$ of $A$ by $$F_p:=\sum_{|T|\le p} A^T.$$ This gives a decreasing filtration: $$\label{e:f}
A=F_0\supset\cdots \supset F_p\supset F_{p+1}\cdots .$$ As in [@ddjo2], it follows from the Decomposition Theorem that the associated graded terms are $$F_p/F_{p+1}=\bigoplus_{|T|=p} D^T.$$
As in Section \[s:A\], we get a coefficient system ${\mathcal {I}}(F_p)$, a cochain complex ${\mathcal {C}}^*(X;{\mathcal {I}}(F_p))$ and corresponding cohomology groups ${\mathcal {H}}^*(X;{\mathcal {I}}(F_p))$. The filtration leads to a filtration of ${\mathcal {H}}^*(X;{\mathcal {I}}(A))$ ($=H^*_c({\mathcal {U}}(\Phi,X))$) by right $G$-modules,
$$\label{e:f2}
{\mathcal {H}}^*(X;{\mathcal {I}}(A))\supset \cdots {\mathcal {H}}^*(X;{\mathcal {I}}(F_p))\supset {\mathcal {H}}^*(X;{\mathcal {I}}(F_{p+1}))\cdots .$$
The Decomposition Theorem implies that $$0{\smash{\mathop{\longrightarrow}\limits^{}}}\,{\mathcal {H}}^*(X;{\mathcal {I}}(F_{p+1})){\smash{\mathop{\longrightarrow}\limits^{}}}\,{\mathcal {H}}^*(X;{\mathcal {I}}(F_p)){\smash{\mathop{\longrightarrow}\limits^{}}}\,
H^*\left(\frac{{\mathcal {C}}^*(X;{\mathcal {I}}(F_p))}{{\mathcal {C}}^*(X;{\mathcal {I}}(F_{p+1}))}\right){\smash{\mathop{\longrightarrow}\limits^{}}} \,0$$ is short exact. From this we deduce the following.
\[t:Gmodule\] Suppose $G$ is a group of automorphisms of $\Phi$. Then the filtration of $H^*_c({\mathcal {U}}(\Phi,X))$ by right $G$-modules has associated graded term, $${\mathcal {H}}^*(X;F_p)/{\mathcal {H}}^*(X;F_{p+1})\cong \bigoplus _{T\in {\mathcal {S}}^{(p)}} H^*(X,X^{S-T})\otimes D^T.$$
**Cocompact $\boldsymbol{G}$-action**. In this final paragraph we assume that there are only finitely many $G$-orbits on $\Phi$ and that $X$ is a finite complex. These hypotheses imply that the quotient space ${\mathcal {U}}(\Phi,X)/G$ is compact and since $X$ is a finite complex, each of the cohomology groups $H^*(X)$ is finitely generated. As a corollary to Theorem \[t:Gmodule\], we have the following.
\[c:fingen\] With the above hypotheses, $H^*_c({\mathcal {U}}(\Phi,X))$ is a finitely generated $G$-module.
The assumption that $G$ has only finitely many orbits of chambers implies that each $A^T$ is finitely generated $G$-module; hence, so is each $D^T$.
Suppose $G$ is chamber transitive, i.e., suppose $\Phi$ consists of a single $G$-orbit. Choose a base chamber ${\varphi}_0$, let $B$ be its stabilizer and for each $T\in {\mathcal {S}}$, let $G_T$ be the stabilzer of the $T$-residue containing ${\varphi}_0$. Then $A$ is the $G$-module ${{\mathbf Z}}(G/B)$ of finitely supported functions on $G/B$ and $A^T$ can be identified with ${{\mathbf Z}}(G/G_T)$. For each $U\supset T$, we have a natural inclusion ${{\mathbf Z}}(G/G_U)\to {{\mathbf Z}}(G/G_T)$ induced by the projection $G/G_T\to G/G_U$ and $D^T$ is the $G$-module formed by dividing out the sum of the images of ${{\mathbf Z}}(G/G_U)$ in ${{\mathbf Z}}(G/G_T)$ for all $U\supset T$.
Here are some further corollaries to Corollary \[c:main\] and Theorem \[t:Gmodule\].
\[c:dim\] Suppose ${\Gamma}\subseteq G$ is a cocompact lattice in $G$. Then $H^*({\Gamma};{{\mathbf Z}}{\Gamma})$ has, in filtration degree $p$, associated graded ${\Gamma}$-module $$\bigoplus _{T\in {\mathcal {S}}^{(p)}} H^*(K,K^{S-T})\otimes D^T.$$ In particular, when ${\Gamma}$ is torsion-free, its cohomological dimension is given by $${\operatorname{cd}}({\Gamma})=\max\{k\mid H^k(K,K^{S-T})\neq 0, \text{for some $T\in {\mathcal {S}}$}\}.$$
A torsion-free group ${\Gamma}$ is an *$n$-dimensional duality group* if $H^*({\Gamma};{{\mathbf Z}}{\Gamma})$ is free abelian and concentrated in dimension $n$. Following [@dm Definition 6.1], we say that the nerve of a Coxeter system has *punctured homology concentrated in dimension $n$* if for all $T\in {\mathcal {S}}$, ${\widetilde}{H}^*(K^{S-T})$ is free abelian and concentrated in dimension $n$. Corollary \[c:main\] gives us a (correct) proof of the following result, stated in [@dm Theorem 6.3].
\[c:duality\] Suppose ${\Gamma}\subseteq G$ is a torsion-free, cocompact lattice in $G$. Then the following are equivalent.
${\Gamma}$ is an $n$-dimensional duality group.
$W$ is an $n$-dimensional virtual duality group.
The nerve of $(W,S)$ has punctured homology concentrated in dimension $n-1$.
[88]{} P. Abramenko and K. Brown, *Approaches to Buildings*, Springer, New York, 2008.
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[, ]{}*Erratum to “The topology at infinity of Coxeter groups and buildings”*, Comment. Math. Helv. **82** (2007), 235–236.
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[^1]: The first author was partially supported by NSF grant DMS 0706259.
[^2]: The second author was partially supported by KBN grant N201 012 32/0718
[^3]: The third author also was partially supported by NSF grant DMS 0706259.
[^4]: The fourth author was partially supported by a Richard King Mellon research grant.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present the transmission spectrum of HAT-P-12b through a joint analysis of data obtained from the *Hubble Space Telescope* Space Telescope Imaging Spectrograph (STIS) and Wide Field Camera 3 (WFC3) and Spitzer, covering the wavelength range 0.3–5.0 $\mu$m. We detect a muted water vapor absorption feature at 1.4 $\mu$m attenuated by clouds, as well as a Rayleigh scattering slope in the optical indicative of small particles. We interpret the transmission spectrum using both the state-of-the-art atmospheric retrieval code SCARLET and the aerosol microphysics model CARMA. These models indicate that the atmosphere of HAT-P-12b is consistent with a broad range of metallicities between several tens to a few hundred times solar, a roughly solar C/O ratio, and moderately efficient vertical mixing. Cloud models that include condensate clouds do not readily generate the sub-micron particles necessary to reproduce the observed Rayleigh scattering slope, while models that incorporate photochemical hazes composed of soot or tholins are able to match the full transmission spectrum. From a complementary analysis of secondary eclipses by Spitzer, we obtain measured depths of $0.042\%\pm0.013\%$ and $0.045\%\pm0.018\%$ at 3.6 and 4.5 $\mu$m, respectively, which are consistent with a blackbody temperature of $890^{+60}_{-70}$ K and indicate efficient day–night heat recirculation. HAT-P-12b joins the growing number of well-characterized warm planets that underscore the importance of clouds and hazes in our understanding of exoplanet atmospheres.'
author:
- 'Ian Wong, Bj[" o]{}rn Benneke, Peter Gao, Heather A. Knutson, Yayaati Chachan, Gregory W. Henry, Drake Deming, Tiffany Kataria, Graham K. H. Lee, Nikolay Nikolov, David K. Sing, Gilda E. Ballester, Nathaniel J. Baskin, Hannah R. Wakeford, and Michael H. Williamson'
title: 'Optical to near-infrared transmission spectrum of the warm sub-Saturn HAT-P-12'
---
Introduction {#sec:intro}
============
Over the past decade, major improvements in telescope capabilities and advancements in observation and analysis methods have enabled the intensive atmospheric characterization of an increasingly diverse population of exoplanets. Transmission spectroscopy has emerged as a powerful tool in studying the chemical composition of exoplanet atmospheres. By measuring the variations in transit depth as a function of wavelength, this technique directly probes the optically thin portion of the atmosphere along the day–night terminator of these tidally locked planets and is sensitive to various atmospheric components through their absorption signatures in the transmission spectrum.
Transmission spectroscopy has hitherto successfully detected a broad range of chemical species in exoplanet atmospheres [e.g., @madhusudhan2014; @demingseager]. Deriving estimates of the relative abundances of multiple atomic and molecular species yields constraints on more fundamental properties, such as disk-averaged metallicity, C/O ratio, and the temperature–pressure profile along the terminator. However, a large number of recent transmission spectroscopy studies have been confounded by the presence of clouds and hazes [e.g., @singstis]. Even trace amounts of cloud and haze particles can significantly increase the scattering opacity [e.g., @fortney2005; @pont2008], resulting in attenuation of absorption features in the transmission spectrum and reducing the ability to place meaningful constraints on key atmospheric properties, as, for example, in the cases of GJ 436b [@knutson2014] and GJ 1214b [@kreidberg]. Looking ahead to the future, a fuller understanding of the conditions under which clouds and hazes occur will be crucial in the selection of optimal targets with clear atmospheres for intensive observations using the limited time allocation available on next-generation telescopes, such as the James Webb Space Telescope [JWST; e.g., @bean2018; @schlawin2018].
As part of the continuing effort to better understand clouds in exoplanetary atmospheres, we examine in detail the transmission spectrum of HAT-P-12b. This planet is classified as a low-density sub-Saturn with a radius of 0.96 $R_{\mathrm{J}}$ and a mass of 0.21 $M_{\mathrm{J}}$, orbiting a K dwarf (0.73 $M_{\astrosun}$, 0.70 $R_{\astrosun}$, $T_{\mathrm{eff}}=4650$ K, \[Fe/H\]$ =-0.29$) with a period of 3.21 days [@hartman]. Recent measurements of the Rossiter–McLaughlin effect for this system revealed a highly misaligned orbit [$\lambda=-54^{+41}_{-13}$; @mancini]. A previous analysis showed that the near-infrared transmission spectrum was flat, indicating the presence of high-altitude aerosols [@line]. This planet has also been observed in transit at visible wavelengths both from the ground [@mallonn; @alexoudi] and from space [@singstis; @alexoudi], with the latter studies revealing a slope in the optical transmission spectrum indicative of Rayleigh scattering by fine aerosol particles in the upper atmosphere [@barstow].
The basic mechanisms for forming clouds and hazes on both solar system bodies and exoplanets involve either (a) condensation, in which a gaseous species changes phase to a liquid or solid upon becoming locally supersaturated either homogeneously or heterogeneously with the aid of condensation nuclei (e.g., @ackermanmarley2001; @lodders2002; @visscher2006; @helling; @visscher2010; @charnay2018; @gao; @lee2018; see also the reviews by @marley2013 and @helling2018), or (b) photochemistry, induced by ultraviolet irradiation of the planet from the stellar host leading to the destruction of gaseous molecules and polymerization of the photolysis products into fine aerosol particles in the upper atmosphere [e.g., @zahnle2009; @line2011; @moses2011; @venot2015; @lavvas2017; @horst; @kawashima2018; @adams2019]. Much of the detailed microphysics driving aerosol particle formation remains poorly understood, and models typically approximate haze formation using assumed chemical pathways, compositions, and formation efficiencies.
In addition to atmospheric metallicity, surface gravity, and the local temperature, secondary phenomena such as advection of material from the nightside to the dayside [see, for example, the reviews by @showman2010; @hengshowman2015], the interplay between the degree of vertical mixing and particle size [e.g., @parmentier; @zhangshowman], and gravitational settling of particles [e.g., @lunine1989; @marley1999; @ackermanmarley2001; @woitke2003; @helling; @gao2018] can affect the cloud properties at the terminator. The importance of clouds in interpreting and understanding exoplanet atmospheres has led to the development of increasingly complex cloud models incorporating many of the aforementioned chemical and physical processes [e.g., @helling2016; @lee2016; @lavvas2017; @ohno2017; @gao; @kawashima2018; @lines1; @lines2; @helling2019a; @helling2019b; @powell2019; @woitke2019].
Analyzing a planet’s emission spectrum using secondary eclipse observations offers a complementary view of the atmosphere that may peer through the clouds that often obscure transmission spectra. This technique measures the outgoing flux from the planet’s dayside hemisphere and provides independent constraints on dayside temperature, atmospheric metallicity, and cloud coverage. Both numerical models [e.g., @parmentier; @lineparmentier; @lines2; @powell2018; @caldas2019; @helling2019a; @helling2019b] and phase curve observations [e.g., @demory; @shporerhu] suggest that clouds in exoplanet atmospheres are often localized to particular regions in the atmosphere, with incomplete coverage of the dayside hemisphere. In these instances, the planet’s dayside emission spectrum is dominated by flux from the hotter, brighter cloud-free regions of the atmosphere and can yield additional insights into the atmosphere of planets with cloudy terminators, as in the case of HD 189733b [@crouzet2014] and GJ 436b [@morley2017].
In this paper, we analyze new near-infrared transit observations of HAT-P-12b obtained using the Wide Field Camera 3 (WFC3) instrument on the Hubble Space Telescope (HST) in spatial scan mode. Combining these data with previously published transit observations from the HST Space Telescope Imaging Spectrograph (STIS), WFC3, and Spitzer, we derive the transmission spectrum spanning the wavelength range 0.3–5.0 $\mu$m. Our analysis is supplemented by secondary eclipse measurements at 3.6 and 4.5 $\mu$m. In interpreting the results from our analysis, we utilize both atmospheric retrievals and predictions from microphysical cloud models to constrain the atmospheric properties of this planet.
Observations and Data Reduction {#sec:obs}
===============================
In this paper, we analyze a total of eight transit and four secondary eclipse observations obtained using three different instruments that span the wavelength range $0.3-5.0$ $\mu$m. This section provides a general overview of the methodology we use to extract light curves from the raw data for each of the three instruments.
HST WFC3 {#subsec:wfc3}
--------
As part of the Cycle 23 HST program GO-14260 (PI: D. Deming), we obtained time-resolved spectroscopic observations during two transits of HAT-P-12b on UT 2015 December 12 and 2016 August 31 using the G141 grism (1.0–1.7 $\mu$m) on WFC3. Each visit was comprised of five 96 minute HST orbits, with 45 minute gaps in data collection due to Earth occultations. The observations were carried out in spatial scan mode, with the star scanned perpendicularly to the dispersion direction across the detector at a rate of $0\overset{''}{.}03$ s$^{-1}$. In addition, at the start of the first orbit of each visit, we obtained an undispersed direct image of the star using the F139M grism for use in wavelength calibration. Each of the 74 spectra has a total exposure time of 112 s and extends roughly 30 pixels in the spatial direction. With the SPARS25 NSAMP=7 readout mode, each image file consists of seven nondestructive reads of the entire 266$\times$266 pixel subarray. These two scan mode visits have been previously analyzed in @tsiaras2018.
We also include in our analysis an older stare mode transit observation from UT 2011 May 29 (GO-12181; PI: D. Deming) that was analyzed previously in @line and @singstis. This visit consisted of 112 12.8 s exposures over the course of four orbits. Each orbit necessitated five buffer dumps, resulting in $\sim$9 minute gaps interrupting the data collection. There are 16 nondestructive reads of the 512$\times$512 pixel subarray in each image file. When reducing the images, we treat the stare mode data in the same way as the spatial scan mode observations. The observation details for the three WFC3 visits are summarized in Table \[hubbleobservations\].
Starting with the dark- and bias-corrected *\*ima.fits* files produced by the standard WFC3 calibration pipeline, CALWFC3, we proceed with the data reduction using the Python 2–based Exoplanet Transits, Eclipses, and Phase Curves (ExoTEP) pipeline developed by B. Benneke and I. Wong [see also @benneke2017; @benneke2019]. To achieve maximal background subtraction in the extracted spectra, we follow a standard procedure for WFC3 spatial scan image processing [e.g., @deming2013; @kreidberg; @evans2016]: we construct subexposures by subtracting consecutive nondestructive reads and coadd all of the background-subtracted subexposures together to form the full background-corrected data frame.
The spatial extent of each subexposure is determined by calculating the median flux profile for the difference image along the scan direction, i.e., $y$-direction, and locating the pixels where the flux falls to 20% of the maximum value. To form the subexposure, we take the data that lie between these two rows, with an extra buffer of 15 pixels on the top and bottom, while setting all other pixel values to zero. This method ensures that all of the stellar flux collected by the instrument between nondestructive reads is extracted and that the size of the extraction region for a given subexposure (e.g., difference of the third and second reads) remains largely consistent across each visit. The final results are not sensitive to the particular choice of buffer size between 10 and 20 pixels. The background level of each subexposure is set as the median of a 50 column wide region situated sufficiently far from the spectral trace and avoiding the edges of the subarray.
Due to the particular geometry of the WFC3 instrument, the first-order spectrum of the G141 grism is not perfectly parallel to the detector rows. Also, there are significant variations in the length of the spectrum in the dispersion direction across the spatial scan, which results in the wavelength associated with a particular detector column varying from the top to the bottom. Lastly, imperfections in the pointing resets between each exposure lead to small horizontal shifts in the spectra across each visit [e.g., @deming2013; @knutson2014; @kreidberg]. Therefore, the shape of the spectrum on the detector is trapezoidal and slightly inclined relative to the subarray rows. Some previous analyses of WFC3 data have addressed this issue either by aligning the rows of the spectrum via interpolation [@kreidberg] or by deriving correction factors for the published wavelength calibration coefficients [@wilkins].
In the ExoTEP pipeline, we follow the methodology described in detail in @tsiaras and compute the exact wavelength solution across the entire subarray for each exposure. In short, we first determine the position of the star along the $x$-axis of the detector for each exposure by taking the position of the star in the direct undispersed image, adjusting for differences in reference pixel location and subarray size between the direct and spatial scan images, and calculating the horizontal offset of each spectrum relative to the first spectroscopic exposure. The offsets are calculated by computing the centroid of each exposure and measuring the horizontal shift relative to the first exposure of the visit.
Next, assuming that the spatial scan shifts the star position perfectly vertically across the detector, we determine the trace position and the wavelength solution along the trace using the calibration coefficients included in the configuration file *WFC3.IR.G141.V2.5.conf* [@kuntschner] for a range of stellar $y$ positions. After a 2D cubic spline interpolation, we can now calculate the wavelength at every location on the subarray for each exposure. We also utilize this wavelength solution to apply a wavelength-dependent flat-field correction, using the cubic flat-field coefficients listed in the calibration file *WFC3.IR.G141.flat.2.fits* [@kuntschner2; @tsiaras].
The last step in the ExoTEP data reduction process before light-curve extraction is cosmic-ray correction. For each exposure, we calculate the normalized row-added flux template. Next, we flag outliers using $5\sigma$ moving median filters of 10 pixels in width in both the $x$ and $y$ directions. Flagged pixel values are replaced by the value in the template corresponding to its $y$ position, appropriately scaled to match the total flux in its column. The particular parameters of the median filters are manually adjusted by inspecting the final corrected images and checking that all visible outliers have been removed. Due to the narrow vertical spatial profile of the trace in the stare mode images, we only apply the bad pixel correction in the horizontal direction for that visit.
To construct the spectroscopic light curves, we define a 20 nm wavelength grid from 1.10 to 1.66 $\mu$m and determine the spatial boundaries of the patch corresponding to each wavelength bin on the subarray using the previously derived wavelength solution. We calculate the flux within each patch by adding the pixel counts for all pixels that are fully within the patch and then computing the additional contribution from the partial pixels that are intersected by the patch boundaries. For each partial pixel, we integrate a local 2D cubic polynomial interpolation function over the subpixel regions that lie inside and outside of the given patch in order to compute the fraction of the total pixel count lying within the patch. This process ensures that the total flux is conserved and yields a modest reduction in the photometric scatter relative to more conventional extraction methods, which typically smooth the data in the dispersion direction prior to light-curve extraction [e.g., @deming2013; @knutson2014; @tsiaras].
The time stamp for each data point is set to the mid-exposure time. To produce the broadband HST WFC3 light curve (i.e., white light curve), we simply sum the flux from the full set of individual spectroscopic light curves.
[ l m[0.1cm]{} c m[0.1cm]{} c c c ]{} Data Set & & UT Start Date & & $n_{\mathrm{exp}}$^a^ & $t_{\mathrm{int}}$ (s)^b^ & Mode\
WFC3 G141 & & & & & &\
Visit 1 & & 2011 May 29 & & 112 & 12.8 & Stare\
Visit 2 & & 2015 Dec 12 & & 74 & 112 & Scan\
Visit 3 & & 2016 Aug 31 & & 74 & 112 & Scan\
STIS G430L & & & & & &\
Visit 1 & & 2012 Apr 11 & & 34 & 280 & $\dots$\
Visit 2 & & 2012 Apr 30 & & 34 & 280 & $\dots$\
STIS G750L & & & & & &\
Visit 1 & & 2013 Feb 4 & & 34 & 280 & $\dots$\
[**Notes.**]{}
^a^Total number of exposures
^b^Total integration time per exposure
[ l m[0.1cm]{} c m[0.1cm]{} c c c c m[0.1cm]{} c c c ]{} Dataset & & UT Start Date & & $n_{\mathrm{img}}$^a^ & $t_{\mathrm{int}}$ (s)^b^ & $t_{\mathrm{trim}}$ (minutes)^c^ & $r_{0}$^c^ && $r_{1}$^c^ & $r_{\mathrm{phot}}$^c^ & Binning^d^\
3.6 $\mu$m & && && & & && & &\
Transit & &2013 Mar 8 && 8064 & 1.92 & 0 & 2.5 & & $\dots$ & 1.5 & 64\
Eclipse 1 && 2010 Mar 16 && 2097 & 10.4 & 60 & 3.0 && 2.0 & $\sqrt{\widetilde{\beta}}\times 1.3$& 16\
Eclipse 2 && 2014 Apr 15 && 9024 & 1.92 & 30 & 4.0 && 1.0 & $\sqrt{\widetilde{\beta}}\times 1.5$& 16\
4.5 $\mu$m & &&& & & && & & &\
Transit && 2013 Mar 11 && 8064 & 1.92 & 0 & 3.0 & & $\dots$ & 1.6 & 128\
Eclipse 1 && 2010 Mar 26 && 2097 & 10.4 & 45 & 3.5 && 2.5 & $\sqrt{\widetilde{\beta}}+0.7$ & 32\
Eclipse 2 && 2014 May 8 && 9024 &1.92 & 60 & 2.5 && $\dots$ & 2.4 & 128\
[**Notes.**]{}
^a^Total number of images.
^b^Total integration time per image.
^c^Here $t_{\mathrm{trim}}$ is the amount of time trimmed from the start of each time series prior to fitting, $r_{0}$ is the radius of the aperture used to determine the star centroid position, and $r_{1}$ is the radius of the aperture used to compute the noise pixel parameter $\widetilde{\beta}$. The $r_{\mathrm{phot}}$ column denotes how the photometric extraction aperture is defined. All radii are given in units of pixels. When using a fixed aperture, the noise pixel parameter is not needed, so $r_{1}$ is undefined. See text for more details.
^d^Number of data points placed in each bin when binning the photometric series prior to fitting.
HST STIS {#subsec:stis}
--------
We observed three transits of HAT-P-12b with the HST STIS instrument as part of the program GO-12473 (PI: D. Sing). Observations of two transits were carried out using the G430L grating (290–570 nm) on UT 2012 April 11 and 30; the third transit was observed using the G750L grating (550–1020 nm) on UT 2013 February 4. The two gratings used have resolutions of $R=530-1040$ (5.5 and 9.8 Å per 2 pixel resolution element for the G430L and G750L gratings, respectively). Each visit contains a total of 34 science exposures across four HST orbits, with the third orbit occurring during mid-transit. To reduce overhead, data were read out from a 1024$\times$128 subarray with a per-exposure integration time of 280 s. The observational details for the three STIS visits are listed in Table \[hubbleobservations\]. This set of observations has been analyzed in two previous independent studies: @singstis and @alexoudi.
The raw images are flat-fielded using the latest version of CALSTIS. The subsequent data reduction is completed using the ExoTEP pipeline. We remove outlier pixel values in the time series by first computing the median image across each visit and then replacing all pixel values in the individual exposure frames varying by more than $4\sigma$ with the corresponding value in the median image. We apply the wavelength solution provided in the *\*sx1.fits* calibrated files and extract the column-added 1D spectra, choosing the aperture width and whether to subtract the background so as to minimize the scatter in the residuals from the transit light-curve fit [e.g., @deming2013]. In our analysis of the two G430L observations, we utilize 9 and 7 pixel wide apertures, respectively, removing the background for the first visit only; in the case of the G750L transit, we find that extracting spectra from a 7 pixel wide aperture after background subtraction results in the minimum scatter.
Data collected using the G750L grism suffer from a fringing effect, which manifests itself as an interference pattern superposed on the 1D spectrum and is especially apparent at wavelengths longer than 700 nm. Following the methods outlined in previously published analyses of data from this program [e.g., @nikolov; @nikolov2; @singstis], we defringe our data using a fringe flat frame obtained at the end of the G750L science observations.
Lastly, we correct for subpixel wavelength shifts in the dispersion direction across each visit by fitting for the horizontal offsets and amplitude scaling factors that align all extracted spectra with the first one. The normalized broadband light curve is simply the time series of the optimized amplitude scaling factors. To generate the spectroscopic light curves, we collect the flux within 200 and 100 pixel bins for the G430L and G750L observations, respectively. The wavelength bounds corresponding to the 200 pixel bins for the two G430L transit observations differ by less than the characteristic wavelength resolution element (0.55 nm). For the G750L dataset, we also include two narrow wavelength bins centered around the sodium and potassium absorption lines (588.7–591.2 and 770.3–772.3 nm respectively).
Spitzer IRAC {#subsec:spitzer}
------------
Two transits of HAT-P-12b were observed in the 3.6 and 4.5 $\mu$m broadband channels of the Infrared Array Camera (IRAC) on the Spitzer Space Telescope (Program ID 90092; PI: J.-M. D[' e]{}sert). The observations took place on UT 2013 March 8 and 11 and were carried out in subarray mode, which produces 32$\times$32 pixel ($39''\times39''$) images centered on the stellar target. Each transit observation is comprised of 8064 images with a per-exposure effective integration time of 1.92 s.
A set of two secondary eclipse observations, one in each of the two postcryogenic IRAC channels, was obtained on UT 2010 March 16 and 26 (Program ID 60021; PI: H. Knutson). These data consist of 2097 images per passband obtained in full array mode at a resolution of 256$\times$256 pixels ($5\overset{''}{.}2\times5\overset{''}{.}2$) with an effective exposure time of 10.4 s per image. Peak-up pointing was utilized, which entails an initial 30 minute observation prior to the start of the science observation to allow for the stabilization of the telescope pointing. These eclipses were previously analyzed in @todorov. A second set of hitherto unpublished secondary eclipse observations, including one in each channel, was obtained on UT 2014 April 15 and May 8 (Program ID 10054; PI: H. Knutson). These observations were taken in subarray mode with peak-up pointing and contain 9024 images with effective exposure times of 1.92 s.
We extract photometry following the techniques described in detail in previous analyses of postcryogenic Spitzer data [e.g., @knutson2012; @lewis; @todorov; @wong2; @wong3]. Starting with the dark-subtracted, flat-fielded, linearized, and flux-calibrated images produced by the standard IRAC pipeline, we calculate the sky background via a Gaussian fit to the distribution of pixel values, excluding pixels near the star and its diffraction spikes, as well as the problematic top (32nd) row, which has flux values that are systematically lower than the other rows. We also iteratively trim outlier pixel values on a pixel-by-pixel basis using a $3\sigma$ moving median filter across the adjacent 64 images in the time series.
The position of the star on the detector is determined using the flux-weighted centroiding method [e.g., @knutson2012]. The width of the star’s point response function (PRF; i.e., the convolution of the star’s point-spread function and the detector response function) is estimated by computing the noise pixel parameter $\widetilde{\beta}$ [see @lewis for a full discussion]. The stellar position and PRF width are calculated using circular apertures of radius $r_{0}$ and $r_{1}$, respectively, which we vary in 0.5 pixel steps to produce different versions of the extracted photometry. The photometric series can be extracted using both fixed and time-varying circular apertures, where in the case of time-varying apertures, the radii are related to the square root of the noise pixel parameter by either a constant scaling factor or a constant shift [e.g., @wong2; @wong3].
Prior to fitting (see Section \[sec:analysis\]), we can bin the photometric series into various intervals equal to powers of two (i.e., 1, 2, 4, 8, etc. points). To aid in the removal of instrumental systematics, we also experiment with trimming the first 15, 30, 45, or 60 minutes of data from the time series. Before fitting each photometric series with our transit/eclipse light-curve model, we apply an iterative moving median filter of 64 data points in width to remove points with measured fluxes, $x$ or $y$ star centroid positions, or $\sqrt{\widetilde{\beta}}$ values that vary by more than 3$\sigma$ from the corresponding median values. For all Spitzer datasets, the number of removed points is less than 5% of the total number of data points, and slightly altering the width of the median filter does not significantly affect the number of removed points.
For each $Spitzer$ transit or secondary eclipse observation, we determine the optimal aperture and photometric parameters by fitting the various photometric series with the model light curve and selecting the version that minimizes the scatter in the resultant residuals, binned in 5 minute intervals [@wong2; @wong3]. The optimal values are listed in Table \[spitzerphotometry\].
Photometric monitoring for stellar activity {#subsec:monitoring}
-------------------------------------------
High levels of chromospheric activity, which can lead to significant photometric variability and incur wavelength-dependent biases in the measured transmission spectrum [e.g., @rackham], can be displayed by K dwarfs such as HAT-P-12. In particular, the presence of unocculted starspots can impart slope changes to the shape of the transmission spectrum in the optical, affecting the interpretation of the planet’s atmospheric properties.
To characterize the level of stellar activity on HAT-P-12, we obtained Cousins $R$-band photometry of the star using the the Tennessee State University Celestron 14 inch (C14) Automated Imaging Telescope (AIT) located at Fairborn Observatory, Arizona. Differential magnitudes of HAT-P-12 were calculated relative to the mean brightnesses of five constant comparison stars from five to ten co-added consecutive exposures. Details of our observing, data reduction, and analysis procedures with the AIT are described in @sing2015.
A total of 237 successful nightly observations were collected across two observing seasons (season 1: 2011 September 20–2012 June 22; season 2: 2012 September 24–2013 June 26). The individual observations are plotted in Figure \[photovar\]. The seasonal means in differential magnitude are $-0.2689\pm0.0004$ and $-0.2708\pm0.0004$, with corresponding single observation standard deviations of 0.0046 and 0.0042, respectively. These scatter values are comparable to the approximate limit of the measurement precision. This indicates that HAT-P-12 does not show any significant variability.
When performing a periodogram analysis of individual seasonal datasets, we do not retrieve any frequencies that produce amplitudes larger than the seasonal standard deviations. In particular, we do not detect a variability signal with a period near the estimated rotational period of the star [$P_{\mathrm{rot}} \sim44$ days; @mancini]. Such a periodicity in the photometry would be indicative of rotational modulation of weak features on the stellar surface. We therefore conclude that HAT-P-12 is a very quiescent host star.
This conclusion is consistent with the findings from high-resolution spectroscopy of the host star during the initial discovery and characterization of the system [@hartman], which did not detect significant levels of variability suggestive of large starspots across the stellar surface. Analyses of stellar spectra from the Keck High Resolution Echelle Spectrometer (HIRES) and more recently from the High Accuracy Radial Velocity Planet Searcher (HARPS) instrument also indicate low stellar activity as determined by the Ca II H and K lines: $\log(R'_{\mathrm{HK}})=-5.104$ [@knutson2010] and $\log(R'_{\mathrm{HK}})=-4.9$ [@mancini]. In addition, none of the transit light curves analyzed in this work show evidence for occulted spots.
It is notable that HAT-P-12b has an optical transmission spectrum that shows a slope indicative of Rayleigh scattering [@singstis; @alexoudi and this work], while the host star has low stellar activity. This is in contrast to the paradigmatic case of HD 189733b, which has a clear optical scattering slope and an active host star. Thus, HAT-P-12b serves as an important test of whether the transmission slope is related to stellar activity, which could happen in the case of unocculted stellar spots or with enhanced photochemistry as a product of higher stellar far- and near-UV levels.
![Composite $R$-band nightly differential photometry of HAT-P-12 for the 2011–2012 and 2012–2013 observing seasons, obtained with the C14 AIT at Fairborn Observatory. The standard deviation of the data is 0.0046 and 0.0042 for the two seasons, comparable to the measurement precision, indicating that HAT-P-12 shows no significant variability. []{data-label="photovar"}](photometry.pdf){width="\linewidth"}
Analysis {#sec:analysis}
========
We carry out a global analysis of all eight transit light curves (three HST WFC3 G141 visits, two HST STIS G430L visits, one HST STIS G750L visit, and two Spitzer IRAC visits at 3.6 and 4.5 $\mu$m) by simultaneously fitting our transit light-curve model, instrumental systematics models, and photometric noise parameters using the ExoTEP pipeline [@benneke2017; @benneke2019]. We also perform an independent combined fit of the four Spitzer secondary eclipse light curves.
Broadband Light-Curve Fits {#subsec:wlc}
--------------------------
### Instrumental Systematics {#systematics}
Prior to fitting the HST WFC3 light curves, we discard the first orbit, as well as the first two exposures of each orbit, which notably improves the resultant fits. We also remove the 31st and 68th exposures, which were affected by cosmic ray hits, from the second spatial scan mode transit light curve.
Raw uncorrected light curves obtained using the HST WFC3 instrument exhibit well-documented systematic flux variations across the visit, as well as within each individual spacecraft orbit [e.g., @deming2013; @kreidberg]. We model the HST WFC3 instrumental systematics with the following analytical function [e.g., @berta; @kreidberg2015]: $$\label{wfc3systematics}S_{\mathrm{WFC3}}(t) = (c+vt_v)\cdot(1-\exp[-at_{\mathrm{orb}}-b-D(t)]).$$ Here $c$ is a normalization constant, $v$ is the visit-long slope, $a$ and $b$ are the rate constant and amplitude of the orbit-long exponential ramps, and $t_{v}$ and $t_{\mathrm{orb}}$ are the time elapsed since the beginning of the visit and since the beginning of the orbit, respectively. Here $D(t)$ is set to a constant $d$ for points in the first fitted orbit and zero everywhere else, reflecting the observed difference in the ramp amplitude between the first fitted orbit and the subsequent orbits.
We find that stare mode observations exhibit an additional quasi-linear systematic trend across exposures taken between each buffer dump (five per orbit for our visit). We can correct for this trend by appending an extra factor of $(1+dt_{d})$ to Eq. , where $d$ is the linear slope, and $t_{d}$ is the time elapsed since the end of the last buffer dump.
Similar ramp-like instrumental systematics are also apparent in HST STIS raw light curves, albeit with a somewhat different shape. We correct these systematics using a standard analytical model [@sing2008], $$\label{stissystematics}S_{\mathrm{STIS}}(t) = (c+vt_v)\cdot(1+p_1 t_{\mathrm{orb}}+p_2 t_{\mathrm{orb}}^2+p_3 t_{\mathrm{orb}}^3+p_4 t_{\mathrm{orb}}^4),$$ where $c$, $v$, $t_{v}$, and $t_{\mathrm{orb}}$ are defined in the same way as in Eq. , and the coefficients $p_{1-4}$ describe the fourth-order polynomial shape of the orbit-long trend. As with the HST WFC3 light curves, we remove the first orbit, as well as the first two exposures of each orbit prior to fitting.
Raw photometry obtained using the Spitzer IRAC instrument is characterized by short-timescale variations in the measured flux due to small oscillations of the telescope pointing and nonuniform sensitivity of the detector at the subpixel scale. We correct for these intrapixel sensitivity variations by using the modified version of the Pixel Level Decorrelation method [PLD; @pld] described in @benneke2017: $$\label{pld}S_{\mathrm{IRAC}}(t) = 1+\sum\limits_{k=1}^{9}w_{k}\hat{P}_{k}(t_i)+ vt_i.$$
The arrays $\hat{P}_{k}$ represent the pixel counts for the nine pixels located in a $3\times 3$ box centered on the star’s centroid position normalized to sum to unity at each point in the time series. These normalized pixel count arrays are placed into a linear combination with weights $w_{k}$. The last term models a visit-long linear trend, where $v$ is the slope parameter and $t_{i}$ denotes the time elapsed since the beginning of the time series. As with the photometric series, the pixel count arrays can be binned prior to fitting. We optimize for the binning interval and the number of points trimmed from the start of the observation by carrying out individual fits of each IRAC transit light curve (see Section \[subsec:spitzer\]). In the global transit light-curve fit, no additional alterations of the IRAC light curves are needed.
Parameter Instrument Wavelength (nm) Value
--------------------------------------------------- ----------------- ----------------- ---------------------------------
Planet radius, $R_{p}/R_{*}$ STIS G430L 289–570 $0.13798\pm0.00069$
Planet radius, $R_{p}/R_{*}$ STIS G750L 526–1025 $0.13915^{+0.00053}_{-0.00054}$
Planet radius, $R_{p}/R_{*}$ WFC3 G141 920–1800 $0.13743^{+0.00017}_{-0.00016}$
Planet radius, $R_{p}/R_{*}$ IRAC 3.6 $\mu$m 3161–3928 $0.13627^{+0.00074}_{-0.00068}$
Planet radius, $R_{p}/R_{*}$ IRAC 4.5 $\mu$m 3974–5020 $0.1386^{+0.0014}_{-0.0015} $
Transit center time, $T_{0}$ (BJD$_\mathrm{TDB}$) $\dots$ $\dots$ $2,357,368.783203\pm0.000025$
Period, $P$ (days) $\dots$ $\dots$ $3.21305831\pm0.00000024$
Impact parameter, $b$ $\dots$ $\dots$ $0.272^{+0.016}_{-0.017}$
Inclination,^a^ $i$ (deg) $\dots$ $\dots$ $88.655^{+0.090}_{-0.084}$
Relative semimajor axis, $a/R_{*}$ $\dots$ $\dots$ $11.574^{+0.055}_{-0.054}$
[**Note.**]{} ^a^Inclination derived from impact parameter via $b=(a/R_{*})\cos i$.
{width="12cm"}
### Limb Darkening
The ExoTEP pipeline incorporates the Python-based package LDTK [@ldtk] to automatically calculate limb-darkening coefficients. Given the literature values and uncertainties for the stellar parameters ($T_{\mathrm{eff}}=4650\pm60$ K, $\log g=4.61\pm0.01$, $\mathrm{[Fe/H]}=-0.29\pm0.05$; @hartman), this program generates a mean limb-darkening profile and profile uncertainties for each specified bandpass (broadband or spectroscopic) via Monte Carlo sampling of interpolated $50-2600$ nm PHOENIX stellar intensity spectra [@husser] within a $3\sigma$ range in the space of the three stellar parameters.
Subsequent maximum-likelihood optimization returns the best-fit linear, quadratic, or nonlinear limb-darkening coefficients to be used in calculating the transit shape in each bandpass. In our global fit, we find that using the four-parameter nonlinear limb-darkening model yields the lowest residual scatter during ingress and egress, particularly for the high signal-to-noise HST WFC3 spatial scan mode visits.
Because the custom stellar spectra accessed by the LDTK package do not cover wavelengths longer than 2.6 $\mu$m, we set the limb-darkening coefficients for the Spitzer IRAC 3.6 and 4.5 $\mu$m transit light curves to the values computed following the methods described in @sing. These coefficients are tabulated online[^1] for a wide range of ($T_{\mathrm{eff}}$, $\log g$, $z$) values, and we choose the values listed for the set of stellar parameters closest to the literature values for HAT-P-12.
To empirically verify that our choice of fixing limb-darkening coefficients to modeled or tabulated values does not have a significant effect on the measured transmission spectrum, we have experimented with fitting for quadratic limb-darkening coefficients in individual fits of the WFC3 scan mode visits and the broadband Spitzer transit light curves; these visits have either complete transit coverage or the highest per-point precision. We find that the fitted coefficients have large relative uncertainties (20%–70%), i.e., are not well constrained by the data, while being statistically consistent with the corresponding values produced by LDTK or listed in the @sing tables. Crucially, no significant shifts in transit depth occur when switching from fixed limb-darkening coefficients to fitted values.
{width="12cm"}
{width="12cm"}
### Global Fit Results
[ l m[1cm]{} c]{} Wavelength (nm) & & $R_{p}/R_{*}$\
STIS G430L & &\
346–401 & & $0.1418\pm0.0028$\
401–456 & & $0.1405\pm0.0012$\
456–511 & & $0.1387\pm0.0008$\
511–565 & & $0.1390\pm0.0008$\
& &\
STIS G750L & &\
528–577 & & $0.1392\pm0.0011$\
577–626 & & $0.1378\pm0.0013$\
626–675 & & $0.1394\pm0.0009$\
675–723 & & $0.1377\pm0.0008$\
723–772 & & $0.1388\pm0.0007$\
772–821 & & $0.1386\pm0.0019$\
821–870 & & $0.1379\pm0.0013$\
870–919 & & $0.1364\pm0.0015$\
919–968 & & $0.1370\pm0.0021$\
968–1016 & & $0.1369\pm0.0029$\
588.7–591.2 (Na)^a^ & & $0.1357\pm0.0032$\
770.3–772.3(K)^a^ & & $0.1391\pm0.0052$\
& &\
WFC3 G141& &\
1100–1120 & & $0.13666\pm0.00050$\
1120–1140 & & $0.13834\pm0.00047$\
1140–1160 & & $0.13794\pm0.00045$\
1160–1180 & & $0.13744\pm0.00042$\
1180–1200 & & $0.13682\pm0.00036$\
1200–1220 & & $0.13779\pm 0.00044$\
1220–1240 & & $0.13686\pm0.00040$\
1240–1260 & & $0.13761\pm0.00042$\
1260–1280 & & $0.13739\pm0.00044$\
1280–1300 & & $0.13714\pm0.00043$\
1300–1320 & & $0.13733\pm0.00040$\
1320–1340 & & $0.13711\pm0.00038$\
1340–1360 & & $0.13736\pm0.00041$\
1360–1380 & & $0.13798\pm0.00038$\
1380–1400 & & $0.13730\pm0.00038$\
1400–1420 & & $0.13827\pm0.00036$\
1420–1440 & & $0.13817\pm0.00039$\
1440–1460 & & $0.13783\pm0.00039$\
1460–1480 & & $0.13744\pm0.00041$\
1480–1500 & & $0.13754\pm0.00037$\
1500–1520 & & $0.13703\pm0.00050$\
1520–1540 & & $0.13697\pm0.00039$\
1540–1560 & & $0.13667\pm0.00038$\
1560–1580 & & $0.13679\pm0.00040$\
1580–1600 & & $0.13710\pm0.00041$\
1600–1620 & & $0.13741\pm0.00039$\
1620–1640 & & $0.13636\pm0.00043$\
1640–1660 & & $0.13650\pm0.00043$\
[**Note.**]{}
^a^Narrow wavelength bins centered on the alkali (Na and K) absorption lines.
In our pipeline, the transit shape $f(t)$ is calculated using the BATMAN package [@batman]. For the global broadband light-curve analysis, we fit for a separate transit depth ($R_{p}/R_{*}$) in each of the five bandpasses (STIS G430L, STIS G750L, WFC3 G141, IRAC 3.6 $\mu$m, and IRAC 4.5 $\mu$m), along with a single set of transit geometry parameters ($a/R_{*}$, $b$) and transit ephemerides ($T_{0}$, $P$) for all light curves.
The log-likelihood function for our joint light-curve fits is $$\begin{aligned}
\label{logL}\log L = \sum\limits_{V=1}^{N}\Bigg\lbrack -n_{V}&\log {\sqrt{2\pi}} \sigma_{V}\notag \\& - \frac{1}{2}\sum\limits_{i=1}^{n_V}\frac{[D_{V}(t)-S_{V}(t)\cdot f_{V}(t)]^2}{\sigma_{V}^{2}}\Bigg\rbrack,\end{aligned}$$ where the outer summation goes over all $N=8$ visits. For each visit $V$, $n_{V}$ is the number of data points $D_{V}$, $S_{V}$ is the appropriate instrumental systematics model (Section \[systematics\]), $f_{V}$ is the transit light-curve model, and $\sigma_{V}$ is a free photometric noise parameter. We have introduced an independent noise parameter for each visit to account for differences in the level of scatter across the various transit light curves. The best-fit values of the noise parameters establish conservative estimates of the photometric uncertainty on each data point.
The ExoTEP pipeline simultaneously computes the best-fit values and $\pm1\sigma$ uncertainties for all astrophysical and systematics model parameters using the affine-invariant Markov Chain Monte Carlo (MCMC) ensemble sampler `emcee` [@emcee]. To facilitate convergence of the chains, we initialize the global fit with the best-fit values calculated by fitting each transit individually. The global transit fit contains a total of 53 free astrophysical, systematics, and noise parameters. We use $53\times4=212$ walkers and chain lengths of 20,000 steps, discarding the first 60% of each chain when computing the posterior distributions of the fit parameters. To check for convergence, we run the fit five times and ensure that the parameter estimates are consistent across the five runs at better than the $0.1\sigma$ level. The results of our global transit light-curve analysis are listed in Table \[tab:fit\]. Plots of the best-fit transit light curves and their corresponding residuals are shown in Figures \[wfcwlc\]–\[spitzerwlc\].
Our global fits assume a single transit ephemeris across all visits, as well as a common transit depth for visits observed in the same bandpass. To validate this treatment, we also analyze each visit individually in order to compare the best-fit transit timings with the global best-fit transit ephemeris and ensure consistent transmission spectrum shapes among the visits. Figure \[transitcompare\] shows the calculated transit times for individual visits relative to the best-fit global transit ephemeris; only visits with full transit coverage or partial coverage including ingress and egress are included. All of the individual transit times agree with the global ephemeris at better than the $1\sigma$ level, ruling out any statistically significant transit timing variation.
![Observed minus calculated transit time plot showing the best-fit individual WFC3 G141 and IRAC 3.6 and 4.5 $\mu$m transit times (blue points) relative to the best-fit transit ephemeris derived from the global broadband transit fit (black curves). The STIS transit times are not included because the light curves from those visits do not cover ingress or egress, resulting in significantly larger transit time uncertainties.[]{data-label="transitcompare"}](transittime.pdf){width="9cm"}
![Comparison plot showing the transmission spectra derived from the individual visits in the WFC3 G141 (scan mode only) and STIS G430L bandpasses (blue and green points) alongside the corresponding spectra computed from the joint analysis (black points). The individual spectra agree well with the joint spectrum across all wavelengths. []{data-label="spectrumcompare"}](spectracompare.pdf){width="9cm"}
Spectroscopic Light-curve Fits {#subsec:speclc}
------------------------------
When fitting the individual spectroscopic light curves in the STIS G430L, STIS G750L, and WFC3 G141 bandpasses, we fix the transit geometry parameters and transit ephemeris to the best-fit values from the global broadband transit analysis (Table \[tab:fit\]), with the transit depth being the only free astrophysical parameter.
The ExoTEP pipeline offers a choice of three methods for defining the instrumental systematics model for the constituent spectroscopic light curves. The first method utilizes the full systematics model for the corresponding instrument, computing the best-fit instrumental systematics parameters for each spectroscopic light curve independently from the broadband light curve. The other methods apply a common-mode correction to the spectrophotometric series prior to fitting, dividing each series by either (1) the best-fit broadband systematics model [e.g., @kreidberg], or (2) the ratio of the uncorrected broadband photometric series and the best-fit broadband transit model [e.g., @deming2013].
Performing a precorrection on the spectroscopic light curves takes advantage of the more well-defined systematics model derived using the high signal-to-noise broadband light curves. This technique also enables us to use fewer systematics parameters in the individual spectroscopic light-curve fits, which typically results in tighter constraints on the best-fit transit depths. We account for residual systematic flux variations in the spectroscopic light curves using a simplified model, $$\label{specsystematics}S_{\mathrm{spec}}(t) = c+v\cdot(x-x_{0}),$$ which describes a linear function with respect to the measured subpixel shifts $x-x_{0}$ in the dispersion direction relative to the first exposure in the time series, with $c$ and $v$ being the offset and slope parameters, respectively.
To demonstrate consistency in the transmission spectrum shape between separate observations in the same bandpass, we first analyze the spectroscopic light curves of individual visits. Figure \[spectrumcompare\] shows the transmission spectra of the individual WFC3 G141 scan mode and STIS G430L visits plotted with the corresponding spectra derived from the joint analysis. In both cases, there is good agreement between the individual transit depths in each wavelength bin, and the spectrum shapes are consistent across the visits. In particular, each WFC3 G141 scan mode visit spectrum shows a discernible absorption feature at 1.4 $\mu$m. It is also important to note that this feature was not detected in the older stare mode data analyzed in @line, which underscores the significant improvement in sensitivity provided by the scan mode observations.
Using the same log-likelihood expression as in our global broadband transit light-curve fit (Eq. ), we then fit all visits in a given bandpass jointly, letting the systematics model and photometric noise parameters vary independently for each light curve. For the STIS G430L and G750L spectroscopic light curves, in line with similar previous studies [e.g., @singstis], we find that the shapes of the systematic trends vary significantly across the various wavelength bins, necessitating the use of the full systematics model. Meanwhile, the HST WFC3 systematics are largely independent of wavelength and detector position, and we find that the two precorrection strategies described above result in fits of comparable quality. In this paper, we report the best-fit depths derived from using the latter of the two precorrection methods (i.e., dividing the ratio of the uncorrected flux and the best-fit transit model from the broadband light curve).
The results of our spectroscopic light-curve fits are listed in Table \[tab:fit2\]. The best-fit transit light curves and associated residuals are plotted in the Appendix for each of the HST STIS and WFC3 visits. When experimenting with different wavelength bin widths (10–40 nm), we get consistent transmission spectrum shapes. Visual inspection of the systematics-corrected light curves does not reveal any salient outliers or residual uncorrected systematics trends. We combine the transit depths from the spectroscopic light-curve fits with the broadband Spitzer IRAC transit depths to construct the full transmission spectrum of HAT-P-12b, which is plotted in Figure \[spectrum\]. The transit depths for the narrow wavelength bins in the main alkali absorption regions are consistent with the depths measured in the wider bins spanning those regions, indicating a nondetection of the alkali absorption; these data points are not shown in the transmission spectrum plot. The primary features of the transmission spectrum are the Rayleigh slope extending through the optical bandpasses and a small absorption feature around 1.4 $\mu$m indicative of water vapor. These observations together suggest the presence of both uniform clouds and fine-particle scattering in the atmosphere of HAT-P-12b.
The shape of the transmission spectrum at visible wavelengths matches the results of previous analyses of the HST STIS data by @singstis and @alexoudi. It is worth mentioning that an earlier study of HAT-P-12b’s atmosphere using ground-based broadband photometry produced a flat transmission spectrum throughout the visible wavelength range [@mallonn], consistent with an opaque layer of clouds as opposed to Rayleigh scattering. This discrepancy was discussed in @alexoudi and attributed to uncertainties in the inclination and semi-major axis of HAT-P-12b’s orbit, which are correlated with transit depths and can yield wavelength-dependent shifts that alter the apparent transmission spectrum slope in the optical.
When assuming different values of $i$ and $a/R_{*}$ in reanalyzing the @mallonn light curves, @alexoudi were able to recover a discernible Rayleigh scattering slope in the visible transmission spectrum. In our global fit, we take advantage of the well-sampled ingress and egress from the scan mode HST WFC3 and Spitzer light curves to place much narrower constraints on $i$ and $a/R_{*}$ than these earlier studies. Therefore, our results are a robust validation of the previously published reports of a negative slope in the visible transmission spectrum of HAT-P-12b.
{width="\linewidth"}
Secondary Eclipses {#subsec:ecl}
------------------
The eclipse light curve is defined in the same way as a transit light curve but without the limb-darkening effect. We utilize the same modified PLD instrumental systematics model to account for the Spitzer IRAC intrapixel sensitivity variations (Eq. ). For each eclipse observation, we select the optimal aperture, photometric parameters, binning, and trimming by fitting the eclipse light curve individually, fixing the transit geometry parameters ($a/R_{*}$, $b$) and transit ephemerides ($T_{0}$, $P$) to the best-fit values from the global broadband transit light-curve analysis (Table \[tab:fit\]).
When performing individual eclipse fits on the relatively low signal-to-noise data, we facilitate comparison between different versions of the photometry/binning/trimming by fixing the time of eclipse to an orbital phase of 0.5. The orbital phase here is defined relative to the best-fit ephemeris from the global transit fit. To correct for any residual flux ramps at the start of the data, we also experiment with including an exponential factor $(1 - a_{1}e^{-t_{i}/a_{2}})$ in the systematics model, where $a_{1}$ and $a_{2}$ are the amplitude and time constant, respectively, and $t_{i}$ is the time elapsed since the beginning of the time series. Following @wong2 [@wong3], we choose the photometric series that produce the lowest residual scatter. Only for the first 3.6 $\mu$m eclipse dataset does the inclusion of a ramp appreciably improve the fit (i.e., minimizes the value of the Bayesian Information Criterion).
We also carry out a global analysis of all four secondary eclipse observations. In this fit, we allow the instrumental systematics parameters for each dataset to vary independently while assuming common 3.6 and 4.5 $\mu$m eclipse depths and center of eclipse phase as free parameters. The results of our individual and global eclipse fits are listed in Table \[tab:eclipse\]. The raw and systematics-corrected eclipse light curves are shown in Figure \[eclipses\]. From the individual fits, we only find marginal eclipse detections for the full array 3.6 and 4.5 $\mu$m visits ($<1.5\sigma$), while the more recent subarray observations yield more robust detections ($>2.5\sigma$). The best-fit eclipse phase from the combined analysis is consistent with a circular orbit, and the global 3.6 and 4.5 $\mu$m depths are statistically consistent with each of the individual best-fit eclipse depths at better than the $1.1\sigma$ level.
![Left: plots of the two Spitzer 3.6 $\mu$m secondary eclipses, binned in 5 minute intervals. In the top panels, the unbinned photometric series is shown in gray, with the binned data overplotted in black. The middle panels show the corrected light curve with the intrapixel sensitivity effect removed. The best-fit eclipse light curve is overplotted in red. The corresponding residuals from the fit are shown in the bottom panels. The error bars shown are the standard deviation of the residuals from the best-fit light curve, scaled by the square root of the number of points in each 5 minute bin. Right: analogous plots for the two Spitzer 4.5 $\mu$m secondary eclipses.[]{data-label="eclipses"}](eclipses.pdf){width="9cm"}
[ l m[0.1cm]{} c m[0.1cm]{} c]{} Eclipse & & Depth (%) & & Phase\
3.6 $\mu$m & & & &\
Eclipse 1 & & $0.019\pm0.017$ & &$\equiv0.5$^a^\
Eclipse 2 & &$0.064^{+0.017}_{-0.018}$&& $\equiv0.5$\
Global^b^ & &$0.042\pm0.013$ & &$0.5009^{+0.0026}_{-0.0014}$\
4.5 $\mu$m && & &\
Eclipse 1 && $0.032\pm0.024$ && $\equiv0.5$\
Eclipse 2 && $0.066^{+0.027}_{-0.026}$ && $\equiv0.5$\
Global^b^ && $0.045^{+0.017}_{-0.019}$ & &$0.5009^{+0.0026}_{-0.0014}$\
[**Notes.**]{}
^a^The eclipse phase was fixed at 0.5 for all individual eclipse fits, assuming the best-fit orbital ephemeris from the global transit fit (Table \[tab:fit\]).
^b^Computed from a simultaneous fit of all four eclipses.
Atmospheric Retrieval {#sec:disc}
=====================
We simultaneously interpret the full transmission and emission spectra presented in this work to deliver quantitative constraints on the atmosphere of HAT-P-12b using the SCARLET atmospheric retrieval framework [@bennekeseager1; @bennekeseager2; @kreidberg; @knutson2014; @benneke2015; @benneke2019]. Employing SCARLET’s chemically consistent mode, we define the atmospheric metallicity, the C/O ratio, the cloud properties, and the vertical temperature structure as free parameters. SCARLET then determines their posterior constraints by combining a chemically-consistent atmospheric forward model with a Bayesian MCMC analysis. We perform the retrieval analysis with 100 walkers using uniform priors on all the parameters and run the chains well beyond formal convergence to obtain smooth posterior distribution even near the $3\sigma$ contours.
To evaluate the likelihood for a particular set of atmospheric parameters, the SCARLET forward model in chemically consistent mode first computes the molecular abundances in chemical and hydrostatic equilibrium and the opacities of molecules and Mie-scattering clouds [@bennekeseager2]. The elemental composition in the atmosphere is parameterized using the atmospheric metallicity, \[M/H\], and the atmospheric C/O ratio. We employ log-uniform priors, and we consider the line opacities of H$_2$O, CO, and CO$_2$ from HiTemp [@rothman] and CH$_4$, NH$_3$, HCN, H$_2$S, C$_2$H$_2$, O$_2$, OH, PH$_3$, Na, K, TiO, SiO, VO, and FeH from ExoMol [@tennyson], as well as the collision-induced absorption of H$_2$ and He.
Following @benneke2019, we use a three-parameter Mie-scattering cloud description for the retrieval analysis defining the mean particle size $R_{\mathrm{part}}$, the pressure level $P_{\tau=1}$ at which the clouds become optically opaque to grazing starlight at 1.5 $\mu$m, and the scale height of the cloud profile relative to the gas pressure scale height $H_{\mathrm{part}}/H_{\mathrm{gas}}$ as free parameters. All free parameters are allowed to vary independently in the retrieval. When calculating the cloud opacity, the retrieval is agnostic to the particular composition of the spherical cloud particles, considering only their size and vertical distribution; the former is assumed to be a logarithmic Gaussian distribution with a fixed width of $\sigma_{R}=1.5$. This three-parameter cloud description is motivated by the information content of transmission spectra and captures the wavelength-dependent opacities of a wide range of finite-sized cloud particles near the cloud deck in a highly orthogonal way, ideal for retrieval [@benneke2019]. It reduces to Rayleigh hazes in the limit of small particles and a gray cloud deck for large particles while simultaneously allowing for any finite-sized Mie-scattering particles in between. We employ log-uniform priors on the three cloud parameters. Our temperature structure is parameterized using the five-parameter analytic model from @parmentier2014 augmented with a constraint on the plausibility of the total outgoing flux. Given the relatively weak constraints on the atmospheric composition, we conservatively ensure the plausibility of the temperature structure by enforcing that the wavelength-integrated outgoing thermal flux is consistent with the stellar irradiation, a Bond albedo between 0 and 0.7, and heat redistribution values between full heat redistribution across the planet and no heat redistribution. In the retrieval, we parameterize only one temperature structure for both the dayside and the terminator because the retrieved temperature uncertainties are hundreds of K and the precision of the transmission spectrum does not justify additional parameters describing the terminator temperature structure separately.
Finally, high-resolution synthetic transmission and emission spectra are computed using line-by-line radiative transfer and integrated over the appropriate instrument response functions before being compared to the observations. Sufficient wavelength resolution in the synthetic spectra is ensured by repeatedly verifying that the likelihood for a given model is not significantly affected by the finite wavelength resolution ($\Delta\chi^2<0.001$). Reference models are computed at $\frac{\lambda}{\Delta\lambda}=250,000$.
Parameter Value Unit
------------------------------------------------- ------------------------- --------- --
Atmospheric metallicity, $\log M$ $2.43^{+0.33}_{-0.60}$ x solar
Atmospheric C/O ratio $0.48_{-0.37}^{+0.10}$ $\dots$
Mean particle size,^a^ $\log R_{\mathrm{part}}$ $-1.47^{+0.45}_{-0.36}$ $\mu$m
Opacity pressure level,^b^ $\log P_{\tau=1}$ $0.38^{+2.23}_{-1.18}$ mbar
Relative cloud scale height,^c^
$\log H_{\mathrm{part}}/H_{\mathrm{gas}}$ $0.12^{+0.30}_{-0.34}$ $\dots$
: HAT-P-12b Atmospheric Retrieval Results[]{data-label="retrievalresults"}
[**Notes.**]{}
^a^Mean particle size, assuming a logarithmic Gaussian distribution with fixed width of $\sigma_{R}=1.5$
^b^Pressure at transmission optical depth of unity at 1.5 $\mu$m
^c^Scale height of cloud profile relative to gas pressure scale height
{width="\linewidth"}
![The 2D posterior of cloud-top pressure $\log(P_{\tau=1})$ vs. atmospheric metallicity from the atmospheric retrieval. The solid black lines indicate $1\sigma$, $2\sigma$, and $3\sigma$ bounds. The HAT-P-12b transmission spectrum is consistent with both cloudy atmospheres spanning a broad range of metallicities and clear atmospheres with highly enhanced metallicities.[]{data-label="cloudmetal"}](cloudmetal.pdf){width="\linewidth"}
Retrieval results
-----------------
We run a set of retrievals that assume chemical and thermal equilibrium, setting the atmospheric metallicity $\log M$, atmospheric C/O ratio, and cloud properties ($R_{\mathrm{part}}$, $P_{\tau=1}$, and $H_{\mathrm{part}}/H_{\mathrm{gas}}$) as free parameters. The range of representative atmospheric models derived from the retrievals is illustrated in Figure \[spectrum\]. The median atmospheric model is shown by the blue curve, and the $1\sigma$ and $2\sigma$ credible intervals are indicated by the shaded regions. In short, both the observed transmission and emission spectra are well fit across all wavelengths by cloudy atmospheres with cloud-top pressures between 0.2 mbar and 0.4 bar ($1\sigma$ bounds) and supersolar metallicities. The data favor submicron cloud particle sizes, and the posterior spans most of the assumed prior range below $\sim$200 nm. Crucially, the retrieved particle sizes cover the range necessary to produce the Rayleigh scattering in the optical evident in the transmission spectrum, consistent with a previous retrieval of the HAT-P-12b atmosphere [@barstow]. The list of parameter estimates is given in Table \[retrievalresults\].
The full triangle plot displaying all one- and two-parameter marginalized posteriors is shown in Figure \[triangle\]. Of particular interest is the degeneracy between cloud-top pressure and atmospheric metallicity, which is shown separately in Figure \[cloudmetal\]. Overall, the atmospheric metallicity is not well constrained: the L-shaped posterior indicates that while the data are largely consistent with cloudy atmospheres spanning a wide range of supersolar metallicities, clear atmospheres with strongly enhanced metallicities above 100 times solar cannot be ruled out at the $1\sigma$ level. This degeneracy is a common feature in atmospheric retrievals of exoplanet transmission spectra with weak or undetected 1.4 $\mu$m water features [e.g., HAT-P-11b; @fraine2014], where the small magnitude of the water absorption can be caused either by attenuation due to the presence of clouds or by an intrinsically weak absorption from a hydrogen-depleted atmosphere with high mean molecular weight.
Core accretion models predict a trend of increasing bulk metallicity with decreasing planet mass [e.g., @mordasini2012; @fortney2013], and most known gas giant exoplanets have supersolar bulk metallicities [e.g., @thorngren2016]. Meanwhile, the relationship between bulk and atmospheric metallicity is more complex. From planet evolution and interior structure modeling, @thorngren2019 predicted a 95% atmospheric metallicity upper limit of 82.3 for HAT-P-12b, broadly consistent with the results of our atmospheric retrievals and the corresponding bulk metallicity of the planet. Further enrichment of the atmospheric metallicity can result from secondary processes such as core erosion [e.g., @wilson2012; @madhusudhan2016] and accretion of solid material during the late stages of planet formation [e.g., @pollack1996]. The atmospheric metallicity of HAT-P-12b is also comparable to similarly sized sub-Saturn planets, such as WASP-39b (100–200$\times$ solar; @wakeford2017) and WASP-127b (10–40$\times$ solar; @spake2019). Given this context, the elevated metallicity of HAT-P-12b is not entirely unexpected.
Another notable result from the retrievals is the near-solar atmospheric C/O ratio of $0.48_{-0.37}^{+0.10}$, with a $3\sigma$ upper limit at roughly 0.83. The presence of a water vapor absorption feature at 1.4 $\mu$m rules out carbon-dominated atmospheres, because the formation of H$_{2}$O becomes disfavored as C/O approaches unity. The absence of a 1.15 $\mu$m absorption in the WFC3 bandpass comparable in magnitude to the observed 1.4 $\mu$m feature also supports the conclusion of an oxygen-dominated chemistry by eliminating CH$_{4}$ as the molecular species responsible for the near-infrared absorption features [e.g., @benneke2015]. Methane has a strong absorption feature at around 3.3 $\mu$m, so it follows that the relatively low transit depth measured in the Spitzer 3.6 $\mu$m bandpass in comparison with the 4.5 $\mu$m depth likewise points toward a near-solar C/O ratio.
In addition to the chemical and thermal equilibrium retrievals, we run a set of “free" atmospheric retrievals that do not assume chemical or thermal equilibrium; instead, the abundance of each molecular gas species is independently varied, in addition to the previously defined parameters describing the clouds. In these runs, we focus on H$_{2}$O, CH$_{4}$, CO, and CO$_{2}$ as the primary atmospheric components to be constrained. We do not find any notable constraints on the abundances of the carbon-bearing species relative to H$_{2}$O.
{width="\linewidth"} {width="\linewidth"}
Comparison to Microphysical Cloud Models {#sec:carma}
========================================
In addition to the atmospheric retrievals presented in the previous section, we use the Community Aerosol and Radiation Model for Atmospheres (CARMA) to simulate condensation clouds and photochemical hazes in the atmosphere of HAT-P-12b. CARMA is a time-stepping cloud microphysics model that computes the bin-resolved particle size distributions of aerosols as a function of altitude in planetary atmospheres. CARMA treats aerosol formation and evolution as a kinetic processes, with convergence dictated by balancing the rates of particle nucleation, condensational growth and evaporation, coagulation, and transport via sedimentation, advection, and diffusion calculated from classical theories of cloud physics [@pruppacher]. It is thus significantly different from phase equilibrium models, such as @ackermanmarley2001, which do not consider the time evolution of the rates of microphysical processes. The specific physical formalism used in the model is described in full in the Appendix of @gao2018.
{width="\linewidth"}
By comparing the CARMA simulation results to the observations, we hope to gain a more physical understanding of the processes controlling aerosol distributions. In our modeling of the HAT-P-12b atmosphere, we consider both condensate clouds and photochemical hazes separately. We refer the reader to the Appendix for a detailed description of the condensate and aerosol modeling setup in CARMA. For each model run, the temperature–pressure profile of the background atmosphere is set to the best-fit profile from the atmospheric retrieval (Section \[sec:disc\] and Figure \[spectrum\]). Vertical mixing of condensate or haze particles is driven by eddy diffusion, and we consider eddy diffusion coefficient $K_{zz}$ values of $10^{7}$, $10^{8}$, $10^{9}$, and $10^{10}$ cm$^2$ s$^{-1}$. The atmospheric metallicity is set to 10$\times$, 100$\times$, or 1000$\times$ solar; adjusting the metallicity affects the initial abundance of condensate species in the model, as well as the atmospheric scale height.
Given the uncertainties in the specific chemical pathways and efficiencies of haze production, CARMA does not carry out an *ab initio* haze formation calculation but instead sets the haze production rate as a free parameter. We consider haze production rates of $10^{-14}$, $10^{-13}$, and $10^{-12}$ g cm$^{-2}$ s$^{-1}$ at a pressure of 1 $\mu$bar, consistent with the values computed in exoplanet photochemical studies [e.g., @venot2015; @lavvas2017; @kawashima2018; @lines1; @adams2019]. We investigate the impact of different haze compositions on the atmospheric opacity by considering different refractive indices. In particular, we consider both soots, which are expected to survive at the high temperatures of exoplanet atmospheres due to their relatively low volatility, and tholins, which we use as a proxy for lower-temperature organic hazes [@morley2015].
We find that haze models match the observed transmission spectrum much better than condensate cloud models. While many of the condensate cloud models are able to reproduce the shape of the muted water vapor absorption feature at 1.4 $\mu$m, none of them generate the observed steep slope throughout the optical, resulting in reduced $\chi^{2}$ (RCS) values significantly higher than unity. When examining the average particle sizes predicted by the condensate cloud model runs, we find relatively large condensate particles on the order of or exceeding 1 $\mu$m — too large to allow for Rayleigh scattering in the optical. Meanwhile, the haze models readily reproduce the observed Rayleigh scattering slope. In addition, cloud models that can match the amplitude of the 1.4 $\mu$m water feature are too flat to explain the large offset between the two Spitzer points due to the extensive cloud opacity at 3–5 $\mu$m, while the haze opacity falls off with increasing wavelength sufficiently quickly to allow for larger-amplitude molecular features there.
Figure \[RCS\] shows the RCS values for the full grid of tholin and soot haze models. In both cases, the best-performing run (lowest RCS) has an atmospheric metallicity of 100$\times$ solar and a moderate rate of vertical mixing ($K_{zz}= 10^{8}$ cm$^{2}$ s$^{-1}$). For tholins, the observations are best matched when assuming a haze production rate of $10^{-12}$ g cm$^{-2}$ s$^{-1}$, whereas for soot, a lower production rate of $10^{-13}$ g cm$^{-2}$ s$^{-1}$ is preferred, since soots are more absorbing than hazes at the wavelengths of interest [@adams2019]. The model transmission spectra derived from the best-fitting condensate cloud, tholin, and soot models are shown in Figure \[carmamodels\]. Both of the photochemical haze models match the full set of observations and have RCS values below 1. Meanwhile, the lowest-RCS condensate cloud model performs more poorly than even a featureless flat spectrum. When comparing the soot and tholin spectra, the only salient distinguishing feature is at $\sim$6.5 $\mu$m, where the tholin spectrum displays an additional absorption possibly attributable to double-bonded carbon atoms, double-bonded carbon and nitrogen atoms, and single-bonded amine groups [@imanaka2004; @gautier2012].
The size and vertical distributions of the haze particles for the soot and tholin models are shown in Figure \[carmadist\]. The color coding indicates the number density of particles per logarithmic radius bin. For both cases, the haze distribution is dominated by submicron particles, particularly at the lowest pressure levels (below 0.1 mbar), consistent with the observed Rayleigh scattering slope in visible wavelengths. The horizontal dashed lines denote the highest pressure probed by our observations (i.e., optical depth of unity in transmission). Notably, the modeled particle size distributions and the opacity pressure levels are in agreement with the corresponding values $\log R_{\mathrm{part}}$ and $\log P_{\tau=1}$ inferred from the SCARLET retrieval (Table \[retrievalresults\]) to within the $1\sigma$ uncertainties. The demonstrated agreement between the retrieval and the CARMA results serves as an illustrative example of the increasing explanatory power of current aerosol models that incorporate detailed microphysical calculations and account for the opacity contributions from photochemical hazes.
![Size and vertical distribution of haze particles for the best-fit soot and tholin haze models computed by CARMA. The colors indicate the number density of haze particles per logarithmic radius bin. The horizontal dashed white lines show the pressure levels where the optical depth in transmission at a wavelength of 1.5 $\mu$m is unity. In both cases, the hazes are dominated by submicron particles, with the smallest particle sizes in the upper atmosphere. The typical particle radii and opacity pressure levels are consistent with the values from our SCARLET retrieval.[]{data-label="carmadist"}](bestTransDist.png){width="\linewidth"}
Constraints from secondary eclipse measurements
===============================================
The secondary eclipse measurements offer an independent look at the atmosphere of HAT-P-12b. While the transmission spectrum directly probes the day–night terminators, the secondary eclipse depths indicate the total outgoing flux from the dayside hemisphere relative to the star’s flux. In Section \[subsec:ecl\], we calculated depths of $0.042\%\pm 0.013\%$ and $0.045^{+0.017}_{-0.019}\%$ at 3.6 and 4.5 $\mu$m, respectively.
From these values, we can estimate the blackbody brightness temperature of the dayside hemisphere. We account for the uncertainties in the stellar parameters by deriving empirical analytical functions for the integrated stellar flux in the Spitzer bandpasses. This is done by fitting a polynomial in ($T_{\mathrm{eff}}$, \[M/H\], $\log g$) to the calculated stellar flux for a grid of `ATLAS` models [@atlas] spanning the ranges $T_{\mathrm{eff}}=[4000,5000]$ K, $[\mathrm{M/H}]=[-1.0,+0.5]$, and $\log g=[4.5,5.0]$. We then computed the posterior distribution of the dayside brightness temperature using a Monte Carlo sampling method, given priors on the stellar properties from @hartman.
We obtain brightness temperature estimates of $980^{+80}_{-100}$ K at 3.6 $\mu$m and $810^{+90}_{-160}$ K at 4.5 $\mu$m. We also find that both eclipse depths are consistent with a single blackbody temperature of $890^{+60}_{-70}$ K. This estimate is consistent at the $1.1\sigma$ level with the terminator temperature of $1010\pm80$ K previously derived from an analysis of the HST STIS transmission spectrum when assuming Rayleigh scattering [@singstis]. The predicted dayside equilibrium temperature of HAT-P-12b assuming zero albedo is 1150 K if incident energy is reradiated from the dayside only and 970 K if the planet reradiates the absorbed energy uniformly over the entire surface. The relatively low calculated dayside temperature indicates very efficient day–night recirculation of incident energy and possibly a nonzero albedo.
The Spitzer secondary eclipse depths can also provide constraints on atmospheric metallicity. Specifically, the ratio between the 3.6 and 4.5 $\mu$m depths varies systematically with metallicity. From the bottom left panel of Figure \[spectrum\], we see the comparison between the measured depths and model spectra generated by SCARLET. The constraints provided by the Spitzer secondary eclipse depths in the combined transmission and emission spectra retrieval are weak due to the low signal-to-noise of the planetary flux detection as well as the low wavelength resolution of the two broadband points. Examining the model emission spectra, we can see diagnostic features in the 3–5 $\mu$m region that could be adequately probed with even modest wavelength resolution ($R\sim 20-30$). Near-future instruments, such as NIRSpec on the James Webb Space Telescope (JWST), will enable detailed studies of planetary emission spectra spanning the thermal infrared, opening up a new domain for exoplanet atmospheric characterization.
Conclusions
===========
We have presented eight transit observations of the warm sub-Saturn HAT-P-12b obtained from HST and Spitzer. The resulting transmission spectrum from a joint analysis of all transit light curves covers the optical and near-infrared wavelength range from 0.3 to 5.0 $\mu$m. We obtain precise, updated estimates for the orbital parameters of the system.
The main features of the transmission spectrum are a weak water vapor absorption feature at 1.4 $\mu$m and a prominent Rayleigh scattering slope throughout the visible wavelength range with no detected alkali absorption peaks. These features indicate significant cloud opacity in the atmosphere of HAT-P-12b, with a strong contribution from small-particle scattering in the upper atmosphere. The detection of Rayleigh scattering in the transmission spectrum and the low stellar activity of the host star make HAT-P-12b an important test case for evaluating the relationship between optical scattering slopes and stellar activity.
We have complemented our analysis of the transmission spectrum with new fits of secondary eclipse light curves in the 3.6 and 4.5 $\mu$m Spitzer bandpasses, from which we derive the depths $0.042\%\pm0.013\%$ and $0.045\%\pm0.018\%$, respectively. The dayside atmosphere is consistent with a single blackbody temperature of $890^{+60}_{-70}$ K and efficient day–night heat recirculation.
Through a multifaceted approach combining atmospheric retrievals from SCARLET using both transmission and emission spectra with the results of the aerosol microphysics model CARMA, we find that the atmosphere of HAT-P-12b has a near-solar C/O ratio of $0.48_{-0.37}^{+0.10}$ and an atmospheric metallicity that broadly spans the range between several tens and a few hundred times solar. While condensate cloud models produce particles that are too large to reproduce the observed Rayleigh scattering slope, models incorporating photochemical hazes consisting of tholins or soot readily generate submicron particles in the upper atmosphere and match the full range of observations. The aerosol modeling indicates moderate vertical mixing (eddy diffusion coefficient $K_{zz}= 10^{8}$ cm$^{2}$ s$^{-1}$) and opacity pressure levels around 0.1 mbar, consistent with the results of the retrievals.
HAT-P-12b fits within the growing population of well-characterized cooler exoplanets that show evidence for photochemical hazes. The temperature range spanned by these planets allows for the formation of an enormous diversity of condensate species [e.g., @singstis]. The importance of clouds and hazes in interpreting observed transmission spectra and their wide-ranging effects on atmospheric chemistry and dynamics illustrates the need for continued refinement in our understanding of the myriad physical and chemical processes that govern the formation and distribution of condensates in exoplanetary atmospheres. While current state-of-the-art cloud and haze models are becoming more sophisticated and capable of describing observations of individual exoplanets and observed trends in exoplanet cloudiness, there remain significant gaps in our knowledge of the detailed microphysics of *ab initio* aerosol formation and the effects of secondary processes such as vertical mixing.
Our work also underscores the importance of increased spectral resolution in amplifying the explanatory power of both transmission and emission spectroscopy. The broadband Spitzer photometry at 3.6 and 4.5 $\mu$m has provided weak complementary constraints on the more discerning transmission spectra at shorter wavelengths. However, with even moderately increased spectral resolution in the 2–5 $\mu$m region, we can obtain much more precise estimates of atmospheric metallicity and C/O ratio and probe the absorption and emission features of a wide range of major atmospheric species. The capabilities of upcoming space-based telescopes such as JWST in this regard will usher in a new era of exoplanet atmospheric characterization.
This work is based on observations with the NASA/ESA *Hubble Space Telescope*, obtained at the Space Telescope Science Institute (STScI) operated by AURA, Inc. This work is also based in part on observations made with the *Spitzer Space Telescope*, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Program (FP7/2007-2013)/ERC grant agreement No. 336792. Support for this work was also provided by NASA/STScI through grants linked to the HST-GO-12473 and HST-GO-14767 programs. I.W. and P.G. are supported by Heising-Simons Foundation *51 Pegasi b* postdoctoral fellowships. H.A.K. acknowledges support from the Sloan Foundation.
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HST Light Curves
================
The following plots show the spectroscopic light curves for the six HST WFC3 and STIS transit observations. The left panel shows the light curves for each of the wavelength bins, corrected for instrumental systematics and arranged top to bottom in the order listed in Table \[tab:fit2\]. The best-fit transit light curves are overplotted in black. The right panel shows the corresponding residuals in parts per thousand (ppt). The error bars on all data points are set to the best-fit photometric noise parameter.
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Cloud and haze modeling with CARMA
==================================
CARMA was initially developed to investigate aerosol processes on Earth [@toon1979; @turco], and has since been adapted to various solar system bodies [@toon1992; @james; @colaprete; @barth2006; @gao2014; @gao2017] and exoplanets [@gao; @powell2018; @powell2019; @adams2019]. Here we use the exoplanet version of CARMA, which has the ability to model clouds composed of a variety of species predicted by equilibrium chemistry and kinetic cloud formation models, including KCl, ZnS, Na$_{2}$S, MnS, Cr, Mg$_{2}$SiO$_{4}$, Fe, TiO$_{2}$, and Al$_{2}$O$_{3}$ [see, for example, the review in @marley2013]. We refer the reader to @gao and @powell2019 for the relevant material properties of the condensates and the formation pathways we consider. Briefly, we consider homogeneous nucleation for species that can undergo direct phase change (TiO$_2$, Fe, Cr, and KCl) and heterogeneous nucleation for species that form via thermochemical reactions, represented in the gas phase by their limiting species (Al$_2$O$_3$: Al, Mg$_2$SiO$_4$: Mg, MnS: Mn, Na$_2$S: Na, ZnS: Zn; see, for example, @visscher2006, @visscher2010, and @morley2012). Here TiO$_2$ is chosen to be the condensation nuclei of Al$_2$O$_3$, Mg$_2$SiO$_4$, MnS, and Na$_2$S due to its low energy barrier to homogeneous nucleation [e.g., @lee2018], while KCl acts as the condensation nuclei to ZnS, as they both form at lower temperatures than the other condensates. Here Fe and Cr are also allowed to heterogeneously nucleate on TiO$_2$. The resulting cloud particles are either pure, in the case of the homogeneously nucleated particles, or a core surrounded by a mantle, in the case of the heterogeneously nucleated particles. This is a simplification of the mixed-grains formalism of other kinetic cloud models [e.g., @helling2016; @lee2016; @lines2]. Cloud particles of different compositions do not interact, and their size distributions are computed independently of each other, except in the case of condensation nuclei and mantling species; i.e. formation of the latter depletes the former.
Each cloud simulation begins with a background H$_{2}$/He atmosphere devoid of cloud particles, with condensate vapor only at the deepest atmospheric level. We use GGchem [@ggchem] to set the initial mixing ratio of each condensate species at this lower boundary. As the simulation advances, all condensate vapors are mixed upward via eddy diffusion and parameterized by the eddy diffusion coefficient $K_{zz}$, until they either become well mixed in the atmosphere or achieve supersaturation. Particle nucleation and condensation may then occur, provided that the supersaturation is sufficiently large to overcome the nucleation energy barriers of the various condensate species. Cloud particle formation depletes the condensate vapors until their resupply by eddy diffusion from depth is sufficient to balance. We do not explicitly consider any gas chemistry in our modeling. Growth of cloud particles by coagulation and vertical transport of cloud particles proceed until a steady state is reached.
CARMA also models coagulation and vertical transport of photochemical hazes, following the methodology developed in @gao2017 and @adams2019. The detailed chemical pathways and formation efficiencies of exoplanet hazes are much more complex and less understood than those predicted for condensation clouds [@fleury2018; @he2018; @horst], and CARMA does not explicitly model the production of aerosol particles. Instead, we choose to model haze production generically by setting the haze production rate as a free parameter. We assume spherical haze particles with a mass density of 1 g cm$^{-3}$ and a minimum radius of 10 nm; it has been shown that the minimum particle radius does not strongly affect the optical depth at equilibrium [e.g., @adams2019]. We do not consider condensation when modeling hazes.
Haze simulations also begin with a background H$_{2}$/He atmosphere devoid of aerosols. As the simulation advances, 10 nm haze particles are produced at high altitudes, after which they can grow by coagulation and are transported into the deep atmosphere by sedimentation and eddy diffusion. Haze particles are assumed to evaporate at the lower boundary of the model, though it does not impact the resulting transmission spectra, since the lower boundary is set at pressures $>$10 bar.
To generate predicted transmission spectra from the forward-modeled aerosol distributions, we first use the `pymiecoated` tool to compute the extinction efficiency, single scattering albedo, and asymmetry factor of the aerosol particles. The refractive indices for the various aerosol species are compiled from @posch2003, @zeidler2011, @morley2012, @wakefordsing, and @lavvas2017. We then use a standard 1D radiative transfer model to produce the transmission spectra [e.g., @fortney2010]. These spectra are subsequently binned to the resolution of the observations, and the base planet radius is shifted to best fit the observed transmission spectrum. We compute the RCS goodness-of-fit metric to choose the best-performing model runs.
[^1]: `pages.jh.edu/\simdsing3/David_Sing/Limb_Darkening.html`
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Using Kirby Calculus, we explicitly pass from Berge’s R-R descriptions of ten families of knots with lens space surgeries to surgery descriptions on the minimally twisted five chain link (MT5C). Since the MT5C admits a strong involution, we also give the corresponding tangle descriptions.'
address: |
Department of Mathematics, University of Georgia\
Athens, Georgia 30602
author:
- 'Kenneth L. Baker'
bibliography:
- 'MathBiblio.bib'
title: 'Surgery descriptions and volumes of Berge knots II: Descriptions on the minimally twisted five chain link'
---
[^1]
Introduction
============
In [@berge:skwsyls] Berge describes twelve families of knots that admit lens space surgeries. These knots are referred to as [*Berge knots*]{} and appear to comprise all knots in $S^3$ known to have lens space surgeries. Ten of these families, (I)-(VI) being the knots in solid tori with surgeries yielding solid tori and (IX)-(XII) being the ‘sporadic’ knots, are described via R-R diagrams.
\[thm:main\] The knots in Berge’s families (I)-(VI) and (IX)-(XII) admit surgery descriptions on the minimally twisted five chain link.
Figure \[fig:MT5C\] shows the minimally twisted five chain link, MT5C for short. The proof of the theorem is given in §\[sec:proofofmain\]. We use Kirby Calculus (see e.g. [@rolfsen] or [@gompfstipsicz:4makc]) to pass from these R-R diagrams to surgery descriptions on the MT5C. Since the MT5C is strongly invertible, we give the corresponding tangle descriptions of these surgeries in §\[sec:tangles\].
As an immediate consequence of Thurston’s Hyperbolic Dehn Surgery Theorem [@thurston:gt3m] and the fact that the MT5C is a hyperbolic link (e.g. Theorem 5.1 (ii) of [@neumannreid]), our theorem has the following corollary.
Volumes of hyperbolic knots in Berge’s families (I)-(VI) and (IX)-(XII) are bounded above by the volume of the MT5C.
Families (VII) and (VIII) are the knots which lie as essential simple closed curves on the fiber of the trefoil and figure eight knot respectively. In the prequel [@baker:sdavobkI] we show that they contain hyperbolic knots of arbitrarily large volume. Consequentially, they cannot all be all be described by surgery on the MT5C. Nevertheless, each of them admits a surgery description on some minimally twisted chain link.
Acknowledgements
----------------
The author wishes to thank both John Luecke for his direction and many useful conversations and Yuichi Yamada for his comments.
Proof of Theorem {#sec:proofofmain}
================
We prove the theorem by exhibiting a passage from Berge’s R-R diagrams to surgery descriptions on the MT5C, or its reflection. Please refer to [@berge:skwsyls] for Berge’s original descriptions of these knots and the notation conventions. The first step in this passage translates the R-R diagram to a knot on a genus $2$ Heegaard surface for $S^3$ together with surgery instructions on a surrounding link. Half of the correspondence is shown in Figure \[fig:Diagram-to-sfce\]. The other half is obtained by rotating the pictures $180^\circ$ in the plane of the page.
The first six families of Berge knots arise from knots in solid tori with surgeries yielding solid tori. Where relevant we take $n, p, q, r, s, K \in {\ensuremath{\mathbb{Z}}}$, $|ps-qr| = 1$ and $\epsilon = \pm 1$.
Type (I), torus knots
---------------------
We pass from Berge’s diagram (Figure \[fig:I-Diagram\]) to a corresponding realization of these knots $k$ on a genus 2 Heegaard surface via surgeries (Figure \[fig:I-sfce\]). In Figure \[fig:I-sfce\], $\frac{x}{y} = \frac{1}{0}$ corresponds to the meridional filling of $k$ while $\frac{x}{y} = -\frac{1}{1}$ corresponds to the lens space surgery of $k$.
After dropping the Heegaard surface from the picture,we add components with meridional framings, perform isotopies, and do Kirby Calculus as shown in Figure \[fig:I-isotopies\] to get a description of $k$ as surgery on the MT5C.
Type (II), cables about torus knots
-----------------------------------
We pass from Berge’s diagram (Figure \[fig:II-Diagram\]) to a corresponding realization of these knots $k$ on a genus 2 Heegaard surface via surgeries (Figure \[fig:II-sfce\]). In Figure \[fig:II-sfce\], $\frac{x}{y} = \frac{1}{0}$ corresponds to the meridional filling of $k$ while $\frac{x}{y} = -\frac{1}{1}$ corresponds to the lens space surgery of $k$.
After dropping the Heegaard surface from the picture, we add components with meridional framings, perform isotopies, and do Kirby Calculus as shown in Figure \[fig:II-isotopies\] to get a description of $k$ as surgery on the MT5C. Notice that the surgery of $+\frac{1}{1}$ on the component corresponding to $k$ is the lens space surgery. The trivial (i.e. $S^3$) surgery is still $\frac{1}{0}$.
Type (III)
----------
We pass from Berge’s diagram (Figure \[fig:III-Diagram\]) to a corresponding realization of these knots $k$ on a genus 2 Heegaard surface via surgeries (Figure \[fig:III-sfce\]). In Figure \[fig:III-sfce\], $\frac{x}{y} = \frac{1}{0}$ corresponds to the meridional filling of $k$ while $\frac{x}{y} = -\frac{1}{1}$ corresponds to the lens space surgery of $k$. Also, $\frac{r}{s} = \frac{2 \epsilon + (2p + \epsilon) K}{\epsilon + p K}$.
We drop the Heegaard surface to get the first link of Figure \[fig:III-reduced\]. The surgeries on the two parallel components in the first link of Figure \[fig:III-reduced\] can be amalgamated into a surgery on a single component. From here we perform isotopies and Kirby Calculus to arrive at the MT5C. Notice that the surgery of $+\frac{1}{1}$ on the component corresponding to $k$ on the MT5C is the lens space surgery. The trivial (i.e. $S^3$) surgery is still $\frac{1}{0}$.
Type (IV)
---------
We pass from Berge’s diagram (Figure \[fig:IV-Diagram\]) to a corresponding realization of these knots $k$ on a genus 2 Heegaard surface via surgeries (Figure \[fig:IV-sfce\]). In Figure \[fig:IV-sfce\], $\frac{x}{y} = \frac{1}{0}$ corresponds to the meridional filling of $k$ while $\frac{x}{y} = -\frac{1}{1}$ corresponds to the lens space surgery of $k$. Also, $\frac{r}{s} = \frac{2 \epsilon + (2p + \epsilon) K}{\epsilon +p K}$.
After dropping the Heegaard surface from the picture, we add components with meridional framings, perform isotopies, and do Kirby Calculus as shown in Figure \[fig:IV-isotopies\] to get a description of $k$ as surgery on the MT5C. Notice that the surgery of $-\frac{1}{1}$ on the component corresponding to $k$ is the lens space surgery. The trivial (i.e. $S^3$) surgery is still $\frac{1}{0}$.
Type (V)
--------
We pass from Berge’s diagram (Figure \[fig:V-Diagram\]) to a corresponding realization of these knots $k$ on a genus 2 Heegaard surface via surgeries (Figure \[fig:V-sfce\]). In Figure \[fig:V-sfce\], $\frac{x}{y} = \frac{1}{0}$ corresponds to the meridional filling of $k$ while $\frac{x}{y} = -\frac{5}{1}$ corresponds to the lens space surgery of $k$. Also, $\frac{r}{s} = \frac{2 \epsilon + (2p + \epsilon) K}{\epsilon +p K}$.
After dropping the Heegaard surface from the picture, we amalgamate two parallel components and do Kirby Calculus as shown in Figure \[fig:V-isotopies\] to get a description of $k$ as surgery on the MT5C. Notice that the surgery of $+\frac{1}{1}$ on the component corresponding to $k$ is the lens space surgery. The trivial (i.e. $S^3$) surgery is still $\frac{1}{0}$.
Type (VI)
---------
We pass from Berge’s diagram (Figure \[fig:VI-Diagram\]) to a corresponding realization of these knots $k$ on a genus 2 Heegaard surface via surgeries (Figure \[fig:VI-sfce2\]). In Figure \[fig:VI-sfce2\], $\frac{x}{y} = \frac{1}{0}$ corresponds to the meridional filling of $k$ while $\frac{x}{y} = -\frac{5}{1}$ corresponds to the lens space surgery of $k$.
After dropping the Heegaard surface from the picture, we amalgamate the surgeries on two parallel components in the first step of Figure \[fig:VI-isotopies\]. We then continue with Kirby Calculus to get a description of $k$ as surgery on the MT5C. Notice that the surgery of $-\frac{1}{1}$ on the component corresponding to $k$ is the lens space surgery. The trivial (i.e. $S^3$) surgery is still $\frac{1}{0}$.
Sporadic knots type a) and b)
-----------------------------
We pass from Berge’s diagram (Figure \[fig:SpAB-Diagram\]) to a corresponding realization of these knots $k$ on a genus 2 Heegaard surface via surgeries (Figure \[fig:SpAB-sfce\]). In Figure \[fig:SpAB-sfce\], $\frac{x}{y} = \frac{1}{0}$ corresponds to the meridional filling of $k$ while $\frac{x}{y} = \frac{0}{1}$ corresponds to the lens space surgery of $k$. Here $n \in Z$. For type a) knots $(p,p',m,m') = (1,1,2,3)$. For type b) knots $(p,p',m,m') = (2,1,3,2)$.
As shown in Figure \[fig:SpAB-sfce-annulus\], one component bounds a Möbius band. The framing on that component induced by the Möbius band agrees with the framing from the surface.
In Figure \[fig:SpAB-reduced\], we replace the component that bounds the Möbius band with the core curve of the Möbius band and the corresponding surgery instructions. We continue, after an isotopy, with Kirby Calculus in Figure \[fig:SpAB-isotopies\] to get a description of $k$ as surgery on the MT5C.
Sporadic knots type c) and d)
-----------------------------
We pass from Berge’s diagram (Figure \[fig:SpCD-Diagram\]) to a corresponding realization of these knots $k$ on a genus 2 Heegaard surface via surgeries (Figure \[fig:SpCD-sfce\]). In Figure \[fig:SpCD-sfce\], $\frac{x}{y} = \frac{1}{0}$ corresponds to the meridional filling of $k$ while $\frac{x}{y} = +\frac{1}{1}$ corresponds to the lens space surgery of $k$. Here $n \in Z$. For type c) knots $(p,p',m,m') = (4,-3,3,-2)$. For type d) knots $(p,p',m,m') = (3,-5,2,-3)$.
As shown in Figure \[fig:SpCD-sfce-annulus\], one component bounds a Möbius band. The framing on that component induced by the Möbius band agrees with the framing from the surface.
In Figure \[fig:SpCD-reduced\], we replace the component that bounds the Möbius band with the core curve of the Möbius band and the corresponding surgery instructions. We continue, after an isotopy, with Kirby Calculus in Figure \[fig:SpCD-isotopies\] to get a description of $k$ as surgery on the MT5C.
Summary of Berge knots as surgeries on the MT5C. {#subsec:summary}
------------------------------------------------
We conclude the proof of the theorem by summarizing the results above. Figure \[fig:MT5C-BergeI-VI\] shows Berge knot types (I) — (VI) as surgeries on the minimally twisted five chain link or its mirror. Figure \[fig:MT5C-BergeSp\] shows Berge sporadic knot types a) — d) as surgeries on the the minimally twisted five chain link or its mirror. The component corresponding to the Berge knot is shown with a pair of surgeries $(\rho_{S^3}, \rho_{\textrm{LensSp}})$ where $\rho_{S^3}$ is the surgery slope yielding $S^3$ and $\rho_{\textrm{LensSp}}$ is the surgery slope yielding the lens space. Recall also that $n, p, q, r, s, \in {\ensuremath{\mathbb{Z}}}$, $|ps-qr|=1$ and $\epsilon = \pm1$.
Tangles {#sec:tangles}
=======
Because the MT5C is strongly invertible, we may rephrase the minimally twisted chain link surgery descriptions of Berge’s families of knots as tangle descriptions. We include these tangle descriptions as they may be of interest to others and provide an alternate means of verification of the above surgery descriptions.
On the remaining boundary component of each of the the tangles below, a choice of rational tangles to be inserted is indicated. Inserting the first rational tangle (which is the $\infty$ tangle in each case) renders the tangle into the unknot. As the double branched cover of the unknot in $S^3$ is $S^3$ again, this corresponds to the trivial surgery on the Berge knot. Inserting the second rational tangle yields a two-bridge link, a link that may be decomposed into two rational tangles. As the double branched cover of a two-bridge knot is a lens space, this corresponds to the lens space surgery on the Berge knot.
Conventions {#subsec:tangleconv}
-----------
A [*tangle*]{} $(B,t)$ is a pair consisting of a punctured $3$-sphere $B$ and a properly embedded collection $t$ of disjoint arcs and simple closed curves. Two tangles $(B_1, t_1)$ and $(B_2, t_2)$ are homeomorphic if there is a homeomorphism of pairs $$h \colon (B_1, t_1) \to (B_2, t_2).$$
A boundary component $({\ensuremath{\partial}}B_0, t \cap {\ensuremath{\partial}}B_0)$ of a tangle $(B,t)$ is a sphere ${\ensuremath{\partial}}B_0$ together with some finite set of points $p=t \cap {\ensuremath{\partial}}B_0$. Here we consider the situation where $p$ consists of just four points.
Given a sphere $S$ and set $p$ of four distinct points on $S$, a [*framing*]{} of $(S, p)$ is an ordered pair of (unoriented) simple closed curves $(\hat{m}, \hat{l})$ on $S - N(p)$ such that each curve separates different pairs of points of $p$.
Let $(S, p)$ be a sphere with four points with framing $(\hat{m}, \hat{l})$. The double cover of $S$ branched over $p$ is a torus. Single components, say $m$ and $l$, of the lifts of the framing curves $\hat{m}$ and $\hat{l}$ when oriented so that $m \cdot l = +1$ (with respect to the orientation of the torus) form a basis for the torus. Similarly, a basis on a torus induces a framing on the sphere with four points obtained by the quotient of an involution that fixes four points on the torus. See Figure \[fig:tangleframing\].
This implies the correspondence between inserting into a boundary component of a tangle the rational tangle $p/q$ and the $p/q$ Dehn filling of a torus boundary component in the double branched cover of the tangle.
Figure \[fig:tanglelegend\] shows by example the conventions we use for rational tangles. Here, $[a,b, \dots, c]$ denotes the continued fraction $1/(a - 1/(b- 1/( \dots -1/c)))$. Notice that $1 = [1] = [0,-1]$.
$$\begin{array}{c}
\PandocStartInclude{Figures/tanglelegend1.pstex_t}\PandocEndInclude{input}{481}{38}\\
\PandocStartInclude{Figures/tanglelegend2.pstex_t}\PandocEndInclude{input}{482}{38}
\end{array}$$
Tangle descriptions
-------------------
Please refer to [@baker:sdavobkI] for an illustration of the passage between a surgery description on the MT5C to a tangle description. We now present the tangle descriptions obtained from the surgery descriptions in §\[subsec:summary\] above using the tangle conventions in §\[subsec:tangleconv\]. Note that some of the surgery descriptions in §\[subsec:summary\] are on the mirror of the MT5C and these tangle descriptions reflect that accordingly. In each tangle below, where presented with the choice $(\infty, \delta)$ for $\delta \in\{-1,0,+1\}$ inserting the rational tangle $\infty$ yields the unknot and inserting the rational tangle $\delta$ yields a $2$–bridge link.
[^1]: This work was partially supported by a graduate traineeship from the VIGRE Award at the University of Texas at Austin and a VIGRE postdoc under NSF grant number DMS-0089927 to the University of Georgia at Athens.
|
{
"pile_set_name": "ArXiv"
}
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---
author:
- 'Truong Thi Hong Thanh and Duong Quoc Viet\'
date:
-
-
title: |
**TOWARD MIXED MULTIPLICITIES\
AND JOINT REDUCTIONS**
---
Introduction
============
Let $(A, \frak{m})$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$ and infinite residue field $k = A/\mathfrak{m}.$ Let $M$ be a finitely generated $A$-module. Let $J$ be an $\frak
m$-primary ideal, $ I_1,\ldots, I_d$ be ideals of $A.$ Put $I
= I_1\cdots I_d;$ $\overline {M}= M/0_M: I^\infty$ and $$\begin{aligned}
&{\bf n} =(n_1,\ldots,n_d);{\bf k} = (k_1,\ldots,k_d); {\bf
0}=(0,\ldots,0);
\\
&\mathbf{e}_i = (0, \ldots, 1, \ldots, 0)\in \mathbb{N}^{d}
(\text{the }i\text{th coordinate is } 1);\\
& \mathrm{\bf I}= I_1,\ldots,I_d; \mathrm{\bf I}^{[\mathrm{\bf
k}]}= I_1^{[k_1]},
\ldots,I_d^{[k_d]}; \mathbb{I}^{\mathrm{\bf n}}= I_1^{n_1}\cdots I_d^{n_d};
\\
&{\bf n^k}= n_1^{k_1}\cdots n_d^{k_d}; \mathbf{k}!= k_1!\cdots
k_d!; \;|{\bf k}| = k_1 + \cdots + k_d.\end{aligned}$$ Assume that $I
\nsubseteq \sqrt{\mathrm{Ann}M},$ then $\overline {M} \not= 0.$ Set $q=\dim \overline {M}$. Then $\ell\Big(\dfrac{J^{n_0}\mathbb{I}^{\bf
n}M}{J^{n_0+1}\mathbb{I}^{\bf n}M}\Big)$ is a polynomial of total degree $q-1$ for all large enough $n_0, \bf n$ by [@Vi Proposition 3.1] (see [@MV]). Denote by $P$ this polynomial. Write the terms of total degree $q-1$ of $P$ in the form $\sum_{k_0 + |\mathbf{k}| = q - 1}
e(J^{[k_0+1]},\mathrm{\bf I}^{[\mathrm{\bf k}]};
M)\frac{n_0^{k_0}\mathbf{n}^\mathbf{k}}{k_0!\mathbf{k}!},$ then $e(J^{[k_0+1]},
\mathbf{I}^{[\mathbf{k}]}; M)$ are non-negative integers not all zero, and $e(J^{[k_0+1]},
\mathbf{I}^{[\mathbf{k}]}; M)$ is called the [*mixed multiplicity of $M$ with respect to $J, \bf I$ of the type $(k_0+1, \bf k)$*]{} (see e.g. [@MV; @Ve; @Vi]).
It has long been known that the mixed multiplicity of ideals is an important invariant of algebraic geometry and commutative algebra. In past years, mixed multiplicities of ideals have attracted much attention (see e.g. \[3$-$10, 13$-$34\]).
We turn now to some facts of mixed multiplicities $e(J^{[k_0+1]},
\mathbf{I}^{[\mathbf{k}]}; M)$ related to joint reductions. In the case of ideals of dimension $0,$ Risler and Teissier in 1973 [@Te] defined mixed multiplicities and interpreted them as Hilbert-Samuel multiplicities of ideals generated by general elements; Rees in 1984 [@Re] built joint reductions and showed that each mixed multiplicity is the multiplicity of a joint reduction. And under the viewpoint of joint reductions, Rees’s mixed multiplicity theorem can be viewed as an extension of the result of Risler and Teissier. Developing this theorem of Rees, Böger extended to the case of non-$\frak m$-primary ideals (see [@SH]); Swanson proved the converse of Rees’s mixed multiplicity theorem for formally equidimensional rings [@Sw] (see [@SH Theorem 17.6.1]); Dinh-Thanh-Viet [@VDT Theorem 3.1] extended to the case that the ideal $I$ have height larger than $|\mathbf{k}|.$ But whether there is a similar result for arbitrary ideals, is not yet known. This is one of motivations to direct us towards the following question.
[**Question 1:**]{} When are mixed multiplicities equal to the Hilbert-Samuel multiplicity of joint reductions?
Recall that the notion of joint reductions was built in Rees’s work in 1984 [@Re]. And O’Carroll in 1987 [@Oc] proved the existence of joint reductions in the general case. This concept was studied in .
Let $\frak I_i$ be a sequence consisting $k_i$ elements of $I_i$ for all $1 \le i \le d.$ Put ${\bf x} = \frak I_1, \ldots, \frak
I_d$ and $(\emptyset) = 0_A$. Then ${\bf x}$ is called a [*joint reduction*]{} of $\mathbf I$ with respect to $M$ of the type ${\bf
k}=(k_1,\ldots,k_d)$ if $\mathbb{I}^{\mathbf{n} }M =
\sum_{i=1}^d(\frak I_i) \mathbb{I}^{\mathbf{n} - \mathbf{e}_i}M$ for all large $\bf n.$ If $d=1$ then $(\frak I_1)$ is called a [*reduction*]{} of $I_1$ with respect to $M$ [@NR].
The direction towards Question 1 leads us to the following result.
\[thm2.31/1\] Let ${\bf x}$ be a joint reduction of ${\bf I}, J$ with respect to $M$ of the type $({\bf k}, k_0+1)$ with $k_0 + |{\bf k}| = \dim \overline M-1.$ Assume that $\dim {M}/I{M} < \dim {M} - |\bf k|.$ Then ${\bf x}$ is a system of parameters for ${M}$ and $e(J^{[k_0 +1]}, \mathbf{I}^{[\mathbf{k}]}; M) = e(\mathbf{x};
{M}).$
It should be noted that if one omits the assumption $\dim {M}/I{M} < \dim {M} - |\bf k|,$ then in general, $e(J^{[k_0 +1]}, \mathbf{I}^{[\mathbf{k}]}; M)$ can not be the Hilbert-Samuel multiplicity of a joint reduction of ${\bf I}, J$ with respect to $M$ of the type $({\bf k}, k_0+1)$ even when this joint reduction is a system of parameters for $M$ (see Remark \[rm35\]).
Theorem \[thm2.31/1\] not only replaces the condition on the height of the ideal $I$ in the statement of [@VDT Theorem 3.1] by a weaker condition on $I$, but also removes the condition that the joint reduction ${\bf x}$ is a system of parameters for ${M}$ in [@VDT Theorem 3.1]. Theorem \[thm2.31/1\] seems to make the problem of expressing mixed multiplicities into the Hilbert-Samuel multiplicity of joint reductions become closer.
The paper is divided into three sections. Section 2 is devoted to the discussion of some notions and results used in the paper. Section 3 proves the main result. And as interesting consequences of the main result, we get Corollary \[co4.0a\] that is a stronger result than [@VDT Theorem 3.1], and Corollary \[thm2.19v\] which shows that mixed multiplicities are the Hilbert-Samuel multiplicity of weak-(FC)-sequences.
Mixed multiplicities and some relative sequences
================================================
In this section, we recall notions of weak-(FC)-sequences and joint reductions, and give some facts on the relationship between mixed multiplicities and these sequences.
Let $(A, \frak{m})$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$ and infinite residue field $k = A/\mathfrak{m}.$ Let $M$ be a finitely generated $A$-module. Let $J$ be an $\frak
m$-primary ideal, $ I_1,\ldots, I_d$ be ideals of $A.$ Set $I
= I_1\cdots I_d;$ $\overline {M}= M/0_M: I^\infty;$ $q=\dim \overline {M}$ and $$\begin{aligned}
&\mathbf{e}_i = (0, \ldots, \stackrel{(i)}{1}, \ldots, 0); {\bf n} =(n_1,\ldots,n_d); {\bf k} =(k_1,\ldots,k_d); {\bf
0}=(0,\ldots,0)\in \mathbb{N}^{d}; \\
& |{\bf k}| = k_1 + \cdots + k_d; \mathrm{\bf I}= I_1,\ldots,I_d;
\mathbb{I}^{\mathrm{\bf n}}= I_1^{n_1}\cdots I_d^{n_d}; \mathrm{\bf I}^{[\mathrm{\bf
k}]}= I_1^{[k_1]},
\ldots,I_d^{[k_d]}.\end{aligned}$$ And let $I
\nsubseteq \sqrt{\mathrm{Ann}M}.$ Denote by $P(n_0, {\bf n}, J, \mathbf{I}, M)$ the Hilbert polynomial of the function $\ell\Big(\dfrac{J^{n_0}\mathbb{I}^{\bf
n}M}{J^{n_0+1}\mathbb{I}^{\bf n}M}\Big).$ Then recall that $P(n_0, {\bf n}, J, \mathbf{I}, M)$ is a polynomial of total degree $q-1$ by [@Vi Proposition 3.1] (see [@MV]). If one writes the terms of total degree $q-1$ of $P(n_0, {\bf n}, J, \mathbf{I}, M)$ in the form $\sum_{k_0 + |\mathbf{k}| = q - 1}
e(J^{[k_0+1]},\mathrm{\bf I}^{[\mathrm{\bf k}]};
M)\frac{n_0^{k_0}\mathbf{n}^\mathbf{k}}{k_0!\mathbf{k}!},$ then $e(J^{[k_0+1]},
\mathbf{I}^{[\mathbf{k}]}; M)$ is called the [*mixed multiplicity of $M$ with respect to $J, \bf I$ of the type $(k_0+1, \bf k)$*]{} (see e.g. [@MV; @Ve; @Vi]).
Denote by $\bigtriangleup^{(h_0,\mathbf{h})}Q(n_0, \bf n)$ the $(h_0,\bf
h)$-difference of the polynomial $Q(n_0,\bf n).$ And throughout the paper, we put $e(J^{[k_0+1]},
\mathbf{I}^{[\mathbf{k}]}; M) =0 $ if $k_0 + |{\bf k}| > q-1.$
\[re2.00\] Let $e(J^{[k_0+1]},
\mathbf{I}^{[\mathbf{k}]}; M)$ be the mixed multiplicity of $M$ with respect to ideals $J,\mathrm{\bf I}$ of the type $(k_0+1,\mathrm{\bf k}).$ Then $e(J^{[k_0+1]},
\mathbf{I}^{[\mathbf{k}]}; M) = \bigtriangleup^{(k_0,\mathrm{\bf
k})}P(n_0, {\bf n}, J, \mathbf{I}, M).$
The concept of joint reductions of $\mathfrak{m}$-primary ideals was given by Rees [@Re] in 1984. This concept was extended to the set of arbitrary ideals by .
\[de01\] Let $\frak I_i$ be a sequence consisting $k_i$ elements of $I_i$ for all $1 \le i \le d$ and $k_1,\ldots,k_d \ge 0.$ Put ${\bf x} = \frak I_1, \ldots, \frak
I_d$ and $(\emptyset) = 0_A$. Then ${\bf x}$ is called a [*joint reduction*]{} of $\bf I$ with respect to $M$ of the type ${\bf k}
=(k_1,\ldots,k_d)$ if $\mathbb{I}^{\mathbf{n} }M =
\sum_{i=1}^d(\frak I_i) \mathbb{I}^{\mathbf{n} - \mathbf{e}_i}M$ for all large $\bf n.$ If $d=1$ then $(\frak I_1)$ is called a [*reduction*]{} of $I_1$ with respect to $M$ [@NR].
The weak-(FC)-sequence, which was defined in [@Vi], is a kind of superficial sequences, and it is proven to be useful in several contexts (see e.g. [@DMT; @DV; @DQV; @VT; @VT4]).
\[de011\] Set $I
= I_1\cdots I_d.$ An element $x \in I_i$ $(1 \leqslant i \leqslant d)$ is called a [*weak*]{}-(FC)-[*element*]{} of $\bf I$ with respect to $M$ if the following conditions are satisfied:
1. $x{M}\bigcap {\mathbb I}^{\mathbf{n}}{M}
= x {\mathbb I}^{\mathbf{n}-\mathbf{e}_i}{M}$ for all $n_i \gg 0$ and all $n_1,\ldots,n_{i-1},n_{i+1},
\ldots, n_d \geq 0$.
2. $x$ is an $I$-filter-regular element with respect to $M,$ i.e.,$0_M:x \subseteq 0_M: I^{\infty}.$
Let $x_1, \ldots, x_t$ be elements of $A$. For any $0\leqslant i
\leqslant t,$ set $M_i = {M}\big/{(x_1, \ldots, x_{i})M}$. Then $x_1, \ldots, x_t$ is called a [*weak*]{}-(FC)-[*sequence*]{} of $\mathbf{I}$ with respect to $M$ if $x_{i + 1}$ is a weak-(FC)-element of $\mathbf{I}$ with respect to $M_i$ for all $0 \leqslant i \leqslant t - 1$. If a weak-(FC)-sequence consists of $k_i$ elements of $I_i$ $(1 \le i \le d),$ then it is called a weak-(FC)-sequence of the type $(k_1,\ldots,k_d).$
The following remark recalls some important properties of weak-(FC)-sequences.
\[note12\] Let $\frak I_i$ be a sequence of elements of $I_i$ for all $1 \le i \le d$. Assume that $\frak I_1, \ldots, \frak
I_d$ is a weak-(FC)-sequence of $\mathbf{I}$ with respect to $M$. Then $$\label{vttt1}(\frak I_1, \ldots, \frak
I_d)M \bigcap \mathbb{I}^{\mathbf{n}}{M}
= \sum_{i=1}^d(\frak I_i)
\mathbb{I}^{\mathbf{n} - \mathbf{e}_i}M$$ for all large $\bf n$ by [@Vi4 Theorem 3.4 (i)]. And if $x \in I_i$ $(1 \le i \le d)$ is a weak-[(FC)]{}-element of $\mathbf{I}, J$ with respect to $M,$ then $$\label{vttt2}P(n_0, {\bf n}, J, \mathbf{I}, M/xM)= P(n_0, {\bf n}, J, \mathbf{I}, M)-P(n_0, {\bf n}-\mathbf{e}_i, J, \mathbf{I}, M)$$ by [@DV (3)] (or the proof of [@MV Proposition 3.3 (i)]).
The role of weak-(FC) sequences in studying mixed multiplicities is showed by the following note which will be used as a tool in this paper.
\[re2.6aa\] Set $J= I_0.$ Let $x \in I_i$ $(0 \le i \le d)$ be a weak-[(FC)]{}-element of $\mathbf{I}, J$ with respect to $M$. Then by (\[vttt2\]), we get $\dim M/xM:I^\infty \leqslant \dim M/0_M:I^\infty -1$ and
1. $P(n_0, {\bf n}, J, \mathbf{I}, M/xM)=\begin{cases}
\bigtriangleup^{(0,\mathrm{\bf e}_i)}P(n_0, {\bf n}, J,
\mathbf{I}, M) \quad \text{ if }\; 1 \le i \le d\\
\bigtriangleup^{(1,\mathrm{\bf 0})}P(n_0, {\bf n}, J, \mathbf{I},
M) \quad \;\text{ if }\; i = 0.
\end{cases} $
2. If $k_i > 0$ for certain $0 \le i \le d,$ then by Remark \[re2.00\] and (i) we obtain $$e(J^{[k_0+1]}, \mathbf{I}^{[\mathbf{k}]}; M) =\begin{cases}
e(J^{[k_0+1]}, \mathbf{I}^{[\mathrm{\bf k} - \mathbf{e}_i]};
M\big/xM) \;\text{ if }\; 1 \le i \le d\\
e(J^{[(k_0-1)+1]}, \mathbf{I}^{[\mathrm{\bf k}]}; M\big/xM)
\text{ if } \; i = 0.
\end{cases}$$
3. For any $\mathbf{k} \in \mathbb{N}^{d}$, by [@VDT Proposition 2.3] (see [@Vi Remark 1]), there exists a weak-$\mathrm{(FC)}$-sequence of $\mathbf{I}$ with respect to $M$ of the type $\mathbf{k}.$
4. Let $\mathbf{y}$ be a weak-(FC)-sequence of $\mathbf{I}, J$ with respect to $M.$ Then by (\[vttt1\]) in Remark \[note12\], it follows that $\mathbf{y}$ is a joint reduction of $\mathbf{I}, J$ with respect to $M$ if and only if $J^{n_0}\mathbb{I}^{\mathbf{n}}(M/(\mathbf{y})M) = 0$ for all large $n_0, \mathbf{n}.$
5. Let $|{\bf
k}|+k_0 = \dim \overline{M}-1.$ Then $\deg P(n_0, {\bf
n}, J, \mathbf{I}, M) = |{\bf
k}|+k_0 $ by [@Vi Proposition 3.1] (see [@MV]). So $\bigtriangleup^{(k_0+1, \mathrm{\bf k})}P(n_0, {\bf
n}, J, \mathbf{I}, M) = 0.$ Now if ${\bf x}$ is a weak-[(FC)]{}-sequence of $\mathbf{I}, J$ with respect to $M$ of the type $(\mathrm{\bf k}, k_0+1),$ then by (i) we obtain $P(n_0, {\bf n}, J, \mathbf{I},
M/({\bf x})M)=\bigtriangleup^{(k_0+1, \mathrm{\bf k})}P(n_0, {\bf
n}, J, \mathbf{I}, M) = 0.$ Hence by (iv), ${\bf x}$ is a joint reduction of $\mathbf{I}, J$ with respect to $M$ of the type $(\mathrm{\bf k}, k_0+1).$
In order to prove the results of this paper, we need the following facts.
Recall that an ideal $\frak{a}$ of $A$ is called an [*ideal of definition*]{} of $M$ if $\ell_A(M/\frak{a}M)<\infty,$ and a sequence ${\bf y} = y_1, \ldots, y_n$ of elements in $ \frak m$ is called a [*multiplicity system*]{} of $M$ if $(\bf y)$ is an ideal of definition of $M$ (see e.g. [@BH1 Page 192]). Let ${\bf y}$ be a multiplicity system. Then one defines the [*multiplicity symbol*]{} of ${\bf y}$ as follows: if $n=0$, then $\ell_A(M) <\infty$, and set $e(\mathbf{y};M) = \ell_A(M)$. If $n> 0$, set $$e(\mathbf{y};
M) = e(\mathbf{y}'; M\big/y_1M) - e(\mathbf{y}'; 0_{M}:y_1)$$ (see e.g. [@BH1 Definition 4.7.3]). It is well known that $e(\mathbf{y}; M) \not=0$ if and only if $\bf
y$ is a system of parameters for $M$, and in this case, $e(\mathbf{y}; M) = e((\mathbf{y}); M)$ is the Hilbert-Samuel multiplicity of the ideal $(\bf y)$ with respect to $M$ (see e.g. [@BH1]). And $I
= I_1\cdots I_d;$ $\overline {M}= M/0_M: I^\infty;$ $q=\dim \overline {M}.$
\[no4.3a\] Let ${\bf x}$ be a finite sequence of elements in $ \frak m.$ Then we have:
1. $(\mathbf{x})$ is an ideal of definition of $I^mM$ for all large enough $m$ if and only if $(\mathbf{x})$ is an ideal of definition of $\overline {M}$.
2. If $\mathbf{x}$ is a multiplicity system of $\overline{M},$ then $ e(\mathbf{x}; I^mM)
= e(\mathbf{x}; \overline {M})$ for all large enough $m.$
3. Let $\bf x$ be a joint reduction of $\mathbf{I},J$ with respect to $M$ of the type $(
{\bf 0}, k_0+1).$ Then $(\mathbf{x})$ is an ideal of definition of $\overline {M}$ and $e(J^{[k_0+1]}, \mathbf{I}^{[\mathbf{0}]};
M) =e(\mathbf{x}; I^mM) = e(\mathbf{x}; \overline{M})$ for all large enough $m.$ Moreover, $\dim \overline{M} \le k_0+1,$ and equality holds if only if $e(J^{[k_0+1]},
\mathbf{I}^{[\mathbf{0}]}; M) \not=0,$ and in this case, ${\bf x}$ is a system of parameters for $ \overline {M}.$
Because $\sqrt{\mathrm{Ann}[\overline
{M}\big/(\mathbf{x})\overline {M}}]=
\sqrt{\mathrm{Ann}[I^mM\big/(\mathbf{x})I^mM]}$ for all large enough $m$, we have (i). By Artin-Rees Lemma, it implies that $I^mM \cap (0_M: I^\infty) = 0$ for all large enough $m$. Hence $I^m\overline {M} \cong
I^mM$ for all large enough $m$. From this it follows that $
e(\mathbf{x}; I^mM) = e(\mathbf{x}; I^m\overline {M})$ for all large enough $m.$ Therefore we get (ii) since $\dim (\overline {M}\big/I^m\overline {M}) < \dim
\overline {M}.$ The proof of (iii): Since $\bf x$ is a joint reduction of the type $( {\bf 0}, k_0+1),$ $(\bf x)$ is a reduction of $J$ with respect to $I^mM$ for large enough $m.$ Thus ${\bf x}$ is a multiplicity system of $I^mM$ for large enough $m$ since $J$ is $\frak m$-primary, and so of $\overline {M}$ by (i). Hence $\dim \overline{M} \le |{\bf x}|= k_0+1.$ By [@Vi Lemma 3.2 (i)] and (ii) we get $e(J^{[k_0+1]},
\mathbf{I}^{[\mathbf{0}]}; M) =e(\mathbf{x}; I^mM) = e(\mathbf{x};
\overline{M})$. And thus $\dim \overline{M} = k_0+1$ if and only if $e(J^{[k_0+1]}, \mathbf{I}^{[\mathbf{0}]}; M)\not=0,$ in this case, ${\bf x}$ is a system of parameters for $ \overline {M}.$
Mixed multiplicities and multiplicities of joint reductions
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This section proves Theorem \[thm2.19vt\] which is the main theorem of the paper. And as immediate consequences of this theorem, we obtain Corollary \[co4.0a\] which is a stronger result than [@VDT Theorem 3.1], and Corollary \[thm2.19v\] that interpreted mixed multiplicities as Hilbert-Samuel multiplicities of weak-[(FC)]{}-sequences.
To prove the main theorem, we need the following lemmas.
\[le2019\] Let $x, x_1\in I_1$ and $x_2\in J.$ Let $\dim M/IM < \dim M.$ Assume that $x,x_2$ and $x_1,x_2$ are systems of parameters for ${M}$ and are joint reductions of ${\bf I}, J$ with respect to $M.$ Then $$e(x,x_2;M)=e(x_1,x_2;M).$$
Set $$\Pi = \{\frak p \in \mathrm{Min}{M}\mid \dim A/\frak p = \dim
{M}\}.$$ Let $\mathfrak{p} \in \Pi,$ set $B =
A/\mathfrak{p}.$ Since $x_1, x_2$ and $x, x_2$ are both joint reductions of $\mathbf{I}, J$ with respect to $M$ and are systems of parameters for ${M}$, $x_1, x_2$ and $x, x_2$ are both joint reductions of $\mathbf{I}, J$ with respect to $B$, and are systems of parameters for ${B}$ (see e.g [@SH Lemma 17.1.4]). Hence there exists $m \gg 0$ such that $(x_1)$ and $(x)$ are both reductions of $I_1$ with respect to $J^m(C/x_2C),$ here $ C
=\begin{cases} B \;\;\;\;\;\;\;\;\;\;\qquad\text{ if }
d =1\\
(I_2 \cdots I_d)^mB \;\text{ if }\; d >1.\\
\end{cases} $ Since $\dim M/IM < \dim M$, $I =
I_1\cdots I_d \nsubseteq \frak p$ and so $\dim B/C < \dim B.$ Hence since $x_1, x_2$ and $x, x_2$ are systems of parameters for $B,$ it can be verified that $x_1, x_2$ and $x, x_2$ are systems of parameters for $C.$ Since $J$ is $\frak m$-primary, $$\dim
(C/x_2C)/J^m(C/x_2C) = 0.$$ From this it follows that $x_1$ and $x$ are systems of parameters for $J^m(C/x_2C).$ Moreover, since $(x_1)$ and $(x)$ are reductions of $I_1$ with respect to $J^m(C/x_2C),$ by [@NR Theorem 1], we get $e(x;J^m(C/x_2C)) =
e(I_1;J^m(C/x_2C)) = e(x_1; J^m(C/x_2C)).$ Therefore we obtain $e(x;J^m(C/x_2C)) = e(x_1;
J^m(C/x_2C)).$ On the other hand since $\dim
(C/x_2C)/J^m(C/x_2C) = 0,$ we get $e(x; C/x_2C)= e(x;
J^m(C/x_2C))$ and $e(x_1; C/x_2C)= e(x_1; J^m(C/x_2C))$. So $e(x;
C/x_2C)=e(x_1; C/x_2C). $ Remember that $x_1, x_2$ is a system of parameters for $M$. Hence $x_2\notin \frak{q}$ for all $\frak{q}\in \Pi$. Consequently $x_2$ is not a zero divisor of $C$. From this it follows that $e(x; C/x_2C) =
e(x,x_2;C)$ and $$e(x_1; C/x_2C) =e(x_1,x_2;C)$$ by [@AB Page 641, lines 27-28, (D)]. Therefore we have $e(x,x_2;C)=e(x_1,x_2;C).$ And since $\dim B/C < \dim B,$ $e(x,x_2;B)= e(x,x_2;C)$ and $e(x_1,x_2;C)=e(x_1,x_2;B).$ Hence we obtain $$e(x,x_2;B)= e(x_1,x_2;B).$$ Thus for any $\mathfrak{p} \in \Pi,$ $$e(x,x_2; A/\mathfrak{p})
= e(x_1,x_2;A/\mathfrak{p}).$$ Consequently, we get $$\sum_{\mathfrak{p}\in \Pi}
\ell_A(M_{\mathfrak{p}})e(x,x_2; A/\mathfrak{p}) = \sum_{\mathfrak{p}\in
\Pi}\ell_A(M_{\mathfrak{p}}) e(x_1,x_2; A/\mathfrak{p}).$$ On the other hand, $e(x,x_2; {M})
=\sum_{\mathfrak{p}\in \Pi}\ell_A(M_{\mathfrak{p}})e(x,x_2;
A/\mathfrak{p})$ and $$e(x_1,x_2; M)
=\sum_{\mathfrak{p}\in
\Pi}\ell_A(M_{\mathfrak{p}})e(x_1,x_2; A/\mathfrak{p})$$ (see e.g. [@SH Theorem 11.2.4]). Therefore $e(x,x_2; M)= e(x_1,x_2;{M}).$
\[bd32\] Let ${\bf x}$ be a joint reduction of ${\bf I}, J$ with respect to $M$ of the type $({\bf k}, k_0+1)$ with $k_0 + |{\bf k}| = \dim \overline M-1.$ Assume that $\dim {M}/I{M} < \dim {M} - |\bf k|.$ Then ${\bf x}$ is a system of parameters for ${M}$.
Set $n = k_0 + |{\bf k}| +1.$ Let ${\bf x}=x_1,\ldots,x_n$ be a joint reduction of ${\bf I}, J$ with respect to $M$ of the type $({\bf k}, k_0+1)$ with ${\bf x}_{\mathbf{I}} = x_1, \ldots,
x_{|\mathbf{k}| } \subset \mathbf{I}$ and $U = x_{|\mathbf{k}|+1},\ldots,x_n \subset J.$ Then $(U)$ is a reduction of $J$ with respect to $I^m[M/({\bf x}_{\mathbf{I}})M]$ for all large enough $m.$ Since $J$ is $\frak m$-primary, therefore $(U)$ is an ideal of definition of $I^m[M/({\bf x}_{\mathbf{I}})M]$ for all large enough $m.$ Hence $(U)$ is also an ideal of definition of $M/({\bf x}_{\mathbf{I}})M:I^\infty$ by Lemma \[no4.3a\] (i). Since $\dim {M}/I{M} < \dim {M} - |\bf k|$ and $\dim {M}/({\bf x}_{\mathbf{I}}){M} \ge \dim {M} - |\bf k|,$ it follows that if $\frak p \in \mathrm{Min}[{M}/({\bf x}_{\mathbf{I}}){M}]$ such that $\dim A/\frak p = \dim
{M}/({\bf x}_{\mathbf{I}}){M}$ then $I \nsubseteq \frak p.$ Hence $\frak p \in \mathrm{Min}[{M}/({\bf x}_{\mathbf{I}}){M}:I^\infty].$ So $\dim A/\frak p \le \dim {M}/({\bf x}_{\mathbf{I}}){M}:I^\infty.$ Since $(U)$ is an ideal of definition of $M/({\bf x}_{\mathbf{I}})M:I^\infty,$ we get $$\dim {M}/({\bf x}_{\mathbf{I}}){M}:I^\infty \le |U| = k_0+1= \dim \overline{M} - |{\bf k}| \le \dim {M} - |\bf k|.$$ Consequently, we obtain $$\dim {M} - |{\bf k}| \le \dim
{M}/({\bf x}_{\mathbf{I}}){M}=\dim A/\frak p \le \dim {M}/({\bf x}_{\mathbf{I}}){M}:I^\infty \le \dim {M} - |\bf k|.$$ Thus, $$\label{pt-001VT}\begin{aligned}\dim
{M}/({\bf x}_{\mathbf{I}}){M}=\dim {M}/({\bf x}_{\mathbf{I}}){M}:I^\infty =\dim {M} - |\bf k|\end{aligned}$$ and $$\label{pt-001VT0}\begin{aligned}&\{\frak p \in \mathrm{Min}[{M}/({\bf x}_{\mathbf{I}}){M}] \mid \dim A/\frak p = \dim
{M}/({\bf x}_{\mathbf{I}}){M}\}\\ &= \{\frak p \in \mathrm{Min}[{M}/({\bf x}_{\mathbf{I}}){M}:I^\infty] \mid \dim A/\frak p = \dim
{M}/({\bf x}_{\mathbf{I}}){M}:I^\infty\}. \end{aligned}$$ By (\[pt-001VT\]), ${\bf x}_{\mathbf{I}}$ is part of a system of parameters for ${M}$ and $U$ is a system of parameters for ${M}/({\bf x}_{\mathbf{I}}){M}:I^\infty.$ Since $U$ is a system of parameters for ${M}/({\bf x}_{\mathbf{I}}){M}:I^\infty,$ it follows by (\[pt-001VT0\]) that $U$ is a system of parameters for ${M}/({\bf x}_{\mathbf{I}}){M}.$ So ${\bf x}$ is a system of parameters for ${M}.$
\[thm2.19vt\] Let ${\bf x}$ be a joint reduction of ${\bf I}, J$ with respect to $M$ of the type $({\bf k}, k_0+1)$ with $k_0 + |{\bf k}| = \dim \overline M-1.$ Assume that $\dim {M}/I{M} < \dim {M} - |\bf k|.$ Then ${\bf x}$ is a system of parameters for ${M}$ and $$e(J^{[k_0 +1]}, \mathbf{I}^{[\mathbf{k}]}; M) = e(\mathbf{x};
{M}).$$
By Lemma \[bd32\], ${\bf x}$ is a system of parameters for ${M}$. Recall that $n = k_0 + |{\bf k}| +1$ and ${\bf x}=x_1,\ldots,x_n.$ Set $$\Pi = \{\frak p \in \mathrm{Min}{M}\mid \dim A/\frak p = \dim
{M}\}.$$ Note that since ${\bf x}$ is a joint reduction of ${\bf I}, J$ with respect to $M, $ ${\bf x}$ is also a joint reduction of ${\bf I}, J$ with respect to $A/\frak p$ for all $\frak p \in \Pi$ (see e.g [@SH Lemma 17.1.4]). And since $\dim {M}/I{M} < \dim {M} - |\bf k|,$ it follows that $\dim A/(\frak p, I) < \dim A/\frak p - |\bf k|$ for all $\frak p \in \Pi.$ Indeed, assume that $\dim {M}/I{M} = \dim {M}-t,$ then there exist $a_1,\ldots,a_t \in I$ such that $\dim {M}/(a_1,\ldots,a_t){M} = \dim {M}-t.$ So $a_1,\ldots,a_t$ is part of a system of parameters for ${M}.$ Hence $a_1,\ldots,a_t$ is also part of a system of parameters for $A/\frak p.$ Consequently, $$\begin{aligned}&\dim A/(\frak p, I) \le \dim A/(\frak p, a_1,\ldots,a_t) = \dim A/\frak p -t = \dim {M}-t \\&= \dim {M}/I{M} < \dim {M} - |{\bf k}|=\dim A/\frak p - |\bf k|.\end{aligned}$$ So $$\label{pt-001KV}\begin{aligned}
\dim A/(\frak p,I)A < \dim A/\frak p - |{\bf k}|. \end{aligned}$$ And ${\bf x}$ is also a system of parameters for $A/\frak p$ for all $\frak p \in \Pi$ since ${\bf x}$ is a system of parameters for ${M}.$
[**Note 1.**]{} Set $J= I_0.$ Assume that $k_i>0$ and without loss of generality, assume that $x_1\in I_i$ for $0 \le i \le d.$ We will prove that there exists $x\in I_i$ such that $x$ is a weak-(FC)-element and $x, x_2, \ldots,
x_n$ is a joint reduction of $
\mathbf{I}, J$ with respect to $A/\frak p$ of the type $({\bf k}, k_0+1)$ for all $\frak p \in \Pi.$ Indeed, by [@SH Proposition 17.3.2], there exists a Zariski open subset $V$ of $I_i/\frak
mI_i$ such that if $x \in I_i$ with the image $x + \frak mI_i \in V,$ then $x, x_2, \ldots, x_n$ is a joint reduction of $
\mathbf{I}, J$ with respect to $A/\mathfrak{p}$ of the type $({\bf k}, k_0+1)$ for all $\mathfrak{p} \in \Pi.$ And since $x_1 + {\frak m} I_i \in V,$ we have $V \neq \emptyset.$ On the other hand, by [@VDT Proposition 2.3], there exists a non-empty Zariski open subset $W$ of $I_i/\frak mI_i$ such that if $x \in
I_i$ with $x + \frak mI_i \in W,$ then $x$ is a weak-(FC)-element of $\mathbf{I}, J$ with respect to $A/\mathfrak{p}$ for all $\mathfrak{p} \in \Pi.$ Since $V, W$ are non-empty Zariski open, it implies that $V\cap W$ is also a non-empty Zariski open subset of $I_i/\frak
mI_i$. Hence there exists $x \in I_i$ such that $x+ \frak mI_i \in V\cap W,$ then $x$ is a weak-(FC)-element and $x, x_2, \ldots, x_n$ is a joint reduction of $
\mathbf{I}, J$ with respect to $A/\mathfrak{p}$ of the type $({\bf k}, k_0+1)$ for all $\mathfrak{p} \in \Pi.$
Now, we prove by induction on $k_0+|{\bf k}|$ that $$\label{eqtv1-2019}e(J^{[k_0+1]},\mathrm{\bf I}^{[\mathrm{\bf k}]}; M) =
e(\mathbf{x};{M}).$$
If $|{\bf k}| = 0$, then since $\bf x$ is a joint reduction of $
\mathbf{I}, J$ with respect to $M$ of the type $({\bf 0}, k_0+1)$, by Lemma \[no4.3a\] (iii) we get $e(J^{[k_0+1]},
\mathbf{I}^{[\mathbf{0}]}; M) = e(\mathbf{x};{M}/0_M:I^\infty).$ Since $I \nsubseteq \frak p$ for any $\mathfrak{p} \in \Pi,$ it follows that $({M}/0_M:I^\infty)_{\frak p} = M_{\frak p}.$ Hence by (\[pt-001VT0\]), we get $e(\mathbf{x};{M}/0_M:I^\infty)= e(\mathbf{x};{M})$ (see e.g. [@SH Theorem 11.2.4]). So $$e(J^{[k_0+1]},
\mathbf{I}^{[\mathbf{0}]}; M) = e(\mathbf{x};{M}).$$ Hence we also get the proof of (\[eqtv1-2019\]) in the case $k_0+|\mathbf{k}| = 0.$
Consider the case that $|\mathbf{k}| >0.$ And without loss of generality, assume that $k_1>0$ and $x_1\in I_1.$ Recall that $e(J^{[k_0+1]},\mathrm{\bf I}^{[\mathrm{\bf k}]}; M)
=\sum_{\mathfrak{p}\in
\Pi}\ell_A(M_{\mathfrak{p}})e(J^{[k_0+1]},\mathrm{\bf
I}^{[\mathrm{\bf k}]}; A/\mathfrak{p})$ by [@VT1 Theorem 3.2]. Hence to have (\[eqtv1-2019\]), we need to prove that $$\label{eqtv111} e(J^{[k_0+1]},\mathrm{\bf I}^{[\mathrm{\bf k}]}; A/\mathfrak{p}) =
e({\bf x}; A/\mathfrak{p})$$ for any $\mathfrak{p} \in \Pi.$ Let $B= A/\mathfrak{p},$ $\mathfrak{p} \in \Pi.$ By Note 1, there exists $x \in I_1$ such that $x$ is a weak-(FC)-element and $x, x_2,\ldots,x_n$ is a joint reduction of $
\mathbf{I}, J$ with respect to $B.$ Since $x$ is a weak-(FC)-element of $\mathbf{I}, J$ with respect to $B$, by Remark \[re2.6aa\] (ii), $$e(J^{[k_0+1]},
\mathbf{I}^{[\mathbf{k}]}; B)=e(J^{[k_0+1]},
\mathbf{I}^{[\mathbf{k-e}_1]}; B/xB).$$ By (\[pt-001KV\]), $\dim B/IB < \dim B - |{\bf k}|.$ From this it follows that $$\begin{aligned}&\dim (B/xB)/I(B/xB) = \dim B/(x,I)B \le \dim B/IB \\&< \dim B - |{\bf k}|= \dim B/xB - |{\bf k}-{\bf e}_1|.\end{aligned}$$ Consequently, $$\label{eqtv2019}\dim (B/xB)/I(B/xB) < \dim B/xB - |{\bf k}-{\bf e}_1|.$$ Note that since $x, x_2,\ldots,x_n$ is a joint reduction of $
\mathbf{I}, J$ with respect to $B$ of the type $(\mathbf{k}, k_0+1),$ $x_2,\ldots,x_n$ is a joint reduction of $
\mathbf{I}, J$ with respect to $B/xB$ of the type $(\mathbf{k-e}_1, k_0+1).$ Hence by the inductive hypothesis, $$e(J^{[k_0+1]},
\mathbf{I}^{[\mathbf{k-e}_1]}; B/xB) = e(x_2,\ldots,x_n;B/xB).$$ Since $x$ is not a zero divisor of $B,$ $e(x_2,\ldots,x_n; B/xB) = e(x,x_2,\ldots,x_n; B)$ by [@AB Page 641, lines 27-28, (D)]. Consequently, $$\label{eqtv20}e(J^{[k_0+1]},\mathbf{I}^{[\mathbf{k}]}; B)=e(x,x_2,\ldots,x_n; B).$$ So to prove (\[eqtv111\]), we need to show that $$\label{eqtv21}e(x_1,x_2,\ldots,x_n; B)=e(x,x_2,\ldots,x_n; B)$$ via the following cases.
[**Case 1:**]{} $k_0+|\mathbf{k}| =1.$
Then $k_0 =0.$ Hence since $\dim B/IB < \dim B - |{\bf k}| < \dim B$ by (\[pt-001KV\]), we get the proof of (\[eqtv21\]) by Lemma \[le2019\].
[**Case 2:**]{} $k_0+|{\bf k}| > 1$ and $|{\bf k}| = 1$ and $\dim B/IB > 1.$
In this case, $k_0 > 0$ and $n \ge 3,$ $x_1 \in I_1$ and $x_2,\ldots, x_n \in J.$ Set $U=x_2,\ldots, x_n.$ Then $\dim B/(U,I)B \le \dim B/(U)B = 1 < \dim B/IB.$ By [@SH Proposition 17.3.2], $x_2,\ldots,x_n \in J\setminus{\frak m}J.$ Denote by $x_2',\ldots, x_n'$ the images of $x_2,\ldots, x_n$ in $J/{\frak m}J,$ respectively. Since $x_2,\ldots, x_n$ is part of the system of parameters ${\bf x},$ it follows that $x_2',\ldots, x_n'$ is part of a basis of $k$-vector space $J/{\frak m}J.$ So $(U)\cap {\frak m}J = {\frak m}(U)$ (see e.g. [@SH Proposition 8.3.3(1)]). Hence $((U) + {\frak m}J)/{\frak m}J \cong (U)/{\frak m}(U).$ Consequently, we can consider $(U)/{\frak m}(U)$ as a $k$-vector subspace of $J/\frak m
J.$ Now, assume that $P_1,P_2,\ldots, P_t$ are all the prime ideals of $\mathrm{Min}[B/IB]$ such that $$\dim B/IB = \mathrm{Coht} P_j\quad (1 \le j \le t).$$ For each $j = 1, \ldots, t$, let $W_j$ be the image of $P_j \bigcap (U)$ in $J/\frak mJ$. Since $$\dim B/IB > \dim B/(U,I)B,$$ $(U)\setminus \bigcup_{j=1}^tP_j \not= \emptyset$ by Prime Avoidance. Hence $W_1, \ldots, W_t$ are proper $k$-vector subspaces of $(U)/\frak m(U)$ by Nakayama’s lemma. Since $k$ is an infinite field, $$\Lambda = [(U)/\frak m(U)] \setminus \bigcup_{j=1}^tW_j$$ is a non-empty Zariski open subset of $(U)/\frak m(U)$. By [@SH Proposition 17.3.2], there exists a Zariski open subset $\Gamma$ of $J/\frak
mJ$ such that if $x \in J$ with the image $x' \in \Gamma,$ then $x_1, x, \ldots, x_n$ is a joint reduction of $
\mathbf{I}, J$ with respect to $B$ of the type $({\bf k}, k_0+1).$ Remember that $x'_2 \in \Gamma \cap [(U)/\frak m(U)]$ since $x_1, x_2, \ldots, x_n$ is a joint reduction. So $$\Gamma'=\Gamma \cap [(U)/\frak m(U)] \not= \emptyset.$$ Hence $\Gamma' \cap \Lambda \not= \emptyset.$ Now, let $u \in (U)$ such that the image $u'\in \Gamma' \cap \Lambda.$ Then we have $$\dim B/(u,I)B = \dim B/IB -1$$ and $x_1, u, \ldots, x_n$ is a joint reduction of $
\mathbf{I}, J$ with respect to $B.$ In this case, $x_1, u, \ldots, x_n$ is a system of parameters for $B.$ Hence it can be verified that $u',x_3',\ldots,x_n'$ is also a basis of $k$-vector space $(U)/{\frak m}(U).$ Consequently, $(U)= (u,x_3,\ldots,x_n)$ by Nakayama’s lemma. So $$(x_1, x_2,\ldots, x_n)= (x_1,u,x_3,\ldots,x_n) \; \mathrm{and}\; (x, x_2,\ldots, x_n)= (x,u,x_3,\ldots,x_n).$$ Moreover in this case, $\dim B/IB < \dim B -|{\bf k}|$ by (\[pt-001KV\]), and hence $$\begin{aligned} &\dim (B/uB)/I(B/uB)=\dim B/(u, I)B = \dim B/IB -1\\ &< \dim B -|{\bf k}| -1 = \dim B/uB -|{\bf k}|.\end{aligned}$$ Note that $x,x_3,\ldots,x_{n}$ and $x_1,x_3,\ldots,x_{n}$ are joint reductions with respect to $B/uB$ of the type $({\bf k}, k_0).$ Hence by the inductive hypothesis on $k_0 + |{\bf k}|$, we have $$e(x,x_3,\ldots, x_n;B/uB)=e(J^{[(k_0-1)+1]},\mathrm{\bf I}^{[\mathrm{\bf k}]};
B/uB)= e(x_1,x_3,\ldots, x_n;B/uB).$$ On the other hand, since $u$ is not a zero divisor of $B$, by [@AB Page 641, lines 27-28, (D)], we get $$e(x,u,x_3,\ldots, x_n;B) = e(x_1,u,x_3,\ldots, x_n;B).$$ Recall that $(x_1, x_2,\ldots, x_n)= (x_1,u,x_3,\ldots,x_n), (x, x_2,\ldots, x_n)= (x,u,x_3,\ldots,x_n).$ Consequently, $$e(x,x_2,x_3,\ldots, x_n;B) = e(x_1,x_2,x_3,\ldots, x_n;B).$$ Therefore, we obtain the proof of (\[eqtv21\]) in this case.
[**Case 3:**]{} $k_0+|{\bf k}| > 1$ and $|{\bf k}| = 1$ and $\dim B/IB \le 1.$
In this case, $k_0 > 0.$ Consider the case $k_0 = 1.$ Since $x, x_2$ and $x_1, x_2$ are both joint reductions of $\mathbf{I}, J$ with respect to $B/x_3B$, and are systems of parameters for ${B/x_3B},$ and on the other hand, $$\dim (B/x_3B)/I(B/x_3B)=\dim B/(x_3, I)B \leq \dim B/IB\leq 1< 2= \dim B/x_3B,$$ by Lemma \[le2019\], we get $e(x,x_2; B/x_3B) = e(x_1, x_2; B/x_3B)$. Remember that $\bf x$ is a system of parameters for $M$. Hence $x_3\notin \frak{p}$. Consequently $x_3$ is not a zero divisor of $B$. So we have $e(x, x_2,x_3;B)= e(x_1, x_2,x_3;B)$ by [@AB Page 641, lines 27-28, (D)].
If $k_0 > 1,$ then $\dim B = n \ge 4.$ Then $x_2, x_3, x_4, \dots, x_n \in J$ and we have $$\dim (B/x_2B)/I(B/x_2B)=\dim B/(x_2, I)B \le \dim B/IB \le 1 < \dim B/x_2B -|{\bf k}|.$$ Consequently, by the inductive hypothesis on $k_0 + |{\bf k}|$, $$e(x, x_3,\ldots, x_n;B/x_2B)=e(J^{[(k_0-1)+1]},\mathrm{\bf I}^{[\mathrm{\bf k}]};
B/x_2B)= e(x_1,x_3,\ldots, x_n;B/x_2B).$$ Now, recall that $\bf x$ is a system of parameters for $M$. Hence $x_2\notin \frak{p}$. So $x_2$ is not a zero divisor of $B.$ Therefore, $$e(x,x_2,x_3,\ldots, x_n;B) = e(x_1,x_2,x_3,\ldots, x_n;B).$$ We get (\[eqtv21\]).
[**Case 4:**]{} $k_0+|{\bf k}| > 1$ and $|{\bf k}| > 1.$
Then $x_2 \in I_i$ for some $1 \le i \le d.$ Hence by (\[eqtv2019\]), $$\dim (B/x_2B)/I(B/x_2B) < \dim B/x_2B - |{\bf k}-{\bf e}_i|.$$ Therefore, by the inductive hypothesis on $k_0 + |{\bf k}|$, $$e(x, x_3,\ldots, x_n;B/x_2B)=e(J^{[k_0+1]},\mathrm{\bf I}^{[\mathrm{\bf k-e}_i]};
B/x_2B)= e(x_1,x_3,\ldots, x_n;B/x_2B).$$ Hence since $x_2$ is not a zero divisor of $B$, $$e(x,x_2,x_3,\ldots, x_n;B) = e(x_1,x_2,x_3,\ldots, x_n;B).$$ We have (\[eqtv21\]).
From the above cases, we get the proof of (\[eqtv21\]). Consequently, by $(\ref{eqtv20}),$ we get (\[eqtv111\]), that $$e(J^{[k_0+1]},\mathrm{\bf I}^{[\mathrm{\bf k}]}; A/\mathfrak{p}) =
e({\bf x}; A/\mathfrak{p})$$ for all $\mathfrak{p} \in \Pi.$ Hence $$\sum_{\mathfrak{p}\in \Pi}
\ell_A(M_{\mathfrak{p}})e(J^{[k_0+1]},\mathrm{\bf I}^{[\mathrm{\bf
k}]}; A/\mathfrak{p}) = \sum_{\mathfrak{p}\in
\Pi}\ell_A(M_{\mathfrak{p}}) e(\mathbf{x}; A/\mathfrak{p}).$$ On the other hand, $e(\mathbf{x}; {M})
=\sum_{\mathfrak{p}\in \Pi}\ell_A(M_{\mathfrak{p}})e(\mathbf{x};
A/\mathfrak{p})$ (see e.g. [@SH Theorem 11.2.4]) and $$e(J^{[k_0+1]},\mathrm{\bf I}^{[\mathrm{\bf k}]}; M)
=\sum_{\mathfrak{p}\in
\Pi}\ell_A(M_{\mathfrak{p}})e(J^{[k_0+1]},\mathrm{\bf
I}^{[\mathrm{\bf k}]}; A/\mathfrak{p})$$ by [@VT1 Theorem 3.2]. Therefore, we get (\[eqtv1-2019\]) in the case that $|{\bf k}| > 0,$ and hence the proof of (\[eqtv1-2019\]) is complete, that $e(J^{[k_0+1]},\mathrm{\bf I}^{[\mathrm{\bf k}]}; M)= e(\mathbf{x};
{M}).$
\[rm35\] From Theorem \[thm2.19vt\], one may raise a question: Does this theorem hold if $\dim {M}/I{M} \ge \dim {M} - |\bf k|$? Consider the case $M=A$ and $\dim A =2.$ Let $x_1, x_2$ be a system of parameters for $A.$ Set $\frak I = x_1A$ and $J = (x_1, x_2)A.$ Then $J$ is an $\frak{m}$-primary ideal of $A$ and $x_1, x_2$ is a joint reduction of $\frak I, J$ with respect to $A$ of the type $(1, 1)$ since $\frak I^nJ^m = x_1\frak I^{n-1}J^m + x_2\frak I^nJ^{m-1}$ for all $n, m \ge 1.$ In this case, ${\bf k} = (1),$ $k_0 =0$ and $\dim A/\frak IA = 1 = \dim A - |\bf k|.$ It can be verified that $(x_1) \cap \frak I^{n}J^m = x_1\frak I^{n-1}J^m$ for all $n \ge 1; m \ge 0$ and $0_A: x_1 \subset 0_A: \frak I^\infty.$ Hence $x_1$ is a weak-(FC)-element of $\frak{I}, J$ with respect to $A.$ Consequently, by Remark \[re2.6aa\] (ii), we get $e(J^{[1]}, \frak I^{[1]}; A)= e(J^{[1]}, \frak I^{[0]}; A/x_1A).$ Now, since $\frak I = x_1A,$ $\frak I[A/x_1A]= 0.$ So we obtain $e(J^{[1]}, \frak I^{[0]}; A/x_1A)=0$. Thus $e(J^{[1]}, \frak I^{[1]}; A)=0.$ Hence $$e(J^{[1]}, \frak I^{[1]}; A) \not= e(x_1, x_2; A).$$ This shows that Theorem \[thm2.19vt\] does not hold in general if one omits the assumption $\dim {M}/I{M} < \dim {M} - |\bf k|.$
Now, we return [@VDT Theorem 3.1] that covers Rees’s theorem [@Re Theorem 2.4 (i), (ii)] by the following stronger result than [@VDT Theorem 3.1].
\[co4.0a\] Let ${\bf x}$ be a joint reduction of $\mathbf{I}, J$ with respect to $M$ of the type $({\bf k},k_0+1)$ with $k_0 + |{\bf k}| = \dim \overline M-1.$ Assume that $\mathrm{ht}\Big(\dfrac{I+\mathrm{Ann}M}{\mathrm{Ann}M}\Big) >
|\bf k|.$ Then ${\bf x}$ is a system of parameters for ${M}$ and $$e(J^{[k_0 +1]}, \mathbf{I}^{[\mathbf{k}]}; M) = e(\mathbf{x};
{M}).$$
Since $\mathrm{ht}\Big(\dfrac{I+\mathrm{Ann}M}{\mathrm{Ann}M}\Big) >
|\bf k|,$ we have $\dim M/IM < \dim M - |{\bf k|}$. Hence from Theorem \[thm2.19vt\] we get the proof.
Finally, we would like to give the following result which shows that mixed multiplicities are the Hilbert-Samuel multiplicity of weak-(FC)-sequences.
\[thm2.19v\] Let ${\bf x}$ be a weak-$\mathrm{(FC)}$-sequence of $\mathbf{I},
J$ with respect to $M$ of the type $({\bf
k},k_0+1)$ with $k_0 + |{\bf k}| = \dim \overline M-1.$ Assume that $\dim {M}/I{M} < \dim {M} - |\bf k|.$ Then ${\bf x}$ is a system of parameters for ${M}$ and $e(J^{[k_0 +1]}, \mathbf{I}^{[\mathbf{k}]}; M) = e(\mathbf{x};
{M}).$
By Remark \[re2.6aa\] (v), ${\bf x}$ is a joint reduction of $\mathbf{I},
J$ with respect to $M$ of the type $({\bf k},k_0+1).$ Hence the corollary is shown by Theorem \[thm2.19vt\].
[**Acknowledgement:**]{} [Published: 26 October 2019 in Bulletin of the Brazilian Mathematical Society, New Series. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04.2015.01.]{}
[99]{}
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Department of Mathematics, Hanoi National University of Education\
136 Xuan Thuy Street, Hanoi, Vietnam\
Emails: [email protected] and [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Each isometric complex structure on a 2$\ell$-dimensional euclidean space $E$ corresponds to an identification of the Clifford algebra of $E$ with the canonical anticommutation relation algebra for $\ell$ ( fermionic) degrees of freedom. The simple spinors in the terminology of E. Cartan or the pure spinors in the one of C. Chevalley are the associated vacua. The corresponding states are the Fock states (i.e. pure free states), therefore, none of the above terminologies is very good.'
---
[COMPLEX STRUCTURES AND]{}
[THE ELIE CARTAN APPROACH]{}
[TO THE THEORY OF SPINORS]{}
[**Michel DUBOIS–VIOLETTE**]{}
Laboratoire de Physique Théorique et Hautes Energies\
Bâtiment 211, Université Paris XI\
91405 ORSAY Cedex, France\
E-mail: [email protected]
Lecture given at the Second Max Born Symposium\
“Spinors, Twistors and Clifford Algebras"\
held in Wrocław, Poland, Sept. 24–27, 1992.
Introduction
============
In this lecture, we will discuss complex structures and spinors on euclidean space. This is an extension of the algebraic part of a work \[1\] describing a sort of generalization of Penrose and Atiyah–Ward transformations in $2\ell$ dimension. We shall not describe this work here, refering to \[1\], but concentrate the lecture upon the notion of simple spinor of E. Cartan \[2\] (or pure spinor in the terminology of C. Chevalley \[3\]). Many points of this lecture are well known facts and, in some sense, this may be considered as an introductory review. The notations used here are standard, let us just point out that by an euclidean space we mean a [*real*]{} vector space with a positive scalar product and by a Hilbert space we mean a [*complex*]{} Hilbert space.
Isometric Complex Structures
============================
Notations
---------
Let $E$ be an oriented 2$\ell$-dimensional euclidean space ($E\simeq \bbbr^{2\ell}$) with a scalar product denoted by ($\bullet, \bullet$). The dual space $
E^\ast$ of $E$ is also, in a canonical way, an euclidean space and we again denote its scalar product by ($\bullet,
\bullet$). On the complexified space $E^\ast_c = E^\ast
\otimes \bbbc$ of $E^\ast$ one may extend the scalar product of $E^\ast$ in two different ways: Either one extends it by bilinearity and the corresponding bilinear form will again be denoted by ($\bullet, \bullet$) or one extends it by sesquilinearity and the corresponding sesquilinear form will be denoted by $\langle \bullet
\vert \bullet\rangle$. As for any complexified vector space, there is a canonical complex conjugation $\omega
\mapsto \bar\omega$ on $E^\ast$, (an antilinear involution), and the connection between the two scalar products is given by: $$\langle\omega_1 \vert \omega_2\rangle = (\bar \omega_1,
\omega_2), \qquad \forall \omega_1, \omega_2 \in
E^\ast_c.$$
Isometric Complex Structures or Hilbertian Structures
-----------------------------------------------------
Let ${\cal H}(E)$ be the set of isometric complex structures on $E$ or, which is the same, the set of orthogonal antisymmetric endormorphisms of $E$, i.e. $${\cal H}(E) = \{J\in {\rm End} (E) \vert J\in O(E)\
{\rm and}\ J^2=-\bbbone \} =$$ $$= \{J\in {\rm End}(E) \vert J\in O(E)\
{\rm and}\ (X,JY) = -(JX,Y), \quad \forall X,Y\in E\}$$Let $J\in {\cal H}(E)$ and define $$(x+iy) V = xV + yJV, \qquad \forall (x+iy)\in \bbbc,
\qquad \forall V \in E$$ and $$\langle X\vert Y\rangle_J = (X,Y) - i(X,JY), \qquad
\forall X,Y \in E.$$ Equipped with the above structure, $E$ is a $\ell$-dimensional Hilbert space which we denote by $E_J$. For a basis $(e_1,\dots , e_\ell)$ of the complex vector space $E_J$, ($e_1, \dots, e_\ell , Je_1, \dots ,
Je_\ell$) is a basis of $E$ the orientation of which is independent of $(e_1, \dots , e_\ell$) but only depends on $J$. Accordingly, ${\cal H}(E)$ splits in two pieces : ${\cal H}(E) = {\cal H}_+(E) \cup {\cal H}_-(E)$. The orthogonal group $O(E)$ acts transitively on ${\cal H}(E)$ and the subgroup $SO(E)$ of orientation preserving orthogonal transformations acts transitively on ${\cal H}_+(E)$ and on ${\cal H}_-(E)$.\
Thus one has ${\cal H}(E)\simeq O(E) / U(E_J)$ and ${\cal H}_+(E)\simeq SO(E) / U(E_J) \simeq {\cal H}_-(E)$ where $U(E_J)$ is the unitary group of $E_J$ for a fixed $J\in {\cal H}(E)$ (i.e. $U(E_J)\simeq U(\bbbc^\ell)).$ We equip ${\cal H}(E), {\cal H}_+(E)$ and ${\cal H}_-(E)$ with the corresponding manifold structure. In particular, ${\rm dim}_\bbbr{\cal H}(E) = {\rm dim}_\bbbr
{\cal H}_\pm (E) = \ell(2\ell -1) - \ell^2 = \ell(\ell
-1).$
Identification of Dual Spaces
-----------------------------
The dual Hilbert space of $E_J$ can be identified with the Hilbert subspace $\Lambda^{1,0}E_J^\ast$ of $E^\ast_c$ defined by $$\Lambda^{1,0}E^\ast_J = \{\omega\in E^\ast_c \vert
\omega\circ J = i\omega\}.$$ One verifies easily that $\Lambda^{1,0}E_J^\ast$ is maximal isotropic in $E^\ast_c$ for ($\bullet ,
\bullet$) or, which is the same, that $\Lambda^{1,0}E_J^\ast$ is orthogonal to its conjugate $\overline{\Lambda^{1,0}E^\ast_J}=\Lambda^{0,1}E^\ast_J$ in $E^\ast_c$ for $\langle \bullet\vert\bullet\rangle$ (i.e. $E^\ast_c$ is the hilbertian direct sum $\Lambda^{1,0}E^\ast_J \oplus \Lambda^{0,1} E^\ast_J$).\
Conversely if $F \subset E^\ast_c$ is a maximal isotropic subspace for ($\bullet , \bullet$), then there is a unique $J\in {\cal H}(E)$ such that $F=\Lambda^{1,0}E^\ast_J$. It follows that ${\cal H}(E)$ identifies with a complex algebraic submanifold of the grassmannian $G_\ell(E^\ast_c)$ of $\ell$-dimensional subspaces of $E^\ast_c$, $(G_\ell(E^\ast_c)\simeq
G_{\ell,2\ell}(\bbbc))$. In particular, ${\cal H}(E)$ is a compact Kähler manifold of complex dimension $\frac{\ell(\ell-1)}{2}$ and its Kähler metric is given by $ds^ 2=\frac{1}{4} {\rm tr} ((dP^{1,0}_J)^2)$ where $P^{1,0}_J$ is the hermitian projector of $E^\ast_c$ on $\Lambda^{1,0}E^\ast_J$. Notice that one has $\overline{P^{1,0}_J} = P^{0,1}_J = \bbbone - P^{1,0}$.\
Furthermore $\Lambda^{1,0}E^\ast_J$ is the fibre at $J\in
{\cal H}(E)$ of a holomorphic hermitian vector bundle of rank $\ell$ over ${\cal H}(E)$ which we denote by $\Lambda^{1,0}E^\ast$.\
Finally notice that one has the hilbertian sum identifications $$\Lambda^kE^\ast_c = \oplusinf_{r+s=k} \Lambda^{r,s}
E^\ast_J, \qquad \forall J \in {\cal H}(E)$$ where $\Lambda^{r,s}E^\ast_J =
\Lambda^r(\Lambda^{1,0}E^\ast_J) \otimes \Lambda^s
(\overline{\Lambda^{1,0}E^\ast_J})$, (here the tensor product is over $\bbbc$).We denote by $P^{r,s}_J$ the corresponding hermitian projectors.
Examples
--------
One has ${\cal H}_+(\bbbr^2) = \{I_+\}, {\cal
H}_+(\bbbr^4) = \bbbc P^1, {\cal H}_+(\bbbr^6) = \bbbc
P^3$ and, as will be shown below, ${\cal
H}_+(\bbbr^{2\ell}) \subset \bbbc P^{2^{\ell-1}-1}$ but the inclusion is strict for $\ell\geq 4$ as it follows by comparison of the dimensions.
Hodge duality
-------------
On $\Lambda E^\ast$ there is a linear involution, $\ast$, defined by $\ast (\omega^1 \wedge \dots \wedge
\omega^p)=\omega^{p+1} \wedge \dots \wedge \omega^{2\ell}$ for any positively oriented orthonormal basis $(\omega^1,
\dots ,\omega^{2\ell})$. One extends this involution by linearity to $\Lambda E^\ast_c$. One has the following lemma.\
[**Lemma.**]{} [*Let $\Omega$ be an element of $\Lambda^\ell E_c^\ast$. Then one has $\Omega + i^\ell
\ast \Omega =0$, (resp. $\Omega-i^\ell \ast \Omega = 0$), if and only if $P^{0,\ell}_J \Omega = 0$, $\forall J\in
{\cal H}_+(E)$, (resp. $\forall J\in {\cal H}_-(E)).$*]{}\
For $\ell = 2$ (i.e. in dimension 4), this is the basic algebraic lemma for the Penrose–Atiyah–Ward transformation.
The Clifford algebra as C.A.R. algebra
======================================
Definition
----------
We define the Clifford algebra ${\rm Cliff}(E^\ast)$ to be the complex associative $\ast$-algebra with a unit $\bbbone$ generated by the following relations $$[\gamma(\omega_1), \gamma(\omega_2)]_+ = 2(\omega_1,
\omega_2) \bbbone\ {\rm and}\ \gamma(\omega)^\ast =
\gamma(\omega)\ {\rm for}\ \omega, \omega_i\in E^\ast.$$ The $\gamma(\omega), \omega \in E^\ast$, are hermitian generators and $\gamma : E^\ast \rightarrow {\rm Cliff}
(E^\ast)$ is an injective $\bbbr$-linear mapping. One extends $\gamma$ as a $\bbbc$-linear mapping, $\gamma :
E^\ast_c \rightarrow {\rm Cliff}(E^\ast)$, by setting $\gamma(\bar \omega)=\gamma(\omega)^\ast$.
Complex structures and the C.A.R. algebra
-----------------------------------------
Let $J\in {\cal H} (E)$ be given. The algebra ${\rm
Cliff}(E^\ast)$ is generated by the $\gamma(\omega)$ with $\omega\in \Lambda^{1,0}E^\ast_J$ and their adjoints $\gamma(\omega)^\ast = \gamma (\bar\omega)$. In terms of these generators the relations read $$[\gamma(\omega_1), \gamma(\omega_2)]_+=0\ {\rm and}\
[\gamma(\omega_1)^\ast, \gamma(\omega_2)]_+ =
\langle\omega_1 \vert \omega_2\rangle \bbbone, \quad
\forall \omega_i \in \Lambda^{1,0}E^\ast_J.$$ These are the defining relations of the algebra of canonical anticommutation relations (C.A.R. algebra) for $\ell$ (fermionic) degrees of freedom. Thus each $J\in
{\cal H}(E)$ corresponds to an identification of the Clifford algebra with the C.A.R. algebra. Furthermore, the action of the orthogonal group $O(E)$ on ${\cal H}(E)$ corresponds to the Bogolioubov transformations. One has, as well known, ${\rm Cliff} (E^\ast) \simeq
M_{2^\ell}(\bbbc)$.
Spinors and Complex Structures
==============================
Definition
----------
We define a space of spinors associated to $E$ to be a Hilbert space $S$ carrying an irreducible $\ast$-representation of ${\rm Cliff}(E^\ast)$. The spinors are the elements of $S$. Since ${\rm
Cliff}(E^\ast)$ is isomorphic to $M_{2^\ell}(\bbbc)$, $S$ is isomorphic to $\bbbc^{2^\ell}$ and the representation is an isomorphism. We shall identify ${\rm Cliff} (E^\ast)$ with the image of this representation.
The simple spinors of E. Cartan
-------------------------------
Let $\psi\in S$ with $\psi\not= 0$ and set $I_\psi =
\{\omega\in E^\ast_c \vert \gamma(\omega) \psi = 0\}$. If $\omega_1$ and $\omega_2$ are in $I_\psi$, one has $[\gamma(\omega_1), \gamma(\omega_2)]_+ \psi = 2
(\omega_1, \omega_2)\psi = 0$, so $I_\psi$ is an isotropic subspace of $E^\ast_c$ for ($\bullet,
\bullet$).\
If $I_\psi$ is maximal isotropic, i.e. if ${\rm dim}
(I_\psi)=\ell$, then $\psi$ is called a [*simple spinor*]{} by E. Cartan \[2\] or a [*pure spinor*]{} by C. Chevalley \[3\]. We denote by ${\cal F}$ the set of these spinors and by $P({\cal F})$ the corresponding algebraic submanifold of $P(S)=\bbbc P^{2^{\ell}-1}$, (i.e. $P({\cal
F})$ is the set of directions of simple spinors).\
For $\psi \in {\cal F}, I_\psi=I_{\lambda\psi}\quad
\forall \lambda\in \bbbc \backslash \{0\}$, so the maximal isotropic subspace $I_\psi$ of $E^\ast_c$ does only depend on the direction $[\psi]\in P({\cal F})$ of $\psi$. On the other hand we know that there is a unique $J\in
{\cal H}(E)$ such that $I_\psi=\Lambda^{1,0}E^\ast_J$. It follows that one has a mapping of $P({\cal F})$ in ${\cal
H}(E)$ which is in fact an isomorphism of complex manifolds. In the following, we shall identify these manifolds, writing $P({\cal F})={\cal H}(E)$.
The natural line bundle
-----------------------
The restriction to $P({\cal F}) = {\cal H}(E)$ of the tautological bundle of $P(S)$ is a holomorphic hermitian vector bundle of rank one, $L$, over ${\cal H}(E)$. One has $L = {\cal F} \cup \{{\rm the\ zero\ section}\}$.\
As holomorphic hermitian line bundles over ${\cal H}(E)$, one has the following isomorphisms, see in \[1\] : $\Lambda^{\ell,0}E^\ast \simeq L\otimes L$ and $\Lambda^{\frac{\ell(\ell -1)}{2}} T^\ast {\cal H} (E) =
L^{\otimes\ 2 (\ell -1)}$.
Semi-spinors and simple spinors
-------------------------------
To the action of $SO(E)$ on $E^\ast$ corresponds a linear representation of its covering ${\rm Spin}(E)$ in $S$. Under this representation, $S$ splits into two irreductible components $S=S_+ \oplus S_-$ with ${\rm
dim} S_+ = {\rm dim} S_- = 2^{\ell-1}$. The elements of $S_+$ and $S_-$ are called semi-spinors. One the other hand $P({\cal F})={\cal H}(E)$ splits into two transitive parts under the action of $SO(E)$, ${\cal H}(E) = {\cal
H}_+(E) \cup {\cal H}_-(E)$. It follows that ${\cal F} =
{\cal F}_+ \cup {\cal F}_-$ with ${\cal F}_\pm = {\cal F}
\cap S_\pm$ and (with an eventual relabelling in the $\pm$) $P({\cal F}_\pm) = {\cal H}_\pm (E)$. In other words ${\cal F}$ consists of semi-spinors. It turns out that for $\ell\leq 3$ all non vanishing semi-spinors are in ${\cal F}$ (i.e. ${\cal F}_\pm = S_\pm\backslash \{0\})$ but for $\ell \geq 4$ the inclusions ${\cal F}_\pm \subset
S_\pm\backslash \{0\}$ are strict inclusions. For $\ell\geq 4$ ${\cal
H}_+(E)$ is no more a projective space.
Fock States and Simple Spinors
==============================
States on algebras
------------------
Let ${\cal A}$ be an associative complex $\ast$-algebra with a unit $\bbbone$. We recall that a state on ${\cal
A}$ is a linear form $\phi$ on ${\cal A}$ such that $\phi
(X^\ast X) \geq 0,\quad \forall X\in {\cal A}$ and $\phi(\bbbone)=1$. The set of all states on ${\cal A}$ is a convex subset of the dual space ${\cal A}^\ast$ of ${\cal A}$. The extreme points of this convex subset (i.e. which are not convex combinations of two distinct states) are called pure states. To the states on ${\cal
A}$ correspond cyclic $\ast$-representations of ${\cal
A}$ in Hilbert space via the G.N.S. construction; pure states correspond then to irreducible representations.\
Coming back to the case ${\cal A} = {\rm Cliff}(E^\ast)$, we see that to each spinor $\psi\not=0$ corresponds a state $X \mapsto \frac{\langle \psi \vert
X\psi\rangle}{\parallel \psi\parallel^2}$ (its direction ) which is a pure state leading to an irreducible, or simple, representation. This is why the terminology of C. Chevalley or E. Cartan to denote the elements of ${\cal F}$ is somehow misleading. What characterizes the elements of ${\cal F}$ is that the corresponding states (i.e. elements of $P({\cal F}) = {\cal H}(E))$ are Fock states or free states on ${\rm Cliff} (E^\ast)$ (see below); thus the name Fock spinors or free spinors would be better.
Fock states on the Clifford algebra
-----------------------------------
First of all it is clear from above that the elements of ${\cal F}$ are all possible vacua corresponding to the identifications of ${\rm Cliff}(E^\ast)$ with the C.A.R. algebra. It is well known that given a vacuum, the vacuum expectation values factorize and only depend on the “two-point functions" i.e. on the vacuum expectation values of $\gamma(\omega_1)\gamma(\omega_2)$ for $\omega_i\in E^\ast$, (this is the very property of the free states).\
More precisely, a [*Fock state*]{}, (see for instance \[4\]), on ${\rm Cliff}(E^\ast)$ is a pure state $\phi$ satisfying the following (Q.F.) property:
$$(Q.F.) \left\{ \begin{array}{l}
\phi(\gamma(\omega_1) \dots \gamma(\omega_{2n+1}))=0\\
\phi(\gamma(\omega_1) \dots \gamma(\omega_{2n})) =
\sum_{k=2}^{2n}(-1)^k\phi(\gamma(\omega_1)\gamma(\omega_k)).
\phi(\gamma(\omega_2).\buildrel {\rm k \atop ^\vee}
\over .. \gamma(\omega_{2n}))
\end{array}
\right.$$ for $\omega_i\in E^\ast$, (where $\buildrel {\rm k \atop
^\vee} \over .$ means omission of the k$^{\rm th}$ term). From (Q.F.) one sees that $\phi$ is determined by the $\phi(\gamma(\omega_1)\gamma(\omega_2)) =
h(\omega_1,\omega_2)+i\sigma (\omega_1, \omega_2),$ $\omega_i \in E^\ast$, where $h$ and $\sigma$ are real bilinear forms. The defining relations of ${\rm
Cliff}(E^\ast)$ implie that $h(\omega_1,\omega_2)+h(\omega_2, \omega_1)=2(\omega_1,
\omega_2)$ and $\sigma (\omega_1,\omega_2)+\sigma(
\omega_2, \omega_1)=0$. The positivity of $\phi$ is equivalent to $\phi(\gamma(\omega_1+i\omega_2) \gamma
(\omega_1-i\omega_2))\geq 0$ which is equivalent to $h(\omega_1, \omega_2)=(\omega_1, \omega_2)$ and $\sigma(\omega_1,\omega_2)= (A\omega_1, \omega_2)=-
(\omega_1, A\omega_2)$ with $\parallel A \parallel \leq
1$. By polar decomposition, $A=J\vert A\vert$ with $J\in
{\cal H} (E)$ and\
$\vert A\vert \geq 0$ ($\parallel\vert
A\vert \parallel\geq 1$). Then, $\phi$ is pure if and only if $\vert A\vert = 1$. Therefore, $\phi$ is a Fock state iff. it satisfies (Q.F.) and $\phi(\gamma(\omega_1)\gamma(\omega_2)) =
(\omega_1,\omega_2)+ i(J\omega_1,\omega_2),\quad \forall
\omega_i\in E^\ast$, with $J\in {\cal H}(E)$. Thus, the Fock states are parametrized by ${\cal H}(E)=P({\cal
F})$ and, in fact, the set of Fock states is $P({\cal F})$; indeed if $\psi\in {\cal F}$ is such that $I_\psi=\Lambda^{1,0}E^\ast_J$ then one has $$\frac{\langle\psi\vert
\gamma(\omega_1)\gamma(\omega_2)\psi\rangle}{\parallel
\psi\parallel^2} =
(\omega_1,\omega_2)+i(J\omega_1,\omega_2), \quad \forall
\omega_i\in E^\ast$$ and (Q.F.) is satisfied.
Spinors and Fock Space Constructions
====================================
The standard construction of the Fock space for the C.A.R. algebra implies that, for each $J$, $S$ is isomorphic to $$\oplusinf_n \Lambda^{0,n}E^\ast_J.$$ However, there is the vacuum, namely an element of $L_J$, which is hidden here.\
In fact, one has an isomorphism $\Phi$ of hermitian vector bundles over ${\cal H}(E)$ from $$\oplusinf_n \Lambda^{0,n}E^\ast\otimes L$$ onto the trivial bundle with fibre equal to $S$, such that $$\Phi(\omega\wedge\varphi) = \frac{1}{\sqrt{2}}
\gamma(\omega)\Phi(\varphi), \quad \forall
\omega\in\Lambda^{0,1}E^\ast_J\
{\rm and}\ \forall
\varphi \in \oplusinf_n\Lambda^{0,n} E^\ast_J\otimes
L_J.$$ More precisely one has the following $$\left. \begin{array}{lll}
\Phi_J : \oplusinf_p\Lambda^{0,2p} E^\ast_J \otimes L_J
\simeq S_+ & ({\rm resp.}\ S_-)\\
\Phi_J :\oplusinf_p \Lambda^{0,2p+1} E^\ast_J\otimes L_J
\simeq S_- & ({\rm resp.}\ S_+) \end{array}
\right\} \forall J\in {\cal H}_+ (E)\ ({\rm resp.} {\cal
H}_-(E))$$ which gives the identification of semi-spinors.
Bundles of Complex Structures
=============================
Let $M$ be a $2\ell$-dimensional oriented riemannian manifold. The tangent space $T_x(M)$ at $x\in M$ is an oriented $2\ell$-dimensional euclidean space so one can consider the complex manifold ${\cal H}(T_x(M))$ as above. ${\cal H}(T_x(M))$ is the fiber at $x\in M$ of a bundle ${\cal H}(T(M))$ on M which we call the [*bundle of isometric complex structures over*]{} $M$. This bundle is associated to the orthonormal frame bundle so there is a natural connection on it coming from the Levi–Civita connection of $M$. On ${\cal H}(T(M))$, there is a [*natural almost complex structure*]{} defined by the following construction. Let $J_x\in {\cal H} (T_x(M))$, then by horizontal lift, $J_x$ defines a complex structure on the tangent horizontal subspace at $J_x$; on the other hand the tangent vertical subspace at $J_x$ is the tangent space to the complex manifold ${\cal H}(T_x(M))$ so it is naturally a complex vector space, so by taking the direct sum one has a complex structure on the tangent space to ${\cal H}(T(M))$ at $J_x$ and finally, ${\cal H}(T(M))$ becomes an almost complex manifold. It is easy to show that the almost complex manifold ${\cal H}(T(M))$ only depends on the conformal structure of $M$. In particular, the almost complex structure of ${\cal H}(T(M))$ is integrable, i.e. ${\cal H}(T(M))$ is a complex manifold, whenever $M$ is conformally flat. The Penrose and the Atiyah-Ward transformations are obtained, in the four-dimensional case, by lifting to ${\cal H}(T(M))$ various objects living on $M$ (see in \[1\]).\
Let us end this lecture by noticing that the complex manifold ${\cal H} (T(S^{2\ell}))$ identifies with the complex manifold ${\cal H}(\bbbr^{2\ell+2})$ of isometric complex structures on the euclidean space $\bbbr^{2\ell +2}$, \[1\]. So, in particular, by restriction to the positively oriented complex structures one has ${\cal H}_+(T(S^4)) = {\cal H}_+(\bbbr^6) = \bbbc P^3$.
[999]{}
DUBOIS–VIOLETTE M. : 1980, “Structures complexes au–dessus des variétés, applications", Séminaire math. E.N.S. in “Mathématique et Physique", L. Boutet de Montvel, A. Douady and J.L. Verdier, eds. [Progress in Mathematics]{} [**vol. 37**]{}, 1–42.
CARTAN, E. : 1938, Leçons sur la théorie des spineurs I,II, Hermann et Cie.
CHEVALLEY, C. : 1954, “The algebraic theory of spinors", Columbia University Press, Morningside Heights, New York.
MANUCEAU, J. : 1970, “Etude algébrique des états quasi–libres; A. Etats quasi–libres des fermions". In [*Cargèse lectures in Physics*]{} [**vol. 4**]{}, D. Kastler ed., Gordon and Breach.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'An orbital ordering, often observed in Mott insulators with orbital degeneracy, is usually supposed to disappear with doping, e.g. in the ferromagnetic metallic phase of manganites. We propose that the orbital ordering of a novel type may exist in such situation: there may occur ferro. orbital ordering of [*complex orbitals*]{} (linear superposition of basic orbitals $d_{x^2-y^2}$ and $d_{z^2}$ with complex coefficients). Despite the perfect orbital ordering, such state still retains cubic symmetry and thus would not induce any structural distortion. This novel state can resolve many problems in the physics of CMR manganites and can also exist in other doped Mott insulators with Jahn-Teller ions.'
address: |
Laboratory of Solid State Physics, Groningen University,\
Nijenborgh 4, 9747 AG Groningen, The Netherlands
author:
- 'D. Khomskii'
title: 'Novel type of orbital ordering: complex orbitals in doped Mott insulators'
---
Mott insulators often display various types of ordering connected with localized electrons. In particular, in the presence of an orbital degeneracy, very often an orbital ordering occurs besides the magnetic (spin) one. Classical examples are the systems with doubly-degenerate $e_g$-electrons, e.g. KCuF$_3$ or MnF$_3$ . At present special attention is paid to the manganites La$_{1-x}M_x$MnO$_3$ ($M=\rm Ca,Sr$) with the colossal magnetoresistance (CMR). The undoped material LaMnO$_3$ is a Mott insulator with the orbital ordering. What becomes of this ordering in doped materials and which role the orbital degrees of freedom do play in the CMR phase, is a matter of active investigation and of hot debate. On the one hand it is usually assumed that the orbital ordering (at least the long-range one) disappears in a ferromagnetic metallic (FM) phase ($0.16<x<0.5$) displaying CMR. This is in particular concluded from the disappearance of lattice distortion typical for the orbitally ordered states. On the other hand the attempts to explain transport properties in this regime using only electron–spin scattering in the framework of the standard double-exchange model failed [@Millis], which forced Millis and coworker to suggest that the orbital degrees of freedom are still important in this regime, e.g. producing Jahn-Teller polarons. Nevertheless the orbitals were still assumed to be disordered in the FM phase below $T_c$.
Until now in all the numerous studies of the cooperative Jahn-Teller effect and orbital ordering only one class of solutions was considered: occupied orbitals were always assumed to be certain linear combinations of the basic orbitals $d_{z^2}\sim\smash{\frac{1}{\sqrt6}}(2z^2-x^2-y^2)$ and $d_{x^2-y^2}\sim\smash{\frac{1}{\sqrt2}}(x^2-y^2)$ of the form $$|\theta\rangle=\cos\frac\theta2\,|z^2\rangle+\sin\frac\theta2\,|x^2-
y^2\rangle\;,
\label{eq1}$$ i.e. linear superposition with real coefficients. Such orbitals give an asymmetric (quadrupolar) distortion of the electron density at a given site. Consequently, any ordering involving orbitals of the type (\[eq1\]), be it of “ferro” or “antiferro” type, corresponds to a decrease of the symmetry of charge distribution and is accompanied (or driven) by the corresponding lattice distortion.
There exists however yet another possibility: there may exist [*orbital ordering without any lattice distortion*]{}. From the basis set of doubly-degenerate orbitals we can also form a linear superposition of
the type (\[eq1\]) but [*with complex coefficients*]{}. The simplest example is the normalized functions $$|\pm\rangle=\frac{1}{\sqrt2}\bigl(|z^2\rangle\pm
i|x^2-y^2\rangle\bigr)\;.
\label{eq2}$$ We can form an orbital ordering with the occupied orbitals of such type, e.g. the ferro. orbital ordering (FOO) with the orbitals $|{+}\rangle$ at each site. It can be easily checked that the electron charge distribution in this state is perfectly symmetric, it is the same along $x$, $y$ and $z$-directions, so that the net symmetry of a crystal would remain cubic. Correspondingly, the effective electron hopping matrix elements are isotropic, $t^x=t^y=t^z=t^*$, as well as superexchange interaction $J^x=J^y=J^z=2t^{*2}/U$. (Here we have in mind the traditional description of the strongly interacting electrons by the Hubbard-type model which in case of orbital
degeneracy is actually a degenerate Hubbard model with the on-site repulsion $U$ and electron hopping between different orbitals, to be specified below.) We suggest that just such an ordering is realized in the FM phase of CMR manganites. If true, this would imply that both the spin and orbital degeneracies are lifted in the ground state in these systems which consequently are [*ferromagnetic in both the spin and orbital variables*]{} (but with “strange” orbitals (\[eq2\])).
Why have such states not been considered before? Apparently it is connected with two factors. On one hand, in most real materials with Jahn-Teller ions studied up to now there always occurred orbital ordering accompanied by lattice distortion. On the other hand, in the theoretical description of these phenomena only the operators corresponding to an ordering of conventional type appear in the effective Hamiltonian (see below), so that it seemed that other possibilities are never realized in practice . Most probably, indeed only the “real” solutions (\[eq1\]) can be realized
in stoichiometric compounds with Jahn-Teller ions. The situation however may be different in doped materials — and that is what we propose here.
It is well known that the electron (or hole) motion in systems with strongly correlated electrons is hindered by an antiferromagnetic background and is much facilitated if the background ordering becomes ferromagnetic (see e.g. ): in nondegenerate Hubbard model it gives rise to Nagaoka’s ferromagnetism [@Nagaoka], and in systems with two types of electrons — to ferromagnetism due to double exchange. The energy gain in such a ferro.state as compared to the antiferro. or paramagnetic (disordered) one is of the order of the electron bandwidth, or of the hopping matrix element $t$ per doped charge carrier: $\Delta E=-ctx$ ($c$ is some constant of order 1), whereas the energy loss is of the order of $J$ $(\sim t^2/U)\cdot(1-x)$.
Thus, as argued already in [@Nagaoka], the saturated ferromagnetic state is realized for $x>x_c\sim J/t\sim t/U$. For smaller $x$ the inhomogeneous phase separated state with ferromagnetic droplets in an antiferomagnetic matrix can be realized [@Visscher].
In exact analogy to this case we should expect that due to the same mechanism a ferro. orbital ordering will be established in doped strongly interacting Mott insulators with orbital degeneracy. Which particular orbitals would be stabilized at that, may depend on the particular situation; we suggest that it may be the complex orbitals of the type (\[eq2\]).
A convenient way to describe the orbital ordering of double-degenerate $e_g$-orbitals is to introduce pseudospin variables $\tau_i$ such that e.g. the orbital $d_{z^2}$ corresponds to $\tau^z=+\frac12$, and the orthogonal orbital $d_{x^2-y^2}$ — to $\tau^z=-\frac12$. The orbital states (\[eq1\]) considered until now are parametrized by the angle $\theta$ in the $(\tau^z,\tau^x)$-plane. Cubic symmetry is reflected in the $\theta$-plane in $\frac{2\pi}3$ symmetry: the state $\theta=\frac{2\pi}3$ corresponds to the orbital $|x^2\rangle\sim(2x^2-y^2-z^2)$ and $\theta=-\frac{2\pi}3$ — to $|y^2\rangle\sim(2y^2-x^2-z^2)$; these are more or less the orbitals occupied in two sublattices in the undoped LaMnO$_3$. When we dope the system with orbital ordering, the motion of the charge carriers would initially (for small $x$) occur on a background of this orbital ordering. Similar to the hole motion on an antiferromagnetic spin background, the “antiferro” orbital ordering would hinder the motion of a hole and would reduce its bandwidth. One can indeed check that, by making the orbital order ferromagnetic, e.g. occupying the same orbital at each site, for instance $|z^2\rangle$ or $|z^2-\nobreak y^2\rangle$, we would increase the bandwidth and correspondingly decrease kinetic energy of holes; the mathematical details are presented below. However such a FOO seems to contradict experimental observations. Thus e.g.$|z^2\rangle$-ferromagnetism would lead to a tetragonal distortion of the lattice with $c/a>1$ and to a strong anisotropy of transport properties; but nothing like that is observed experimentally. Probably that is why this possibility, which by analogy with the spin case seems quite natural, is never considered for CMR phase of manganites, and it is usually assumed that the orbital ordering
is simply lost in the FM phase leading e.g. to an orbital liquid [@Nagaosa] However if at such a FOO the [*complex*]{} orbitals (\[eq2\]) would be occupied — there would be no contradiction with the structural data, and still the motion of doped charge carriers would be unhindered. This is the main idea of the present paper. In the pseudospin language introduced above the state (\[eq2\]) is an eigenstate of the operator $\tau^y$ which also exists in the algebra of $\tau=\frac12$ operators.
For strongly interacting electrons with one electron per site one can describe the effective spin and orbital interaction by the Hamiltonian which schematically has the form $${\cal H}=\sum J_s(\vec s_i\vec
s_j)+J_\tau(\tau_i\tau_j)+J_{s\tau}(\vec s_i\vec
s_j)(\tau_i\tau_j)\,,
\label{eq3}$$ where the first and the third terms are due to superexchange, $J_{exch}\sim J_s\sim J_{s\tau}\sim t^2/U$, and in the second term also the Jahn-Teller interaction of degenerate orbitals with the lattice contribute, $J_\tau=J_{exch}+J_{\rm JT}$ The interaction (\[eq3\]) is in general anisotropic with respect to $\tau$-operators and usually it does not contain terms with $\tau^y$. Consequently $\tau^y$-states do not appear in the mean-field approximation which is nearly always used to treat insulators with orbital degeneracy. But it does not mean that such states cannot appear in certain situations, e.g. for doped systems.
Using the specific form of the $e_g$-orbitals and of the corresponding hopping integrals between different orbitals in different directions, one can easily calculate the one-electron band structure for different types of orbital ordering. Denoting the hopping between $z^2$-orbitals for a pair along $z$-direction as $t$, we have: $t^z_{z^2,z^2}=t$, $t^{x,y}_{z^2,z^2}=t/4$, $t^{x,y}_{x^2-y^2,x^2-y^2}=3t/4$, $t^{x,y}_{z^2,x^2-y^2}=\pm\sqrt3\,t/4$, $t^x_{z^2,x^2}=t^{x,y}_{x^2,y^2}=t/2$, etc., see e.g.
Generally speaking, in contrast to spin case (see for discussion and references), an antiferro. orbital ordering, e.g. that in LaMnO$_3$, does not completely suppress the hole motion even if one ignores quantum effects: the hole can always hop to a neighbouring site without destroying orbital order along its trajectory. This is connected with the fact that the nondiagonal hopping in orbital channel is in general allowed, and pseudospin projection $\tau^z$ is not conserved during hopping: the nondiagonal hopping integrals $t_{z^2,x^2-y^2}$ are in general nonzero. Thus the hole can here move without leaving a “trace” of wrong spins [@Bulaevskii] and, consequently, there will be no “confinement”. However the bandwidth of this coherent motion without disturbing the background orbital ordering is reduced as compared to the bandwidth for FOO. Thus, using the values of hopping integrals $t_{\alpha,\beta}^{\langle
ij\rangle}$ given above, one obtains for the undoped LaMnO$_3$ (alternation of $x^2$ and $y^2$ orbitals) the spectrum $$\textstyle
\varepsilon(k)=-2\left[\frac12t(\cos k_x+\cos
k_y)+\frac14t\cos k_z\right]
\label{eq4*}$$ so that the minimum energy of the hole (the bottom of the band) will be $\varepsilon_{\it min}=\varepsilon(k{=}0)=-2.5t$. If however we would make a FOO, e.g. occupying at each site $z^2$-orbital, the spectrum will be $$\textstyle
\varepsilon^{z^2\hbox{\scriptsize\it-ferro}}(k)=-2t\left[\frac14(\cos
k_x+\cos
k_y)+\cos k_z\right]
\label{eq5*}$$ and $\varepsilon^{z^2\hbox{\scriptsize\it-ferro}}_{\it min}=-3t$. Thus we indeed see that the energy gain in the FOO state is of order $\sim t$. The same minimum energy is reached also for $(x^2-y^2)$-ferro.ordering and for FOO with any orbital of the type (\[eq1\]). From (\[eq1\]) one can easily obtain that $$\textstyle
t_{\theta,\theta}^z=\langle\theta|\hat
t\,^z|\theta\rangle=t\cos^2\frac\theta2;\qquad
t_{\theta,\theta}^{x\,{/}\,y}=\frac
t4(\cos\frac\theta2\pm\sqrt3\,\sin\frac\theta2)^2\;,
\label{eq9n}$$ And one obtains that the bottom of the spectrum with the hopping integrals given by (\[eq9n\]) does not depend on $\theta$ and coincides with the value $-3t$.
But exactly the same $\varepsilon_{\it min}$ is also reached for the $\tau^y$-ferromagnetism, where the state (\[eq2\]) is occupied at each site. Using the values of $t$’s presented above, one obtains that $t_{\tau^y,\tau^y}=\frac12t$ in all three directions, and $$\varepsilon^{\tau^y\hbox{\scriptsize\it-ferro}}(k)=-t(\cos k_x+\cos
k_y+\cos k_z)$$ so that the spectrum is indeed isotropic, and the minimum energy is also equal to $-3t$.
Note that, similarly to the nondegenerate Hubbard model, we can use the simple dispersion relations (\[eq4\*\]), (\[eq5\*\]) only for the electron motion which does not destroy the background ordering. The energy gain in the FOO state is again, similarly to the Nagaoka case [@Nagaoka], $\sim tx$, and the energy loss is $\sim J$ (in orbital sector $\sim
J_\tau+J_{s\tau}$). Here again we obtain the totally ferromagnetic state both in spins and in orbitals for $x>x_c\sim J/t$, and one can have phase separation for $x<x_c$.
As we saw above, for small $x$ different FOO states are equivalent as to the gain in kinetic energy. On the other hand we can see that $\tau^y$-ferro. state is favourable as to the loss of the effective exchange energy (\[eq3\]). Indeed, the effective Hamiltonian (\[eq3\]) contains the orbital operators in combinations $\tau^z_i \tau^z_j$ , $\tau^x_i \tau^x_j$ and $\tau^z_i \tau^x_j$ with positive (antiferromagnetic) coeficients $J_{\tau}$, $J_{s\tau}$ . In the ground state of the undoped system a certain spin and orbital order is realised which minimizes the total energy — and it will be an antiferro.ordering of real orbitals (\[eq1\]). If however we [*force*]{} our system to be ferromagnetic both in spins and in orbitals, e.g., as argued above, by doping, we strongly increase this exchange energy — and we can minimize this energy loss by chosing FOO of $\tau^y$-type.
One can check directly that the superexchange part of the energy is indeed smaller in the $\tau^y$-FOO state as compared e.g. with the FOO state of real orbitals. Thus, taking the standard expression for the exchange integrals $J=2t^2/U$ and calculating exchange constants with the proper values of $t$, we obtain e.g. that for the $z^2$-FOO the energy of the ferromagnetic state (per site) in mean-field approximation can be written in the form: $$E_{\it
ferro}^{z^2\hbox{\scriptsize-FOO}}=\frac94\frac{t^2}{U}
+ C\;,
\label{eq11}$$ where $C = - 3 \frac{t^2}{U}$. The same value of $E_{\it ferro}$ we would obtain for $(x^2-y^2)$-FOO and for all other FOO states with arbitrary state $|\theta\rangle$ of the type (\[eq1\]): again one can easily show that with the values of hopping integrals $t_{\theta\theta}^\alpha$ (\[eq9n\]) the magnetic energy of both ferro- and antiferromagnetic states does not depend on $\theta$ and is given by (\[eq11\]). Similar treatment of the $\tau^y$-FOO state shows however that the net exchange energy in this state is lower: $$E_{\it
ferro}^{\tau^y\hbox{\scriptsize-FOO}}=\frac32\frac{t^2}{U}\,
+ C\;,$$ where $C$ is the same as in Eq.(\[eq11\]). Similarly, the intersite Jahn-Teller energy in FOO state with real orbitals is positive, $\sim J_{\rm JT}$, whereas it is zero for the $\tau^y$-FOO state; again from this point of view the $\tau^y$-FOO state wins against
all other possible FOO states with real orbitals.
As the electronic energy of doped charge carriers coincides for the states (\[eq1\]) and (\[eq2\]), and the magnetic energy of the latter is lower, one can conclude that the proposed ferromagnetic state with $\tau^y$-FOO (occupation of complex orbitals (\[eq2\])) will be the best among all possible ferro. ordered states — and these, we argue, may be stabilized by the usual Nagaoka mechanism. This state can possibly be realized in the ferromagnetic metallic phase of CMR manganites. The finite band filling would increase electron kinetic energy somewhat faster for $\tau^y$-ordering as compared e.g. to $z^2$-FOO or $(x^2-y^2)$-FOO (the energy spectrum in the latter cases is anisotropic and the density of states at the band edge is higher, cf. [@vandenBrink]). Consequently the $\tau^y$-FOO may be destabilized at still higher doping levels.
The state with $\tau^y$-FOO need not be necessarily homogeneous: as in the pure spin case, for small doping there may exist a tendency towards phase separation due to an instability of the homogeneous canted spin state [@Visscher; @Yunoki]. But, as follows from the arguments given above, the structure of the hole-rich regions can well be of $\tau^y$-FOO type.
>From the point of view of symmetry the proposed $\tau^y$-FOO corresponds to the $A_{2g}$ (or $\Gamma_2$)-representation of the cubic group [@Abragam]. The corresponding order parameter is $$T_{xyz}=\langle{\cal S}l_xl_yl_z\rangle\,,
\label{new2}$$ where $\langle\;\rangle$ is the ground state average, ${\cal S}$ denotes
symmetrization and $l_x$, $l_y$, $l_z$ are the corresponding components of the momentum operator $\hat l=2$. This follows from the group symmetry, and we checked by direct calculation that it is the lowest nonzero average in our state. One can show that in the basis $|z^2\rangle$, $|x^2-y^2\rangle$ $T_{xyz}$ is indeed proportional to the $\tau^y$-Pauli matrix. From eq. (\[new2\]) we see that the order parameter in our state is a magnetic octopole, cf. . It is in principle possible, although not easy, to observe such ordering experimentally. Probably the most promising would be the resonance experiments like NMR or circular X-Ray dichroism. One can also show that this state should have a piezomagnetism.
It is not apriori clear whether this $\tau^y$-FOO and the ferromagnetic spin ordering would occur at the same temperature. The direct product of corresponding irreducible representations $A_2$ and $T_1$ does not contain unit representation $A_1$ [@Abragam], so that there will be in general no linear coupling between these order parameters. However if the rhombohedral distortion $T_{2g}$, present in manganites at high temperatures (S. W. Cheong, private communication), would still exist at low temperatures, then there will exist linear coupling of these three types of ordering ($A_2\times T_1\times
T_2\in A_1$) and the $\tau^y$-FOO and the ferromagnetic ordering will occur at the same $T_c$. This could help to resolve the problem pointed out in Ref. [@Millis], that the change of resistivity through $T_c$ in purely spin case is too small: one would get an extra scattering in a “para” phase — an orbital-disorder scattering — in addition to the spin-disorder one. (Alternatively one can speak of the increase of resistivity above $T_c$ not due to orbital fluctuations, but, again in analogy with the spin case , due to the effective narrowing of the band in the orbitally disordered phase.)
Summarizing, we suggest in this paper that, similarly to magnetic ordering, doping of orbitally degenerate Mott insulators containing Jahn-Teller ions could stabilize the ferro. orbital ordering of special type, the occupied orbitals being “complex” — i.e. the linear superpositions of basic orbitals $d_{x^2-y^2}$ and $d_{z^2}$ with complex coefficients. Such orbitals, despite perfect ordering, do not induce any structural distortion. At the same time the motion of charge carriers on such ordered background is completely free, and there is no extra band narrowing; it is just this factor that favours such ferro. orbital ordering. Thus the origin of the ferro. orbital ordering in doped systems is the same as that of the Nagaoka ferromagnetism in a partially-filled Hubbard model. The ferro. ordering of complex orbitals gives the same band energy as the ordering of conventional real ones but has lower exchange and Jahn-Teller energy; therefore the ferro. ordering of complex orbitals may be preferable. This idea may explain the main properties of the colossal magnetoresistance manganites in the most interesting ferromagnetic metallic concentration range: in this picture they are perfectly ordered with respect to both spin and orbitals, but with the [*complex*]{} orbitals occupied at each site. In particular, this could help to explain the sharp drop of resistivity below $T_c$, with the crystal structure remaining undistorted. Similar states may in principle exist also in other doped Mott insulators with Jahn-Teller ions which constitute a large interesting class of magnetic materials with very specific properties.
I am grateful to G. A. Sawatzky and M. V. Mostovoy for useful discussions. I am also grateful do D. Cox who pointed out a possible importance of the rhombohedral distortion in manganites. This work was supported by the Netherlands Foundation FOM, by the European Network OXSEN and by the project INTAS–97 0963.
[9]{}
K. I. Kugel and D. I. Khomskii, Sov. Phys.—JETP [**37**]{}, 725 (1973); Sov. Phys.—Uspekhi, [**25**]{}, 231 (1982).
A. J. Millis, P. B. Littlewood and B. I. Schraiman, Phys. Rev. Lett. [**74**]{}, 5144 (1995).
The only works where such possibility was considered are two old papers by L. I. Korovin and E. K. Kudinov, Sov. Phys. Solid State [**15**]{}, 826 (1973); [**16**]{}, 1666 (1975), who carried out general symmetry classification of possible solutions and pointed out that the solutions with orbitals of the type (\[eq2\]) are in general allowed.
Recently similar treatment was independently carried out in connection with the properties of Ce$_x$La$_{1-x}$Bi$_6$ by R. Shiina, H. Shiba and P. Thalmeier, J. Phys. Soc. Jap. [**66**]{}, 1741 (1997) and by H. Kusunose and Y. Kuramoto, Phys. Rev. B [**59**]{}, 1902 (1999).
D. I. Khomskii and G. A. Sawatzky, Solid State Comm. [**102**]{}, 87 (1997).
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P. B. Visscher, Phys. Rev. B [**10**]{}, 943 (1974).
S. Ishihara, M. Yamanaka and N. Nagaosa, Phys. Rev. B [**56**]{}, 686 (1997).
L. N. Bulaevskii, E. L. Nagaev and D. I. Khomskii, Sov. Phys.—JETP [**27**]{}, 836 (1968).
J. van den Brink and D. I. Khomskii, Phys. Rev. Lett. [**82**]{}, 1016 (1999).
A. Moreo, S. Yunoki and E. Dagotto, Science [**283**]{}, 2034 (1999); D. Arovas and F. Guinea, Phys. Rev. B [**58**]{}, 9150(1998); M. Kagan, D. Khomskii and M. Mostovoy, Eur. Phys. J. B [**22**]{}, 217 (1999).
A. Abragam and B. Bleaney, “Electronic paramagnetic resonance of transition ions,” Clarendon Press, Oxford (1970).
W. F. Brinkman and T. M. Rice, Phys. Rev. B [**2**]{}, 1324 (1970).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'For almost half of the one hundred year history of Einstein’s theory of general relativity, Strong Cosmic Censorship has been one of its most intriguing conjectures. The SCC conjecture addresses the issue of the nature of the singularities found in most solutions of Einstein’s gravitational field equations: Are such singularities generically characterized by unbounded curvature? Is the existence of a Cauchy horizon (and the accompanying extensions into spacetime regions in which determinism fails) an unstable feature of solutions of Einstein’s equations? In this short review article, after briefly commenting on the history of the SCC conjecture, we survey some of the progress made in research directed either toward supporting SCC or toward uncovering some of its weaknesses. We focus in particular on model versions of SCC which have been proven for restricted families of spacetimes (e.g., the Gowdy spacetimes), and the role played by the generic presence of Asymptotically Velocity Term Dominated behavior in these solutions. We also note recent work on spacetimes containing weak null singularities, and their relevance for the SCC conjecture.'
author:
- 'James Isenberg[^1]'
bibliography:
- 'bibliography.bib'
title: On Strong Cosmic Censorship
---
Introduction {#Intro}
============
Ever since the 1916 discovery of the Schwarzschild solution, singularities have played a major role in general relativity. During the first fifty years following Einstein’s proposal of general relativity in 1915, singularities appeared primarily as a mathematical feature of most of the explicit solutions of Einstein’s gravitational field equations found in that period: In the Friedmann-Lemaître-Robertson-Walker (FLRW), the Kasner, the Taub, and the Kerr solutions, as well as that of Schwarzschild, it was found that if the coordinates are extended to their natural limits, the metrics of these solutions either blow up or go to zero, thereby “going singular". Such behavior was familiar to physicists in solutions of the more familiar theories such as Maxwell’s theory of electromagnetism[^2]. However in Maxwell’s theory, many solutions without singularities were well-known as well. With singular solutions appearing to dominate in general relativity, many asked if in fact they are a generic feature of the theory, or rather just an artifact of the symmetries of the known solutions.
The issue of the genericity of singularities in solutions of Einstein’s equations was a major topic of research in general relativity during the 1960s. A number of approaches to explore this issue were pursued; the one which (in a certain sense) led to a definitive answer, is the one based on the dynamics of congruences of causal geodesic paths. Using this approach, Hawking and Penrose proved a number of results (see Section \[HPSing\]) which, suitably interpreted, claim that singularities occur in generic solutions. These results, which are often collectively labeled as the “Hawking-Penrose Singularity Theorems", lead to this claim *if* one interprets “singularity" to be causal geodesic incompleteness (CGI), and *if* one interprets the hypotheses of the Hawking-Penrose theorems as corresponding to a generic class of solutions.
While causal geodesic incompleteness indicates some degree of pathology in a spacetime with this characteristic, the features accompanying CGI vary widely from spacetime to spacetime. For example, in the FLRW and the Kasner solutions, curvature and tidal force blowups accompany the CGI; while in the Taub-NUT extension of the Taub spacetime, the curvature is bounded, but the presence of a Cauchy horizon is closely tied to causal geodesic incompleteness. In terms of what an observer sees along his or her worldline, those proceeding toward the singularity in an FLRW or Kasner spacetime are stretched and crushed to death, while those heading for the singularity in a Taub-NUT spacetime may enter a region in which the ability to predict the future from a known set of initial conditions breaks down.
Whether or not one or the other of these behaviors is “physically preferable" or in some sense “less singular", it would be very interesting to determine which happens more often. Penrose addressed this question almost fifty years ago, proposing his Strong Cosmic Censorship (SCC) conjecture. Roughly stated (as was the case in its early form), the SCC conjecture claims that in solutions of Einstein’s equations, curvature blowup generically accompanies causal geodesic incompleteness.
Both because of its mathematical elusiveness and because of its somewhat compelling physical implications (should we expect to be crushed, or might we be able to “go back in time"?), the Strong Cosmic Censorship conjecture has been viewed as one of the central questions of mathematical relativity. Despite this strong interest, SCC remains unresolved, and one might argue that very little of the work which has been done to date tells us anything directly regarding whether SCC holds or not. Much of this work has been devoted to the study (and proof) of model versions of Strong Cosmic Censorship, which are either restricted to families of solutions characterized by each spacetime in the family admitting a nontrivial isometry group of dimension one or higher (e.g., solutions which are spatially homogeneous, or are invariant under a spatially acting $T^2$ or $U(1)$ isometry group), or are restricted to sets of solutions which are small perturbations of known solutions (e.g., perturbations of an FLRW spacetime [@RS]). Since SCC is fundamentally about behavior which is generic in the set of *all* solutions of Einstein’s equations, and since these isometry-based families are effectively of measure zero in this set, a model proof of SCC in one of these families is not directly related to the validity of the full SCC conjecture. However, it is hoped that these model studies do allow researchers to develop tools that could be useful in the study of the full conjecture.
We begin this report on the Strong Cosmic Censorship conjecture with a brief review (in Section \[HPSing\]) of what the Hawking and Penrose theorems explicitly tell us regarding the prevalence of singularities—in the sense of geodesic incompleteness—in solutions of Einstein’s equations. Next, in Section \[Penrose\], we discuss Penrose’s original conception of Strong Cosmic Censorship. In doing this, we also comment on his conception of Weak Cosmic Censorship (WCC), noting that SCC does not imply WCC, and WCC does not imply SCC.
In these earliest formulations, Penrose did not seek to state either conjecture in a rigorous way. We provide an example of a rigorous statement of SCC in Section \[PoldGowdy\], where we discuss a model Strong Cosmic Censorship theorem for the limited case of Polarized Gowdy spacetimes. This discussion introduces the idea of studying SCC by stating and proving model SCC-type theorems for limited families of spacetimes. We continue in this direction in Section \[TGowdy\], where we discuss some of the ideas used in Ringström’s significantly deeper and more intricate proof of a model SCC theorem for all ($T^3$) Gowdy spacetimes.
One of the key steps in the proofs of these model SCC theorems, both for the polarized and for the general Gowdy spacetimes, is the demonstration that generic $T^3$ Gowdy spacetimes exhibit *asymptotically velocity-term dominated* (AVTD) behavior. We discuss in Section \[PoldGowdy\] what AVTD behavior is, and how it can be useful in studying SCC. Then in Section \[AVTD\], we discuss evidence for (and against) the presence of AVTD behavior in a number of families of spacetime solutions of Einstein’s equations.
In his early discussions of SCC, one of Penrose’s main arguments for the conjecture was that “blue shift" effects would tend to disrupt the formation of a Cauchy horizon inside black holes. Christodoulou, Dafermos, and others have developed this idea into an approach for studying SCC which has recently yielded considerable insight. A key feature of this approach is the possible development of *weak null singularities* inside black holes. We discuss in Section \[Blue\] what these are, their possible stability, and the implications for SCC if indeed weak null singularities are a stable feature of solutions of Einstein’s equations. We conclude with comments on the fruitful role the Strong Cosmic Censorship has played in mathematical relativity.
The Hawking-Penrose Singularity Theorems {#HPSing}
========================================
The notion of a singularity is difficult to pin down in general relativity because, in contrast to Maxwell’s theory or Navier-Stokes’ theory for which there is an a priori fixed background spacetime on which to check whether or not the fields are bounded, for Einstein’s theory the spacetime on which the fields are defined is not fixed. Sets on which the fields blow up can be removed or added; hence the presence or absence of unbounded fields in a given solution is a malleable feature. As well, the boundedness of the gravitational field in a given solution can depend strongly on the choice of frame and the choice of coordinates. These difficulties led researchers during the 1960s to settle on causal geodesic incompleteness of an inextendible spacetime[^3] $(M^{n+1}, g, \Psi)$ to be the criterion for labeling that spacetime as singular. (See [@Ger] for further discussion and justification of this criterion.)
If one decides to use CGI as the mark of a singular solution of the Einstein equations, the study of congruences of geodesic paths in them is a natural way to determine if singularities are a prevalent feature of such solutions. One of the key tools for studying geodesic congruences is the Raychaudhuri equation, which (for a surface-orthogonal timelike congruence) takes the form $$\label{Raychaud}
\nabla_U \Theta = - \frac 14 \Theta^2 - \Sigma_{\alpha \beta} \Sigma ^{\alpha \beta}-R_{\alpha \beta}U^{\alpha} U^{\beta};$$ here $U$ is the vector field tangent to the congruence, $\nabla_U$ is the directional derivative along $U$, $\Theta:=\nabla_\alpha U^\alpha $ is the expansion of the congruence, $\Sigma_{\alpha \beta} := \frac 12 (\nabla_\alpha U_\beta + \nabla_\beta U_\alpha)- \frac 14 \Theta g_{\alpha \beta} $ is the shear of the congruence, and $R_{\alpha \beta}$ is the Ricci curvature tensor of the spacetime.
As stated here, this equation has nothing to do with whether or not the spacetime containing the geodesic congruence is a solution of Einstein’s equations; it is purely a geometric consequence of tracing over the definition of the Riemann curvature tensor. However, if one uses the Einstein field equations in the form $$\label{RicEinsteqs}
R_{\alpha \beta} = \kappa (T_{\alpha \beta} - \frac 12 g_{\alpha \beta} T) + \Lambda g_{\alpha \beta}$$ to replace the Ricci tensor in the Raychaudhuri equation , and if one presumes that the stress-energy tensor $T_{\alpha \beta}$ (together with the cosmological constant $\Lambda$, and Newton’s constant $\kappa$) satisfies the positivity condition[^4] $$\label{strongenergycond}
\kappa(T_{\alpha \beta} W^\alpha W^{\beta} - T^\beta_\beta W_\alpha W^\alpha ) + \Lambda W_\alpha W^\alpha>0$$ for any timelike vector field $W$, then the Raychaudhuri equation tells us that if the congruence expansion $\Theta$ is non-zero at any point $p$ for some congruence of surface-orthogonal paths, then $\Theta$ must blow up in finite proper (affine) time either to the future (if $\Theta(p)>0$) or to the past (if $\Theta(p)<0$) of $p$. This result is one of the primary tools used for proving most of the Hawking-Penrose singularity theorems.
A wide variety of different results proven during the 1960s are collectively known as the Hawking-Penrose singularity theorems. Just about all of them have a particular characteristic form: They show that a spacetime $(M^{n+1}, g, \Psi)$ must be causal geodesically incomplete (and hence “singular") so long as that spacetime satisfies a set of conditions including each of the following: (i) a causality condition (e.g., the spacetime admits no closed causal paths); (ii) a regularity condition (e.g., the spacetime is smooth); (iii) an “energy condition" (e.g., the spacetime satisfies ); (iv) a curvature “generic condition" (e.g., every causal geodesic in the spacetime contains at least one point at which $$\label{generic}
V_{[\gamma} R_{\alpha] \mu \nu [\beta}V_{\delta] }V^\mu V^\nu \neq 0,$$ with $V^{\mu}$ the vector tangent to the geodesic, and with $V_{[\alpha} W_{\beta]}$ indicating index skew-symmetrization); and (v) a boundary/initial condition (e.g., the spacetime admits a closed achronal hypersurface).
The following archetypal example of such a theorem, proven by Hawking and Penrose in 1970 [@HP70] follows this pattern closely:
\[HPThm\] If a spacetime $(M^{3+1},g,\Psi)$ with stress-energy $T^{\alpha \beta}$ is a smooth solution of Einstein’s equations, if it contains no closed timelike paths, if it satisfies the strong energy condition , if the inequality holds at least somewhere along every one of its causal geodesic paths, and if it admits either a closed achronal hypersurface or a closed trapped surface, then the spacetime cannot be causal geodesically complete.
Does this, or any other such theorem, show that “generic" solutions are CGI, or even that “physically reasonable" solutions are generically CGI? Of course deciding this one way or the other depends on what one means in using this terminology, and how such meaning compares with the conditions contained in the hypothesis of Theorem \[HPThm\]. In exploring the behavior of solutions of Einstein’s equations, it is not generally considered to be overly restrictive to eliminate those solutions which fail to satisfy a causality condition or fail to be differentiable in some appropriate sense.[^5] The genericity of the other three conditions is less convincing, however: While the strong energy condition does hold for Einstein-vacuum as well as Einstein-Maxwell solutions, it fails for solutions with negative cosmological constant (the sign which is needed for simple cosmological models with accelerated expansion). One may reasonably choose to focus on spacetimes containing a closed achronal hypersurface or even a closed Cauchy surface; but if one is interested in asymptotically flat solutions, the presumption that there is an embedded trapped surface is somewhat restrictive[^6]. As for the condition that inequality hold somewhere along every causal geodesic path, although this is often labeled “the generic condition" by those using it, there is no particular evidence one way or the other that this condition is indeed generic.
Whether or not it follows from the singularity theorems that solutions of Einstein’s equation are generically CGI, this issue is not crucial in considering Strong Cosmic Censorship. SCC is concerned with generic behavior in spacetimes which contain incomplete causal geodesics, not whether the CGI property itself is generic.
Penrose’s Cosmic Censorship Conjectures {#Penrose}
=======================================
Soon after proving the first of the Hawking-Penrose singularity theorems, Penrose began discussing ideas which evolved into the cosmic censorship conjectures. The first appearance of these ideas in the literature was in [@Pen68] and [@Pen69] in the late 1960s. While these references do not present a definitive statement of SCC, they do provide an intuitive formulation:
\[Intuitive Version of Strong Cosmic Censorship\] Globally hyperbolic spacetime solutions of Einstein’s equations generically cannot be extended as solutions past a Cauchy horizon[^7].
Simultaneous with his discussions of SCC, Penrose proposed a second, very different, but equally intriguing conjecture. Labeled Weak Cosmic Censorship (WCC), this conjecture takes the following intuitive form:
\[Intuitive Version of Weak Cosmic Censorship\] In generic asymptotically flat spacetime solutions of Einstein’s equations, singularities are contained within black hole horizons.
Our focus in this review is on SCC, not WCC. We mention the latter here primarily to emphasize the fact that neither conjecture (if proven) implies the other–they are logically distinct. We also note that the shared name “cosmic censorship" pertains more aptly to WCC than to SCC: Weak cosmic censorship proposes that “naked singularities"–those visible to far away observers–do not occur generically (they are forbidden by the “cosmic censor").
One key shared feature of SCC and WCC is that both concern the behavior of *generic* solutions. While this term must be made precise before either conjecture can be proven, even in rough form the implication is clear that the existence of solutions with Cauchy horizons does not disprove SCC, and the existence of asymptotically flat solutions with singularities to the causal past of asymptotic observers does not disprove WCC. This important feature invalidates the majority of the proffered counterexamples to both WCC and SCC which have appeared in the literature.
A Model SCC Theorem: Polarized Gowdy Spacetimes {#PoldGowdy}
===============================================
One way to explore evidence favoring or disfavoring a comprehensive conjecture such as SCC is to study if a suitably adapted form of it is valid for special families of solutions. Presuming that these families are essentially of measure zero in the space of all solutions, such studies can neither prove nor disprove the conjecture. However, in attempting to prove or disprove model versions of SCC (“model-SCC") in special families (such as the Gowdy spacetimes) one can develop ideas, techniques, and scenarios which might ultimately be useful in determining if in fact the SCC conjecture holds. Of course, one must also keep in mind that the lessons learned in proving model-SCC for a given family could instead be misleading, as we discuss below.
While model versions of SCC have been proven for larger families of spacetimes, the family of polarized Gowdy solutions provides a very good example of the rigorous formulation and proof of such a result. Hence, we discuss some of the details of this case here.
The polarized Gowdy spacetimes are solutions of the vacuum Einstein’s equations which are characterized by the following geometric features: 1) Each solution admits an effective $T^2$ isometry group acting spatially (hence there are two independent commuting everywhere-spacelike Killing fields). 2) The two Killing fields can be aligned orthogonally everywhere (this is the “polarizing condition"; without it, one has a general Gowdy spacetime). 3) The Killing field “twists", which take the form $X \wedge Y \wedge dX$ and $X \wedge Y \wedge dY$ for $X$ and $Y$ labeling the one-forms corresponding to the Killing fields, vanish. 4) The spacetimes admit compact Cauchy surfaces.
These conditions allow for three possible spacetime manifolds: $T^3 \times R^1, S^2\times S^1 \times R^1$, and $S^3 \times R^1$ (Lens spaces may also replace the 3-sphere; however the analysis is no different for such replacements). While a model-SCC has been proven for all of these cases [@CIM], to avoid unnecessary detail here, we restrict our discussion here to the $T^3 \times R^1$ case.
For polarized Gowdy spacetimes, coordinates may be chosen so that the metric can be written as follows: $$\label{PolGowdyMetric}
g=e^{\frac{(\tau+\lambda)}{2}}(-e^{-2\tau}d\tau^2 +d\theta^2) +e^{-\tau} (e^P dx^2 +e^{-P}dy^2).$$ Here $\tau \in R^1$ and $(\theta, x,y)$ are coordinates on the 3-torus, with the orbits of the $T^2$ isometry group corresponding to 2-surfaces of constant $\tau$ and constant $\theta$. In terms of the metric functions $P$ and $\lambda$ (which are functions of $\theta$ and $\tau$ only), the vacuum Einstein equations take the form $$\begin{aligned}
\label{Peqn}
\partial_{\tau \tau} P &= e^{-2\tau}\partial_{\theta \theta} P,\\
\label{lambdatau}
\partial_\tau \lambda &= (\partial_\tau P)^2 +e^{-2\tau} (\partial_\theta P)^2,\\
\label{lambdatheta}
\partial_\theta \lambda &= 2 \partial_\tau P \partial_\theta P.\end{aligned}$$ It is readily apparent from the form of these equations that the initial value problem for the polarized Gowdy spacetimes is well-posed. In particular, one sees that for any choice of a smooth pair of functions $P(\theta,0) =p(\theta)$ and $\partial_\tau P(\theta,0)=\pi(\theta)$ satisfying the integrability condition $\int_{S^1} \pi \partial_\theta p
d\theta =0$ on the circle, the wave equation admits a unique (maximally extended) solution $P(\theta, \tau)$; and for any choice of a constant $\lambda(0,0)=\ell$ together with the solution for $P(\theta, \tau)$, equation can be integrated to produce initial data $\lambda(\theta,0)$ for $\lambda$, after which can be used to evolve to a unique (maximally extended) solution $\lambda(\theta, \tau)$. Letting $\Pi_{pol}$ denote the space of initial data sets (a pair of functions on the circle plus a constant, with the functions satisfying the integrability condition noted above) for the polarized Gowdy spacetimes, and noting that the evolution of $(P(\theta, \tau), \lambda(\theta, \tau))$ from a set of initial data as just described corresponds to the unique maximal globally hyperbolic spacetime development [@CB-G] of that data set, we may state a model SCC theorem for these solutions as follows:
\[Model-SCC Theorem for Polarized Gowdy Spacetimes\] \[SCCPoldGowdy\] There exists an open dense subset (in $C^\infty$ topology) $\hat \Pi_{pol}$ of $\Pi_{pol}$ such that the maximal globally hyperbolic spacetime development of any data set in $\hat \Pi_{pol}$ is inextendible.
The key to proving this theorem is to first show that the AVTD property holds for all polarized Gowdy spacetimes; then to use this result to show that there is a homeomorphism from $\Pi_{pol}$ to a space $\mathcal Q_{pol}$ consisting of sets of asymptotic data which characterize the behavior of the solution approaching the singularity; and finally to show that for an open and dense subset $\hat {\mathcal Q}_{pol}$ of $\mathcal Q_{pol}$, the curvature blows up in a neighborhood of the singularity, hence preventing extension across a Cauchy horizon in that singular region. In addition, one must show that in the expanding direction of these $T^3$ polarized Gowdy solutions[^8], the spacetimes are geodesically complete and nonsingular.
Since AVTD behavior plays such a central role in the study of SCC for these as well as other spacetimes discussed below, it is useful to describe the property and how it is verified for these spacetimes in a bit more detail. To define AVTD behavior for a family of solutions of Einstein’s equations, we need to first determine an associated VTD system of equations. In the case of the polarized Gowdy solutions, written in areal coordinates, the associated VTD equations are the same as the full set of Einstein’s equations -, but with the spatial derivatives dropped from the first two equations[^9] $$\begin{aligned}
\label{PeqnV}
\partial_{\tau \tau} \tilde P &= 0,\\
\label{lambdatauV}
\partial_\tau \tilde \lambda &= (\partial_\tau \tilde P)^2 ,\\
\label{lambdathetaV}
\partial_\theta \tilde \lambda &= 2 \partial_\tau \tilde P \partial_\theta \tilde P.\end{aligned}$$ Noting that the singularity for these spacetimes occurs as $\tau$ approaches $ + \infty$, we define a particular polarized Gowdy solution $(P(\theta, \tau), \lambda(\theta, \tau))$ to have AVTD behavior if there exists a solution $(\tilde P(\theta, \tau), \tilde \lambda(\theta, \tau))$ of the VTD equations such that the solution of the full system rapidly approaches the VTD solution for large $\tau$. In fact for the polarized Gowdy solutions, it has been shown [@CIM] that there exists a constant $C$ such that $|P(\theta, \tau) - \tilde P(\theta, \tau)|< C e^{-2 \tau}$ and $|\lambda (\theta, \tau) - \tilde \lambda (\theta, \tau)|< C e^{-2 \tau}$.
The polarized Gowdy VTD equations - are simple enough that it is very easy to determine the form of the sets of asymptotic data which comprise the space $\mathcal Q_{pol}$. Since the general solution to is $\tilde P(\theta, \tau) = v(\theta) \tau +\phi (\theta)$ for an arbitrary pair of (smooth) functions $v$ and $\phi$ which satisfy the integrability condition $\int_{S^1}v \frac{d \phi}{d\theta}d\theta =0$ on the circle, and since the solution $\tilde \lambda (\theta, \tau)$ is readily obtained by simply integrating and with a single specified constant, the space $\mathcal Q_{pol}$ of asymptotic data consists of choices of $v(\theta)$ and $\phi(\theta)$ satisfying the integrability condition, plus a constant.
The proof that all solutions of the polarized Gowdy equations do exhibit AVTD behavior in the sense described above is a relatively straightforward consequence of the verification [@IM90] that each of the sequence of energy functionals $$\label{PolGowdyEnergies}
E_k = \sum_{j \leq k-1} \int_{S^1} [ \frac{1}{2}(\partial^j_\theta \partial_\tau P)^2 + \frac{1}{2}(\partial^{j+1}_\theta P)^2 ] d \theta$$ monotonically decays in time $\tau$. This monotonicity, together with Sobolev embedding, allows one to control the growth of $P$ and its derivatives, from which the convergence result and the consequent verification of AVTD behavior follow. The bijectivity and continuity of the map from $\Pi_{pol}$ to $\mathcal Q_{pol}$ readily follow from these estimates as well.
How do we infer results concerning the generic inextendibility of polarized Gowdy solutions from these AVTD results? As shown in [@IM90], if one writes the general solutions to the polarized Gowdy equations in the form of solutions of the VTD equations plus strongly controlled remainder terms, one can calculate the spacetime curvature polynomial scalars (including the Kretschmann scalar) in terms of the asymptotic data $(v(\theta), \phi(\theta))$, and determine that these invariants fail to blow up along an observer path approaching the limiting spatial coordinate $\theta_0$ *only* if $v^2(\theta_0)=1$, $\frac{d v}{d\theta}(\theta_0)=0$, and $\frac{d^2v}{d\theta^2}(\theta_0)=0$ all hold. The collection of solutions which cannot be extended past the singularity at $\tau \rightarrow \infty$ because of curvature blowup corresponds to all sets of asymptotic data in $\mathcal Q_{pol}$ which do not satisfy these conditions; clearly this set is open and dense among the set of all solutions.
In closing this discussion of the verification of a model version of Strong Cosmic Censorship for the polarized Gowdy spacetimes, we note that while the proof of these results relies heavily on the verification that the AVTD property holds for these spacetimes, and while the AVTD property is defined and verified with respect to a particular choice of coordinates (areal coordinates here), neither the statement of Theorem \[SCCPoldGowdy\] nor its validity depends on a choice of coordinates.
A Model SCC Theorem: $T^3$ Gowdy Spacetimes {#TGowdy}
===========================================
The difference between the polarized Gowdy spacetimes and the general Gowdy spacetimes is the presence in the latter of an extra (off-diagonal) metric coefficient in the Killing field orbits; specifically, in its $T^3$ version, the metric takes the form $$\label{GowdyMetric}
g=e^{\frac{(\tau+\lambda)}{2}}(-e^{-2\tau}d\tau^2 +d\theta^2) +e^{-\tau} [e^P dx^2 + 2 e^P Q dxdy +(e^{-P} +e^PQ^2) dy^2],$$ and the vacuum Einstein field equations take the form $$\begin{aligned}
\label{GPeqn}
\partial_{\tau \tau} P &= e^{-2\tau}\partial_{\theta \theta} P + e^{2P}(\partial_\tau Q^2 -e^{-2\tau} \partial_\theta Q^2),\\
\label{Qeqn}
\partial_{\tau \tau}Q &= e^{-2\tau}\partial_{\theta \theta} Q -2(\partial_\tau P \partial_\tau Q- e^{-2\tau} \partial _\theta P \partial _\theta Q),\\
\label{Glambdatau}
\partial_\tau \lambda &= (\partial_\tau P)^2 +e^{-2\tau} (\partial_\theta P)^2 +e^{2P}(\partial_ \tau Q^2 +e^{-2 \tau} \partial_\theta Q^2) ,\\
\label{Glambdatheta}
\partial_\theta \lambda &= 2 (\partial_\tau P \partial_\theta P + e^{2P} \partial _\tau Q \partial_\theta Q).\end{aligned}$$ We note that the addition of the dynamical variable $Q(\theta, \tau)$ clearly complicates the dynamics of the Gowdy spacetimes, but still leaves the function $\lambda(\theta, \tau)$ in a subsidiary role: One can solve for $P(\theta, \tau)$ and $ Q(\theta, \tau)$ independently of $\lambda$, and then obtain the latter by integrating and then . We note as well that, as in the polarized Gowdy case, an initial data set $(P(\theta,0), \partial_\tau P(\theta,0), Q(\theta,0), \partial_\tau Q(\theta,0), \lambda(0,0))=(p(\theta), \pi(\theta), q(\theta), \xi(\theta), \ell)$ must satisfy an integrability condition $\int_{S^1} (\pi \partial_\theta p +e^{2p} \xi \partial_\theta q )d\theta =0$. We let $\Pi$ denote the space of all such data sets.
The extra field variable and the extra terms in the field equations result in the dynamics of the Gowdy solutions being considerably more complicated than that of the polarized Gowdy. One can, however, still prove a model-SCC theorem for the general Gowdy $T^3$ solutions.[^10][@RingSCC09]
\[Model-SCC Theorem for $T^3$ Gowdy Spacetimes\] \[SCCGowdy\] There exists a subset $\tilde \Pi$ which is open with respect to the $C^1 \times C^0$ topology in $\Pi$, and dense in the this space with respect to the $C^\infty$ topology, such that the maximal globally hyperbolic spacetime development of any data set in $\tilde \Pi$ is $C^2$ inextendible.
We note that the statement of the model-SCC theorem for $T^3$ Gowdy spacetimes is very similar to that for polarized Gowdy solutions. As well, the source of the inextendibilty of both sorts of spacetimes is very similar: In both cases, one proves that the solutions are geodesically complete in the expanding direction ($\tau \rightarrow -\infty$), and that they generically have curvature blowups in the direction toward the shrinking (singular) direction ($\tau \rightarrow \infty$).
The proof of Theorem \[SCCGowdy\] is, however, considerably more difficult than that of Theorem \[SCCPoldGowdy\]. In a phenomenological sense, the source of this difficulty can be seen in a characteristic behavior which was first observed in Gowdy solutions in the numerical simulations of these spacetimes carried out by Berger and Moncrief in [@BM93] in the early 1990s. While their simulations did indicate the presence of AVTD behavior in the $T^3$ Gowdy spacetimes, they also found that very pronounced *spikes* in the graphs of the metric fields often develop in the evolving spacetimes in a way which could in principle interfere with the asymptotic behavior expected in a spacetime characterized by AVTD behavior, should the spikes become very prevalent. A key feature of Ringström’s beautiful work [@Ring04; @Ring05; @RingSCC09] on the Gowdy spacetimes, work which culminated in a proof of Theorem \[SCCGowdy\], is the very careful treatment of these spikes in their many different forms.
Indeed, an essential part of what distinguishes $\tilde \Pi$ from $\Pi$ is the control of the formation of spikes. In solutions which evolve from data in $\tilde \Pi$, only a finite number of spikes develop. As a result, AVTD-type asymptotic behavior is observed along generic timelike paths approaching the singularity. Furthermore, along such paths, the asymptotic velocity $$\label{AsymptVeloc}
V(\theta) := \lim_{\tau \rightarrow \infty} [\partial_\tau P^2 (\theta, \tau) + e^{2P} \partial_\tau Q^2 (\theta, \tau)]^{\frac{1}{2}}$$ is well-defined [@Ring06a]. This quantity, which generalizes the asymptotic data function $v(\theta)$ used in working with polarized Gowdy spacetimes, determines whether or not the curvature is bounded along a timelike path which approaches a specified value of the coordinate $\theta$: The curvature blows up along such a path so long as the value of $V$ at that point is not one. This condition $V(\theta) \neq 1$ is found to hold generically, from which it follows that there can be no extensions across a Cauchy horizon (in the $\tau \rightarrow \infty$ direction) in the spacetimes corresponding to $\tilde \Pi$ data. Ringström proceeds to show that $\tilde \Pi$ is an open and dense subset of $\Pi$. This result, together with his verification that the $\tilde \Pi$ solutions are geodesically complete in the $\tau \rightarrow - \infty$ direction, proves Theorem \[SCCGowdy\].
Evidence for AVTD Behavior in More General Families of Spacetimes {#AVTD}
=================================================================
The $T^3$ Gowdy spacetimes constitute the least restrictive family of solutions of the vacuum Einstein’s equations for which a model SCC theorem has been proven. Such theorems have been proven for more restrictive families of solutions, such as those which are spatially homogeneous (and therefore have a three-dimensional isometry group) [@Ren94; @Chr-Ren95]. They have also been proven for a number of families of spacetimes satisfying various Einstein-matter equations, including polarized Gowdy solutions of the Einstein-Maxwell equations [@Nun-Ren09], as well as solutions of the Einstein-Vlasov equations with $T^2$-symmetry [@Daf-Ren06], spherical symmetry, or hyperbolic symmetry [@Daf-Ren07]. Our main interest here is on what we know and what we conjecture for vacuum solutions with less restrictive conditions than the $T^3$ Gowdy solutions.
In proving model SCC theorems for both the polarized and the general ($T^3$) Gowdy spacetimes, AVTD behavior plays an important role. Hence, in looking for more general families of spacetimes for which such theorems may hold, it is useful to determine if there are such families for which AVTD behavior is known to be present. This is the case for three families of vacuum solutions: the polarized (and half-polarized) $T^2$-symmetric spacetimes, the polarized (and half-polarized) $U(1)$-symmetric spacetimes, plus general spacetimes in $(10+1)$-dimensions.
It has not been proven for any of these families of solutions that AVTD behavior is to be found in every member of the family, or in some open and dense subset of the full family. However, for the polarized $T^2$-symmetric solutions as well as for the polarized $U(1)$-symmetric solutions, there is strong evidence for AVTD behavior based on numerical simulations [@Lim], [@BM98]. For all three families, it has been proven using Fuchsian methods that there are at least some solutions (a collection parametrized by the free choice of certain functions, in each case) with AVTD behavior.
To illustrate how the Fuchsian approach works, we focus on the application of these techniques to the polarized $T^2$-symmetric solutions. Like the Gowdy solutions, the $T^2$-symmetric solutions are characterized by a 2-torus isometry group acting spatially. For the latter family, however, the twist constants do not vanish. As a consequence, the metrics necessarily take a more complicated form, which we write as follows: $$\label{PolT2Metric}
g = e^{2(\eta -U)} ( -\alpha dt^2 + d\theta^2 )
+ e^{2U} dx^2
+ e^{-2U} t^2 ( dy + G d\theta )^2.$$ We note that if $\alpha =1$ and if $G$ vanishes in , these metrics reduce to polarized Gowdy metrics. It is convenient for the discussion of the Fuchsian analysis that we use a slightly different form of metric parametrization here than that used above in the discussion of the Gowdy solutions; in particular, we replace $\tau$ by the time coordinate $t:= e^{-\tau}$, so that the singularity occurs at $t=0$, and we also make small changes in the choice of the metric coefficients (replacing $P$ and $\lambda$ by the closely related $U$ and $\eta$).
The vacuum Einstein field equations for the polarized $T^2$-symmetric spacetimes take the form $$\begin{aligned}
\label{T2U}
\partial_{tt}U +\frac{1}{t} \partial_tU &= \alpha \partial_{\theta \theta}U +\frac{1}{2}\partial_\theta \alpha \partial_\theta U +\frac{1}{2\alpha} \partial_t \alpha \partial_t U,\\
\label{T2etat}
\partial_t \eta &=t \partial_t U^2 + t \alpha \partial_\theta U^2 +\frac{e^{2\eta}}{4t^3} \alpha K^2,\\
\label{T2etax}
\partial_\theta \eta &=2 t \partial_t U \partial_\theta U - \frac{\partial_\theta \alpha}{2 \alpha},\\
\label{T2alpha}
\partial_t \alpha &= - \frac{e^{2 \eta}}{t^3} \alpha^2 K^2,\\
\label{T2G}
\partial_t G &= e^{2 \eta} \sqrt{\alpha}K t^{-3},\end{aligned}$$ where $K$ designates the non-vanishing twist constant which distinguishes these spacetimes from the Gowdy solutions[^11]. Comparing the features of this system of equations with those of the Gowdy equations - above (and the corresponding polarized Gowdy equations), we notice a key difference: While the Gowdy equations for $\lambda$ are semi-decoupled from those for $P$ and $Q$, here the system is fully coupled (apart from the equation for $G$). In a rough sense, this coupling is responsible for making the analysis of the polarized $T^2$-symmetric solutions more difficult (and consequently more interesting) than that of the polarized Gowdy solutions.
As noted above, numerical simulations strongly indicate that generic polarized $T^2$-symmetric solutions are AVTD in a neighborhood of their singularities at $t \rightarrow 0$. These simulations also suggest that the development of spikes, which complicate the dynamics of the general Gowdy solutions, plays at most a very minor role in the dynamics of polarized $T^2$-symmetric spacetimes. While no theorem concerning the generic presence of AVTD behavior in these spacetimes has yet been proven, Fuchsian methods have been used to show that there are polarized $T^2$-symmetric solutions with AVTD behavior. Roughly speaking, the way this works is as follows.
Writing out the polarized $T^2$-symmetric VTD equations[^12] as $$\begin{aligned}
\label{T2UV}
\partial_{tt}U +\frac{1}{t} \partial_tU &= \frac{1}{2\alpha} \partial_t \alpha \partial_t U,\\
\label{T2etatV}
\partial_t \eta &=t \partial_t U^2 + +\frac{e^{2\eta}}{4t^3} \alpha K^2,\\
\label{T2etaxV}
\partial_\theta \eta &=2 t \partial_t U \partial_\theta U - \frac{\partial_\theta \alpha}{2 \alpha},\\
\label{T2alphaV}
\partial_t \alpha &= - \frac{e^{2 \eta}}{t^3} \alpha^2 K^2,\\
\label{T2GV}
\partial_t G &= e^{2 \eta} \sqrt{\alpha} K t^{-3}, \end{aligned}$$ we first verify that for a general collection of functions $k(\theta)$, $U_*(\theta)$, $\eta_*(\theta)$, $\alpha_*$ and $G_*(\theta)$ (which we call collectively the “asymptotic data"), the following are asymptotically solutions of this VTD system: $$\begin{aligned}
\label{VTDSolns}
\hat U(\theta, t)&=\frac 12(1-k(\theta))\log t+U_{*}(\theta),\\
\hat \eta(\theta, t)&=\frac 14(1-k(\theta))^2\log t+\eta_*(\theta),\\
\hat \alpha(\theta, t)&=\alpha_*(\theta),\\
\label{VTDSolnss}
\hat G(\theta,t)&=G_*(\theta).\end{aligned}$$ Next, we express the unknown metric coefficients as sums of these (function-parametrized) VTD solutions plus remainder-field terms $\tilde U(\theta,t), \tilde \eta(\theta,t), \tilde \alpha(\theta, t)$, and $\tilde G(\theta,t)$, $$\begin{aligned}
\label{AVTDExpan}
U(\theta, t)&=\hat U (\theta, t) + \tilde U(\theta, t),\\
\eta(\theta, t)&=\hat \eta(\theta, t) + \tilde \eta(\theta,t),\\
\alpha(\theta, t)&=\hat \alpha (\theta,t) + \tilde \alpha (\theta, t),\\
G(\theta,t)&=\hat G(\theta, t)+\tilde G(\theta, t), \end{aligned}$$ and we substitute these expressions into the polarized $T^2$-symmetric Einstein vacuum equations -. We thus obtain a ($k(\theta)$, $U_*(\theta)$, $\eta_*(\theta)$, $\alpha_*(\theta)$ $G_*(\theta)$)-parametrized PDE system for the remainder-field terms.
The idea is to show that for each suitable choice of the asymptotic data, there exists (for $t$ sufficiently close to zero) a unique solution to the remainder-field PDE system, and to show moreover that all of the remainder fields $\tilde U(\theta,t), \tilde \eta(\theta,t), \tilde \alpha(\theta, t)$, and $\tilde G(\theta,t)$ included in this unique solution approach zero as $t$ approaches zero. If one can do this, it follows that the polarized $T^2$-symmetric spacetime composed from the specified asymptotic data together with the resulting remainder-field solution is AVTD.
If the asymptotic data functions are all real analytic, it is relatively straightforward to determine conditions on this data which are sufficient for the existence of a remainder-field solution with the desired decay. So long as one can write the remainder-field PDE system collectively in the form $$\label{AnalytFuchs}
t\partial_t \Phi + M(\theta) \Phi = t^\epsilon F(\theta, t, \Phi, \partial_\theta \Phi),$$ where the vector field $\Phi$ includes as its components all of the remainder fields and their $\theta$ derivatives, where the matrix $M$ (whose explicit form depends on the asymptotic data) must satisfy certain positivity conditions, where $\epsilon$ is a positive constant, and where the function F (also depending on the asymptotic data) is continuous in $t$, is analytic in all of its other arguments, and extends continuously to $t=0$, then indeed a unique solution satisfying the desired properties exists. As shown in [@IK99], so long as $\alpha_*(\theta)$ is positive, so long as $k(\theta)$ satisfies certain inequalities, and so long as the asymptotic data collectively satisfy an integrability condition (derived from the constraint equation ), then the remainder field equations can be written in this form, with $M$ and $F$ satisfying the conditions listed above. Thus one verifies that there exists a parametrized set of real analytic polarized $T^2$-symmetric solutions which exhibit AVTD behavior.
Although Fuchsian techniques were originally developed to work with real analytic solutions of PDE systems with real analytic coefficients (see, e.g., [@Kich]), they have since been adapted (by Rendall in [@Ren00], and by Ames, Beyer, LeFloch, and the author in [@ABIL]) to apply to PDE systems and solutions of those systems with much less assumed regularity. These adaptations (to date) require one to work with a more restricted class of PDE systems, such as those which are quasilinear and symmetric hyperbolic and take the form $$\label{QlinSymHyp}
S(\theta, t, \Psi)t \partial_t \Psi + T(\theta, t, \Psi) t \partial_\theta\Psi +N(\theta, t, \Psi) \Psi =E(\theta, t, \Psi),$$ where $\Psi(\theta, t)$ is a vector-valued function representing the collection of fields and their first order derivatives. As discussed in [@ABIL], so long as a number of technical conditions are satisfied by the matrix functions $S, T, N$ and $E$, both in general and for certain choices of $\Psi$ as “asymptotic data" $\hat \Psi(\theta, t)$, then it follows that for those choices of the asymptotic data, there exist unique solutions $\Psi=\hat \Psi +\tilde \Psi$ of with the remainder terms $\tilde \Psi$ decaying to zero as $t \rightarrow 0$.
Both for smooth PDE coefficients and asymptotic data, and for less regular choices of the coefficients and the asymptotic data (as specified by certain choices of weighted Sobolev spaces[^13]), these adapted Fuchsian techniques have been used to find parametrized classes of polarized $T^2$-symmetric solutions (of the stated regularity) with AVTD behavior near the $t=0$ singularity [@ABIL]. In doing this, one chooses $\hat \Psi(\theta, t)$ to correspond to the choices - of $\hat U(\theta,t), \hat \eta(\theta,t), \hat \alpha(\theta, t)$, and $\hat G(\theta,t)$ discussed above, which asymptotically approach solutions of the VTD equations -.
As is the case for the real analytic solutions, it is not yet known if these solutions with AVTD behavior represent anything more than a set of measure zero among all polarized $T^2$-symmetric solutions. Numerical simulations suggest that indeed AVTD behavior may be prevalent, if not generic [@Lim], but nothing of this nature has been proven.
Fuchsian methods have been used to prove that other families of solutions of Einstein’s equations include at least some solutions with AVTD behavior. The earliest such results pertained to the $T^3$ Gowdy solutions: Kichenassamy and Rendall used Fuchsian techniques to prove the existence of real analytic Gowdy solutions with AVTD behavior in [@KR98], and Rendall did the same for smooth Gowdy solutions in [@Ren00]. This work of course presaged Ringström’s proof that generic $T^3$ Gowdy solutions exhibit AVTD behavior. For the $T^2$-symmetric spacetimes, Fuchsian methods have been used to show not only that there are polarized solutions with AVTD behavior, but that there are“half-polarized" solutions with this behavior as well [@CI07]. Half-polarized solutions allow the presence of a non vanishing $Q(\theta,t) dxdy$ term in the expression for the metric ; however the dynamics of this term is strongly restricted, with one non vanishing function in the asymptotic data controlling it, as opposed to the two functions ($k$ and $U_*$) in the asymptotic data which control $U$.
The Gowdy spacetimes and the $T^2$-symmetric solutions are all characterized by the very restrictive assumption that they each admit a spatially-acting two-dimensional isometry group. Loosening this restriction to the admission of a spatially-acting isometry group of only one dimension, one finds that Fuchsian methods can indeed be used to prove that among these $U(1)$-symmetric spacetimes, there are some which show AVTD behavior. As for the $T^2$-symmetric spacetimes, AVTD behavior has been shown to exist only in $U(1)$-symmetric solutions which are either polarized or half-polarized[@IM02; @CBIM04]. We note as well that all $U(1)$-symmetric spacetimes shown thus far to exhibit AVTD behavior are real analytic.
It is expected that using Fuchsian methods we will be able to show that there are smooth polarized $U(1)$-symmetric solutions with AVTD behavior. Based on evidence from numerical simulations [@BM00], it is *not*, however, expected that $U(1)$-symmetric solutions without any polarization restriction will be found which exhibit AVTD behavior. The same is true for $T^2$-symmetric solutions [@BIW01].
There has been significant speculation that for families of solutions more general than those discussed thus far, while AVTD behavior may not be found, one may find “Mixmaster" behavior instead. Roughly speaking, a solution shows Mixmaster behavior near its singularity if observers approaching the singularity do not each see his/her own Kasner-like behavior[^14], but rather each sees his/her own Bianchi type IX solution[^15]. The Mixmaster evolution is characterized by an infinite succession of episodic Kasner-type evolutions, each of which is ultimately disrupted by a short-lived “bounce", followed by a transition to the next Kasner episode.
The prediction that Mixmaster behavior is likely to be seen generically in spacetime solutions less restricted than those which exhibit AVTD behavior is based partly on numerical simulations (as cited above [@BIW01; @BM00]), partly on the pioneering work of Belinskii, Khalatnikov and Lifschitz (BKL)[@BKL70], and partly on more recent speculative studies [@Dd08]. On the other hand, others have argued that the prevalence of spikes in the evolution of these spacetimes with more intricate dynamics strongly indicate that the conjecture of generic Mixmaster behavior is very unlikely to hold. This issue remains to be settled.
We do note that, should it be shown that Mixmaster behavior characterizes the behavior of generic solutions near their singularities, in light of the fairly good understanding we have of the evolution of the curvature in Bianchi Type IX solutions, such a result could be a very useful tool for the study of Strong Cosmic Censorship.
We close this section by noting that, if instead of vacuum solutions one considers spacetimes satisfying the Einstein equations with certain stiff fluids or scalar fields coupled in, then AVTD behavior *is* found. This was shown using Fuchsian methods (applied to real analytic solutions) by Andersson and Rendall in [@AR01]. More recently, Rodnianski and Speck have shown [@RS] that the presence of AVTD behavior is in fact *stable* among these Einstein-scalar field or Einstein-stiff fluid solutions. We also note the work [@DHRW] which uses Fuchsian techniques to prove that there are vacuum solutions of dimension 11 or higher which show AVTD behavior.
Blue Shift Effects, Weak Null Singularities, and the Nature of SCC {#Blue}
==================================================================
While some of the original motivation for believing in the validity of the Strong Cosmic Censorship conjecture came from the conviction that a respectable theory of the gravitational field should not (generically) allow for physical spacetimes in which one’s ability to predict the future from knowledge about the past breaks down, Penrose also based his view that SCC should hold on his assessment of the “blue shift effect" on Cauchy horizons in black hole interiors. Reissner-Nordstom as well as Kerr black hole interiors contain Cauchy horizons; if the existence of these structures were found to be stable under generic perturbations, then SCC would be refuted. However, Penrose reasoned that any small perturbation of an astrophysical system evolving towards a Reissner-Nordstrom or a Kerr spacetimes would “fall" into the developing black hole, and in doing so would become strongly blue-shifted and consequently would become powerful enough to destroy the Cauchy horizon. This blue shift effect would therefore “save" Strong Cosmic Censorship.
Early, somewhat heuristic, explorations of this blue shift effect (by Hiscock, Israel, Poisson, Ori, and others) have suggested a surprising scenario: that perturbations of charged black hole solutions would contain null surfaces across which *continuous extensions of the metric* could be made, but *continuous extensions of the curvature* could not be made. That is, the Reissner-Nordstrom Cauchy horizons, according to this scenario, are stable in a certain $C^0$ sense, but not in a $C^2$ sense.
Remarkably, a wide range of subsequent studies very strongly support this scenario. In the first of these works (done twelve years ago), Dafermos [@D03; @D05] has shown that for any asymptotically Euclidean spherically symmetric initial data set with non vanishing charge whose Einstein-Maxwell development forms a black hole, the maximal globally hyperbolic development does admit a non-empty null surface across which a continuous extension of the metric can be carried out. Further (with certain technical assumptions), he proves [@D05] that these $C^0$-type Cauchy horizons are generically singular, with the curvature and the Hawking mass blowing up; moreover, the metric extensions do not admit locally square integrable Christoffel symbols. These properties have led to these null surfaces being labeled *weak null singularities*.
Considerations of this work raise three important questions: Do weak null singularities exist in spacetimes which are not spherically symmetric? If so, might they characterize generic perturbations of Reissner-Nordstrom and Kerr (and Kerr-Newman) black holes and their interiors? If weak null singularities are stable, does this constitute a proof that Strong Cosmic Censorship is false?
The first question is answered by very recent work by Luk [@Luk], which proves that the maximal developments of certain sets of characteristic initial data (which are *not* required to admit any isometries) always contain weak null singularities. While the second question has not been fully resolved in general, Dafermos and Luk claim that they can show that *if* the stability of the exterior structure of the Kerr solutions can be proven[^16], then the stability of the internal Cauchy horizon, as a weak null singularity, would follow as a corollary.
Should we now conclude that if Kerr is proven to be stable (in the exterior sense), then the Strong Cosmic Censorship conjecture is false? This becomes a matter of interpretation. Since its inception, Strong Cosmic Censorship has been an imprecise and malleable conjecture. To formulate it explicitly, one needs to answer all of the following questions: What are “generic solutions"? Is the primary issue whether or not the curvature is bounded in the neighborhood of the singularity? Or is the issue whether or not spacetime metric extensions can be carried out? If the stability of extensions is the key, then does it matter whether the curvature as well as the metric can be extended? Does it matter if the singularity which forms is spacelike or null?
Resolving all of these questions is important if one wants a single statement of the Strong Cosmic Censorship conjecture, to confirm or refute. On the other hand, it may be more useful to consider several different versions of the conjecture, and ultimately determine that there are some reasonable forms of SCC that are true, and others which are not.
The primary purpose of the Strong Cosmic Censorship conjecture has always been to stimulate interesting questions and studies in mathematical relativity. For this purpose, it has certainly been successful.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was partially supported by NSF grant PHY-1306441 at the University of Oregon. I thank the Simons Center for hospitality during the course of the writing of this review, and I thank Beverly Berger, Mihalis Dafermos, Ellery Ames and Florian Beyer for useful conversations. I also thank the referee for useful comments.
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[^1]: Department of Mathematics and Institute of Theoretical Science, University of Oregon. [email protected]
[^2]: Physicists in the 1800s worried about the singularity at the origin of the Coulomb solution of Maxwell’s equations. These worries were quieted by the recognition of the quantum nature of “point sources" for electromagnetic fields, such as the electron.
[^3]: Here $M^{n+1}$ is a spacetime manifold, $g$ is a Lorentz-signature spacetime metric, and $\Psi$ collectively represents the non-gravitational fields
[^4]: This positivity condition has been labeled the “strong energy condition".
[^5]: The spacetime should be sufficiently differentiable so that geodesic incompleteness does not arise simply because the spacetime is not smooth enough to admit a geodesic congruence.
[^6]: See, however, the work of Christodoulou [@Chris] and of Klainerman and Rodnianski [@KlainRod] in which conditions on initial data are given which guarantee that a trapped surface will form in the spacetime development of that data.
[^7]: A Cauchy horizon in a spacetime $(M,g)$ is a null hypersurface which divides the spacetime into a region which is globally hyperbolic, and a region which is not.
[^8]: In terms of the areal coordinates for the $T^3$ Gowdy spacetimes, which we use here in writing the metric in the form , the singularity occurs at $\tau \rightarrow \infty$, and the spacetime expands with decreasing $\tau$. The $S^3$ and $S^2\times S^1$ Gowdy spacetimes are singular both to the future and the past.
[^9]: Spatial derivatives are dropped from the first two equations, since the idea is that spatial derivatives are dominated by temporal derivatives. In the third equation, , there are no temporal derivatives, so the spatial derivatives are not neglected.
[^10]: For the polarized Gowdy solutions, one can prove a model-SCC theorem for all allowed Gowdy topologies; for the general Gowdy solutions, such has been proven only for the $T^3$ case.
[^11]: Without loss of generality in studying these spacetimes, one may set one of the twist constants to zero, labeleing the remaining one as $K$.
[^12]: These equations are clearly obtained by dropping the terms with spatial derivatives in the system -, *except* for in the constraint equation , which contains no temporal derivatives.
[^13]: The weighting pertains to the decay of functions as they approach $t=0$.
[^14]: We recall that the VTD equations involve the dropping of all spatial-derivative terms in the Einstein evolution equations; hence the metric evolution seen by each observer is the same as the evolution of a spacetime with a spatially-acting $T^3$-isometry group, which corresponds to the Kasner spacetime.
[^15]: These are the solutions with $SU(2)$ acting transitively on space-like slices.
[^16]: There is, of course, a very large amount of mathematical effort currently being directed toward proving the stability of Kerr solutions.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We derive cosmological constraints on the masses of generic scalar fields which decay only through gravitationally suppressed interactions into unstable gravitinos and ordinary particles in the supersymmetric standard model. For the gravitino mass 100GeV-1TeV, the scalar masses should be larger than 100TeV to keep the success of big-bang nucleosynthesis if no late-time entropy production dilutes the gravitino density.'
---
c ł u v Ł ¶ §
ø
\#1\#2\#3[[Nucl. Phys.]{} [**[B\#1]{}**]{} (19\#2) \#3]{} \#1\#2\#3[[Phys. Lett.]{} [**[B\#1]{}**]{} (19\#2) \#3]{} \#1\#2\#3[[Phys. Rev.]{} [**[D\#1]{}**]{} (19\#2) \#3]{} \#1\#2\#3[[Phys. Rev. Lett.]{} [**\#1**]{} (19\#2) \#3]{} \#1\#2\#3[[Prog. Theor. Phys.]{} [**\#1**]{} (19\#2) \#3]{}
UT-812\
TU-544, RCNS-98-06\
April 1998
0.8cm
[**Gravitino Overproduction\
through Moduli Decay**]{}
1.2cm
M. Hashimoto$^{1}$
[^1], Izawa K.-I.$^1$, M. Yamaguchi$^2$ and T. Yanagida$^1$\
1.5cm [*$^1$Department of Physics, University of Tokyo,\
Tokyo 113-0033, Japan*]{}\
Introduction
============
Superstring theories have infinitely degenerate supersymmetric vacua which are continuously connected by massless scalar fields, called moduli. These moduli fields are generally expected to acquire their masses $m_\v$ of the order of the gravitino mass $m_{3/2}$ once supersymmetry breaking effects are included. The moduli decay into two gravitinos if the masses of moduli are larger than $2m_{3/2}$.
In this paper, we consider the decay of the moduli into two gravitinos and discuss its cosmological consequences. It is known [@Wei; @Kaw] that the gravitino with the mass 100GeV-1TeV decays soon after the nucleosynthesis and the decay product destroys light nuclei produced in the early universe. We see that the moduli decay tends to produce too many gravitinos to keep the success of big-bang nucleosynthesis. We derive stringent constraints on the moduli masses to avoid this disaster such as $m_\v {\raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept
\hbox{$>$}\ }}100{\rm TeV}$.
We stress that this constraint is applicable to generic scalar fields that decay only through gravitationally suppressed interactions as long as their masses are larger than the threshold of two-gravitino decay channel.
The Interaction
===============
We assume one modulus field throughout the paper, though the generalization to the case of many moduli is straightforward. We set the gravitational scale $2.4 \times 10^{18}$GeV equal to unity.
The relevant terms in the supergravity Lagrangian [@Wes] which describe the decay of a modulus $\v$ into gravitinos $\chi$ is given by = \^[klmn]{}[|]{}\_k[|]{}\_l [1 ø4]{}(K\_v\_mv-K\_[v\^\*]{}\_mv\^[\*]{})\_n - e\^[K ø2]{}(W\^\*\_a\^[ab]{}\_b + W[|]{}\_a [|]{}\^[ab]{}[|]{}\_b), \[LAG\] where $W$ denotes the superpotential and we choose the field $\v$ so that its vacuum expectation value vanishes: $\langle \v \rangle = 0$. With our definition of $\v$, the Kähler potential $K$ generically contains linear terms, $K = c\v + c^*\v^* + \v \v^* + \cdots$, where the coefficient $c$ is of order one.
The Decay
=========
Let us begin by discussing the decay rate of the modulus. When $m_\v \gg 2m_{3/2}$, the order of the decay width of $\v$ into gravitinos is given by (v) \~|c|\^2m\_[3/2]{}\^2m\_, where $c$ denotes the coefficient of the $\v$ term in the Kähler potential. Here, we have used Eq.(\[LAG\]), the gravitino equation of motion and $\langle e^{K/2}W \rangle = m_{3/2}$.
On the other hand, the order of the decay width of $\v$ into radiation is given by (v) \~Nm\_v\^3, where $N$ is the number of the decay channels. Hence the branching ratio of the decay into gravitinos turns out to be B\_\~[|c|\^2 øN]{}( [m\_[3/2]{} øm\_v]{} )\^2.
The modulus $\v$ starts damped oscillation when the Hubble scale $H$ becomes comparable to its mass $m_\v$. The initial amplitude of the coherent oscillation is expected to be of order one in the Planck unit. Then the modulus density $\r_\v$ dominates the universe at the decay time since $\G_\v \ll H \sim m_\v$. The reheat temperature after the modulus decay is given by T\_R \~N\_\*\^[-[1 ø4]{}]{}, where $N_*$ denotes the degrees of freedom at the temperature $T_R$.
This implies that the gravitino number density $n_{3/2}$ produced through the modulus decay at the decay time is given by \~[|c|\^2N\_\*\^[-[1 ø4]{}]{} ø]{} [m\_[3/2]{}\^2 ø[m\_v]{}\^[3 ø2]{}]{}, where $s$ denotes the entropy density and we have used B\_\_\~m\_[v]{}n\_[3/2]{}, \_\~N\_\* T\_R\^4, s \~N\_\* T\_R\^3. Namely, the modulus mass is given by \~( [|c|\^2N\_\*\^[-[1 ø4]{}]{} ø]{} [m\_[3/2]{}\^2 øy\_[3/2]{}]{} )\^[2 ø3]{}, \[MAS\] where $y_{3/2} = n_{3/2}/s$.
Gravitinos are also produced by the scattering processes of the thermal radiations after the modulus decay. The contribution to the gravitino number density is given by [@Kaw] \~10\^[-3]{} T\_R \~10\^[-3]{} N\_\*\^[-[1 ø4]{}]{} m\_v\^[3 ø2]{}. \[THE\]
The Bound
=========
In the previous section, we have estimated $y_{3/2} = n_{3/2}/s$ at the decay time of the modulus $\v$. We may derive cosmological constraints on $y_{3/2}$ from the observation of the present universe since the estimated value $y_{3/2}$ itself yields the value of $n_{3/2}/s$ at the time of the gravitino decay.
Stringent constraints are implemented to keep the successful predictions of the big-bang nucleosynthesis provided the gravitino with the mass 100GeV-1TeV mainly decays into a photon and a photino [@Kaw]: y\_[3/2]{} [-2.truept]{}10\^[-15]{}-10\^[-13]{}. By means of Eq.(\[MAS\]), we obtain m\_[-2.truept]{}100 [TeV]{}.
On the other hand, the constraints due to gravitinos produced by the scattering processes of the thermal radiations read as follows: [-2.truept]{}(10\^[7]{}-10\^[9]{})[TeV]{}, where we have used Eq.(\[THE\]).
Conclusion
==========
We have derived the constraint $100{\rm TeV} {\raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept
\hbox{$<$}\ }}m_\v {\raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept
\hbox{$<$}\ }}10^9{\rm
TeV}$ on the moduli masses $m_\v$ in the case of the unstable gravitino[^2] with the mass 100GeV-1TeV. This may have obvious implications for mechanisms of the moduli stabilization. Here, the moduli may be regarded as generic scalar fields that decay only through gravitationally suppressed interactions as long as their masses are larger than the threshold of two-gravitino decay channel.
In the course of the analysis, we assumed that no entropy production has diluted the modulus and gravitino densities since the modulus density once dominated the universe. In fact, entropy production may evade the constraints. New inflation and thermal inflation are possible candidates of enough entropy production. Without such inflationary dilution, the moduli masses are severely constrained. On the other hand, if the moduli masses lie in the region $100 {\rm TeV} {\raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept
\hbox{$<$}\ }}m_\v {\raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept
\hbox{$<$}\ }}10^9 {\rm TeV}$, the reheat temperature of the cosmological inflation could be very high since the density of gravitinos produced just after the inflation is diluted substantially by the decay of moduli.
Acknowledgements {#acknowledgements .unnumbered}
================
The work of M.Y. was supported in part by the Grant–in–Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan No. 09640333.
[99]{}
S. Weinberg, ;\
D.V. Nanopoulos, K.A. Olive and M. Srednicki, ;\
M.Yu. Khlopov and A.D. Linde, ;\
J. Ellis, J.E. Kim and D.V. Nanopoulos, ;\
R. Juszkiewicz, J. Silk and A. Stebbins, ;\
J. Ellis, D.V. Nanopoulos and S. Sarkar, ;\
M. Kawasaki and K. Sato, ;\
J. Ellis, G.B. Gelmini, J.L. Lopez, D.V. Nanopoulos and S. Sarkar, .
M. Kawasaki and T. Moroi, .
J. Wess and J. Bagger, [*Supersymmetry and Supergravity*]{} (Princeton University Press, 1992).
H. Pagels and J.R. Primack, ;\
J. Ellis, D.V. Nanopoulos and S. Sarkar, ;\
V.S. Berezinsky, ;\
T. Moroi, H. Murayama and M. Yamaguchi, .
[^1]: Research Fellow of the Japan Society for the Promotion of Science.
[^2]: On the other hand, no stringent constraint on the moduli masses is implemented due to the moduli decay in the case of the lighter gravitino $m_{3/2} {\raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept
\hbox{$<$}\ }}10{\rm GeV}$ [@Pag].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We describe the [*digest2*]{} software package, a fast, short-arc orbit classifier for small Solar System bodies. The [*digest2*]{} algorithm has been serving the community for more than 13 years. The code provides a score, ${D_2}{}$, which represents a pseudo-probability that a tracklet belongs to a given Solar System orbit type. [*digest2*]{} is primarily used as a classifier for Near-Earth Object (NEO) candidates, to identify those to be prioritized for follow-up observation. We describe the historical development of [*digest2*]{} and demonstrate its use on real and synthetic data. We find that [*digest2*]{} can accurately and precisely distinguish NEOs from non-NEOs. At the time of detection, 14% of NEO tracklets and 98.5% of non-NEOs tracklets have ${D_2}{}$ below the critical value of ${D_2}{}=65$. 94% of our simulated NEOs achieved the maximum ${D_2}=100$ and 99.6% of NEOs achieved ${D_2}{}\ge65$ at least once during the simulated 10-year timeframe. We demonstrate that ${D_2}{}$ varies as a function of time, rate of motion, magnitude and sky-plane location, and show that NEOs tend to have lower ${D_2}{}$ at low Solar elongations close to the ecliptic. We use our findings to recommend future development directions for the [*digest2*]{} code.'
author:
- Sonia Keys
- Peter Vereš
- 'Matthew J. Payne'
- 'Matthew J. Holman'
- Robert Jedicke
- 'Gareth V. Williams'
- Tim Spahr
- 'David J. Asher'
- Carl Hergenrother
title: 'The [*digest2*]{} NEO Classification Code'
---
Introduction {#SECN:INTRO}
============
Near-Earth Objects (NEOs), defined as any small Solar System body having perihelion less than 1.3${\,\mathrm{au}}$, are of significant interest for a number of reasons. These include planetary defense, spacecraft missions, commercial development, and investigations into the origin and evolution of the Solar System. The NEO discovery rate has risen rapidly in recent decades, driven by the 1998 Congressional mandate to discover 90% of NEOs larger than 1km [the Spaceguard Survey @Morrison92], and the subsequent 2005 Congressional mandate to discover 90% of NEOs larger than 140m (George E. Brown, Jr. NEO Survey Act [^1]).
Solar System objects are typically identified through their apparent motion in a series of images spanning minutes to hours. Given such a sequence of exposures, it is straightforward to identify “tracklets” [@Kubica07], sets of detections that are consistent with an object with a fixed rate of motion. A related term is an *arc*, which may contain any number of tracklets, possibly from different observatories. The time span from first to last observation is termed the *arc-length*.
It is straightforward to determine whether a tracklet represents a known object with a well defined orbit. In that case, the observations coincide with the predicted positions for the known object. Otherwise, the tracklet likely represents a newly discovered object. It is then of interest to determine what type of orbit that object has, and in particular, if it is a new NEO. Given limited telescopic resources, it is not possible to re-observe all objects to immediately determine their orbits. It is therefore essential to identify which objects are more likely to be NEOs and prioritize those for further observation.
Although a tracklet does not uniquely determine an orbit, characteristic sky-plane motion can be used to infer a possible orbit type [e.g. @1996AJ....111..970J]. The tool currently used for this is [*digest2*]{}[^2]. The [*digest2*]{} code uses only a single motion vector, derived from a tracklet or short arc, to identify all possible elliptical orbits consistent with that motion. This set of possible orbits is then divided into disjoint orbital categories. Given a population model that represents the number of solar system objects of each type, we can estimate the likelihood that the tracklet represent a member of each category.
With a suitable threshold, [*digest2*]{} can serve as an NEO binary classifier. Tracklets with high ${\emph{digest2}}{}$ scores, ${D_2}{}$, are posted on the Near-Earth Object Confirmation Page (NEOCP[^3]). These objects are prioritized by the NEO follow-up community for additional observations.
In this manuscript we describe the [*digest2*]{} code. The code as implemented has been used by the community for a number of years. Our primary goal is to describe and document the major elements of [*digest2*]{} and the practical consequences of its design and implementation. Although we highlight possible areas of improvement, these are left for future work.
Methodology {#s:METHOD}
===========
The observations from a short-arc tracklet directly constrain the position, $(\alpha, \delta)$, and motion, $(\dot\alpha,\dot\delta)$, of the asteroid in the sky, but the topocentric radial-distance, $\rho$, and radial-velocity, $\dot\rho$, between the asteroid and the observer are essentially unconstrained. This is the fundamental challenge of orbit determination.
An *assumed* $\rho$ allows a heliocentric position to be derived (see Appendix \[app:admiss\]), and with the assumption of a topocentric $\dot\rho$, a heliocentric velocity can also be calculated. @Milani04 refer to the “admissible region” as being the set of $(\rho,\dot\rho)$ for which the resultant orbit is heliocentrically bound (elliptical w.r.t. the Sun).
Many authors have addressed the problem of short-arc orbit determination and constraint [e.g. @Vaisala39; @Marsden85; @Bowell89; @1990acm..proc...19B; @Marsden91; @Tholen2000; @Milani04; @Milani05; @Oszkiewicz09; @Spoto18 and discussions therein]. Of particular relevance to [*digest2*]{}, in the late 1980s and early 1990s, R. H. McNaught of the Siding Spring Observatory undertook an extensive effort to determine allowable orbital element ranges and class-specific object probabilities based on two observations and a magnitude [[[Pangloss]{}]{} @McNaught99]. The method proceeded by stepping through a range of possible topocentric distances and angles (between the velocity vector and the line of sight) which were consistent with the observations and bound to the sun. [[Pangloss]{}]{} used a population model to assign weights to different orbit classes. The [[Pangloss]{}]{} code provided the foundation for the [*digest2*]{} code described in this manuscript.
We review two related methods for addressing the problem of short-arc orbits. The first concerns the technique referred to as “statistical ranging” [@Virtanen01], in which two observations are selected from the tracklet, thus fixing $(\alpha,\dot\alpha,\delta,\dot\delta)$, and then topocentric ranges at each of the epochs of the two observations are randomly chosen and a corresponding orbit computed from the admissible region that is compatible with the observational tracklet data. This process is repeated over many random topocentric distances to generate a set of compatible orbits.
The second method refers to “systematic ranging”. This method was introduced by @Chesley05 and is described in detail in @Farnocchia15. In this method, a systematic raster over $(\rho,\dot\rho)$ space is performed, generating an orbit for each and comparing to *all* tracklet observations. The RMS of the fit residuals for each $(\rho,\dot\rho)$ point indicates the quality of the fit, and $\chi^2$ probabilities can be used to derive confidence regions in $(\rho,\dot\rho)$ space.
While [*digest2*]{} is not directly derived from either @Virtanen01 or @Chesley05 (and the [[Pangloss]{}]{} ranging code, from which [*digest2*]{} is derived, predates both), the [*digest2*]{} code uses the same fundamental approach that is common to both statistical and systematic ranging techniques: sets of bound heliocentric orbits are generated that all satisfy the short-arc observations.
Historical Development of [*digest2*]{} {#s:HIST}
---------------------------------------
As described above, the early origins of the [*digest2*]{} were the [[Pangloss]{}]{} code developed by R. McNaught. The current version of [*digest2*]{} evolved from a FORTRAN 77 code (“`223.f`”) employed in @1996AJ....111..970J, and then further developed by C. Hergenrother and T. Spahr from the [[Pangloss]{}]{} code of R. McNaught. It accepted observation files in the MPC’s 80-character format[^4], encoded the vector solution described in section \[s:SolnAlgo\], employed the parabolic limit described in appendix \[app:admiss\], and used the model population look-up described in section \[s:POPN\].
Since the first publication,[*digest2*]{} has undergone many incremental improvements. In Appendix \[app:HIST\], we provide a detailed description of the key changes to the [*digest2*]{}.
Computational elements of [*digest2*]{} {#s:COMP}
=======================================
In this section we describe the main components of the current version of [*digest2*]{} and explain the algorithmic choices in its development. We emphasize that we are providing a factual report of how the code was implemented, and hence how the code has been operating for more than a decade. Significant improvements are possible in future versions, but the focus of this paper is documenting the development, features, and performance of the existing code.
We discuss [*digest2*]{}’s key algorithmic steps in Sections \[s:EndPoint\] to \[s:score\], and then summarize them in Section \[s:AlgSumm\] and Algorithm Table \[alg: essential\].
Endpoint Synthesis {#s:EndPoint}
------------------
The algorithm starts by reading tracklet and population-model data from input files. Since version 11 (June 2011), [*digest2*]{} considers all observations in the arc and synthesizes end points to define a motion vector, with the aim of improving the robustness of [*digest2*]{} against both bad observations and statistical observational error. Figure \[fig:tracklet\] demonstrates a number of cases handled by the code.
[*Short-Arcs:* ]{} Panel A illustrates relatively short arcs, with all observations from the same site. At the top, both observations in the two-observation-arc are used as-is. For the second arc, a more typical tracklet with four observations, a great circle linear motion vector is fit to the observations. Endpoints are synthesized near the 17th and 83rd percentile of the observations in the arc, instead of near the extreme endpoints. This is not to account for positional uncertainty, but rather is an approximate method to account for cases in which excessive measurement error or poor astrometry can significantly shift the endpoint, and thus change the shape of the admissible region.[^5]. For the third arc, synthesizing endpoints near the 17th and 83rd percentile yields synthetic observations within the tracklet. The fourth arc shows significant curvature, with the synthesized linear motion vector exhibiting significant residuals w.r.t all observations.
[*Space-based observations[^6]:* ]{} As illustrated in Panel B, space-based observations often show strong curvature on the sky due to parallax as the spacecraft orbits the earth. In this case interpolation on a great circle fit is not meaningful. If any observations in an arc are space-based, two actual observations are selected, near the 17th and 83rd percentile.
[*Longer-Arcs:* ]{} Panel C shows cases of longer arcs and/or arcs with multiple observing sites. Here, two sub-arcs are selected where each sub-arc is < 3 hours and all detections in a sub-arc are from the same site. These two sub-arcs are then separately reduced to yield the motion vector end points. In the top arc of Panel C, sub-arcs can be selected to use all available observations. A separate great circle linear motion fit is then applied to each sub-arc and observations are synthesized, again near the 17th and 83rd percentiles. The second arc illustrates how the conditions that sub-arcs be < 3 hours and all from the same site may leave a number of observations unused. The third arc shows a case where the 83rd percentile is outside of the end sub-arc. In this case no point is synthesized outside the sub-arc, rather the first observation of the sub-arc is used as-is.
### Dithering and Observational Uncertainty {#s:Dithering}
Once a two-point tracklet has been constructed as described in Section \[s:EndPoint\], the end-points are varied within the uncertainties, $\sigma$, of the input detections. The nominal end-points are varied by adding $\pm0.5\sigma$ to either (or both) of the RA and Dec values of the initial end points, producing a total of 9 variant tracklets: the initial one, plus the additional 8 variations illustrated in Figure \[fig:dither\].
The astrometric uncertainties used for $\sigma$ can be defined per observatory code in the [*digest2*]{} configuration file. The current MPC settings of uncertainties and keywords are presented in Table \[t:tabUnc\] of Appendix \[app:tables\].
While this approach has the effect of producing a range of different motion vectors, it should be emphasized that this method of dealing with astrometric uncertainty is *not* ideal from a statistical viewpoint, omitting occasional astrometry that is worse than our statistics, or significantly better due to improved astrometric catalog or measurement. Improvements expected in the near future will employ astrometric uncertainties directly reported with the submitted astrometry on an individual detection basis to handle observational uncertainty.
Orbit Solution {#s:SolnAlgo}
--------------
Given a nominated observer-object distance, $D$, and a nominated angle, $\alpha$, between the observer-object unit vector, $\vec{d}$, and the object velocity vector, a specific orbit can be computed (Appendix \[app:admiss\]). The $H$-magnitude can be calculated for each orbit using the geocentric vector $\vec{\Delta}=D\,\vec{d}$ and the heliocentric distance of the object, $\vec{r}$, to get the object phase angle $\Phi$, which, with the apparent magnitude $V$, gives $H$ in the IAU adopted H-G magnitude system @1989aste.conf..524B.
Population Model used by [*digest2*]{} {#s:POPN}
--------------------------------------
The [*digest2*]{} code requires a population model against which tracklet-derived-orbits can be compared, representing the number of solar system objects in a variety of dynamical categories. [*digest2*]{} uses two different population models. The first is a full population model including all objects down to a given diameter. The second is the difference between the full population model and the known orbit catalog, representing the undiscovered population.
Desig. RMS Int NEO N22 N18 Other Possibilities
NE00030 0.15 100 100 36 0
NE00199 0.56 98 98 17 0 (MC 2) (JFC 1)
NE00269 0.42 24 23 4 0 (MC 7) (Hun 3) (Pho 15) (MB1 <1) (Han <1) (MB2 41) (MB3 5) (JFC 1)
### Full Population Model {#s:POPN:Full}
[*digest2*]{} uses the term “raw” to indicate when the full population model is used to score tracklet-derived orbits. The current version of [*digest2*]{} uses the Pan-STARRS Synthetic Solar System Model (S3M) of @Grav11, consisting of over 14 million simulated Keplerian orbits.
[*digest2*]{} bins S3M into 15 different orbit classes (Table \[t:classes\] in Appendix \[app:tables\]). The model uses bins in perihelion ($q$), eccentricity ($e$), inclination ($i$) and absolute magnitude ($H$). Binning in perihelion, rather than semi-major axis $a$, enables a bin cut at $q = 1.3$ to directly distinguish NEO orbits from non-NEO orbits. The binning is non-uniform and provides enhanced resolution in higher density regions of parameter space, while simultaneously reducing the number of empty and near-empty bins in low density regions. The bin boundaries and counts for the full population are depicted in Figures \[fig:Population\] in Appendix \[app:bins\]. The binned population and model reduction is done by the [[muk]{}]{} program[^7].
### Unknown Population {#s:POPN:Unknown}
[*digest2*]{} also allows a comparison of tracklet-derived orbits against the likely population of “undiscovered” or “unknown” objects, referred to as the “no ID” model. An argument can be made that “unknown” objects can be more accurately scored against a population model that excludes objects typical of those that are already known.
Given a binned version of a full population model (as described in Section \[s:POPN:Full\]), the desired “unknown” population model is constructed by reducing bin populations by the number of cataloged objects with well determined orbits. Given a metric for orbit quality, an orbit is selected from a catalog *of known objects* and the orbit is graded as to whether it would likely be identified. If the catalog orbit is identifiable, the count of objects in the appropriate bin within the *full, modelled* population can then be reduced. Repeating this procedure for all well-known objects reduces the full population model to the “unknown” population model. This model reduction and bin selection is performed by the above-mentioned program [[muk]{}]{}.
The known catalog currently used by [[muk]{}]{} is `astorb.dat`[^8] @1994IAUS..160..477B, @astorb. Its orbit quality metric is `astorb` field 24 [@1993Icar..104..255M], corresponding to the peak ephemeris uncertainty over a ten year period. The Peak ephemeris uncertainty is observationally motivated, offers a direct comparison regarding how secure the object is against getting lost, and is the most generic way to compress the information of the covariance matrix into a scalar. [[muk]{}]{} considers such an orbit to be determined well enough that its identification with a tracklet would be trivial. For each of these orbits, the corresponding model bin is decremented by one, unless the bin population reaches zero.
A binned population model divided into bins for “raw” and “no id” populations is distributed with the [*digest2*]{} source code and updated when a new version of [*digest2*]{} code is available.
Searching for bins {#s:bin search}
------------------
Given an orbit calculated as described in Section \[s:SolnAlgo\], the next step is to assign the orbit to the appropriate $(q,e,i,H)$ population-bin. The algorithm searches over a range of distances, $D$, and angles, $\alpha$, within the admissible region (Section \[s:METHOD\]). There are a number of possible approaches to selecting a set of points for evaluation within this region. [*digest2*]{} does *not* use a $\chi^2$ (or RMS) approach, nor does it generate a fixed grid of points, but instead utilizes a binary search approach (See Appendix \[app:bin\_search\]).
Class Score {#s:score}
-----------
Given the set of bins identified in Section \[s:bin search\], a “class score” is calculated over those bins. These scores are calculated for any of the classes listed in Section \[s:POPN\] that are selected by the user at run-time as being of interest.
For a specified orbit class of interest, a sum $\Sigma_{class}$ is accumulated based on the modeled population consistent with the orbit class. Another sum, $\Sigma_{other}$, is accumulated for the modeled population consistent with orbits [*not*]{} of the class. The class score $S_c$ is given by $$\label{score_function}
S_c = 100 \frac{\Sigma_{class}}{\Sigma_{class} + \Sigma_{other}}$$
Therefore, the [*digest2*]{} score, ${D_2}{}$, ranges from 0 to 100 in any given class. The example in Figure \[f:verbatimA\] demonstrates a range of ${D_2}{}$ values for three tracklets of different objects, after execution of the compiled ${\emph{digest2}}{}$ program[^9] with its supplied sample input files[^10].
The output generated by [*digest2*]{} lists scores for tracklets on individual rows. The number of displayed columns as well as the configuration of [*digest2*]{} can be controlled using keywords in the configuration file (See Table \[t:keywords\] of Appendix \[app:tables\]). The first object (NE00030) appears to be an NEO and an “Interesting” object based on ${D_2}{}=100$. However, its scores for being a large NEO ($N22$, $N18$) are rather low, so this object is therefore most likely a small NEO with $H>22$. The second object ($NE00199$) also has large NEO scores, however, its RMS is large, suggesting either astrometry with large errors or deviation from a great circle motion due to a close encounter with the Earth. The last two columns, in the parentheses, suggest that there is a small probability that this object is a Mars Crosser or a Jupiter Family Comet. The third object ($NE00269$) also has a large RMS and its NEO scores are low (${D_2}{}<25$). Based on other orbit classes computed for $NE00269$, this object is most likely a Central Main-Belt asteroid.
Algorithmic Summary {#s:AlgSumm}
-------------------
In Algorithm Table \[alg: essential\], we summarize the key algorithmic steps employed in [*digest2*]{}, These are: (1) the reduction of the tracklet observations to a single motion vector (Line \[ess:two pos\]), (2) the sampling of the distance, $D$, and velocity angle, $\alpha$ (Lines \[ess:dist loop\]-\[ess:angle loop\]), (3) the generation of orbital elements and population comparison (Lines \[ess:elements\]-\[ess:pop\]), leading to (4) the accumulation of a score (Lines \[ess:accumulate\]-\[ess:score\]).
Open observation file \[ess:load\_obs\] Load binned population model into memory \[ess:load\_popn\]
As described in Section \[s:SolnAlgo\], [*digest2*]{} generates a large number of different observer-object distances ($D$) and angles ($\alpha$) between the observer-object unit vector and the object velocity vector, each pair of $(D,\alpha$) yielding different keplerian orbital elements. The distance, $D$, and angle $\alpha$, are selected in order to construct a bound heliocentric orbit.
For each candidate orbit, the elements index a bin (Section \[s:bin search\]) of a population model of the Solar System (Section \[s:POPN\]). Indexed bin populations then represent populations of modeled objects consistent with the input arc. For a dynamic class of interest, such as NEOs, population sums can be accumulated that allow score calculation (Section \[s:score\]). A score indicates a quasi-likelihood that the input arc represents an object of the class of interest.
[*digest2*]{} score analysis {#s:Classifier}
============================
The primary use of the [*digest2*]{} code has been as a *binary classifier* for NEOs, in which a tracklet’s NEO ${D_2}$ score[^11] is compared to a critical value, $D_{2,crit}$[^12] Tracklets with $D_{2} > D_{2,crit}$ are considered by the MPC to be eligible for submission to the Near-Earth Object Confirmation Page (NEOCP). The rationale and history for selecting a given score is described later at the end of Section \[SECN:RES:ACCURACY\]. Submitting newly discovered tracklets to the NEOCP allows them to be prioritized for follow-up by the community, allowing their arcs to be extended, and a more precise orbit to be determined for them, ultimately determining whether the object is indeed an NEO.
The value of $D_{2,crit}$ used to decide whether a tracklet is admissible to the NEOCP has varied since the introduction of [*digest2*]{}. This has changed both the *number* of tracklets admitted to the NEOCP and the *purity* of the tracklets (what fraction are NEOs) on the NEOCP. The detailed effects of these changes are discussed at length in Section \[SECN:RES:ACCURACY\].
Application of [*digest2*]{} to Individual Objects {#s:IND}
--------------------------------------------------
=0.11cm
------------- ------ ----- --------- -------- -------- --------- ---------
$q$ $e$ $i$
Designation H PHA $[au]$ $-$ $[deg]$
(198752) 19.8 No NEO Amor $1.21$ $0.52$ $1.75$
2015 DP155 21.5 Yes NEO Amor $1.02$ $0.22$ $5.38$
2012 HN40 20.4 No NEO Apollo $0.89$ $0.67$ $14.39$
2012 HZ33 20.4 Yes NEO Apollo $0.95$ $0.21$ $23.88$
2006 EW1 16.0 No non-NEO MB $2.80$ $0.12$ $6.03$
2000 ED68 16.4 No non-NEO MB $1.89$ $0.40$ $25.40$
2013 BX45 17.6 No non-NEO MC $1.56$ $0.40$ $29.51$
2017 BG123 19.5 No non-NEO HUN $1.60$ $0.13$ $23.29$
2014 QC158 15.9 No non-NEO HIL $3.12$ $0.22$ $14.19$
2012 RW6 13.1 No non-NEO TRO $4.87$ $0.06$ $16.24$
------------- ------ ----- --------- -------- -------- --------- ---------
: Individual NEOs (top) and non-NEOs (bottom) selected for detailed study in Section \[s:IND\].
\[Digest\_nonNEOs\]
We provide illustrative examples of the [*digest2*]{} score for a number of representative objects from the MPC database. The objects and their orbital classifications are listed in Table \[Digest\_nonNEOs\].
Because the observing cadence for object tracklets submitted to the MPC is rather sparse, we generated daily ephemerides and synthesized tracklets for our sample over a period of 10 years. For each object, the orbit was numerically integrated and an ephemeris was generated for two epochs per night, separated by 20 minutes. The pair of points within a night is then transformed into Geocentric (RA,Dec) coordinates, and used to define a tracklet for each night. Each tracklet is then used as an input to the [*digest2*]{} code, and a score generated.
All of the NEOs plotted in Figure \[fig:individual\] have a maximum [*digest2*]{} score $\sim 100$, but all exhibit periods during which their scores are significantly lower, and all spend some time during which their score is $<65$. The *non*-NEOs in Figure \[fig:individual\] have a wide range of behaviors. Some, such as the MBA “2006 EW1” always display low [*digest2*]{} scores. In contrast, some objects such as the Mars-crosser “2013 BX45” spend sizable fractions of the time with a score $>65$.
Because a critical [*digest2*]{} score $D_{2,crit}=65$ is used by the MPC to post objects to the [[NEOCP]{}]{} and prioritize them for observational follow-up, it follows from Figure \[fig:individual\] that, (a) an NEO could fall below $D_{2,crit}$ for some fraction of the time it is visible, and hence “miss-out” on being sent to the [[NEOCP]{}]{}, and (b) non-NEOs can gain scores higher than $D_{2,crit}$, and hence be “unnecessarily” sent to the [[NEOCP]{}]{}. In the following sections we quantify the frequency of such scenarios.
In Figure \[fig:IndHeat\] we plot ${D_2}{}$ as a function of ecliptic coordinates (with respect to opposition) and the rate of motion. The plotted positions are for the time of visibility depicted in Figure \[fig:individual\]. It is clear from the middle plot of Figure \[fig:IndHeat\] that the value of the [*digest2*]{} score is strongly dependent on the rate-of-motion and latitude. While this *generally* serves to distinguish NEOs from non-NEOs, we can see that some of the non-NEOs spend some fraction of their time with relatively high rates-of-motion and/or high-ecliptic latitudes. This inevitably leads to confusion between NEO and non-NEO, as evidenced by the high ${D_2}{}$ scores for some of the non-NEOs.
Application of [*digest2*]{} to Populations {#s:POP}
-------------------------------------------
### Data Sets and Simulations {#s:Sets}
[lllll]{}
$MPC_{D,11}$ & Real (All) & 1-Year & 14,003&14,003\
$MPC_{D,17}$ & Real (All) & 1-Year &18,715& 18,715\
$MPC_{A,17}$ & Real (All) & 1-Month &374,406&143,983\
$Sim$ & Synthetic (NEO) & 10-Years &$3.8\times10^6$&16,230\
LSST & Synthetic (NEO, MB) & 9-days &$14.3\times10^6$&$1.4\times10^6$\
Granvik & Synthetic (NEO) & 1-Year &618,951&14,458\
To investigate the behaviour of ${D_2}{}$ for a large population, we computed ${D_2}{}$ for the data sets listed in Table \[t:Data\_Sets\]. The data sets $MPC_{D,11}$ and $MPC_{D,17}$ contain all *discovery* tracklets[^13] submitted to the MPC in 2011 and 2017 respectively, thus containing tracklets of all types of orbit. We use both $MPC_{D,11}$ and $MPC_{D,17}$ data sets because in 2011 the MPC used $D_{2,Crit}=50$ for posting to the NEOCP, whereas in 2017 the MPC used $D_{2,Crit}=65$, allowing us insight into the effects of different value of $D_{2,Crit}$ used in the past.
$MPC_{A,17}$ contains a month’s worth of *all* tracklets submitted to the MPC, containing all types of submitted orbit. We selected January 2017 as a time when Jupiter’s Trojans were observable near opposition. We only selected tracklets fainter than 18 magnitude, to work in the regime of newly discovered objects.
The $Sim$ data set was obtained by integrating the known catalog of NEOs (as of July 2018) with $H>18$ for 10 years, starting on January 1, 2008 and deriving a daily ephemeris from the Geocenter, providing a two-detection tracklet spanning 1-hour for ${D_2}{}$ computation. To simulate accessibility for ground-based telescopes, we constrained the maximum V-band magnitude to 22.5 and distance from opposition to be within $\pm100$ degrees in longitude and $\pm70$ degrees in latitude. 90% of objects in the catalog were observable within the $Sim$ data.
The $LSST$ data set contains tracklets from 9-nights worth of a simulated LSST survey [@Veres17]. The data set contains NEOs from @Granvik18 and Main-Belt asteroids from @Grav11.
The $Granvik$ data set contains NEO tracklets for the $H>18$ objects in the synthetic population of @Granvik18 that we propagated for one calendar year and then used the resulting ephemerides to compute the value of ${D_2}{}$ on a nightly basis. We employed the same observability criteria for the limiting V-band magnitude and the opposition-centered ecliptic coordinates as we used on the $Sim$ data set.
### Distribution of [*digest2*]{} Scores {#s:DIST}
The overall distribution of scores for the $MPC_{A,17}$ data set can be seen in Figure \[fig:month\_A\]. This data set contains various object types. We plot histograms of the ${D_2}{}$ score at the top, and fractional cumulative distributions below. Most Main-Belt asteroids have ${D_2}\sim0$, and most NEOs have ${D_2}\sim100$. However, the long tail of NEOs with low ${D_2}{}$ is interesting because those with ${D_2}<D_{2,Crit}$ are not recognized as NEOs and will not be placed onto the NEOCP for follow-up. In the $MPC_{A,17}$ data set, 14% (98.5%) of NEO (non-NEO) tracklets had ${D_2}<D_{2,Crit}$. Figure \[fig:month\_A\] shows that Trojans and Mars-Crossers are objects that “mimic NEOs”, as these two populations of object have the highest fraction with ${D_2}>D_{2,Crit}$. Trojans dominate numerically for $55<{D_2}<90$, while Mars-Crossers contribute equally across the entire range of ${D_2}{}$ values.
### Comparison of NEO [*digest2*]{} Scores Across Data Sets {#s:NEOs}
\[f:NEO\_d2\] ![ Cumulative distribution of [*digest2*]{} scores for all *NEO* data sets. The blue ($MPC_{A,17}$) line is reproduced from Figure \[fig:month\_A\]. While most NEO tracklets have ${D_2}{}=100$ at any given time, the distribution of smaller ${D_2}{}$ differs between data sets. In the unbiased $Sim$ sample about 30% of NEO tracklets have ${D_2}{}<D_{2,Crit}$. ](d2_datasets.png "fig:"){width="\columnwidth"}
Figure \[f:NEO\_d2\] shows the cumulative distribution of ${D_2}{}$ for the NEOs generated within the data sets of Table \[t:Data\_Sets\]. The data sets differ at small ${D_2}{}$, introducing differing biases. The *discovery* data sets ($MPC_{D,11}$, $MPC_{D,17}$) have the smallest fraction of tracklets in the low ${D_2}{}$ range and clearly display the effects of imposing a $D_{2,Crit}$-cut-off on objects submitted to the NEOCP.
The data sets containing *all* submitted tracklets ($MPC_{A,11}$) or simulated tracklets ($Granvik$, $LSST$, and $Sim$) have increased ratios of low-scoring NEOs because (a) they are not subject to the $D_{2,Crit}$ threshold placed on objects sent to the NEOCP, and (b) they contain objects observed repeatedly. The largest fraction of low scoring tracklets comes from the $Sim$ data set, which may be a consequence of the long-term simulation window (10-years) and our optimistic nightly cadence over the entire visible night sky, allowing it to contain objects with large synodic periods, and objects that only occasionally have ${D_2}>D_{2,Crit}$. The $Sim$ data set has a wide range of NEOs: 28% of all visible tracklets had ${D_2}{}<65$ at any given time. However, we note that within the data set, some NEOs are visible for only a single one night, while others are visible for many months.
### Detailed Analysis of [*digest2*]{} scores for NEOs {#s:NEOs_details}
We use the $Sim$ data set to examine the frequency with which NEOs and their subclasses (Amor, Apollo and Aten type orbits, PHAs[^14] and low-MOID[^15] (<0.05 AU) NEOs) have tracklets with low values of ${D_2}$ .
![ Cumulative histogram of maximum, mean and minimum [*digest2*]{} score per object for NEOs and their orbital subclasses from $Sim$ data set. Only 0.4% of NEOs never reach $D>65$.[]{data-label="FIG:PVa"}](min_D2.png){width="\columnwidth"}
Figure \[FIG:PVa\] shows the cumulative histograms of the minimum, mean and maximum ${D_2}{}$ per object from the $Sim$ data set, and the differences in ${D_2}{}$ seen between the different orbit classes. NEOs can, at times, have low scores: 53% of NEOs have ${D_2}{} < 65$ at some point while observable. But in general, NEOs (and their subclasses) achieve high ${D_2}{}$ in almost all cases: 94% of NEOs reach ${D_2}{}=100$ (Table \[t:scoresSim\]). This ratio is highest for objects that come close to the Earth or the Sun ( Aten, Apollo and low-MOID classes). The ratio is smallest in the case of Amors, only 87% of which ever reach ${D_2}{}=100$, because Amors remain distant and often mimic Main-belt motion when near aphelion. *Only 0.4% of NEOs never reach $D>65$ while visible.*
------------ ------------------ ------------------ -------------------
Orbit type min ${D_2}{}>65$ max ${D_2}{}<65$ max ${D_2}{}=100$
(%) (%) (%)
NEO 46.7 0.4 93.9
PHA 22.8 0.4 99.2
Aten 65.9 0.2 99.6
Apollo 46.4 0.2 99.5
Amor 43.8 0.6 86.4
Low-MOID 59.9 0.1 99.5
------------ ------------------ ------------------ -------------------
: Fraction of NEOs that achieved a given ${D_2}{}$ in the $Sim$ data set.
\[t:scoresSim\]
To understand the NEOs with low ${D_2}{}$, in Figure \[fig:ecliptic\_grid\] we plot the rate-of-motion and sky-plane location (in ecliptic coordinates centered at opposition ) of each tracklet from the $Sim$ data set. There are two regions in the morning and evening sky where NEOs tend to have lower ${D_2}{}$. NEOs in these regions are moving very slowly and mimic the Main-belt rate. These regions are coincident with so-called “sweet-spots” at low solar elongations, that are the only regions of the sky in which many NEOs become observable [e.g. @STOKES03; @CS04; @Boattini07]. There is also an area near opposition where low digest scores are possible for NEOs moving with rates of motion similar to those of MBAs. Interestingly, if the object moves even slower, e.g. below 0.1 degree per day, its score is very high. All fast moving objects (rates above $\sim0.8$ degrees/day), and all objects at large observed ecliptic latitudes, have ${D_2}{}=100$.
{width="\textwidth"}
Accuracy, Precision and the NEOCP {#SECN:RES:ACCURACY}
---------------------------------
To understand how accurate ${D_2}{}$ is when used as an NEO classifier, we can calculate the value of ${D_2}{}$ for tracklets from a population of known objects (both NEOs and non-NEOs). We can then compare ${D_2}{}$ to an imposed critical value $D_{2,Crit}$, and calculate for the population the rate of “True Positives”($TP$), “False Positives” ($FP$), “False Negatives” ($FN$), and “True Negatives” ($TN$). Repeating this gives the values of these quantities as a function of $D_{2,Crit}$. We have summarized these quantities (and some additional metrics) in Table \[t:EM\_Def\].
[l|rr]{}
$D_2 >= D_{2,Crit}$ & True Positives, $TP$ & False Positives, $FP$\
$D_2 < D_{2,Crit}$ & False Negatives, $FN$ & True Negatives, $TN$\
\
\
\
\
![ [**Top Left:**]{} $MPC_{D,11}$, All discovery tracklets from 2011; [**Top Right:**]{} $MPC_{D,17}$, All discovery tracklets from 2017; [**Bottom Left:**]{} $MPC_{A,17}$, All tracklets from January 2017. [**Bottom Right:**]{} $LSST$, All tracklets from LSST sim. Each panel plots the False-Positive-Rate (FPR, blue), the False-Negative-Rate (FNR, orange), the Precision (Green), and the Accuracy (red). The precision is a measure of the purity of the tracklets that will end-up on the [[NEOCP]{}]{}. The “kink” that is particularly obvious in the “Precision” measure for $MPC_{D,17}$ is driven by the $D_{2,Crit}=65$ NEOCP threshold. []{data-label="f:PREC"}](NEOratio_.png){width="\columnwidth"}
As described in Section \[s:Classifier\], unknown tracklets with ${D_2}{}>D_{2,Crit}$ are submitted to the NEOCP, increasing the likelihood of follow-up observations being obtained and hence of the object being confirmed as real. The precision, $PRE$, is a measure of the purity of the tracklets that will end-up on the NEOCP (i.e. what fraction of the tracklets on the NEOCP will actually be NEOs).
In Figure \[f:PREC\] we plot the *False Positive Rate*, *False Negative Rate*, *Precision*, and *Accuracy* as functions of the critical [*digest2*]{} score, $D_{2,Crit}$. We do this for both real ($MPC_{D,11}$, $MPC_{D,17}$ and $MPC_{A,17}$) and synthetic ($LSST$) data sets. These plots show that the *discovery* data sets ($MPC_{D,11}$ and $MPC_{D,17}$) both display “kinks” in the various performance metrics at the values of $D_{2,Crit}$ used at the time of submission. Above the threshold, most NEOs are followed-up and orbits computed. However, NEOs below the threshold at the time of first observation are less likely to be immediately followed-up. In contrast, no selection effects are imposed on either the $MPC_{A,17}$ or $LSST$ data sets, hence no kinks appear.
We find that the false negative rate is the largest in the $MPC_{A,17}$ and $LSST$ data sets, essentially because tracklets with ${D_2}{}<{D_2}{}_{,crit}$ are absent from the other ($MPC_{D,11} \& MPC_{D,17}$) data-sets.
We note that the value of $D_{2,Crit}=65$ was chosen with the aim of generating approximately half of the objects on the NEOCP being NEOs. In 2010, the $D_{2,Crit}$ was lowered to $D_{2,Crit}=50$ based on the request of the community and availability of more follow-up and discovery assets. After a year, due to the lack of follow-up of the low-scoring NEO candidates, MPC increased $D_{2,Crit}$ back to $D_{2,Crit}=65$ in mid-2012. It is clear from both the curves in Figure \[f:PREC\] and by @Veres2018 that this choice was successful in generating the approximate desired level of precision. If members of the community wish to advocate for changes to the adopted value of $D_{2,Crit}=65$, they should weigh-up the pros and cons of such a change: Increasing the value of $D_{2,Crit}$ will cause the false positive rate to drop (cutting down on the number of non-NEOs “unnecessarily” followed-up), but the false negative rate will rise (hence losing NEOs).
Nuances of Practical [*digest2*]{} Usage {#s:practical}
----------------------------------------
There are a number of practical details relating to the algorithmic design of [*digest2*]{} that should be considered when using the [*digest2*]{} code in practice.
### Effects of Random-Choice of $\alpha$ {#s:random}
The [*digest2*]{} code uses a pseudo-Monte Carlo method to generate a range of variant orbits (see Appendix \[app:admiss\]). By default, the pseudo random number generator is seeded randomly. Therefore, even though the admissible region is the same, the sampling of available orbits could hit different population bins during an independent instance of the program. We demonstrate an extreme example of the ${D_2}{}$ variation using a real NEO candidate (*P10Gj15*) that was observed on January 17, 2018 as a 3-detection tracklet. The tracklet had ${D_2}{}=66$ and therefore it was posted to the NEOCP. When we re-ran [*digest2*]{} 1000-times on the tracket, the distribution of ${D_2}{}$ in the left-panel of Figure \[fig:two\_scores\_alpha\] shows that on many occasions the ${D_2}{}$ value was below 65.
To understand this result, we show two specific runs, one leading to ${D_2}=66$ and one to ${D_2}{}=60$ in the center of Figure \[fig:two\_scores\_alpha\], demonstrating that the sampled orbits inhabit almost identical regions of phase space. However, counting the bins hit by generated variant orbits in the two runs (illustrated in the right-hand side of Figure \[fig:two\_scores\_alpha\]), we see that different population bins were hit, giving different scores. In the case of $P10Gj15$, further follow-up observations allowed the orbit determination of the object that became announced as $2018\,BE1$ - a Hungaria-class asteroid.
We repeated this analysis for 1000 randomly selected tracklets from the $Granvik$ NEO data set, running the ${\emph{digest2}}{}$ code 100-times for each. We found that 82% of the tracklets had no change in the integer value of ${D_2}{}$. In particular, tracklets with ${D_2}{}\sim0$ or ${D_2}{}\sim100$ had almost no variation in ${D_2}{}$. Only a small fraction of tracklets, 0.7%, had a variation $\Delta{D_2}{}>3$. Most of the varying ${D_2}{}$-tracklets were in the range $40<{D_2}<60$, the region where both NEO and Main-Belt populations seem to be probable for a given velocity vector.
The small variation is caused by the binned-population model and the algorithmic design. Future versions of the code should reduce/eliminate this effect.
We emphasize that if the user wishes to suppress these variations, the keyword *repeatable* can be added to the configuration file. The random angle $\alpha$ is then reseeded with a constant value for each tracklet, yielding repeatable scores in independent runs.
### Effects of Observational Uncertainty {#s:obs err}
As illustrated in Section \[s:Dithering\], “dithered” tracklets are constructed from two end-points or two generated points of the tracklet. The dithering itself is governed by the expected astrometric uncertainty. The default astrometric uncertainty used by [*digest2*]{} is 1.0 arcsecond. However, in our work and at the MPC, astrometric uncertainties are assigned in the ${\emph{digest2}}{}$ configuration file, using the long-term average for each observatory code (Table \[t:tabUnc\], Appendix \[app:tables\]).
To demonstrate the effect of changing the assumed uncertainty on a single tracklet, we again chose the Pan-STARRS tracklet $P10Gj15$. Figure \[fig:uncObj\] shows how ${D_2}{}$ depends on the assumed uncertainty. The variation in score is driven by the same effect demonstrated in Section \[s:random\] (Figure \[fig:two\_scores\_alpha\]): varying the uncertainty changes the population bins included in the tracklet score calculation. In the example illustrated, when the astrometric uncertainty is overestimated, $P10Gj15$ ${D_2}{}$ leads to more variant orbits that hit the Main-Belt bins.
To study the significance of the assumed astrometric uncertainties for a large data set, we selected 30,000 NEOs from the $Granvik$ data set, generated two-detection tracklets with 0.2“ astrometric errors and computed ${D_2}{}$ using three different assumed astrometric uncertainties: underestimated (0.05”), nominal (0.2“) and overestimated (1.0”). Note, that the analysis was undertaken with the [*digest2*]{} keyword “repeatable” set to avoid variation due to angle $\alpha$ randomness.
When the astrometric uncertainty is overestimated by 0.8“ (underestimated by 0.15”), the value of ${D_2}{}$ varies by an amount $\Delta{D_2}{}\approx-1$ ( $\Delta{D_2}{}\approx+0.2$). This effect is particularly obvious for intermediate values of ${D_2}{}$ (20 to 80), where overestimation (underestimation) lead to $\Delta{D_2}{}\approx-5$ ( $\Delta{D_2}{}\approx+1.0$). ${D_2}{}$ values near zero and 100 do not show any variation. Overall there was no change for 90% of the “underestimated” and 78% of the “overestimated” tracklets.
### Great Circle Departure {#s:GCR}
As described in Section \[s:Dithering\], the [*digest2*]{} algorithm uses a pair of synthetic detections derived from the set of observed positions. In a typical short arc tracklet spanning $\sim 1$hour, the motion is mostly linear. However, when the object is close to the Earth, the effect of diurnal parallax for the topocentric observer can cause discernible curvature that is seen as a deviation from the great circle, even within one hour. Using the keyword “rms” will cause the [*digest2*]{} code to output the root-mean-square of the great circle residuals for the positions reported in a tracklet (if the tracklet has $\ge3$ detections). If the $rms$ score is within the expected astrometric uncertainties, then, linear motion is a good approximation for the [*digest2*]{} method. A large $rms$ can be due to either bad astrometry, or to diurnal parallax.
To test the effect of great circle deviations we selected close approach events from the CNEOS web site[^16] for asteroids that approached the Earth within $0.2{\,\mathrm{au}}$ during 2006-2018. We generated 16,800 events for 11,828 NEOs. We generated 4-detection tracklets spanning 1 hour for each event. We selected the Pan-STARRS1 $F51$ observatory code as the topocentric position and “smeared” the generated astrometry by an expected astrometric uncertainty of 0.2 arc-second. Only tracklets $>60$ degrees from the Sun were considered.
Figure \[f:rms1\] illustrates the rate of motion, the $rms$ of the great-circle-residuals, and the geocentric distance of each tracklet. As expected, the $rms$ increases with decreasing geocentric distance. In addition, the correlation between rate of motion and $rms$, as well as between rate of motion and geocentric distance is obvious. Low scoring NEOs had $rms<0.2"$, their rate of motion was consistent with the rate of Main-belt asteroids and were distant at the same time.
Earth quasi-satellites, co-orbitals and Trojans {#coorbitals}
-----------------------------------------------
Although the Moon is the only known natural satellite of the Earth, temporarily captured NEOs, NEOs in 1:1 mean-motion resonance with the Earth, Earth Trojans or proposed minimoons[@2012Icar..218..262G; @minimoons] can follow geocentric orbits for days or months. (469219) 2016 HO3 is the closest and most stable Earth’s quasi-satellite [@2016MNRAS.462.3441D]. The only Earth Trojan discovered so far is 2010 TK7[@2011Natur.475..481C; @2012ApJ...760L..12M]. We selected a small group of known quasi-satellites or co-orbitals[^17], generated daily ephemerides from 2000 until 2020, created 20-minute tracklets and simulated sky-plane visibility down to +22.5 $V-band$ magnitude. Their sky-plane motion and [*digest2*]{} score are shown in Figure \[fig:coorbitals\]. The analyzed co-orbitals typically display large valuess of ${D_2}{}$, unless they are in the low-scoring “sweet-spots”. The overall distribution of ${\emph{digest2}}$ scores is similar to $MPC_{A,17}$ and $Granvik$ from Figure \[f:NEO\_d2\] in Section \[s:NEOs\]. The only known Earth Trojan 2010 TK7 has a ${\emph{digest2}}{}$ value oscillating between 93 and 100, and $40\%$ of its tracklets had $DD{}=100$.
{width="0.9\columnwidth"}
NEOs at low Solar elongations {#nearSun}
-----------------------------
Most of our analyses were focused on NEOs observed in favorable locations of the sky within 100 degrees of the opposition (e.g. Figure \[fig:ecliptic\_grid\]). In Section \[s:NEOs\_details\] we identified regions near the ecliptic roughly 100 to 60 degrees from the opposition where the mean [*digest2*]{} score is relatively low. However, some orbit types, such as Atens or Atiras, rarely or never cross the opposition region. Searches close to the Sun are required to find these objects. Ground-based observations near the Sun are limited by large airmass and associated target brightness losses, as well as limited observing time at dawn or dusk. Low Solar elongation observations are ideal for space-based observatories. Figure \[fig:closeToSun\] shows the mean [*digest2*]{} score for the $Sim$ data set in the Sun-centered ecliptic coordinates. Interestingly, closer to the Sun [*digest2*]{} is almost always above 65 and Atiras have particularly very large ${D_2}{}$ values. We demonstrated that ${\emph{digest2}}{}$ score can identify NEOs in low-solar elongations.
{width="\textwidth"}
Interstellar Objects and [*digest2*]{}
--------------------------------------
The interstellar object (ISO) 1I/’Oumuamua [@2017MPEC....U..181B] was discovered by the Pan-STARRS NEO survey and its first tracklet was reported to the NEOCP as a ${D_2}{}=100$ tracklet. Subsequent follow-up quickly revealed its hyperbolic orbit. However, without a large NEO ${\emph{digest2}}{}$ score, it would never have appeared on the NEOCP or been followed-up. As mentioned earlier, one of the key constraints of [*digest2*]{} are orbits bound to the Sun. Previous work [@Engelhardt] showed that the NEO [*digest2*]{} score could be used to identify ISOs, demonstrating that 2/3 of ISOs in a simulated Pan-STARRS survey achieved ${D_2}{}>90$ at some time while visible.
Figure \[FIG:oumuamua\] shows the ${D_2}{}$, rate of motion and magnitude of all reported 1I/’Oumuamua tracklets. From the time of discovery to November 12, 2017, 1I/’Oumuamua had ${D_2}{}=100$ at all times, and in addition, 1I/’Oumuamua had ${D_2}{}=100$ for the entire time it was visible to the main NEO surveys ($V{\lower.7ex\hbox{$\;\stackrel{\textstyle<}{\sim}\;$}}22$).
![ [*digest2*]{} score, mean V-band magnitude and rate of motion of 1I/’Oumuamua tracklets. Dashed vertical lines display epochs when object was observed by Hubble Space Telescope. 1I/’Oumuamua has a high [*digest2*]{} score for the entire time it was visible to surveys ($V{\lower.7ex\hbox{$\;\stackrel{\textstyle<}{\sim}\;$}}22$). \[FIG:oumuamua\] ](oumu.png){width="\columnwidth"}
Discussion {#s:DISC}
==========
We have described the algorithm underlying the [*digest2*]{} code, as well as its usage to calculate the score of a short-arc tracklet with respect to a given orbit class. We focused on the *NEO* [*digest2*]{} score, ${D_2}{}$, as used by the Minor Planet Center (MPC) to classify the likelihood that submitted sets of detections are NEOs. We showed that ${D_2}{}$ changes for each object as it moves across the sky and its brightness and rate of motion vary. For NEOs, the maximum ${D_2}{}$ reaches the extreme value of ${D_2}{}=100$ for more than 93% of NEOs, and over $99\%$ of NEOs have $D_{2,crit}>65$ at some point while visible. Thus, the vast majority of NEOs have sufficiently high ${D_2}{}$ to allow them to be posted to the NEOCP and be rapidly followed-up by observers around the world.
While the [*digest2*]{} code identifies short-arc NEOs correctly in the vast majority of cases, there are a small number of cases when ${D_2}{}$ is below the critical threshold, currently set to $D_{2,crit}>65$. We identify the two locations on the sky, on the ecliptic and at low solar elongations, where the mean ${D_2}{}$ is significantly lower than elsewhere on the sky. This is due to the observing geometry and rates of motion of the objects, that cause them to be confused with Main-Belt objects.
We also demonstrate that without the “repeatable” parameter the [*digest2*]{} code can generate varying output for the same tracklet due to its random selection of variant orbits. However, the variation in ${D_2}{}$ is small and does not affect scores near extremes (${D_2}{}=0$ and ${D_2}{}=100$). The MPC uses [*digest2*]{} without the “repeatable” parameter to avoid bias.
Future directions {#s:DISC:Future}
-----------------
As we move into an era of large surveys that can obtain huge volumes of high-precision observations and utilize highly-precise stellar catalogues, it is clear that the [*digest2*]{} code must evolve. Future developments to the code will include:
- It will be possible to characterize observational uncertainties on a per-observation basis, as provided for in the ADES format[^18].
- A python wrapper and API to provide simplified usage.
- A new NEO population model, such as the high-fidelity @Granvik18 model, will be required. Catalog of known orbits used for unknown population reduction would be generated more frequently.
- The discrete population bins (in $q,e,i, H$) need to be replaced with a smooth 4-dimensional function, perhaps also incorporating additional orbital elements to allow improved discrimination of (e.g.) Hildas and Jupiter Trojans.
- Replacing the synthetic two-detection approach with a robust statistical treatment of all submitted detections, allowing for a significantly improved handling of NEOs that exhibit non-linear motion during close-approach.
- Further tests and improvements of the code are needed for space-based NEO surveys, such the proposed NEOCam mission [@2015AJ....149..172M].
- Replacing the $astorb$ known orbit catalog and its orbit quality metric [@1993Icar..104..255M] with the MPC’s orbit catalog[^19] and orbit uncertainty parameter[^20].
We emphasize that [*digest2*]{} constitutes an evolving code-base and that we welcome reports from the community of any bugs found, or of suggestions for future directions[^21].
We are grateful to Rob McNaught for his insights regarding the history of the [[Pangloss]{}]{}code, and for his many clarifications regarding the historical development of ranging.
MJH and MJP gratefully acknowledge NASA grants NNX12AE89G, NNX16AD69G, and NNX17AG87G, as well as support from the Smithsonian 2015-2017 Scholarly Studies programs. DJA acknowledges the N. Ireland Dept. for Communities support at Armagh.
We are saddened to report that during the course of the work on this manuscript, lead author Sonia Keys died on 13 August 2018 at the age of 57, after a long struggle with cancer. Sonia worked as a commercial software developer from 1983 to 2001 and then began working with the Astronomical Society of Kansas City, specializing in the tracking of Near-Earth Objects (NEOs). Her skill and reliability with NEO astrometry caught the attention of the Minor Planet Center (MPC), and that led to a position as an astronomer and software developer for the MPC. During her tenure at the MPC, Sonia was the lead developer of the [*digest2*]{} software and was the point-of-contact for the many people in the NEO community that utilized it for over a decade. We deeply regret Sonia’s passing, and we hope that we have completed this manuscript in a manner she would have approved of. We hope this publication and note will serve as a small token of the esteem in which we held her. Sonia was the conscience of the MPC. Her clarity of thought, her willingness to ask difficult questions, and her tenacity will be sorely missed.
Variant Orbits {#app:admiss}
==============
![ [**Top:**]{} Vector algebra for solving state vectors according to Appendix \[app:admiss\]. [**$E_i$ and $A_i$ are the positions of the Earth and the asteroid at times $t_i$.**]{} [**Bottom:**]{} Because only the transverse component of the velocity vector is visible to the observer, the true velocity vector could be in the dotted area between $A_2$ and $A_{2}^*$, that represents all possible solutions for bound orbits. $\vec{\Delta_{i}}$ represent topocentric and $\vec{r_{i}}$ the heliocentric vectors of the object and $\vec{R_{i}}$ the heliocentric vectors of the observer at time $t_i$. $\vec{s}$ is directed along the velocity vector of the object. []{data-label="fig:vec"}](diagram5.png){width="10cm"}
In the following algorithmic description, all line references relate to Algorithm \[alg: essential\] of Section \[s:COMP\].
The input file for the [*digest2*]{} program contains astrometry of short-arc tracklets, such as those illustrated in Appendix \[app:Hungaria tracklet\]. The file can contain one or many tracklets (Line \[ess:load\_obs\]). After the input data are loaded, binned population model is loaded into memory as well (Line \[ess:load\_popn\], Section \[s:POPN\]). Each tracklet is subsequently treated individually (Line \[ess:read arc\]):
Two observations of an arc are insufficient for a determination of its orbit, or when the observation arc is too short. However, even very short arcs with few observations can be used to derive a range of possible orbits and assess the probability of the asteroid being in a certain region of the Solar system. Because each tracklet has two or more detections (N), as described in Section \[s:EndPoint\], two detections either selected ($N=2$) or generated ($N>2$).
The two observables, or detections, $A_1$ and $A_2$ of the same object (Line \[ess:two pos\]), are defined by $A_{t_i}$=\[$\alpha_i, \delta_i, V_i$\], where $\alpha_i$ is the right ascension, $\delta_i$ the declination, $V_i$ is the apparent magnitude at the time of observation $t_{i}$.
Because the tracklet consists of $N\geq2$ data points with unique photometry, either due to photometric errors or rotation of the object, the mean tracklet magnitude $V_1$ is computed (line \[ess:V mag\]) in the Johnson-Cousins V-band photometric system. The transformation to V-band is simple [@Veres2018], the correction is $-0.8$ if the reported band was $B$, or $+0.4$ for other bands, except of $V$, where no correction is applied. If no photometry is provided, $V_1=21$ is assumed as a typical magnitude limit of the current asteroid surveys.
We can easily transform the observation \[$\alpha_i, \delta_i$\] to its topocentric cartesian unit vector (Figure \[fig:vec\]) $\vec{d_i}=[x_i,y_i,z_i]$:
$$(x_i , y_i , z_i) =
(\cos{\alpha_i} \cos{\delta_i},\,\,\,
\sin{\alpha_i} \cos{\delta_i},\,\,\,\sin{\delta_i}
)$$
For further analysis, we will need to transform unit vectors from equatorial to ecliptic coordinate system (See Figure \[fig:vec\]): $$\vec{d_i}= R_{\epsilon}^T \cdot \vec{d_i}$$
where $R_{\epsilon}$ is the rotation matrix defined by obliquity of the ecliptic $\epsilon$:
$$R_{\epsilon}=
\begin{pmatrix}
1 & 0 & 0\\
0 & \cos{\epsilon} & \sin{\epsilon}\\
0 & -\sin{\epsilon} & \cos{\epsilon}\\
\end{pmatrix}$$
To derive the position state vector $\vec{r}$, we have to *assume* a topocentric distance $D$ to the asteroid. In line \[ess:dist vec\] of Algorithm \[alg: essential\], Sun-observer vectors (labeled $\vec{R_1}, \vec{R_2}$ in Figure \[fig:vec\]) in ecliptic Cartesian coordinates are computed by local sidereal time and observer site parallax constants. Observer’s topocentric positions are loaded from MPC-managed list of observatory codes[^22]. Also the observer-object unit vectors $\vec{d_1}, \vec{d_2}$ are computed in ecliptic Cartesian coordinates.
Then, the topocentric vector to the first observation, $\vec{\Delta_1}$, will be a scalar multiple of $D$ and the unit vector $\vec{d_1}$. $D$ is one of the values selected recursively (Appendix \[app:bin\_search\]) between the minimum (0.05au) and maximum (100au) distances within the allowed admissible region:
$$\vec{\Delta_1}=D\,\vec{d_1}$$
Knowing the position vectors of the observer with respect to the Sun at the time of the two observations ($R_1$ and $R_2$) in ecliptic coordinates, we can easily derive the heliocentric state vector $\vec{r_1}$ as:
$$\vec{r_1}=\vec{R_1}+\vec{\Delta_1}$$
After the position vectors are derived, the algorithm continues in investigation of the velocity vectors. The observed motion of the asteroid on the celestial sphere is a projection of the actual velocity vector to the tangent plane. While we do not know the angle between the tangent plane and the velocity vector $\vec{v_1}$, we can make progress by computing the possible ranges of angles (Figure \[fig:vec\] - bottom) under the assumption that the object is bound to the Sun. At a given distance from the Sun, $r_1$, an orbit is bound if its specific orbital energy $\epsilon<0$. We will find the angles of the $\vec{v_1}$ for the parabolic orbits from:
$$\epsilon=\frac{v_1^2}{2}-\frac{U}{r_1} = 0
\label{epsilon}$$
where $U=k^2$ is the Gaussian gravitational constant squared[^23].
To find $\vec{v_1}$ one has to solve triangle $E_2, A_1, A_2$ on figure \[fig:vec\]. The velocity vector will be derived as $\vec{v_1}=(\vec{\Delta_2}-\vec{\Delta_{21}})/(t_2-t_1)$.
The angle $\theta$ (Figure \[fig:vec\] - bottom) can be derived by computing the intermediate vector $\vec{\Delta_{21}}$ from vector difference of $\vec{r_1}$ and $\vec{R_2}$ and the dot product of known unit vector $\vec{d_2}$ and derived $\vec{\Delta_{21}}$:
$$\begin{aligned}
\begin{split}
\vec{\Delta_{21}}=\vec{r_1}-\vec{R_2}\\
\cos{\theta}=\frac{\vec{\Delta_{21}}\cdot\vec{d_2}}{{|\Delta_{21}| |1|}}
\end{split}\end{aligned}$$
Then, by using the cosine rule, substituting $v_1=s_1/(t_2-t_1)$ and using equation (\[epsilon\]), we can derive the length of the vector $\Delta_2$ as follows. Line \[ess:angle lim\] of Algorithm \[alg: essential\] involves solving a quadratic that gives two values for $\alpha$ corresponding to state vectors $\vec{s}$ and $\vec{s^*}$ that represent parabolic orbits:
$$\begin{aligned}
\begin{split}
{\Delta_{21}}^2+{\Delta_{2}}^2-{s_{1}}^2-2{{\Delta_{21}}\Delta_{2}}\cos{\theta}=0\\
{\Delta_{2}^2}-2{{\Delta_{2}}{\Delta_{21}}}\cos{\theta}+\left({\Delta_{21}}^2-2\frac{U{(t_2-t_1)}^2}{{r_{1}}}\right)=0
\end{split}
\label{roots}\end{aligned}$$
Equation (\[roots\]) has two roots for $\Delta_2$ after doing simple substitution (A, B, C) solving:
$$\begin{aligned}
\begin{split}
(A,B,C) &= (1 , \,\,\, -2\Delta_2\cos{\theta} , \,\,\, {\Delta_{21}}^2-2\frac{U{(t_2-t_1)}^2}{{r_{1}}})
\\
\Delta_2&=\frac{- B \pm \sqrt{B^2-4AC}}{2 A}
\end{split}\end{aligned}$$
Two roots of $\Delta_2$ lead to the extreme solutions of angles at which the vector ${v_1}$ of an asteroid is equal to escape velocity:
Based on figure \[fig:vec\] and cosine rules and law of sines:
$$\begin{aligned}
\begin{split}
s_1&=\sqrt{\Delta_{2}^2+\Delta_{21}^2-2\Delta_{21}\Delta_{2}\cos{\theta}}\\
\cos{\alpha}&=\frac{{s_{1}}^2 + {\Delta_{21}}^2 - {\Delta_{2}}^2}{2{s_1}{\Delta_{21}}} \\
\sin{\alpha}&= \frac{{\Delta_2} \sin{\theta}}{{s_1}} \\
\alpha&=\arctan{\left(\frac{\sin{\alpha}}{\cos{\alpha}}\right)}
\end{split}\end{aligned}$$
Having the topocentric ($\vec{r_1}$) and heliocentric ( $\vec{\Delta_1}$) distance, the mean V-band magnitude $V_1$ and geometry ($\alpha$), we are now able to compute the object’s absolute magnitude $H$ (Line \[ess:H mag\]) of Algorithm \[alg: essential\] by equation:
$$H=V_1-5\log(\Delta_1 r_1) +f(\Phi_1, G)$$
where the phase angle $\Phi_1$ is the angle between $\vec{\Delta_1}$ and $\vec{r_1}$ and $G$ is the slope parameter. The form of the phase function $f(\Phi,G)$ is based on @1989aste.conf..524B and $G=0.15$ is assumed.\
In the next step (Line \[ess:angle loop\]) of Algorithm \[alg: essential\], the algorithm picks angles randomly for a selected topocentric distance $D$ in a range between the two extremes of $\alpha$ that represent a bound elliptical orbit.
With any valid $\alpha$ and $D$, we can derive the velocity vector, again by using the rule of sines (Figure \[fig:vec\] - bottom):
$$\begin{aligned}
\begin{split}
\Delta_2=\frac{\Delta_{21}\sin{\alpha}}{\sin{(\pi-\alpha-\theta)}}
\end{split}\end{aligned}$$
The heliocentric velocity vector $\vec{v}$ in ecliptic coordinates is $$\vec{v}=\frac{{\Delta_2} \vec{d_2} - \vec{\Delta_{21}} }{t_2-t_1}$$
Having $\vec{r_1}$, $\vec{v}$ at a given epoch, we have the orbit defined through its state vector (Line \[ess:state vec\] of Algorithm \[alg: essential\]) which is easy to convert into a set of Keplerian elements (Line \[ess:elements\] of Algorithm \[alg: essential\]).
The entire procedure is repeated for a range of assumed distances, $D_j$ that lead to set of *unique* position vectors, $\vec{r_j}$, and velocity vectors, $\vec{v_j}$. Example of variant orbits generated by [*digest2*]{} in a phase space of $\alpha$ and $\Delta_1$ is shown in Figure \[fig:Trojan\].
![Grid of bound variant orbits generated by [*digest2*]{} for an NEO (left), Main Belt (center) and Centaur (right) tracklet. The asterisks depict the true position.[]{data-label="fig:Trojan"}](neo_grid.png "fig:") ![Grid of bound variant orbits generated by [*digest2*]{} for an NEO (left), Main Belt (center) and Centaur (right) tracklet. The asterisks depict the true position.[]{data-label="fig:Trojan"}](mb_grid.png "fig:") ![Grid of bound variant orbits generated by [*digest2*]{} for an NEO (left), Main Belt (center) and Centaur (right) tracklet. The asterisks depict the true position.[]{data-label="fig:Trojan"}](cen_grid.png "fig:")
[*digest2*]{} Software
======================
Detailed Historical Evolution {#app:HIST}
-----------------------------
In Section \[s:HIST\] we provided a brief outline of the historical development of the [*digest2*]{} algorithm. In Table \[t:Versions\] of this appendix, we provide the interested reader with a more detailed, chronologically-ordered description of the key changes made to the [*digest2*]{} code.
[l|r|l]{}\[hpt\]
\[t:Versions\] pre-2002 & 0.0223 & Original FORTRAN 77 code (“`223.f`”) built on [[Pangloss]{}]{}\
Aug 2005 & 0.1 & MIT-style license, catalog-reduced model.\
Sep 2005 & 0.2 & Bin search algorithm (App. \[app:bin\_search\])\
Feb 2010 & 0.3 & Pan-STARRS S3M model, multiple orbit classes, improved scoring (§ \[s:POPN\]). Parallelized across multiple CPU cores.\
Feb 2010 & 0.4 & Supported space-based observations.\
Feb 2010 & 0.5 & Great circle RMS output. Stand-alone GCR utility.\
Mar 2010 & 0.6 & Improved stand-alone GCR utility.\
Nov 2010 & 0.7 & Orbit classes H18, H22, bug fixes.\
Apr 2011 & 0.8 & More flexible output, new config options. Removed stand-alone GCR utility, bug fixes.\
Apr 2011 & 0.9 & Bug fixes.\
Apr 2011 & 0.10 & Bug fixes.\
May 2011 & 0.11 & Endpoint synthesis (§ \[s:EndPoint\]). Observational Error (§ \[s:obs err\]).\
May 2012 & 0.12 & Bug fixes.\
Jun 2012 & 0.13 & Bug fixes.\
Jan 2013 & 0.14 & Bug fixes.\
Feb 2014 & 0.15 & Allow more obscodes.\
Feb 2015 & 0.16 & Command line option to limit parallelism.\
Jun 2015 & 0.17 & CSV model, bug fixes.\
Jun 2015 & 0.18 & Minor usability improvements. Bug fixes.\
Aug 2017 & 0.19 & Bug fixes, including a serious uninitialized data problem.\
Software Availability and Usage {#app:software}
-------------------------------
The [*digest2*]{} source code and the documentation is freely available at <https://bitbucket.org/mpcdev/digest2/overview>. The download section contains a zip archive of [*digest2*]{} and the population model tar achive *d2model*.
The synthetic solar system model binning and reduction source code *MUK* is freely available at <https://bitbucket.org/mpcdev/d2model/src/master/muk.c>.
The [*digest2*]{} code can be executed directly after its compilation using the supplied input observation file (sample.obs) and configuration file (MPC.config) as follows:
digest2 -c MPC.config sample.obs
Further detailed instructions for the operation of [*digest2*]{} can be found in <https://bitbucket.org/mpcdev/digest2/src/master/OPERATION.md>.
Binary Search Algorithm for Population Bins {#app:bin_search}
===========================================
The initial algorithm (version 0.1) for constructing class scores was as in Algorithm \[alg 0.1\].
\[alg 0.1\] … score $\gets 100 * neo / (neo + mb)$ \[01:score\]
On line \[01:int\] the function $Interesting$ tests if candidate elements are in the class of interest. The single class of interest defined in [*digest2*]{} versions-0.1 and -0.2 corresponded to the Minor Planet Center’s definition of “interesting” which included not just NEOs ($q < 1.3$) but also high eccentricity ($e > 0.5$) and high inclination ($i > 40$) orbits which would also lead to a discovery MPEC. The current code uses the classes described in Table \[t:classes\] of Appendix \[app:tables\].
Version-0.2 contained an improved score formula. The goal of the new score formula was to change the computation of the variables neo and mb (Algorithm \[alg 0.1\] lines \[01:acc neo\] and \[01:acc mb\]) to be more like, $$\begin{aligned}
neo &= \text{ sum of bin populations over all represented NEO bins}\\
mb &= \text{ sum of bin populations over all represented non-NEO bins},
\end{aligned}
\label{e:binary1}$$ where a best effort is made to locate all the population bins represented by the input arc. The reasons for this change ultimately stem from a desire to (a) avoid double-counting population bins, and (b) avoid omitting population bins due to the use of overly large distance and/or angle step-sizes in Algorithm \[alg 0.1\].
Per-class binning enables enhancements to the bin tagging and scoring algorithms. In the current version of [*digest2*]{}, separate bin tags are accumulated per class. Further, for each class, two sets of tags are accumulated per class. An in-class tag is set for a bin when a candidate orbit meets the class definition. A separate out-of-class tag is set otherwise. Two population sums are then computed for each class, a sum of bin populations with in-class tags and sum of bin populations with out-of-class tags. The first represents the sum of class bin populations where a candidate orbit of class was found. The second represents the sum of out-of-class bin populations where a candidate orbit out of class was found.
Computations at a single distance follow algorithm \[alg: essential\] steps \[ess:dist vec\] through \[ess:H mag\], but then call a recursive function to subdivide the angle space between $\alpha_1$ and $\alpha_2$.
At each call, a state vector and then orbital elements are computed, but then rather than accessing the indicated bin population, the corresponding tag is simply checked (in the distance-specific tags). If the tag was already set, subdivision ends and the recursive function returns. If the tag was not set, then the angle space is subdivided further and the recursive function is called for each subdivision.
The distance, $D$, is subdivided similarly with a separate recursive function. On each call, computations are performed for a single distance as just described, then the distance-specific tags are checked. If all bins tagged at that distance had previously been tagged in the overall result, the recursive function simply returns. If new bins were tagged, they are merged into the overall result, the distance space is subdivided further, and the recursive (distance) function is called for each subdivision. After all subdivision functions return, the formulas above for $neo$ and $mb$ are computed as described in Algorithm \[tag acc\].
Tabulated Values and Settings for [*digest2*]{} {#app:tables}
===============================================
In Table \[t:tabUnc\] we list the assumed astrometric uncertainties adopted in the latest version of the [*digest2*]{} code. The values are set in the file [MPC.config]{} and can be altered by the user.
=0.11cm
------------- --------------- ------------- -------------
observatory astrometric observatory astrometric
code uncertainty code uncertainty
106 0.4“&D29&0.5”
291 0.4“&E12&0.5”
568 0.1“&F51&0.2”
691 0.4“&F52&0.2”
703 0.7“&G96&0.3”
704 0.7“&H15&0.5”
A50 0.5“&J75&0.4”
C51 0.7 other 1.0"\
------------- --------------- ------------- -------------
: Astrometric uncertainties currently used by [*digest2*]{} by the MPC and [*digest2*]{} code, based on long-term statistics[^24].
\[t:tabUnc\]
[*digest2*]{} is configurable with keywords that allow control of the output and setup of some input variables. The keywords are set in the same file (MPC.config) as the astrometric uncertainties (Table \[t:tabUnc\]). The available keywords are described in Table \[t:keywords\].
[l|l]{} keyword & meaning\
headings & show heading\
noheadings & hide heading\
rms & show residual RMS from linear motion along a great circle in arc-seconds\
norms & do not show RMS\
raw & show raw [*digest2*]{} (Section \[s:POPN:Full\])\
noid & show noid [*digest2*]{} (Section \[s:POPN:Unknown\])\
repeatable & random seed is defined and always the same (Section \[s:random\])\
random & pseudo-random generation of $\alpha$ and resulting [*digest2*]{} (Section \[s:random\])\
obserr & fixed astrometric uncertainty set to a value in arc seconds\
poss & show other non-zero resulting scores in addition to specified classes [*digest2*]{}\
any orbit class(es) defined in Table \[t:classes\] &e.g. NEO\
user defined astrometric uncertainty (arc seconds) per MPC observatory code&e.g. obserrF51=0.2\
\
\[t:keywords\]
The orbit classes used by [*digest2*]{} are defined by the orbital elements and absolute magnitudes listed in Table \[t:classes\].
[l|l|l]{} orbit class & abbreviation & definition in keplerian elements and absolute magnitude\
MPC Interesting & Int & q < 1.3 || e >= 0.5 || i >= 40 || Q > 10\
Near-Earth Object & NEO & q < 1.3\
Large Near-Earth Object & N18 & q < 1.3 & H<18.5\
Intermediate-size Near-Earth Objects & N22 & q < 1.3 & H<22.5\
Mars-Crossers&MC & q < 1.67 & q >= 1.3 & Q > 1.58\
Hungarias & Hun &a < 2 & a > 1.78 & e < 0.18 & i > 16 & i < 34\
Phocaeas &Pho & a < 2.45 & a > 2.2 & q > 1.5 & i > 20 & i < 27\
Inner Main Belt & MB1 & q > 1.67 & a < 2.5 & a > 2.1 & i < ((a - 2.1) / 0.4) \* 10 + 7\
Pallas family & Pal & a < 2.8 & a > 2.5 & e < 0.35 & i > 24 & i < 37\
Hansas &Han& a < 2.72 & a > 2.55 & e < 0.25 & i > 20 & i < 23.5\
Central Main Belt &MB2 &a < 2.8 & a > 2.5 & e < 0.45 & i < 20\
Outer Main Belt & MB3 & e < 0.4 & a > 2.8 & a < 3.25 & i < ((a - 2.8) / 0.45) \* 16 + 20\
Hildas & Hil & a > 3.9 & a < 4.02 & i < 18 & e < 0.4\
Jupiter Trojans & JTr & a > 5.05 & a < 5.35 & e < 0.22 & i < 38\
Jupiter Family Comets & JFC & q > 1.3 & $T_{J} > 2$ & $T_{J} < 3$\
\
\
\[t:classes\]
Population bins {#app:bins}
===============
In Figure \[fig:Population\] we illustrate the population model used by [*digest2*]{} (see Section \[s:POPN\] above for further description).
\[fig:Population\]
Example Tracklet {#app:Hungaria tracklet}
================
We provide an example of a tracklet taken from a a typical Hungaria-group asteroid. The data is presented in a standardized 80-column format[^25] - Minor Planet Packed Designation (K18B01E), Mode of Observation (“C” for CCD), Time of observations (Year, Month, Day), Right Ascension (Hours, Minutes, Seconds) and Declination (Degrees, Minutes, Seconds), Magnitude, Band (“w”), Catalog Code (“U”), Packed Reference (“$\sim$2VXl”) and Observatory Code (“F51”).
K18B01E* C2018 01 17.42780 08 01 58.347+41 29 48.20 21.4 wU~2VXlF51
K18B01E C2018 01 17.43919 08 01 57.028+41 29 44.58 21.2 wU~2VXlF51
K18B01E C2018 01 17.46196 08 01 54.365+41 29 37.55 21.2 wU~2VXlF51
natexlab\#1[\#1]{}\[1\][[\#1](#1)]{} \[1\][doi: [](http://doi.org/#1)]{} \[1\][[](http://ascl.net/#1)]{} \[1\][[](https://arxiv.org/abs/#1)]{}
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[^1]: Section 321 of the NASA Authorization Act of 2005 (Public Law No. 109-155)
[^2]: There is no known history of a digest1; this is not version 2 of a program. Also nothing is known about the origin or meaning of the name. The program name is simply [*digest2*]{}.
[^3]: <http://www.minorplanetcenter.net/iau/NEO/toconfirm_tabular.html>
[^4]: https://minorplanetcenter.net/iau/info/ObsFormat.html
[^5]: The justification for the particular choice of 17th and 83rd percentiles is unknown to the surviving authors. Moreover, we emphasize that future versions of the [*digest2*]{} code will improve on the handling of measurement uncertainty and tracklet construction.
[^6]: [**<https://minorplanetcenter.net/iau/info/SatelliteObs.html>**]{}
[^7]: [[muk]{}]{} is available from <https://bitbucket.org/mpcdev/d2model/src/master/muk.c>
[^8]: `astorb.dat`, and is available from <ftp://ftp.lowell.edu/pub/elgb/astorb.dat.gz>
[^9]: <https://bitbucket.org/mpcdev/digest2/overview>
[^10]: <https://bitbucket.org/mpcdev/digest2/downloads/>
[^11]: Unless otherwise indicated, in the remainder of this work we will understand ${D_2}{}$ to refer to the NEO “no id” (NID) [*digest2*]{} score.
[^12]: Since 2012, $D_{2,crit}=65$.
[^13]: A *discovery* tracklet is the first reported tracklet of an object
[^14]: Potentially Hazardous Asteroids have MOID $\leq0.05{\,\mathrm{au}}$ and $H\geq22$.
[^15]: Minimum Orbit Intersection Distance with the Earth.
[^16]: <https://cneos.jpl.nasa.gov/ca/>
[^17]: \(3753) Cruithne, (164207), (277810), (419624), (469219), 2002 AA29, 2006 RH120, 2010 TK7
[^18]: <https://minorplanetcenter.net/iau/info/ADES.html>
[^19]: <https://minorplanetcenter.net/iau/MPCORB.html>
[^20]: <https://minorplanetcenter.net/iau/info/UValue.html>
[^21]: At the time of submission, the [*digest2*]{} team are currently diagnosing the cause of a rare output-formatting error that appears to afflict approximately 1 in every 50,000 evaluation attempts when running [*digest2*]{} in a multi-core environment. We anticipate that the solution of this issue will lead to a future version release.
[^22]: <https://minorplanetcenter.net/iau/info/ObservatoryCodes.html>
[^23]: $k=0.01720209895\,au^{3/2}\,day^{-1}\,(solar\,mass)^{-1/2}$
[^24]: <https://minorplanetcenter.net/iau/special/residuals.txt>
[^25]: <https://minorplanetcenter.net/iau/info/ObsFormat.html>
|
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---
abstract: 'We present a two-step procedure to probe hotspots of plasmon-enhanced Raman scattering with carbon nanotubes. Dielectrophoretic deposition places a small carbon nanotube bundle on top of plasmonic Au nanodimer. After ’pre-characterising’ both the nanotubes and dimer structure, we subsequently use the tip of an AFM to push the bundle into the plasmonic hotspot located in the $25\,$nm wide dimer gap, characterize its location inside the gap, and observe the onset of plasmon-enhanced Raman scattering. Evidence for the activation of the carbon nanotube’s double-resonant D-mode by the near-field of the plasmonic hotspot is discussed.'
address: 'School of Materials, The University of Manchester, Manchester M13 9PL, UK'
author:
- 'Sebastian Heeg[^1], Nick Clark, Aravind Vijayaraghavan'
title: 'Probing hotspots of plasmon-enhanced Raman scattering by nanomanipulation of carbon nanotubes'
---
[*Keywords: plasmon-enhanced Raman scattering, carbon nanotubes, plasmonic hotspot, nano-manipulation*]{}\
Introduction
============
Metallic nanostructures enable the localization of electromagnetic waves into nanoscale volumes far smaller than the wavelength of light at optical frequencies [@Maier:2007wq; @Novotny:2012tia]. The electromagnetic wave couples to the collective excitation of electrons in the metal nanostructure called localized surface plasmon resonance (LSPR). This leads to a strong enhancement of the local near-field at the metal surface. When two nanostructures are closely spaced, coupling of the confined LSPRs (at certain polarizations) leads to a collective excitation across both, generating particularly strong field enhancement in the gap between them. Such a region is termed plasmonic hotspot. The field enhancement varies considerably inside and around such a plasmonic hotspot and may differ by several order of magnitude over tens of nanometers [@LeRu:2012fp; @Maier:2007wq; @Novotny:2012tia; @Gramotnev:2010jk].
One of the most spectacular applications of LSPRs is plasmon-enhanced Raman scattering (PERS), where the incident light and that inelastically scattered from an object inside a plasmonic hotspot is greatly enhanced [@Fleischmann:1974jh; @Willets:2007ko]. PERS has enabled the detection of single molecules and offers great potential for studying the fundamental aspects of light-matter interaction as well as biological and chemical sensing applications. [@Kneipp:1997ua; @Nie:1997dl; @ISI:000241099100013; @Roelli:2015bv; @Mueller:2016kya; @Schmidt:2016ek; @Jorio:2017bl; @2016Sci...354..726B; @Sharma:2012vp; @Halas:2011cz; @Stockmann:2015ie]
A recurring problem in quantifying PERS are small geometric variations between plasmonic structures of the same design, which may strongly alter the enhancement experienced by a Raman scatterer that is nominally located at the same position inside a corresponding plasmonic hotspot [@LeRu:2009id; @Darby:2015kn]. It is therefore desirable to probe the PERS enhancement at different locations in and around one plasmonic hotspot using the same Raman scatterer. *Kusch et al.* recently suggested to read out the PERS enhancement at a plasmonic hotspot via the Raman signal of a sharp Si-tip of a scanning near-field optical microscope [@Kusch:2017de]. The spatial extension of the tip, however, masks subtle variations of the enhancement on the nanometer scale. Similar restrictions apply to molecules, the traditional probe for PERS, because their exact location and orientation inside a plasmonic hotspot are impossible to control and, more importantly, cannot be altered. A nanoscale Raman scatterer that probes the enhancement in and around a plasmonic hotspot is therefore highly desirable.
Carbon nanotubes (CNTs) overcome some of these limitations and have recently emerged as a promising alternative to probe and quantify plasmon-enhanced Raman scattering on the nanoscale as their one-dimensional nature allows us to experimentally obtain the exact location and orientation within a plasmonic hotspot [@Assmus:2007fw; @Scolari:2008kt; @Cancado:2009vu; @Heeg:2014cn; @Heeg:2014kx; @Bauml:2017td; @Paradiso:2015jx; @Mueller:2017hb]. Interfacing CNTs with rationally designed hotspots, on the other hand, remains a challenge. We recently demonstrated the benefit of using directed dielectrophoretic assembly (DEP) to place carbon nanotubes precisely into the plasmonic hotspot at the $\sim25\,$nm gap of gold nanodimers, Fig. \[FIG:DEP\_YIELD\](a), and observed a $10^3-10^4$ enhancement of the Raman intensity [@Heeg:2014cn; @Heeg:2014kx]. More importantly, CNTs are perhaps the only nanoscale object that is able to probe PERS enhancement at different locations in a plasmonic hotspot, as the location and orientation of a nanotube can be purposefully alterted by nanomanipulation. [@Falvo1997fa; @Hertel:1998gs]
Here we demonstrate a two-step scheme that combines dielectrophoretic deposition of carbon nanotubes with tip-based nanomanipulation to probe plasmon-enhanced Raman scattering outside and inside the plasmonic hotspot of a gold nanodisc dimer. With the tip of an atomic force microscope (AFM) we push a small carbon nanotube bundle, placed on one of the discs forming the dimer by dielectrophoresis, into the dimer gap. We determine the position of the CNT bundle inside the hotspot and verify the onset of plasmon-enhanced Raman scattering by an $100$-fold Raman enhancement of the nanotube’s G-mode, an inverted polarization behaviour and the spatial localization of the Raman signal after nano-manipulation. Beyond quantifying plasmon-enhanced Raman scattering, the proposed scheme will enable the realization of nanotube-nanoplasmonic experimental systems that were previously not accessable.
![(a) Schematic of CNT at the plasmonic hotspot in a nanodimer gap. (b) AC voltage drives the CNTs in solution to deposit between the electrodes, where the plasmonic dimers are located. Based on Ref. [@Heeg:2014cn], DEP has a $<5\%$ yield for nanotubes passing through the gap, as in (c), and $\sim25\%$ for nanotubes crossing one of the discs forming the dimer in (d).[]{data-label="FIG:DEP_YIELD"}](PushT_Fig1_v3.pdf){width="8.5cm"}
Experimental
============
The sample fabrication and dielectrophoretic deposition process follows the procedures described in Refs. [@Heeg:2014cn] and [@Heeg:2014kx]. In short, arrays of dimers and electrodes for DEP were fabricated on a SiO$_2$ ($90\,$nm) on Si substrate by electron beam lithgraphy using a LEO 1530 Gemini FEG SEM and a Raith Elphy Plus Lithography System. Metallization was carried out by evaporating $5\,$nm Cr + $40\,$ nm Au followed by lift-off in an ultrasonic bath. The dimers are located between electrode pairs (distance $\sim1\,\upmu$m) with sharp tips for directed dielectrophoretic deposition. During directed dielectrophoretic deposition, a droplet of ultrapure, unsorted CNTs (www.nanointegris.com) in aqueous solution is placed on top of the arrays and a kHz AC voltage is applied between the electrodes [@Krupke:2003cl; @Vijayaraghavan:2009gv]. Initial AFM characterization after DEP was performed using an Park Systems XE 150 AFM. Nanomechanical manipulation was performed using a Bruker Dimension Icon AFM using Bruker TAP150A probes with cantilevers with a nominal stiffness of $5\,$N/m. All imaging was performed in non-contact mode. Raman characterization was performed using Horiba Yobin Ivon XploRa ($\lambda=532\,$nm) and Witec alpha300 ($\lambda=633\,$nm) single grating spectrometers using $100\times$ objectives (NA $0.9$), and piezo stages for spatial mapping. To measure the polarization, we rotated the sample by $90^{\circ}$. We used integration times up to $10\,$s and laser powers between $1\,$mW ($532\,$nm) and $50\,\upmu$W ($633\,$nm). The luminescence background from the Au nanostructures was subtracted for all Raman spectra.
Results and Discussion
======================
We fabricated arrays of dimer structures consisting of two closely spaced nanodiscs by electron beam lithography on top of a SiO$_2$/Si substrate (see Methods). The dimer’s dimensions (cylinders with diameter $\sim100\,$nm, height $45\,$nm, separated by a gap of $\sim25\,$nm) were chosen to provide resonant enhancement for $\lambda=633\,$nm excitation polarized along the dimer axis [@Heeg:2013di; @Heeg:2014cn; @Wasserroth:2018bm]. The strongest coupling to the high-intensity near-field occurs for a nanotube placed inside the plasmonic hotspot at the nanoscale gap of the dimer [@Maier:2007wq; @Novotny:2012tia]. We realized such an interface for plasmon-enhanced Raman scattering of CNTs using directed dielectrophoretic deposition from solution as described in detail in Refs. [@Heeg:2014cn] and [@Heeg:2014kx]. Dielectrophoretic forces drive the nanotube to deposit between electrodes where we have placed the plasmonic dimers, c.f. Fig. \[FIG:DEP\_YIELD\](b). Ideally the CNT connects the electrode tips in a straight line and passes through the gap as shown in Fig. \[FIG:DEP\_YIELD\](c). This ideal configuration, however, has a yield below $5\%$. A scenario where the nanotube does not connect the electrode tips directly and crosses one of the discs forming the dimer as shown Fig. \[FIG:DEP\_YIELD\](d) is much more likely ($\sim25\%$) but does not provide significant Raman enhancement. It is therefore the ideal starting point for the two step procedure proposed in this work, because it allows us to characterize the CNT and its Raman signatures before it is interfaced with the plasmonic hotspot at the dimer gap.
![(a) AFM phase image of CNT-B overlaid with its integrated G-peak intensity for $532\,$nm and P$_Y$. The scale bar is $500\,$nm and the intensity normalized to the maximum intensity. (b) Raman spectrum acquired at the dimer. D-mode and G-mode of CNT-B and the second order Si peak are labelled in the spectrum.[]{data-label="FIG:CNT-B_532"}](PushT_Fig2_v3.pdf){width="8.5cm"}
We show an AFM phase image of such an initial scenario in Fig. \[FIG:CNT-B\_532\](a). A small CNT-bundle (height $\sim 7\,$nm, labelled CNT-B) crosses the left disc of a plasmonic dimer. The phase image is overlaid with a spatial map of the nanotube G-band just below $1600\,$cm$^{-1}$, the most prominent Raman feature in CNTs [@Thomsen:2007vc]. The corresponding Raman spectrum, taken with the laser centered on the dimer, is shown in Fig. \[FIG:CNT-B\_532\](b). The D-mode at $\sim1350\,$cm$^{-1}$ indicates the presence of defects. It also appears in Raman spectra of CNTs dropcasted from the as purchased solution on a bare SiO$_2$/Si substrate and is therefore not caused by DEP [@Heeg:2014cn; @Heeg:2014kx]. Both the energy of the excitation ($\lambda=532\,$nm) and its polarization P$_Y$ (perpendicular to the dimer) prevent plasmonic enhancement from the dominant dipolar LSPR of the dimer.
The spatial distribution of the integrated G-mode intensity $I_G(P_Y)$ in Fig. \[FIG:CNT-B\_532\](a) matches that of CNT-B. We attribute the increased intensity around the dimer disc to a maximized overlap between CNT-B and the laser spot in combination with minor plasmonic enhancement from the disc’s LSPR or due to electric field line crowding at the disc edges [@Maier:2007wq]. We did not observe a considerable signal for P$_X$, as the strong antenna effect in CNTs suppresses Raman scattering for polarizations perpendicular to the tube axis [@Thomsen:2007vc; @Reich:2009wo]. For an excitation wavelength that overlaps with the dimer resonance ($\lambda=633\,$nm), no Raman signal within our detection threshold was observed for both P$_X$ and P$_Y$.
![(a) AFM topography of CNT-B crossing the left dimer disc. (b) AFM height profiles connecting the dots of the corresponding colour in (a). The grey dashed line indicates the lateral x-position of CNT-B on top and next to the dimer dics. (c) The downward force F$_D$ holds down the tip while it is moved across the dimer, thereby moving CNT-B into the plasmonic hotspot at the gap. (d) AFM topography after nanomanipulation. (e) Height profiles confirming that CNT is located in the gap.[]{data-label="FIG:PUSHIT"}](PushT_Fig3_v2.pdf){width="8.5cm"}
Before we discuss the nano-manipulation of CNT-B, we investigate more closely the topography of CNT-B and its interface with the dimer structure. A high-resolution AFM topography image is shown in Fig. \[FIG:PUSHIT\](a). It confirms that CNT-B crosses the left nanodisc. Height profiles parallel to the dimer axis are shown in Fig. \[FIG:PUSHIT\](b). They connect the dots of corresponding colour in Fig. \[FIG:PUSHIT\](a), and are offset by $5\,$nm with respect to each other for clarity. On top of the dimer (green), CNT-B sits close to the edge of the left disc as indicated by the dashed vertical line. The AFM tip is insufficiently sharp to probe between the dimers. The tip sides as depicted in Fig. \[FIG:PUSHIT\](b) are angled at $15^{\circ}$ deg (left) and $25^{\circ}$ (right) to the vertical. This leads to a slight asymmetry in the observed height profile at the sides of the discs. Close to the dimer edge (red), the CNT-B adheres to the disc wall and does not show a topographic feature. The tip, on the other hand, reaches to the bottom of the dimer gap. At distances of $20\,$nm (blue) and $100\,$nm (orange) away from the dimer, CNT-B adheres to the substrate at the same lateral x-position.
To move CNT-B into the gap, we ramped the tip into a chosen point on the dimer disc - to the left of CNT-B/dashed lines in Fig. \[FIG:PUSHIT\](b,c) - until a deflection setpoint of $50\,$nm was reached. This value corresponds to a downward force $F_D$ of approximately $200-300\,$nN. While maintaining the downward force, the tip was slowly ($50\,$nm/s) moved laterally $~150\,$nm across the dimer gap to a second chosen point on the right nanodiscs, bottom Fig. \[FIG:PUSHIT\](c), and retracted. The tip was then immediately used in tapping mode to image the dimer and CNT, Fig. \[FIG:PUSHIT\](d). We achieved the same resolution as in Fig. \[FIG:PUSHIT\](a) prior to the nanomanipulation – the observed profile of CNTB is the real profile convoluted with the tip apex profile – which shows that the tip did not measureably blunt while in contact with the surface.
The topography in Fig. \[FIG:PUSHIT\](d) reveals that CNT-B was moved to the right such that it passes through the dimer gap. We will now deduce the exact configuration of the nanotube in the gap by comparing the corresponding height profiles in Fig. \[FIG:PUSHIT\](e) to those take before moving CNT-B in Fig. \[FIG:PUSHIT\](b). On top of the dimer (green), CNT-B has been removed from previous location (dashed line). At the dimer edge (red), CNT-B appears as a feature in the gap at a height of $\sim25\,$nm, preventing the tip from imaging the gap region below. Around $20\,$nm away from the dimer (blue), CNT-B has been shifted to the right to accommodate for its new location at the dimer gap without building up strain. Further away from the dimer (gap), no changes in the tube’s position have occured.
The observed topography before and after nanomanipulation indicates that CNT-B is moved to a position suspended at a height of $25\,$nm in the gap. At first, the tip pushes CNT-B across the left disc’s surface, thereby overcoming adhesion of the CNT-B on the sides of the disc and on the substrate. The tip then enters the dimer gap to maintain F$_D$ and carries the tube in the gap. The tip then retracts from the gap while the nanotube slips off and remains inside the gap. The position at CNT-B at a height of around $25\,$nm is in good agreement with the penetration depth of our tip with $25^{\circ}$ side angle and a gap width of $25\,$nm. Note that during imaging in tapping mode, along the dimer axis the tip only reaches $10\,$nm into the gap, Fig. \[FIG:PUSHIT\](e), and is therefore unable to probe CNT-B. This limitation also applies for mapping out the enhancement arising from a plasmonic dimer through the Raman signal of a Si-tip mounted on a scanning probe microscope as recently suggested by *Kusch et al.* [@Kusch:2017de].
. The scale bar in (b) and (c) is $200\,$nm.[]{data-label="FIG:RAMAN_ENH"}](PushT_Fig4_v2.pdf){width="8.5cm"}
The observation of plasmon-enhanced Raman scattering from CNT-B after nanomanipulation confirms that we have successfully interfaced CNT-B with the plasmonic hotspot in the dimer gap. Figure \[FIG:RAMAN\_ENH\](a) shows Raman spectra of CNT-B measured on the dimer with $\lambda=633\,$nm for both P$_X$ (red) and P$_Y$ (blue). The Raman intensity for P$_X$ dominates the spectrum, and is a clear sign of enhancement due to near-field from the dipolar LSPR of the dimer structure. The signal is much stronger than for the non-resonant case ($\lambda=532\,$nm), where we did not observe any signal for P$_X$. It is also stronger than for P$_Y$, along the CNT-axis, compare Fig. \[FIG:CNT-B\_532\](b). We have previously identified this inverted polarization behaviour as a clear sign of plasmonic enhancement from CNTs in dimer hotspots [@Heeg:2014cn; @Heeg:2014kx].
The spatial distribution of the G-mode signal for P$_X$ is shown in Fig. \[FIG:RAMAN\_ENH\](b). It is strongly localized in space and shows the point-like character of the Raman enhancement arising from hotspot the dimer gap. Compared to P$_X$, the relative magnitude of the enhanced signal for P$_Y$ and its localization, Fig. \[FIG:RAMAN\_ENH\](c), may seem surprising. It originates from minor plasmonic enhancement that couples favourably to CNT-B because the near field is polarized parallel to the axes of the tubes forming CNT-B. This is in stark contrast to the Raman intensity for P$_X$ that scales only with the projection of the near field polarization on the tube axis [@Heeg:2013di; @Heeg:2014cn], explaining the relative intensities in Fig. \[FIG:RAMAN\_ENH\](c). From the inverted polarization behaviour and signal localization presented in Figs. \[FIG:RAMAN\_ENH\](b) and (c), we estimate $2.7\times10^2$ as the lower limit of the enhancement factor of the G-mode, details see Ref. [@Heeg:2014cn].
The D-mode is the dominant feature in our PERS spectra, Fig. \[FIG:RAMAN\_ENH\](b). The ratio $I_D/I_G$ is typically regarded as a measure for the defect concentration in nanoscale graphitic material [@Thomsen:2007vc]. For CNT-B, it increased from $0.43$ ($532\,$nm, P$_Y$) before the nanomanipulation to $0.61$ ($633\,$nm, P$_Y$) and $1.22$ ($633\,$nm, P$_X$) in the presence of enhancement. For bulk quantities of our CNT starting material and for small CNTs bundles deposited directly into the dimer gap by DEP, the $I_D/I_G$ ratio increases upon changing the excitation wavelength from $532\,$nm to $633\,$nm, see Ref. [@Heeg:2014cn]. Therefore, the increase in $I_D/I_G$ for P$_Y$ is intrinsic to the CNTs used in this work and not caused by nanomanipulation. Did moving CNT-B, however, induce structural defects at tube segments now located at the plasmonic hotspot, which could explain the increase in $I_D/I_G$ from $0.61$ ($633\,$nm, P$_Y$) to $1.22$ ($633\,$nm, P$_X$)? Previous studies showed that moving CNTs by AFM does not damage the nanotubes [@Duan:2007gv; @Mussnich:2015iq]. *Yano* and co-workers moved carbon nanotubes with an AFM tip and subsequently characterized them by tip-enhanced Raman spectroscopy [@Yano:2013ft]. They did not observe any defects at the manipulated tube segments while probing with a spatial resolution of $20\,$nm, comparable to the localized nature of the enhancement in this work. As we were moving a bundle of CNTs, any stretching of the bundle due to nanomanipulation can be relaxed by interfacial sliding between tubes in the bundle which occurs at a much lower force than the introduction of structural defects within the nanotube [@Zhang:2012ht]. We further exclude radiation-induced damage because we used low powers ($50\,\mu$W) when acquiring the PERS spectra at $\lambda=633\,$nm.
In the light of these studies, we argue that the increase in the relative D-mode intensity in our PERS spectra is not caused by structural defects. Instead, it is a consequence of the strongly localized nature of the near-field in the plasmonic hotspot. A recent study observed a strong D-mode for defect-free graphene interfaced with a single plasmonic hotspot [@Ikeda:2013bs; @Wasserroth:2018bm]. It was suggested that the confinement of the light fields in space provides the necessary momentum to excite non-vertical optical transitions. This activates the double-resonant Raman process that gives rise to the D-mode in graphene without requiring a real defect for momentum conservation [@Thomsen:2000wf; @Reich:2004fb]. As the D-mode in carbon nanotubes has the same origin as in graphene [@Thomsen:2007vc; @Reich:2009wo], we argue that the strong D-mode in the PERS spectra of CNT-B also arises from the localized near-field at the hotspot. This interpretation explains the factor two difference in the experimentally observed $I_D/I_G$ ratios for P$_Y$ and P$_X$. The near-field is less localized for P$_Y$ than it is for P$_X$, resulting in a smaller – or more likely negligible – plasmon-induced contribution to the D-mode.
The two-step approach combining DEP and nanomanipulation presented here will find use beyond probing PERS from plasmonic dimers. It will allow to place carbon nanotubes into truly nanoscale gaps ($\leq5\,$nm) of bowtie antennas or plasmonic nanoclusters, which provide the electric field strengths necessary to observe phenomena such as the field gradient effect in PERS [@Ye:2012dv; @Kollmann:2014bba; @Aikens:2013gm]. The low yield of placing CNT directly in such narrow gaps by DEP makes this technique impractical. Using DEP to carry the nanotubes close to the hotspots followed by nanomanipulation as described here, however, will achieve the desired interface.
Conclusions
===========
In conclusion, we suggest directed dielectrophoretic deposition of carbon nanotubes followed by tip-based nanomanipulation to probe plasmon-enhanced Raman scattering from nanoscale plasmonic hotspots. With the tip of an AFM, we pushed a small carbon nanotube bundle $25\,$nm into the gap of an Au nanodimer. Its location at the hotspot was confirmed directly by AFM and indirectly by plasmon-enhanced Raman scattering. We observed a $10^2$ enhancement of the G-mode that was strongly localized in space in combination with an inverted polarization behaviour. Given the ability to characterize the nanotube beforehand, the strong D-mode after nanomanipulation was interpreted as a signature of plasmonic enhancement rather than being caused by structural defects while moving the nanotubes. Our two-fold scheme will allow the reliable placement of CNTs into nanoscale gaps of plasmonic structures and thereby enable experiments previsouly not accessable.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors acknowledge valuable discussions with S. Reich and thank A. Oikonomou for assistance in the sample fabrication. This research was supported by the Engineering and Physical Sciences Research Council (EPSRC) grants EP/K016946/1 (SH, AV) and EP/G03737X/1 (NC, AV).
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[^1]: Present address: Photonics Laboratory, ETH Zürich, 8093 Zürich, Switzerland
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'One of the main theories for explaining the formation of spiral arms in galaxies is the stationary density wave theory. This theory predicts the existence of an age gradient across the arms. We use the stellar cluster catalogues of the galaxies NGC 1566, M51a, and NGC 628 from the Legacy Extragalactic UV Survey (LEGUS) program. In order to test for the possible existence of an age sequence across the spiral arms, we quantified the azimuthal offset between star clusters of different ages in our target galaxies. We found that NGC 1566, a grand–design spiral galaxy with bisymmetric arms and a strong bar, shows a significant age gradient across the spiral arms that appears to be consistent with the prediction of the stationary density wave theory. In contrast, M51a with its two well–defined spiral arms and a weaker bar does not show an age gradient across the arms. In addition, a comparison with non–LEGUS star cluster catalogues for M51a yields similar results. We believe that the spiral structure of M51a is not the result of a stationary density wave with a fixed pattern speed. Instead, tidal interactions could be the dominant mechanism for the formation of spiral arms. We also found no offset in the azimuthal distribution of star clusters with different ages across the weak spiral arms of NGC 628.'
author:
- |
F. Shabani,$^{1}$[^1] E.K. Grebel,$^1$ A. Pasquali,$^1$ E. D’Onghia,$^{2,3}$ J.S. Gallagher III,$^2$ A. Adamo,$^4$ M. Messa,$^4$ B.G. Elmegreen,$^5$ C. Dobbs,$^6$ D.A. Gouliermis,$^{7,8}$ D. Calzetti,$^9$ K. Grasha,$^9$ D.M. Elmegreen,$^{10}$ M. Cignoni,$^{11,12,13}$ D.A. Dale,$^{14}$ A. Aloisi,$^{15}$ L.J. Smith,$^{16}$ M. Tosi,$^{13}$ D.A. Thilker,$^{17}$ J.C. Lee,$^{15,18}$ E. Sabbi,$^{15}$ H. Kim,$^{19}$ and A. Pellerin$^{20}$\
$^1$Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstr. 12–14, 69120 Heidelberg, Germany\
$^2$Dept. of Astronomy, University of Wisconsin- Madison, 475 N. Charter Street, Madison, WI 53076–1582, USA\
$^3$Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA\
$^{4}$Dept. of Astronomy, The Oskar Klein Centre, Stockholm University, Stockholm, Sweden\
$^{5}$IBM Research Division, T.J. Watson Research Center, Yorktown Hts., NY, USA\
$^{6}$School of Physics and Astronomy, University of Exeter, Exeter, United Kingdom\
$^{7}$Zentrum für Astronomie der Universität Heidelberg, Institut für Theoretische Astrophysik, Albert-Ueberle-Str.2, 69120 Heidelberg, Germany\
$^{8}$Max Planck Institute for Astronomy, Königstuhl17, 69117 Heidelberg, Germany\
$^{9}$Dept. of Astronomy, University of Massachusetts – Amherst, Amherst, MA 01003, USA\
$^{10}$Dept. of Physics and Astronomy, Vassar College, Poughkeepsie, NY, USA\
$^{11}$Dept. of Physics, University of Pisa, Largo B. Pontecorvo 3, 56127, Pisa, Italy\
$^{12}$INFN, Largo B. Pontecorvo 3, 56127, Pisa, Italy\
$^{13}$INAF - Osservatorio Astrofisico e di Scienza dello Spazio, Bologna, Italy\
$^{14}$Dept. of Physics and Astronomy, University of Wyoming, Laramie, WY, USA\
$^{15}$Space Telescope Science Institute, Baltimore, MD, USA\
$^{16}$European Space Agency/Space Telescope Science Institute, Baltimore, MD, USA\
$^{17}$Dept. of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD, USA\
$^{18}$Visiting Astronomer, Spitzer Science Center, Caltech. Pasadena, CA, USA\
$^{19}$Gemini Observatory, Casilla 603, La Serena, Chile\
$^{20}$Dept. of Physics and Astronomy, State University of New York at Geneseo, Geneseo, NY, USA\
\
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: Search For Star Cluster Age Gradients Across Spiral Arms of Three LEGUS Disk Galaxies
---
\[firstpage\]
galaxies: spiral galaxies: structure galaxies: indiviual: NGC 1566, M51, NGC 628
Introduction {#Introduction}
============
Understanding how spiral patterns form in disk galaxies is a long–standing issue in astrophysics. Two of the most influential theories to explain the formation of spiral structure in disk galaxies are named stationary density wave theory and swing amplification. The stationary density wave theory poses that spiral arms are static density waves [@Lindblad; @LS64]. In this scenario spiral arms are stationary and long–lived. The swing amplification proposes instead that spiral structure is the local amplification in a differentially rotating disk [@G; @JT; @SC; @Sellwood; @E11; @Elena13]. According to this theory indiviual spiral arms would fade away in one galactic year and should be considered transient features. Numerical experiments suggest that non–linear gravitational effects would make spiral arms fluctuate in density locally but be statistically long–lived and self–perpetuating [@Elena13].
To complicate the picture there is the finding that many galaxies in the nearby universe are grand–design, bisymmetric spirals. These galaxies may show evidence of a galaxy companion, suggesting that the perturbations induced by tidal interactions could induce spiral features in disks by creating localized disturbances that grow by swing amplification [@k; @B3; @Ga; @Elena16; @P16]. Some studies have been devoted to explore galaxy models with bar–induced spiral structure [@conto] and spiral features explained by a manifold [@conto; @A]. It is also possible that a combination of these models is needed to describe the observed spiral structure. We refer the interested reader to comprehensive reviews of different theories of spiral structure in [@DB14] and to [@Shu] for detailed explanations of the origin of spiral structure in stationary density wave theory.
The longevity of spiral structure can be tested observationally. In fact, in the stationary density wave theory, spiral arms are density waves moving with a single constant angular pattern speed. The angular speed of stars and gas equals the pattern speed at the corotation radius. Inside the corotation radius, material rotates faster than the spiral pattern. When the gas enters the higher–density region of spiral arms, it may experience a shock which may lead to star formation [@R69]. Consequently, the stars born in the molecular clouds in spiral arms eventually overtake the arms and move away from the spiral patterns as they age. This drift causes an age gradient across the spiral arms. If spiral arms have a constant angular speed, then we expect to find the youngest star clusters near the arm on the trailing side, and the oldest star clusters further away from the spiral arms inside the corotation radius [e.g., @M09]. Outside the corotation radius, the spiral pattern moves faster than the gas and leads to the opposite age sequence.
@DP10 carried out numerical simulations of the age distribution of star clusters in four different spiral galaxy models, including a galaxy with a fixed pattern speed, a barred galaxy, a flocculent galaxy, and an interacting galaxy. The results of their simulations show that in a spiral galaxy with a constant pattern speed or in a barred galaxy, a clear age sequence across spiral arms from younger to older stars is expected. In the case of a flocculent spiral galaxy, no age gradient can be observed in their simulation. Also in the case of an interacting galaxy, a lack of an age gradient as a function of azimuthal distance from the spiral arms is predicted. A simulation of an isolated multiple–arm barred spiral galaxy was performed by [@grand], who explored the location of star particles as a function of age around the spiral arms. Their simulation takes into account radiative cooling and star formation. They found no significant spatial offset between star particles of different ages, suggesting that spiral arms in such a spiral galaxy are not consistent with the long–lived spiral arms predicted by the static or stationary density wave theory. In a recent numerical study, [@D17] looked in detail at the spatial distribution of stars with different ages in an isolated grand–design spiral galaxy. They found that star clusters of different ages are all concentrated along the spiral arms without a clear age pattern.
A simple test of the stationary density wave theory consists of looking for a colour gradient from blue to red across spiral arms due to the progression of star formation. It is important to note that this method can be affected by the presence of dust. Several observational studies have tried to test the stationary density wave theory by looking for colour gradients across the spiral arms. In an early study of the ($B-V$) colours and total star formation rates in a sample of spiral galaxies with and without grand design patterns, [@Bruce86] found no evidence for an excess of star formation due to the presence of a spiral density wave, and explained the blue spiral arm colours as a result of a greater compression of the gas compared to the old stars, with star formation following the gas. [@M09] studied the colour gradients across the spiral arms of 13 SA and SAB galaxies. Ten galaxies in their sample present the expected colour gradient across their spiral arms.
A number of observational studies have used the age of stellar clusters in nearby galaxies as a tool to test the stationary density wave theory. @S09 studied the spatial distribution of 1580 stellar clusters in the interacting, grand–design spiral M51a from Hubble Space Telescope (HST) $UBVI$ photometry. They found no spatial offset between the azimuthal distribution of cluster samples of different age. Their results indicate that most of the young (age < 10 Myr) and old stellar clusters (age > 30 Myr) are located at the centers of the spiral arms. [@k10] also mapped the age of star clusters as a function of their location in M51a using HST data and found no clear pattern in the location of star clusters with respect to their age. Both above studies suggest that spiral arms are not stationary, at least for galaxies in tidal interaction with a companion. In order to study the spatial distribution of star–forming regions, [@Sanchez] produced an age map of six nearby grand–design and flocculent spiral galaxies. Only two grand–design spiral galaxies in their sample presented a stellar age sequence across the spiral arms as expected from stationary density wave theory.
In galaxies where spiral arms are long–lived and stationary as predicted by the static density wave theory, one would expect to find an angular offset among star formation and gas tracers of different age within spiral arms [@R69]. The majority of observational studies of the spiral density wave scenario have tried to examine such an angular offset [@vogel; @Rand]. [@Tam8] detected an angular offset between HI (a tracer of the cold dense gas) and 24 $\rm \mu$m emission (a tracer of obscured star formation) in a sample of 14 nearby disk galaxies. An angular offset between CO (a tracer of molecular gas) and H$\rm \alpha$ (a tracer of young stars) was detected for 5 out of 13 spiral galaxies observed by [@Egusa]. In another observational work, [@Foyle] tested the angular offset between different star formation and gas tracers including HI, $\rm H_{2}$, 24 $\rm \mu $m, UV (a tracer for unobscured young stars) and 3.6 $\rm \mu$m emission (a tracer of the underlying old stellar population) for 12 nearby disk galaxies. They detected no systematic trend between the different tracers. Similarly, [@F13] found no significant angular offset between H$\rm \alpha$ and UV emission in NGC 4321. [@L13] found a large angular offset between CO and H$\rm \alpha$ in M51a while no significant offsets have been found between HI, 21 cm, and 24 $\rm \mu$m emissions. These searches for offsets are based on the assumption that the different tracers represent a time sequence of the way a moving density wave interacts with gas and triggers star formation. [@Elmegreen2014] used the S$^4$G survey [@Sheth] and discovered embedded clusters inside the dust lanes of several galaxies with spiral waves, suggesting that star formation can sometimes start quickly.
In a recent observational study, [@S17] carried out a detailed investigation of a spiral arm segment in M51a. They measured the radial offset of the star clusters of different ages (< 3 Myr, and 3–10 Myr) and star formation tracers (HII regions and 24 $\rm \mu$m) from their nearest spiral arm. No obvious spatial offset between star clusters younger and older than 3 Myr was found in M51a. They also found no clear trend in the radial offset of HII regions and 24 $\mu$m. Similarly, [@chandar17] compared the location of star clusters with different ages (< 6 Myr, 6–30 Myr, 30–100 Myr, 100–400 Myr, and > 400 Myr) with the spiral patterns traced by molecular gas, dust, young and old stars in M51a. They found cold molecular gas and dark dust lanes to be located along the inner edge of the arms while the outer edge is defined by the old stars (traced with 3.6 $\rm \mu$m) and young star clusters. The observed sequence in the spiral arm of M51a is in agreement with the prediction from stationary density wave theory. [@chandar17] also measured the spatial offset between molecular gas, young (< 10 Myr) and old star clusters (100–400 Myr) in the inner (2.0–2.5 kpc) and outer (5.0–5.5 kpc) spiral arms in M51a. They found an azimuthal offset between the gas and star clusters in the inner spiral arm zone, which is consistent with the spiral density wave theory. In the outer spiral arms, the lack of such a spatial offset suggests that the outer spiral arms do not have a constant pattern speed and are not static. [@chandar17] found no star cluster age gradient along four gas spurs (perpendicular to the spiral arms) in M51a.
In conclusion, there have been numerous observational studies aiming to test the longevity of the spiral structure. In many cases, the conclusions show conflicting results and the nature of spiral arms is still an open question.
The main goal of this study is to test whether spiral arms in disk galaxies are static and long–lived or locally changing in density and locally transient. This work is based on the Legacy ExtraGalactic UV Survey (LEGUS)[^2] observations obtained with HST [@C15]. The paper is organized as follows: The survey and the sample galaxies are described in § \[The LEGUS Galaxy Samples\]. The selection of the star cluster samples is presented in § \[s3\]. We investigate the spatial distribution together with clustering of the selected clusters in § \[location\]. In § \[Azimutahl distribution\], we describe the results and analysis and how we measure the spatial offset of our star clusters across spiral arms. In § \[2arms\] we discuss whether the two spiral arms of our target galaxies have the same nature. In § \[chandra\], we use a non–LEGUS star cluster catalogue to measure the spatial offset of star clusters in M51a and we present our conclusions in § \[Summary\].
The sample galaxies {#The LEGUS Galaxy Samples}
===================
LEGUS is an HST Cycle 21 Treasury programme that has observed 50 nearby star–forming dwarf and spiral galaxies within 12 Mpc. High– resolution images of these galaxies were obtained with the UVIS channel of the Wide Field Camera Three (WFC3), supplemented with archival Advanced Camera for Surveys (ACS) imaging when available, in five broad band filters, $NUV\,(F275W)$, $U \,(F336W)$, $B \,(F438W)$, $V \,(F555W)$, and $I \,(F814W)$. The pixel scale of these observations is $ \rm 0.04^{\arcsec} \, pix^{-1}$. A description of the survey, the observations, the image processing, and the data reduction can be found in [@C15].
Face–on spiral galaxies with prominent spiral structures are interesting candidates to study stationary density wave theory. Therefore, three face–on spiral galaxies, namely NGC 1566, M51a, and NGC 628 were selected from the LEGUS survey for our study. The morphology, distance, corotation radius, and the pattern speed of each galaxy are listed in Table \[tab:properties of galaxies\]. The UVIS and ACS footprints of the pointings (red and yellow boxes, respectively) overlaid on Digitized Sky Survey (DSS) images of the galaxies are shown in Fig. \[fig:galaxies\] together with their HST red, green, and blue colour composite mosaics.
Galaxy Morphology D \[Mpc\] $\rm M_{\star} \, (M_{\sun})$ SFR (UV) $\rm(M_{\sun} \, yr^{-1}) $ $\rm R_{cr}$ \[$\mathrm{kpc}$\] $ \rm \Omega_{p}$ \[$\rm km\, s^{-1}\, \rm kpc^{-1}$\] Ref
---------- ------------ ----------- ------------------------------- -------------------------------------- --------------------------------- -------------------------------------------------------- ----- -- -- -- -- -- -- --
NGC 1566 SABbc 18 $\rm 2.7\times 10^{10}$ 2.026 10.6 23$\pm$2 1
M51a SAc 7.6 $\rm 2.4\times 10^{10}$ 6.88 5.5 38$\pm$7 2
NGC 628 SAc 9.9 $\rm 1.1\times 10^{10}$ 3.6 7 32$\pm$2 3
Column 1, 2: Galaxy name and morphological type as listed in the NASA Extragalactic Database (NED)\
Column 3: Distance\
Column 4: Stellar mass obtained from the extinction–corrected B–band luminosity\
Column 5: Star formation rate calculated from the GALEX far–UV, corrected for dust attenuation\
Column 6: Co–rotation radius\
Column 7: Pattern speed\
Column 8: References for the co–rotation radii and pattern speeds: 1- [@A04], 2- [@z4], 3- [@Sakhibov]\
NGC 1566
--------
NGC 1566, the brightest member of the Dorado group, is a nearly face–on (inclination = $\rm 37.3^{\circ}$) barred grand–design spiral galaxy with strong spiral structures [@Debra2]. The distance of NGC 1566 in the literature is uncertain and varies between 5.5 and 21.3 Mpc. In this study, we revised the distance of 13.2 Mpc listed in [@C15] and adopted a distance of 18 Mpc [@sabbi]. NGC 1566 has been morphologically classified as an SABbc galaxy because of its intermediate–strength bar. It hosts a low–luminosity active galactic nucleus (AGN) [@Combes]. The star formation rate and stellar mass of NGC 1566 are $ \rm 2.0 \, M_{\sun }yr^{-1}$and $\rm 2.7 \times 10^{10} \, M_{\sun }$, respectively within the LEGUS field of view [@sabbi].Two sets of spiral arms can be observed in NGC 1566. The inner arms connect with the star–forming ring at 1.7 kpc [@S15], which is covered by the LEGUS field of view (see Fig. \[fig:galaxies\], top panel). The outer arms beyond 100 arcseconds (corresponding to 8 kpc ) are weaker and smoother than the inner arms.
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[0.5]{} {width="1\linewidth"}
[0.50]{} {width="1\linewidth"}
[0.50]{} {width="1\linewidth"}
[0.50]{} {width="1\linewidth"}
[0.50]{} {width="1\linewidth"}
M51a
----
M51a (NGC 5194) is a nearby, almost face–on (inclination = $\rm 22 ^{\circ} $) spiral galaxy located at a distance of 7.6 Mpc [@Tonry]. It is a grand design spiral galaxy morphologically classified as SAc with strong spiral patterns [@Debra2]. M51a is interacting with a companion galaxy, M51b (NGC 5195). M51a has a star formation rate and a stellar mass of $\rm 6.9 \, M_{\sun } yr^{-1}$and $\rm 2.4 \times 10^{10}M_{\sun }$, respectively [@Lee; @Both]. Five UVIS pointings in total were taken through LEGUS observations: 4 pointings cover the center, the north–east, and the south–west regions of M51a, and one covers the companion galaxy M51b.
NGC 628
-------
NGC 628 (M74) is the largest galaxy in its group. This nearby galaxy is seen almost face–on ($\rm i = 25.2 ^{\circ}$) and is located at a distance of 9.9 Mpc [@Oliver]. It has no bulge [@cor] and is classified as a SAc spiral galaxy. Its star formation rate and stellar mass obtained from the extinction–corrected B–band luminosity are $ \rm 3.6 \, M_{\sun } yr^{-1}$and $\rm 1.1 \times 10^{10}M_{\sun }$, respectively [@Lee; @Both]. NGC 628 is a multiple–arm spiral galaxy [@Debra] with two well–defined spiral arms. It has weaker spiral patterns than NGC 1566 and M51a [@Debra2]. The LEGUS UVIS observations of NGC 628 consist of one central and one east pointing that were combined into a single mosaic for the analysis.
Stellar cluster samples {#s3}
=======================
Selection from star cluster catalogues
--------------------------------------
In this section, we provide a detailed explanation of the process adopted to select star cluster candidates in our target galaxies. A general description of the standard data reduction of the LEGUS sample can be found in [@C15]. A careful and detailed description of the cluster extraction, identification, classification, and photometry is given in [@Angela17] and [@messa]. Stellar cluster candidates were extracted with SExtractor [@Bertin] in the five standard LEGUS filters. The resulting cluster candidate catalogues include sources with a $V$–band concentration index (CI)[^3] larger than the CI of star–like sources, which are detected in at least two filters with a photometric error $\leq$ 0.3 mag. The photometry of sources in each filter was corrected for the Galactic foreground extinction [@Schlafly]. In order to derive the cluster physical properties such as age, mass, and extinction, the spectral energy distribution (SED) of the clusters was fitted with Yggdrasil stellar population models [@Z11]. The uncertainties derived in the physical parameters of the star clusters are on average $\rm 0.1\, \rm dex$ [@Angela17]. For some of the LEGUS galaxies, star cluster properties were also estimated based on a Bayesian approach, using the Stochastically Lighting Up Galaxies (SLUG) code [@sila]. A detailed and complete explanation of the Bayesian approach can be found in [@krumholz].
Each source in the stellar cluster catalogue that is brighter than -6 mag in the $V$–band, and detected in at least four bands, has been morphologically classified via visual inspection by three independent members of the LEGUS team [@katie15; @Angela17]. The inspected clusters were divided into four morphological classes: Class 1 contains compact, symmetric, and centrally concentrated clusters. Class 2 includes compact clusters with a less symmetric light distribution, Class 3 represents less compact and multi–peak cluster candidates with asymmetric profiles, and Class 4 consists of unwanted objects like single stars, multiple stars, or background sources. Unclassified objects were labeled as Class 0.
In addition, a machine–learning (ML) approach was tested to morphologically classify the stellar clusters in an automated fashion. A forthcoming paper (Grasha et al., in prep.) will present the ML code that was used for cluster classification in the LEGUS survey and the degree of agreement with human classification. An initial comparison between human and ML classification in M51a was already discussed by @messa.
For our analysis, we use stellar cluster properties estimated with Yggdrasil deterministic models based on the Padova stellar libraries (see @Z11 for details) with solar metallicity, the Milky Way extinction curve [@Cardeli], and the [@Kroupa] stellar initial mass function (IMF). We also selected clusters based on human visual classification for NGC 628, a combination of human and machine learning classification in NGC 1566, and only machine learning for M51a. Star clusters classified as Class 4 and Class 0 are excluded from our analysis. Among our target galaxies, there is a total number of 1573, 3374, and 1262 star cluster candidates classified as Class 1, 2, and 3 in NGC 1566, M51a, and NGC 628, respectively.
A detailed description of the properties of the final cluster catalogues of M51a and NGC 628 and their completeness can be found in @messa and @Angela17.
[0.50]{} ![Distribution of ages and masses of the star clusters (class 1, 2, and 3) in NGC 1566, M51a, and NGC 628. The colours represent different age bins: blue (the young sample), green (the intermediate–age sample), red (the old sample), and black (excluded star clusters). The number of clusters in each sample is shown in parentheses. The horizontal dotted lines in NGC 1566 show the applied mass cut of 5000 $\rm M_{\sun}$ up to the age of 100 Myr and $\rm 10^{4} \, \rm M_{\sun}$ up to the age of 200 Myr. The applied mass cut of 5000 $\rm M_{\sun}$ up to the age of 200 Myr in M51a and NGC 628 are also by horizontal dotted lines. The solid black line shows the 90% completeness limit of 23.5 mag in the $V$–band in NGC 1566 and the magnitude cut of $\rm M_{V}$ = -6 mag in M51a, and NGC 628, respectively.[]{data-label="fig:age_mass"}](age_mass_NGC1566.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![Distribution of ages and masses of the star clusters (class 1, 2, and 3) in NGC 1566, M51a, and NGC 628. The colours represent different age bins: blue (the young sample), green (the intermediate–age sample), red (the old sample), and black (excluded star clusters). The number of clusters in each sample is shown in parentheses. The horizontal dotted lines in NGC 1566 show the applied mass cut of 5000 $\rm M_{\sun}$ up to the age of 100 Myr and $\rm 10^{4} \, \rm M_{\sun}$ up to the age of 200 Myr. The applied mass cut of 5000 $\rm M_{\sun}$ up to the age of 200 Myr in M51a and NGC 628 are also by horizontal dotted lines. The solid black line shows the 90% completeness limit of 23.5 mag in the $V$–band in NGC 1566 and the magnitude cut of $\rm M_{V}$ = -6 mag in M51a, and NGC 628, respectively.[]{data-label="fig:age_mass"}](age_mass_M51.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![Distribution of ages and masses of the star clusters (class 1, 2, and 3) in NGC 1566, M51a, and NGC 628. The colours represent different age bins: blue (the young sample), green (the intermediate–age sample), red (the old sample), and black (excluded star clusters). The number of clusters in each sample is shown in parentheses. The horizontal dotted lines in NGC 1566 show the applied mass cut of 5000 $\rm M_{\sun}$ up to the age of 100 Myr and $\rm 10^{4} \, \rm M_{\sun}$ up to the age of 200 Myr. The applied mass cut of 5000 $\rm M_{\sun}$ up to the age of 200 Myr in M51a and NGC 628 are also by horizontal dotted lines. The solid black line shows the 90% completeness limit of 23.5 mag in the $V$–band in NGC 1566 and the magnitude cut of $\rm M_{V}$ = -6 mag in M51a, and NGC 628, respectively.[]{data-label="fig:age_mass"}](age_mass_NGC628.pdf "fig:"){width="1\linewidth"}
Galaxy age (Myr) < 10 10 $ \leq $ age (Myr) < 50 50 $\leq $ age (Myr) $ \leq $ 200
---------- ------------------- ------------------------------- ----------------------------------- --
NGC 1566 392 679 124
M51a 361 441 979
NGC 628 77 111 302
Selection of star clusters of different ages {#selection of star clusters of different ages}
--------------------------------------------
In this study, we use the age of star clusters in our galaxy sample as a tool to find a possible age gradient across the spiral arms predicted by the stationary density wave theory. Therefore, we group star clusters into three different cluster samples according to their ages.
The estimated physical properties of star clusters based on the Yggdrasil deterministic models are inaccurate for low–mass clusters [@krumholz]. A comparison between the deterministic approach based on Yggdrasil models and the Bayesian approach with SLUG models presented by [@krumholz] suggests that the derived cluster properties are uncertain at cluster masses below 5000 $\rm M_{\sun}$. We adopted the same mass cut–off and for NGC 628 and M51a in our analysis. Using the luminosity corresponding to this mass, namely $\rm M_{V}$ = $-6$ mag ($\rm m_{V}$ = 23.4 and 23.98 mag for NGC 628 and M51a, respectively) results in an age completeness limit of $\rm \leq 200\, \rm Myr$. In @Angela17 and @messa the magnitude cut at $\rm M_{V}$ < $-6$ mag is a more conservative limit than the magnitude limit corresponding to 90% of completeness in the recovery of sources. We have tested our results using different mass cuts as well as by removing any constraint on the limiting mass, and we have not observed any significant change in the age distributions of the clusters as a function of azimuthal distances. Thus, the results presented in § \[Azimutahl distribution\] and § \[2arms\] are robust against uncertainties in the determination of cluster physical properties.
NGC 1566 is the most distant galaxy within our LEGUS sample. Due to the large distance of this galaxy, the 90% completeness limit ($\rm m_{V}$ = 23.5 mag) is significantly brighter than $\rm M_{V}$ = $-6$ mag. Therefore, in order to select star clusters in NGC 1566, we used the 90% completeness limit and a= mass cut of 5000 $\rm M_{\sun}$ for the cluster ages up to 100 Myr and $\rm 10^{4} \rm M_{\sun}$ for the 100–200 Myr old star clusters (see Fig. \[fig:age\_mass\]). Applying these two criteria reduced our cluster samples from 1573 to 1195 clusters for NGC 1566, from 3374 to 1781 clusters for M51a, and from 1262 to 490 for NGC 628.
Then, we selected three cluster samples of different ages for each galaxy as follows:
$\bullet$
: “Young” star clusters: age (Myr) < 10
$\bullet$
: “Intermediate–age” star clusters: 10 $\rm \leq$ age (Myr) < 50
$\bullet$
: “Old” star clusters: 50 $\rm \leq$ age (Myr) $\leq$ 200
The number of star clusters in the “young”, “intermediate–age”, and “old” samples is shown in Tab. \[tab:cluster sample\].
Fig. \[fig:age\_mass\] displays the age–mass diagram of star clusters in NGC 1566, M51a, and NGC 628. The young, the intermediate–age, and the old star cluster samples are shown in blue, green, and red colors, respectively. The excluded star clusters (due to the mass cut) are shown in black. The horizontal and vertical dotted lines show the applied mass cut of $ \rm 5000\, \rm M_{\sun}$ and its corresponding completeness limit at a stellar age of $ 200\, \rm Myr$, respectively.
Spatial distribution and clustering of star clusters {#location}
====================================================
In Fig. \[fig:clusters\], we plot the spatial distribution of star clusters of different ages in the galaxies NGC 1566, M51a, and NGC 628. The young, intermediate–age, and old stellar cluster samples are shown in blue, green, and red, respectively. In general, we observe a similar trend in our target galaxies: First, the young and the intermediate–age star clusters mostly populate the spiral arms rather than the interarm regions. This is particularly evident for NGC 1566 and M51a, which show strong and clear spiral structures in young and intermediate–age star clusters. Second, the old star clusters are less clustered and more widely spread compared to the young and intermediate–age star cluster samples.
Our findings are similar to other literature results on the spatial distribution of star clusters of different ages: [@D17], using LEGUS HST data found that in NGC 1566 the 100 Myr old star clusters clearly trace the spiral arms while in NGC 628 star clusters older than 10 Myr show only weak spiral structures. [@chandar17], using other HST data observed that M51a shows weak spiral structure in older star clusters (>100 Myr).
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Clustering of star clusters has been observationally investigated for a number of local star–forming galaxies [e.g., @Efremov; @EE]. In a detailed study of clustering of the young stellar population in NGC 6503 based on the LEGUS observations, [@d15] found that younger stars were more clustered compared to the older ones. [@katie15] investigated the spatial distribution of the star clusters in NGC 628 from the LEGUS sample. Their findings confirmed that the degree of the clustering increases with decreasing age. More recently, @grasha17a studied the hierarchical clustering of young star clusters in a sample of six LEGUS galaxies. Their results suggested that the youngest star clusters are strongly clustered and the degree of clustering quickly drops for clusters older than 20 Myr and the galactic shear appears to drive the largest sizes of the hierarchy in each galaxy @grasha17b.
Adopting a similar approach as [@katie15], we use the two–point correlation function to test whether or not the clustering distribution of the clusters in our selected age bins shows the expected age dependence. The two–point correlation function $\rm \omega (\theta)$ is a powerful statistical tool for quantifying the probability of finding two clusters with an angular separation $\rm \theta$ against a random, non–clustered distribution [@peebles]. Here we use the Landy–Szalay [@LS] estimator, which has little sensitivity to the presence of edges and masks in the data: $$\omega(\theta) = \frac{r (r-1)}{n (n-1)}\frac{DD}{RR} - \frac{(r-1)}{n}\frac{DR}{RR}+1,$$ where $ n$ and $r$ are the total number of data and random points, respectively. $ DD$, $ RR$, and $ DR$ are the total numbers of data–data, random–random, and data–random pair counts with a separation $\rm \theta \pm d\theta$, respectively. We construct a random distribution of star clusters that has the same sky coverage and masked regions (e.g., the ACS chip gap) as the images of each galaxy.
Fig. \[fig:two\_point\] displays the two–point correlation function for the star clusters in different age bins as defined for our galaxy samples. The blue, green, and red colours represent the young, intermediate–age, and old star cluster samples in each galaxy, respectively. The error bars on the two–point correlation function were estimated using a bootstrapping method with 1000 bootstrap resamples.
The general distribution of the star cluster samples in the target galaxies shows a similar trend: Independent of the presence of spiral arms, young clusters show hierarchical structure, whilst the old star clusters show a non–clustered, smooth distribution.
[0.50]{} ![ two–point correlation function for the star cluster samples of different ages as a function of angular distance (arcseconds) in NGC 1566, M51a, and NGC 628. The young, intermediate–age, and old star cluster samples are shown in blue, green, and red, respectively. The error bars were computed based on a bootstrapping method. The number of star clusters in each age bin are listed in parentheses.[]{data-label="fig:two_point"}](2p_NGC1566.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![ two–point correlation function for the star cluster samples of different ages as a function of angular distance (arcseconds) in NGC 1566, M51a, and NGC 628. The young, intermediate–age, and old star cluster samples are shown in blue, green, and red, respectively. The error bars were computed based on a bootstrapping method. The number of star clusters in each age bin are listed in parentheses.[]{data-label="fig:two_point"}](2p_M51.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![ two–point correlation function for the star cluster samples of different ages as a function of angular distance (arcseconds) in NGC 1566, M51a, and NGC 628. The young, intermediate–age, and old star cluster samples are shown in blue, green, and red, respectively. The error bars were computed based on a bootstrapping method. The number of star clusters in each age bin are listed in parentheses.[]{data-label="fig:two_point"}](2p_NGC628.pdf "fig:"){width="1\linewidth"}
Are the spiral arms static density waves? {#Azimutahl distribution}
=========================================
As discussed in § \[Introduction\], the stationary density wave theory foresees that the age of stellar clusters inside the corotation radius increases with increasing distance from the spiral arms. In other words, we expect to find a shift in the location of stellar clusters with different ages.
In order to test whether the distribution of star clusters of different ages in our target galaxies agrees with the expectations from the stationary density wave theory, we need to quantify the azimuthal offset between star clusters of different ages.
Spiral arm ridge lines definition
---------------------------------
First of all, we need to locate the spiral arms of our galaxy sample. We wish to define a specific location in each spiral arm so we can measure the relative positions of the star clusters in a uniform way. We use the dust lanes for this purpose because they are narrow and well–defined on optical images.
As gas flows into the potential minima of a density wave, it gets compressed and forms dark dust lanes in the inner part of the spiral arms, where star formation is then likely to occur [@R69]. We have used the $B$–band images for this purpose since most of the emission is due to young OB stars and dark obscuring dust lanes can be better identified in this band.
To better define the average positions of the dust lanes, we used a Gaussian kernel (with a 10 pixels sigma) to smooth the images, reduce the noise, and enhance the spiral structure. In the smoothed images the dust lanes are clearly visible as dark ridges inside the bright spiral arms. We defined these dark spiral arm ridge lines manually. For the remainder of this paper, we refer to the southern arm and northern arm as “Arm 1” and “Arm 2”, respectively. Fig. \[fig:arms\] presents the defined spiral arm ridge lines (red lines) overplotted on the smoothed $B$–band images of NGC 1566, M51a, and NGC 628.
[0.50]{} ![The location of spiral arm ridge lines is shown by red lines overplotted on the smoothed $B$—band images of NGC 1566, M51a, and NGC 628. We refer to the southern arm and northern arm as “Arm 1” and “Arm 2”, respectively. The two black dashed circles in each panel mark the onset of the bulge and the location of the co–rotation radius of the galaxies. The horizontal bar in the lower left corner denotes a length scale of 2 kpc. North is up and East to the left.[]{data-label="fig:arms"}](arms_NGC1566.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![The location of spiral arm ridge lines is shown by red lines overplotted on the smoothed $B$—band images of NGC 1566, M51a, and NGC 628. We refer to the southern arm and northern arm as “Arm 1” and “Arm 2”, respectively. The two black dashed circles in each panel mark the onset of the bulge and the location of the co–rotation radius of the galaxies. The horizontal bar in the lower left corner denotes a length scale of 2 kpc. North is up and East to the left.[]{data-label="fig:arms"}](arms_M51.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![The location of spiral arm ridge lines is shown by red lines overplotted on the smoothed $B$—band images of NGC 1566, M51a, and NGC 628. We refer to the southern arm and northern arm as “Arm 1” and “Arm 2”, respectively. The two black dashed circles in each panel mark the onset of the bulge and the location of the co–rotation radius of the galaxies. The horizontal bar in the lower left corner denotes a length scale of 2 kpc. North is up and East to the left.[]{data-label="fig:arms"}](arms_NGC628.pdf "fig:"){width="1\linewidth"}
Measuring azimuthal offset
--------------------------
Knowing the position of star clusters and spiral arm ridge lines in our target galaxies allowed us to measure the azimuthal distance of a star cluster from its closest spiral arm, assuming that it rotates on a circular orbit.
We limited our analysis to the star clusters located in the disk where spiral arms exist. The disk of a galaxy can be defined by its rotation curve. The rotational velocity increases when moving outwards from the central bulge–dominated part and becomes flat in the disk–dominated part of the galaxy. We derived a radius of 2 kpc for the bulge–dominated part of our galaxies using the rotation curves of [@k2000] for NGC 1566, [@sofi2; @sofi1] for M51a, and [@combes] for NGC 628. Furthermore, we limited our analysis to star clusters located inside the corotation radius. If stationary density waves are the dominant mechanism driving star formation in spiral galaxies we expect to find an age gradient from younger to older clusters inside the corotation radius. The bulge–dominated region and co–rotation radius of each galaxy are shown in Fig. \[fig:arms\]. The adopted corotation radii of the galaxies are listed in Tab. \[tab:properties of galaxies\].
Fig. \[fig:hist\] (left panels) shows the normalized distribution of the azimuthal distance of star clusters in the three age bins from their closest spiral arm ridge line in NGC 1566, M51a, and NGC 628. The error bars in each sample were calculated by dividing the square root of the number of clusters in each bin by the total number of clusters. We note that an azimuthal distance of zero degrees shows the location of the spiral arm ridge lines and not the center of the arms. Positive (negative) azimuthal distributions indicate that a cluster is located in front of (behind) the spiral arm ridge lines. Blue, green, and red colours represent the young, intermediate–age, and old star cluster samples, respectively.
Fig. \[fig:hist\] (right panels) shows the cumulative distribution function of star clusters as a function of the azimuthal distance. In order to test whether the samples come from the same distribution, we used a two–sample Kolmogorov–Smirnov test (hereafter K–S test). Since we aim at finding the age gradient in front of the spiral arms, the K–S test was only calculated for star clusters with positive azimuthal distances. The probability that two samples are drawn from the same distribution (p–values) and the maximum difference between pairs of cumulative distributions (D) are listed in Tab \[tab3\].
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In the case of NGC 1566 (Fig. \[fig:hist\], top), we see that the young and intermediate–age star cluster samples are peaking close to the location of the spiral arm ridge lines (azimuthal distance of 0–5 degrees) while the old sample peaks further away from the ridge lines (azimuthal distances of 5–10 degrees). The derived p–values are lower than the test’s significance level (0.05) of the null hypothesis, i.e., that the two samples are drawn from the same distribution. As a consequence, our three star cluster samples are unlikely to be drawn from the same population. A clear age gradient across the spiral arms can be observed in NGC 1566, which is in agreement with the expectation from stationary density wave theory. The existence of such a pattern supports the picture of an age sequence in the model of a grand–design spiral galaxy and a barred galaxy suggested by [@DP10; @dimit17].
No obvious age gradient from younger to older is seen in the azimuthal distributions of the star cluster samples in M51a (Fig. \[fig:hist\], middle). What is remarkable here is that the older star clusters are located closer to the spiral arm ridge lines than the young and intermediate–age star clusters. The K–S test indicates that the probability that the young star cluster sample is drawn from the same distribution as the intemediate-age and old star cluster samples is more than 10%. The derived p–value for the intermediate–age and old cluster samples is lower than the significance level of the K–S test and rejects the null hypothesis that the two samples are drawn from the same distribution. The lack of an age pattern is consistent with the observed age trend for an interacting galaxy, modeled based on M51a, suggested by [@DP10]. Our result is compatible with a number of observational studies have found no indication for the expected spatial offset from the stationary density wave theory in M51a [@S09; @k10; @Foyle; @S17].
There is no evident trend in the azimuthal distribution of star clusters in NGC 628 (Fig. \[fig:hist\], bottom). The majority of the young star clusters tends to be located further away from the ridge lines (azimuthal distance of 20–25 degrees). The calculated p–values from the K–S test are larger than 0.05, which suggests weak evidence against the null hypothesis. As a result, the three young, intermediate–age, and old star cluster samples are drawn from the same distribution. The absence of an age gradient across the spiral arms in NGC 628 is consistent with a simulated multiple arm spiral galaxy by [@grand].
The origin of two spiral arms {#2arms}
=============================
An observational study by [@Egusa17], based on measuring azimuthal offsets between the stellar mass (from optical and near–infrared data) and gas mass distributions (from CO and HI data) in two spiral arms of M51a, suggest that the origin of these spiral arms differs. One spiral arm obeys the stationary density wave theory while the other does not.
In another recent study of M51a, [@chandar17] quantified the spatial distribution of star clusters with different ages relative to different segments of the two spiral arms of M51a traced in the 3.6 $\mu$m image. They observed a similar trend for the western and eastern arms: the youngest star clusters (< 6 Myr) are found near the spiral arm segments, and the older clusters (100–400 Myr) show an extended distribution.
In this section, we test whether measuring the azimuthal offset of star cluster samples from each spiral arm individually leads to different results. We assume that a star cluster whose distance from Arm 1 is smaller than its distance from Arm 2 belongs to Arm 1 and vice versa.
Fig. \[fig:age\_hist\] shows the normalized distribution of ages of star clusters associated with Arm 1 (shown in red) and Arm 2 (shown in blue) in each of the galaxies. No significant differences between the age distribution of star clusters belonging to the two spiral arms in our target galaxies can be observed. Also, the K–S test indicates that the age distributions of star clusters relative to Arm 1 and Arm 2 in each galaxy are drawn from the same population.
[0.50]{} ![The distribution of the age of star clusters associated with Arm 1 (red) and Arm 2 (blue) in NGC 1566, M51a, and NGC 628. The number of star clusters relative to the Arm 1 and Arm 2 is listed in parantheses.[]{data-label="fig:age_hist"}](NGC1566_arm_clusters.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![The distribution of the age of star clusters associated with Arm 1 (red) and Arm 2 (blue) in NGC 1566, M51a, and NGC 628. The number of star clusters relative to the Arm 1 and Arm 2 is listed in parantheses.[]{data-label="fig:age_hist"}](M51_arm_clusters.pdf "fig:"){width="1\linewidth"}
[0.50]{} ![The distribution of the age of star clusters associated with Arm 1 (red) and Arm 2 (blue) in NGC 1566, M51a, and NGC 628. The number of star clusters relative to the Arm 1 and Arm 2 is listed in parantheses.[]{data-label="fig:age_hist"}](NGC628_arm_clusters.pdf "fig:"){width="1\linewidth"}
In Fig. \[fig:hist-arms\] we compare the normalized azimuthal distribution of the three young, intermediate–age, and old star cluster samples relative to Arm 1 (left panels) and Arm 2 (right panels) in our target galaxies. As before, our analysis was limited to the star clusters positioned in the disk and inside the corotation radius of our target galaxies.
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The upper panels of Fig. \[fig:hist-arms\] exhibit a noticeable age gradient across both spiral arms of NGC 1566. The young star clusters are highly concentrated towards the location of Arm 1 and Arm 2 while the older ones are peaking further away from the two spiral arms.
The second row panels of Fig. \[fig:hist-arms\] show the azimuthal distance of star cluster samples across the two arms of M51a. This galaxy displays an offset in the location of young and old star clusters across Arm 1. The young star clusters culminate close to Arm 1 (at azimuthal distances of 2–6 degrees) while the old ones are positioned further away (at azimuthal distances of 6–10 degrees). Even though M51a shows an age gradient across the Arm 1 at first glance, the K–S test does not imply significant differences between the young and old star cluster samples (all derived p–values are larger than the test’s significance level). We do not observe any shift in the azimuthal distribution of the star cluster samples across Arm 2 in M51a.
In the case of NGC 628, no obvious age gradient across Arm 1 and Arm 2 is observed (the lower panels of Fig. \[fig:hist-arms\]). It is important to note that our results are inconclusive for the young star clusters associated with Arm 2 due to the small number statistics. Hence, we also explored the change in the azimuthal distribution of the star clusters by including clusters with masses < 5000 $\rm M_{\sun}$ and ages > 200 Myr. The observed differences are not significant and the general trend is the same as before.
Thus, measuring the azimuthal distance of the star clusters from the two individual spiral arms in each galaxy suggests that the two spiral arms of our target galaxies may have the same physical origin.
Comparison with the non–LEGUS cluster catalogue of M51 {#chandra}
======================================================
---------- ------ --------------------------- ------ --------------------------- ------ ---------------------------
p–value D p–value D p–value
NGC 1566 0.15 $ \rm 3.78\times 10^{-3}$ 0.31 $\rm 2.88 \times 10^{-5}$ 0.26 $\rm 6.19 \times 10^{-5}$
M51a 0.15 0.10 0.13 0.10 0.17 $\rm 2.4 \times 10^{-3}$
NGC 628 0.21 0.49 0.47 0.10
---------- ------ --------------------------- ------ --------------------------- ------ ---------------------------
In this section, we use the [@chandar16] catalogue (hereafter CH16 catalogue) to measure the azimuthal offsets of star clusters with different ages in M51a and to compare the results with our analysis based on the LEGUS catalogue. We caution that the south–eastern region of M51a is not covered by the LEGUS observations. We also investigated whether our results are biased due to the absence of star clusters from that region.
[@chandar16] provided a catalogue of 3816 star clusters in M51a based on HST ACS/WFC2 images obtained the equivalents of $UBVI$ and H$\rm \alpha$ filters. [@messa] compared the age distributions of star clusters in common between the LEGUS and CH16 catalogue. They observed that a large number of young star clusters (age < 10 Myr) in [@chandar16] have a broad age range (age: 1–100 Myr) in the LEGUS catalogue. They argued that the discrepancies in the estimated ages are due to the use of different filter combinations.
In Fig. \[fig:age\_mass\_chandar\], we show the distribution of ages and masses of star clusters in M51a from the CH16 catalogue. In order to be able to compare our results, we considered a mass–limited sample with masses > 5000 $\rm M_{\sun}$ and ages < 200 Myr and selected the same age bins as before: The young (< 10 Myr), intermediate–age (10–50 Myr), and old star cluster samples (50–200 Myr).
![The distribution of ages and masses of the 3816 star clusters in M51a, based on the CH16 catalogue. The young (<10 Myr ), intermediate–age (10–50 Myr), and old (50–200 Myr) star clusters are shown in blue, green, and red, respectively. The black points indicate excluded star clusters due to the applied mass cut and the imposed completeness limit. The number of clusters in each sample is listed in parentheses. The horizontal and vertical dotted lines show the applied mass cut of 5000 $\rm M_{\sun}$ and the corresponding detection completeness limit of 200 Myr, respectively. []{data-label="fig:age_mass_chandar"}](age_mass_chandar.pdf){width="49.00000%"}
In Fig. \[fig:clusters-ch16\], we plot the spatial distribution of the young, intermediate–age, and old star clusters based on the CH16 catalogue in M51a. As we can see, M51a displays a very clear and strong spiral pattern in the young star clusters. The intermediate–age star clusters tend to be located along the spiral arms while the old ones are more scattered and populate the inter–arm regions. Recently, [@chandar17] using the CH16 catalogue found that the youngest star clusters (< 6 Myr) are concentrated in the spiral arms (defined based on 3.6 $\mu$m observations). The older star clusters (6–100 Myr) are also found close to the spiral arms but they are more dispersed, and the spiral structure is not clearly recognisable in older star clusters (> 400 Myr).
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In order to quantify the possible spatial offset in the location of the three young, intermediate–age, and old star cluster samples from the CH16 catalogue across the spiral arms, we computed the normalized azimuthal distribution and corresponding cumulative distribution function of the star cluster samples in Fig. \[fig:azi\_ch\]. We applied our analysis only to the star clusters positioned in the disk and inside the co–rotation radius of M51a (2.0–5.5 kpc). Our result demonstrates that the three young, intermediate–age, and old star cluster samples peak at an azimuthal distance of 6 degrees from the location of the spiral arms. We observe no obvious offsets between the azimuthal distances of the three star cluster age samples in M51a. [@chandar17], using the same cluster catalogue, quantified the azimuthal offset of molecular gas (from PAWS and HERACLES) and young (<10 Myr) and intermediate–age (100–400 Myr) star clusters in the inner (2–2.5 kpc) and outer (5–5.5 kpc) annuli of the spiral arms. They found that in the inner annuli the young star clusters show an offset of 1 kpc from the molecular gas while there is no offset between the molecular gas and young and old star clusters in the outer portion of the spiral arms.
Adopting the CH16 catalogue, we found that there is no noticeable age gradient across the spiral arms of M51a, which is in agreement with our finding based on the LEGUS star cluster catalogue.
DISCUSSION AND CONCLUSIONS {#Summary}
==========================
The stationary density wave theory predicts that the age of star clusters increases with increasing distance away from the spiral arms. Therefore, a simple picture of the stationary density wave theory leads to a clear age gradient across the spiral arms. In this study, we are testing the theory that spiral arms are static features with constant pattern speed. For this purpose, we use the age and position of star clusters relative to the spiral arms.
We use high–resolution imaging observations obtained by the LEGUS survey [@C15] for three face–on LEGUS spiral galaxies, NGC 1566, M51a, and NGC 628. We have measured the azimuthal distance of the LEGUS star clusters from their closest spiral arm to quantify the possible spatial offset in the location of star clusters of different ages (< 10 Myr, 10–50 Myr, and 50–200 Myr) across the spiral arms. We found that the nature of spiral arms in our target galaxies is not unique. The main results are summarized as follows:
- Our detailed analysis of the azimuthal distribution of star clusters indicates that there is an age sequence across spiral arms in NGC 1566. NGC 1566 shows a strong bar and bisymmetric arms typical of a massive self–gravitating disk [@Elena15]. We speculate that when disks are very self–gravitating the bar and the two–armed features dominate a large part of the galaxy, producing an almost constant pattern speed. The observed trend is also in agreement with what was found by [@DP10] in simulations of a grand design and a barred spiral galaxy.
- We find no age gradient across the spiral arms of M51a. This galaxy shows less strong arms and a weaker bar and hence a less self–gravitating disk. The absence of an age sequence in M51a indicates that the grand–design structures of this galaxy are not the result of a steady–state density wave, with a fixed pattern speed and shape, as in the early analytical models. More likely, the spiral is a density wave that is still changing its shape and amplitude with time in reaction to the recent tidal perturbations. A possible mechanism to explain the formation and presence of grand–design structures in spiral galaxies is an interaction with a nearby companion [@Toomre; @72; @k; @B3]. Since such an interaction is obviously occurring in M51a, tidal interactions could be the dominant mechanism for driving its spiral patterns. [@DP10] simulated M51a with an interacting companion (M51b), and observed no age gradient across the tidally induced grand–design spirals arms. Our findings are consistent with the results of several other observational studies, which did not find age gradients as expected from the spiral density wave theory in M51a [@S09; @k10; @Foyle; @S17].
- NGC 628 is a multiple–arm spiral galaxy with weak spiral arms consistent with a pattern speed decreasing with radius and multiple corotation radii. In this case we find no significant offset among the azimuthal distributions of star clusters with different ages, which is consistent with the swing amplification theory. The lack of such an age offset is in agreement with an earlier analysis of NGC 628 [@Foyle], and consistent with the spatial distribution of star clusters with different ages in the simulated multiple–arm spiral galaxy by [@grand].
Acknowledgements {#acknowledgements .unnumbered}
================
This work is based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555. These observations are associated with program 13364. Support for Program 13364 was provided by NASA through a grant from the Space Telescope Science Institute.
This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
A.A. acknowledges the support of the Swedish Research Council (Vetenskapsr[å]{}det) and the Swedish National Space Board (SNSB). D.A.G kindly acknowledges financial support by the German Research Foundation (DFG) through programme GO 1659/3–2.
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\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: https://legus.stsci.edu
[^3]: the magnitude difference between apertures of radius 1 pixel and 3 pixels
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Grebinski and Kucherov (1998) and Alon et al. (2004-2005) studied the problem of learning a hidden graph for some especial cases, such as hamiltonian cycle, cliques, stars, and matchings, which was motivated by some problems in chemical reactions, molecular biology and genome sequencing. The present study aimed to present a generalization of this problem. Graphs $G$ and $H$ were considered, by assuming that $G$ includes exactly one defective subgraph isomorphic to $H$. The purpose is to find the defective subgraph by performing the minimum non-adaptive tests, where each test is an induced subgraph of the graph $G$ and the test is positive in the case of involving at least one edge of the defective subgraph $H$. We present an upper bound for the number of non-adaptive tests to find the defective subgraph by using the symmetric and high probability variation of Lovász Local Lemma. Finally, we present a non-adaptive randomized algorithm to find defective subgraph by at most ${3\over 2}$ times of this upper bound with high probability. [**Keywords:**]{} [ Group testing on graphs, Non-adaptive algorithm, Combinatorial search, Learning a hidden subgraph.]{}'
---
[**Non-adaptive Group Testing on Graphs**]{}\
Introduction
============
In the classic *group testing* problem which was first introduced by Dorfman [[@dorfman1943]]{}, there is a set of $n$ items including at most $d$ defective items. The purpose of this problem is to find the defective items with the minimum number of tests. Every test consists of some items and each test is positive if it includes at least one defective item. Otherwise, the test is negative. There are two types of algorithms for the group testing problem, *adaptive* and *non-adaptive*. In adaptive algorithm, the outcome of previous tests can be used in the future tests and in non-adaptive algorithm all tests perform at the same time and the defective items are obtained by considering the results of all tests.
Regarding some extensions of classical group testing, we can refer to *group testing on graphs*, *complex group testing*, *additive model*, *inhibitor model*, etc. (see [[@du2000combinatorial-group-testing; @du2006pooling-design-group-testing; @survey-group-testing-2008]]{} for more information). Aigner [[@Aigner1986-search-problem-on-graphs]]{} proposed the problem of group testing on graphs, in which we look for one defective edge of the given graph $G$ by performing the minimum adaptive tests, where each test is an induced subgraph of the graph $G$ and the test is positive in the case of involving the defective edge. In the present paper, the problem of *non-adaptive group testing on graphs* was considered by assuming that there is one defective subgraph (not necessarily induced subgraph) of $G$ isomorphic to a graph $H$ and our purpose is to find the defective subgraph with minimum number of non-adaptive tests. Each test $F$ is an induced subgraph of $G$ and the test result is positive if and only if $F$ includes at least one edge of the defective subgraph. This is a generalization of the problem of *non-adaptive learning a hidden subgraph* studied in [[@alon-learning-subgraph; @alon-learning-matching]]{}. In the problem of learning hidden graph, the graph $G$ is a complete graph. In other words, let ${\cal H}$ be a family of labeled graphs on the set $V = \{1, 2, . . . , n\}$. In this problem the goal is to reconstruct a hidden graph $H \in {\cal H}$ by minimum number of tests, where a test ${\cal F} \subset V$ is positive if the subgraph of $H$ induced by ${\cal F}$, contains at least one edge. Otherwise the test is negative.
The problem of learning a hidden graph was emphasized in some models as follows:
*K-vertex model*: In this model, each test has at most k vertices.
*Additive model*: Based on this model, the result of each test $F$ is the number of edges of $H$ induced by $F$. This model is mainly utilized in bioinformatics and was studied in [[@2005-Bouvel-grebinski-survey; @2000-Grebinski-kucherov-additive]]{}.
*Shortest path test*: In this model, each test ${u,v}$ indicates the length of the shortest path between $u$ and $v$ in the hidden graph and if no path exists, return $\infty$. More information about this model and the result is given in [[@2007-learning-edge-counting-srivastava]]{}. Further, this model is regarded as a canonical model in the evolutionary tree literature [[@1989-Hein-hidden-tree-additive-distance; @2003-king-learning-tree-distance; @2007-Reyzin-learning-tree-distance-longest-path]]{}.
There are various families of hidden graphs to study. However, a large number of recent studies have focused on hamiltonian cycles and matchings [[@alon-learning-matching; @2001-Beigel-genome-shotgun-sequencing; @Grebinski1998-learning-hamiltonian-cycle]]{}, stars and cliques [[@alon-learning-subgraph]]{}, graph of bounded degree [[@2005-Bouvel-grebinski-survey; @2000-Grebinski-kucherov-additive]]{}, general graphs [[@2008-adaptive-learning-hidden-graph-per-edge; @2005-Bouvel-grebinski-survey] ]{}. Here, we present a short survey of known results on these problems by using adaptive and non-adaptive algorithms.
Grebinski and Kucherov [[@Grebinski1998-learning-hamiltonian-cycle]]{} suggested an adaptive algorithm to learn a hidden Hamiltonian cycle by $2n lg n$ tests, which achieves the information lower bound for the number of tests needed. Further, Chang et al.[[@2011-learning-hidden-graph-threshold]]{} could improve their results to $(1+o(1))n\log n$.
Alon et al. [[@alon-learning-matching]]{} proposed an upper bound $({1\over 2}+o(1)){n \choose 2}$ on learning a hidden matching using non-adaptive tests. Bouvel et al. [[@2005-Bouvel-grebinski-survey]]{} developed an adaptive algorithm to learn a hidden matching with at most $(1+o(1))n\log n$ tests. In addition, Change et al. [[@2011-learning-hidden-graph-threshold]]{} improved their result to $(1+o(1)){n\log n\over 2}$.
Alon and Asodi [[@alon-learning-subgraph]]{} developed an upper bound $O(n\log^2 n)$ on learning a hidden clique using non-adaptive tests. Also they proved an upper bound $k^3\log n$ on learning a hidden $K_{1,k}$ using non-adaptive tests. Bouvel et al. [[@2005-Bouvel-grebinski-survey]]{} presented two adaptive algorithms to learn hidden star and hidden clique with at most $2n$ tests. Change et al. [[@2011-learning-hidden-graph-threshold]]{} improved their results on learning hidden star and hidden clique to $(1+o(1))n$ and $n+\log n$, respectively.
Grebinski and Kucherov [[@2000-Grebinski-kucherov-additive]]{} gave tight bound of $\theta(dn)$ and $\theta({n^2\over \log n})$ non-adaptive tests on learning a hidden d-degree-bounded and general graphs in additive model, respectively. Angluin and Chen [[@2008-adaptive-learning-hidden-graph-per-edge]]{} proved that a hidden general graph can be identified in $12m log n$ tests through using adaptive algorithm where $m$ (unknown) is the number of edges in the hidden graph. This bound is tight up to a constant factor for classes of non-dense graphs.
Group testing can be implemented in finding pattern in data, DNA library screening, and so on (see [[@du2000combinatorial-group-testing; @du2006pooling-design-group-testing; @Macula2004-finding-patterns-in-data; @2000-survey-group-testing]]{} for an overview of results and more applications). Learning hidden graph, especially hamiltonian cycle and matchings, is mostly applied in genome sequencing, DNA physical mapping, chemical reactions and molecular biology (see [[@2008-adaptive-learning-hidden-graph-per-edge; @2011-learning-hidden-graph-threshold; @Grebinski1998-learning-hamiltonian-cycle; @Sorokin1996-application-physical-mapping]]{} for more information about these applications). Regarding the present study, the main motive behind investigating the problem of non-adaptive group testing on graphs is the application of this problem in chemical reactions. In chemical reactions, we are dealing with a set of chemicals, some pairs of which may involve a reaction. Moreover, before testing, we know some pairs have no reaction. When some chemicals are combined in one test, a reaction takes place if and only if at least one pair of the chemicals reacts in the test. The present study aimed to identify which pairs are reacted using as few tests as possible. Therefore, we can reformulate this problem as follows. Suppose that there are $n$ vertices and two vertices $u$ and $v$ are adjacent if and only if two chemicals $u$ and $v$ may involve a reaction. The reaction of each pair of the chemicals indicates a defective edge and finding all there types of pairs is equal to find the defective subgraph. As we know some pairs have no reaction, the graph $G$ is not necessarily a complete graph.
Notation
========
Throughout this paper, we suppose that $H$ is a subgraph of $G$ with $k$ edges. Moreover, we assume that $G$ contains exactly one defective subgraph isomorphic to $H$. We denote the maximum degree of $H$ by $\Delta=\Delta(H)$. Also, $G[X]$ denotes the subgraph of $G$ induced by $X \cap V(G)$ and for any vertex $v \in G$, $N_H(v)$ stands for the set of neighbours of the vertex $v$ in the graph $H$. Hereafter, we assume that the subgraph $H$ has no isolated vertex, because in the problem of group testing on graphs, just edges are defective.
Main result
===========
For $1\leq l \leq t$, let ${\cal F}_l$ be a random test obtained by choosing each vertex of $V(G)$ independently with probability $p$. For simplicity of notation we write $F_i$ as an induced subgraph of $G$ on vertices of ${\cal F}_i$.
Throughout this paper, let $H_1,H_2,\ldots,H_m$ be all the subgraphs of $G$ isomorphic to $H$. Let $C$ be a random $t \times m$ matrix such that for any $l$ and $j$, where $1\leq j \leq m$ and $1\leq l \leq t$, if $E(F_l \cap H_j) \neq \varnothing$, then $C_{lj}=1$; otherwise, $C_{lj}=0$. The $l$th row of this matrix corresponds to the test $F_l$ and the $j$th column corresponds to the subgraph $H_j$. For any $i,j,l$, where $1\leq i \neq j \leq m$ and $1\leq l \leq t$, define the event $A_{i,j}^l$ to be the set of all matrices $C$ such that $C_{li}= C_{lj}$. Also, define the event $A_{i,j}$ to be the set of all matrices $C$ such that for every $l$, $1\leq l \leq t$, we have $C_{li}= C_{lj}$. In other words, if the event $A_{i,j}^l$ occurs, then the test $F_l$ cannot distinguish between $H_i$ and $H_j$. Also, if the event $A_{i,j}$ occurs, then for every $l$ such that $1\leq l \leq t$, the test $F_l$ cannot distinguish between $H_i$ and $H_j$. So if in the matrix $C$ each pair of columns is different, then none of the bad events $A_{i,j}$ occur and we can find the defective subgraph. So we would like to bound the probability that none of the bad events $A_{i,j}$ occur. In such cases, when there is some relatively small amount of dependence between events, one can use a powerful generalization of the union bound, known as the Lovász Local Lemma. The main device in establishing the Lovász Local Lemma is a graph called the dependency graph. Let $A_1,A_2,\ldots,A_n$ be events in an arbitrary probability space. A graph $D=(V ,E)$ on the set of vertices $V=\{1,2,\ldots ,n \}$ is a dependency graph for events $A_1,A_2,\ldots,A_n$ if for each $1\leq i \leq n$ the event $A_i$ is mutually independent of all the events $\{A_j :\{i,j\} \notin E \}$. We state the Lovász Local Lemma as follows.
[[@2008-probabilistic-method] ]{} [(]{}Lovász Local Lemma, Symmetric Case[)]{}. Suppose that $A_1,A_2,\ldots ,A_n$ are events in a probability space with $Pr(A_i)\leq p$ for all i. If the maximum degree in the dependency graph of these events is $d$, and if $ep(d+1)\leq 1$, then $$Pr \Big( \displaystyle \bigcap_{i=1}^n \overline{A_i} \Big) > 0,$$ where $e$ is the basis of the natural logarithm.
To find the maximum degree in the dependency graph of the events $A_{i,j}$, we define the parameter $r_G(H)$ as follows. Set $r_G(H, H_i)$ is the number of subgraphs of $G$ isomorphic to $H$ have common vertex with $H_i$, i.e., $r_G(H,H_i)=|\{H_j : 1\leq j \leq m, j\neq i, V(H_i)\cap V(H_j)\neq \varnothing \}|$. Also, define $$r_G(H)=\displaystyle \max_{1\leq i\leq m} r_G(H, H_i).$$
In Theorem \[main-theorem\], we show that in the aforementioned random matrix each pair of columns is different with positive probability. More precisely, in this theorem, we prove there is a $t\times m$ matrix $C$ such that for every $i$ and $j$, there is a number $l$ such that $1\leq l\leq t$ and $C_{li}\neq C_{lj}$. It happens if $E(F_l \cap H_i)=\varnothing , E(F_l \cap H_j)\neq \varnothing$ or $E(F_l \cap H_i)\neq \varnothing , E(F_l \cap H_i)=\varnothing$. So if $H_i$ is the defective subgraph, then for every non-defective subgraph $H_j$, there exists a test $F_l$ such that $E(F_l \cap H_i)\neq E(F_l \cap H_j)$. So all the tests $F_1,\ldots,F_t$ can distinguish between the defective subgraph $H_i$ and every non-defective subgraph $H_j$. Therefore, by this matrix we can find the defective subgraph.
\[main-theorem\] Let $H$ be the defective subgraph of $G$ and $H_1,H_2,\ldots,H_m$ be all the subgraphs of $G$ isomorphic to $H$. There are $t$ induced subgraph $F_1,\ldots,F_t$ of $G$ such that for each pair of $H_i$ and $H_j$, at least one of $F_1,\ldots,F_t $ can distinguish between $H_i$ and $H_j$, where $k=|E(H)|$, $\Delta=\Delta (H)$, $$t=1 + \left\lceil\frac{\ln (4 e r_G(H) ) + \ln m }{ \ln {1 \over 1 - P_{k,\Delta}} }\right\rceil ,$$ $P_{k,\Delta}= \frac{1}{2k \Delta }
\left(1-\frac{1}{ 2 \Delta } \right)^{2 \Delta - 1}
\left( 1-\sqrt{{1 \over 2k \Delta }} \left(1-\frac{1}{ 2 \Delta } \right)^{ \Delta - 1} \right)^{2\Delta -2}$, and $e$ is the basis of the natural logarithm.
In order to prove Theorem \[main-theorem\], first we should find the probability that tests $F_1,F_2,\ldots,F_t$, distinguish between each pair of subgraphs $H_i$ and $H_j$. Thus, finding the upper bound for the probability of occurring the bad event $A_{i,j}$ is essential. Accordingly, we should find the lower bound of probability that the random test $F_l$ can distinguish between two subgraphs $H_i$ and $H_j$. In the next theorem, based on some following lemmas, we show that the probability of distinguishing between $H_i$ and $H_j$ has the minimum value whenever $V(H_i)=V(H_j)$ and $|E(H_i)\setminus E(H_j) |=1$.
\[lower bound-A-[i,j]{}\^l\] Let $k=|E(H)|$ and $\Delta=\Delta (H)$. For every $1\leq i \neq j\leq m $ and $1 \leq l \leq t$, we have $$\label{probability-distinguish-dense}
Pr \left( \overline{A_{i,j}^l} \right) \geq 2p^2 (1-p)^{ 2 \Delta } (1- \epsilon),$$ where $p=\sqrt{ { \epsilon \over k } } \left( 1- \epsilon \right)^{ \Delta -1}.$
\[lovasz-independent-set\] Let $T$ be a graph with $n$ vertices, $k$ edges, and maximum degree $\Delta$. Pick, randomly and independently, each vertex of $T$ with probability $p$, where $p=\sqrt{ \frac{\epsilon}{k} } (1-\frac{\epsilon}{k})^{(\Delta -1)}$. If $F$ is the set of all chosen vertices, then $T[F]$ has no edges, with probability at least $1-\epsilon$.
To prove this lemma, we need high probability variation of Lovász Local Lemma.
\[high-probability-lovasz\][[@combinatorial-array-stinson]]{} Let $B_1,B_2,\ldots ,B_k$ be events in a probability space. Suppose that each event $B_i$ is independent of all the events $B_j$ but at most d. For $1 \leq i \leq k$ and $0 < \epsilon <1$, if $Pr(B_i) \leq {\epsilon \over k }(1- {\epsilon \over k} )^d$, then $Pr \Big( \displaystyle \bigcap_{i=1}^k \overline{B_i} \Big) > 1- \epsilon $.
**Proof of Lemma \[lovasz-independent-set\].** Let $E(T) =\{ e_1,e_2,\ldots,e_k \}$. For $1\leq i \leq k$, we define $B_i$ to be the event that $e_i \in E(T[F])$, so $Pr(B_i)=p^2$. Since vertices are chosen randomly and independently, the event $B_i$ is independent of the event $B_j$ if and only if edges $e_i$ and $e_j$ have no common vertex. So the maximum degree of the dependency graph is at most $2 ( \Delta -1 ) $. Since $p^2 \leq {\epsilon \over k }\left( 1- {\epsilon \over k} \right)^ {2 (\Delta -1)}$, by Lemma \[high-probability-lovasz\], $Pr\Big(\displaystyle \bigcap_{i=1}^k \overline{B_i} \Big) > 1- \epsilon$. Hence, $T[F]$ has no edges, with probability at least $1-\epsilon$.\
To find the probability of distinguishing between $H_i$ and $H_j$ and then prove Theorem \[lower bound-A-[i,j]{}\^l\], we consider following three cases,
1. $V(H_i)=V(H_j)$, $|E(H_i)\setminus E(H_j) |=1$.
2. $|V(H_i)\setminus V(H_j)| \geq 1$.
3. The induced subgraph on $V(H_i) - V(H_j)$ has at least one edge.
\[lem-similar1\] If $V(H_i)=V(H_j)$ and $|E(H_i)\setminus E(H_j) |=1$, then $$Pr\big( E(F_l \cap H_i)\neq \varnothing , E(F_l \cap H_j) = \varnothing \big) \geq p^2 (1-p)^{2 \Delta } (1- \epsilon),$$ where $p=\sqrt{ { \epsilon \over k } }\left( 1- \epsilon \right)^{ \Delta - 1}$.
Let $e=\{ u,v \}\in E(H_i)\setminus E(H_j)$. Consider the induced subgraph $H'$ of $G$, where $V(H')=V(H_j) \setminus \Big( u \cup v \cup N_{H_j}(u) \cup N_{H_j}(v) \Big)$. Note that if $u,v \in {\cal F}_l $ and $H_j\cap F_l$ has no edges of $H_j$, then $E(F_l \cap H_i)\neq \varnothing $ and $E(F_l \cap H_j) = \varnothing $. Also, one can see that $u,v \in {\cal F}_l $ and $H_j[F_l]$ has no edges if the following events hold
1. $u,v \in {\cal F}_l $,
2. $N_{H_j}(u) \cap {\cal F}_l = \varnothing$ and $ N_{H_j}(v) \cap {\cal F}_l= \varnothing $,
3. $H'[F_l]$ has no edges.
It is straightforward to check that the aforementioned events are independent. Also, one can see that the event $u,v \in {\cal F}_l $ occurs with probability $p^2$. Since $ | N_{H_j}(u) \cup N_{H_j}(v) | \leq 2 \Delta $, $$\begin{array}{rcl}
Pr \left(N_{H_j}(u) \cap {\cal F}_l = \varnothing , N_{H_j}(v) \cap {\cal F}_l = \varnothing \right) & = & \\
Pr \Big( {\cal F}_l \cap \left( N_{H_j}(u) \cup N_{H_j}(v) \setminus \{u, v\} \right) =\varnothing \Big)
& \geq & (1-p)^{2\Delta}.
\end{array}$$
Set $E(H')=k'$. If $k' = 0 $, then $F_l \cap H'$ has no edges. So $Pr \big( E( F_l \cap H' )= \varnothing \big) = 1 $. Suppose that $k' \geq 1$. Since $k\geq k'$, we have $p^2={\epsilon \over k}(1-\epsilon)^{2 \Delta -2} \leq {\epsilon \over k'}(1-{\epsilon \over k'})^{2\Delta - 2 }$. Each vertex of the induced subgraph $H'$ is chosen with probability $p$. So by Lemma \[lovasz-independent-set\], the induced subgraph on ${\cal F}_l \cap V(H')$ has no edges, with probability at least $1- \epsilon$. In other words, $Pr \big( E(F_l \cap H')= \varnothing \big) \geq 1- \epsilon $. Since the events are independent, we have $$Pr \big(E(F_l \cap H_i)\neq \varnothing , E(F_l \cap H_j) = \varnothing \big) \geq
p^2 (1-p)^{2 \Delta } (1- \epsilon),$$ as desired.
\[lem-similar2\] If $|V(H_i)\setminus V(H_j)| \geq 1$, then $$Pr\Big( E(F_l \cap H_i)\neq \varnothing , E(F_l \cap H_j) = \varnothing \Big) \geq p^2 (1-p)^{\Delta } (1- \epsilon),$$ where $p=\sqrt{ { \epsilon \over k } }\left( 1- \epsilon \right)^{ \Delta -1}$.
Since $H$ has no isolated vertex, there exists at least one edge $e=\{ u,v\} \in E(H_i)\setminus E(H_j)$. Let $v \in V(H_i) \cap V(H_j) $ and $u \in V(H_i) \setminus V(H_j)$. Suppose that $H'$ is an induced subgraph of $H_j$, where $V(H')=V(H_j) \setminus \left( v \cup N(v) \right)$. Set $|E(H')|=k'$. Similar to the proof of Lemma \[lem-similar1\], $E(F_l \cap H_i)\neq \varnothing $ and $E(F_l \cap H_j) = \varnothing $ if the following independent events hold
1. $u,v \in {\cal F}_l $,
2. $ N_{H_j}(v) \cap {\cal F}_l= \varnothing $,
3. $H'[F_l]$ has no edges.
Since $ |N_{H_j}(v)| \leq \Delta$, the probability that $N_{H_j}(v) \cap {\cal F}_l =\varnothing$ is at least $(1-p)^{\Delta } $. The rest of proof is similar to Lemma \[lem-similar1\], so $$Pr \Big(E(F_l \cap H_i)\neq \varnothing , E(F_l \cap H_j) = \varnothing \Big) \geq
p^2 (1-p)^{ \Delta } (1- \epsilon),$$
as desired.
\[lem-similar3\] If the induced subgraph on $V(H_i)\setminus V(H_j)$ has at least one edge, then $$Pr\Big( E(F_l \cap H_i)\neq \varnothing , E(F_l \cap H_j) = \varnothing \Big) \geq p^2 (1- \epsilon),$$ where $p=\sqrt{ { \epsilon \over k } }\left( 1- { \epsilon } \right)^{ \Delta - 1}$.
Let $e=(u,v) \in E(H_i)\setminus E(H_j)$. If the following independent events hold
1. $u,v \in {\cal F}_l$,
2. $H_j[F_l]$ has no edges,
then $E(F_l \cap H_i)\neq \varnothing $ and $E(F_l \cap H_j) = \varnothing $. Since $p^2 = { \epsilon \over k } \left( 1- { \epsilon } \right)^{2 \Delta -2}
\leq { \epsilon \over k } \left( 1- { \epsilon \over k } \right)^{2 \Delta -2}$, by Lemma \[lovasz-independent-set\], $Pr\left( E(F_l \cap H_j) = \varnothing \right) \geq 1- \epsilon $. Also one can see that $$Pr\big(E(F_l \cap H_i)\neq \varnothing \big) \geq Pr\big(e \in E(F_l) \big) = Pr(u,v \in {\cal F}_l )= p^2.$$ Consequently, $
Pr \big(E(F_l \cap H_i)\neq \varnothing , E(F_l \cap H_j) = \varnothing \big) \geq p^2 (1- \epsilon).
$
**Proof of Theorem \[lower bound-A-[i,j]{}\^l\].** Let $E(H_i)\cap E(H_j)= \{f_1,f_2,\ldots,f_r \}$ and $E(H_i) \setminus E(H_j)= \{e_1,e_2,\ldots,e_{k-r}\}$. As previously(at the first of this paper ) mentioned, the event $ \overline{ A_{i,j}^l }$ occurs if and only if $E(F_l \cap H_i)\neq \varnothing , E(F_l \cap H_j)= \varnothing$ or $E(F_l \cap H_j)\neq \varnothing , E(F_l \cap H_i)= \varnothing$. It is easy to check that $$Pr\left(\overline{ A_{i,j}^l }\right)=
Pr\Big(E(F_l \cap H_i)\neq \varnothing , E(F_l \cap H_j)= \varnothing\Big)+
Pr\Big(E(F_l \cap H_j)\neq \varnothing , E(F_l \cap H_i)= \varnothing\Big).$$ In the following we prove $Pr(E(F_l \cap H_i)\neq \varnothing , E(F_l \cap H_j)= \varnothing)\geq p^2 (1-p)^{ 2 \Delta } (1- \epsilon)$ and with the completely similar proof we can prove $Pr(E(F_l \cap H_j)\neq \varnothing , E(F_l \cap H_i)= \varnothing)\geq p^2 (1-p)^{ 2 \Delta } (1- \epsilon)$.
It is easy to check, for every $1\leq q \leq k - r $, $$Pr\Big( E(F_l \cap H_i)\neq \varnothing , E(F_l \cap H_j)=\varnothing \Big)
\geq Pr\Big(e_q\in E(F_l \cap H_i) , E(F_l \cap H_j)=\varnothing \Big).$$ So to find the lower bound for this probability, we need to consider the following three cases,
1. $V(H_i)=V(H_j)$, $|E(H_i)\setminus E(H_j) |=1$.
By Lemma \[lem-similar1\], it is clear $$Pr\big( E(F_l \cap H_i)\neq \varnothing , E(F_l \cap H_j) = \varnothing \big) \geq p^2 (1-p)^{2\Delta } (1- \epsilon).$$
2. $|V(H_i)\setminus V(H_j)| \geq 1$.
By Lemma \[lem-similar2\], we have $$Pr\big( E(F_l \cap H_i)\neq \varnothing , E(F_l \cap H_j) = \varnothing \big)
\geq p^2 (1-p)^{\Delta } (1- \epsilon)
\geq p^2 (1-p)^{2 \Delta } (1- \epsilon).$$
3. The induced subgraph on $V(H_i) - V(H_j)$ has at least one edge.
By Lemma \[lem-similar3\], $$Pr\big( E(F_l \cap H_i)\neq \varnothing , E(F_l \cap H_j) = \varnothing \big)
\geq p^2 (1 - \epsilon ) \geq p^2 (1-p)^{ 2 \Delta } (1- \epsilon).$$
So for every $1 \leq i\neq j \leq m$ and $1 \leq l \leq t$, $Pr \left( \overline{A_{i,j}^l} \right) \geq 2p^2 (1-p)^{ 2 \Delta } (1- \epsilon)$.\
In order to prove Theorem \[main-theorem\], we present an upper bound for the probability of occurring the bad events $A_{i,j}$ for every $1\leq i\neq j \leq m$.
\[theorem-upper-bound-A-[i,j]{}\] Let $k=|E(H)|$ and $\Delta=\Delta (H)$. For every $1\leq i \neq j\leq m $, we have $$\label{upper bound-A-{i,j}}
Pr(A_{i,j}) \leq (1-P_{k,\Delta})^t,$$ where $P_{k,\Delta}= \frac{1}{2k \Delta }
\left(1-\frac{1}{ 2 \Delta } \right)^{2 \Delta - 1}
\left( 1-\sqrt{{1 \over 2k \Delta }} \left(1-\frac{1}{ 2 \Delta } \right)^{ \Delta - 1} \right)^{2\Delta -2}$.
[ Since ${\cal F}_1, {\cal F}_2, \ldots , {\cal F}_t \subset V(G)$ are chosen randomly and independently, the events $A_{i,j}^1, \ldots , A_{i,j}^t$ are mutually independent. So $$Pr(A_{i,j})= Pr(A_{i,j}^l)^t.$$ By the definition of $\overline{ A_{i,j}^l }$ and Theorem \[lower bound-A-[i,j]{}\^l\], we have $Pr \left( \overline{ A_{i,j}^l } \right)
\geq 2 p^2 (1-p)^{ 2 \Delta } (1- \epsilon).$ Acoording to $p=\sqrt{ { \epsilon \over k } }\left( 1- { \epsilon } \right)^{ \Delta - 1}$, we set $\epsilon ={3\over \Delta}$ to almostly maximazie the lower bound of good events $\overline{A_{i,j}^l}$. So $Pr \left( \overline{A_{i,j}^l} \right) \geq P_{k,\Delta}$, where $$P_{k,\Delta}= \frac{6}{k \Delta }
\left(1-\frac{3}{ \Delta } \right)^{2 \Delta - 1}
\left( 1-\sqrt{{3 \over k \Delta }} \left(1-\frac{3}{ \Delta } \right)^{ \Delta - 1} \right)^{2\Delta }.$$ Therefore, $Pr(A_{i,j})= Pr(A_{i,j}^l)^t \leq (1-P_{k,\Delta})^t.$ ]{}
Now, we can prove Theorem \[main-theorem\].
**Proof of Theorem \[main-theorem\].** By Theorem \[theorem-upper-bound-A-[i,j]{}\], for every $1\leq i\neq j \leq m$, $Pr( A_{i,j} )\leq (1 - P_{k,\Delta})^t$. Now we prove that if $t > \frac{\ln (4 e r_G(H) ) + \ln m }{ \ln {1 \over 1 - P_{k,\Delta}} }$, then by Lovász Local Lemma, with positive probability no event $A_{i,j}$ occurs. We construct the dependency graph whose vertices are the events $A_{i,j}$, where $1\leq i,j \leq m$. Two events $A_{i,j}$ and $A_{i',j'}$ are adjacent if and only if $ \Big( V(H_i) \cup V(H_j) \Big) \cap \Big( V(H_{i'}) \cup V(H_{j'}) \Big) \neq \varnothing $. Remember that $r_G(H)=\displaystyle \max_i r_G(H, H_i)$, where $r_G(H, H_i)$ is the number of subgraphs of $G$ isomorphic to $H$ including common vertex with $H_i$. For the fixed $A_{i,j}$, there are at most $r_G(H)$ subgraph $H_{i'}$ isomorphic to $H$ such that $V(H_i)\cap V(H_{i'}) \neq \varnothing$. We can choose $H_{j'}$ with $m-1$ ways. So it is easy to check that the maximum degree in the dependency graph is at most $4r_G(H)(m-1)$. Accordingly, if $$t >\frac{\ln (4 e r_G(H) ) + \ln m }{ \ln {1 \over 1 - P_{k,\Delta}} },$$ then $e \left(1- P_{k,\Delta} \right)^t \big(4 r_G(H) (m-1) + 1 \big) < 1 $, and by Lov' asz Local Lemma $$Pr \Big(\displaystyle \bigcap_{i,j} \overline{ A_{i,j} } \Big) >0.$$ Therefore, if $t=1 + \lceil\frac{\ln (4 e r_G(H) ) + \ln m }{ \ln {1 \over 1 - P_{k,\Delta}} }\rceil $, then with positive probability no event $A_{i,j}$ occurs. Thus, there is $t$ tests ${\cal F}_1, {\cal F}_2,\ldots, {\cal F}_t$ that can distinguish between each pair of $H_i$ and $H_j$.\
We can obtain $t=1+\lceil\frac{2 \ln m}{\ln {1 \over {1 - P_{k,\Delta}} }} \rceil$ if we use union bound. In fact the Lovász Local Lemma is better when the dependencies between events are rare. Based on this theorem there are $t$ tests which distinguish between each pair of $H_i$ and $H_j$ with positive probability. However, an algorithm is essential to find these tests with high probability if we are interested in finding these tests.
\[Algorithm\] Let $H$ be the defective subgraph of $G$ with $k$ edges. If $t={\ln {m^2\over \delta}\over \ln {1\over 1 -P_{k,\Delta}}} $, we can find this defective subgraph by $t$ tests with probability at least $1-\delta$, where $\Delta=\Delta (H)$ and $P_{k,\Delta}= \frac{1}{2k \Delta }
\left(1-\frac{1}{ 2 \Delta } \right)^{2 \Delta - 1}
\left( 1-\sqrt{{1 \over 2k \Delta }} \left(1-\frac{1}{ 2 \Delta } \right)^{ \Delta - 1} \right)^{2\Delta -2}$.
[For events $A_{i,j}$ where $1\leq i<j\leq m$, by Theorem \[theorem-upper-bound-A-[i,j]{}\] and union bound we know $Pr\big(\bigcup A_{i,j} \big)\leq m^2 (1-P_{k,\Delta})^t$. Thus, this upper bound becomes close to zero if $t$ is large enough. It is easy to check if $t = {\ln {m^2\over \delta}\over \ln {1\over 1 -P_{k,\Delta}}}$, then $m^2 (1-P_{k,\Delta})^t = \delta$. In other words, we can distinguish between each pair of $H_i$ and $H_j$ with probability at least $1-\delta$ if we choose tests randomly and independently. ]{}
For simplicity suppose $m$, the number of subgraph isomorphic to $H$, is more than $n$. Suppose for big $m$, $\delta={1\over m}$. Therefore, we can find the defective subgraph with $\frac{3 \ln m}{\ln {1 \over {1 - P_{k,\Delta}} }}$ tests with high probability.
Concluding remarks
==================
In the present paper we assume that the graph $G$ includes few edges since the Lovász Local Lemma is more powerful when the dependencies between events are rare. In the graph $G$ with $O(n^2)$ edges, the parameter $r_G(H)$ is high, which is better to use the union bound. In this case, we can find the defective subgraph with $t=1+\lceil\frac{2 \ln m}{\ln {1 \over {1 - P_{k,\Delta}} }} \rceil$ non-adaptive tests. Finally, if we consider dense and spars defective subgraph separately, we can obtain a little better upper bound for the number of tests in the case of spars defective subgraph.
Acknowledgement {#acknowledgement .unnumbered}
===============
The present study is a part of Hamid Kameli’s Ph.D. Thesis. The author would like to express his deepest gratitude to Professor Hossein Hajiabolhassan to introduce a generalization of learning a hidden subgraph problem and for his invaluable comments and discussion.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
The inherent probabilistic nature of the biochemical reactions, and low copy number of species can lead to stochasticity in gene expression across identical cells. As a result, after induction of gene expression, the time at which a specific protein count is reached is stochastic as well. Therefore events taking place at a critical protein level will see stochasticity in their timing. First–passage time (FPT), the time at which a stochastic process hits a critical threshold, provides a framework to model such events. Here, we investigate stochasticity in FPT. Particularly, we consider events for which controlling stochasticity is advantageous. As a possible regulatory mechanism, we also investigate effect of auto–regulation, where the transcription rate of gene depends on protein count, on stochasticity of FPT. Specifically, we investigate for an optimal auto-regulation which minimizes stochasticity in FPT, given fixed mean FPT and threshold.
For this purpose, we model the gene expression at a single cell level. We find analytic formulas for statistical moments of the FPT in terms of model parameters. Moreover, we examine the gene expression model with auto–regulation. Interestingly, our results show that the stochasticity in FPT, for a fixed mean, is minimized when the transcription rate is independent of protein count. Further, we discuss the results in context of lysis time of an *E. coli* cell infected by a $\lambda$ phage virus. An optimal lysis time provides evolutionary advantage to the $\lambda$ phage, suggesting a possible regulation to minimize its stochasticity. Our results indicate that there is no auto–regulation of the protein responsible for lysis. Moreover, congruent to experimental evidences, our analysis predicts that the expression of the lysis protein should have a small burst size.
author:
- 'Khem Raj Ghusinga$^{1}$ and Abhyudai Singh$^{2}$[^1][^2]'
bibliography:
- 'references.bib'
title: '**Optimal first–passage time in gene regulatory networks** '
---
Introduction
============
Gene expression is the process of *transcription* of genetic information to mRNAs, and *translation* of each mRNA to proteins. As the copy number of species involved in the process is small, the probabilistic nature of biochemical reactions reflects as stochastcity in gene expression [@Blake_noise_2003; @Raser_noiseocc_2005; @ArjunRaj_nnc_2008; @Munsky_noisetoregulation_2012; @Kaern_TP_2005; @AbhiMohammad_var_2013].
Stochasticity in gene expression has an important role in several cellular functions. For example, it can lead genetically identical cells to different cell–fates [@Losick_cellfate_2008; @Arkin_ska_1998; @Weinberger_lentiviral_2005; @Veening_bistability_2008; @Hasty_switchamp_2000; @abhi_transcriptionalburstingHIV_2010]. This helps the cells in responding to the ever–changing environment [@Eldar_functional_2010; @Kussell_pd_2005; @Balaban_bactpersist_2004; @Murat_survival_2008]. On the other hand, stochasticity in expression of housekeeping genes can lead to diseased states [@Kemkemer_incnoise_2002; @Cook_modeling_1998; @Bahar_incvariation_2006], and needs to be minimized [@Lehner_noiseminm_2008; @Fraser_noiseminm_2004]. Accordingly, different regulatory mechanisms are employed to control stochastic fluctuations [@UriAlon_nm_2007; @Becskei_engineeringstability_2000; @ElSamad_regulated_2006; @Swain_attenuatestochasticity_2004; @Orrell_control_2004; @abhi_fbstrength_2009; @Tao_effectoffb_2007; @abhi_mRNA_2011]. Auto–regulation wherein transcription rate is a function of protein count is an example of one such mechanism. Its effect on stochascticity in gene expression has been a subject of several studies [@abhi_fbstrength_2009; @Tao_effectoffb_2007; @abhi_mRNA_2011].
After onset of gene expression, its stochasticity consequently manifests into stochasticity in the time at which a certain protein level is reached. This implies that the timing of a cellular event which triggers at a critical protein level is stochastic in nature [@Amir_noisetiming_2007; @Murugan_fluctuation_2011]. For instance, lysis time for an *E. coli* cell infected by a $\lambda$ phage virus is stochastic. Lysis of the cell takes place when holin, the protein responsible for lysis, reaches a critical threshold [@White_holintriggering_2011; @JohnDennehy_lst_2011; @Abhi_ltv_2014].
Further, it has been suggested that optimality in lysis time provides evolutionary advantage to $\lambda$ phage virus [@INWang_fitness_2006; @Wang_evolutiontiming_1996; @Heineman_optimal_2007; @Shao_adsorptionoptimal_2008; @Bonachela_optimallysis_2014]. This indicates that there could be some regulation of gene expression to ensure lysis at the optimal time, with minimum stochastic fluctuations. In this work, we study stochasticity in first–passage time (FPT), the time it takes for the protein count to reach a fixed threshold for the first time [@redner2001guide], at a single–cell level. We investigate the effect of auto-regulation of transcription on stochasticity of FPT. In particular we seek answer to the question: given the mean FPT (corresponding to optimal lysis time, for example), what auto-regulatory feedback will lead to minimum stochasticity in the FPT?
We first formulate an unregulated gene expression model assuming transcription, translation, and mRNA degradation while considering proteins to be stable. Along the lines of [@Abhi_ltv_2014], we find expressions for statistical moments of FPT for this model, and discuss their implications with respect to minimizing variance in FPT for given mean FPT. Next, we introduce auto-regulation in the above model and derive the moments for FPT. Then, we deduce the expression for optimal feedback function that minimizes the variance in FPT for a given mean. We show that a negative or positive feedback always results into higher variance in first passage time for a given mean than the case when there is no feedback. The results are validated by carrying out simulations. Also, various notations used in this work are tabulated in Table I.
----------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------
$k_m$ Transcription rate for unregulated gene expression model.
$k_p$ Translation rate for both unregulated, and regulated gene expression. models
$\gamma_m$ mRNA degradation rate for both unregulated, and regulated gene. expression models
$B_i$ Burst size after $i^{th}$ transcriptional event.
$\mu$ Parameter of geometric distribution corresponding to. protein bursts
$b$ Mean of protein burst size.
$P(t)$ Protein count at time $t$.
$P_{i}$ Protein count after $i^{th}$ burst.
$k_m(P_i)$ Transcription rate for auto-regulated gene expression model after $i^{th}$ transcription event.
$X$ Threshold for protein count.
$N$ Minimum number of transcription events for protein count to reach the threshold $X$.
$T_i$ Waiting time for $i^{th}$ transcription event.
$Y \sim \exp(\alpha)$ $Y$ is an Exponential random variable with parameter $\alpha$. The probability density function of $Y$ is given by $f_Y(y)=\alpha e^{-\alpha y},\;y\geq 0$.
$f_N(n)$ Probability mass function for minimum number of transcription events to reach the threshold $X$.
$f_{P_i}(j)$ Probability mass function for protein count after $i$ transcription events.
$\left< .\right>$ Expectation operator.
Var Variance.
$k_{\max}$ Maximum possible transcription rate in model with feedback implemented using Hill function .
$r$ Fraction of transcription rate $k_{\max}$ that corresponds to minimum transcription rate in model with feedback implemented using Hill function.
$H$ Hill coefficient.
$c$ Coefficient proportional to binding efficiency; decides when half rate concentration is reached.
----------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------
: Description of notations used in this work
\[tab:notations\]
First–Passage Time For Gene Expression Model Without Regulation
===============================================================
In this section, we formulate a stochastic gene expression model (as shown in Fig. \[fig:geneexpression\]). Then, we define the FPT for this model and derive expressions for its statistical moments. We also discuss the implications of these expressions in context of minimizing variance of FPT, for fixed mean and threshold.
Model Formulation
-----------------
In the model under consideration transcription of mRNAs from the gene occurs at a rate $k_m$, translation of proteins from each mRNA occurs at a rate $k_p$, and each mRNA degrades at a rate $\gamma_m$. The time interval between two transcription events is exponentially distributed. We assume proteins to be stable as the lysis protein in $\lambda$ phage, i.e. holin, is stable [@Shao_holinstable_2009]. To further simplify the model, we assume each mRNA molecule degrades instantaneously after producing a burst of random number of protein molecules [@Friedman_pd_2006; @ShahrezaeVahid_ad_2008; @Paulsson_sge_2005; @Berg_statfluct_1978]. Consistent with experimental, and theoretical evidences; we assume that protein burst follows a geometric distribution, and the mean burst size is given by $b=k_p/\gamma_m$ [@YuXiao_pom_2006; @Elgart_connecting_2011]. Thus, the simplified model considers gene expression wherein each burst event (equivalent to transcription event) occurs at an exponentially distributed time with parameter $k_m$, and size of burst follows a geometric distribution with mean $b$.
Let us denote the size of $i^{th}$ burst by random variable $B_i$ and the parameter of its distribution by $\mu$. The probability mass function, therefore, can be written as [@Degroot_prob_2012]: $$\label{eqn:goemetricburstpmf}
\text{Pr}(B_i=k)=\mu \left(1-\mu \right)^k,\,\mu \in (0,1],\,k \in \{0,1,2..\}.$$
The mean burst size, $b$, can be expressed as [@Degroot_prob_2012]: $$\left<B_i\right>=b = \frac{1-\mu}{\mu}.$$ Further, let protein count after $n$ transcription events be denoted as $P_n$. It can be expressed as a sum of random variables $B_i$: $$\label{eqn:proteincount}
P_n=\sum_{i=1}^{n} B_i.$$ Being sum of independent and identically distributed geometric random variables, $P_n$ has a negative binomial distribution with parameters $n$ and $\mu$ [@Papoulis_prob_2002]. The probability mass function of $P_n$, denoted as $f_{P_n}(j)$, can be expressed as [@Papoulis_prob_2002]: $$\label{eqn:proteincountpmf}
f_{P_n}(j)=\text{Pr}\left(\sum_{i=1}^{n}B_i = j \right) = {n+j-1 \choose n-1}\mu^n\left(1-\mu\right)^j.$$
Also, the cumulative distribution function is given by [@Spiegel_scham_1992]: $$\label{eqn:proteincountcdf}
\text{Pr} \left(\sum_{i=1}^{n}B_i \leq j \right) = 1- I_{1-\mu}(j+1,n),$$ where $I_{1-\mu}(j+1,n)$ is regularized incomplete beta function: $$I_{1-\mu}(j+1,n) = \sum_{l=j+1}^{n+j} {n+j \choose l} (1-\mu)^l \mu^{j+n-l},$$ and satisfies the following property: $$\label{eqn:Iproperty}
I_{1-\mu}(j+1,n) = 1-I_{\mu}(n,j+1).$$ We have determined the distribution for protein population. Next, we defined the first–passage time (FPT) for the protein count to reach a certain threshold.
Expression for First Passage Time
---------------------------------
For a random process corresponding to protein count, $P(t)$, with $P(0)=0$, the first passage time (FPT), for a threshold $X$ is defined as: $$\label{eqn:firstpassagetime}
FPT:= \inf \{t: P(t) \geq X \}, \quad X \in \{1, 2, 3, ...\}.$$ Because in our model, the protein count changes only when a burst occurs (or equivalently, a transcription event occurs); we can calculate the minimum number of transcription events, $N$, it takes for the protein count to reach the threshold $X$ and define the FPT as sum of inter–burst arrival times. This has been depicted in Fig. \[fig:proteincount\].
Let the time between ${i-1}^{th}$ and $i^{th}$ bursts be denoted by random variable $T_i$, then: $$\label{eqn:fptdef}
FPT=\sum_{i=1}^{N}T_i,$$ $$\label{eqn:Ndefinition}
N=\inf \left( n: P_n \geq X\right), \quad n \in \{1,2,...\}, \;\; X \geq 1.$$ Note that in Eq. , $T_i$ are independent, and identically distributed exponential random variables with parameter $k_m$. We denote this by $T_i \sim \exp(k_m)$. Also, each of $T_i$ is independent of $N$.
Using standard results from probability theory, one may write [@Ross_prob_2010]:
$$\begin{aligned}
\left< FPT \right> &= \left<N \right> \left<T_i \right> \label{eqn:FPTMeanNOFB}, \\
\text{Var}(FPT) & = \left<N\right>\text{ Var }(T_i) + \text{ Var } (N) \left<T_i\right>^2 \label{eqn:FPTVarNOFB}.\end{aligned}$$
It can be noted that to determine statistical moments of FPT in Eq. –, we need to derive expressions for first two moments of $T_i$, and $N$.
### First Two Moments of $N$
The cumulative distribution function for $N$ defined in Eq. can be written as:
$$\begin{aligned}
\text{Pr} (N \leq n) &= \text{Pr} \left(P_n \geq X \right), \\
&= 1- \text{Pr}\left(P_n \leq X-1 \right).\end{aligned}$$
Since $\displaystyle P_n$ is a negative binomial distribution, we have:
$$\begin{aligned}
\text{Pr} (N \leq n) &= 1-\left(1-I_{1-\mu}(X,n)\right), \\
&= I_{1-\mu}(X,n) \label{eqn:cdfN1}.\end{aligned}$$
Using the property of incomplete beta function mentioned in Eq. , we get: $$\text{Pr} (N \leq n)=1-I_{\mu}(n,X) \label{eqn:cdfN}.$$
Comparing with Eq. and Eq. , the probability mass function corresponding to Eq. can be written as:
First two statistical moments of the distribution in Eq. are given by [@Papoulis_prob_2002]: $$\begin{aligned}
\left<N\right> &= \frac{\mu X}{1-\mu}+1=\frac{X}{b}+1, \label{eqn:meanN}\\
\text{Var}(N)&=\left<N^2\right>-\left<N\right>^2=\frac{\mu X}{(1-\mu)^2}=\frac{X}{b}\frac{1+b}{b}. \label{eqn:varN}\end{aligned}$$
### First Two Moments of $T_i$
Since $T_i \sim \exp(k_m)$, its statistical moments are given by:
$$\begin{aligned}
\left< T_i \right> &= \frac{1}{k_m} \label{eqn:TimeanIID},\\
\text{Var}(T_i) &= \left< T_i^2 \right>-\left< T_i \right>^2 = \frac{1}{k_m^2}=\left< T_i \right>^2\label{eqn:TivarIID}.\end{aligned}$$
We now have expressions for first two moments of $T_i$, and $N$. The expressions for first two moments of FPT in terms of model parameters can, therefore, be written as: $$\begin{aligned}
\left<FPT\right> &= \left(\frac{X}{b}+1\right)\frac{1}{k_m} \approx \frac{X}{bk_m}, \label{eqn:FPTmeanmodel}\\
\text{Var}(FPT) &= \frac{X(2b+1)+b^2}{b^2k_m^2}\approx \frac{X}{b^2k_m^2}(1+2b), \label{eqn:FPTvarmodel}\end{aligned}$$ where the approximations are valid when $X \gg b$. It can be observed a smaller mean burst size $b$ would result in smaller variance of FPT. The mean FPT can be kept fixed by a commensurate change in the transcription rate, $k_m$. Therefore, the variance can independently be reduced by a lower mean burst size $b=k_p/\gamma_m$. This means adopting a high transcription rate $k_m$, and a low translation rate $k_p$ (and/or having a higher degradation rate $\gamma_m$ for the mRNAs) results in a lower variance in FPT without affecting its mean.
Further, we note that by using $\text{Var}(T_i)=\left<T_i\right>^2$ from Eq. , we can deduce the following relationship between $\left<FTP\right>$ and $\left<FPT^2\right>$ from Eq. and Eq. : $$\label{eqn:RelnFPTMomentsNoFB}
\left<FPT^2\right> = \frac{\left<FPT\right>^2}{\left<N\right>^2}\left<N^2\right>+\frac{\left<FPT\right>^2}{\left<N\right>}.$$ We shall use above relationship in the later part of the paper while deriving expression of the auto-regulation function that minimizes variance in FPT, for given mean FPT.
Next, we introduce auto-regulation of transcription rate by the protein count to investigate how the expressions for statistical moments of FPT change.
Introducing Auto-regulation in Gene Expression Model
====================================================
To investigate the effect of auto-regulation on statistical moments of FPT, we assume that transcription rate is a function of protein count, i.e., it changes after each transcription event. We denote the transcription rate after arrival of $i^{th}$ burst as $k_m(P_{i})$. Similar to previous section, we need to derive expression for moments of inter–burst arrival times $T_i$, and minimum number of transcription events $N$ in order to derive the expression for FPT moments defined in Eq. .
We note that the translation burst size is independent of the transcription rate. Therefore, distribution of $N$ to reach a certain threshold $X$ is same as gene expression model without any regulation discussed in previous section. However, distribution of each $T_i$ is different and depends upon corresponding rate of transcription.
We derive expressions for first two moments of each $T_i$ to find analytical forms of first two moments of FPT.
Inter–burst arrival time for auto-regulatory gene expression model
------------------------------------------------------------------
It may be noted that if protein count after any burst event is known, arrival time for the next burst will be exponentially distributed. Therefore, the distribution of each $T_i$ can be modelled as a conditional exponential distribution. More specifically, we can write: $$T_i \sim \exp \left(k_m(P_{i-1})| P_{i-1}\right),$$
where $T_i$, and $P_{i-1}$ respectively denote the arrival time for $i^{th}$ burst and protein count after the ${i-1}^{th}$ burst.
The expressions for mean and variance of $T_i$ can be calculated as follows.
### Mean
Before arrival of the first burst, there are no protein molecules, i.e., $P_{i-1}=0$ for $i=1$. Therefore, we can write the mean for arrival time for the first burst as: $$\label{eqn:meanT1}
\left<T_1\right>=\frac{1}{k_m(0)}.$$ For $i \in \{2, 3, 4...\}$, the corresponding arrival times would be conditionally exponential, implying:
$$\begin{aligned}
\left<T_i|P_{i-1}=j\right> &=\frac{1}{k_m(j)}, \\
\implies \left<T_i\right>&=\sum_{j=0}^{\infty} \frac{1}{k_m(j)} \text{Pr}\left(P_{i-1}=j\right), \\
&=\sum_{j=0}^{\infty} \frac{1}{k_m(j)} f_{P_{i-1}}(j). \label{eqn:meanTi}\end{aligned}$$
### Second Order Moments
Adopting similar approach as above, we derive the expressions for second order moments of $T_i$. For $i=1$, we have:
$$\label{eqn:secondmomentT1}
\left<T_1^2\right> = \frac{2}{k_m^2(0)}.$$
For $ i \in \{2,3,4...\}$: $$\begin{aligned}
\left<T_i^2|P_{i-1}=j\right> &=\frac{2}{k_m^2(j)}, \\
\implies \left<T_i^2\right>&=\sum_{j=0}^{\infty} \frac{2}{k_m^2(j)} \text{Pr}\left(P_{i-1}=j\right), \\
&=\sum_{j=0}^{\infty} \frac{2}{k_m^2(j)} f_{P_{i-1}}(j). \label{eqn:secondmomentTi}\end{aligned}$$
Therefore the expression for variance of $T_1$: $$\label{eqn:varT1}
\text{Var} (T_1)=\frac{1}{k_m^2(0)}=\left<T_1\right>^2.$$ For $i \in \{2,3,4,...\}$, the expression for $\text{Var}(T_i)$ will be $$\label{eqn:varTi}
\text{Var}(T_i)=\sum_{j=0}^{\infty} \frac{2}{k_m^2(j)} f_{P_{i-1}}(j) - \left[ \sum_{j=0}^{\infty} \frac{1}{k_m(j)} f_{P_{i-1}}(j) \right]^2.$$
Moreover, we have following relationship first two moments of the random variable $1/k_m(P_{i-1})$: $$\sum_{j=0}^{\infty} \frac{1}{k_m^2(j)} f_{P_{i-1}}(j) \geq \left[ \sum_{j=0}^{\infty} \frac{1}{k_m(j)} f_{P_{i-1}}(j) \right]^2,$$ which alongwith Eq. , and yields: $$\label{eqn:varmeanrelnTi}
\text{Var}(T_i)\geq \left< T_i\right>^2.$$ We note that the equality above holds for $i=1$. We will use it in later part of the paper while deducing the expression for optimal auto-regulation that leads to minimum variance in the FPT for fixed mean.
Having derived the expressions for moments of inter–bursts arrival times, we see how the introduction of auto-regulation influences the expressions for FPT moments.
FPT for auto-regulatory gene expression model
---------------------------------------------
We present the expressions for statistical moments of FPT in theorem–proof format. In developing the proofs, we make use of the fact that each $T_i$ will be independent of $N$. Also, $T_i$ are independent of each other. However, they are not identically distributed like the unregulated gene expression case discussed in previous section.
\[thm:meanFPT\] For the FPT defined in Eq. , the mean FPT is given by following expression: $$\left<FPT\right>=\sum_{n=1}^{\infty} \sum_{i=1}^{n}\left<T_i\right> f_N(n) \label{eqn:FPTMeanI},$$ where $\displaystyle f_N(n)$ is defined in Eq. , $\displaystyle \left< T_i \right>$ is given by Eq. , and $\displaystyle \left< N \right>$ is given by Eq. .
To prove the result, we first find conditional expectation given $N=n$ then we have:
$$\begin{aligned}
\left<FPT|N=n\right>&=\left<\sum_{i=1}^{n}T_i\right>, \\
&=\sum_{i=1}^{n}\left<T_i\right>.\end{aligned}$$
Unconditioning above expression with respect to $N$: $$\begin{aligned}
\left<FPT\right> &= \sum_{n=1}^{\infty} \sum_{i=1}^{n}\left<T_i\right> Pr(N=n), \\
&= \sum_{n=1}^{\infty} \sum_{i=1}^{n}\left<T_i\right> f_N(n). \end{aligned}$$
This completes the proof.
For the FPT defined in Eq. , the variance of FPT is given by the following expression: where $\displaystyle f_N(n)$ is defined in Eq. , $\displaystyle \left< N \right>$ is given by Eq. , $\displaystyle \left< N^2 \right>$ can be deduced from Eq. , $\displaystyle \left< T_i \right>$ is given by Eq. , and $\text{Var}(T_i)$ is given by Eq. , .
Since expression for $\left<FPT\right>$ is known and given by Eq. , we need to find expression for $\left<FPT^2\right>$, in order to find expression for variance of FPT.
Using the definition of first passage time in Eq. , we have:
$$\begin{aligned}
\left<FPT^2|N= n\right> &= \left<\sum_{i=1}^{n}\sum_{j=1}^{n}T_i T_j\right>, \\
&= \left<\sum_{i=1}^{n}T_i^2+\sum_{i=1}^{n}\sum_{j=1 \neq i}^{n}T_i T_j\right>\end{aligned}$$
Since $T_i^2$ are independent of each other, and $T_j$ are independent of $T_i$ for each $j\neq i$; we can write: $$\begin{aligned}
\left<FPT^2|N= n\right> &= \sum_{i=1}^{n}\left<T_i^2\right> + \sum_{i=1}^{n}\sum_{j=1\neq i}^{n}\left<T_iT_j\right>, \\
&= \sum_{i=1}^{n}\left<T_i^2\right> + \sum_{i=1}^{n}\sum_{j=1\neq i}^{n}\left<T_i\right>\left<T_j\right>.\end{aligned}$$ Using $\text{Var}(T_i)=\left<T_i^2\right>-\left<T_i\right>^2$, we have: $$\left<FPT^2|N= n\right> = \sum_{i=1}^{n} \text{Var}\left(T_i\right) + \left(\sum_{i=1}^{n}\left<T_i\right>\right)^2.$$
Unconditioning with respect to $N$, expression for $\left<FPT^2\right>$ becomes: $$\left<FPT^2\right> = \sum_{n=1}^{\infty}\left(\sum_{i=1}^{n} \text{Var}\left(T_i\right) + \left(\sum_{i=1}^{n}\left<T_i\right>\right)^2 \right)f_N(n).
\label{eqn:expectFPTsquared}$$ Therefore, using Eq. , and Eq. ; expression for $\text{Var}(FPT)$ becomes: This completes the proof.
So far we have developed analytical expressions for mean and variance of FPT when there is an auto-regulatory feedback to transcription rate from protein count. In the next section, we make use of these expressions to deduce the optimal auto-regulation function to minimize the variance of FPT assuming fixed mean FPT.
Minimizing Variance in First Passage Time for Given Mean
========================================================
Parameter Unit Positive feedback Negative feedback No feedback
------------ -------------------------------------- ------------------- ------------------- -------------
$k_{\max}$ mRNA produced per minute 19.35 84 10
$k_p$ protein produced per mRNA per minute 2.65 2.65 2.65
$\gamma_m$ per minute 0.3 0.3 0.3
$X$ molecules 5000 5000 5000
$r$ - 0.05 0.05 -
$c$ per molecule 0.002 0.002 -
$H$ - 2 2 -
\[tab:parameters\]
In this section, we find expression for the auto-regulatory feedback function, $k_m(P_{i-1}),\;\; i \in \{1,2,3,...\}$ that gives minimum variance in FPT, given the mean FPT and event threshold are fixed. The result is presented in form of a theorem.
Let the first passage time be defined as Eq. , and its mean and variance, respectively, given by Eq. and Eq. . Then, the optimal function to minimize the variance of FPT for a given mean of FPT will be constant, given by following expression: $$\label{eqn:optfunc}
k_m\left(P_{i-1}\right)=\frac{\left<N\right>}{\left<FPT\right>}, \quad \forall i \in \{1, 2, 3, ...\},$$ where $\left<N\right>$ denotes the minimum number of transcription events required to reach the FPT threshold, and is given by Eq. .
We assume that each burst event adds a perturbation to transcription rate, i.e., $1/k_m(P_{i-1})$ can be written as: $$\label{eqn:assumedoptimalfunction}
\frac{1}{k_m(P_{i-1})}:=\frac{\left< FPT \right>}{\left<N\right>}+\delta_i,$$ where $\delta_i$ is perturbation corresponding to transcription rate after $i-1^{th}$ burst. To prove the result, we shall prove that the variance of FPT for given mean will minimize when $\delta_i=0$.
Recalling the expression for $\left<FPT\right>$ from Eq. : $$\left<FPT\right> = \sum_{n=1}^{\infty} \sum_{i=1}^{n}\left<T_i\right> f_N(n).$$
Using expressions in Eqs. , , we can deduce the expressions for $\left<T_i\right>$ as: $$\label{eqn:optimalTiform}
\left<T_i\right>=\frac{\left< FPT \right>}{\left<N\right>}+\epsilon_i,$$ where $\epsilon_i$ is related with $\delta_i$ by following expression: $$\label{eqn:epsdelrelation}
\epsilon_i:=\sum_{j=1}^{\infty}\delta_i f_{P_{i-1}}(j)=\left<\delta_i\right>.$$ Substituting expression for $\left<T_i\right>$ from Eq. , we have:
$$\begin{aligned}
\left<FPT\right> &= \sum_{n=1}^{\infty} \sum_{i=1}^{n} \left(\frac{\left<FPT\right>}{\left<N\right>}+\epsilon_i\right) f_N(n), \\
&= \sum_{n=1}^{\infty} \left(\frac{\left<FPT\right>}{\left<N\right>}n + \sum_{i=1}^{n} \epsilon_i \right)f_N(n), \\
& = \frac{\left<FPT\right>}{\left<N\right>}\sum_{n=1}^{\infty}nf_N(n) + \sum_{n=1}^{\infty} \sum_{i=1}^{n}\epsilon_i f_N(n).\end{aligned}$$
Since $\displaystyle \sum_{n=1}^{\infty}nf_N(n)=\left<N\right>$, we have: $$\label{eqn:epsiloncondition}
\sum_{n=1}^{\infty} \sum_{i=1}^{n}\epsilon_i f_N(n)=0.$$ Note that for a fixed mean FPT, minimizing the variance of FPT and minimizing the second order moment $\left<FPT^2\right>$ are equivalent.
Now, we consider the expression for $\left<FPT^2\right>$, and use expression in Eq. to deduce the desired optimal function. From Eq. , we have: $$\left<FPT^2\right>=\sum_{n=1}^{\infty}\left(\sum_{i=1}^{n} \text{Var}\left(T_i\right) + \left(\sum_{i=1}^{n}\left<T_i\right>\right)^2 \right)f_N(n).$$ Substituting value of $\displaystyle \left<T_i\right>$ from Eq. , we get following expression for $\displaystyle \left<FPT^2\right>$: Further simplifying and using relation obtained in Eq. yields: Using Eq. in Eq. :
$$\begin{aligned}
\left<FPT^2\right> &\geq \frac{\left<FPT\right>^2}{\left<N\right>^2}\left<N^2\right>\nonumber \\ &+\sum_{n=1}^{\infty}\left(\sum_{i=1}^{n} \left<T_i\right>^2 + \left(\sum_{i=1}^{n}\epsilon_i \right)^2\right)f_N(n), \label{eqn:meanFPTsqrd_II}\\
\implies \left<FPT^2\right> &\geq \frac{\left<FPT\right>^2}{\left<N\right>^2}\left<N^2\right>+\frac{\left<FPT\right>^2}{\left<N\right>} \nonumber \\ &+ \sum_{n=1}^{\infty}\left(\sum_{i=1}^{n} \epsilon_i^2 + \left(\sum_{i=1}^{n}\epsilon_i \right)^2\right)f_N(n). \label{eqn:meanFPTsqrd_III}\end{aligned}$$
Further, we note that in above expression if $\epsilon_i=0$ (or equivalently $\left<\delta_i\right>=0$), the expression minimizes and reduces to: $$\begin{aligned}
\label{eqn:FPTsqFPTIneq}
\left<FPT^2\right> \geq \frac{\left<FPT\right>^2}{\left<N\right>^2}\left<N^2\right>+\frac{\left<FPT\right>^2}{\left<N\right>}.\end{aligned}$$ Recalling Eq. , we observe that equality in above expression holds for unregulated gene expression case, which essentially means $\delta_i=0$. This proves the desired result.
In this section, we proved that having no auto-regulation of transcription rate provides minimum stochasticity in the FPT, if mean FPT and event threshold are kept fixed. However, since our analysis simplified the gene expression model to burst–limit, we are interested in validating whether it is true if we don’t make an approximation. In the next section, we discuss the computer simulations we carried out for this purpose.
Simulation Results
==================
In order to verify the result deduced in previous section, we carried out Monte Carlo simulations using Gillespie’s algorithm [@Gillespie_ess_1977]. We did not specifically assume that production of protein is in geometric bursts with parameter $b$. Instead, we assumed a non–zero half–life for mRNA thereby relaxing the burst approximation.
To simulate, we considered three separate cases: no feedback, negative feedback and positive feedback. The positive feedback is implemented using Hill function as follows: $$k_m(j)=k_{\max} \left(r + (1-r) \frac{(jc)^H}{1+(jc)^H} \right),$$ where $k_{\max}$ is maximum transcription rate, $r$ represents minimum transcription rate as the fraction of $k_{\max}$, $H$ denotes the Hill coefficient while $c$ is coefficient proportional to the binding affinity (when $j=1/c,\; k_m(j) =k_{max}/2$).
Similarly, the negative feedback is implemented using following function: $$k_m(j)=k_{\max} \left(r + (1-r) \frac{1}{1+(jc)^H} \right).$$
We carried out the simulations for several sets of parameters assuming a fixed event threshold. Rest of the model parameters were chosen to keep the mean FPT approximately equal. In all of them, we found that no–feedback case has minimum variance in FPT.
In Table \[tab:parameters\], we present one set of such parameters. We assumed the event threshold $X=5000$. Other parameters are chosen in a way that the mean FPT $\approx$ $60$ minutes.
Simulation results for 10000 realizations are shown in Fig. \[fig:simulationsresults\]. We note that the variance is minimum in no–feedback case, validating our theoretical claims for this set of parameter values.
Discussion
==========
In this work, we studied stochasticity in event timing at a single cell level. We considered a standard gene expression model without protein degradation. Next, we formulated the FPT problem for this model and derived the formulas for statistical moments of FPT. Further, we introduced auto-regulation in the gene expression wherein the transcription rate is a function of protein count. We derived the formulas for moments of FPT in this case as well, and demonstrated that for a given mean of FPT, the variance in FPT is minimized when there is no auto-regulation of gene expression. The result was verified with simulations as well.
The result can be connected to the $\lambda$ phage lysis time. Due to existence of optimal lysis time [@INWang_fitness_2006; @Wang_evolutiontiming_1996], the phage would possibly like to kill the cell at that time with as much precision as possible. Thus, it should resort to a strategy that would minimize the lysis time variance and hence have no protein–dependent feedback regulation of transcription rate in the expression of holin. In expression from late promoter in $\lambda$ phage, which produces holin, has no evidence of a regulation [@Ptashne_GeneticSwitch_1991; @Oppenheim_switch_2005].
Recalling that in no auto–regulation case too, the variance of FPT can be independently decreased by lowering the mean burst size $b$. Other studies also reveal that in case of $\lambda$ phage, the burst size is indeed small [@JohnDennehy_lst_2011; @INWang_fitness_2006]. Also, antiholin, another protein expressed from the same promoter that expresses holin, binds to holin to decrease the effective burst size [@David_holinantiholin_2000; @Abhi_ltv_2014].
In this paper, there is an underlying assumption of protein being stable. In future work, we plan to use a gene expression model with protein degradation, and carry out a similar analysis. This can be further extended to more generalized gene expression models wherein the promoter can also switch between *on* and *off* states [@ShahrezaeVahid_ad_2008; @abhi_transcriptionalburstingHIV_2010].
Acknowledgment {#acknowledgment .unnumbered}
==============
AS is supported by the National Science Foundation Grant DMS-1312926, University of Delaware Research Foundation (UDRF) and Oak Ridge Associated Universities (ORAU).
[^1]: $^{1}$Khem Raj Ghusinga is with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE, USA 19716. [[email protected]]{}
[^2]: $^{2}$Abhyudai Singh is with Faculty of Electrical and Computer Engineering, Biomedical Engineering, Mathematical Sciences, University of Delaware, Newark, DE, USA 19716. [[email protected]]{}
|
{
"pile_set_name": "ArXiv"
}
|
This work began while both authors shared the hospitality of Centre de Recherches Mathématiques (Montréal) during the theme year “Analysis in Number Theory” (first semester of 2006). We would like to thank C. David, H. Darmon and A. Granville for their invitation. This work was essentially completed in september 2006 in CIRM (Luminy) at the occasion of J.-M. Deshouillers’ sixtieth birthday. We would like to wish him the best.
Introduction and statement of the results
=========================================
Description of the families of $L$-functions studied
----------------------------------------------------
The purpose of this paper is to compute various statistics associated to low-lying zeros of several families of symmetric power $L$-functions in the level aspect. First of all, we give a short description of these families. To any primitive holomorphic cusp form $f$ of prime level $q$ and even weight[^1] $\kappa\geq 2$ (see § \[sec\_autoback\] for the automorphic background) say $f\in\prim{\kappa}{q}$, one can associate its $r$-th symmetric power $L$-function denoted by $L(\sym^rf,s)$ for any integer $r\geq 1$. It is given by an explicit absolutely convergent Euler product of degree $r+1$ on $\Re{s}>1$ (see § \[sec\_sympow\]). The completed $L$-function is defined by $$%
\Lambda(\sym^rf,s)\coloneqq\left(q^{r}\right)^{s/2}L_\infty(\sym^rf,s)L(\sym^rf,s)%$$ where $L_\infty(\sym^rf,s)$ is a product of $r+1$ explicit $\fGamma_{\R}$-factors (see § \[sec\_sympow\]) and $q^r$ is the arithmetic conductor. We will need some control on the analytic behaviour of this function. Unfortunately, such information is not currently known in all generality. We sum up our main assumption in the following statement. \[hypohypo\]
The function $\Lambda\left(\sym^rf,s\right)$ is a *completed $L$-function* in the sense that it satisfies the following *nice* analytic properties:
- it can be extended to an holomorphic function of order $1$ on $\C$,
- it satisfies a functional equation of the shape $$%
\Lambda(\sym^rf,s)=\epsilon\left(\sym^rf\right)\Lambda(\sym^rf,1-s)$$ where the sign $\epsilon\left(\sym^rf\right)=\pm 1$ of the functional equation is given by $$\label{valueofsign}%
\epsilon\left(\sym^rf\right)\coloneqq%
\begin{cases}
+1 & \text{if $r$ is even},\\
\epsilon_f(q)\times\epsilon(\kappa,r) & \text{otherwise}
\end{cases}$$ with $$%
\epsilon(\kappa,r)\coloneqq i^{\left(\frac{r+1}{2}\right)^2(\kappa-1)+\frac{r+1}{2}}=%
\begin{cases}
i^{\kappa} & \text{if $\;r\equiv 1\pmod{8}$,} \\
-1 & \text{if $\;r\equiv 3\pmod{8}$}, \\
-i^{\kappa} & \text{if $\;r\equiv 5\pmod{8}$}, \\
+1 & \text{if $r\;\equiv 7\pmod{8}$}
\end{cases}$$ and $\epsilon_f(q)=\pm 1$ is defined in and only depends on $f$ and $q$.
Hypothesis $\Nice(r,f)$ is known for $r=1$ (E. Hecke [@MR1513069; @MR1513122; @MR1513142]), $r=2$ thanks to the work of S. Gelbart and H. Jacquet [@GeJa] and $r=3,4$ from the works of H. Kim and F. Shahidi [@KiSh1; @KiSh2; @Ki].
We aim at studying the low-lying zeros for the family of $L$-functions given by $$\mathcal{F}_r\coloneqq%
\bigcup_{\text{$q$ prime}}\left\{L(\sym^rf,s), f\in\prim{\kappa}{q}\right\}$$ for any integer $r\geq 1$. Note that when $r$ is even, the sign of the functional equation of any $L(\sym^rf,s)$ is constant of value $+1$ but when $r$ is odd, this is definitely not the case. As a consequence, it is very natural to understand the low-lying zeros for the subfamilies given by $$\mathcal{F}_r^{\epsilon}\coloneqq%
\bigcup_{\text{$q$ prime}}\left\{L(\sym^rf,s), \;f\in\prim{\kappa}{q}, \epsilon\left(\sym^rf\right)=\epsilon\right\}$$ for any odd integer $r\geq 1$ and for $\epsilon=\pm 1$.
Symmetry type of these families
-------------------------------
One of the purpose of this work is to determine the symmetry type of the families $\mathcal{F}_r$ and $\mathcal{F}_r^{\epsilon}$ for $\epsilon=\pm 1$ and for any integer $r\geq 1$ (see § \[sec\_densres\] for the background on symmetry types). The following theorem is a quick summary of the symmetry types obtained.
\[thm\_A\]Let $r\geq 1$ be any integer and $\epsilon=\pm 1$. We assume that hypothesis $\Nice(r,f)$ holds for any prime number $q$ and any primitive holomorphic cusp form of level $q$ and even weight $\kappa\geq 2$. The symmetry group $G(\mathcal{F}_r)$ of $\mathcal{F}_r$ is given by $$G(\mathcal{F}_r)=\begin{cases}%
Sp & \text{ if $r$ is even,}\\%
O & \text{ otherwise.}%
\end{cases}%$$ If $r$ is odd then the symmetry group $G(\mathcal{F}_r^\epsilon)$ of $\mathcal{F}_r^\epsilon$ is given by $$G(\mathcal{F}_r^\epsilon)=\begin{cases}%
SO(\mathrm{even}) & \text{ if $\epsilon=+1$,}\\%
SO(\mathrm{odd}) & \text{ otherwise.}%
\end{cases}$$
\[remark2\] It follows in particular from the value of $\epsilon\left(\sym^rf\right)$ given in that, if $r$ is even, then $\sym^rf$ has not the same symmetry type than $f$ and, if $r$ is odd, then $f$ and $\sym^rf$ have the same symmetry type if and only if $$\begin{aligned}
r \equiv 1 \pmod{8}\; \text{ and }\; \kappa\equiv 0\pmod{4}%
\shortintertext{or}%
r \equiv 5 \pmod{8}\; \text{ and }\; \kappa\equiv 2\pmod{4}%
\shortintertext{or}%
r \equiv 7 \pmod{8}.%\end{aligned}$$
Note that we do not assume any Generalised Riemann Hypothesis for the symmetric power $L$-functions.
In order to prove theorem \[thm\_A\], we compute either the (signed) asymptotic expectation of the one-level density or the (signed) asymptotic expectation of the two-level density. The results are given in the next two sections in which $\epsilon=\pm 1$, $\nu$ will always be a positive real number, $\Phi, \Phi_1$ and $\Phi_2$ will always stand for even Schwartz functions whose Fourier transforms $\widehat{\Phi}, \widehat{\Phi_1}$ and $\widehat{\Phi_2}$ are compactly supported in $[-\nu,+\nu]$ and $f$ will always be a primitive holomorphic cusp form of prime level $q$ and even weight $\kappa\geq 2$ for which hypothesis $\Nice(r,f)$ holds. We refer to § \[sec\_proba\] for the probabilistic background.
### (Signed) asymptotic expectation of the one-level density
The *one-level density* (relatively to $\Phi$) of $\sym^rf$ is defined by $$D_{1,q}[\Phi;r](f)%
\coloneqq%
\sum_{\rho,\;\Lambda(\sym^rf,\rho)=0}%
\Phi\left(\frac{\log{\left(q^r\right)}}{2i\pi}\left(%
\Re{\rho}-\frac{1}{2}+i\Im{\rho}%
\right)\right)%$$ where the sum is over the non-trivial zeros $\rho$ of $L(\sym^rf,s)$ with multiplicities. The *asymptotic expectation* of the one-level density is by definition $$\lim_{\substack{q\;\text{prime} \\ q\to+\infty}}%
\smashoperator[r]{%
\sum_{f\in\prim{\kappa}{q}}%
}\omega_q(f)D_{1,q}[\Phi;r](f)$$ where $\omega_q(f)$ is the harmonic weight defined in and similarly the *signed asymptotic expectation* of the one-level density is by definition $$\lim_{\substack{q\;\text{prime} \\ q\to+\infty}}%
2%
\smashoperator[r]{%
\sum_{\substack{f\in\prim{\kappa}{q} \\ \epsilon\left(\sym^rf\right)=\epsilon}}%
}
\omega_q(f)D_{1,q}[\Phi;r](f)$$ when $r$ is odd.
\[thm\_B\]Let $r\geq 1$ be any integer and $\epsilon=\pm 1$. We assume that hypothesis $\Nice(r,f)$ holds for any prime number $q$ and any primitive holomorphic cusp form of level $q$ and even weight $\kappa\geq 2$ and also that $\theta$ is admissible (see hypothesis $\Hy_2(\theta)$ page ). Let$$%
\nu_{1,\mathrm{max}}(r,\kappa,\theta)\coloneqq%
\left(1-\frac{1}{2(\kappa-2\theta)}\right)\frac{2}{r^2}.%$$ If $\nu<\nu_{1,\mathrm{max}}(r,\kappa,\theta)$ then the asymptotic expectation of the one-level density is $$\widehat{\Phi}(0)+\frac{(-1)^{r+1}}{2}\Phi(0).$$ Let $$%
\nu_{1,\mathrm{max}}^\epsilon(r,\kappa,\theta)\coloneqq%
\inf{\left(\nu_{1,\mathrm{max}}(r,\kappa,\theta),\frac{3}{r(r+2)}\right).}%$$ If $r$ is odd and $\nu<\nu_{1,\mathrm{max}}^\epsilon(r,\kappa,\theta)$ then the signed asymptotic expectation of the one-level density is $$%
\widehat{\Phi}(0)+\frac{(-1)^{r+1}}{2}\Phi(0).%$$
\[remark4\] The first part of Theorem \[thm\_B\] reveals that the symmetry type of $\mathcal{F}_r$ is $$%
G(\mathcal{F}_r)=%
\begin{cases}
Sp & \text{if $r$ is even,}\\%
O & \text{if $r=1$,}\\%
SO(\mathrm{even}) \text{ or } O \text{ or } SO(\mathrm{odd}) & \text{if $r\geq 3$ is odd.}%
\end{cases}%$$ We cannot decide between the three orthogonal groups when $r\geq 3$ is odd since in this case $\nu_{1,\mathrm{max}}(r,\kappa,\theta)<1$ but the computation of the two-level densities will enable us to decide. Note also that we go beyond the support $[-1,1]$ when $r=1$ as Iwaniec, Luo & Sarnak [@IwLuSa] (Theorem 1.1) but without doing any subtle arithmetic analysis of Kloosterman sums. Also, A. G[ü]{}loglu in [@Gu Theorem 1.2] established some density result for the same family of $L$-functions but when the weight $\kappa$ goes to infinity and the level $q$ is fixed. It turns out that we recover the same constraint on $\nu$ when $r$ is even but we get a better result when $r$ is odd. This can be explained by the fact that the analytic conductor of any $L(\sym^rf,s)$ with $f$ in $\prim{\kappa}{q}$ which is of size $$%
q^r\times%
\begin{cases}%
\kappa^{r} & \text{if $r$ is even}\\%
\kappa^{r+1} & \text{otherwise}%
\end{cases}%$$ is slightly larger in his case than in ours when $r$ is odd.
\[remark5\] The second part of Theorem \[thm\_B\] reveals that if $r$ is odd and $\epsilon=\pm 1$ then the symmetry type of $\mathcal{F}_r^\epsilon$ is $$%
G(\mathcal{F}_r^{\epsilon})=%
SO(\mathrm{even}) \text{ or } O \text{ or } SO(\mathrm{odd}).%$$ Here $\nu$ is always strictly smaller than one and we are not able to recover the result of [@IwLuSa Theorem 1.1] without doing some arithmetic on Kloosterman sums.
### Sketch of the proof
We give here a sketch of the proof of the first part of Theorem \[thm\_B\] namely we briefly explain how to determine the asymptotic expectation of the one-level density assuming that hypothesis $\Nice(r,f)$ holds for any prime number $q$ and any primitive holomorphic cusp form of level $q$ and even weight $\kappa\geq 2$ and also that $\theta$ is admissible. The first step consists in transforming the sum over the zeros of $\Lambda(\sym^rf,s)$ which occurs in $D_{1,q}[\Phi;r](f)$ into a sum over primes. This is done *via* some Riemann’s explicit formula for symmetric power $L$-functions stated in Proposition \[explicit\] which leads to $$D_{1,q}[\Phi;r](f)=\widehat{\Phi}(0)+\frac{(-1)^{r+1}}{2}\Phi(0)+P_q^1[\Phi;r](f)+\sum_{m=0}^{r-1}(-1)^mP_q^2[\Phi;r,m](f)+o(1)$$ where $$\label{eq_cneser}
P_{q}^1[\Phi;r](f)\coloneqq-\frac{2}{\log{\left(q^r\right)}}\sum_{\substack{p\in\prem \\ p\nmid q}}\lambda_{f}\left(p^r\right)\frac{\log{p}}{\sqrt{p}}\widehat{\Phi}\left(\frac{\log{p}}{\log{\left(q^r\right)}}\right).$$ The terms $P_q^2[\Phi;r,m](f)$ are also sums over primes which look like $P_{q}^1[\Phi;r](f)$ but can be forgotten in first approximation since they can be thought as sums over squares of primes which are easier to deal with. The second step consists in averaging over all the $f$ in $\prim{\kappa}{q}$. While doing this, the asymptotic expectation of the one-level density $$\widehat{\Phi}(0)+\frac{(-1)^{r+1}}{2}\Phi(0)$$ naturally appears and we need to show that $$-\frac{2}{\log{\left(q^r\right)}}\sum_{\substack{p\in\prem \\ p\nmid q}}\left(\sum_{f\in\prim{\kappa}{q}}\omega_q(f)\lambda_{f}\left(p^r\right)\right)\frac{\log{p}}{\sqrt{p}}\widehat{\Phi}\left(\frac{\log{p}}{\log{\left(q^r\right)}}\right)$$ is a remainder term provided that the support $\nu$ of $\Phi$ is small enough. We apply some suitable trace formula given in Proposition \[iwlusatr\] in order to express the previous average of Hecke eigenvalues. We cannot directly apply Peterson’s trace formula since there may be some old forms of level $q$ especially when the weight $\kappa$ is large. Nevertheless, these old forms are automatically of level $1$ since $q$ is prime and their contribution remains negligible. So, we have to bound $$-\frac{4\pi i^\kappa}{\log{\left(q^r\right)}}\sum_{\substack{p\in\prem \\ p\nmid q}}\sum_{\substack{c\geq 1 \\
q\mid c}}\frac{S(1,p^r;c)}{c}J_{\kappa-1}\left(\frac{4\pi\sqrt{p^r}}{c}\right)\frac{\log{p}}{\sqrt{p}}\widehat{\Phi}\left(\frac{\log{p}}{\log{\left(q^r\right)}}\right)$$ where $S(1,p^r;c)$ is a Kloosterman sum and which can be written as $$-\frac{4\pi i^\kappa}{\log{\left(q^r\right)}}\sum_{\substack{c\geq 1 \\
q\mid c}}\sum_{m\geq 1}a_m\frac{S(1,m;c)}{c}g(m;c)$$ where $$a_m\coloneqq\un_{[1,q^{r^2\nu}]}(m)\frac{\log{m}}{rm^{1/(2r)}}\times\begin{cases}
1 & \text{if $m=p^r$ for some prime $p\neq q$}, \\
0 & \text{otherwise}
\end{cases}$$ and $$g(m;c)\coloneqq J_{\kappa-1}\left(\frac{4\pi\sqrt{m}}{c}\right)\widehat{\Phi}\left(\frac{\log{m}}{r\log{\left(q^r\right)}}\right).$$ We apply the large sieve inequality for Kloosterman sums given in proposition \[sieve\]. It entails that if $\nu\leq 2/r^2$ then such quantity is bounded by $$\ll_\epsilon q^{\left(\frac{\kappa-1}{2}-\theta\right)(r^2\nu-2)+\epsilon}+q^{\left(\frac{\kappa}{2}-\theta\right)r^2\nu-\left(\kappa-\frac{1}{2}-2\theta\right)+\epsilon}.$$ This is an admissible error term if $\nu<\nu_{1,\mathrm{max}}(r,\kappa,\theta)$. We focus on the fact that we did any arithmetic analysis of Kloosterman sums to get this result. Of course, the power of spectral theory of automorphic forms is hidden in the large sieve inequalities for Kloosterman sums.
### (Signed) asymptotic expectation of the two-level density
The *two-level density* of $\sym^rf$ (relatively to $\Phi_1$ and $\Phi_2$) is defined by $$%
D_{2,q}[\Phi_1,\Phi_2;r](f)%
\coloneqq%
\sum_{\substack{(j_1,j_2)\in\mathcal{E}(f,r)^2\\ j_1\neq\pm j_2}}%
\Phi_1\left(\widehat{\rho}_{f,r}^{(j_1)}\right)%
\Phi_2\left(\widehat{\rho}_{f,r}^{(j_2)}\right).%$$ For more precision on the numbering of the zeros, we refer to § \[sec\_explicit\]. The *asymptotic expectation* of the two-level density is by definition $$\lim_{\substack{q\;\text{prime} \\ q\to+\infty}}%
\smashoperator[r]{%
\sum_{f\in\prim{\kappa}{q}}%
}%
\omega_q(f)D_{2,q}[\Phi_1,\Phi_2;r](f)%$$ and similarly the *signed asymptotic expectation* of the two-level density is by definition $$\lim_{\substack{q\;\text{prime} \\ q\to+\infty}}%
2%
\smashoperator[r]{%
\sum_{\substack{f\in\prim{\kappa}{q} \\ \epsilon\left(\sym^rf\right)=\epsilon}}%
}
\omega_q(f)D_{2,q}[\Phi_1,\Phi_2;r](f)%$$ when $r$ is odd and $\epsilon=\pm 1$.
\[thm\_C\] Let $r\geq 1$ be any integer and $\epsilon=\pm 1$. We assume that hypothesis $\Nice(r,f)$ holds for any prime number $q$ and any primitive holomorphic cusp form of level $q$ and even weight $\kappa\geq 2$. If $\nu<1/r^2$ then the asymptotic expectation of the two-level density is $$\begin{gathered}
\left[%
\widehat{\Phi_1}(0)%
+%
\frac{(-1)^{r+1}}{2}\Phi_1(0)%
\right]%
\left[%
\widehat{\Phi_2}(0)%
+%
\frac{(-1)^{r+1}}{2}\Phi_2(0)%
\right]%
\\
+2\int_\R\abs{u}\widehat{\Phi_1}(u)\widehat{\Phi_2}(u)\dd u %
-2\widehat{\Phi_1\Phi_2}(0)%
+\left((-1)^{r}+\frac{%
%\delta(2\nmid r)%
\un_{2\N+1}(r)
}{2}\right)\Phi_1(0)\Phi_2(0).%\end{gathered}$$ If $r$ is odd and $\nu<1/(2r(r+2))$ then the signed asymptotic expectation of the two-level density is $$\begin{gathered}
\left[%
\widehat{\Phi_1}(0)%
+%
\frac{1}{2}\Phi_1(0)%
\right]%
\left[%
\widehat{\Phi_2}(0)%
+%
\frac{1}{2}\Phi_2(0)%
\right]%
\\%
+2\int_\R\abs{u}\widehat{\Phi_1}(u)\widehat{\Phi_2}(u)\dd u%
-2\widehat{\Phi_1\Phi_2}(0)%
-\Phi_1(0)\Phi_2(0)%
\\%
+%
%\delta(\epsilon=-1)%
\un_{\{-1\}}(\epsilon)
\Phi_1(0)\Phi_2(0).%\end{gathered}$$
\[remark6\] We have just seen that the computation of the one-level density already reveals that the symmetry type of $\mathcal{F}_r$ is $Sp$ when $r$ is even. The asymptotic expectation of the two-level density also coincides with the one of $Sp$ (see [@KaSa Theorem A.D.2.2] or [@Mil Theorem 3.3]). When $r\geq 3$ is odd, the first part of Theorem \[thm\_C\] together with a result of Katz & Sarnak (see [@KaSa Theorem A.D.2.2] or [@Mil Theorem 3.2]) imply that the symmetry type of $\mathcal{F}_r$ is $O$.
The second part of Theorem \[thm\_C\] and a result of Katz & Sarnak (see [@KaSa Theorem A.D.2.2] or [@Mil Theorem 3.2]) imply that the symmetry type of $\mathcal{F}_r^{\epsilon}$ is as in Theorem \[thm\_A\] for any odd integer $r\geq 1$ and $\epsilon=\pm 1$.
In order to prove Theorem \[thm\_C\], we need to determine the *asymptotic variance* of the one-level density which is defined by $$\lim_{\substack{q\;\text{prime} \\ q\to+\infty}}%
\smashoperator[r]{%
\sum_{f\in\prim{\kappa}{q}}%
}%
\omega_q(f)\left(D_{1,q}[\Phi;r](f)-%
\sum_{g\in\prim{\kappa}{q}}\omega_q(g)D_{1,q}[\Phi;r](g)\right)^2$$ and the *signed asymptotic variance* of the one-level density which is similarly defined by $$\lim_{\substack{q\;\text{prime} \\ q\to+\infty}}%
2%
\smashoperator[r]{%
\sum_{\substack{f\in\prim{\kappa}{q} \\ \epsilon\left(\sym^rf\right)=\epsilon}}%
}%
\omega_q(f)\left(D_{1,q}[\Phi;r](f)-%
2%
\smashoperator[r]{%
\sum_{\substack{g\in\prim{\kappa}{q} \\ \epsilon\left(\sym^rg\right)=\epsilon}}%
}%
\omega_q(g)D_{1,q}[\Phi;r](g)\right)^2$$ when $r$ is odd and $\epsilon=\pm 1$.
\[thm\_D\] Let $r\geq 1$ be any integer and $\epsilon=\pm 1$. We assume that hypothesis $\Nice(r,f)$ holds for any prime number $q$ and any primitive holomorphic cusp form of level $q$ and even weight $\kappa\geq 2$. If $\nu<1/r^2$ then the asymptotic variance of the one-level density is $$%
2\int_\R\abs{u}\widehat{\Phi}^2(u)\dd u.$$ If $r$ is odd and $\nu<1/(2r(r+2))$ then the signed asymptotic variance of the one-level density is $$2\int_\R\abs{u}\widehat{\Phi}^2(u)\dd u.%$$
Asymptotic moments of the one-level density
-------------------------------------------
Last but not least, we compute the *asymptotic $m$-th moment* of the one-level density which is defined by $$\lim_{\substack{q\;\text{prime} \\ q\to+\infty}}%
\smashoperator[r]{%
\sum_{f\in\prim{\kappa}{q}}%
}%
\omega_q(f)\left(D_{1,q}[\Phi;r](f)%
-\sum_{g\in\prim{\kappa}{q}}\omega_q(g)D_{1,q}[\Phi;r](g)\right)^m$$ for any integer $m\geq 1$.
\[thm\_F\] Let $r\geq 1$ be any integer and $\epsilon=\pm 1$. We assume that hypothesis $\Nice(r,f)$ holds for any prime number $q$ and any primitive holomorphic cusp form of level $q$ and even weight $\kappa\geq 2$. If $m\nu<4\left/(r(r+2))\right.$ then the asymptotic $m$-th moment of the one-level density is $$\begin{cases}
%\begin{dcases}
0 & \text{if $m$ is odd,}\\%
2\int_\R\abs{u}\widehat{\Phi}^2(u)\dd u\times\frac{m!}{2^{m/2}\left(\frac{m}{2}\right)!} & \text{otherwise.}%
\end{cases}
%\end{dcases}$$
This result is another evidence for mock-Gaussian behaviour (see [@MR2166468; @HuRu; @HuRu2] for instance).
We compute the first asymptotic moments of the one-level density. These computations allow to compute the asymptotic expectation of the first level-densities [@MR2166468 §1.2]. We will use the specific case of the asymptotic expectation of the two-level density and the asymptotic variance in § \[sec\_twoandvar\].
Let us sketch the proof of Theorem \[thm\_F\] by explaining the origin of the main term. We have to evaluate $$\label{eq_lasomme}%
\sum_{%
\substack{%
0\leq\ell\leq m \\ %
0\leq\alpha\leq\ell%
}%
}%
\binom{m}{\ell}\binom{\ell}{\alpha}R(q)^{\ell-\alpha}%
\Eh[q]\left(P_q^1[\Phi;r]^{m-\ell}P_q^2[\Phi;r]^{\alpha}\right)%$$ where $P_q^1[\Phi;r]$ has been defined in , $$%
P_q^2[\Phi;r](f)=%
-\frac{2}{\log(q^r)}%
\sum_{j=1}^{r}%
(-1)^{r-j}%
\sum_{%
\substack{%
p\in\prem\\%
p\nmid q%
}
}
\lambda_f\left(p^{2j}\right)%
\frac{\log p}{p}\widehat{\Phi}\left(\frac{2\log p}{\log(q^r)}\right)%$$ and $R(q)$ satisfies $$%
R(q)=O\left(\frac{1}{\log q}\right).%$$ The main term comes from the contribution $\ell=0$ in the sum . Using a combinatorial lemma, we rewrite this main contribution as $$%
\frac{(-2)^m}{\log^m{(q^r)}}%
\sum_{s=1}^m%
\sum_{\sigma\in P(m,s)}%
\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}%
\Eh[q]\left(%
\prod_{u=1}^s%
\lambda_f\left(\widehat{p}_{i_{u}}^r\right)^{\varpi^{(\sigma)}_u}%
\right)%$$ where $P(m,s)$ is the set of surjective functions $$%
\sigma \colon \{1,\dotsc,\alpha\}\twoheadrightarrow \{1,\dotsc,s\}%$$ such that for any $j\in\{1,\dotsc,s\}$, either $\sigma(j)=1$ or there exists $k<j$ such that $\sigma(j)=\sigma(k)+1$ and for any $j\in\{1,\dotsc,s\}$ $$%
\varpi_j^{(\sigma)}\coloneqq\#\sigma^{-1}(\{j\}).%$$ $\left(\widehat{p}_i\right)_{i\geq 1}$ stands for the increasing sequence of prime numbers different from $q$. Linearising each $\lambda_f\left(\widehat{p}_{i_{u}}^r\right)^{\varpi^{(\sigma)}_u}$ in terms of $\lambda_f\left(\widehat{p}_{i_{u}}^{j_u}\right)$ with $j_u$ runs over integers in $[0,r\varpi^{(\sigma)}_u]$ and using a trace formula to prove that the only $\sigma\in P(m,s)$ leading to a principal contribution satisfy $\varpi^{(\sigma)}_j=2$ for any $j\in\{1,\dotsc,s\}$, we have to estimate $$\label{eq_homo}%
\frac{(-2)^m}{\log^m{(q^r)}}%
\sum_{s=1}^m%
\sum_{\substack{%
\sigma\in P(m,s)\\%
\forall j\in\{1,\dotsc,s\}, \varpi^{(\sigma)}_j=2%
}
}
\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}%
\prod_{u=1}^s%
\frac{%
\log^2{(\widehat{p}_{i_u})}%
}{%%
\widehat{p}_{i_u}%
}%
\widehat{\Phi}^2\left(%
\frac{%
\log\widehat{p}_{i_{u}}%
}{\log{(q^r)}}%
\right).$$ This sum vanishes if $m$ is odd since $$%
\sum_{j=1}^s\varpi_j^{(\sigma)}=m$$ and it remains to prove the formula for $m$ even. In this case, and since we already computed the moment for $m=2$, we deduce from that the main contribution is $$%
\Eh[q](P_q^1[\Phi;r]^2)\times%
\#\left\{%
\sigma\in P(m,m/2) \colon \varpi^{(\sigma)}_j=2\ (\forall j)%
\right\}$$ and we conclude by computing $$%
\#\left\{%
\sigma\in P(m,m/2) \colon \varpi^{(\sigma)}_j=2\ (\forall j)%
\right\}%
=%
\frac{m!}{2^{m/2}\left(\frac{m}{2}\right)!}.$$ Proving that the other terms lead to error terms is done by implementing similar ideas, but requires – especially for the double products (namely terms implying both $P_q^1$ and $P_q^2$) – much more combinatorial technicalities.
Organisation of the paper
-------------------------
Section \[autoproba\] contains the automorphic and probabilistic background which is needed to be able to read this paper. In particular, we give here the accurate definition of symmetric power $L$-functions and the properties of Chebyshev polynomials useful in section \[momentt\]. In section \[technical\], we describe the main technical ingredients of this work namely large sieve inequalities for Kloosterman sums and Riemann’s explicit formula for symmetric power $L$-functions. In section \[one\], some standard facts about symmetry groups are given and the computation of the (signed) asymptotic expectation of the one-level density is done. The computations of the (signed) asymptotic expectation, covariance and variance of the two-level density are done in section \[two\] whereas the computation of the asymptotic moments of the one-level density is provided in section \[momentt\]. Some well-known facts about Kloosterman sums are recalled in appendix \[klooster\].
We write $\prem$ for the set of prime numbers and the main parameter in this paper is a prime number $q$, whose name is the level, which goes to infinity among $\prem$. Thus, if $f$ and $g$ are some $\C$-valued functions of the real variable then the notations $f(q)\ll_{A}g(q)$ or $f(q)=O_A(g(q))$ mean that $\abs{f(q)}$ is smaller than a “constant” which only depends on $A$ times $g(q)$ at least for $q$ a large enough prime number and similarly, $f(q)=o(1)$ means that $f(q)\rightarrow 0$ as $q$ goes to infinity among the prime numbers. We will denote by $\epsilon$ an absolute positive constant whose definition may vary from one line to the next one. The characteristic function of a set $S$ will be denoted $\un_S$.
Automorphic and probabilistic background {#autoproba}
========================================
Automorphic background {#sec_autoback}
----------------------
### Overview of holomorphic cusp forms
In this section, we recall general facts about holomorphic cusp forms. A reference is [@Iw]. We write $\gGamma_{0}(q)$ for the congruence subgroup of level $q$ which acts on the upper-half plane $\pk$. A holomorphic function $f\colon\pk\mapsto\C$ which satisfies $$%
\forall\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}\in\gGamma_{0}(q),\forall z\in\pk,\quad f\left(\frac{az+b}{cz+d}\right)=(cz+d)^\kappa f(z)$$ and vanishes at the cusps of $\gGamma_{0}(q)$ is a *holomorphic cusp form* of level $q$, even weight $\kappa\geq 2$. We denote by $\cusp{\kappa}{q}$ this space of holomorphic cusp forms which is equipped with the Peterson inner product $$%
\scal{f_{1}}{f_{2}}_q\coloneqq%
\int_{\quotientgauche{\gGamma_0(q)}{\pk}}y^{\kappa}f_{1}(z)\overline{f_{2}(z)}\frac{\dd x\dd y}{y^{2}}.$$ The Fourier expansion at the cusp $\infty$ of any such holomorphic cusp form $f$ is given by $$%
\forall z\in\pk,\quad f(z)=\sum_{n\geq1}\psi_{f}(n)n^{(\kappa-1)/2}e(nz)$$ where $e(z)\coloneqq\exp{(2i\pi z)}$ for any complex number $z$. The *Hecke operators* act on $\cusp{\kappa}{q}$ by $$%%
T_{\ell}(f)(z)
\coloneqq%
\frac{1}{\sqrt{\ell}}\sum_{\substack{ad=\ell\\ (a,q)=1}}\sum_{0\leq b<d}f\left(\frac{az+b}{d}\right)%
\qquad%$$ for any $z\in\pk$. If $f$ is an eigenvector of $T_\ell$, we write $\lambda_f(\ell)$ the corresponding eigenvalue. We can prove that $T_\ell$ is hermitian if $\ell\geq 1$ is any integer coprime with $q$ and that $$\label{compo}
T_{\ell_{1}}\circ T_{\ell_{2}}=\sum_{\substack{d\mid(\ell_{1},\ell_{2})\\(d,q)=1}}T_{\ell_{1}\ell_{2}\left/d^{2}\right.}$$ for any integers $\ell_1, \ell_2\geq 1$. By Atkin & Lehner theory [@AtLe], we get a splitting of $\cusp{\kappa}{q}$ into $\anc{\kappa}{q}\oplus^{\perp_{\scal{\cdot}{\cdot}_q}}\nouv{\kappa}{q}$ where $$\begin{aligned}
\anc{\kappa}{q} & \coloneqq \Vect_{\C}\left\{f(qz), f\in\cusp{\kappa}{1}\right\}\cup\cusp{\kappa}{1}, \\
\nouv{\kappa}{q} & \coloneqq \left(\anc{\kappa}{q}\right)^{\perp_{\scal{\cdot}{\cdot}_q}}\end{aligned}$$ where “o” stands for “old” and “n” for “new”. Note that $\anc{\kappa}{q}=\{0\}$ if $\kappa<12$ or $\kappa=14$. These two spaces are $T_{\ell}$-invariant for any integer $\ell\geq 1$ coprime with $q$. A *primitive* cusp form $f\in\nouv{\kappa}{q}$ is an eigenfunction of any operator $T_\ell$ for any integer $\ell\geq 1$ coprime with $q$ which is new and arithmetically normalised namely $\psi_{f}(1)=1$. Such an element $f$ is automatically an eigenfunction of the other Hecke operators and satisfies $\psi_{f}(\ell)=\lambda_{f}(\ell)$ for any integer $\ell\geq 1$. Moreover, if $p$ is a prime number, define $\alpha_{f}(p)$, $\beta_{f}(p)$ as the complex roots of the quadratic equation $$\label{quadra}
X^{2}-\lambda_{f}(p)X+\epsilon_{q}(p)=0$$ where $\epsilon_{q}$ denotes the trivial Dirichlet character of modulus $q$. Then it follows from the work of Eichler, Shimura, Igusa and Deligne that $$%
\abs{\alpha_{f}(p)}, \abs{\beta_{f}(p)}\leq 1$$ for any prime number $p$ and so $$\label{individualh}
\forall\ell\geq 1,\quad
\abs{\lambda_{f}(\ell)}\leq\tau(\ell).$$ The set of primitive cusp forms is denoted by $\prim{\kappa}{q}$. It is an orthogonal basis of $\nouv{\kappa}{q}$. Let $f$ be a holomorphic cusp form with Hecke eigenvalues $\left(\lambda_{f}(\ell)\right)_{(\ell,q)=1}$. The composition property entails that for any integer $\ell_{1}\geq 1$ and for any integer $\ell_{2}\geq 1$ coprime with $q$ the following multiplicative relations hold: $$\begin{aligned}
\label{compoeigen}
\psi_f(\ell_1)\lambda_f(\ell_2) &= %
\sum_{\substack{d\mid(\ell_1,\ell_2)\\ (d,q)=1}}\psi_f\left(\ell_{1}\ell_{2}\left/d^{2}\right.\right), \\
\psi_f(\ell_1\ell_2) &= %
\sum_{\substack{d\mid(\ell_1,\ell_2)\\ (d,q)=1}}\mu(d)\psi_f\left(\ell_1/d\right)\lambda_f\left(\ell_2/d\right)\end{aligned}$$ and these relations hold for any integers $\ell_{1}, \ell_{2}\geq 1$ if $f$ is primitive. The adjointness relation is $$\label{adjointness}
\lambda_{f}(\ell)=\overline{\lambda_{f}(\ell)}, \quad \psi_{f}(\ell)=\overline{\psi_{f}(\ell)}$$ for any integer $\ell\geq 1$ coprime with $q$ and this remains true for any integer $\ell\geq 1$ if $f$ is primitive. We need two definitions. The harmonic weight associated to any $f$ in $\cusp{\kappa}{q}$ is defined by $$\label{eq_facomeg}%
\omega_q(f)\coloneqq\frac{\fGamma(\kappa-1)}{(4\pi)^{\kappa-1}\scal{f}{f}_q}.%$$ For any natural integer $m$ and $n$, the $\Delta_q$-symbol is given by $$\label{eq_deltasymb}
\Delta_q(m,n)\coloneqq\delta_{m,n}+%
2\pi i^{\kappa}\sum_{\substack{c\geq 1 \\ q\mid c}}\frac{S(m,n;c)}{c}J_{\kappa-1}\left(\frac{4\pi\sqrt{mn}}{c}\right)$$ where $S(m,n;c)$ is a Kloosterman sum defined in appendix \[Kloos\] and $J_{\kappa-1}$ is a Bessel function of first kind defined in appendix \[Bessel\].The following proposition is *Peterson’s trace formula*.
\[prop\_orth\] If $\orth{\kappa}{q}$ is any orthogonal basis of $\cusp{\kappa}{q}$ then $$\label{tr1}
\sum_{f\in\orth{\kappa}{q}}\omega_q(f)\psi_f(m)\psi_f(n)=\Delta_q(m,n)$$ for any integers $m$ and $n$.
H. Iwaniec, W. Luo & P. Sarnak proved in [@IwLuSa] a useful variation of Peterson’s trace formula which is an average over only primitive cusp forms. This is more convenient when there are some old forms which is the case for instance when the weight $\kappa$ is large. Let $\nu$ be the arithmetic function defined by $$\nu(n)\coloneqq n\prod_{p\mid n}\left(1+1/p\right)$$ for any integer $n\geq 1$.
\[iwlusatr\]If $\left(n,q^2\right)\mid q$ and $q\nmid m$ then $$%
\label{tr2}
\sum_{f\in \prim{\kappa}{q}}\omega_q(f)\lambda_f(m)\lambda_f(n)=%
\Delta_q(m,n)-\frac{1}{q\nu((n,q))}\sum_{\ell\mid q^\infty}\frac{1}{\ell}\Delta_1\left(m\ell^2,n\right).$$
The first term in is exactly the term which appears in whereas the second term in will be usually very small as an old form comes from a form of level $1$! Thus, everything works in practice as if there were no old forms in $\cusp{\kappa}{q}$.
### Chebyshev polynomials and Hecke eigenvalues
Let $p\neq q$ a prime number and $f\in\prim{\kappa}{q}$. The multiplicativity relation leads to $$%
\sum_{r\geq 0}\lambda_f(p^r)t^r%
=%
\frac{1}{1-\lambda_f(p)t+t^2}.$$ It follows that $$\label{eq_ams}
\lambda_f(p^r)=X_r\left(\lambda_f(p)\right)%$$ where the polynomials $X_r$ are defined by their generating series $$%
\sum_{r\geq 0}X_r(x)t^r%
=%
\frac{1}{1-xt+t^2}.$$ They are also defined by $$%%
X_r(2\cos\theta)=\frac{\sin{((r+1)\theta)}}{\sin{(\theta)}}.$$ These polynomials are known as the Chebyshev polynomials of second kind. Each $X_r$ has degree $r$, is even if $r$ is even and odd otherwise. The family $\{X_r\}_{r\geq 0}$ is a basis for $\Q[X]$, orthonormal with respect to the inner product $$%
\scalst{P}{Q}\coloneqq\frac{1}{\pi}%
\int_{-2}^{2}P(x)Q(x)\sqrt{1-\frac{x^2}{4}}\dd x.$$ In particular, for any integer $\varpi\geq 0$ we have $$\label{eq_lintch}
X_r^\varpi=\sum_{j=0}^{r\varpi}x(\varpi,r,j)X_j$$ with $$\label{eq_valx}
x(\varpi,r,j)\coloneqq \scalst{X_r^\varpi}{X_j}%
=\frac{2}{\pi}\int_0^\pi\frac{\sin^\varpi{((r+1)\theta)}\sin{((j+1)\theta)}}{\sin^{\varpi-1}{(\theta)}}\dd\theta.%$$ The following relations are useful in this paper $$\label{propriox}
x(\varpi,r,j)=\begin{cases}
1 & \text{if $j=0$ and $\varpi$ is even,} \\
0 & \text{if $j$ is odd and $r$ is even,} \\
0 & \text{if $j=0$, $\varpi=1$ and $r\geq 1$.}
\end{cases}$$
### Overview of $L$-functions associated to primitive cusp forms
Let $f$ in $\prim{\kappa}{q}$. We define $$%
L(f,s)\coloneqq\sum_{n\geq 1}\frac{\lambda_f(n)}{n^s}=%
\prod_{p\in\prem}\left(1-\frac{\alpha_f(p)}{p^s}\right)^{-1}\left(1-\frac{\beta_f(p)}{p^s}\right)^{-1}$$ which is an absolutely convergent and non-vanishing Dirichlet series and Euler product on $\Re{s}>1$ and also $$%
L_\infty(f,s)\coloneqq\fGamma_{\R}\left(s+(\kappa-1)/2\right)\fGamma_{\R}\left(s+(\kappa+1)/2\right)$$ where $\fGamma_{\R}(s)\coloneqq\pi^{-s/2}\fGamma\left(s/2\right)$ as usual. The function $$\Lambda(f,s)\coloneqq q^{s/2}L_\infty(f,s)L(f,s)$$ is a *completed $L$-function* in the sense that it satisfies the following *nice* analytic properties:
- the function $\Lambda(f,s)$ can be extended to an holomorphic function of order $1$ on $\C$,
- the function $\Lambda(f,s)$ satisfies a functional equation of the shape $$%
\Lambda(f,s)=i^{\kappa}\epsilon_f(q)\Lambda(f,1-s)$$ where $$\label{eq_signe}
\epsilon_f(q)=-\sqrt{q}\lambda_f(q)=\pm 1.$$
### Overview of symmetric power $L$-functions {#sec_sympow}
Let $f$ in $\prim{\kappa}{q}$. For any natural integer $r\geq 1$, the *symmetric $r$-th power* associated to $f$ is given by the following Euler product of degree $r+1$ $$%
L(\sym^rf,s)\coloneqq\prod_{p\in\prem}L_p(\sym^rf,s)$$ where $$%
L_p(\sym^rf,s)\coloneqq%
\prod_{i=0}^r\left(1-\frac{\alpha_f(p)^{i}\beta_f(p)^{r-i}}{p^{s}}\right)^{-1}$$ for any prime number $p$. Let us remark that the local factors of this Euler product may be written as $$%
L_p(\sym^rf,s)=\prod_{i=0}^r\left(1-\frac{\alpha_f(p)^{2i-r}}{p^s}\right)^{-1}$$ for any prime number $p\neq q$ and $$%
L_q(\sym^rf,s)=1-\frac{\lambda_f(q)^r}{q^{s}}=1-\frac{\lambda_f(q^r)}{q^{s}}$$ as $\alpha_f(p)+\beta_f(p)=\lambda_f(p)$ and $\alpha_f(p)\beta_f(p)=\epsilon_q(p)$ for any prime number $p$ according to . On $\Re{s}>1$, this Euler product is absolutely convergent and non-vanishing. We also defines [@CoMi (3.16) and (3.17)] a local factor at $\infty$ which is given by a product of $r+1$ Gamma factors namely $$%
L_\infty(\sym^rf,s)\coloneqq%
\prod_{0\leq a\leq(r-1)/2}%
\fGamma_{\R}\left(s+(2a+1)(\kappa-1)/2\right)%
\fGamma_{\R}\left(s+1+(2a+1)(\kappa-1)/2\right)$$ if $r$ is odd and $$%
L_\infty(\sym^rf,s)\coloneqq%
\fGamma_{\R}(s+\mu_{\kappa,r})%
\prod_{1\leq a\leq r/2}%
\fGamma_{\R}\left(s+a(\kappa-1)\right)%
\fGamma_{\R}\left(s+1+a(\kappa-1)\right)$$ if $r$ is even where $$%
\mu_{\kappa,r}\coloneqq\begin{cases}
1 & \text{if } r(\kappa-1)/2 \text{ is odd,} \\
0 & \text{otherwise.} \\
\end{cases}$$ All the local data appearing in these local factors are encapsulated in the following completed $L$-function $$%
\Lambda(\sym^rf,s)\coloneqq\left(q^{r}\right)^{s/2}L_\infty(\sym^rf,s)L(\sym^rf,s).$$ Here, $q^r$ is called the arithmetic conductor of $\Lambda(\sym^rf,s)$ and somehow measures the size of this function. We will need some control on the analytic behaviour of this function. Unfortunately, such information is not currently known in all generality. Our main assumption is given in hypothesis $\Nice(r,f)$ page . Indeed, much more is expected to hold as it is discussed in details in [@CoMi] namely the following assumption is strongly believed to be true and lies in the spirit of Langlands program.
There exists an automorphic cuspidal self-dual representation, denoted by $\sym^r\pi_f=\otimes'_{p\in\prem\cup\{\infty\}}\sym^r\pi_{f,p}$, of $GL_{r+1}\left(\A_{\Q}\right)$ whose local factors $L\left(\sym^r\pi_{f,p},s\right)$ agree with the local factors $L_p\left(\sym^rf,s\right)$ for any $p$ in $\prem\cup\{\infty\}$.
Note that the local factors and the arithmetic conductor in the definition of $\Lambda\left(\sym^rf,s\right)$ and also the sign of its functional equation which all appear without any explanations so far come from the explicit computations which have been done *via* the local Langlands correspondence by J. Cogdell and P. Michel in [@CoMi]. Obviously, hypothesis $\Nice(r,f)$ is a weak consequence of hypothesis $\sym^{r}(f)$. For instance, the cuspidality condition in hypothesis $\sym^{r}(f)$ entails the fact that $\Lambda\left(\sym^rf,s\right)$ is of order $1$ which is crucial for us to state a suitable explicit formula. As we will not exploit the power of automorphic theory in this paper, hypothesis $\Nice(r,f)$ is enough for our purpose. In addition, it may happen that hypothesis $\Nice(r,f)$ is known whereas hypothesis $\sym^{r}f$ is not. Let us overview what has been done so far. For any $f$ in $\prim{\kappa}{q}$, hypothesis $\sym^{r}f$ is known for $r=1$ (E. Hecke), $r=2$ thanks to the work of S. Gelbart and H. Jacquet [@GeJa] and $r=3,4$ from the works of H. Kim and F. Shahidi [@KiSh1; @KiSh2; @Ki].
Probabilistic background {#sec_proba}
------------------------
The set $\prim{\kappa}{q}$ can be seen as a probability space if
- the measurable sets are all its subsets,
- the *harmonic probability measure* is defined by $$%
\muh[q](A)\coloneqq\sumh_{f\in A}1\coloneqq\sum_{f\in A}\omega_q(f)$$ for any subset $A$ of $\prim{\kappa}{q}$.
Indeed, there is a slight abuse here as we only know that $$\label{eq_mesasy}
\lim_{\substack{q\in\prem \\ q\to+\infty}}\muh[q]\left(\prim{\kappa}{q}\right)=1$$ (see remark \[rem\_moyun\]) which means that $\muh[q]$ is an “asymptotic” probability measure. If $X_q$ is a measurable complex-valued function on $\prim{\kappa}{q}$ then it is very natural to compute its *expectation* defined by $$%
\Eh[q]\left(X_q\right)\coloneqq\sumh_{f\in\prim{\kappa}{q}}X_q(f),$$ its *variance* defined by $$%
\Vh[q]\left(X_q\right)\coloneqq\Eh[q]\left(\left(X_q-\Eh[q]\left(X_q\right)\right)^2\right)$$ and its *$m$-th moments* given by $$%
\Mh[q,m]\left(X_q\right)\coloneqq\Eh[q]\left(\left(X_q-\Eh[q]\left(X_q\right)\right)^m\right)$$ for any integer $m\geq 1$. If $X\coloneqq\left(X_q\right)_{q\in\prem}$ is a sequence of such measurable complex-valued functions then we may legitimely wonder if the associated complex sequences $$%
\left(\Eh[q]\left(X_q\right)\right)_{q\in\prem},\quad %
\left(\Vh[q]\left(X_q\right)\right)_{q\in\prem},\quad %
\left(\Mh[q,m]\left(X_q\right)\right)_{q\in\prem}$$ converge as $q$ goes to infinity among the primes. If yes, the following general notations will be used for their limits $$%
\Eh[\infty]\left(X\right),\quad %
\Vh[\infty]\left(X\right),\quad%
\Mh[\infty,m]\left(X\right)$$ for any natural integer $m$. In addition, these potential limits are called *asymptotic expectation*, *asymptotic variance* and *asymptotic $m$-th moments* of $X$ for any natural integer $m\geq 1$.
For the end of this section, we assume that $r$ is *odd*. We may remark that the sign of the functional equations of any $L(\sym^rf,s)$ when $q$ goes to infinity among the prime numbers and $f$ ranges over $\prim{\kappa}{q}$ is not constant as it depends on $\epsilon_f(q)$. Let $$%
\primeps{\kappa}{q}\coloneqq\left\{f\in \prim{\kappa}{q}, \epsilon(\sym^rf)=\epsilon\right\}$$ where $\epsilon=\pm 1$. If $f\in\primpair{\kappa}{q}$, then $\sym^rf$ is said to be *even* whereas it is said to be *odd* if $f\in\primimpair{\kappa}{q}$. It is well-known that $$%
\lim_{\substack{q\in\prem \\
q\to +\infty}}\muh[q]\left(\left\{f\in\prim{k}{q} \colon \epsilon_f(q)=\epsilon\right\}\right)=\frac{1}{2}.$$ Since $\epsilon(\sym^rf)$ is $\epsilon_q(f)$ up to a sign depending only on $\kappa$ and $r$ (by hypothesis $\Nice(r,f)$), it follows that $$\label{eq_sign}
\lim_{\substack{q\in\prem \\
q\to +\infty}}\muh[q]\left(\primeps{\kappa}{q}\right)=\frac{1}{2}.$$ For $X_q$ as previous, we can compute its *signed expectation* defined by $$%
\Eheps[q]\left(X_q\right)\coloneqq%
%\frac{1}{2}\sumh_{f\in\primeps{\kappa}{q}}X_q(f),
2\sumh_{f\in\primeps{\kappa}{q}}X_q(f),$$ its *signed variance* defined by $$%
\Vheps[q]\left(X_q\right)\coloneqq\Eheps[q]\left(\left(X_q-\Eheps[q]\left(X_q\right)\right)^2\right)$$ and its *signed $m$-th moments* given by $$%
\Mheps[q,m]\left(X_q\right)\coloneqq\Eheps[q]\left(\left(X_q-\Eheps[q]\left(X_q\right)\right)^m\right)$$ for any natural integer $m\geq 1$. In case of existence, we write $\Eheps[\infty](X)$, $\Vheps[\infty](X)$ and $\Mheps[\infty,m](X)$ for the limits which are called *signed asymptotic expectation*, *signed asymptotic variance* and *signed asymptotic moments*. The signed expectation and the expectation are linked through the formula $$\begin{aligned}
\notag
\Eheps[q](X_q)%
&=%
2\sumh_{f\in\prim{\kappa}{q}}\frac{1+\epsilon\times\epsilon(\sym^rf)}{2}X_q(f)
\\
\label{eq_removeps}
&=%
\Eh[q](X_q)-\epsilon\times\epsilon(\kappa,r)\sqrt{q}\sumh_{f\in\prim{\kappa}{q}}\lambda_f(q)X_q(f).\end{aligned}$$
Main technical ingredients of this work {#technical}
=======================================
Large sieve inequalities for Kloosterman sums
---------------------------------------------
One of the main ingredients in this work is some large sieve inequalities for Kloosterman sums which have been established by J.-M. Deshouillers & H. Iwaniec in [@DeIw] and then refined by V. Blomer, G. Harcos & P. Michel in [@BlHaMi]. The proof of these large sieve inequalities relies on the spectral theory of automorphic forms on $GL_2\left(\A_{\Q}\right)$. In particular, the authors have to understand the size of the Fourier coefficients of these automorphic cusp forms. We have already seen that the size of the Fourier coefficients of holomorphic cusp forms is well understood but we only have partial results on the size of the Fourier coefficients of Maass cusp forms which do not come from holomorphic forms. We introduce the following hypothesis which measures the approximation towards the *Ramanujan-Peterson-Selberg conjecture*.
\[hyp\_RPS\] If $\pi\coloneqq\otimes'_{p\in\prem\cup\{\infty\}}\pi_{p}$ is any automorphic cuspidal form on $GL_{2}(\A_\Q)$ with local Hecke parameters $\alpha_{\pi}^{(1)}(p)$, $\alpha_{\pi}^{(2)}(p)$ at any prime number $p$ and $\mu_{\pi}^{(1)}(\infty)$, $\mu_{\pi}^{(2)}(\infty)$ at infinity then $$%
\forall j\in\{1,2\},\quad %
\abs{\alpha_{\pi}^{(j)}(p)}\leq p^{\theta}$$ for any prime number $p$ for which $\pi_p$ is unramified and $$%
\forall j\in\{1,2\},\quad%
\abs{%
\Re{\left(\mu_{\pi}^{(j)}(\infty)\right)}
}
\leq\theta$$ provided $\pi_{\infty}$ is unramified.
We say that $\theta$ is *admissible* if $\Hy_2(\theta)$ is satisfied.
The smallest admissible value of $\theta$ is currently $\theta_{0}=\frac{7}{64}$ thanks to the works of H. Kim, F. Shahidi and P. Sarnak [@KiSh2; @Ki]. The Ramanujan-Peterson-Selberg conjecture asserts that $0$ is admissible.
\[def\_propS\]Let $T\colon\mathbb{R}^3\to\R^+$ and $(M,N,C)\in(\mathbb{R}\setminus\{0\})^{3}$, we say that a smooth function $h\colon\R^3\to\R^3$ satisfies the property $\prp(T;M,N,C)$ if there exists a real number $K>0$ such that $$\begin{gathered}
\forall(i,j,k)\in\N^3, \forall(x_1,x_2,x_3)\in%
\left[\frac{M}{2},2M\right]\times\left[\frac{N}{2},2N\right]\times\left[\frac{C}{2},2C\right], \\
x_1^ix_2^jx_3^k\frac{\partial^{i+j+k} h}%
{\partial x_1^i\partial x_2^j\partial x_3^k}(x_1,x_2,x_3)\leq %
KT(M,N,C)\left(1+\frac{\sqrt{MN}}{C}\right)^{i+j+k}.\end{gathered}$$
With this definition in mind, we are able to write the following proposition which is special case of a large sieve inequality adapted from the one of Deshouillers & Iwaniec [@DeIw Theorem 9] by Blomer, Harcos & Michel [@BlHaMi Theorem 4].
\[sieve\]Let $q$ be some positive integer. Let $M, N, C\geq 1$ and $g$ be a smooth function satisfying property $\prp(1;M,N,C)$. Consider two sequences of complex numbers $(a_m)_{m\in[M/2,2M]}$ and $(b_n)_{n\in[N/2,2N]}$. If $\theta$ is admissible and $MN\ll C^2$ then $$\begin{gathered}
%
\sum_{\substack{c\geq 1 \\ q\mid c}}%
\sum_{m\geq 1}\sum_{n\geq 1}a_mb_n\frac{S(m,\pm n;c)}{c}g(m,n;c)%
\\%
\ll_{\epsilon}%
(qMNC)^\epsilon%
\left(\frac{C^2}{MN}\right)^{\theta}%
\left(%
1+\frac{M}{q}%
\right)^{1/2}%
\left(%
1+\frac{N}{q}%
\right)^{1/2}%
\norm{a}_2\norm{b}_2%\end{gathered}$$for any $\epsilon>0$.
We shall use a test function. For any $\nu>0$ let us define $\Schwartz_\nu(\R)$ as the space of even Schwartz function $\Phi$ whose Fourier transform $$%
\widehat{\Phi}(\xi)\coloneqq \mathcal{F}[x\mapsto\Phi(x)](\xi)\coloneqq%
\int_{\R}\Phi(x)e(-x\xi)\dd x%$$ is compactly supported in $[-\nu,+\nu]$. Thanks to the Fourier inversion formula: $$\label{eq_fouinv}
\Phi(x)=\int_{\R}\widehat{\Phi}(\xi)e(x\xi)\dd x=%
\mathcal{F}[\xi\mapsto\widehat{\Phi}(\xi)](-x),$$ such a function $\Phi$ can be extended to an entire even function which satisfies $$\label{estim}
\forall s\in\C,\quad %
\Phi(s)\ll_n\frac{\exp{(\nu\abs{\Im{s}})}}{(1+\abs{s})^n}%$$ for any integer $n\geq 0$.The version of the large sieve inequality we shall use several times in this paper is then the following.
\[usefulsieve\]Let $q$ be some prime number, $k_1, k_2>0$ be some integers, $\alpha_1, \alpha_2, \nu$ be some positive real numbers and $\Phi\in\Schwartz_{\nu}(\R)$. Let $h$ be some smooth function satisfying property $\prp(T;M,N,C)$ for any $1\leq M\leq q^{k_1\alpha_1\nu}$, $1\leq N\leq q^{k_2\alpha_2\nu}$ and $C\geq q$. Let $\left(a_{p}\right)_{\substack{p\in\prem\\ p\leq q^{\alpha_1\nu}}}$ and $\left(b_{p}\right)_{\substack{p\in\prem\\ p\leq q^{\alpha_2\nu}}}$ be some complex numbers sequences. If $\;\theta$ is admissible and $\nu\leq2\left/(k_1\alpha_1+k_2\alpha_2)\right.$ then $$\begin{gathered}
%
\sum_{\substack{c\geq 1\\ q\mid c}}%
\sum_{\substack{p_1\in\prem\\ p_1\nmid q}}%
\sum_{\substack{p_2\in\prem\\ p_2\nmid q}}%
a_{p_1}b_{p_2}\frac{S(p_1^{k_1},p_2^{k_2};c)}{c}%
h\left(p_1^{k_1},p_2^{k_2};c\right)%
\widehat{\Phi}\left(\frac{\log p_1}{\log(q^{\alpha_1})}\right)%
\widehat{\Phi}\left(\frac{\log p_2}{\log(q^{\alpha_2})}\right)%
\\%
\ll%
%\\%
q^{\epsilon}%
\sumsh_{\substack{%
1\leq M\leq q^{\nu\alpha_1k_1}\\%
1\leq N\leq q^{\nu\alpha_2k_2}\\%
C\geq q/2%
}}%
\left(1+\sqrt{\frac{M}{q}}\right)%
\left(1+\sqrt{\frac{N}{q}}\right)%
\left(\frac{C^2}{MN}\right)^\theta%
T(M,N,C)%
%\\%
%\times%
\norm{a}_2\norm{b}_2\end{gathered}$$ where $\sharp$ indicates that the sum is on powers of $\sqrt{2}$. The constant implied by the symbol $\ll$ depends at most on $\epsilon$, $k_1$, $k_2$, $\alpha_1$, $\alpha_2$ and $\nu$.
Define $\left(\widehat{a}_m\right)_{m\in\N}$, $\left(\widehat{b}_n\right)_{n\in\N}$ and $g(m,n;c)$ by $$\begin{aligned}
%
\widehat{a}_m &\coloneqq a_{m^{1/k_1}}\un_{\prem^{k_1}}(m)\un_{[1,q^{\nu\alpha_1k_1}]}(m)\\%
\widehat{b}_n &\coloneqq b_{n^{1/k_1}}\un_{\prem^{k_1}}(n)\un_{[1,q^{\nu\alpha_1k_1}]}(n)\\%
g(m,n;c) &\coloneqq h(m,n,c)%
\widehat{\Phi}\left(\frac{\log m}{\log(q^{\alpha_1k_1})}\right)%
\widehat{\Phi}\left(\frac{\log n}{\log(q^{\alpha_2k_2})}\right).\end{aligned}$$ Using a smooth partition of unity, as detailed in § \[unity\], we need to evaluate $$\label{eq_vtmp}
\sumsh_{\substack{%
1\leq M\leq q^{\nu\alpha_1k_1}\\%
1\leq N\leq q^{\nu\alpha_2k_2}\\%
C\geq q/2%
}}%
T(M,N,C)%
\sum_{\substack{c\geq 1\\ q\mid c}}\sum_{m\geq 1}\sum_{n\geq 1}%
\widehat{a}_m%
\widehat{b}_n%
\frac{S(m,n;c)}{c}%
\frac{g_{M,N,C}(m,n;c)}{T(M,N,C)}.%$$Since $\nu\leq2\left/(\alpha_1k_1+\alpha_2k_2)\right.$, the first summation is restricted to $MN\ll C^2$ hence, using proposition \[sieve\], the quantity in is $$%
\ll%
\norm{a}_2\norm{b}_2q^\epsilon%
\sumsh_{\substack{%
1\leq M\leq q^{\nu\alpha_1k_1}\\%
1\leq N\leq q^{\nu\alpha_2k_2}\\%
C\geq q/2%
}}%
T(M,N,C)%
\left(1+\sqrt{\frac{M}{q}}\right)%
\left(1+\sqrt{\frac{N}{q}}\right)%
\left(\frac{C^2}{MN}\right)^\theta.%$$
Riemann’s explicit formula for symmetric power $L$-functions {#sec_explicit}
------------------------------------------------------------
In this section, we give an analog of Riemann-von Mangoldt’s explicit formula for symmetric power $L$-functions. Before that, let us recall some preliminary facts on zeros of symmetric power $L$-functions which can be found in section 5.3 of [@IwKo]. Let $r\geq 1$ and $f\in \prim{\kappa}{q}$ for which hypothesis $\Nice(r,f)$ holds. All the zeros of $\Lambda(\sym^rf,s)$ are in the critical strip $\{s\in\C\colon 0<\Re{s}<1\}$. The multiset of the zeros of $\Lambda(\sym^rf,s)$ counted with multiplicities is given by $$%
\left\{%
\rho_{f,r}^{(j)}=\beta_{f,r}^{(j)}+i\gamma_{f,r}^{(j)} \colon j\in\mathcal{E}(f,r)
\right\}$$ where $$%
\mathcal{E}(f,r)\coloneqq%
\begin{cases}
\Z & \text{if $\sym^rf$ is odd}\\
\Z\setminus\{0\} & \text{if $\sym^rf$ is even.}
\end{cases}$$ and $$\begin{aligned}
\beta_{f,r}^{(j)} & = \Re{\rho_{f,r}^{(j)}}, \\
\gamma_{f,r}^{(j)} & = \Im{\rho_{f,r}^{(j)}}\end{aligned}$$ for any $j\in\mathcal{E}(f,r)$. We enumerate the zeros such that
1. the sequence $j\mapsto\gamma_{f,r}^{(j)}$ is increasing
2. we have $j\geq 0$ if and only if $\gamma_{f,r}^{(j)}\geq 0$
3. we have $\rho_{f,r}^{(-j)}=1-\rho_{f,r}^{(j)}$.
Note that if $\rho_{f,r}^{(j)}$ is a zero of $\Lambda(\sym^rf,s)$ then $\overline{\rho_{f,r}^{(j)}}$, $1-\rho_{f,r}^{(j)}$ and $1-\overline{\rho_{f,r}^{(j)}}$ are also some zeros of $\Lambda(\sym^rf,s)$. In addition, remember that if $\sym^rf$ is odd then the functional equation of $L(\sym^rf,s)$ evaluated at the critical point $s=1/2$ provides a trivial zero denoted by $\rho_{f,r}^{(0)}$. It can be shown [@IwKo Theorem 5.8] that the number of zeros $\Lambda(\sym^rf,s)$ satisfying $\abs{\gamma_{f,r}^{(j)}}\leq T$ is $$\label{eq_nbzero}
\frac{T}{\pi}\log{\left(\frac{q^rT^{r+1}}{(2\pi e)^{r+1}}\right)}+O\left(\log(qT)\right)$$ as $T\geq 1$ goes to infinity. We state now the *Generalised Riemann Hypothesis* which is the main conjecture about the horizontal distribution of the zeros of $\Lambda(\sym^rf,s)$ in the critical strip.
For any prime number $q$ and any $f$ in $\prim{\kappa}{q}$, all the zeros of $\Lambda(\sym^rf,s)$ lie on the critical line $\left\{s\in\C \colon \Re{s}=1/2\right\}$ namely $\beta_{r,f}^{(j)}=1/2$ for any $j\in\mathcal{E}(f,r)$.
We *do not* use this hypothesis in our proofs.
Under hypothesis $\GRH(r)$, it can be shown that the number of zeros of the function $\Lambda(\sym^rf,s)$ satisfying $\abs{\gamma_{f,r}^{(j)}}\leq 1$ is given by $$%
\frac{1}{\pi}\log{\left(q^r\right)}(1+o(1))$$ as $q$ goes to infinity. Thus, the spacing between two consecutive zeros with imaginary part in $[0,1]$ is roughly of size $$\label{meanspacing}
\frac{2\pi}{\log{\left(q^r\right)}}.$$ We aim at studying the local distribution of the zeros of $\Lambda(\sym^rf,s)$ in a neighborhood of the real axis of size $1/\log q^r$ since in such a neighborhood, we expect to catch only few zeros (but without being able to say that we catch only one[^2]). Hence, we normalise the zeros by defining $$%
\widehat{\rho}_{f,r}^{(j)}%
\coloneqq %
\frac{\log{\left(q^r\right)}}{2i\pi}\left(%
\beta_{f,r}^{(j)}-\frac{1}{2}+i\gamma_{f,r}^{(j)}
\right).$$ Note that $$%
\widehat{\rho}_{f,r}^{(-j)}=-\widehat{\rho}_{f,r}^{(j)}.$$
Let $f\in\prim{\kappa}{q}$ for which hypothesis $\Nice(r,f)$ holds and let $\Phi\in\Schwartz_\nu(\R)$. The *one-level density* (relatively to $\Phi$) of $\sym^rf$ is $$\label{eq_defdens}
D_{1,q}[\Phi;r](f)%
\coloneqq%
\sum_{j\in\mathcal{E}(f,r)}%
\Phi\left(\widehat{\rho}_{f,r}^{(j)}\right).$$
To study $D_{1,q}[\Phi;r](f)$ for any $\Phi\in\Schwartz_\nu(\R)$, we transform this sum over zeros into a sum over primes in the next proposition. In other words, we establish an explicit formula for symmetric power $L$-functions. Since the proof is classical, we refer to [@IwLuSa §4] or [@Gu §2.2] which present a method that has just to be adapted to our setting.
\[explicit\] Let $r\geq 1$ and $f\in \prim{\kappa}{q}$ for which hypothesis $\Nice(r,f)$ holds and let $\Phi\in\Schwartz_\nu(\R)$. We have $$%
D_{1,q}[\Phi;r](f)=%
E[\Phi;r]+P_q^1[\Phi;r](f)+\sum_{m=0}^{r-1}(-1)^mP_q^2[\Phi;r,m](f)+O\left(\frac{1}{\log{\left(q^r\right)}}\right)$$ where $$\begin{aligned}
E[\Phi;r] & \coloneqq \widehat{\Phi}(0)+\frac{(-1)^{r+1}}{2}\Phi(0), \\
P_{q}^1[\Phi;r](f) & \coloneqq %
-\frac{2}{\log{\left(q^r\right)}}\sum_{\substack{p\in\prem \\ p\nmid q}}%
\lambda_{f}\left(p^r\right)\frac{\log{p}}{\sqrt{p}}\widehat{\Phi}\left(\frac{\log{p}}{\log{\left(q^r\right)}}\right), \\
P_q^2[\Phi;r,m](f) & \coloneqq %
-\frac{2}{\log{\left(q^r\right)}}\sum_{\substack{p\in\prem \\ p\nmid q}}%
\lambda_f\left(p^{2(r-m)}\right)\frac{\log{p}}{p}\widehat{\Phi}\left(\frac{2\log{p}}{\log{\left(q^r\right)}}\right)\end{aligned}$$ for any integer $m\in\{0,\ldots,r-1\}$.
Contribution of the old forms
-----------------------------
In this short section, we prove the following useful lemmas.
\[lem\_delun\] Let $p_1$ and $p_2\neq q$ be some prime numbers and $a_1$, $a_2$, $a$ be some nonnegative integers. Then $$%%
\sum_{\ell\mid q^{\infty}}\frac{\Delta_1\n(\ell^2p_1^{a_1},p_2^{a_2}q^a)}{\ell}%
\ll%
%%
\frac{1}{q^{a/2}}%
%$$ the implied constant depending only on $a_1$ and $a_2$.
Using proposition \[prop\_orth\] and the fact that $\orth{\kappa}{1}=\prim{\kappa}{1}$, we write $$\begin{aligned}
\Delta_1\n(\ell^2p_1^{a_1},p_2^{a_2}q^a)%
&=%
\sumh_{f\in\prim{\kappa}{1}}\lambda_f\n(\ell^2p_1^{a_1})\lambda_f\n(p_2^{a_2}q^a)\\
&\ll%
\label{eq_ici}
\sumh_{f\in\prim{\kappa}{1}}\abs{\lambda_f\n(\ell^2p_1^{a_1})}\cdot\abs{\lambda_f\n(p_2^{a_2})}\cdot\abs{\lambda_f(q^a)}.\end{aligned}$$ By Deligne’s bound we have $$\label{eq_sansq}
\abs{\lambda_f\n(\ell^2p_1^{a_1})}\cdot\abs{\lambda_f\n(p_2^{a_2})}%
\leq%
\tau(\ell^2p_1^{a_1})\tau(p_2^{a_2})%
\leq (a_1+1)(a_2+2)\tau(\ell^2).$$ By the multiplicativity relation and the value of the sign of the functional equation , we have $$\label{eq_avecq}
\abs{\lambda_f(q^a)}\ll%
%%
\frac{1}{q^{a/2}}.%
%$$ We obtain the result by reporting and in and by using and $$%
\sum_{\ell\mid q^{\infty}}\frac{\tau(\ell^2)}{\ell}=\frac{1+1/q}{(1-1/q)^2}\ll 1.$$
\[deltaestimate\] Let $m,n\geq 1$ be some coprime integers. Then, $$%
\Delta_q(m,n)-\delta(m,n)\ll%
%\begin{dcases}
\begin{cases}
\frac{(mn)^{1/4}}{q}\log\left(\frac{mn}{q^2}\right) & \text{if $mn>q^2$}\\
\frac{(mn)^{( \kappa-1)/2}}{q^{\kappa-1/2}} %
\leq \frac{(mn)^{1/4}}{q} & \text{if $mn\leq q^2$.}
%\end{dcases}
\end{cases}$$
This is a direct consequence of the Weil-Estermann bound and lemma \[lem\_picard\].
\[lem\_sumlambdafq\] For any prime number $q$, we have $$%
\sqrt{q}\sumh_{f\in\prim{\kappa}{q}}\lambda_f(q)\ll\frac{1}{q^{\delta_\kappa}}$$ where $$%%
\delta_\kappa\coloneqq%
%\begin{dcases*}
\begin{cases}
\frac{\kappa-1}{2} & \text{if $\kappa\leq 10$ or $\kappa=14$} \\ %
\frac{5}{2} & \text{otherwise.}%
%\end{dcases*}
\end{cases}$$
Let $\mathcal{K}=\{\kappa\in 2\N \colon 2\leq\kappa\leq 14,\, \kappa\neq 12\}$. By proposition \[iwlusatr\], we have $$\label{eq_termzero}%
\sumh_{f\in\prim{\kappa}{q}}\lambda_f(q)=%
\Delta_q(1,q)-%
\frac{\delta(\kappa%
%%
\notin%
%%
\mathcal{K})}{q\nu(q)}%
\sum_{\ell\mid q^\infty}\frac{\Delta_1(\ell^2,q)}{\ell}.$$ The term $\delta(\kappa\notin\mathcal{K})$ comes from proposition \[prop\_orth\] with the fact that there is no cusp forms of weight $\kappa\in\mathcal{K}$ and level $1$. Lemma \[deltaestimate\] gives $$\label{eq_termun}
\Delta_q(1,q)\ll \frac{1}{q^{\kappa/2}}$$ and lemma \[lem\_delun\] gives $$\label{eq_termdeux}%
\sum_{\ell\mid q^\infty}\frac{\Delta_1(\ell^2,q)}{\ell}\ll\frac{1}{%
%%
\sqrt{q}%
%%
}.$$ Since $\nu(q)>q$, the result follows from reporting and in .
\[rem\_moyun\] In a very similar fashion, one can prove that $$\label{eq_moyun}
\muh[q]\left(\prim{\kappa}{q}\right)%
=\Eh[q](1)%
=1+O\left(\frac{1}{q^{\gamma_\kappa}}\right).$$ where $$%
\gamma_\kappa\coloneqq%
%\begin{dcases*}
\begin{cases}
\kappa-\frac{1}{2} & \text{if $\kappa\leq 10$ or $\kappa=14$} \\ %
1 & \text{otherwise.}%
%\end{dcases*}
\end{cases}$$ Corollary \[lem\_sumlambdafq\], and imply $$\label{eq_mbun}
\Eheps[q](1)%
=1+O\left(\frac{1}{q^{\beta_\kappa}}\right)$$ where $$%
\beta_\kappa\coloneqq
\begin{cases}
%\begin{dcases*}
\frac{\kappa-1}{2} & \text{if $\kappa\leq 10$ or $\kappa=14$} \\ %
1 & \text{otherwise.}%
%\end{dcases*}
\end{cases}$$
A direct consequence of lemma \[lem\_delun\] is the following one.
\[lem\_old\] Let $\alpha_1,\alpha_2,\beta_1,\beta_2,\gamma_1,\gamma_2,w$ be some nonnegative real numbers. Let $\Phi_1$ and $\Phi_2$ be in $\Schwartz_\nu(\R)$. Then, $$\begin{gathered}
\sum_{\substack{p_1\in\prem\\ p_1\nmid q}}%
\sum_{\substack{p_2\in\prem\\ p_2\nmid q}}%
\frac{\log p_1}{p_1^{\alpha_1}}\frac{\log p_2}{p_2^{\alpha_2}}%
\widehat{\Phi_1}\left(\frac{\log p_1}{\log{\left(q^{\beta_1}\right)}}\right)%
\widehat{\Phi_2}\left(\frac{\log p_2}{\log{\left(q^{\beta_2}\right)}}\right)%
\sum_{\ell\mid q^\infty}\frac{\Delta_1(\ell^2p_1^{\gamma_1},p_2^{\gamma_2}q^w)}{\ell}%
\\
\ll%
q^{\delta\nu-w/2+\varepsilon}\end{gathered}$$ with $\delta$ given in table \[tab\_vald\].
$]0,1]$ $[1,+\infty[$
--------------- ------------------------------------------- -----------------------
$]0,1]$ $\beta_1(1-\alpha_1)+\beta_2(1-\alpha_2)$ $\beta_2(1-\alpha_2)$
$[1,+\infty[$ $\beta_1(1-\alpha_1)$ $0$
: Values of $\delta$[]{data-label="tab_vald"}
Linear statistics for low-lying zeros {#one}
=====================================
Density results for families of $L$-functions {#sec_densres}
---------------------------------------------
We briefly recall some well-known features that can be found in [@IwLuSa]. Let $\mathcal{F}$ be a family of $L$-functions indexed by the arithmetic conductor namely $$%
\mathcal{F}=\bigcup_{Q\geq 1}\mathcal{F}(Q)$$ where the arithmetic conductor of any $L$-function in $\mathcal{F}(Q)$ is of order $Q$ in the logarithmic scale. It is expected that there is a symmetry group $G(\mathcal{F})$ of matrices of large rank endowed with a probability measure which can be associated to $\mathcal{F}$ such that the low-lying zeros of the $L$-functions in $\mathcal{F}$ namely the non-trivial zeros of height less than $1/\log{Q}$ are distributed like the eigenvalues of the matrices in $G(\mathcal{F})$. In other words, there should exist a symmetry group $G(\mathcal{F})$ such that for any $\nu>0$ and any $\Phi\in\Schwartz_\nu(\R)$, $$\begin{gathered}
\lim_{Q\to+\infty}\frac{1}{\mathcal{F}(Q)}%
\sum_{\pi\in\mathcal{F}(Q)}\sum_{\substack{%
0\leq\beta_\pi\leq 1 \\
\gamma_\pi\in\R \\
L\left(\pi,\beta_\pi+i\gamma_\pi\right)=0%
}}
\Phi\left(\frac{\log{Q}}{2i\pi}\left(\beta_\pi-\frac{1}{2}+i\gamma_\pi\right)\right) \\
=%
\int_{\R}\Phi(x)W_1(G(\mathcal{F}))(x)\dd x\end{gathered}$$ where $W_1(G(\mathcal{F}))$ is the one-level density of the eigenvalues of $G(\mathcal{F})$. In this case, $\mathcal{F}$ is said to be of *symmetry type* $G(\mathcal{F})$ and we said that we proved a *density result* for $\mathcal{F}$. For instance, the following densities are determined in [@KaSa]: $$\begin{aligned}
W_1(SO(\mathrm{even}))(x) &= 1+\frac{\sin{(2\pi x)}}{2\pi x},\\
W_1(O)(x) &= 1+\frac{1}{2}\delta_0(x),\\
W_1(SO(\mathrm{odd}))(x) &= 1-\frac{\sin{(2\pi x)}}{2\pi x}+\delta_0(x),\\
W_1(Sp)(x) &= 1-\frac{\sin{(2\pi x)}}{2\pi x}\end{aligned}$$ where $\delta_0$ is the Dirac distribution at $0$. According to Plancherel’s formula, $$%
\int_{\R}\Phi(x)W_1(G(\mathcal{F}))(x)\dd x=\int_{\R}\widehat{\Phi}(x)\widehat{W}_1(G(\mathcal{F}))(x)\dd x$$ and we can check that $$\begin{aligned}
\widehat{W}_1(SO(\mathrm{even}))(x) &= \delta_0(x)+\frac{1}{2}\eta(x),\\
\widehat{W}_1(O)(x) &= \delta_0(x)+\frac{1}{2},\\
\widehat{W}_1(SO(\mathrm{odd}))(x) &= \delta_0(x)-\frac{1}{2}\eta(x)+1,\\
\widehat{W}_1(Sp)(x) &= \delta_0(x)-\frac{1}{2}\eta(x)\end{aligned}$$ where $$%
\eta(x)\coloneqq
\begin{cases}
1 & \text{if $\abs{x}<1$,} \\
\frac{1}{2} & \text{if $x=\pm 1$,} \\
0 & \text{otherwise.}
\end{cases}$$ As a consequence, if we can only prove a density result for $\nu\leq 1$, the three orthogonal densities are indistinguishable although they are distinguishable from $Sp$. Thus, the challenge is to pass the natural barrier $\nu=1$.
Asymptotic expectation of the one-level density
-----------------------------------------------
The aim of this part is to prove a density result for the family $$%
\mathcal{F}_r\coloneqq\bigcup_{q\in\prem}\left\{L(\sym^rf,s), f\in \prim{\kappa}{q}\right\}$$ for any $r\geq 1$ which consists in proving the existence and computing the asymptotic expectation $\Eh[\infty]\left(D_{1}[\Phi;r]\right)$ of $D_1[\Phi;r]\coloneqq\left(D_{1,q}[\Phi;r]\right)_{q\in\prem}$ for any $r\geq 1$ and for $\Phi$ in $\Schwartz_\nu(\R)$ with $\nu>0$ as large as possible in order to be able to distinguish between the three orthogonal densities if $r$ is small enough. Recall that $E[\Phi;r]$ has been defined in proposition \[explicit\].
\[density1\] Let $r\geq 1$ and $\Phi\in\Schwartz_\nu(\R)$. We assume that hypothesis $\Nice(r,f)$ holds for any prime number $q$ and any $f\in \prim{\kappa}{q}$ and also that $\theta$ is admissible. Let $$%
\nu_{1,\mathrm{max}}(r,\kappa,\theta)\coloneqq\left(1-\frac{1}{2(\kappa-2\theta)}\right)\frac{2}{r^2}.$$ If $\nu<\nu_{1,\mathrm{max}}(r,\kappa,\theta)$ then $$%
\Eh[\infty]\left(D_{1}[\Phi;r]\right)=E[\Phi;r].$$
We remark that $$\begin{aligned}
{2}
\label{eq_pqcn}
\nu_{1,\mathrm{max}}(r,\kappa,\theta_0) &= \left(1-\frac{16}{32\kappa-7}\right)\frac{2}{r^2}& \geq \frac{82}{57r^2}, \\
\nu_{1,\mathrm{max}}(r,\kappa,0) &= \left(1-\frac{1}{2\kappa}\right)\frac{2}{r^2}& \geq \frac{3}{2r^2} \end{aligned}$$ and thus $\nu_{1,\mathrm{max}}(1,\kappa,\theta_0)>1$ whereas $\nu_{1,\mathrm{max}}(r,\kappa,\theta_0)\leq 1$ for any $r\geq 2$.
\[rem\_symtyp\] Note that $$%
E[\Phi;r]=\int_{\R}\widehat{\Phi}(x)\left(\delta_0(x)+\frac{(-1)^{r+1}}{2}\right)\dd x.$$ Thus, this theorem reveals that the symmetry type of $\mathcal{F}_r$ is $$%
G(\mathcal{F}_r)=
\begin{cases}
Sp & \text{if $r$ is even,} \\
O & \text{if $r=1$,} \\
SO(\mathrm{even}) \text{ or } O \text{ or } SO(\mathrm{odd}) & \text{if $r\geq 3$ is odd.}
\end{cases}$$ Some additional comments are given in remark \[remark4\] page .
The proof is detailed and will be a model for the next density results. According to proposition \[explicit\] and , we have $$\begin{gathered}
\label{explicitaverage}
\Eh[q]\left(D_{1,q}[\Phi;r]\right)=%
E[\Phi;r]%
+\Eh[q]\left(P_q^1[\Phi;r]\right) \\
+\sum_{m=0}^{r-1}(-1)^m\Eh[q]\left(P_q^2[\Phi;r,m]\right)+O\left(\frac{1}{\log{\left(q^r\right)}}\right).\end{gathered}$$ The first term in is the main term given in the theorem. We now estimate the second term of *via* the trace formula given in proposition \[iwlusatr\]. $$\label{eq_pu}
\Eh[q]\left(P_q^1[\Phi;r]\right)=%
\PP{q,\mathrm{new}}1[\Phi;r]+\PP{q,\mathrm{old}}1[\Phi;r]$$ where $$\begin{aligned}
\PP{q,\mathrm{new}}1[\Phi;r] &= %
-\frac{2}{\log{\left(q^r\right)}}\sum_{\substack{p\in\prem \\ p\nmid q}}\Delta_q(p^r,1)\frac{\log{p}}{\sqrt{p}}%
\widehat{\Phi}\left(\frac{\log{p}}{\log{\left(q^r\right)}}\right), \\
\PP{q,\mathrm{old}}1[\Phi;r] &= \frac{2}{q\log{\left(q^r\right)}}%
\sum_{\ell\mid q^\infty}\frac{1}{\ell}\sum_{\substack{p\in\prem \\ p\nmid q}}\Delta_1(p^r\ell^2,1)%
\frac{\log{p}}{\sqrt{p}}\widehat{\Phi}\left(\frac{\log{p}}{\log{\left(q^r\right)}}\right).\end{aligned}$$ Let us estimate the new part which can be written as $$\begin{gathered}
\PP{q,\mathrm{new}}1[\Phi;r]=%
-\frac{2%
(2\pi i^\kappa)
}{\log{\left(q^r\right)}}\sum_{\substack{c\geq 1 \\ q\mid c}}%
\sum_{p\in\prem}\left(\frac{\log{p}}{\sqrt{p}}\delta_{q\nmid p}\un_{\left[1,q^{r\nu}\right]}(p)\right)%
\frac{S(p^r,1;c)}{c} \\
\times J_{\kappa-1}\left(\frac{4\pi\sqrt{p^r}}{c}\right)\widehat{\Phi}\left(\frac{\log{p}}{\log{\left(q^{r}\right)}}\right).\end{gathered}$$ Thanks to , the function $$%
h(m;c)\coloneqq J_{\kappa-1}\left(\frac{4\pi\sqrt{m}}{c}\right)%$$satisfies hypothesis $\prp(T;M,1,C)$ with $$%
T(M,1,C)=\left(1+\frac{\sqrt{M}}{C}\right)^{1/2-\kappa}%
\left(\frac{\sqrt{M}}{C}\right)^{\kappa-1}.$$Hence, if $\nu\leq 2/r^2$ then corollary \[usefulsieve\] leads to $$\begin{aligned}
%
\label{eq_plusgrand}%
\PP{q,\mathrm{new}}1[\Phi;r]%
&\ll_\epsilon%
q^\epsilon%
\sumsh_{\substack{1\leq M\leq q^{\nu r^2}\\ C\geq q/2}}%
\left(1+\sqrt{\frac{M}{q}}\right)%
\left(\frac{\sqrt{M}}{C}\right)^{\kappa-1-2\theta}%
\\%
&\ll_\epsilon%
q^\epsilon%
\sumsh_{1\leq M\leq q^{\nu r^2}}%
\left(\frac{M^{\frac{\kappa-1}{2}-\theta}}{q^{\kappa-1-2\theta}}+%
\frac{M^{\frac{\kappa}{2}-\theta}}{q^{\kappa-\frac{1}{2}-2\theta}}\right)%\end{aligned}$$thanks to . Summing over $M$ *via* leads to $$\label{eq_pun}
\PP{q,\mathrm{new}}1[\Phi;r]\ll_\epsilon %
q^{\left(\frac{\kappa-1}{2}-\theta\right)(r^2\nu-2)+\epsilon}+%
q^{\left(\frac{\kappa}{2}-\theta\right)r^2\nu-\left(\kappa-\frac{1}{2}-2\theta\right)+\epsilon}$$ which is an admissible error term if $\nu<\nu_{1,\mathrm{max}}(r,\kappa,\theta)$. According to lemma \[lem\_old\] (with $\alpha_2=+\infty$) we have $$\label{eq_puo}
\PP{q,\mathrm{old}}1[\Phi;r]\ll_\epsilon q^{\frac{r\nu}{2}-1+\epsilon}$$ which is an admissible error term if $\nu<2/r$. Reporting and in we obtain $$\label{eq_psum}
\Eh[q]\left(P^1_q[\Phi;r]\right)\ll\frac{1}{q^{\delta_1}}$$ for some $\delta_1>0$ (depending on $\nu$ and $r$) as soon as $\nu<\nu_{1,\mathrm{max}}(r,\kappa,\theta)$. We now estimate the third term of . If $0\leq m\leq r-1$ then the trace formula given in proposition \[iwlusatr\] implies that $$\label{eq_pd}
\Eh[q]\left(P_q^2[\Phi;r,m]\right)=\PP{q,\mathrm{new}}2[\Phi;r,m]+\PP{q,\mathrm{old}}2[\Phi;r,m]$$ where $$\begin{aligned}
\PP{q,\mathrm{new}}2[\Phi;r,m] &= %
-\frac{2}{\log{\left(q^r\right)}}\sum_{\substack{p\in\prem \\ p\nmid q}}%
\Delta_q\left(p^{2(r-m)},1\right)\frac{\log{p}}{p}\widehat{\Phi}%
\left(\frac{\log{\left(p^2\right)}}{\log{\left(q^r\right)}}\right), \\
\PP{q,\mathrm{old}}2[\Phi;r,m] &= \frac{2}{q\log{\left(q^r\right)}}%
\sum_{\ell\mid q^\infty}\frac{1}{\ell}\sum_{\substack{p\in\prem \\ p\nmid q}}%
\Delta_1\left(p^{2(r-m)}\ell^2,1\right)\frac{\log{p}}{p}\widehat{\Phi}%
\left(\frac{\log{\left(p^2\right)}}{\log{\left(q^r\right)}}\right).\end{aligned}$$ Let us estimate the new part which can be written as $$\begin{gathered}
\PP{q,\mathrm{new}}2[\Phi;r,m]=-\frac{2%
(2\pi i^\kappa)%
}{\log{\left(q^r\right)}}\sum_{\substack{c\geq 1 \\
q\mid c}}\sum_{p\in\prem}\left(\frac{\log{p}}{\sqrt{p}}\delta_{q\nmid p}\un_{\left[1,q^{\frac{r\nu}{2}}\right]}(p)\right)\frac{S\left(p^{2(r-m)},1;c\right)}{c} \\
\times \frac{1}{\sqrt{p}}J_{\kappa-1}\left(\frac{4\pi\sqrt{p^{2(r-m)}}}{c}\right)%
\widehat{\Phi}\left(%
\frac{\log p}{\log q^{r/2}}
\right).\end{gathered}$$ The function $$%
h(m,c)\coloneqq J_{\kappa-1}\left(\frac{4\pi\sqrt{m}}{c}\right)\times\frac{1}{m^{1/(4(r-m))}}%$$satisfies hypothesis $\prp(T;M,1,C)$ with $$%
T(M,1,C)=\left(1+\frac{\sqrt{M}}{C}\right)^{1/2-\kappa}%
\left(\frac{\sqrt{M}}{C}\right)^{\kappa-1}%
\frac{1}{M^{1/(4(r-m))}}.$$Hence, if $\nu\leq 2/r^2$ then corollary \[usefulsieve\] leads to $$%
\PP{q,\mathrm{new}}2[\Phi;r,m]%
\ll_\epsilon%
q^\epsilon%
\sumsh_{\substack{M\leq q^{\nu r(r-m)}\\ C\geq q/2}}%
\frac{1}{(M)^{1/(4r-4m)}}%
\left(\frac{\sqrt{M}}{C}\right)^{\kappa-1-2\theta}%
\left(1+\sqrt{\frac{M}{q}}\right).%$$ This is smaller than the bound given in and hence is an admissible error term if $\nu<\nu_{1,\mathrm{max}}(r,\kappa,\theta)$. According to lemma \[lem\_old\], we have $$\label{eq_pdo}
\PP{q,\mathrm{old}}2[\Phi;r]\ll_\epsilon %
q%
^{-1+\epsilon}.$$ We obtain $$\label{eq_psdm}
\Eh[q]\left(P^2_q[\Phi;r%
,m%
]\right)\ll\frac{1}{q^{\delta_2}}$$ for some $\delta_2>0$ (depending on $\nu$ and $r$) as soon as $\nu<\nu_{1,\mathrm{max}}(r,\kappa,\theta)$. Finally, reporting and in , we get $$\label{eq_mhe}
\Eh[q]\left(D_{1,q}[\Phi;r]\right)=%
E[\Phi;r]+O\left(\frac{1}{\log q}\right).$$
Signed asymptotic expectation of the one-level density
------------------------------------------------------
In this part, we prove some density results for subfamilies of $\mathcal{F}_r$ on which the sign of the functional equation remains constant. The two subfamilies are defined by $$%%
\mathcal{F}_r^{\epsilon}\coloneqq\bigcup_{q\in\prem}\left\{L(\sym^rf,s), f\in \primeps{\kappa}{q}\right\}.%$$ Indeed, we compute the asymptotic expectation $\Eheps[\infty]\left(D_{1}[\Phi;r]\right)$.
\[density2\] Let $r\geq 1$ be an odd integer, $\epsilon=\pm 1$ and $\Phi\in\Schwartz_\nu(\R)$. We assume that hypothesis $\Nice(r,f)$ holds for any prime number $q$ and any $f\in \prim{\kappa}{q}$ and also that $\theta$ is admissible. Let $$%%
\nu_{1,\mathrm{max}}^\epsilon(r,\kappa,\theta)\coloneqq\inf{\left(\nu_{1,\mathrm{max}}(r,\kappa,\theta),\frac{3}{r(r+2)}\right).}%$$ If $\nu<\nu_{1,\mathrm{max}}^\epsilon(r,\kappa,\theta)$ then $$%%
\Eheps[\infty]\left(D_{1}[\Phi;r]\right)=E[\Phi;r].%$$
Some comments are given in remark \[remark5\] page .
By , we have $$\label{eq_fmnr}%%
\Eheps[q]\left(D_{1,q}[\Phi;r]\right)%
= %
\Eh[q]\left(D_{1,q}[\Phi;r]\right)-%
\epsilon\times\epsilon(k,r)\sqrt{q}\Eh[q]\left(\lambda_.(q)D_{1,q}[\Phi;r]\right).
%\epsilon\times\epsilon(k,r)\sqrt{q}\sumh_{f\in\prim{\kappa}{q}}\lambda_f(q)D_{1,q}[\Phi;r](f).%$$ The first term is the main term of the theorem thanks to theorem \[density1\]. According to proposition \[explicit\] and corollary \[lem\_sumlambdafq\], the second term (without the epsilon factors) is given by $$\begin{gathered}
\label{start}
\sqrt{q}\Eh[q]\left(\lambda_.(q)P_q^1[\Phi;r]\right) \\%
+\sqrt{q}\sum_{m=0}^{r-1}(-1)^m\Eh[q]\left(\lambda_.(q)P_q^2[\Phi;r,m]\right)+O\left(\frac{1}{\log{\left(q^r\right)}}\right).%\end{gathered}$$ Let us focus on the first term in knowing that the same discussion holds for the second term with even better results on $\nu$. We have $$\label{eq_resun}%%
\sqrt{q}\Eh[q]\left(\lambda_.(q)P_q^1[\Phi;r]\right)=\sqrt{q}\PP{q,\mathrm{new}}1[\Phi;r]+\sqrt{q}\PP{q,\mathrm{old}}1[\Phi;r]%$$ where $$\begin{aligned}
\PP{q,\mathrm{new}}1[\Phi;r]%
&=%
-\frac{2}{\log{\left(q^r\right)}}\sum_{\substack{p\in\prem \\
p\nmid q}}\Delta_q\left(p^{r}q,1\right)\frac{\log{p}}{\sqrt{p}}\widehat{\Phi}\left(\frac{\log{p}}{\log{\left(q^r\right)}}\right), \\%
\PP{q,\mathrm{old}}1[\Phi;r]%
&=%
\frac{2}{q\nu(q)\log{\left(q^r\right)}}\sum_{\ell\mid q^\infty}\frac{1}{\ell}\sum_{\substack{p\in\prem \\%
p\nmid q}}\Delta_1\left(p^{r}\ell^2,q\right)\frac{\log{p}}{\sqrt{p}}\widehat{\Phi}\left(\frac{\log{p}}{\log{\left(q^r\right)}}\right).%\end{aligned}$$ Lemma \[lem\_old\] implies $$\label{eq_rqpuqo}%
\sqrt{q}\PP{q,\mathrm{old}}1[\Phi;r]\ll q^{(\nu r-4)/2}%$$ which is an admissible error term if $\nu<4/r$. The new part is given by $$%
\PP{q,\mathrm{new}}1[\Phi;r]%
=%
-\frac{2%
(2\pi i^\kappa)
}{\log{\left(q^r\right)}}%
\sum_{\substack{c\geq 1 \\%
q\mid c}}\sum_{\substack{p\in\prem \\%
q\nmid p}}\frac{\log{p}}{\sqrt{p}}\frac{S\left(p^{r}q,1;c\right)}{c}%
J_{\kappa-1}\left(\frac{4\pi\sqrt{p^{r}q}}{c}\right)%
\widehat{\Phi}\left(\frac{\log{\left(p\right)}}{\log{\left(q^{r}\right)}}\right).%$$ and can be written as $$-\frac{2(2\pi i^\kappa)}{\log{\left(q^r\right)}}\sum_{\substack{c\geq 1 \\
q\mid c}}\sum_{m\geqslant 1}\widehat{a}_m\frac{S(m,1;c)}{c}J_{\kappa-1}\left(\frac{4\pi\sqrt{m}}{c}\right)\widehat{\Phi}\left(\frac{\log{\left(m/q\right)}}{\log{(q^{r^2})}}\right)$$ where $$\widehat{a}_m\coloneqq\un_{[1,q^{1+\nu r^2}]}\begin{cases}
0 & \text{if $q\nmid m$ or $m\neq p^rq$ for some $p\neq q$ in $\mathcal{P}$}, \\
\frac{\log{p}}{\sqrt{p}} & \text{if $m=p^rq$ for some $p\neq q$ in $\mathcal{P}$.}
\end{cases}$$ Thus, if $\nu\leq 1/r^2$ then we obtain $$%
\PP{q,\mathrm{new}}1[\Phi;r,m]%
\ll_\epsilon%
q^\epsilon%
\sumsh_{\substack{M\leq q^{1+\nu r^2}\\ C\geq q/2}}%
\left(\frac{\sqrt{M}}{C}\right)^{\kappa-1-2\theta}%
\left(1+\sqrt{\frac{M}{q}}\right)%$$ as in the proof of corollary \[usefulsieve\]. Summing over $C$ *via* gives $$%
\PP{q,\mathrm{new}}1[\Phi;r%
,m%
]\ll_\epsilon %
q^\epsilon\sumsh_{M\leq q^{1+r^2\nu}}%
\left(\frac{M^{\frac{\kappa-1}{2}-\theta}}{q^{\kappa-1-2\theta}}+%
\frac{M^{\frac{\kappa}{2}-\theta}}{q^{\kappa-\frac{1}{2}-2\theta}}\right).$$ Summing over $M$ *via* leads to $$\label{eq_rqpuqn}
\PP{q,\mathrm{new}}1[\Phi;r%
,m%
]\ll_\epsilon %
q^{\left(\frac{\kappa-1}{2}-\theta\right)r^2\nu-(\frac{\kappa-1}{2}-\theta)+\epsilon}+%
q^{\left(\frac{\kappa}{2}-\theta\right)r^2\nu-\left(\frac{\kappa-1}{2}-\theta\right)+\epsilon}$$ which is an admissible error term if $\nu<\frac{1}{r^2}\left(1-\frac{1}{\kappa-2\theta}\right)$.
Quadratic statistics for low-lying zeros {#two}
========================================
Asymptotic expectation of the two-level density and asymptotic variance {#sec_twoandvar}
-----------------------------------------------------------------------
\[def\_tld\] Let $f\in\prim{\kappa}{q}$ and $\Phi_1$, $\Phi_2$ in $\Schwartz_\nu(\R)$. The *two-level density* (relatively to $\Phi_1$ and $\Phi_2$) of $\sym^rf$ is $$%
D_{2,q}[\Phi_1,\Phi_2;r](f)%
\coloneqq%
\sum_{\substack{(j_1,j_2)\in\mathcal{E}(f,r)^2\\ j_1\neq\pm j_2}}%
\Phi_1\left(\widehat{\rho}_{f,r}^{(j_1)}\right)
\Phi_2\left(\widehat{\rho}_{f,r}^{(j_2)}\right).$$
In this definition, it is important to note that the condition $j_1\neq j_2$ *does not imply* that $\widehat{\rho}_{f,r}^{(j_1)}\neq \widehat{\rho}_{f,r}^{(j_2)}$. It only implies this if the zeros are simple. Recall however that some $L$-functions of elliptic curves (hence of modular forms) have multiple zeros at the critical point .
The following lemma is an immediate consequence of definition \[def\_tld\].
\[lem\_tld\] Let $f\in\prim{\kappa}{q}$ and $\Phi_1$, $\Phi_2$ in $\Schwartz_\nu(\R)$. Then, $$\begin{gathered}
D_{2,q}[\Phi_1,\Phi_2;r](f)=%
D_{1,q}[\Phi_1;r](f)D_{1,q}[\Phi_2;r](f)%
-2D_{1,q}[\Phi_1\Phi_2;r](f)\\
+%
%\delta\left(\epsilon\left(\sym^rf\right)=-1\right)%
\un_{\primimpair{\kappa}{q}}(f)%
\times\Phi_1(0)\Phi_2(0).\end{gathered}$$
We first evaluate the product of one-level statistics on average.
\[lem\_conesson\]Let $r\geq 1$. Let $\Phi_1$ and $\Phi_2$ in $\Schwartz_\nu(\R)$. We assume that hypothesis $\Nice(r,f)$ holds for any prime number $q$ and any $f\in \prim{\kappa}{q}$ and also that $\theta$ is admissible. If $\;\nu<1/r^2$ then $$%
\Eh[\infty]\left(%
D_{1}[\Phi_1;r]D_{1}[\Phi_2;r]%
\right)%
=%
E[\Phi_1;r]E[\Phi_2;r]+%
2\int_\R\abs{u}\widehat{\Phi_1}(u)\widehat{\Phi_2}(u)\dd u.$$
Since theorem \[density1\] implies that $$\begin{gathered}
\Eh[\infty]\left(%
D_{1}[\Phi_1;r]D_{1}[\Phi_2;r]%
\right)%
-%
E[\Phi_1;r]E[\Phi_2;r]
=\\
\Eh[\infty]\left(%
D_{1}[\Phi_1;r]D_{1}[\Phi_2;r]%
\right)
-
\Eh[\infty]\left(D_{1}[\Phi_1;r]\right)%
\Eh[\infty]\left(D_{1}[\Phi_2;r]\right),\end{gathered}$$ lemma \[lem\_conesson\] reveals that the term $$%
\Ch[\infty]\left(D_{1}[\Phi_1;r],D_{1}[\Phi_2;r]\right)\coloneqq %
2\int_\R\abs{u}\widehat{\Phi_1}(u)\widehat{\Phi_2}(u)\dd u%$$ measures the dependence between $D_{1}[\Phi_1;r]$ and $D_{1}[\Phi_2;r]$. This term is the *asymptotic covariance* of $D_{1}[\Phi_1;r]$ and $D_{1}[\Phi_2;r]$. In particular, taking $\Phi_1=\Phi_2$, we obtain the asymptotic variance.
\[th\_variance\] Let $\Phi\in\Schwartz_{\nu}(\R)$. If $\nu<1/r^2$ then the asymptotic variance of the random variable $D_{1,q}[\Phi;r]$ is $$%
\Vh[\infty]\left(D_1[\Phi;r]\right)=2\int_\R\abs{u}\widehat{\Phi}^2(u)\dd u.$$
From proposition \[explicit\], we obtain $$\begin{gathered}
\label{eq_prodD}
\Eh[q]\left(D_{1,q}[\Phi_1;r]D_{1,q}[\Phi_2;r]\right)%
=%
E[\Phi_1;r]E[\Phi_2;r]+\Ch[q]%
\\
+\sum_{\substack{(i,j)\in\{1,2\}^2\\ i\neq j}}%
\sum_{m=0}^{r-1}(-1)^m\Eh[q]\left(P_q^1[\Phi_i;r]P_q^2[\Phi_j;r,m]\right)%
\\
+\sum_{m_1=0}^{r-1}\sum_{m_2=0}^{r-1}(-1)^{m_1+m_2}\Eh[q]\left(P_q^2[\Phi_1;r,m_1]P_q^2[\Phi_2;r,m_2]\right)%
+O\left(\frac{1}{\log{\left(q^r\right)}}\right)\end{gathered}$$ with $$%
\Ch[q]\coloneqq \Eh[q]\left(P_q^1[\Phi_1;r]P_q^1[\Phi_2;r]\right).$$ The error term is evaluated by use of theorem \[density1\] and equations , and . We first compute $\Ch[q]$. Using proposition \[iwlusatr\], we compute $\Ch[q]=E^n-4E^o$ with $$E^n\coloneqq%
\frac{4}{\log^2{\left(q^r\right)}}%
\sum_{\substack{p_1\in\prem\\ p_1\nmid q}}%
\sum_{\substack{p_2\in\prem\\ p_2\nmid q}}%
\frac{\log p_1}{\sqrt{p_1}}\frac{\log p_2}{\sqrt{p_2}}%
\widehat{\Phi_1}\left(\frac{\log p_1}{\log{\left(q^r\right)}}\right)%
\widehat{\Phi_2}\left(\frac{\log p_2}{\log{\left(q^r\right)}}\right)%
\Delta_q(p_1^r,p_2^r)$$ and $$\begin{gathered}
E^o\coloneqq%
\frac{1}{q\log^2{\left(q^r\right)}}%
\\%
\times%
\sum_{\substack{p_1\in\prem\\ p_1\nmid q}}%
\sum_{\substack{p_2\in\prem\\ p_2\nmid q}}%
\frac{\log p_1}{\sqrt{p_1}}\frac{\log p_2}{\sqrt{p_2}}%
\widehat{\Phi_1}\left(\frac{\log p_1}{\log{\left(q^r\right)}}\right)%
\widehat{\Phi_2}\left(\frac{\log p_2}{\log{\left(q^r\right)}}\right)%
\sum_{\ell\mid q^\infty}\frac{\Delta_1(\ell^2p_1^r,p_2^r)}{\ell}.\end{gathered}$$ By definition of the $\Delta$-symbol, we write $E^n=E^n_{\mathrm{p}}+\frac{8\pi i^{\kappa}}{\log^2{\left(q^r\right)}}E^n_{\mathrm{e}}$ with $$%
E^n_{\mathrm{p}}\coloneqq%
\frac{4}{\log^2{(q^r)}}%
\sum_{\substack{p\in\prem\\ p\nmid q}}%
\frac{\log^2 p}{p}\left(\widehat{\Phi_1}\widehat{\Phi_2}\right)%
\left(\frac{\log p}{\log{(q^r)}}\right)%$$ and $$\begin{gathered}
E^n_{\mathrm{e}}\coloneqq%
\sum_{\substack{c\geq 1\\ q\mid c}}
\sum_{\substack{p_1\in\prem\\ p_1\nmid q}}%
\sum_{\substack{p_2\in\prem\\ p_2\nmid q}}%
\frac{\log p_1}{\sqrt{p_1}}\frac{\log p_2}{\sqrt{p_2}}%
\widehat{\Phi_1}\left(\frac{\log p_1}{\log{\left(q^r\right)}}\right)%
\widehat{\Phi_2}\left(\frac{\log p_2}{\log{\left(q^r\right)}}\right)%
\\
\times%
\frac{S(p_1^r,p_2^r;c)}{c}%
J_{\kappa-1}\left(\frac{4\pi\sqrt{p_1^rp_2^r}}{c}\right).\end{gathered}$$ We remove the condition $p\nmid q$ from $E^n_{\mathrm{p}}$ at an admissible cost and obtain, after integration by parts, $$\label{eq_epn}
E^n_{\mathrm{p}}=2\int_\R\abs{u}\widehat{\Phi_1}(u)\widehat{\Phi_2}(u)\dd u%
+O\left(\frac{1}{\log^2{(q^r)}}\right).$$Using corollary \[usefulsieve\], we get $$\label{eq_een}
E^n_{\mathrm{e}}\ll\frac{1}{\log^2{\left(q^r\right)}}$$ as soon as $\nu\leq 1/r^2$. Finally, using lemma \[lem\_old\], we see that $E^o$ is an admissible error term for $\nu<1/r$ so that equations and lead to $$\label{eq_en}
\Ch[q]=2\int_\R\abs{u}\widehat{\Phi_1}(u)\widehat{\Phi_2}(u)\dd u%
+O\left(\frac{1}{\log^2{\left(q^r\right)}}\right).$$ Let $\{i,j\}=\{1,2\}$. We prove next that each $\Eh[q]\left(P^1_q[\Phi_i;r]P^2_q[\Phi_j;r,m]\right)$ is an error term when $\nu<1/r^2$. Using proposition \[iwlusatr\] and lemma \[lem\_old\] we have $$\begin{gathered}
\Eh[q]\left(P^1_q[\Phi_i;r]P^2_q[\Phi_j;r,m]\right)%
=%
\frac{8\pi i^\kappa}{\log^2{(q^r)}}%
\sum_{\substack{c\geq 1\\ q\mid c}}%
\sum_{\substack{p_1\in\prem\\ p_1\nmid q}}%
\sum_{\substack{p_2\in\prem\\ p_2\nmid q}}%
\frac{\log p_1}{\sqrt{p_1}}\frac{\log p_2}{p_2}%
\widehat{\Phi_i}\left(\frac{\log p_1}{\log{\left(q^r\right)}}\right)%
\\
\times%
\widehat{\Phi_j}\left(\frac{\log p_2}{\log{\left(q^{r/2}\right)}}\right)%
\frac{S(p_1^r,p_2^{2r-2m};c)}{c}%
J_{\kappa-1}\left(\frac{4\pi\sqrt{p_1^rp_2^{2r-2m}}}{c}\right)
+O\left(\frac{1}{\log{\left(q^r\right)}}\right)^2.\end{gathered}$$ We use corollary \[usefulsieve\] to conclude that$$\label{eq_utile}
\Eh[q]\left(P^1_q[\Phi_i;r]P^2_q[\Phi_j;r,m]\right)
\ll\frac{1}{\log q}$$ when $\nu<1/r^2$. Finally, $\Eh[q]\left(P^2_q[\Phi_1;r,m_1]P^2_q[\Phi_2;r,m_2]\right)$ is shown to be an error term in the same way.
Using lemmas \[lem\_tld\] and \[lem\_conesson\], theorem \[density1\], hypothesis $\Nice(r,f)$ and remark \[rem\_moyun\], we prove the following theorem.
\[thm\_twodens\] Let $r\geq 1$. Let $\Phi_1$ and $\Phi_2$ in $\Schwartz_\nu(\R)$. We assume that hypothesis $\Nice(r,f)$ holds for any prime number $q$ and any $f\in \prim{\kappa}{q}$ and also that $\theta$ is admissible. If $\;\nu<\nu_{2,\mathrm{max}}(r,\kappa,\theta)$ then $$\begin{gathered}
\Eh[\infty]\left(D_2[\Phi_1,\Phi_2;r]\right)%
=%
\left[%
\widehat{\Phi_1}(0)%
+%
\frac{(-1)^{r+1}}{2}\Phi_1(0)%
\right]%
\left[%
\widehat{\Phi_2}(0)%
+%
\frac{(-1)^{r+1}}{2}\Phi_2(0)%
\right]%
\\
+2\int_\R\abs{u}\widehat{\Phi_1}(u)\widehat{\Phi_2}(u)\dd u
-2\widehat{\Phi_1\Phi_2}(0)%
+\left((-1)^{r}+%
\frac{%
%\delta(2\nmid r)%
\un_{2\N+1}(r)
}{2}\right)%
\Phi_1(0)\Phi_2(0).%\end{gathered}$$
Some comments are given in remark \[remark6\] page .
Signed asymptotic expectation of the two-level density and signed asymptotic variance
-------------------------------------------------------------------------------------
In this part, $r$ is *odd*.
\[lem\_signedconesson\] Let $\Phi_1$ and $\Phi_2$ in $\Schwartz_\nu(\R)$. If $\nu<1/(2r^2)$ then $$%
\Eheps[\infty]\left(%
D_{1}[\Phi_1;r]D_{1}[\Phi_2;r]%
\right)%
=%
E[\Phi_1;r]E[\Phi_2;r]+%
2\int_\R\abs{u}\widehat{\Phi_1}(u)\widehat{\Phi_2}(u)\dd u.$$
By theorem \[density2\] and lemma \[lem\_signedconesson\] we have $$\begin{gathered}
\Eheps[\infty]\left(%
D_{1}[\Phi_1;r]D_{1}[\Phi_2;r]%
\right)%
-%
E[\Phi_1;r]E[\Phi_2;r]
=\\
\Eheps[\infty]\left(%
D_{1}[\Phi_1;r]D_{1}[\Phi_2;r]%
\right)
-
\Eheps[\infty]\left(D_{1}[\Phi_1;r]\right)%
\Eheps[\infty]\left(D_{1}[\Phi_2;r]\right).\end{gathered}$$ Thus, $$%
\Cheps[\infty]\left(D_{1}[\Phi_1;r],D_{1}[\Phi_2;r]\right)\coloneqq %
2\int_\R\abs{u}\widehat{\Phi_1}(u)\widehat{\Phi_2}(u)\dd u$$ is the *signed asymptotic covariance* of $D_{1}[\Phi_1;r]$ and $D_{1}[\Phi_2;r]$. In particular, taking $\Phi_1=\Phi_2$, we obtain the signed asymptotic variance.
Let $\Phi\in\Schwartz_{\nu}(\R)$. If $\nu<1/(2r^2)$ then the signed asymptotic variance of $D_1[\Phi;r]$ is $$%
\Vheps[\infty]\left(D_1[\Phi;r]\right)=2\int_\R\abs{u}\widehat{\Phi}^2(u)\dd u.$$
From proposition \[explicit\] and , we obtain $$\begin{gathered}
\label{eq_resu}
\Eheps[q]\left(D_{1,q}[\Phi_1;r]D_{1,q}[\Phi_2;r]\right)%
=%
E[\Phi_1;r]E[\Phi_2;r]%
+\Cheps[q]%
\\%
+%
\sum_{\substack{(i,j)\in\{1,2\}^2\\ i\neq j}}%
\sum_{m=0}^{r-1}(-1)^m\Eheps[q]\left(P_q^1[\Phi_i;r]P_q^2[\Phi_j;r,m]\right)%
\\
+\sum_{m_1=0}^{r-1}\sum_{m_2=0}^{r-1}(-1)^{m_1+m_2}\Eheps[q]\left(P_q^2[\Phi_1;r,m_1]P_q^2[\Phi_2;r,m_2]\right)%
+O\left(\frac{1}{\log{\left(q^r\right)}}\right)\end{gathered}$$ with $$%
\Cheps[q]\coloneqq \Eheps[q]\left(P_q^1[\Phi_1;r]P_q^1[\Phi_2;r]\right).$$Assume that $\nu<1/r^2$. Then equations , and proposition \[iwlusatr\] lead to $$\label{eq_valCheps}%
\Cheps[q]=2\int_{\R}\abs{u}\widehat{\Phi_1}(u)\widehat{\Phi_2}(u)\dd u%
-\epsilon\times\epsilon(\kappa,r)(G^n-4G^o)$$ with $$%%
G^n\coloneqq \frac{4\sqrt{q}}{\log^2{\left(q^r\right)}}%
\sum_{\substack{p_1\in\prem\\ p_1\nmid q}}%
\sum_{\substack{p_2\in\prem\\ p_2\nmid q}}%
\frac{\log p_1}{\sqrt{p_1}}%
\frac{\log p_2}{\sqrt{p_2}}%
\widehat{\Phi_1}\left(\frac{\log p_1}{\log{\left(q^r\right)}}\right)%
\widehat{\Phi_2}\left(\frac{\log p_2}{\log{\left(q^r\right)}}\right)%
\Delta_q\left(p_1^rq,p_2^r\right)$$ and $$\begin{gathered}
%
G^o\coloneqq \frac{1}{\sqrt{q}\log^2{\left(q^r\right)}}%
\\%
\times%
\sum_{\substack{p_1\in\prem\\ p_1\nmid q}}%
\sum_{\substack{p_2\in\prem\\ p_2\nmid q}}%
\frac{\log p_1}{\sqrt{p_1}}%
\frac{\log p_2}{\sqrt{p_2}}%
\widehat{\Phi_1}\left(\frac{\log p_1}{\log{\left(q^r\right)}}\right)%
\widehat{\Phi_2}\left(\frac{\log p_2}{\log{\left(q^r\right)}}\right)%
\sum_{\ell\mid q^\infty}%
\frac{\Delta_q\left(\ell^2p_1^r,p_2^rq\right)}{\ell}.\end{gathered}$$ Lemma \[deltaestimate\] implies that if $\nu<1/(2r^2)$ then $$\label{eq_majoGn}
G^n\ll
\frac{q^{\nu r[r(\kappa-1)+1]/2}}{q^{(\kappa-1)/2}}
%\frac{q^{-1/4+2\nu r(r/4+1/2)}}{\log^2{\left(q^r\right)}}$$ hence $G^n$ is an error term as soon as $\nu\leq 1/(2r^2)$. Lemma \[lem\_old\] implies $$\label{eq_majoGo}
G^o\ll q^{-3/2+\nu r+\epsilon}$$ which is an error term. Reporting equations and in we obtain $$\label{eq_unos}
\Cheps[\infty]=%
2\int_{\mathbb{R}}\abs{u}\widehat{\Phi_1}(u)\widehat{\Phi_2}(u)\dd u%$$ for $\nu\leq 1/(2r(r+2))$. Next, we prove that each $\Eheps[q]\left(P_q^1[\Phi_i;r]P_q^2[\Phi_j;r,m]\right)$ is an error term as soon as $\nu\leq 1/(2r^2)$. From equations and , we obtain $$\begin{gathered}
\label{eq_signedsecond}
\Eheps[q]\left(P_q^1[\Phi_i;r]P_q^2[\Phi_j;r,m]\right)%
=\\
-\epsilon\times\epsilon(\kappa,r)\sqrt{q}\sumh_{f\in\prim{\kappa}{q}}%
\lambda_f(q)P_q^1[\Phi_i;r]P_q^2[\Phi_j;r,m]%
+O\left(\frac{1}{\log q)}\right).%\end{gathered}$$ We use proposition \[iwlusatr\] and lemmas \[lem\_old\] and \[deltaestimate\] to have $$\begin{gathered}
\label{eq_signedinter}
\sqrt{q}\sumh_{f\in\prim{\kappa}{q}}%
\lambda_f(q)P_q^1[\Phi_i;r]P_q^2[\Phi_j;r,m]%
\ll\\%
\frac{q^{\nu r(2r-m+2)/4-1/4}}{\log^2q}+%
\frac{q^{(\nu r-1)/2+\epsilon}}{\log q}.%\end{gathered}$$ It follows from and that $$\label{eq_dos}
\Eheps[\infty]\left(P_q^1[\Phi_i;r]P_q^2[\Phi_j;r,m]\right)=0$$ for $\nu\leq 1/(2r(r+1))$. In the same way, we have, for $\nu$ in the previous range, $$\label{eq_tres}
\Eheps[\infty]\left(P_q^2[\Phi_1;r,m_1]P_q^2[\Phi_2;r,m_2]\right)=0.$$ Reporting , and in , we have the announced result.
Using lemmas \[lem\_tld\], \[lem\_signedconesson\], theorem \[density2\], hypothesis $\Nice(r,f)$ and , we prove the following theorem.
\[thm\_signedtwodens\] Let $f\in\prim{\kappa}{q}$ and $\Phi_1$, $\Phi_2$ in $\Schwartz_\nu(\R)$. If $\nu<1/(2r(r+1))$ then $$\begin{gathered}
\Eheps[\infty]\left(D_2[\Phi_1,\Phi_2;r]\right)%
=%
\left[%
\widehat{\Phi_1}(0)%
+%
\frac{1}{2}\Phi_1(0)%
\right]%
\left[%
\widehat{\Phi_2}(0)%
+%
\frac{1}{2}\Phi_2(0)%
\right]%
\\
+2\int_\R\abs{u}\widehat{\Phi_1}(u)\widehat{\Phi_2}(u)\dd u
-2\widehat{\Phi_1\Phi_2}(0)%
-\Phi_1(0)\Phi_2(0)%
\\
+%
%\delta(\epsilon=-1)%
\un_{\{-1\}}(\epsilon)
\Phi_1(0)\Phi_2(0).\end{gathered}$$
Remark \[rem\_symtyp\] together with theorem \[thm\_signedtwodens\] and a result of Katz & Sarnak (see [@KaSa Theorem A.D.2.2] or [@Mil Theorem 3.2]) imply that the symmetry type of $\mathcal{F}_r^{\epsilon}$ is as in table \[tab\_symty\]. Some additional comments are given in remark \[remark2\] page .
even odd
------ ------ ---------------------
$-1$ $SO(\mathrm{odd})$
$1$ $Sp$ $SO(\mathrm{even})$
: Symmetry type of $\mathcal{F}_r^{\epsilon}$[]{data-label="tab_symty"}
First asymptotic moments of the one-level density {#momentt}
=================================================
In this section, we compute the asymptotic $m$-th moment of the one level density namely $$\Mh[\infty,m]\left(D_{1,q}[\Phi;r]\right)\coloneqq\lim_{\substack{q\in\mathcal{P} \\ q\to+\infty}}\Mh[q,m]\left(D_{1,q}[\Phi;r]\right)$$ where $$\Mh[q,m]\left(D_{1,q}[\Phi;r]\right)=%
\Eh[q]%
\left(%
\left(D_{1,q}[\Phi;r]-\Eh[q](D_{1,q}[\Phi;r])\right)^m
\right)%$$ for $m$ small enough (regarding to the size of the support of $\Phi$). The end of this section is devoted to the proof of theorem \[thm\_F\]. Note that we can assume that $m\geq 3$ since the work has already been done for $m=1$ and $m=2$. Thanks to equation and proposition \[explicit\], we have $$\begin{aligned}
\Mh[q,m]\left(D_{1,q}[\Phi;r]\right) & = %
\sum_{\ell=0}^m\binom{m}{\ell}\Eh[q]\left(P_q^1[\Phi;r]^{m-\ell}\left(P_q^2[\Phi;r]+%
O\left(\frac{1}{\log q}\right)\right)^\ell\right) \\
\label{eq_evmom} \\
& = \sum_{\substack{0\leq\ell\leq m \\
0\leq\alpha\leq\ell}}\binom{m}{\ell}\binom{\ell}{\alpha}R(q)^{\ell-\alpha}%
\Eh[q]\left(P_q^1[\Phi;r]^{m-\ell}P_q^2[\Phi;r]^{\alpha}\right)\end{aligned}$$ where $$\begin{aligned}
P_q^2[\Phi;r](f) & \coloneqq %
-\frac{2}{\log(q^r)}%
\sum_{j=0}^{r-1}(-1)^j%
\sum_{\substack{p\in\prem\\ p\nmid q}}%
\lambda_f\left(p^{2(r-j)}\right)\frac{\log p}{p}\widehat{\Phi}\left(\frac{2\log p}{\log(q^r)}\right) \\
\label{eq_tard} & = %
-\frac{2}{\log(q^r)}\sum_{j=1}^{r}(-1)^{r-j}\sum_{\substack{p\in\prem\\ p\nmid %
q}}\lambda_f\left(p^{2j}\right)\frac{\log p}{p}\widehat{\Phi}\left(\frac{2\log %
p}{\log(q^r)}\right)\end{aligned}$$ and $R$ is a positive function satisfying $$%
R(q)\ll\frac{1}{\log q}.$$ Thus, an asymptotic formula for $\Mh[q,m]\left(D_{1,q}[\Phi;r]\right)$ directly follows from the next proposition.
\[moments\] Let $r\geq 1$ be any integer. We assume that hypothesis $\Nice(r,f)$ holds for any prime number $q$ and any primitive holomorphic cusp form of level $q$ and even weight $\kappa$. Let $\alpha\geq 0$ and $\ell\geq 0$ be any integers.
- If $\;\alpha\geq 1$ and $\;\alpha\nu<4/r^2$ then $$\Eh[q]\left(P_q^2[\Phi;r]^{\alpha}\right)=O\left(\frac{1}{\log{q}}\right).$$
- If $\;1\leq\alpha\leq\ell\leq m-1$ and $(\alpha+m-\ell)\nu<4/(r(r+2))$ then $$\Eh[q]\left(P_q^1[\Phi;r]^{m-\ell}P_q^2[\Phi;r]^{\alpha}\right)=O\left(\frac{1}{\log{q}}\right).$$
- If $\;\alpha\geq 1$ and $\;\alpha\nu<4/(r(r+2))$ then $$\Eh[q]\left(P_q^1[\Phi;r]^{\alpha}\right)=\begin{cases}
O\left(\frac{1}{\log^2{(q)}}\right) & \text{if $\alpha$ is odd,} \\
2\int_\R\abs{u}\widehat{\Phi}^2(u)\dd u\times\frac{\alpha!}{2^{\alpha/2}\left(\frac{\alpha}{2}\right)!}+O\left(\frac{1}{\log^2{(q)}}\right) & \text{otherwise}.
\end{cases}$$
One some useful combinatorial identity
--------------------------------------
In order to use the multiplicative properties of Hecke eigenvalues in the proof of proposition \[moments\], we want to reorder some sums over many primes to sums over distinct primes. We follow the work of Hughes & Rudnick [@HuRu §7] (see also [@MR2166468] and the work of Soshnikov [@Sos00]) to achieve this. Let $P(\alpha,s)$ be the set of surjective functions $$%
\sigma \colon \{1,\dotsc,\alpha\}\twoheadrightarrow \{1,\dotsc,s\}%$$ such that for any $j\in\{1,\dotsc,\alpha\}$, either $\sigma(j)=1$ or there exists $k<j$ such that $\sigma(j)=\sigma(k)+1$. This can be viewed as the number of partitions of a set of $\alpha$ elements into $s$ nonempty subsets. By definition, the cardinality of $P(\alpha,s)$ is the Stirling number of second kind [@Sta97 §1.4]. For any $j\in\{1,\dotsc,s\}$, let $$%
\varpi_j^{(\sigma)}\coloneqq\#\sigma^{-1}(\{j\}).%$$ Note that $$\label{eq_trivcond}%
\varpi_j^{(\sigma)}\geq 1 \quad\text{ for any $1\leq j\leq s$}\qquad\text{ and }\qquad
\sum_{j=1}^s\varpi_j^{(\sigma)}=\alpha.$$ The following lemma is lemma 7.3 of [@HuRu §7].
\[primesdistincts\] If $g$ is any function of $m$ variables then $$\sum_{j_1,\dotsc,j_m}g\left(x_{j_1},\dotsc,x_{j_m}\right)=\sum_{s=1}^m\sum_{\sigma\in P(m,s)}\sum_{\substack{i_1,\dotsc,i_s \\
\text{distinct}}}g\left(x_{i_{\sigma(1)}},\dotsc,x_{i_{\sigma(m)}}\right).$$
Proof of the first bullet of proposition \[moments\]
----------------------------------------------------
By the definition , we have $$\begin{gathered}
\label{eq_mais}
\Eh[q]\left(P_q^2[\Phi;r]^\alpha\right)=\frac{(-2)^\alpha}{\log^\alpha{(q^r)}}\sum_{1\leq j_1,\dotsc,j_\alpha\leq r}(-1)^{\alpha r-(j_1+\dotsc+j_\alpha)} \\
\times\sum_{\substack{p_1,\dotsc,p_\alpha\in\prem\\ q\nmid p_1\dotsc p_\alpha}}\left(\prod_{i=1}^\alpha\frac{\log p_i}{p_i}\widehat{\Phi}\left(\frac{2\log p_i}{\log(q^r)}\right)\right)\Eh[q]\left(\prod_{i=1}^\alpha\lambda_f\left(p_i^{2j_i}\right)\right).\end{gathered}$$ Writing $\{\widehat{p}_i\}_{i\geq 1}$ for the increasing sequence of prime numbers except $q$, we have $$\begin{gathered}
\label{eq_renum}
\sum_{\substack{p_1,\dotsc,p_\alpha\in\prem\\ q\nmid p_1\dotsc p_\alpha}}\left(\prod_{i=1}^\alpha\frac{\log p_i}{p_i}\widehat{\Phi}\left(\frac{2\log p_i}{\log(q^r)}\right)\right)\Eh[q]\left(\prod_{i=1}^\alpha\lambda_f\left(p_i^{2j_i}\right)\right) \\
=\sum_{i_1,\dotsc,i_\alpha}\left(\prod_{\ell=1}^\alpha\frac{\log\widehat{p}_{i_\ell}}{\widehat{p}_{i_\ell}}\widehat{\Phi}\left(\frac{2\log\widehat{p}_{i_\ell}}{\log(q^r)}\right)\right)\Eh[q]\left(\prod_{\ell=1}^\alpha\lambda_f\left(\widehat{p}_{i_\ell}^{2j_\ell}\right)\right).\end{gathered}$$ Using lemma \[primesdistincts\], we rewrite the right sum in as $$\begin{gathered}
\label{eq_sosru}
\sum_{s=1}^\alpha\sum_{\sigma\in P(\alpha,s)}\sum_{\substack{k_1,\dotsc,k_s\\\text{distinct}}}\left(\prod_{i=1}^\alpha\frac{\log\widehat{p}_{k_{\sigma(i)}}}{\widehat{p}_{k_{\sigma(i)}}}\widehat{\Phi}\left(\frac{
2\log\widehat{p}_{k_{\sigma(i)}}}{\log(q^r)}\right)\right)\Eh[q]\left(\prod_{i=1}^\alpha\lambda_f\left(\widehat{p}_{k_{\sigma(i)}}^{2j_i}\right)\right) \\
=\sum_{s=1}^\alpha\sum_{\sigma\in P(\alpha,s)}\sum_{\substack{k_1,\dotsc,k_s\\\text{distinct}}}\left(\prod_{u=1}^s\left(\frac{\log\widehat{p}_{k_{u}}}{\widehat{p}_{k_{u}}}\widehat{\Phi}\left(\frac{
2\log\widehat{p}_{k_{u}}}{\log(q^r)}\right)\right)^{\varpi^{(\sigma)}_u}\right)\Eh[q]\left(\prod_{\substack{1\leq u\leq s \\
1\leq j\leq r}}\lambda_f\left(\widehat{p}_{k_{u}}^{2j}\right)^{\varpi_{u,j}^{(\sigma)}}\right)\end{gathered}$$ where $$\varpi_{u,j}^{(\sigma)}\coloneqq\#\{1\leq i\leq\alpha, \sigma(i)=u, j_i=j\}$$ for any $1\leq u\leq s$ and any $1\leq j\leq r$. Now, we show that $$\begin{gathered}
\label{s<alpha}
\sum_{s=1}^{\alpha-1}\sum_{\sigma\in P(\alpha,s)}\sum_{\substack{k_1,\dotsc,k_s\\\text{distinct}}}\left(\prod_{u=1}^s\left(\frac{\log\widehat{p}_{k_{u}}}{\widehat{p}_{k_{u}}}\widehat{\Phi}\left(\frac{
2\log\widehat{p}_{k_{u}}}{\log(q^r)}\right)\right)^{\varpi^{(\sigma)}_u}\right)\Eh[q]\left(\prod_{\substack{1\leq u\leq s \\
1\leq j\leq r}}\lambda_f\left(\widehat{p}_{k_{u}}^{2j}\right)^{\varpi_{u,j}^{(\sigma)}}\right) \\
\ll\log^{\alpha-1}{(q)}.\end{gathered}$$ For $s<\alpha$ and $\sigma\in P(\alpha,s)$, we use together with to obtain that the left-hand side of the previous equation is bounded by $$\sum_{s=1}^{\alpha-1}\sum_{\sigma\in P(\alpha,s)}%
\sum_{\substack{k_1,\dotsc,k_s\\\text{distinct}}}%
\prod_{u=1}^s%
\left(%
\frac{%
\log\widehat{p}_{k_{u}}%
}{%
\widehat{p}_{k_{u}}%
}%
\abs{%
\widehat{\Phi}\left(%
\frac{%
2\log\widehat{p}_{k_{u}}%
}{\log(q^r)}%
\right)%
}%
\right)^{\varpi_u^{(\sigma)}}.%$$ Since $s<\alpha$, equation implies that $\varpi_u^{(\sigma)}>1$ for some $1\leq u\leq s$. These values lead to convergent, hence bounded, sums. Let $$%
d^{(\sigma)}\coloneqq\#\left\{1\leq u\leq s \colon \varpi_u^{(\sigma)}=1\right\}\in\{0,\dotsc,\alpha-1\},%$$ then $$\begin{gathered}
\sum_{s=1}^{\alpha-1}\sum_{\sigma\in P(\alpha,s)}%
\sum_{\substack{k_1,\dotsc,k_s\\\text{distinct}}}%
\prod_{u=1}^s%
\left(%
\frac{%
\log\widehat{p}_{k_{u}}%
}{%
\widehat{p}_{k_{u}}%
}%
\abs{%
\widehat{\Phi}\left(%
\frac{%
2\log\widehat{p}_{k_{u}}%
}{\log(q^r)}%
\right)%
}%
\right)^{\varpi_u^{(\sigma)}} \\
\ll%
\sum_{s=1}^{\alpha-1}\sum_{\sigma\in P(\alpha,s)}%
\sum_{\substack{k_1,\dotsc,k_d\\\text{distinct}}}%
\prod_{u=1}^{d^{(\sigma)}}%
\left(%
\frac{%
\log\widehat{p}_{k_{u}}%
}{%
\widehat{p}_{k_{u}}%
}%
\abs{%
\widehat{\Phi}\left(%
\frac{%
2\log\widehat{p}_{k_{u}}%
}{\log(q^r)}%
\right)%
}%
\right)%
\ll\log^{\alpha-1}{(q)}.\end{gathered}$$ We have altogether $$\begin{gathered}
\label{eq_warwick}
\Eh[q]\left(P_q^2[\Phi;r]^\alpha\right)=\frac{(-2)^\alpha}{\log^\alpha{(q^r)}}\sum_{1\leq j_1,\dotsc,j_\alpha\leq r}(-1)^{\alpha r-(j_1+\dotsc+j_\alpha)} \\
\times\sum_{\substack{k_1,\dotsc,k_\alpha\\\text{distinct}}}\left(\prod_{u=1}^\alpha\left(\frac{\log\widehat{p}_{k_{u}}}{\widehat{p}_{k_{u}}}\widehat{\Phi}\left(\frac{
2\log\widehat{p}_{k_{u}}}{\log(q^r)}\right)\right)\right)\Eh[q]\left(\lambda_f\left(\prod_{u=1}^\alpha\widehat{p}_{k_{u}}^{2j_u}\right)\right) \\
+O\left(\frac{1}{\log{q}}\right)\end{gathered}$$ since the only element of $P(\alpha,\alpha)$ is the identity function. By lemmas \[lem\_delun\] and \[deltaestimate\], we have $$%
\Eh[q]\left(\lambda_f\left(\prod_{u=1}^\alpha\widehat{p}_{k_{u}}^{2j_u}\right)\right)%
\ll%
\frac{1}{q}\prod_{u=1}^\alpha \widehat{p}_{k_{u}}^{j_u/2}\log{\widehat{p}_{k_{u}}}$$ hence the first term in the right-hand side of is bounded by a negative power of $q$ as soon as $\alpha\nu r^2 < 4$.
Proof of the third bullet of proposition \[moments\]
----------------------------------------------------
By proposition \[explicit\], we have $$\label{eq_EhPqU}
\Eh[q](P_q^1[\Phi;r]^\alpha)%
=%
\frac{(-2)^\alpha}{\log^\alpha{(q^r)}}%
\sum_{\substack{p_1,\dotsc,p_\alpha\in\prem\\ p_1,\dotsc,p_\alpha\nmid q}}%
\left(\prod_{i=1}^\alpha\frac{\log p_i}{\sqrt{p_i}}\widehat{\Phi}\left(\frac{\log p_i}{\log q^r}\right)\right)%
\Eh[q]\left(\prod_{i=1}^\alpha\lambda_f\left(p_i^r\right)\right).%$$ Using lemma \[primesdistincts\], we rewrite equation as $$\begin{aligned}
\Eh[q](P_q^1[\Phi;r]^\alpha)%
&=%
\frac{(-2)^\alpha}{\log^\alpha{(q^r)}}%
\sum_{s=1}^\alpha%
\sum_{\sigma\in P(\alpha,s)}%
\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}%
\left(\prod_{j=1}^\alpha%
\left(\frac{\log\widehat{p}_{i_{\sigma(j)}}}{\sqrt{\widehat{p}_{i_{\sigma(j)}}}}%
\widehat{\Phi}\left(\frac{\log\widehat{p}_{i_{\sigma(j)}}}{\log{(q^r)}}\right)\right)\right)%
\\%
&%
\phantom{%
=\frac{(-2)^\alpha}{\log^\alpha{(q^r)}}%
\sum_{s=1}^\alpha%
\sum_{\sigma\in P(\alpha,s)}%
\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}%
}%
\times%
\Eh[q]\left(\prod_{j=1}^\alpha\lambda_f\left(\widehat{p}_{i_{\sigma(j)}}^r\right)\right)%
\\%
&=%
\frac{(-2)^\alpha}{\log^\alpha{(q^r)}}%
\sum_{s=1}^\alpha%
\sum_{\sigma\in P(\alpha,s)}%
\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}%
\left(%
\prod_{u=1}^s%
\left(%
\frac{%
\log\widehat{p}_{i_u}%
}{%
\sqrt{%
\widehat{p}_{i_u}%
}%
}%
\widehat{\Phi}\left(%
\frac{%
\log\widehat{p}_{i_{u}}%
}{\log q^r}%
\right)%
\right)^{\varpi^{(\sigma)}_u}%
\right)%
\\%
&%
\label{eq_ewrite}
\phantom{%
=
\frac{(-2)^\alpha}{\log^\alpha{(q^r)}}%
\sum_{s=1}^\alpha%
\sum_{\sigma\in P(\alpha,s)}%
\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}%
}
\times%
\Eh[q]\left(%
\prod_{u=1}^s%
\lambda_f\left(\widehat{p}_{i_{u}}^r\right)^{\varpi^{(\sigma)}_u}%
\right).\end{aligned}$$ It follows from and that $$%%
\lambda_f\left(\widehat{p}_{i_{u}}^r\right)^{\varpi^{(\sigma)}_u}%
=%
\sum_{j_u=0}^{r\varpi^{(\sigma)}_u}%
x(\varpi^{(\sigma)}_u,r,j_u)\lambda_f\left(\widehat{p}_{i_{u}}^{j_u}\right).$$ Since $u\neq v$ implies that $\widehat{p}_{i_{u}}\neq\widehat{p}_{i_{v}}$, equation becomes $$\begin{gathered}
\label{eq_drole}
\Eh[q](P_q^1[\Phi;r]^\alpha)%
=%
\frac{(-2)^\alpha}{\log^\alpha{(q^r)}}%
\sum_{s=1}^\alpha%
\sum_{\sigma\in P(\alpha,s)}%
\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}%
\left(%
\prod_{u=1}^s%
\left(%
\frac{%
\log\widehat{p}_{i_u}%
}{%
\sqrt{%
\widehat{p}_{i_u}%
}%
}%
\widehat{\Phi}\left(%
\frac{%
\log\widehat{p}_{i_{u}}%
}{\log{(q^r)}}%
\right)%
\right)^{\varpi^{(\sigma)}_u}%
\right)%
\\%
\times%
\sum_{\substack{j_1,\dotsc,j_s \\
0\leq j_u\leq r\varpi^{(\sigma)}_u}}\left(%
\prod_{u=1}^sx(\varpi^{(\sigma)}_u,r,j_u)%
\right)%
\Eh[q]\left(%
\lambda_f\left(%
\prod_{u=1}^s\widehat{p}_{i_u}^{j_u}%
\right)%
\right).\end{gathered}$$ Using proposition \[iwlusatr\] and lemmas \[deltaestimate\] and \[lem\_delun\], we get $$%
\Eh[q]\left(%
\lambda_f\left(%
\prod_{u=1}^s\widehat{p}_{i_u}^{j_u}%
\right)%
\right)%
=%
\prod_{u=1}^s\delta_{j_u,0}+O\left(%
\frac{1}{q}\prod_{u=1}^s\widehat{p}_{i_u}^{j_u/4}\log\widehat{p}_{i_u}
\right)%$$ hence $$\label{eq_cesttpte}
\Eh[q](P_q^1[\Phi;r]^\alpha)=\TP+O(\TE)%$$ with $$\label{eq_deftp}
\TP\coloneqq%
\frac{(-2)^\alpha}{\log^\alpha{(q^r)}}%
\sum_{s=1}^\alpha%
\sum_{\sigma\in P(\alpha,s)}%
\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}%
\prod_{u=1}^s%
\left(%
\frac{%
\log\widehat{p}_{i_u}%
}{%
\sqrt{%
\widehat{p}_{i_u}%
}%
}%
\widehat{\Phi}\left(%
\frac{%
\log\widehat{p}_{i_{u}}%
}{\log{(q^r)}}%
\right)%
\right)^{\varpi^{(\sigma)}_u}%
x(\varpi^{(\sigma)}_u,r,0)$$ and $$\label{eq_defte}
\TE\coloneqq%
\frac{1}{q\log^\alpha{(q^r)}}%
\sum_{s=1}^\alpha%
\sum_{\sigma\in P(\alpha,s)}%
\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}%
\prod_{u=1}^s%
\left(%
\widehat{p}_{i_u}^{(r-2)/4}\log^2\widehat{p}_{i_u}%
\abs{%
\widehat{\Phi}\left(%
\frac{%
\log\widehat{p}_{i_{u}}%
}{\log{(q^r)}}%
\right)%
}
\right)^{\varpi^{(\sigma)}_u}.%$$ We have $$\label{eq_majte}
\TE=%
\frac{1}{q\log^\alpha{(q^r)}}%
\left(%
\sum_{\substack{p\in\prem\\ p\nmid q}}p^{(r-2)/4}\log^2p%
\abs{%
\widehat{\Phi}\left(%
\frac{\log p}{\log{(q^r)}}%
\right)%
}%
\right)^\alpha%
\ll%
q^{\alpha r\nu(r+2)/4-1}$$ so that, $\TE$ is an error term as soon as $$\label{eq_conddeux}%
\alpha r\nu(r+2)<4.$$ We assume from now on that this condition is satisfied. According to (recall that $r\geq 1$), we rewrite as $$\label{eq_tpsimpl}
\TP=%
\frac{(-2)^\alpha}{\log^\alpha{(q^r)}}%
\sum_{s=1}^\alpha%
\sum_{\sigma\in P^{\geq 2}(\alpha,s)}%
\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}%
\prod_{u=1}^s%
\left(%
\frac{%
\log\widehat{p}_{i_u}%
}{%
\sqrt{%
\widehat{p}_{i_u}%
}%
}%
\widehat{\Phi}\left(%
\frac{%
\log\widehat{p}_{i_{u}}%
}{\log{(q^r)}}%
\right)%
\right)^{\varpi^{(\sigma)}_u}%
x(\varpi^{(\sigma)}_u,r,0)$$ where $$%
P^{\geq 2}(\alpha,s)\coloneqq%
\left\{%
\sigma\in P(\alpha,s) \colon %
\forall u\in\{1,\dotsc,s\}, %
\varpi^{(\sigma)}_u\geq 2%
\right\}.$$ Moreover, if for at least one $\sigma$ and at least one $u$ (say $u_0$) we have $\varpi^{(\sigma)}_u\geq 3$, then $$\begin{gathered}
\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}%
\prod_{u=1}^s%
\left(%
\frac{%
\log\widehat{p}_{i_u}%
}{%
\sqrt{%
\widehat{p}_{i_u}%
}%
}%
\widehat{\Phi}\left(%
\frac{%
\log\widehat{p}_{i_{u}}%
}{\log{(q^r)}}%
\right)%
\right)^{\varpi^{(\sigma)}_u}%
x(\varpi^{(\sigma)}_u,r,0)%
\\%
\ll%
\left(%
\sum_{\substack{p\in\prem\\ p\leq q^{r\nu}}}\frac{\log^3{(p)}}{p^{3/2}}%
\right)%
\prod_{\substack{u=1\\ u\neq u_0}}^s
\left(%
\sum_{\substack{p_u\in\prem\\ p_u\leq q^{r\nu}}}\frac{\log^2{(p_u)}}{p_u}%
\right)%
\\%
\ll\label{eq_setm}%
(\log q)^{2s-2}.\end{gathered}$$ But, from , we deduce $$%
2s\leq\sum_{j=1}^s\varpi^{(\sigma)}_j=\alpha%$$ hence $(\log q)^{2s-2}\ll (\log q)^{\alpha-2}$. Reinserting this in and the result in , we obtain $$\begin{gathered}
\label{eq_tpplussimpl}
\TP=%
\frac{(-2)^\alpha}{\log^\alpha{(q^r)}}%
\sum_{s=1}^\alpha%
\sum_{\sigma\in P^{2}(\alpha,s)}%
\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}%
\prod_{u=1}^s%
\left(%
\frac{%
\log\widehat{p}_{i_u}%
}{%
\sqrt{%
\widehat{p}_{i_u}%
}%
}%
\widehat{\Phi}\left(%
\frac{%
\log\widehat{p}_{i_{u}}%
}{\log q^r}%
\right)%
\right)^{\varpi^{(\sigma)}_u}%
x(\varpi^{(\sigma)}_u,r,0) \\
+O\left(\frac{1}{\log^2{(q)}}\right)\end{gathered}$$ where $$%
P^{2}(\alpha,s)\coloneqq%
\left\{%
\sigma\in P(\alpha,s) \colon %
\forall u\in\{1,\dotsc,s\}, %
\varpi^{(\sigma)}_u= 2%
\right\}.$$ From , and , we deduce $$\begin{gathered}
\Eh[q](P_q^1[\Phi;r]^\alpha)%
=%
\frac{(-2)^\alpha}{\log^\alpha{(q^r)}}%
\sum_{s=1}^\alpha%
\sum_{\sigma\in P^{2}(\alpha,s)}%
\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}%
\prod_{u=1}^s%
\frac{%
\log^2{(\widehat{p}_{i_u})}%
}{%%
\widehat{p}_{i_u}%
}%
\widehat{\Phi}^2\left(%
\frac{%
\log\widehat{p}_{i_{u}}%
}{\log{(q^r)}}%
\right) \\
+O\left(\frac{1}{\log^2{(q)}}\right)\end{gathered}$$ since $x(2,r,0)=1$ according to . Note in particular that, according to the previous sum is zero if $\alpha$ is odd. Thus, we can assume now that $\alpha$ is even and get $$\begin{gathered}
\label{eq_ouf}
\Eh[q](P_q^1[\Phi;r]^\alpha)%
=%
\frac{(-2)^\alpha}{\log^\alpha{(q^r)}}%
\sum_{\sigma\in P^{2}(\alpha,\alpha/2)}%
\sum_{\substack{i_1,\dotsc,i_{\alpha/2}\\\text{distinct}}}%
\prod_{u=1}^{\alpha/2}%
\frac{%
\log^2{(\widehat{p}_{i_u})}%
}{%%
\widehat{p}_{i_u}%
}%
\widehat{\Phi}^2\left(%
\frac{%
\log\widehat{p}_{i_{u}}%
}{\log{(q^r)}}%
\right) \\
+O\left(\frac{1}{\log^2{(q)}}\right).\end{gathered}$$ However, summing over all the possible $(i_1,\dotsc,i_{\alpha/2})$ instead of the one with distinct indices reintroduces convergent sums that enter the error term because of the $1/\log^\alpha{(q^r)}$ factor. It follows that becomes: $$\begin{gathered}
\label{eq_ooo}
\Eh[q](P_q^1[\Phi;r]^\alpha)%
=%
\left[%
\frac{4}{\log^2{(q^r)}}%
\sum_{p\in\prem}%
\frac{\log^2{(p)}}{p}%
\widehat{\Phi}^2\left(%
\frac{\log p}{\log{(q^r)}}%
\right)%
\right]^{\alpha/2}%
\#P^{2}(\alpha,\alpha/2) \\
+O\left(\frac{1}{\log^2{(q)}}\right).\end{gathered}$$ Taking $m=2$ (we already proved that the second moment is finite, see section \[sec\_twoandvar\]) and reinserting the result in implies that $$%%
\Eh[q](P_q^1[\Phi;r]^\alpha)%
=%
\Eh[q](P_q^1[\Phi;r]^2)%
\#P^{2}(\alpha,\alpha/2)+O\left(\frac{1}{\log^2{(q)}}\right).$$ We conclude by computing $$%
\#P^{2}(\alpha,\alpha/2)=\frac{\alpha!}{2^{\alpha/2}\left(\frac{\alpha}{2}\right)!}.$$ (see [@Stan00 Example 5.2.6 and Exercise 5.43]).
Proof of the second bullet of proposition \[moments\]
-----------------------------------------------------
We mix the two techniques which have been used to prove the first and third bullets of proposition \[moments\]. We get following the same lines and thanks to lemma \[primesdistincts\] $$\begin{gathered}
\Eh[q]\left(P_q^1[\Phi;r]^{m-\ell}P_q^2[\Phi;r]^\alpha\right)=%
\frac{(-2)^{\alpha+m-\ell}}{\log^{\alpha+m-\ell}{(q^r)}}%
\sum_{1\leq j_1,\dotsc,j_\alpha\leq r}(-1)^{\alpha r-(j_1+\dotsc+j_\alpha)}%
\sum_{s=1}^{\alpha+m-\ell} \\
\times%
\sum_{\sigma\in%
P(\alpha+m-\ell,s)}\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}%
\prod_{u=1}^{s}%
\left(%
\frac{\log^{\varpi^{(\sigma,1)}_{u}+\varpi^{(\sigma,2)}_{u}}%
{\left(\widehat{p}_{i_u}\right)}}{\widehat{p}_{i_{u}}^{\varpi^{(\sigma,1)}_{u}/2+\varpi^{(\sigma,2)}_{u}}}%
\widehat{\Phi}\left(\frac{\log\widehat{p}_{i_{u}}}{\log(q^r)}\right)^{\varpi^{(\sigma,1)}_{u}}%
\widehat{\Phi}\left(\frac{2\log\widehat{p}_{i_{u}}}{\log(q^r)}\right)^{\varpi^{(\sigma,2)}_{u}}%
\right) \\
\times %
\Eh[q]\left(\prod_{u=1}^s\left(\lambda_f\left(\widehat{p}_{i_{u}}^{r}\right)^{\varpi^{(\sigma,1)}_{u}}%
\prod_{j=1}^r\lambda_f\left(\widehat{p}_{i_{u}}^{2j}\right)^{\varpi^{(\sigma,2)}_{u,j}}\right)\right)\end{gathered}$$ where $$\begin{aligned}
\varpi^{(\sigma,1)}_{u} & \coloneqq \#\left\{i\in\left\{1,\dotsc,m-\ell\right\},\; \sigma(i)=u\right\}, \\
\varpi^{(\sigma,2)}_{u} & \coloneqq \#\left\{i\in\left\{1,\dotsc,\alpha\right\},\; \sigma(m-\ell+i)=u\right\}, \\
\varpi^{(\sigma,2)}_{u,j} & \coloneqq %
\#\left\{i\in\left\{1,\dotsc,\alpha\right\},\; \sigma(m-\ell+i)=u \text{ and } j_{i}=j\right\}
%\#\left\{i\in\left\{1,\dotsc,\alpha\right\},\; \sigma(m-\ell+i)=u \text{ and } j_{m-\ell+i}=j\right\}\end{aligned}$$ for any $1\leq u\leq s$, any $1\leq j\leq r$ and any $\sigma\in P(\alpha+m-\ell,s)$. Note that these numbers satisfy $$\label{prop1}
\sum_{u=1}^s\left(\varpi^{(\sigma,1)}_{u}+\varpi^{(\sigma,2)}_{u}\right)=m-\ell+\alpha$$ and $$\label{prop2}
\sum_{j=1}^r\varpi^{(\sigma,2)}_{u,j}=\varpi^{(\sigma,2)}_{u}$$ for any $1\leq u\leq r$ and any $\sigma\in P(\alpha+m-\ell,s)$ by definition. They also satisfy $$\label{prop3}
\forall\sigma\in P(\alpha+m-\ell,s), \forall u\in\left\{1,\dotsc,s\right\}, \quad \varpi^{(\sigma,1)}_{u}+\varpi^{(\sigma,2)}_{u}\geq 1$$ since any $\sigma\in P(\alpha+m-\ell,s)$ is surjective and $$\label{prop4}
\forall\sigma\in P(\alpha+m-\ell,s), \forall i\in\left\{1,2\right\},\exists u_{i,\sigma}\in\left\{1,\dotsc,s\right\}, \quad \varpi^{(\sigma,i)}_{u_{i,\sigma}}\geq 1$$ since $\alpha\geq 1$ and $m-\ell\geq 1$. The strategy is to estimate individually each term of the $\sigma$-sum. Thus, we fix some integers $j_1,\dotsc,j_\alpha$ in $\left\{1,\dotsc,r\right\}$, some integer $s$ in $\left\{1,\dotsc,r\right\}$ and some application $\sigma$ in $P(\alpha+m-\ell,s)$.[*:* $\quad\mathit{\forall u\in\left\{1,\dotsc,s\right\}, \;\varpi^{(\sigma,1)}_{u}/2+\varpi^{(\sigma,2)}_{u}\leq 1.}$]{}Let us remark that if $\varpi^{(\sigma,2)}_{u}=1$ for some $1\leq u\leq s$ then there exists a unique $1\leq j_{i_u}\leq r$ depending on $\sigma$ such that $\varpi^{(\sigma,2)}_{u,j_{i_u}}=1$ and $\varpi^{(\sigma,2)}_{u,j}=0$ for any $1\leq j\neq j_{i_u}\leq r$ according to . Thus, $$\begin{gathered}
\prod_{u=1}^s\left(\lambda_f\left(\widehat{p}_{i_{u}}^{r}\right)^{\varpi^{(\sigma,1)}_{u}}\prod_{j=1}^r\left(\lambda_f\left(\widehat{p}_{i_{u}}^{2j}\right)^{\varpi^{(\sigma,2)}_{u,j}}\right)\right)=\lambda_f\left(\prod_{\substack{1\leq u\leq s\\
\left(\varpi^{(\sigma,1)}_{u},\varpi^{(\sigma,2)}_{u}\right)=(2,0)}}\widehat{p}_{i_{u}}^{r\varpi^{(\sigma,1)}_{u}/2}\right) \\
\times\lambda_f\left(\prod_{\substack{1\leq u\leq s\\
\left(\varpi^{(\sigma,1)}_{u},\varpi^{(\sigma,2)}_{u}\right)=(2,0)}}\widehat{p}_{i_{u}}^{r\varpi^{(\sigma,1)}_{u}/2}\prod_{\substack{1\leq u\leq s\\
\left(\varpi^{(\sigma,1)}_{u},\varpi^{(\sigma,2)}_{u}\right)=(1,0)}}\widehat{p}_{i_{u}}^{r\varpi^{(\sigma,1)}_{u}}\prod_{\substack{1\leq u\leq s\\
\left(\varpi^{(\sigma,1)}_{u},\varpi^{(\sigma,2)}_{u}\right)=(0,1)}}\widehat{p}_{i_{u}}^{2j_{i_u}\varpi^{(\sigma,2)}_{u,j_{i_u}}}\right)\end{gathered}$$ where the two integers appearing in the right-hand side of the previous equality are different according to . Consequently, proposition \[iwlusatr\] and lemmas \[deltaestimate\] and \[lem\_delun\] enable us to assert that $$\begin{gathered}
\Eh[q]\left(\prod_{u=1}^s\left(\lambda_f\left(\widehat{p}_{i_{u}}^{r}\right)^{\varpi^{(\sigma,1)}_{u}}\prod_{j=1}^r\left(\lambda_f\left(\widehat{p}_{i_{u}}^{2j}\right)^{\varpi^{(\sigma,2)}_{u,j}}\right)\right)\right)
\ll%
\frac{1}{q}%
\prod_{\substack{1\leq u\leq s\\ \left(\varpi^{(\sigma,1)}_{u},\varpi^{(\sigma,2)}_{u}\right)=(2,0)}}%
\frac{\log{\widehat{p}_{i_{u}}}}{\widehat{p}_{i_{u}}^{-r\varpi^{(\sigma,1)}_{u}/4}}%
\\
\times%
\prod_{\substack{1\leq u\leq s\\ \left(\varpi^{(\sigma,1)}_{u},\varpi^{(\sigma,2)}_{u}\right)=(1,0)}}%
\frac{\log{\widehat{p}_{i_{u}}}}{\widehat{p}_{i_{u}}^{-r\varpi^{(\sigma,1)}_{u}/4}}%
\prod_{\substack{1\leq u\leq s\\ \left(\varpi^{(\sigma,1)}_{u},\varpi^{(\sigma,2)}_{u}\right)=(0,1)}}%
\frac{\log{\widehat{p}_{i_{u}}}}{\widehat{p}_{i_{u}}^{-r\varpi^{(\sigma,2)}_{u}/2}}.%\end{gathered}$$ Note that, in this first case, the right hand term is $$%
\frac{1}{q}%
\prod_{u=1}^s%
\frac{%
\log{\widehat{p}_{i_{u}}}%
}%
{%
\widehat{p}_{i_{u}}^{-r(\varpi^{(\sigma,1)}_{u}/4+\varpi^{(\sigma,2)}_{u}/2)}%
}$$ hence the contribution of these $\sigma$’s to $\Eh[q]\left(P_q^1[\Phi;r]^{m-\ell}P_q^2[\Phi;r]^\alpha\right)$ is bounded by $$%
\frac{q^\epsilon}{q}
\left(%
\sum_{p\leq q^{\nu r}}\frac{1}{p^{1/2-r/4}}%
\right)^{m-\ell}%
\left(%
\sum_{p\leq q^{\nu r/2}}\frac{1}{p^{1-r/2}}%
\right)^{\alpha}%
\ll%
q^{%
\nu r/4[(m-\ell)(r+2)+\alpha r]-1+\epsilon%
}.$$ This is an admissible error term as long as $\nu r/4[(m-\ell)(r+2)+\alpha r]<1$.
[*:* $\quad\mathit{\exists
u_\sigma\in\left\{1,\dotsc,s\right\},
\;\varpi^{(\sigma,1)}_{u_\sigma}/2+\varpi^{(\sigma,2)}_{u_\sigma}>1.}$]{}
According to and , if $1\leq u\leq s$ and $1\leq j\leq r$ then $$\lambda_f\left(\widehat{p}_{i_{u}}^r\right)^{\varpi^{(\sigma,1)}_u}=\sum_{k_{u,1}=0}^{r\varpi^{(\sigma,1)}_u}x(\varpi^{(\sigma,1)}_u,r,k_{u,1})\lambda_f\left(\widehat{p}_{i_{u}}^{k_{u,1}}\right)$$ and $$\lambda_f\left(\widehat{p}_{i_{u}}^{2j}\right)^{\varpi^{(\sigma,2)}_{u,j}}=%
\sum_{k_{u,j,2}=0}^{j\varpi^{(\sigma,2)}_{u,j}}%
x(\varpi^{(\sigma,2)}_{u,j},2j,2k_{u,j,2})%
\lambda_f\left(\widehat{p}_{i_{u}}^{2k_{u,j,2}}\right)$$ since $x(\varpi^{(\sigma,2)}_{u,j},2j,k_{u,j,2})=0$ if $k_{u,j,2}$ is odd (see ). Then, one may remark that $$\prod_{1\leq j\leq r}\lambda_f\left(\widehat{p}_{i_{u}}^{2k_{u,j,2}}\right)=\sum_{\ell_u=0}^{K_u}y_{\ell_u}\lambda_f\left(\widehat{p}_{i_{u}}^{2\ell_u}\right)$$ for some integers $y_{\ell_u}$ and where $K_u\coloneqq\sum_{1\leq j\leq r}k_{u,j,2}$ for any $1\leq u\leq s$. All these facts lead to $$\begin{gathered}
\Eh[q]\left(P_q^1[\Phi;r]^{m-\ell}P_q^2[\Phi;r]^\alpha\right)=\frac{(-2)^{\alpha+m-\ell}(-1)^{\alpha r}}{\log^{\alpha+m-\ell}{(q^r)}}\sum_{1\leq j_1,\dotsc,j_\alpha\leq r}(-1)^{j_1+\dotsc+j_\alpha}\sum_{s=1}^{\alpha+m-\ell} \\
\times\sum_{\sigma\in P(\alpha+m-\ell,s)}\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}\prod_{u=1}^{s}\left(\frac{\log^{\varpi^{(\sigma,1)}_{u}+\varpi^{(\sigma,2)}_{u}}{\left(\widehat{p}_{i_u}\right)}}{\widehat{p}_{i_{u}}^{\varpi^{(\sigma,1)}_{u}\left/2\right.+\varpi^{(\sigma,2)}_{u}}}\widehat{\Phi}\left(\frac{\log\widehat{p}_{i_{u}}}{\log(q^r)}\right)^{\varpi^{(\sigma,1)}_{u}}\widehat{\Phi}\left(\frac{
2\log\widehat{p}_{i_{u}}}{\log(q^r)}\right)^{\varpi^{(\sigma,2)}_{u}}\right) \\
\times\sum_{\substack{0\leq k_{1,1}\leq r\varpi^{(\sigma,1)}_1 \\
\vdots \\
0\leq k_{s,1}\leq r\varpi^{(\sigma,1)}_s}}\sum_{\substack{0\leq k_{1,1,2}\leq \varpi^{(\sigma,2)}_{1,1} \\
\vdots \\
0\leq k_{s,1,2}\leq\varpi^{(\sigma,2)}_{s,1}}}\dotsc\sum_{\substack{0\leq k_{1,r,2}\leq r\varpi^{(\sigma,2)}_{1,r} \\
\vdots \\
0\leq k_{s,r,2}\leq r\varpi^{(\sigma,2)}_{s,r}}}\sum_{\substack{0\leq\ell_1\leq K_1 \\
\vdots \\
0\leq\ell_s\leq K_s}} \\
\times\prod_{u=1}^s\left(x\left(\varpi^{(\sigma,1)}_u,r,k_{u,1}\right)y_{\ell_u}\prod_{j=1}^r\left(x\left(\varpi^{(\sigma,2)}_{u,j},2j,2k_{u,j,2}\right)\right)\right) \\
\times\Eh[q]\left(\lambda_f\left(\prod_{u=1}^s\widehat{p}_{i_{u}}^{k_{u,1}}\right)\lambda_f\left(\prod_{u=1}^s\widehat{p}_{i_{u}}^{2\ell_u}\right)\right).\end{gathered}$$ Proposition \[iwlusatr\] and lemmas \[deltaestimate\] and \[lem\_delun\] enable us to assert that $$\Eh[q]\left(\lambda_f\left(\prod_{u=1}^s\widehat{p}_{i_{u}}^{k_{u,1}}\right)\lambda_f\left(\prod_{u=1}^s\widehat{p}_{i_{u}}^{2\ell_u}\right)\right)=\prod_{u=1}^s\delta_{k_{u,1},2\ell_u} \\
+O\left(\frac{1}{q}\prod_{u=1}^s\widehat{p}_{i_{u}}^{k_{u,1}/4+\ell_u/2}\log{\widehat{p}_{i_{u}}}\right)$$ and we can write $$\Eh[q]\left(P_q^1[\Phi;r]^{m-\ell}P_q^2[\Phi;r]^\alpha\right)=\TP+O(\TE)%$$ with $$\begin{gathered}
\label{endddd}
\TP\coloneqq\frac{(-2)^{\alpha+m-\ell}(-1)^{\alpha r}}{\log^{\alpha+m-\ell}{(q^r)}}\sum_{1\leq j_1,\dotsc,j_\alpha\leq r}(-1)^{j_1+\dotsc+j_\alpha}\sum_{s=1}^{\alpha+m-\ell} \\
\times\sum_{\sigma\in P(\alpha+m-\ell,s)}\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}\prod_{u=1}^{s}\left(\frac{\log^{\varpi^{(\sigma,1)}_{u}+\varpi^{(\sigma,2)}_{u}}{\left(\widehat{p}_{i_u}\right)}}{\widehat{p}_{i_{u}}^{\varpi^{(\sigma,1)}_{u}\left/2\right.+\varpi^{(\sigma,2)}_{u}}}\widehat{\Phi}\left(\frac{\log\widehat{p}_{i_{u}}}{\log(q^r)}\right)^{\varpi^{(\sigma,1)}_{u}}\widehat{\Phi}\left(\frac{
2\log\widehat{p}_{i_{u}}}{\log(q^r)}\right)^{\varpi^{(\sigma,2)}_{u}}\right) \\
\times\sum_{\substack{0\leq k_{1,1,2}\leq \varpi^{(\sigma,2)}_{1,1} \\
\vdots \\
0\leq k_{s,1,2}\leq\varpi^{(\sigma,2)}_{s,1}}}\dotsc\sum_{\substack{0\leq k_{1,r,2}\leq r\varpi^{(\sigma,2)}_{1,r} \\
\vdots \\
0\leq k_{s,r,2}\leq r\varpi^{(\sigma,2)}_{s,r}}}\sum_{\substack{0\leq\ell_1\leq r\min{\left(\varpi^{(\sigma,1)}_{1}/2,\varpi^{(\sigma,2)}_{1}\right)} \\
\vdots \\
0\leq\ell_s\leq r\min{\left(\varpi^{(\sigma,1)}_{s}/2,\varpi^{(\sigma,2)}_{s}\right)}}} \\
\times\prod_{u=1}^{s}\left(x\left(\varpi^{(\sigma,1)}_u,r,2\ell_u\right)y_{\ell_u}\prod_{j=1}^r\left(x\left(\varpi^{(\sigma,2)}_{u,j},2j,2k_{u,j,2}\right)\right)\right)\end{gathered}$$ and $$\begin{gathered}
\TE\coloneqq%
\frac{1}{q\log^{\alpha+m-\ell}{(q^r)}}%
\\%
\times%
\sum_{s=1}^{\alpha+m-\ell}%
\sum_{\sigma\in P(\alpha+m-\ell,s)}%
\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}
\prod_{u=1}^{s}\log^{\varpi^{(\sigma,1)}_{u}+\varpi^{(\sigma,2)}_{u}+1}{(\widehat{p}_{i_u})}
\widehat{p}_{i_u}^{\left(r/2-1\right)\left(\varpi^{(\sigma,1)}_{u}/2+\varpi^{(\sigma,2)}_{u}\right)}%
\\%
\times%
\left\vert\widehat{\Phi}\left(\frac{\log\widehat{p}_{i_{u}}}{\log(q^r)}\right)\right\vert^{\varpi^{(\sigma,1)}_{u}}%
\left\vert\widehat{\Phi}\left(\frac{2\log\widehat{p}_{i_{u}}}{\log(q^r)}\right)\right\vert^{\varpi^{(\sigma,2)}_{u}}\end{gathered}$$ which is bounded by $O_\epsilon\left(q^{(\alpha+m-\ell)\nu r^2/4-1+\varepsilon}\right)$ for any $\varepsilon>0$ and is an admissible error term if $(\alpha+m-\ell)\nu<4/r^2$. Estimating $\TP$ is possible since we can assume that $\sigma$ satisfies the following additional property. If $\varpi_u^{(\sigma,2)}=0$ for some $1\leq u\leq s$ then $\varpi_u^{(\sigma,1)}>1$. Let us assume on the contrary that $\varpi_u^{(\sigma,1)}\leq 1$ which entails $\varpi_u^{(\sigma,1)}=1$ according to . Then, $$x\left(\varpi^{(\sigma,1)}_u,r,2\ell_u\right)=x\left(1,r,0\right)=0$$ since $\ell_u=0$ and according to . Thus, the contribution of the $\sigma$’s which do not satisfy this last property vanishes. As a consequence, the sum over the distinct $i_1,\dotsc,i_s$ is bounded by $$\begin{gathered}
\sum_{\substack{i_1,\dotsc,i_s\\\text{distinct}}}\prod_{\substack{1\leq u\leq s \\
\left(\varpi_u^{(\sigma,1)},\varpi_u^{(\sigma,2)}\right)=(2,0)}}\left(\frac{\log^{2}{\left(\widehat{p}_{i_u}\right)}}{\widehat{p}_{i_{u}}}\left\vert\widehat{\Phi}\left(\frac{\log\widehat{p}_{i_{u}}}{\log(q^r)}\right)\right\vert^{2}\right) \\
\times\prod_{\substack{1\leq u\leq s \\
\left(\varpi_u^{(\sigma,1)},\varpi_u^{(\sigma,2)}\right)=(0,1)}}\left(\frac{\log{\left(\widehat{p}_{i_u}\right)}}{\widehat{p}_{i_{u}}}\left\vert\widehat{\Phi}\left(\frac{2\log\widehat{p}_{i_{u}}}{\log(q^r)}\right)\right\vert\right) \\
\times\prod_{\substack{1\leq u\leq s \\
\varpi_u^{(\sigma,1)}/2+\varpi_u^{(\sigma,2)}>1}}\left(\frac{\log^{\varpi^{(\sigma,1)}_{u}+\varpi^{(\sigma,2)}_{u}}{\left(\widehat{p}_{i_u}\right)}}{\widehat{p}_{i_{u}}^{\varpi^{(\sigma,1)}_{u}\left/2\right.+\varpi^{(\sigma,2)}_{u}}}\left\vert\widehat{\Phi}\left(\frac{\log\widehat{p}_{i_{u}}}{\log(q^r)}\right)\right\vert^{\varpi^{(\sigma,1)}_{u}}\left\vert\widehat{\Phi}\left(\frac{
2\log\widehat{p}_{i_{u}}}{\log(q^r)}\right)\right\vert^{\varpi^{(\sigma,2)}_{u}}\right)\end{gathered}$$ which is itself bounded by $O\left(\log^{A_\sigma}{(q)}\right)$ where the exponent is given by $$\begin{gathered}
A_\sigma\coloneqq 2\#\left\{1\leq u\leq s, \varpi_u^{(\sigma,2)}=0 \text{ and } \varpi_u^{(\sigma,1)}/2+\varpi_u^{(\sigma,2)}\leq 1\right\} \\
+\#\left\{1\leq u\leq s, \varpi_u^{(\sigma,2)}=1 \text{ and } \varpi_u^{(\sigma,1)}/2+\varpi_u^{(\sigma,2)}\leq 1\right\}<m-\ell+\alpha.\end{gathered}$$ The last inequality follows from (see and the additional property of $\sigma$) $$m-\ell+\alpha=A_\sigma+\sum_{\substack{1\leq u\leq s \\
\varpi_u^{(\sigma,1)}/2+\varpi_u^{(\sigma,2)}>1}}\left(\varpi_u^{(\sigma,1)}+\varpi_u^{(\sigma,2)}\right).$$ Thus, the contribution of the $\TP$ term of these $\sigma$’s to $\Eh[q]\left(P_q^1[\Phi;r]^{m-\ell}P_q^2[\Phi;r]^\alpha\right)$ is bounded by $O\left(\log^{-1}{(q)}\right)$.
Analytic and arithmetic toolbox {#klooster}
===============================
On smooth dyadic partitions of unity {#unity}
------------------------------------
Let $\psi\colon\R_+\to\R$ be any smooth function satisfying $$\psi(x)=\begin{cases}
0 & \text{if } 0\leq x\leq 1, \\
1 & \text{if } x>\sqrt{2}
\end{cases}$$ and $x^j\psi^{(j)}(x)\ll_j 1$ for any real number $x\geq 0$ and any integer $j\geq 0$. If $\rho\colon\R_+\to\R$ is the function defined by $$\rho(x)\coloneqq\begin{cases}
\psi(x) & \text{if } 0\leq x\leq\sqrt{2}, \\
1-\psi\left(\frac{x}{\sqrt{2}}\right) & \text{otherwise}
\end{cases}$$ then $\rho$ is a smooth function compactly supported in $[1,2]$ satisfying $$x^j\rho^{(j)}(x)\ll_j 1\quad\text{ and }\quad\sum_{a\in\mathbb{Z}}\rho\left(\frac{x}{\sqrt{2}^a}\right)=1$$ for any real number $x\geq 0$ and any integer $j\geq 0$.
If $F\colon\R_+^n\to\R$ is a function of $n\geq 1$ real variables then we can decompose it in $$F=\sum_{a_1\in\mathbb{Z}}\ldots\sum_{a_n\in\mathbb{Z}}F_{A_1,\cdots,A_n}$$ where $A_i\coloneqq\sqrt{2}^{a_i}$ and $$F_{A_1,\cdots,A_n}(x_1,\cdots,x_n)\coloneqq\prod_{i=1}^n\rho_{A_i}(x_i)F(x_1,\cdots,x_n)$$ with $\rho_{A_i}(x_i)\coloneqq\rho\left(x_i\left/A_i\right.\right)$ is a smooth function compactly supported in $[A_i,2A_i]$ satisfying $x_i^j\rho_{A_i}^{(j)}(x_i)\ll_j 1$ for any real number $x_i\geq 0$ and any integer $j\geq 0$. Let us introduce the following notation for summation over powers of $\sqrt{2}$ : $$%
\sumsh_{A\leq M\leq B}f(M)\coloneqq%
\sum_{%
\substack{n\in\N\\ A\leq 2^{n/2}\leq B}%
}%
f\left(%
2^{n/2}%
\right).%$$
We will use such smooth dyadic partitions of unity several times in this paper and we will also need these natural estimates in such contexts $$\label{dyadic1}
\sumsh_{M\leq M_1}
M^\alpha%
\ll %
M_1^\alpha%$$ for any $\alpha, M_1>0$ and $$\label{dyadic2}
%%
\sumsh_{M\geq M_0}%
\frac{1}{M^{\alpha}}%
\ll%
\frac{1}{M_0^\alpha}%
%%$$ for any $\alpha, M_0>0$.
On Bessel functions {#Bessel}
-------------------
The Bessel function of first kind and order a integer $\kappa\geq 1$ is defined by $$%
\forall z\in\C, \quad J_\kappa(z)\coloneqq\sum_{n\geq 0}\frac{(-1)^n}{n!(\kappa+n)!}\left(\frac{z}{2}\right)^{\kappa+2n}.$$
It satisfies the following estimate (founded in [@MR1915038 Lemma C.2]), valid for any real number $x$, any integer $j\geq 0$ and any integer $\kappa\geq 1$: $$\label{bessel}
\left(\frac{x}{1+x}\right)^jJ_\kappa^{(j)}(x)\ll_{j,\kappa}\frac{1}{\left(1+x\right)^{\frac{1}{2}}}\left(\frac{x}{1+x}\right)^\kappa$$ for any real number $x$, any integer $j\geq 0$ and any integer $\kappa\geq 1$. The following useful lemma follows immediately.
\[lem\_picard\]Let $X>0$ and $\kappa\geq 1$, then $$%
\sum_{d>0}\frac{\tau(d)}{\sqrt{d}}%
\abs{J_{\kappa}\left(\frac{X}{d}\right)}%
\ll%
\begin{cases}%
X^{1/2}\log X & \text{if $\;X>1$,}\\
X^{\kappa} & \text{if $\;0<X\leq 1$.}
\end{cases}$$
Basic facts on Kloosterman sums {#Kloos}
-------------------------------
For any integer $m,n, c\geq 1$, the Kloosterman sum is defined by $$%
S(m,n;c)\coloneqq%
\sum_{\substack{x\mod (c) \\ (x,c)=1}}%
e\left(\frac{mx+n\overline{x}}{c}\right)%$$ where $\overline{x}$ stands for the inverse of $x$ modulo $c$. We recall some basic facts on these sums. The Chinese remainder theorem implies the following multiplicativity relation $$\label{eq_crt}%
S(m,n;qr)%
=%
S(m\overline{q}^2,n;r)%
S(m\overline{r}^2,n;q)%$$ valid as soon as $(q,r)=1$. Here, $\overline{q}$ (resp. $\overline{r}$) is the inverse of $q$ (resp. $r$) modulo $r$ (resp. $q$). If $p$ and $q$ are two prime numbers, $\gamma\geq 1$ and $r\geq 1$ then, from and [@55.0703.02 (2.312)] we obtain $$\label{eq_klzero}%
S\left(p^{\gamma}q,1;qr\right)%
=%%
\begin{cases}%
-S\left(p^{\gamma}\overline{q},1;r\right) & \text{if $(q,r)=1$, }\\%
0 & \text{otherwise.}%
\end{cases}%$$The Weil-Estermann inequality [@Es] is $$\label{weil}%
\abs{S(m,n;c)}\leq\sqrt{(m,n,c)}\tau(c)\sqrt{c}. %$$ \[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
[10]{}
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, *Über [M]{}odulfunktionen und die [D]{}irichletschen [R]{}eihen mit [E]{}ulerscher [P]{}roduktentwicklung. [I]{}*, Math. Ann. **114** (1937), no. 1, 1–28.
, *Über [M]{}odulfunktionen und die [D]{}irichletschen [R]{}eihen mit [E]{}ulerscher [P]{}roduktentwicklung. [II]{}*, Math. Ann. **114** (1937), no. 1, 316–351.
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, *Mock-[G]{}aussian behaviour for linear statistics of classical compact groups*, J. Phys. A **36** (2003), no. 12, 2919–2932, Random matrix theory.
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, *Functorial products for [${\rm GL}\sb 2\times{\rm GL}\sb 3$]{} and the symmetric cube for [${\rm GL}\sb 2$]{}*, Ann. of Math. (2) **155** (2002), no. 3, 837–893, With an appendix by Colin J. Bushnell and Guy Henniart.
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[^1]: In this paper, the weight $\kappa$ is a *fixed* even integer and the level $q$ goes to infinity among the prime numbers.
[^2]: We refer to Miller [@mil02a] and Omar [@oma00] for works related to the “first” zero.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'V. deHeij, R. Braun, and W.B. Burton'
date: 'Received mmddyy/ accepted mmddyy'
title: 'An all–sky study of compact, isolated high–velocity clouds'
---
Introduction
============
Since the discovery of the high–velocity clouds by Muller et al. ([@muller63]), different explanations, each with its own characteristic distance scale, have been proposed. It is likely that not all of the anomalous–velocity represents a single phenomenon, in a single physical state. Determining the topology of the entire population of anomalous–velocity is not a simple matter, and the task is all the more daunting to carry out on an all–sky basis because of disparities between the observational survey material available from the northern and southern hemispheres. The question of distance remains the most important, because the principal physical parameters depend on distance: mass varying as $d^2$, density as $d^{-1}$, and linear size directly as $d$. Most of the emission at anomalous velocities is contributed from extended complexes containing internal sub-structure but embedded in a common diffuse envelope, with angular sizes up to tens of degrees. Such structures include the Magellanic Stream of debris from the Galaxy/LMC interaction and several HVC complexes, most notably complexes A, C, and H. The complexes are few in number but dominate the flux observed.
The Magellanic Stream comprises gas stripped from the Large Magellanic Cloud, either by the Galactic tidal field or by the ram–pressure of the motion of the LMC through the gaseous halo of the Galaxy. It therefore will be located at a distance comparable to that of the Magellanic Cloud, i.e. some $50\rm\;kpc$ (see e.g. Putman & Gibson [@putm99]). The distance to Complex A has been constrained by van Woerden et al. ([@woer99]) and then more tightly by Wakker ([@wakker01]) to lie within the distance range $8<d<10$ kpc. If, as seems plausible, the other large complexes also lie at distances ranging from several to some 50 kpc, they will have been substantially affected by the radiation and gravitational fields of the Milky Way.
Another category of anomalous high–velocity clouds are the compact, isolated high–velocity clouds discussed by Braun & Burton ([@braun99]). CHVCs are distinct from the HVC complexes in that they are sharply bounded in angular extent at very low column density limits, i.e. below 1.5$\times$10$^{18}$cm$^{-2}$ (deHeij et al. [@deheij02]). This is an order of magnitude lower than the critical column density of about 2$\times$10$^{19}$cm$^{-2}$, where the ionized fraction is thought to increase dramatically due to the extragalactic radiation field. For this reason, these objects are likely to provide their own shielding to ionizing radiation. Although not selected on the basis of angular size, such sharply bounded objects are found to be rather compact, with a median angular size of less than 1 degree.
An analogy of the CHVC ensemble with that of the dwarf galaxy population in the Local Group is suggestive, and illustrates the hypothesis that is under discussion here. Some few Local Group dwarf galaxies also extend over large angles. The Sgr Dwarf Spheroidal discovered by Ibata et al. ([@ibat94]) spans some $40\deg$; presumably it was once a rather conventional dwarf, but its current proximity to the Milky Way accounts for its large angular size. This proximity has fundamentally distorted its shape, and will determine its further evolution. The streams of stars found in the halo of the Galaxy by Helmi et al. ([@helmi99]) probably represent even more dramatic examples of the fate which awaits dwarf galaxies which transgress into the sphere of the Galaxy’s dominance. Analogous stellar streams have been found in M31 by Ibata et al. ([@ibat01]) and by Choi et al. ([@choi02]), as well as in association with Local Group dwarf galaxies by Majewski et al. ([@maje00]), indicating that accretion (and subsequent stripping) of satellites is an ongoing process. If a selection were to be made of the dwarf galaxy population in the Local Group on the basis of angular size, the few large–angle systems which would be selected, namely the LMC and SMC, and the Sgr dwarf, and — depending on the flexibility of the selection criteria — perhaps the ill–fated coherent stellar streams in the Galactic halo, would represent systems nearby, of large angular extent, and currently undergoing substantial evolution. Those systems selected on the basis of being compact and isolated, on the other hand, would represent dwarf galaxies typically at substantial distances and typically at a more primitive stage in their evolution. Regarding distance and evolutionary status they would differ from the nearby, extended objects, although at some earlier stage the distinction would not have been relevant.
The possibility that some of the high–velocity clouds might be essentially extragalactic has been considered in various contexts by, among others, Oort ([@oort66], [@oort70], [@oort81]), Verschuur ([@vers75]), Eichler ([@eich76]), Einasto et al. ([@eina76]), Giovanelli ([@giov81]), Bajaja et al. ([@baja87]), Wakker & van Woerden ([@wakker97]), Braun & Burton ([@braun99]), and Blitz et al. ([@blitz99]). It is interesting to note that the principal earlier arguments given against a Local Group deployment, most effectively in the papers cited above by Oort and Giovanelli, were based on the angular sizes of the few large complexes and on the predominance of negative velocities in the single hemisphere of the sky for which substantial observational data were then available. The more complete data available now, however, show about as many features at positive velocities as at negative ones. It is also interesting to note that the papers by Eichler and by Einasto et al. cited above consider distant high–velocity clouds as possible sources of matter, including dark matter, fueling continuing evolution of the Galaxy. Blitz et al. ([@blitz99]) revived the suggestion that high–velocity clouds are the primordial building blocks fueling galactic growth and evolution, and argued that the extended complexes owe their angular extent to their proximity.
Braun & Burton ([@braun99]) identified CHVCs as a subset of the anomalous–velocity gas that might be characteristic of a single class of HVCs, whose members plausibly originated under common circumstances and share a common subsequent evolutionary history. They emphasized the importance of extracting a homogenous sample of independently confirmed objects from well–sampled, high–sensitivity surveys. The spatial and kinematic distributions of the CHVCs were found by Braun & Burton to be consistent with a dynamically cold ensemble spread throughout the Local Group, but with a net negative velocity with respect to the mean of the Local Group galaxies. They suggested that the CHVCs might represent the low–circular–velocity dark matter halos predicted by Klypin et al. ([@klyp99]) and Moore et al. ([@moor99]) in the context of the hierarchical structure paradigm of galactic evolution. These halos would contain no, or only a few, stars; most of their visible matter would be in the form of atomic hydrogen. Although many of the halos would already have been accreted into the Galaxy or M31, some would still populate the Local Group, either located in the far field or concentrated around the two dominate Local Group galaxies. Those passing close to either the Milky Way or M31 would be ram–pressure stripped of their gas and tidally disrupted by the gravitational field. Near the Milky Way, the tidally distorted features would correspond to the high–velocity–cloud complexes observed.
The quality and quantity of survey material is important to interpretation of the CHVC population, which is a global one. The observational data entering this analysis involved merging two catalogs of CHVCs, one based on the material in the Leiden/Dwingeloo Survey (LDS) of Hartmann & Burton ([@hartmann97]), and the other based on the Parkes All–Sky Survey (HIPASS) described by Barnes et al. ([@barnes]). Both of these surveys were searched for anomalous–velocity features using the algorithm described by deHeij et al. ([@deheij02], Paper I). This algorithm led to the LDS catalog of deHeij et al. for the CHVCs at declinations north of $-30\deg$, and to the HIPASS catalog of Putman et al. ([@putman02a]) for those at $\delta<0\deg$. The surveys overlap in the declination range $+2\deg <\delta < -30\deg$, allowing estimates of the relative completeness of the catalogs. We are able to predict how a survey with the LDS parameters would respond to the CHVCs detected by HIPASS, and vice versa. In the subsequent simulations, we are able to [*sample the simulated material as if it were being observed*]{} by one of these surveys.
This paper is organized as follows. We describe the application of the algorithm in §\[sec:algorithm\]. In §\[sec:selection\] we discuss various observational selection effects, and indicate how to account for these. We address obscuration by our own Galaxy in §\[sec:obscur\], the consequences of the differing observational parameters of the LDS and HIPASS data in §\[sec:obspara\], and the resulting differing degrees of completeness of the LDS and HIPASS catalogs in §\[sec:complete\]. We discuss the observable all–sky properties of the CHVC ensemble in §\[sec:allsky\], including the spatial deployment (in §\[sec:spatial\]), the kinematic deployment (in §\[sec:velocity\]), and the distributions of flux and angular size (in §\[sec:over\]). We then attempt to reproduce these properties by considering models, first based on Local Group distributions as discussed in §\[sec:model\] and then on distributions within an extended Galactic Halo as discussed in §\[sec:simplemodel\]; these simulations are sampled [*as if being observed*]{} with the LDS and HIPASS programs. We discuss the conclusions that can be drawn from this analysis in §\[sec:conclusion\].
Observations representing CHVCs over the entire sky {#sec:data}
===================================================
Identification criteria {#sec:algorithm}
-----------------------
A full description of the cloud extraction algorithm is given in Paper I; the most salient aspects of the algorithm are the following:
- All pixels in the HIPASS and LDS material above the $1.5 \sigma$ level (as appropriate to the particular survey) were assigned to a local intensity maximum; each pixel was assigned to the same maximum as its brightest neighboring pixel. If the local maximum was brighter than 3 $\sigma$, then the local maximum and the pixels which were assigned to it were considered to constitute a cloudlet.
- Adjacent cloudlets were merged into clouds if the brightest enclosing contour for the two cloudlets either exceeds 40% of the brighter peak or if the brightness exceeds 10$\sigma$.
- Those merged cloudlets for which the peak temperature exceeds 5 $\sigma$ were deemed clouds, and were entered in the catalog pending further consideration of their deviation velocity, as described below. The value of the noise that was used for this selection was determined locally, whereas for the other steps a preset noise value was used.
To minimize the influence of noise on the peak detection, the data were first smoothed along both the spatial and spectral axes. The [RMS]{} fluctation level noted above refers to that within the smoothed data. Once the relevant pixels were assigned to a cloud, unsmoothed data were used to determine the cloud properties.
The Paper I search algorithm led to the identification of sub–structure within extended anomalous–velocity cloud complexes as well as to the identification of sharply bounded, isolated sources. Because of the importance to the present analysis of selecting only isolated objects, we comment on the determination of the degree of isolation. In order to determine the degree of isolation of the clouds that were found by the algorithm, velocity–integrated images were constructed of $10^\circ$ by $10^\circ$ fields centered on each general catalog entry. The range of integration in these moment maps extended over the velocities of all of the pixels that were assigned to a particular cloud. The CHVC classification then depended on the column density distribution at the lowest significant contour level (about 3$\sigma$) of 1.5$\times$10$^{18}$cm$^{-2}$. We demanded that this contour satisfy the following criteria: (1) that it be closed, with its greatest radial extent less than the $10\deg$ by $10\deg$ image size; and (2) that it not be elongated in the direction of any nearby extended emission. Since some subjectivity was involved in this assessment, two of the authors (VdH and RB) each independently carried out a complete classification of all sources in the HIPASS and LDS catalogs. Identical classification was given to about 95% of the sample, and consensus was reached on the remaining 5% after re–examination.
A slightly different criterion for isolation was employed by Putman et al. [@putman02a] in their analysis of the HIPASS sample of HVCs. Rather than employing the column density contour at a fixed minimum value to make this assessment, they employed the contour at 25% of the peak for each object. Since the majority of detected objects are relatively faint, with a peak column density near 12$\sigma$, the two criteria are nearly identical for most objects. Only for the brightest $\sim$10% of sources might the resulting classifications differ. We have reclassified the entire HIPASS HVC catalog with the absolute criteria above, and have determined identical classifications for 1800 of the 1997 objects listed. Given that the agreement in classification is better than 90%, we have chosen to simply employ the Putman et al. classifications in the current study. In this way the analysis presented here can be reproduced from these published sources.
High–velocity clouds are recognized as such by virtue of their anomalous velocities. Although essentially any physical model of these objects would predict that they also occur at the modest velocities characteristic of the conventional gaseous disk of the Milky Way, at such velocities the objects would not satisfy our criterion for isolation. In our analysis, only anomalous–velocity objects with a deviation velocity greater than $70\rm\;km\;s^{-1}$ were considered. As defined by Wakker ([@wakker90]), the deviation velocity is the smallest difference between the velocity of the cloud and any Galactic velocity, measured in the Local Standard of Rest reference frame, allowed by a conventional kinematic model in the same direction. The kinematic and spatial properties of the conventional Galactic HI were described by a thin gaseous disk whose properties of volume density, vertical scale–height, kinetic temperature, and velocity dispersion, remain constant within the solar radius; at larger galactocentric distances the gaseous disk flares and warps, as described in Paper I, following Voskes & Burton ([@voskes]). The gas exhibits circular rotation with a flat rotation curve constant at $220\rm\;km\;s^{-1}$. Synthetic spectra were calculated for this model Galaxy, and then deviation velocities were measured from the extreme–velocity pixels in these spectra for which the intensity exceeded $0.5\rm\;K$. Selection against objects at $V_{\rm dev}< 70$ in the LSR frame introduces systematic effects, as discussed in the following subsection.
Although compactness was not explicitly demanded of these isolated objects, the 67 CHVCs found in the LDS survey and the 179 CHVCs found in the HIPASS data have a small median angular size, amounting to less than $1^\circ$ FWHM.
Selection effects and completeness {#sec:selection}
----------------------------------
The CHVC samples used in this analysis were not extracted from a single, homogeneous set of data, nor are they free of selection effects, nor are they complete. We discuss below how we attempt to recognize and, insofar as possible, to account for some inevitable limitations.
### Systematic consequences of obscuration by in the Milky Way {#sec:obscur}
The inevitable obscuration that follows from our perspective immersed in the gaseous disk of the Milky Way, and which motivates the use of the deviation velocity, will discriminate against some CHVC detections. We may extend an analogy of the optical Zone of Avoidance to the 21–cm regime. The optical Zone of Avoidance refers to extinction of light by dust in the Milky Way, and thus traces a band with irregular borders but roughly defined by $|b|<5^\circ$; kinematics are irrelevant in the optical case, since the absorption is broad–band. The HI searches for galaxies in the optical Zone of Avoidance carried out by, among others, Henning et al. ([@henn98]) in the north, and by Henning et al. ([@henn00]) and Juraszek ([@jura00]) in the south, were confined to $|b|<5\deg$.
In the 21–cm regime, extinction due to high–optical–depth foreground is largely negligible, but confusion due to line blending occurs at all latitudes. The analogous zone refers to a certain range in velocity, of varying width depending on $l$ and on $b$, but present to some extent everywhere: near zero LSR velocity, the “Zone of Avoidance" covers the entire sky. The nearby LSB galaxy Cep I was discovered during CHVC work (Burton et al. [@burt99]); although it is at a relatively substantial latitude, $b=8\fdg0$, its velocity of $V_{\rm hel}= 58$ locates it within the obscuration zone. Because of the strong dependence on velocities measured with respect to the Local Standard of Rest, the zone of obscuration is distorted upon transformation to a different kinematic reference frame. (We note that the Magellanic Stream and the HVC complexes plausibly also discriminate against CHVC detections, but because these extended features are smaller in scale and more confined in velocity than the Galaxy, we do not consider them further here.)
The relationship between the different velocity reference systems used to characterize the CHVC kinematics is given by the equations below. $$\label{eqn:vlsr}
v_{\rm LSR}=v_{\rm HEL}+9\cos(l)\cos(b)+12\sin(l)\cos(b)+7\sin(b)$$ $$v_{\rm GSR}=v_{\rm LSR}+0\cos(l)\cos(b)+220\sin(l)\cos(b)+0\sin(b)$$ $$v_{\rm LGSR}=v_{\rm GSR}-62\cos(l)\cos(b)+40\sin(l)\cos(b)-35\sin(b)$$ Note that a typographical error is present (the sign of the coefficient of $sin(b)$) in the version of Eqn.\[eqn:vlsr\] that is published in Braun & Burton ([@braun99]).
The influence of the obscuration by the modeled Galaxy is illustrated by Fig. \[fig:calcskydistr\], where the integral $\int\exp(-(V -
\mu)^2 / 2\sigma^2) dV$ is plotted. The range of integration extends over all velocities which deviate more than $70\rm\;km\;s^{-1}$ from any Local Standard of Rest (LSR) velocity allowed by the Galactic model described above; $\mu$ is the average velocity and $\sigma$ is the standard deviation of the test clouds. The panels in Fig. \[fig:calcskydistr\] show the fraction of a population of clouds, homogeneously distributed on the sky and with a Gaussian velocity distribution relative to a particular reference frame, that are not obscured by virtue of being coincident with emission from the Milky Way. The upper panel of the figure represents a model in which the Gaussian velocity distribution is with respect to the Local Standard of Rest frame, with an average velocity of $-50\rm\;km\;s^{-1}$ and dispersion of $240\rm\;km\;s^{-1}$, in rough agreement with the measured CHVC values in this frame. In this case the obscuration is simply proportional to the velocity width of the obscuring emission. The obscuration at high latitudes is quite uniform since the infalling population is always displaced from =0 , where the obscuring gas resides. The middle panel presents a model wherein the Gaussian velocity distribution is with respect to the Galactic Standard of Rest (GSR) frame with an average velocity of $-50\rm\;km\;s^{-1}$ and dispersion of $110\rm\;km\;s^{-1}$, in rough agreement with the CHVC values in this frame. The low–latitude obscuration is similar to that in the LSR model, although more strongly modulated since the velocity dispersion is smaller. The high–latitude obscuration is quite strongly modulated since the infall velocity in the GSR frame overlaps with =0 in the plane approximately perpendicular to the direction of rotation, $(l,b)~=~(90^\circ,0^\circ)$. Broad apparent maxima in unobscured object density are centered near $l=90\deg$ and $l=270\deg$. The lower panel presents a model in which the Gaussian velocity distribution is with respect to the Local Group Standard of Rest (LGSR), again using an average velocity of $-50\rm\;km\;s^{-1}$ and dispersion of $110\rm\;km\;s^{-1}$. The pattern of obscuration is very similar to that of the GSR case, although the maxima in unobscured object density are slightly shifted with respect to $b=0\deg$. These results indicate that caution must be exercised in interpreting apparent spatial concentrations of detected objects without properly accounting for the distortions introduced by the Zone of Avoidance.
We have also considered how the measured statistics of a distribution, namely the mean velocity and dispersion, are influenced by the non–completeness caused by obscuration. Figure \[fig:obscure\] shows the distribution of the errors in the average velocity and dispersions for 1000 simulations, each involving 200 test clouds; one set of simulations was run with the GSR as the natural reference frame, and a second set was run with the LGSR as the natural frame. After removing the test clouds that have an LSR deviation velocity less than $70\rm\;km\;s^{-1}$, the velocity dispersion of the simulated ensemble was measured for both the GSR and the LGSR velocity systems, and compared with what would have been determined if there had been no obscuration by the Galaxy.
The upper left–hand panel in Fig. \[fig:obscure\] refers to test objects with a Gaussian distribution in $V_{\rm GSR}$ with a dispersion of $115\rm\;km\;s^{-1}$ and average of $-50\rm\;km\;s^{-1}$. The measured dispersion exceeds the true one by $9\rm\;km\;s^{-1}$, whereas a more negative average velocity is inferred by $12\rm\;km\;s^{-1}$. The lower left–hand panel is based upon test samples with a Gaussian distribution in $V_{\rm LGSR}$ with a dispersion of $105\rm\;km\;s^{-1}$ and an average of $-55\rm\;km\;s^{-1}$. The differences between the measured and true dispersion and average velocity, of $6\rm\;km\;s^{-1}$ and $-5\rm\;km\;s^{-1}$, respectively, are smaller than for the GSR system. From the 200 clouds which were in the input ensemble, an average of 80 were removed because of obscuration in the GSR model and only 60 in the LGSR model, indicating that the statistical properties of the LGSR model are somewhat better preserved in this case. The particular population attributes chosen above for the GSR and LGSR systems were chosen to match the observed parameters in these systems, as shown below in §\[sec:allsky\].
Another question that can be addressed with these simulations is whether it might be possible to distinguish between a GSR and an LGSR CHVC population based on a significant difference in the statistical properties. We assessed this by taking 500 populations of 200 objects in both the GSR and LGSR frames, each with a dispersion of 110 and an average velocity of $-$50 . Each of these 1000 populations was analyzed in both the GSR and LGSR frames, both before and after decimation by obscuration. The results are shown in the right–hand panels of Fig. \[fig:obscure\] for differences in velocity dispersion (relative to the GSR versus LGSR frames) and mean velocity, respectively. The measured differences in velocity dispersion and mean velocity of our CHVC sample (from §\[sec:allsky\]) are plotted in these panels as dashed lines. The model results for mean velocity differences form a continuous cloud, for which it is impossible to distinguish between the actual reference frame of the model population. The model results for velocity dispersion differences, on the other hand, are separated into two distinct clouds. The velocity dispersion of each model population is minimized in its own reference frame with a variance of only a few , while the dispersions within the GSR and LGSR frames are separated by about 20 , both before and after obscuration. The measured difference in velocity dispersion of the CHVC sample relative to the GSR and LGSR frames, of 16 , is more consistent with an LGSR reference frame.
### Consequences of the differing observational parameters of the LDS and HIPASS {#sec:obspara}
Because the LDS and the HIPASS data do not measure the sky with the same limiting sensitivities, angular resolutions, velocity resolutions, or velocity coverages, the northern population of CHVCs will be differently sampled than the southern one. In particular, the maximum depth of the two samples will be different since the surveys have different limiting fluxes. We describe below how we identify, and compensate for, the differing properties of the two catalogs; we also describe how we sample the simulations using the selection criteria corresponding to the observations. A detailed comparison of objects detected in the two surveys is made in Paper I.
The LDS covered the sky north of declination $-30^\circ$ (the actual declination cut-off varied between $-32$ and $-28^\circ$); the angular resolution of the 25–m Dwingeloo telescope was $36^\prime$. The effective velocity coverage of the LDS extends over LSR velocities from $-450\rm\;km\;s^{-1}$ to $+400\rm\;km\;s^{-1}$, resolved into channels $1.03\rm\;km\;s^{-1}$ wide. The formal rms sensitivity is 0.07 K per 1.0 channel. Stray radiation has been removed as described by Hartmann et al. ([@hartmann96]). Due to the presence of radio frequency interference, it was important that the reality of all CHVC candidates that were identified in the LDS be independently confirmed. Although interference in the LDS often had the shape of extremely narrow–band signals that are easily recognized as artificial, some types of interference were indistinguishable from naturally occurring features. The reality of the CHVC candidates was either confirmed by the identification of the candidates with objects in independent published material, or by new observations made with the Westerbork Synthesis Radio Telescope, operating as a collection of 14 single dishes.
The HIPASS program covered the sky south of declination $+2^\circ$. The survey has been reduced in such a way that emission which extends over more than $2^\circ$ was filtered out. To recover a larger fraction of the extended emission, the part of the survey which covers LSR velocities ranging from $-700\rm\;km\;s^{-1}$ to $+500\rm\;km\;s^{-1}$ was re–reduced using the [minmed5]{} method described by Putman ([@putman]), before production of the Putman et al. ([@putman02a]) catalog. The HIPASS data were gridded with lattice points separated by $4^\prime$ with an angular resolution of $15\farcm5$. The HIPASS velocity resolution after Hanning smoothing is $26.4\rm\;km\;s^{-1}$, thus substantially coarser than the 1.03 of the LDS. The HIPASS sensitivity for such a velocity resolution is 10 mK for unresolved sources. Because the observing procedure involved measuring each line of sight five times in order to reach the full sensitivity, all HIPASS sources have effectively been confirmed after median gridding.
Figure \[fig:unity\] shows that the LDS and the HIPASS reflect differing measures of the CHVC properties, because of their differing observational properties. The panel in the upper left of this figure contrasts the observed velocity widths of the LDS and HIPASS samples. The velocity FWHM measured in the LDS ranges from about 20 to some 40 , with a median of about 25 . Only for a few sources were values as low as 5 measured. The relatively high median FWHM likely indicates that most of the observed in the CHVCs is in the form of warm neutral medium. High–resolution observations of a sample of ten CHVCs made with the $3\farcm5$ resolution afforded by the Arecibo telescope (Burton et al. [@burt01]) showed warm halos to be a common property of these objects. On the other hand, the median HIPASS velocity width is about 35 FWHM. We can demonstrate that the two observed FWHM distributions are consistent with the same object population by convolving the LDS distribution with the HIPASS velocity resolution. The resulting distribution agrees well with that measured in the HIPASS.
The panel in the upper right of Fig. \[fig:unity\] shows histograms of the angular sizes of the cataloged CHVCs, determined from velocity integrated images of each cloud. A contour was drawn at the intensity of half the peak column density of the cloud. After fitting an ellipse to this contour, the size of the cloud was measured as the average of the minor and major axes. It is clear from these distributions that many of the CHVCs are resolved by HIPASS, but that this is rarely the case for the LDS. Some CHVCs in the LDS catalog were only detected in a single spectrum — giving the peak in the histogram at $0\fdg4$, which is an artifact of the sub–Nyquist LDS sky sampling. After convolving the HIPASS distribution with the LDS beam a more similar distribution of sizes is found, although there remains a small excess of relatively large objects in the north.
The panel on the lower left of Figure \[fig:unity\] shows the flux distribution for the CHVCs detected by HIPASS and the LDS, respectively. An excess of faint sources is present in the HIPASS sample, even after compensation for the lower LDS sensitivity (as outlined below). Conversely, the LDS may have a small excess of bright objects. If semi-isolated objects are considered (i.e. the :HVC and ?HVC categories discussed by Putman et al. [@putman02a] and deHeij et al. [@deheij02]) as in the panel on the lower right of Fig. \[fig:unity\], these differences remain, with the adjusted HIPASS sample showing an excess of faint sources in the south and the LDS sample showing a small excess of brighter sources in the north.
### Completeness and uniformity of the CHVC samples {#sec:complete}
The finite sensitivity of the LDS and HIPASS observations results in sample incompleteness at low flux levels in both surveys. The different sensitivities of the two surveys will bias the derived sky–distribution, average velocity, and velocity dispersion towards the more sensitively observed hemisphere, namely the southern one. To compensate for this bias, the objects found in the southern hemisphere were weighted with the likelihood that they would be detected by a survey with the LDS properties. For this likelihood we use the relation plotted in Fig. \[fig:completeness\], following deHeij et al. ([@deheij02]), who assess the degree of completeness of the LDS catalog as a function of limiting peak brightness from a comparison of the detection rates of cataloged external galaxies over the range $-30^\circ<\delta<90^\circ$, and from a comparison with the HIPASS catalog of Putman et al. ([@putman02a]) for the range $-30^\circ<\delta<0^\circ$. To incorporate plausible uncertainties in this relation, the calculations have also been done for a fictional survey 25% more sensitive, and for one 25% less sensitive, than the LDS, as indicated by the dashed lines in the figure.
Table \[table:nsource\] lists the number of sources with a minimum peak brightness temperature for the northern hemisphere, as observed by the LDS, and for the southern one, as observed by HIPASS. Due to the differences in spectral and spatial resolution, the LDS and HIPASS measure different peak temperatures for the same cloud. For all clouds that are observed in both surveys, the median of the temperature ratio as measured in HIPASS and LDS is 1.5 (deHeij et al. [@deheij02]). Applying this temperature scaling to the HIPASS data provides very good agreement with the external galaxy completeness curve of Fig. \[fig:completeness\] for declinations $-30\deg$ to $0\deg$. However, over the entire HIPASS declination range, the compensated HIPASS data show a strong excess in the source detection rate for sources with an LDS peak temperature in the range 0.2 to 0.4 K. According to Fig. \[fig:completeness\], the LDS completeness for these sources should exceed 80%. Therefore the difference in the numbers of relatively faint CHVCs detected by HIPASS and LDS indicates an asymmetry in the distribution upon the sky, with about a factor of two more occurring in the southern hemisphere than in the north. Reducing the sensitivity of the LDS survey by 25% does not change this conclusion.
The CHVC tabulation is probably not incomplete as a consequence of the velocity–range limits of the observational material. Although the part of the LDS that was searched only extended over the range $-450 <
V_{\rm LSR} < +350$ , deHeij et al. ([@deheij02]) plausibly did not miss many (if any) clouds because of this limited interval. The high–velocity feature with the most extreme negative velocity yet found is that discovered by Hulsbosch ([@hulsbosch78]) at $=-466$. (This object is listed in Paper I as ?HVC$110.6\!-\!07.0\!-\!466$: being incompletely sampled in velocity, it does not meet the stringent isolation criteria for the CHVC category, and so does not enter this analysis further.) The Wakker & van Woerden ([@wakk91c]) tabulation, which relied on survey data covering the range $-900 < V_{\rm LSR} < +750$ , found no high–velocity cloud at a more negative velocity. The HIPASS search by Putman et al. ([@putman02a]) sought anomalous–velocity emission over the range $-700 < V_{\rm LSR} < +1000$ . Of the HIPASS CHVCs cataloged by Putman et al., only 10 have $V_{\rm LSR}<-300$, but the most extreme negative velocity is $-353$ , for CHVC$125.1\!-\!66.4\!-\!353$. Regarding the positive–velocity extent of the ensemble, we note that only 7 objects in the HIPASS catalog have $V_{\rm LSR}$ greater than $+300$ , and only one has a velocity greater than $350$ , namely CHVC$258.2\!-\!23.9\!+\!359$. All of the 7 CHVCs with substantial positive velocities are near $(l,b)~=~(270^\circ,0^\circ)$, where Galactic rotation contributes to a high positive LSR velocity. Since this extended region has a negative declination, it is sampled with the wider velocity coverage of HIPASS, rather than that of the LDS. In view of these detection statistics, we consider it unlikely that the velocity–range limits of either the LDS or of the HIPASS have caused a significant number of CHVCs to be missed. In other words, the true velocity extent, as well as the non–zero mean in the LSR frame, of the anomalous–velocity ensemble are well represented by the observed extrema of $-466$ and $+359$ .
The strong concentration of faint CHVCs with an extreme variation in their radial velocity in the direction of the south Galactic pole was already noted by Putman et al. ([@putman02a]). A complete model for the all-sky distribution of objects will need to reproduce the enhancement in numbers as well as local velocity dispersion in this direction. Much of the north–south detection asymmetry for faint CHVCs remains even after excluding all objects with a Galactic latitude less than $-65^\circ$, as we discuss in detail below.
[c|ccc]{} minimum & $ N_ {\rm LDS}$ & $ N_{\rm HIPASS}$ & $
N_{\rm HIPASS}$\
$T_{\rm peak}\rm\;[K]$ & $>T_{\rm peak}$ & $>T_{\rm peak}$ & $>1.5\,T_{\rm peak}$ 1.0 & 3 & 5 & 3 0.5 & 9 & 24 & 9 0.4 & 12 & 37 & 16 0.3 & 20 & 56 & 29 0.2 & 30 & 85 & 56 0.1 & 38 & 160 & 115
All–sky spatial, kinematic, and column density properties of the CHVC ensemble {#sec:allsky}
==============================================================================
We show in this section the basic observational data for the all–sky properties of the CHVCs; specifically, the deployment in position and velocity as well as the perceived size and column density distributions. These basic properties constitute the observables against which the simulations described in the following sections are tested.
Distribution of CHVCs on the sky {#sec:spatial}
---------------------------------
Figure \[fig:skydistr\] shows the all–sky distribution of the cataloged CHVCs superimposed on the integrated emission observed in the range $-450<$$<+400$ , but with $>70$ . The LDS catalog and data are used in the north and the HIPASS catalog and data in the south, with a solid line marking the demarcation at $\delta=0^\circ$ separating the LDS from the HIPASS material. Red symbols indicate positive LSR velocities and black symbols negative velocities[^1] The much higher object density observed in the southern hemisphere is quite striking, as is the absence of diffuse emission in the HIPASS [minmed5]{} data. We comment further below on the extent to which the CHVC density is a consequence of the differing observational parameters, especially that of sensitivity.
To get a better impression of the CHVC clustering and distribution on the sky, an average density field is constructed; this smoothed field is more appropriate for comparison with simulated fields, which, as indicated below, are similarly smoothed. A field of delta functions at the CHVC locations was convolved with a Gaussian with a dispersion of $20^\circ$. The total flux of each delta function is set to unity for the LDS sources and to the value of the likelihood that such a particular CHVC would be observed in an LDS–like survey for the HIPASS sources. Changes in the likelihood relation do not change the overall picture of the CHVC concentrations; only the contrasts of the overdensity regions with respect to the average changes.
Figure \[fig:skydistr\] shows that the projected density of CHVCs displays a number of local enhancements. The three most prominent of these occur in the southern hemisphere, and were previously noted by Putman et al. ([@putman02a]) as Groups 1 through 3. Group 1 is concentrated at the south Galactic pole and extends from about $b=-60\deg$ to $-90^\circ$. It is remarkable for possessing a local velocity dispersion in excess of 150 , about twice that seen in any other part of the sky. This region is bisected by a portion of the Magellanic Stream and is also spatially coextensive with the nearest members of the Sculptor group of galaxies (with D$\sim$1.5 Mpc). Group 2 is located near $(l,b) \sim (280\deg,-15\deg)$, with an extent of about $30\deg$. This concentration is approximately in the direction of the leading arm of the Magellanic Clouds but is also near the Local Group anti-barycenter direction, where the Blitz et al. ([@blitz99]) model predicts an enhancement of high–velocity clouds. Group 3 is centered near $(l,b) \sim (30\deg,-15\deg)$, a region that Wakker & van Woerden ([@wakk91c]) have identified with the GCN (Galactic Center Negative velocity) population. The most diffuse concentration, which we label Group 4, is in the northern sky near $(l,b) \sim (115\deg,-30\deg)$, approximately coinciding with the Local Group barycenter. The Blitz et al. ([@blitz99]) model also predicts an enhancement of high–velocity clouds here, albeit a stronger one than observed. Likewise the mini–halo simulations of Klypin et al. ([@klyp99]), Moore et al. ([@moor99], [@moor01]), and Putman & Moore ([@putman02c]) predict a strongly enhanced density of low mass objects around the major galaxies of the Local Group, in particular toward M31, which lies close to the barycenter direction. We comment further on the expected strength of such an enhancement in the observed distribution below.
Distribution of CHVCs in velocity {#sec:velocity}
----------------------------------
The kinematic properties of the CHVC population provide an important constraint that must be reproduced by a successful model of the phenomenon. The kinematic distribution is plotted against Galactic longitude and latitude, for the Local, Galactic, and Local Group kinematic reference frames in Figs. \[fig:glonv\] and \[fig:glatv\], respectively. The CHVCs are confined within a kinematic envelope narrower in extent than the spectral coverage of the surveys; we stressed above in §\[sec:complete\] that this confinement is not a selection effect; it is one of the global kinematic properties of the ensemble which must be accounted for.
Table \[table:velostat\] shows that the ensemble of clouds has a lower velocity dispersion in both the GSR and LGSR systems, compared to that measured in the LSR frame, suggesting that either the Galaxy or the Local Group might be the natural reference system of the CHVCs. By measuring the dispersion in the LSR frame, one introduces the solar motion around the Galactic center into the velocities, which results in a higher dispersion.
The CHVC groups noted in the previous subsection can also be identified in the $(l,V)$ and $(b,V)$ distributions. Group 1 is best seen in Fig. \[fig:glatv\] where it gives rise to the very broad velocity extent in both the GSR and LGSR frames for $b<-60^\circ$. Group 2, on the other hand, is best seen in Fig. \[fig:glonv\], centered near $l=280^\circ$. This group has a positive mean velocity in the GSR frame. Only by going to the LGSR frame does the mean group velocity approach zero. Group 3 is evident in both Figs. \[fig:glonv\] and \[fig:glatv\]. This concentration is seen near $l = 30^\circ$ and has a remarkably high negative velocity of about $-200$ in both the GSR and LGSR frames. Group 4 can also be distinguished near $l=115^\circ$ in Fig. \[fig:glonv\]. This group also retains a large negative velocity in both the GSR and LGSR frames.
Table \[table:velostat\] gives the all–sky statistical parameters of the CHVC ensemble, calculated by weighting the HIPASS objects with the likelihood that they would be observed in an LDS–like survey. The variation of these parameters with the (flux–dependent) relative weighting of the HIPASS sub–sample is explored by considering both 25% higher and lower relative sensitivity. Although the dispersion is not affected strongly by the weighting given to the HIPASS sub–sample, the mean velocity becomes increasingly negative as the fainter HIPASS sub–sample receives a higher relative weight.
[c|cccc]{} & CHVCs & CHVCs & L.G. galaxies & L.G. galaxies\
reference frame & $<$velocity$>$ & dispersion & $<$velocity$>$ & dispersion\
& () & () & () & () & $-33$ & $253$ & & LSR & $-45$ & $238$ & $-57$ & $196$ & $-59$ & $240$ & & & $-58$ & $128$ & & GSR & $-63$ & $128$ & $-22$ & $104$ & $-69$ & $126$ & & & $-57$ & $114$ & & LGSR & $-60$ & $112$ & $+4$ & $79$ & $-65$ & $110$ & &
CHVCs near the galactic equator display the horizontal component of their space motion. Figure \[fig:glatv\] shows that the radial motions at low $|b|$ are at least as large as those at high latitudes, and furthermore that the CHVC distribution does not avoid the Galactic equator, and that substantial positive–velocity amplitudes, as well as negative–velocity ones, are observed. Large horizontal motions as well as high positive velocities are difficult to account for in terms of a galactic fountain model (e.g. Shapiro & Field [@shap76], Bregman [@breg80]). Similarly, CHVCs located near the galactic poles offer unambiguous information on the vertical, $z$, component of their space motion. The vertical motions are substantial, with positive velocities approximately equal in number and amplitude to negative velocities; the vertical motions are of approximately the same amplitude as the horizontal ones. This situation also is incompatible with the precepts of the fountain model, which predicts negative $V_z$ velocities for material returning in a fountain flow. Furthermore, the values of $V_z$ are predicted to not exceed the velocity of free fall, of some 200 . In fact, amplitudes substantially larger than the free–fall value are observed.
Several aspects of the spatial and kinematic topology of the class are difficult to account for if the CHVCs are viewed as a Milky Way population, in particular if they are viewed as consequences of a galactic fountain; these same aspects would seem to discourage a revival of several of the mechanisms suggested earlier for a Milky Way population of high–velocity clouds (reviewed, for example, by Oort [@oort66]), including ejection from the Galactic nucleus, association with a Galactic spiral arm at high latitude, and ejection following a nearby supernova explosion. We note that the spatial deployment plotted Fig. \[fig:skydistr\] shows no preference for the Galactic equator, nor for the longitudes of the inner Galaxy expected to harbor most of the disruptive energetic events. CHVCs do not contaminate the terminal–velocity locus in ways which would be expected if they pervaded the Galactic disk; this observation constrains the clouds either to be an uncommon component of the Milky Way disk, confined to the immediate vicinity of the Sun, or else to be typically at large distances beyond the Milky Way disk. We note also that the lines of sight in the directions of each of the low $|b|$ CHVCs traverse some tens of kpc of the disk before exiting the Milky Way: unless one is prepared to accept these CHVCs as boring through the conventional disk at hypersonic speeds (for which there is no evidence), and atypical in view of the cleanliness of the terminal–velocity locus, then their distance is constrained to be large. We note further that some of the CHVC objects are moving with velocities in excess of a plausible value of the Milky Way escape velocity (cf. Oort [@oort26]).
Figure \[fig:vfield\] shows the average velocity field and velocity dispersion field, which is constructed in the same way as the average density field. A field of delta functions was convolved with a Gaussian with a dispersion of $20^\circ$. The flux of each delta function was set equal to the measured CHVC velocity and multiplied by the likelihood that the CHVC would be observed in an LDS–like survey. The convolved image was then normalized by the density field. For the velocity dispersion field, a gridded distribution of squared velocity was similarly generated and the velocity dispersion was calculated from the square root of the mean squared velocity less the mean velocity squared, $\sigma=\sqrt(<V^2>-<V>^2)$. The velocity dispersion field was blanked where the normalized density was below the mean, since insufficient objects otherwise contribute to the measurement of local dispersion.
Kinematic patterns in the LSR velocity field are dominated by the contribution of Galactic rotation. After removing the contribution of Galactic rotation by changing to the GSR reference frame, the following characteristics of the kinematics of the groups are evident. Relative minima of $V_{\rm GSR} =-100$ to $-175$ are seen in the directions of Groups 3 and 4, and a relative maximum of $V_{\rm GSR} =
+45\rm\;km\;s^{-1}$ is seen in the vicinity of Group 2. Transforming to the LGSR frame generally lowers the magnitude of these kinematic properties (except in the case of Group 3 which becomes more negative in velocity) although they are all still present. The relative velocities of Groups 2, 3, and 4 fit into a coherent global pattern shared by much of the CHVC population, consisting of a strong gradient in the GSR and LGSR velocity that varies from strongly negative below the Galactic plane in the first and second quadrants to near zero in the third and fourth quadrants near the plane.
The distribution of velocity dispersion is not as strongly effected by the choice of reference frame since it is a locally defined quantity. The exception to this rule is near $l=0\deg$, where there are large gradients in the velocity field, leading to larger apparent dispersions when sampled with our smoothing kernel. Group 1 is remarkable for its extremely high velocity dispersion, exceeding that of Groups 2–4 by a factor of two or more. It is plausible that the Group 1 concentration represents a somewhat different phenomenon than the remainder of the CHVC sample, as we discuss further below.
Summary of the basic observables of the CHVC ensemble {#sec:over}
-----------------------------------------------------
In the preceding sub–sections we have attempted to correct for the differing detection levels in the northern and southern hemisphere data to produce a spatially unbiased estimate of the CHVC distributions in position and velocity. However, when making comparisons with model calculations it is possible to explicitly take account of the differing resolutions and sensitivity of the data in the north and south, obviating the need to re–weight portions of our sample in advance. The basic observables from our all–sky sample of CHVCs, without re–weighting, are shown in Fig. \[fig:dataoverview\] relative to the GSR frame. The top row of panels represents the density, velocity, and velocity-dispersion fields, just as in Figs. \[fig:skydistr\] and \[fig:vfield\], except that the HIPASS sub–sample has not been re-weighted relative to the LDS. Smoothed versions of the $(l,V)$ and $(V,b)$ plots shown in Figs. \[fig:glonv\] and \[fig:glatv\] are shown in the middle panels of Fig. \[fig:dataoverview\] to facilitate comparison with the model distributions discussed below. The distribution of delta functions was convolved with a Gaussian with a dispersion of $20^\circ$ in angle and 20 in velocity. Composite histograms of the peak column density and angular size distributions for the whole sky are shown in the lower panels of Fig. \[fig:dataoverview\]. Since the LDS and HIPASS survey resolutions are different (as discussed above in §\[sec:obspara\]) these observables have a different physical implication in the two hemispheres, but again, these differences can be accounted for explicitly in the comparison with model distributions.
A Local Group population model for the CHVC ensemble {#sec:model}
====================================================
Determining Local Group membership for nearby galaxies is not a trivial undertaking. The well–established members of the Local Group have been used to define a mass–weighted Local Group Standard of Rest, corresponding to a solar motion of $V_\odot = 316\rm\;km\;s^{-1}$ towards $l=93^\circ,\;b=-4^\circ$ (Karachentsev & Makarov ([@karachentsev96]). The $1\sigma$ velocity dispersion of Local Group galaxies with respect to this reference frame is about 60 (Sandage [@sandage86]). A plot of heliocentric velocity versus the cosine of the angular distance between the solar apex and the galaxy in question, as shown in Fig. \[fig:vhel\], has often been used to assess the likelihood of Local Group membership (e.g. van den Bergh [@berg94]) when direct distance estimates have not been available. Local Group galaxies tend to lie within about one sigma of the line defined by the solar motion in the LGSR reference frame. Braun & Burton ([@braun99]) pointed out how the original LDS CHVC sample followed this relationship, although offset with a significant infall velocity. The all–sky CHVC sample has been plotted in this way in Fig. \[fig:vhel\]. Both the sample size and sky coverage are significantly enhanced relative to what was available to Braun & Burton. A systematic trend now becomes apparent in the CHVC kinematics. While the CHVCs at negative $cos(\theta)$ (predominantly in the southern hemisphere) tend to lie within the envelope defined by the LGSR solar apex and the Local Group velocity dispersion, the CHVCs at positive $cos(\theta)$ have a large negative offset from this envelope. Obscuration by Galactic may well be important in shaping this trend. Only by analyzing realistic model populations and subjecting them to all of the selection and sampling effects of the existing surveys can meaningful conclusions be drawn.
Recently several simulations have been performed to test the hypothesis that the CHVCs are the remaining building blocks of the Local Group. Putman & Moore ([@putman02c]) compared the results of the full N–body simulation described by Moore et al. ([@moor01]) with various spatial and kinematic properties of the CHVC distribution, as well as with properties of the more general HVC phenomenon, without regard to object size and degree of isolation. Putman & Moore were led to reject the Local Group deployment of CHVCs, for reasons which we debate below. Blitz et al. ([@blitz99]) performed a restricted three body analysis of the motion of clouds in the Local Group. In their attempt to model the HVC distributions, Blitz et al. modeled the dynamics of dark matter halos in the Local Group and found support for the Local Group hypothesis when compared qualitatively with the deployment of a sample of anomalous–velocity containing most HVCs, but excluding the large complexes (including the Magellanic Stream) for which plausible or measured distance constraints are available. Assuming that 98% of the Local Group mass is confined to the Milky Way and M31 and their satellites, Blitz et al. described the dynamics of the Local Group in a straightforward manner. Driven by their mutual gravity and the tidal field of the neighboring galaxies, the Milky Way and M31 approach each other on a nearly radial orbit. The motion of the dark matter halos which fill the Local Group was determined by the gravitational attraction of the Milky Way and M31, and the tidal field of the neighboring galaxies. All halos which ever got closer than a comoving distance of 100 kpc from the Milky Way or M31 center were removed from the sample. Blitz et al. describe how their simulations account for several aspects of the HVC observations. We follow here the modeling approach of Blitz et al., but judge the results of our simulations against the properties of the CHVC sample, viewing the simulated data [*as if it were observed*]{} with the LDS and HIPASS surveys.
Model description
-----------------
We use a test particle approach similar to that used by Blitz et al. ([@blitz99]) to derive the kinematic history of particles as a function of their current position within the Local Group, but combine this with an assumed functional form (rather than simply a uniform initial space density) to describe the number density distribution of the test particles. The density function contains a free parameter which sets the degree of concentration of the clouds around the Milky Way and M31. By determining a best fit of the models we are able to constrain the values of this concentration parameter, and thereby constrain the distance to the CHVCs. The fits depend upon the derived velocity field and the properties of the clouds.
The density fields which we use are a sum of two Gaussian distributions, centered on the Galaxy and M31, respectively. As a free parameter we use the radial dispersion of these distributions. This free parameter has the same value for both the concentration around the Galaxy and that around M31. The ratio of the central densities of the distributions at M31 and the Galaxy must also be specified. We set this ratio equal to the mass ratio of the two galaxies. The Gaussian dispersions which are used in the models range from 100 kpc to 2 Mpc; i.e. the distributions range from very tightly concentrated around the galaxies to an almost homogeneous filling of the Local Group.
The CHVC kinematics are simulated by tracking the motions of small test particles within the gravitational field of the Milky Way, the M31 system, and the nearby galaxies. Both the description of the tidal field that is produced by the nearby galaxies, and the properties of the Galaxy–M31 orbit, are taken from the analysis of Raychaudhury & Lynden–Bell ([@raychaudhury]), who studied the influence of the tidal field on the motion of the Galaxy and M31, deriving the dipole tidal field from a catalog of galaxies compiled by Kraan–Korteweg ([@kraan]). The motions of the Galaxy and M31 are determined in a simulation. In this simulation M31 and the Galaxy are released a short time after the Big Bang. The initial conditions are tuned in such a way that the relative radial velocity and position correspond with the values currently measured.
We track not only the motions of M31 and the Galaxy, but also the motions of a million test particles. The test particles are, together with M31 and the Milky Way, released a short time after the Big Bang with a velocity equal to that of the Hubble flow. Initially, the test particles homogenously fill a sphere with a comoving radius of 2.5 Mpc. Their motions are completely governed by the gravitational field of M31 and the Milky Way and by the tidal field of the nearby galaxies. The result of the simulation at the present age of the Universe is used to define the kinematic history as a function of current 3–D position within the Local Group for our simulated CHVC populations. For every 3–D position where an object is to be placed by our assumed density distribution, we simply assign the kinematic history from the test particle in the kinematic simulation which ended nearest to that 3–D position. The most important aspects of the kinematic history are merely the final space velocity vector as well as the parameters of the closest approach of the test particle to M31 and the Milky Way, where the effects ram pressure and tidal stripping will be assessed.
The parameters which determine the outcome of the simulation are the Hubble constant, $H$, the average density of the Universe, $\Omega_0$, the total mass of the Local Group, $M_{\rm LG}$, (=$M_{\rm M31}$+$M_{\rm
MW}$), and the mass–to–light ratio of the nearby galaxies, which make up the tidal field. The Hubble constant and the average density of the Universe not only set the age of the Universe, but also the initial velocities of the test particles. Further evolution is independent of these parameters. The evolution is set by the values of the tidal field and the masses of M31 and the Galaxy. Values for all these parameters were taken from Raychaudhury and Lynden–Bell ([@raychaudhury]), namely: $H = 50 \rm\;km\;s^{-1}$, $M_{\rm LG} =
4.3\times10^{12}\rm\;M_\odot$, a mass–to–light ratio of 60, $\Omega_0
= 1$, and $M_{\rm M\,31} / M_{ \rm MW} = 2$. Whereas the mutual gravitational attraction between the Milky Way and M31 is described by a [point–mass potential]{}, the gravitational attraction of these galaxies on the test particles is described by an isochrone potential of the form $$\Phi_{\rm iso} (r) = - \frac{{\rm G} \; M}{r_0 + \sqrt{{r_0}^2 + r^2}},$$ where $M$ is the total mass of the galaxy and $r_0$ is a characteristic radius; $r_0$ is set such that the rotation velocity as derived from the potential equals the measured one at the edge of the unwarped disk, i.e. $V_{\rm circ}^{\rm MW} (12 \rm\;kpc) = 220\;km\;s^{-1}$ and $V_{\rm circ}^{\rm M\,31} (16 \rm\;kpc) = 250\;km\;s^{-1}$. Figure \[fig:modelfield\] shows the average velocity field superposed on density contours for a Gaussian distribution of the test particles, characterized by a dispersion of $200\rm\,kpc$. The ellipsoidal turn–around surface of the Local Group can be seen where the velocity vectors approach zero length at radii near 1.2 Mpc. The velocity field is approximately radial at large distances from both M31 and the Galaxy, but becomes more complex at smaller radii.
Before we can simulate the way in which a Local Group population of clouds would be observed by a HIPASS– or LDS–like survey, we have to set the properties of the test clouds. To do so, we assume that the clouds are isothermal gas spheres, with each such cloud located inside a dark–matter halo. Given the temperature of the gas and the potential in which it resides, the density profile follows from the relation $$\label{eq:density}
n(r) = n_0 \cdot \exp
\left(
- \frac{\rm {m}_{\rm HI} } { {\rm k} \, T_{\rm eff} }
\left[
\Phi(r) - \Phi(0)
\right]
\right),$$ where $\rm{m}_{\rm HI}$ is the mass of the hydrogen atom, $\Phi(r)$ is the potential at a distance $r$ from the cloud center, and $T_{\rm
eff}$ is an effective gas temperature. Since in addition to the thermal pressure there is also rotational support of a gas cloud against the gravitational attraction of the dark–matter halo, an effective temperature is used which incorporates both processes. In general, the average energy of an atom equals $\frac{1}{2} {\rm k}T$ per motional degree of freedom, so we have defined the effective temperature such that $$\label{eq:temperature}
\frac{3}{2} {\rm k} \, T_{\rm eff} =
\frac{3}{2} {\rm k} \, T_{\rm kin} + \frac{1}{2} {\rm m}_{\rm HI}
V^2_{\rm circ},$$ where $T_{\rm kin}$ is the gas kinetic temperature, taken to be 8000 K, and $V_{\rm circ}$ is a characteristic rotation velocity.
The description of the gravitational potential of the dark matter halo follows that of Burkert ([@burkert]), who was able to fit a [*universal*]{} function to the rotation curves of four different dwarf galaxies. The shape of the function is completely set by the amount of dark matter in the core, $M_0$. The potential, which is derived from the rotation curve, has the form $$\begin{aligned}
\Phi(r) -\Phi(0) & = & -\pi \; {\rm G} \; \rho_0 \; r_0^2 \; \{ \\
& & 2 \left( 1 + \frac{r_0}{r} \right) \cdot
\ln\left( 1 + r / r_0 \right) - \\
& & 2 \left( 1 + \frac{r_0}{r} \right) \cdot
\arctan \left( r / r_0 \right) - \\
& & \emptybox{2} \left( 1 - \frac{r_0}{r} \right)
\cdot \ln \left( 1 + (r / r_0)^2 \right) \; \},\end{aligned}$$ where the core radius, $r_0$, and the central density, $\rho_0$, are set by the relations $$\begin{aligned}
r_0 & = & 3.07 \; \left( \frac{M_0}{10^9\rm\;M_\odot}
\right)^{3/7} \rm\;kpc \\
{\rm and} \\
\rho_0 & = & 1.46 \cdot 10^{-24} \; \left( \frac{M_0}{10^9\rm\;M_\odot}
\right)^{-2/7} \rm\;g\;cm^{-3}. \\\end{aligned}$$ The circular–velocity rotation curve has a maximum value $$V_{\rm circ}^{\rm max} = 48.7 \; \left( \frac{M_0}{10^9 \rm M_\odot}
\right)^{(2/7)}\rm\;km\;s^{-1}.$$ We use $V_{\rm circ}^{\rm max}$ as a parameter for $V_{\rm circ}$ in Eq. \[eq:temperature\]. Because the total mass corresponding to the given potential is infinite, the dark matter mass is characterised either by the core mass, $M_0$, or by the virialized mass of the halo, $M_{\rm vir}$. We adopt a total dark–matter mass of $M_{\rm vir}$ for each object. According to Burkert ([@burkert]), these are related by $M_{\rm vir} = 5.8\;M_0$.
Although we could use Eq. \[eq:density\] directly to determine the predicted 21–cm images of a CHVC given an mass, we instead chose to approximate the corresponding column–density distribution by a Gaussian, in order to enable faster evaluation of the simulation. Two parameters specify the Gaussian, namely the central density, $n_0$, and the FWHM, derived as follows. The volume–density distribution given by Eq. \[eq:density\], can be closely approximated with an exponential form with a matched scale–length, $h_e$, defined by $n(h_e)~=~n(0)/e$. The column density distribution can then be expressed in an analytic form, containing a modified Bessel function of order 1 (e.g. Burton et al. [@burt01]). This analytic representation of the column–density distribution is then approximated by a Gaussian of the same halfwidth from, FWHM$=2.543 h_e$. Knowing the mass of the gas cloud and its FWHM, the central density of the Gaussian can be determined.
To get the amount of mass in the cloud, we adopt the relations between the dark–matter mass and baryonic mass for galaxies. For normal, massive galaxies there is approximately ten times as much dark matter as there is baryonic matter. Chiu et al. ([@chiu]) show that this ratio depends on the total mass. Whereas the mass spectrum of the dark–matter halos in their simulation has the form $n(M_{\rm
dark}) \propto M_{\rm dark}^{-2}$, the baryonic mass spectrum has the form $n(M_{\rm HI}) \propto M_{\rm HI}^{-1.6}$. The difference in slope is due to the ionizing extragalactic radiation field. The lowest mass halos are simply unable to retain their ionized baryonic envelopes, which have a kinetic temperature of $10^4$K. The slopes of both the baryonic and the dark–matter distributions completely determine the mass dependency of the ratio between dark matter and baryonic matter. Furthermore, given the fact that the ratio equals 10 for objects with a baryonic mass of $10^9\rm\;M_\odot$, the ratio is set for all masses. Although the simulation of Chiu et al. gives a value of $-1.6$ for the baryonic mass spectrum slope, we explore a range of values for the mass spectrum. The most appropriate value is then obtained by fitting the models to the observations.
For the sake of simplicity, we assume that the baryonic mass of each simulated cloud is entirely in the form of . In fact, a significant mass fraction will be in the form of ionized gas. It is likely that the mass fraction of ionized gas will increase toward lower masses such that below some limiting mass the objects would be fully ionized. A realistic treatment of the ionized mass fraction was deemed beyond the scope of this study. However, we do comment further on the implications of this simplifying assumption where appropriate.
The definitions of the velocity and density fields of the test objects in the Local Group, together with their properties, resulted in simulated CHVC populations which could be sampled with the observational parameters of an LDS– or HIPASS–like survey. The free parameters, defined above, were allowed to take the following values:
- The Gaussian dispersion of the density distributions centered on the Galaxy and M31 can range from 100 kpc to 2 Mpc.
- The slope of the mass spectrum, $\beta$, not only sets the number ratio of the less–massive with respect to the more–massive objects, it also determines the dark–to–baryonic mass ratios. The slope is allowed to range over the values $-2.0, -1.9, -1.8, -1.7,
-1.6, -1.4, -1.2, -1.0$.
- $M_1$ is the highest mass which is allowed for clouds in the simulations. The logarithm of this mass is allowed the values $6.0, \;6.5,\;\ldots\;, \;9.0$. It was found necessary to introduce this upper mass cut–off since otherwise high column densities $\sim~10^{21}$cm$^{-2}$ were predicted, such as observed in actively star-forming galaxies, but unlike what is found in the CHVC population.
Figure \[fig:chvcmodel\] shows the basic cloud properties as a functions of mass. As indicated above, the dark–matter fraction as function of mass is determined by $\beta$, the slope of the mass spectrum, so three curves are shown in each panel, corresponding to $\beta = -1.2$, $-1.6$, and $-2.0$, respectively. The typical object size and internal velocity dispersion increases only slowly with mass, from about 1.5 to 4 kpc, and 10 to 20 , respectively, between $M_{\rm HI}=10^5$ and $10^8$ M$_{\odot}$. The central volume density varies much more dramatically with mass, as does the peak column density. Note that the peak column densities of simulated clouds only exceed N$_{\rm HI}~>~10^{19}$cm$^{-2}$ for M$_{\rm
HI}~>~10^{5.5}$ M$_\odot$ and $\beta$ in the range $-1.6$–$-2.0$. It is critical that peak column densities of this order are achieved in long-lived objects, since this is required for self-shielding from the extragalactic ionizing radiation field (eg. Maloney [@maloney93], Corbelli and Salpeter [@corbelli93]).
In order to compare the simulation results with the observational data in the most effective way, and thereby constrain the model parameters, we created a single CHVC catalog from the HIPASS and LDS ones. Thirty-eight CHVCs at $\delta \ge 0^\circ$ were extracted from the deHeij et al. ([@deheij02]) LDS catalog, and 179 at $\delta <
0^\circ$ from the Putman et al. ([@putman02a]) HIPASS one. A large concentration of faint sources (Group 1 noted above) with an exceptionally high velocity dispersion is detected toward the Galactic south pole. Because this overdensity may well be related to the nearby Sculptor group of galaxies (see Putman et al. [@putman02a]), the 53 CHVCs at $b < -65^\circ$ were excluded from comparison with the simulations.
A simulation was run for each set of model parameters. We chose a position for each object, in agreement with the spatial density distribution of the ensemble, but otherwise randomly. The velocity is given by the velocity field described above. The mass of the test cloud is randomly set in agreement with the given power–law mass distribution between the specified upper mass limit and a lower mass limit described below. The physical size and linewidth of each object follow from the choice of $\beta$. Once all these parameters were set, we determined the observed peak column density and angular size. Objects in the northern hemisphere were convolved with a beam appropriate to the LDS, while those at $\delta < 0\deg$ were convolved with the HIPASS beam. Simulated objects were considered detected if the peak brightness temperature exceeded the detection threshold of the relevant survey, i.e. the LDS for objects at $\delta >0\deg$, and HIPASS otherwise. Furthermore, in order for a test object to be retained as detected its deviation velocity was required to exceed $70\rm\;km\;s^{-1}$ in the LSR frame, and its Galactic latitude to be above $b = -65^\circ$ (for consistency with the exclusion of Group 1 from our CHVC sample noted above). In addition, as we describe below, each simulated cloud should be stable against both tidal disruption and ram–pressure stripping by the Milky Way and M31. We continued simulating additional objects following this prescription until the number of detected model clouds was equal to the number of CHVCs in our observed all–sky sample.
Before carrying out each simulation with a power–law distribution of masses we began by determining an effective lower mass limit, $M_0$, to the objects that should be considered. This was necessary to avoid devoting most of the calculation effort to objects too faint to be detected in any case. As a first guess we took $M_0=0.5 M_1$. A sub-sample of twenty objects in the mass range $M_0$–$M_1$ was simulated which were deemed stable to both disruption and stripping and were detectable with the relevant survey parameters. Generating twenty detectable objects typically required evaluating of order 500 test objects. Given the total number of test objects needed to generate this observable sub–sample and the slope of the mass distribution function, it is possible to extrapolate the number of required test objects in other intervals of mass belonging to this same distribution. The predicted number of test objects in the interval $0.67 M_0$ to $M_0$ was simulated. If at least one of these was deemed detectable, then the lower mass limit was replaced with $0.67 M_0$ and the procedure outlined above was repeated. This process continued until no detectable object was found in the mass interval $0.67 M_0$ to $M_0$. Tests carried out with better number statistics, involving a sub–sample size of 180 objects and requiring a minimum of nine detections in the lowest mass interval, demonstrated that this procedure was robust.
Figure \[fig:obsdistance\] shows the distance out to which simulated CHVCs of a given mass can be detected with the HIPASS survey. Both the linewidth and spatial FWHM are dependent on the dark–matter fraction, as illustrated in Fig. \[fig:chvcmodel\], so separate curves are shown for $\beta = -1.2$, $-1.6$, and $-2.0$. A limiting case is provided by $\beta = -1.2$ which is extremely dark–matter dominated for low mass. In this case the objects are so spatially extended (about 5 kpc FWHM) and have such a high linewidth (about 60 FWHM) that they fall below the HIPASS detection threshold for log($M_{\rm HI}) < \sim6.4$. More plausible linewidths and spatial FWHM follow for $\beta = -1.6$ and $-2.0$. Such objects are sufficiently concentrated that they can still be detected, even when highly resolved in the HIPASS data.
The clouds are regarded stable against ram–pressure stripping if the gas pressure at the center of the cloud exceeds the ram pressure, $P_{\rm ram} = n_{\rm halo} \cdot V^2$, for a cloud moving with velocity $V$ through a gaseous halo with density $n_{\rm halo}$. Because both the gaseous halo density and the cloud velocity are the highest if the distance of the cloud to the galaxy is the smallest, the stability against ram–pressure stripping was evaluated at closest approach. We therefore kept track of the closest approach of each test–particle to the Galaxy and to M 31, while simulating the velocity field of the Local Group. We used a density profile for the Galactic halo which is an adaptation of the model of Pietz et al. ([@pietz]), derived to explain the diffuse soft X–ray emission as observed by ROSAT. Whereas their model is flattened towards the Galactic plane, we simply use a spherical density distribution, in which the radial profile is equal to the Galactic plane density profile of Pietz et al. The density at a distance $r$ from the Galactic center is given by $$n(r) = n_0 \cdot \left(
\frac{\cosh(r_\odot / h)} {\cosh(r / h)}
\right)^2,$$ where $n_0=0.0013\,{\rm cm}^{-3}$ is the central density, $h=12.5$kpc is the scalelength of the distribution, and $r_\odot=8.5$kpc is the radius of the solar orbit around the Galactic center. According to this model, the total mass in the Galactic halo is $1.5\times10^9\rm\;M_\odot$. To describe the halo around M 31, we use the same expression and the same parameter values except for $n_0$, for which we use a value twice the Galactic one. Figure \[fig:ramdistance\] shows an example of the calculated distance at which a cloud of a particular mass will be stripped. The clouds in this example are assumed to have a relative velocity of $200\rm\;km\;s^{-1}$ with respect to the Galactic halo.
A cloud will be tidally disrupted if the gravitational tidal field of either the Galaxy or M 31 exceeds the self–gravity of the cloud. We consider a cloud stable if $$\label{eq:tidal}
\frac{M_{\rm dark}}{\sigma^2} \ge
\left| \frac{\rm d^2}{{\rm d} r^2} \Phi_{\rm iso} (r) \right|
\cdot \sigma,$$ where $\sigma$ is the spatial dispersion of the Gaussian describing the distribution in the cloud, $M_{\rm dark}(r\le\sigma)$ is the core mass of the dark matter halo, and $|{\rm d}^2 \Phi_{\rm iso} (r) / {\rm d} r^2|$ is the tidal force of either the Galaxy or M 31. Solving the equation for $r$ shows that only the least massive clouds with the lowest dark–matter fractions are likely to suffer from tidal disruption. If the slope of the mass distribution is as steep as $-$2.0, then clouds with an mass less than $10^5\rm\;M_\odot$ are tidally disrupted at distances of about 60 kpc, as shown in Fig. \[fig:tidalfield\]. For $M_{\rm
HI} > 2\times10^5\rm\;M_\odot$, or $\beta~>~-2$, the clouds are stable. Changing the radius at which Eq. \[eq:tidal\] is evaluated from $1 \sigma$ to $2 \sigma$, does not dramatically change this result.
Results of the Local Group simulations
--------------------------------------
Before searching for a global best fit, we determined the range of parameter values over which at least a moderately good representation of the observed data was possible with the simulated data. To quantify the degree of agreement between the simulated size–, column–density and velocity–FWHM distributions with the observations, we used a $\chi^2$–test taken from §14.3 of [*Numerical Recipes*]{} (Press et al. [@press]). The size–, column–density, and velocity–FWHM distributions of a simulation were considered reasonable if $\chi^2
({\rm size})<3$, $\chi^2 ({\rm N_{HI}})<5$, and $\chi^2 ({\rm
FWHM})<5$. The incorporation of the spatial and kinematic information was done by comparing the modeled $(l,b)$, $(l,V_{\rm GSR})$, and $(V_{\rm GSR},b)$ distributions with those observed. We used the two-dimensional K–S test described in §14.7 of [*Numerical Recipes*]{} to make this comparison. The fits were considered acceptable if $\tilde\chi^2 (l,b)$, $\tilde\chi^2 (l,V_{\rm GSR})$, and $\tilde\chi^2
(V_{\rm GSR},b)$ where all less than 0.3.
Table \[table:model\] shows which part of the parameter space produces moderately good fits. The best fits have a Gaussian dispersion between $150$ and $250\rm\;kpc$, an upper mass cut-off between $10^{6.5}$ and $10^{8.0}\rm\;M_\odot$ and a slope of the mass distribution of $-1.7$ to $-1.9$.
\#1[$10^{#1}\rm\;M_\odot$]{}
$M_1=$
-------------- -------- ----------------------- -------- -------- -------- -- --
$\beta=-1.2$
$-1.4$
$-1.6$
$-1.7$ $\sigma_d$=150200 kpc 150250 150200 150200
$-1.8$ 150250 150250 150250
$-1.9$ 200250 200250
$-2.0$
Each simulation contains a relatively small number of detectable objects, namely the same number of objects as in our all–sky CHVC catalog. Therefore the $\chi^2$ values are prone to shot–noise. By performing a larger number of simulations for a specific combination of free parameters, we are able to determine a more representative value of $\chi^2$ for each model. The most promising combinations of parameter values, i.e. the entries in Table \[table:model\], were repeated 35 times to reduce the shot–noise, and the average results and their dispersions are shown in Table \[table:bestfit\]. The range of resulting fit quality due purely to this shot–noise is illustrated in Fig. \[fig:qualmodel\], which shows model data with the highest and lowest $\chi^2$ values from a sequence of 35 simulations.
The best overall fits are fairly well–constrained to lie between $\sigma_d=150$ and $200\rm\;kpc$, with an upper mass cut–off of about $10^{7}$ to $10^{7.5}\rm\;M_\odot$ and a slope of the mass distribution of $-1.7$ to $-1.8$. Comparison with Fig. \[fig:chvcmodel\] suggests that populations of these types have sufficiently high peak column densities that they can provide self-shielding to the extragalactic ionizing radiation field for M$_{\rm HI}~>~10^{5.5}\rm\;M_\odot$. The results of simulations \# 9 and\# 3 from Table \[table:bestfit\] are shown in Fig. \[fig:model01\] and Fig. \[fig:model02\], respectively. A single instance of each simulation has been used in the subsequent figures that had $\chi^2$ values consistent with the ensemble average.
------- ------------ ------------------- --------- ---------------------- ------------------------ ---------------------- --------------------- ------------------------------ ---------------------
Model $\sigma_d$ $M_1$ $\beta$ $\chi^2({\rm size})$ $\chi^2({\rm N_{HI}})$ $\chi^2({\rm FWHM})$ $\tilde\chi^2(l,b)$ $\tilde\chi^2(l,V{\rm GSR})$ $\tilde\chi^2(V{\rm
GSR},b)$
\# (kpc) $({\rm M}_\odot$)
1 150 $10^{6.5}$ $-1.7$ 2.0$\pm$0.6 1.9$\pm$0.3 4.0$\pm$0.4 0.29$\pm$0.03 0.25$\pm$0.02 0.29$\pm$0.03
2 150 $10^{7.0}$ $-1.7$ 2.2$\pm$0.5 2.3$\pm$0.4 3.5$\pm$0.4 0.27$\pm$0.03 0.23$\pm$0.03 0.28$\pm$0.03
3 150 $10^{7.5}$ $-1.7$ 2.6$\pm$0.6 2.7$\pm$0.4 3.9$\pm$0.6 0.25$\pm$0.04 0.21$\pm$0.03 0.25$\pm$0.03
4 150 $10^{8.0}$ $-1.7$ 3.0$\pm$0.6 3.0$\pm$0.5 3.9$\pm$0.5 0.24$\pm$0.03 0.21$\pm$0.03 0.24$\pm$0.02
5 150 $10^{7.0}$ $-1.8$ 2.0$\pm$0.5 2.4$\pm$0.5 4.5$\pm$0.6 0.27$\pm$0.04 0.23$\pm$0.02 0.28$\pm$0.03
6 150 $10^{7.5}$ $-1.8$ 2.4$\pm$0.5 2.6$\pm$0.5 4.1$\pm$0.5 0.25$\pm$0.04 0.21$\pm$0.03 0.24$\pm$0.02
7 150 $10^{8.0}$ $-1.8$ 2.7$\pm$0.6 2.7$\pm$0.4 4.0$\pm$0.6 0.24$\pm$0.03 0.21$\pm$0.03 0.24$\pm$0.03
8 200 $10^{6.5}$ $-1.7$ 2.4$\pm$0.5 2.9$\pm$0.3 4.9$\pm$0.6 0.29$\pm$0.04 0.28$\pm$0.02 0.30$\pm$0.02
9 200 $10^{7.0}$ $-1.7$ 2.3$\pm$0.5 2.9$\pm$0.3 4.1$\pm$0.5 0.28$\pm$0.04 0.24$\pm$0.02 0.29$\pm$0.03
10 200 $10^{7.5}$ $-1.7$ 2.9$\pm$0.6 3.0$\pm$0.5 4.8$\pm$0.7 0.26$\pm$0.04 0.22$\pm$0.03 0.26$\pm$0.03
11 200 $10^{8.0}$ $-1.7$ 3.0$\pm$0.5 3.1$\pm$0.5 4.9$\pm$0.6 0.26$\pm$0.04 0.22$\pm$0.03 0.25$\pm$0.03
12 200 $10^{7.0}$ $-1.8$ 2.2$\pm$0.5 3.6$\pm$0.5 4.2$\pm$0.5 0.28$\pm$0.04 0.24$\pm$0.03 0.29$\pm$0.03
13 200 $10^{7.5}$ $-1.8$ 2.5$\pm$0.6 3.7$\pm$0.4 3.8$\pm$0.5 0.27$\pm$0.04 0.22$\pm$0.03 0.27$\pm$0.03
14 200 $10^{8.0}$ $-1.8$ 2.7$\pm$0.5 3.7$\pm$0.5 3.6$\pm$0.5 0.26$\pm$0.04 0.22$\pm$0.02 0.26$\pm$0.04
15 200 $10^{7.5}$ $-1.9$ 2.3$\pm$0.5 4.0$\pm$0.5 4.8$\pm$0.7 0.26$\pm$0.04 0.22$\pm$0.03 0.27$\pm$0.04
16 200 $10^{8.0}$ $-1.9$ 2.4$\pm$0.5 4.0$\pm$0.5 4.3$\pm$0.8 0.26$\pm$0.05 0.22$\pm$0.02 0.26$\pm$0.04
17 250 $10^{7.0}$ $-1.7$ 2.3$\pm$0.6 4.3$\pm$0.6 4.8$\pm$0.7 0.28$\pm$0.04 0.24$\pm$0.02 0.29$\pm$0.03
18 250 $10^{7.0}$ $-1.8$ 2.2$\pm$0.5 5.2$\pm$0.6 4.1$\pm$0.5 0.26$\pm$0.04 0.23$\pm$0.02 0.27$\pm$0.03
19 250 $10^{7.5}$ $-1.8$ 2.5$\pm$0.6 5.1$\pm$0.6 3.8$\pm$0.5 0.25$\pm$0.03 0.22$\pm$0.02 0.26$\pm$0.03
20 250 $10^{8.0}$ $-1.8$ 2.7$\pm$0.4 5.3$\pm$0.7 3.6$\pm$0.6 0.25$\pm$0.04 0.22$\pm$0.02 0.25$\pm$0.03
21 250 $10^{7.5}$ $-1.9$ 2.3$\pm$0.5 5.6$\pm$0.6 4.8$\pm$0.6 0.26$\pm$0.03 0.22$\pm$0.02 0.26$\pm$0.03
22 250 $10^{8.0}$ $-1.9$ 2.5$\pm$0.6 5.6$\pm$0.6 4.3$\pm$0.6 0.25$\pm$0.03 0.22$\pm$0.02 0.25$\pm$0.03
------- ------------ ------------------- --------- ---------------------- ------------------------ ---------------------- --------------------- ------------------------------ ---------------------
Both of these Local Group simulations succeed reasonably well in reproducing the observed kinematic and population characteristics of the CHVC sample as summarized in Fig. \[fig:dataoverview\]. The CHVC concentrations, named Groups 2, 3, and 4 above, while not reproduced in detail, have counterparts in the simulations which arise from the combination of Gaussian density distributions centered on the Galaxy and M31, together with the Local Group velocity field, population decimation by disruption effects, and the foreground obscuration. A notable success of these simulations is their good reproduction of the smoothed velocity field, including both the numerical values and the location of minima and maxima.
One aspect of the observed CHVC distributions which can not be reproduced accurately by our simulations is the distribution of observed linewidth. The model objects are assumed to contain only the warm component of with a minimum linewidth corresponding to a 8000 K gas. The profiles are then further broadened by the contribution of rotation as indicated in Eq. \[eq:temperature\]. The actual objects are known to have cool core components (e.g. Braun and Burton [@braun01], Burton et al. [@burt01]) which can contribute a significant fraction of the mass and consequently lead to narrower observed line profiles. This shortcoming of the model distributions is illustrated in the central panel of Fig. \[fig:qualmodel\]. The narrow–linewidth tail of the observed distributions can never be reproduced by the models. The best–fitting models can only succeed in reproducing the median value and high-velocity tail of the distribution.
In order to better isolate the effect of foreground obscuration from the intrinsic distribution properties of the simulation itself we show the unobscured version of model \#9 in Fig. \[fig:model01no\]. Comparison of Figs. \[fig:model01\] and \[fig:model01no\] illustrates how the foreground obscuration from the Zone of Avoidance modifies the distribution of object density. The location of apparent object concentrations are shifted and density contrasts are enhanced. The comparison also reveals that the large gradient in the smoothed GSR velocity field is an intrinsic property of the Local Group model and not simply an artifact of the Galactic obscuration. Substantial negative velocities ($<-100$ ) are predicted in the direction of M31 (which effectively defines the Local Group barycenter), while slightly positive velocities are predicted in the anti-barycenter direction, just as observed.
A better appreciation of the physical appearance of these Local Group models is provided in Fig. \[fig:mod3d\], where two perpendicular projections of the model \#9 population are displayed. The $(x,y)$ plane in the figure is the extended Galactic plane, with the Galaxy centered at $(x,y)=(0,0)$ with the positive $z$ axis corresponding to positive $b$. The intrinsic distribution of objects is an elongated cloud encompassing both the Galaxy and M31, which is dominated in number by the M31 concentration. The objects that have at some point in their history approached so closely to either of these galaxies that their would not survive the ram–pressure or tidal stripping are indicated by the filled black circles. Cloud disruption appears to have been substantially more important in the M31 concentration than for the Galaxy. The objects that are too faint to have been detected by the LDS or HIPASS observations, depending on declination, are indicated by grey circles. [*The bulk of the M31 sub–concentration is not detected in our CHVC sample for two reasons*]{}: (1) these objects have a larger average distance than the objects in the Galactic sub–concentration, and (2) the M31 sub–concentration is located primarily in the northern celestial hemisphere, where the lower LDS sensitivity compromises detection. We will return to this point below.
Those objects which are obscured by the distribution of the Galaxy are indicated in Fig. \[fig:mod3d\] by open red circles. Somewhat counter–intuitively, the consequences of obscuration are not concentrated toward the Galactic plane, but instead occur in the plane perpendicular to the LGSR solar apex direction $(l,b)=(93^\circ,-4^\circ)$. This can be understood by referring back to our discussion in §\[sec:obscur\] and the illustration in Fig. \[fig:calcskydistr\]. Obscuration from the position– and velocity–dependent Zone of Avoidance is most dramatic when the kinematic properties of a population result in overlap with $=0$ , since this can occur over a large solid angle, while the Galactic plane is relatively thin.
The various processes which influence the observed distributions are further quantified in Table \[table:modstat\]. Matching the detected sample size of 163 CHVCs above $b=-65^\circ$, required the simulation of some 6300 objects in the case of models \#9 and \#3. About three quarters of the simulated populations were classified as disrupted due to ram–pressure or tidal stripping; while 80% of the remaining objects were deemed too faint to detect with the LDS (in the north) or HIPASS (in the south). Obscuration by Galactic eliminated about one half of the otherwise detectable objects. The total masses involved in these two model populations were $4.3\times10^9$M$_\odot$ and $6.4\times10^9$M$_\odot$, respectively. In both cases, about 75% of this mass had already been consumed by M31 and the Galaxy via cloud disruption, leaving only 25% still in circulation, although distributed over some 1200 low–mass objects.
---------------------------- ---------- ------------------- ---------- -------------------
Fate of input CHVCs
number $M_{\rm HI}$ number $M_{\rm HI}$
of CHVCs (10$^8$M$_\odot$) of CHVCs (10$^8$M$_\odot$)
Total number 6281 43 6310 64
Disrupted by ram or tide 4759 31 5178 50
Too faint to be detected 1220 6.5 831 4.4
Detectable if not obscured 302 5.4 301 10
Unobscured by ZoA 173 3.3 172 6.5
Unobscured, not at SGP 163 3.2 163 6.4
---------------------------- ---------- ------------------- ---------- -------------------
The crucial role of survey sensitivity in determining what is seen of such Local Group cloud populations is also illustrated in Figs. \[fig:mod01glonv\] and \[fig:mod01glatv\]. The red symbols in these figures indicate objects detectable with the relevant LDS or HIPASS sensitivities, while the black symbols indicate those that remain undetected due to either limited sensitivity or obscuration. If these models describe the actual distribution of objects, then the prediction is that future deeper surveys will detect large numbers of objects at high negative LSR velocities in the general vicinity (about $60\times60^\circ$) of M31. To make this prediction more specific, we have imagined the sensitivity afforded by the current HIPASS survey in the south extended to the entire northern hemisphere. Fig. \[fig:mod01hom\] illustrates the prediction. A high concentration of about 250 faint newly detected CHVCs is predicted in the Local Group barycenter direction once HIPASS sensitivity is available.
A Galactic Halo population model for the CHVC ensemble {#sec:simplemodel}
======================================================
In the previous section we have outlined a physical model for self–gravitating, dark–matter dominated CHVCs evolving in the Local Group potential. While that model was quite successful in describing the global properties of the CHVC phenomenon, we noted that some aspects of the observed kinematic and spatial deployment were strongly influenced by the effects of obscuration by foreground Galactic and that, furthermore, the sensitivity limitations of the currently available survey material preclude tightly constraining the characteristic distances. In this section we consider to what extent a straightforward model in which the CHVCs are distributed throughout an extended halo centered on the Galaxy might also satisfy the observational constraints. We consider such a Galactic Halo model ad hoc in the sense that it lacks the physical motivation that the hierarchical structure paradigm affords the Local Group model.
We consider a spherically symmetric distribution of clouds, centered on the Galaxy. The radial density profile of the population is described by a Gaussian function, with its peak located at the Galactic center and its dispersion to be specified as a free parameter of the simulations. The mass distribution is given by a power–law the slope of which is a free parameter. Different values are allowed for the lowest mass in the simulation. The density distribution of an individual cloud is also described by a Gaussian function. The central volume density is the same for all clouds in a particular simulation. Given the mass and central density of an object, the spatial FWHM of the distribution follows. For the velocity FWHM we have simply adopted the thermal linewdth of an 8000 K gas of 21 .
Each simulated cloud is “observed” with the parameters corresponding to the LDS observations, if it is located in the northern celestial hemisphere, but with the HIPASS parameters if it is located in the southern hemisphere. Clouds are removed from the simulation if they are too faint to be detected. To include the effects of obscuration by the Milky Way, the velocity field of the clouds must be specified. The population is considered in the Galactic Standard of Rest system, where it is distributed as a Gaussian with a mean velocity of $-50\rm\;km\;s^{-1}$ and dispersion of $110\rm\;km\;s^{-1}$. These values follow directly from the observed parameters summarized in Table \[table:velostat\] after correction for obscuration as in Fig. \[fig:obscure\]. Clouds with a deviation velocity (as defined in §\[sec:obscur\]) less than $70\rm\;km\;s^{-1}$ are removed. Additional clouds that pass the selection criteria are simulated until their number equals the number of CHVCs actually observed.
We performed the simulations with the following values for the four parameters that describe the distance, mass, and spatial extent of the population.
- The spatial dispersion of the cloud population. Values range from $10\rm\;kpc$ to $2\rm\;Mpc$; specifically we consider the values of 10, 15, 20, …50, 60, 70, …100, 150, 200, …500, 1000, 2000 kpc.
- The slope of the mass distribution, $\beta$. Values for $\beta$ were $-2.0, -1.8, \ldots, -0.8$.
- The lowest mass, $M_0$, allowed in a simulation. Values for $M_0$ were $10^2$, $10^3$, $10^4$, or $10^5\rm\;M_\odot$.
- The central gas density in the clouds, $n_0$. Values for $n_0$, which remained constant for a single run, were $3\times10^{-3}$, $1\times10^{-2}$, $3\times10^{-2}$, $0.1$, and $0.3\rm\;cm^{-3}$.
The only measured quantities which can usefully be compared to the models are the distributions of angular sizes and peak column densities. This is because the average kinematics in these simulations have already been defined to match the data by our choice of the mean velocity and its dispersion. Given four free model parameters for each simulation and only two distributions to determine the degree of agreement between simulations and observations, it is clear that the problem is under–determined. We can only hope to constrain the range of reasonable parameters in the four–dimensional parameter space.
In order to assess the degree of agreement between the simulation outcomes and the observations, we use a $\chi^2$–test from §14.3 of [*Numerical Recipes*]{}, (Press et al. [@press]). The size and column density distributions of the models and the data are compared. A simulation was considered acceptable if $\chi^2 ({\rm size}) < 5$ and $\chi^2 (N_{\rm HI}) < 5$. Figure \[fig:qualsimplemodel\] shows examples of the range of fit quality that was deemed acceptable for both the column density and size distributions.
Table \[table:simplemodel\] lists the parameter combinations that produce formally acceptable results, and shows that for each $M_0$ value the acceptable solutions are concentrated around a line. The solutions range from nearby models, for which the central density is of the order of $0.1\rm\;cm^{-3}$, the mass slope is $-2.0$, and the characteristic distance is several tens of kpc, to more distant models, having a central density of $0.01\rm\;cm^{-3}$, a mass slope of $-1.4$, and characteristic distances of several hundreds of kpc. Since column density is simply the product of depth and density this coupling of distance to central density is easily understood.
\#1[10\^[\#1]{}cm\^[-3]{}]{}
Simulations with $M_0=10^2$:
$n_0=3\ten{-3}$ $1\ten{-2}$ $3\ten{-2}$ $1\ten{-1}$ $3\ten{-1}$
-------------- ---------------------- ------------- ------------- ------------- -------------
$\beta=-1.0$
$-1.2$ $\sigma_d=$150300kpc 3545 1020
$-1.4$ 200350 45100 1540 15
$-1.6$ 200350
$-1.8$ 500
$-2.0$
Simulations with $M_0=10^3$:
$n_0=3\ten{-3}$ $1\ten{-2}$ $3\ten{-2}$ $1\ten{-1}$ $3\ten{-1}$
-------------- ---------------------- ------------- ------------- ------------- -------------
$\beta=-1.0$
$-1.2$ 70100
$-1.4$ $\sigma_d=$100400kpc 45100 1540 15
$-1.6$ 200 50, 90 1525 1015
$-1.8$ 1015 1015
$-2.0$ 10 1015
Simulations with $M_0=10^4$:
$n_0=3\ten{-3}$ $1\ten{-2}$ $3\ten{-2}$ $1\ten{-1}$ $3\ten{-1}$
-------------- ---------------------- ------------- ------------- ------------- -------------
$\beta=-1.0$
$-1.2$ $\sigma_d=$100250kpc
$-1.4$ 150450 50100 5070
$-1.6$ 400 45100 4050
$-1.8$ 4060 3045
$-2.0$ 4045 3045
Simulations with $M_0=10^5$:
$n_0=3\ten{-3}$ $1\ten{-2}$ $3\ten{-2}$ $1\ten{-1}$ $3\ten{-1}$
-------------- ------------------- ------------- ------------- ------------- -------------
$\beta=-1.0$
$-1.2$ $\sigma_d=250$kpc
$-1.4$ 250450 150
$-1.6$ 150300 100150
$-1.8$ 150200 100150
$-2.0$ 90150
Overviews of two of the best–fitting models of the Galactic Halo type are given in Figs. \[fig:abestsimplemodel\] and \[fig:bbestsimplemodel\]. Figure \[fig:abestsimplemodel\] shows a cloud population with 30 kpc dispersion, while the population in Fig. \[fig:bbestsimplemodel\] has a dispersion of 200 kpc. These figures can be compared with Fig. \[fig:dataoverview\], showing the situation actually observed. Despite there being almost a factor of ten difference in the average object distance for these two models, they produce similar distributions of observables, which are to a large extent determined by the effects of obscuration. Relative to the observed CHVC sample shown in Fig. \[fig:dataoverview\], the density distributions of these models are more uniformly distributed on the sky. The average velocity fields are also more symmetric about $ b=0\deg $, lacking the extreme negative excursion toward $(l,b)=(125^\circ,-30^\circ)$ seen in the CHVC population, that produces a large gradient in the $(V_{\rm GSR},b)$ plot.
Discussion and conclusions {#sec:conclusion}
==========================
The effects of both obscuration by the gaseous disk of the Galaxy and the limited sensitivity of currently available surveys have important consequences for the observed properties of the HVC phenomenon. We have identified those consequences in this paper. Obscuration leads to apparent localized enhancements of object density, as well as to systematic kinematic trends that need not be inherent to the population of CHVCs. A varying resolution and sensitivity over the sky substantially complicates the interpretation of the observed distributions. Taking account of both these effects in a realistic manner is crucial to assessing the viability of models for the origin and deployment of the anomalous–velocity H[i]{}. Our discussion leads to specific predictions for the numbers and kinematics of faint CHVCs which can be tested in future surveys.
Galactic Halo models
--------------------
As shown in §\[sec:simplemodel\], a straightforward empirical model in which CHVCs are dispersed throughout an extended halo centered on the Galaxy does not provide the means to discriminate between distances typical of the Galactic Halo and those of the Local Group. Comparable fit quality is realized for distance dispersions ranging from about 30 to 300 kpc. In addition to requiring a relatively large number of free parameters, such empirical models beg a number of serious physical questions. In the first instance: how is it that clouds can survive at all in a low–pressure, high–radiation–density environment without the pressure support given by a dark halo? Presumably such “naked” Galactic Halo clouds would either be very short–lived or require continuous replenishment, since the timescales for reaching thermal and pressure equilibrium are only about 10$^7$ years (Wolfire et al. [@wolf95]). Realistic assessment of such a scenario must await more detailed simulations that track the long–term fate of gas, for example after tidal stripping from the LMC/SMC, within the Galactic Halo. Only by including more physics will it be possible to reduce the number of free parameters and determine meaningful constraints on this type of scenario. This class of model also suffers from a number of shortcomings in describing the observed distributions, namely that the object density enhancement coupled with high negative velocities seen in the Local Group barycenter direction are not reproduced.
The Galactic Halo simulations returned formally acceptable values of characteristic distance as low as some 30 kpc. There is, however, a growing body of independent evidence based on high–resolution imaging of a limited number of individual CHVCs that such nearby distances do not apply. Braun & Burton ([@braun00]) discussed evidence from Westerbork synthesis observations of rotating cores in ?HVC204.2$+$29.8$+$075 (using the deHeij et al. [@deheij02] notation for a semi-isolated source) whose internal kinematics could be well modeled by rotation curves in flattened disk systems within cold dark matter halos as parameterized by Navarro et al. ([@navarro97]), if at a distance of at least several hundred kpc. Similar distances were indicated for ?HVC115.4$+$13.4$-$260 on the basis of dynamical stability and crossing–time arguments regarding the several cores observed with different systemic velocities, but embedded in a common diffuse envelope. The WSRT data for CHVC125.3$+$41.3$-$205 likewise supported distances of several hundred kpc, based on a volume–density constraint stemming from the observed upper limit to the kinetic temperature of 85K. Burton et al. ([@burt01]) found evidence in Arecibo imaging of ten CHVCs for exponential edge profiles of the individual objects: the outer envelopes of the CHVCs are not tidally truncated and thus are likely to lie at substantial distances from the Milky Way. For plausible values of the thermal pressure at the core/halo interface, these edge profiles support distance estimates which range between 150 and 850 kpc.
Local Group models
------------------
The Local Group deployment models of §\[sec:model\] offer a more self–consistent and physically motivated scenario for the CHVC population. Dark–matter halos provide the gravitational confinement needed to produce a two–phase atomic medium with cool condensations within warm envelopes, and provide in addition the necessary protection against ram–pressure and tidal stripping to allow long–term survival. The kinematics of the population follow directly from an assumed passive evolution within the Local Group potential. While three free parameters (the distance scalelength, the mass function slope, and the upper mass cut–off) were then tuned to explore consistency with the observations, only the distance was effectively a “free” parameter. The mass function slopes of the best fits have values of $-1.7$ to $-1.8$, in rough agreement with the value of $-1.6$ favored by Chiu et al. ([@chiu]) for the distribution of the baryonic masses in their cosmological simulations. The somewhat steeper slopes and therefore larger baryonic fractions favored by our model fits might be accomodated by recondensation onto the dark–matter halos at later times.
The upper mass cut–off introduced in the Local Group models can also be externally constrained. In addition to satisfying the observational demand that no column densities exceeding a few times 10$^{20}$cm$^{-2}$ are seen in the CHVC population (consistent with the absence of current internal star formation), there is the observed lower limit of about 3$\times10^7$M$_\odot$ for the mass seen in a large sample of late–type dwarf galaxies (Swaters [@swat99]). The upper mass cut–off favored by the simulations, of about $10^7$M$_\odot$, is essentially unavoidable given these two constraints.
The spatial Gaussian dispersion which is favored by these simulations is quite tightly constrained to lie between about 150 and 200 kpc. The implication for the distribution of object distances is illustrated in Fig. \[fig:mod01dist\] in the form of a histogram of the detected objects from model \#9. The distribution has a broad peak extending from about 200 to 450 kpc with a few outliers extending out to 1 Mpc due primarily to the M31 sub–population. The filled circles in the figure are the distance estimates for individual CHVCs found by Braun & Burton ([@braun00]) and Burton et al. ([@burt01]). Although very few in number, these estimates appear consistent with the model distribution, also peaking in number near 250 kpc.
We have made the simplifying assumption that the baryonic matter in our model clouds is exclusively in the form of , rather than being partially ionized. It is reassuring that the best-fitting models have peak column densities which are sufficiently high that the objects should be self-shielding to the extragalactic ionizing radiation field for M$_{\rm HI}~>~10^{5.5}$M$_\odot$ as noted above. Since the neutral component requires a power–law slope of about $-1.7$ to fit the data, it seems likely that the total baryonic mass distribution might follow an even steeper distribution, since the mass fraction of ionized gas will increase toward lower masses.
The Local Group mass function
-----------------------------
An interesting question to consider is whether the extrapolated mass distributions of our Local Group CHVC models can also account for the number of galaxies currently seen. In Fig. \[fig:lgall\] we plot the mass distribution of objects in one of the best–fitting Local Group models, model \#9 of Table \[table:bestfit\]. The thin–line histogram gives the mass distribution of the model population after accounting for the effects of ram–pressure and tidal stripping. The thick–line histogram gives the observed CHVC distribution that results from applying the effects of Galactic obscuration and sensitivity limitations appropriate to the LDS and HIPASS properties in the northern and southern hemispheres, respectively. The hatched histogram gives the inferred total baryonic (plus stellar) mass distribution of the Local Group galaxies tabulated by Mateo ([@mateo]), assuming a stellar mass–to–light ratio of $M/L_B =
3$M$_\odot$/L$_\odot$. M31 and the Galaxy, with baryonic masses of some 10$^{11}$M$_\odot$, are not included in the plot. The diagonal line in the figure has the slope of the model mass function of $\beta=-1.7$. The figure demonstrates that the low–mass populations of these models are roughly in keeping with what is expected from the number of massive galaxies together with a constant mass function slope of about $\beta=-1.7$. At intermediate masses, 10$^7$–10$^{8.5}$M$_\odot$, there is a small deficit of cataloged Local Group objects relative to this extrapolated distribution, while at higher masses there is a small excess. Conceivably this may be the result of galaxy evolution by mergers.
It is important to note that the distribution of objects shown in Fig. \[fig:lgall\] is only the current relic of a much more extensive parent population. As shown in Table \[table:modstat\], about 75% of the CHVC population in these models is predicted to have been disrupted by ram pressure or tidal stripping over a Hubble time, contributing about $3\times10^9$ M$\odot$ of baryons to the Local Group environment and the major galaxies.
The M31 population of CHVCs
---------------------------
One of the most suggestive attributes of the CHVC population in favor of a Local Group deployment is the modest concentration of objects which are currently detected in the general direction of M31, i.e. in the direction of the Local Group barycenter. These objects have extreme negative velocities in the GSR reference frame. While this is a natural consequence of the Local Group models it does not follow from the empirical Galactic halo models, nor is it a consequence of obscuration by Galactic H[i]{}. Putman & Moore ([@putman02c]) have made some comparisons between numerical simulations of dark matter mini–halos in the Local Group with the $(l,V_{\rm LGSR})$ distributions of HVCs and CHVCs, and were led to reject the possibility of CHVC deployment throughout the Local Group. Our discussion here has shown that such comparisons require taking explicit account of detection thresholds in the available survey observations, as well as of the vagaries of obscuration caused by the Zone of Avoidance. The Putman & Moore investigation did not take these matters into account. The modest apparent amplitude of the M31 concentration relative to the Galactic population as seen with present survey sensitivities provides the best current constraints on the global distance scale of the CHVC ensemble. There follows a testable prediction, namely that with increased sensitivity a larger fraction of the M31 population of CHVCs should be detected. This prediction was made explicit in Fig. \[fig:mod01hom\], where one of our model distributions was shown as it would have been detected if HIPASS sensitivity were available in the northern sky. For that particular model, some 250 additional detected objects are predicted, of which the majority are concentrated in the $60\times60^\circ$ region centered on M31. The ongoing HIJASS survey of the sky north of $\delta=25^\circ$ (Kilborn [@kilborn]), which is being carried out using the 76–m Lovell Telescope at Jodrell Bank to about the same velocity coverage, angular resolution, and sensitivity as the HIPASS effort, should allow this prediction to be tested.
The Sculptor Group lines of sight
---------------------------------
We have omitted the part of the sky around the south Galactic pole in our fitting of Local Group models to the observations, because of the extreme velocity dispersions measured in this direction. The nearest external group of galaxies, the Sculptor Group, is located in the direction of the south Galactic pole. If the CHVCs are distributed around the major Local Group galaxies, then plausibly the same sort of objects could be present in the Sculptor Group. Putman et al. ([@putman02a]) mention detection of clouds in the direction of the southern part of the Sculptor Group. Because no similar clouds were detected in the northern part of this Group, they consider it unlikely that this concentration of CHVCs is associated with the Sculptor Group. We note, however, that rather than being a spherical concentration of galaxies, the Sculptor Group has an extended filamentary morphology, which ranges in distance from $1.7\rm\;Mpc$ in the south to $4.4\rm\;Mpc$ in the north. Putman et al. assumed that the HIPASS sensitivity would allow detection of masses of $7\times10^6\rm\;M_\odot$ throughout the Sculptor Group. But in Fig. \[fig:obsdistance\] we show the actual distance out to which HIPASS can detect masses given a realistic cloud model and detection threshold: even the most massive and rare objects in our simulated distributions, with $M_{\rm HI}$ = $10^7\rm\;M_\odot$, can only be detected out to $2.5\rm\;Mpc$. It is therefore only the near portion of the Sculptor filament that might be expected to show any enhancement in CHVC density with the currently available sensitivities.
Predicted CHVC populations in other galaxy groups
-------------------------------------------------
It is also interesting to consider whether the simulated Local Group model populations would be observable in external galaxy groups at even larger distances. In Fig. \[fig:mod01mass\] we show one of our best–fitting Local Group models, model \#9 of Table \[table:bestfit\], projected onto a plane as in Fig. \[fig:mod3d\]. In Fig. \[fig:mod3d\], the surviving clouds were distinguished by flux; in Fig. \[fig:mod01mass\], the distinction is by mass. We indicate with grey dots those objects that were deemed to have been disrupted by ram–pressure or tidal stripping. The red and black dots indicate the remaining objects in the population, with the red dots representing objects that exceed $M_{\rm HI}$ = 3$\times10^6$M$_\odot$ and the black dots those that fall below this mass limit. The choice of a limiting mass of $M_{\rm
HI}$=3$\times10^6$M$_\odot$ over a linewidth of 35 was made to represent what might be possible for a deep survey of an external galaxy group. In this example, some 95 objects occur which exceed this mass limit distributed over a region of some 1.5$\times$1.0 Mpc extent. For a limiting mass of $M_{\rm HI}$=5$\times10^6$M$_\odot$ over 35 , the number drops to 45. It is clear that a very good mass sensitivity will be essential to detecting such potential CHVC populations in external galaxy groups. Current searches for such populations, reviewed by Braun & Burton ([@braun01]), have generally not reached a sensitivity as good as even $M_{\rm
HI}$=$10^7\rm\;M_\odot$ over 35 , so it is no surprise that such distant CHVCs have not yet been detected.
The Westerbork Synthesis Radio Telescope is operated by the Netherlands Foundation for Research in Astronomy, under contract with the Netherlands Organization for Scientific Research.
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[^1]: Several of the figures in this paper make color-coded distinctions; although the printed journal and electronic versions will display the color coding, black-and-white printouts will not, of course.
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{
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author:
- |
[^1]\
University of Iowa, Iowa City, IA 52242-1479, U.S.A.\
E-mail:
title: 'Measurement of the energy flow in a large $\eta$ range and forward jets at LHC at $\sqrt{s} =$ 0.9 TeV, 2.36 TeV and 7 TeV'
---
Introduction
============
At very large center-of-mass energies, the momentum fraction of the proton carried by the partons in the hard scattering (x1, x2) can become very small. In such a small-x region, parton densities might become very large and the probability of more than one partonic interaction per event increases as depicted in Fig 1. This approach is described in the models of multiparton interactions (MPI) [@MPI], however, the details of the energy dependence of MPI cross sections is not well known yet.
![(a) Radiated partons coming from a parton shower. (b) More than one partonic interaction chain.[]{data-label="fig:1"}](PartonShower.png "fig:"){width="35.00000%"}![(a) Radiated partons coming from a parton shower. (b) More than one partonic interaction chain.[]{data-label="fig:1"}](MultiPartonShower.png "fig:"){width="35.00000%"}\
(a) (b)
The measurement of the underlying event using the energy flow in the forward region is complementary to the measurement of the underlying event in the central region, thus can provide additional input to the determination of the parameters for the MPI models [@MPI2]. Forward jet production at the Large Hadron Collider (LHC) is also an ideal process to investigate small-x QCD effects [@smallx]. In this study, a first measurement of forward jets and the energy flow in the forward region (3.15 $< |\eta| <$ 4.9) of the Compact Muon Solenoid (CMS) detector is presented.
CMS Hadron Forward Calorimeter
==============================
CMS [@CMS] is a general-purpose detector designed to run at the highest luminosity provided by the CERN LHC. Two hadron forward calorimeter (HF$\pm$), one on each side of the CMS interaction point (IP), at about $\pm$11 m, cover the very forward angles of CMS, in the pseudorapidity range $3 < |\eta| < 5$. Due to the severe radiation environment around beam pipe, HF calorimeters are built with radiation hard quartz fibers embedded in steel absorbers. The signal in HF is produced in the form of Cerenkov light generated by the showering particles passing through quartz fibers. Half of the fibers run over the full depth of the absorber whereas the other half, read out separately, start at a depth of 22 cm from the front face of the detector. This structure makes it possible to distinguish showers generated by electrons and photons from those generated by hadrons. The objectives with the HF detector are to improve the measurement of the transverse energy and to measure very forward jets. The performance and thechnical details of the HF calorimeters can be found elsewhere [@HF].
Data Samples and Event Selection
================================
The data from $pp$ collisions, which were collected with the CMS detector at $\sqrt{s} =$ 0.9 TeV and 2.36 TeV in the fall of 2009 and at $\sqrt{s} =$ 7 TeV in March 2010 were used in this analysis.
Two subsystems, the Beam Pick-up Timing for eXperiments (BPTX) and the Beam Scintillator Counters (BSC) [@CMS; @BSC], were used in the trigger of the detector readout. The two BPTX devices, which are located around the beam pipe at a distance of $\pm$175 m from the IP, are designed to provide precise information on the structure and timing of the LHC beams, with a time resolution better than 0.2 ns. The two BSCs, consisting of a set of 16 scintillator tiles, are located along the beam line on each side of the IP at a distance of $\pm$10.86 m and they provide information on hits and coincidence signals with a time resolution of 3 ns. The events analyzed in this study were selected by requiring the trigger signal in the BSC counters to be in coincidence with BPTX signals from both beams. This requirement refers to the minimum bias trigger condition and indicates that there was activity in the forward regions, furthermore, at least one primary vertex was required to be reconstructed from at least 5 tracks with a $z$ distance to the interaction point with $|z|\leq$ 15 cm. These requirements ensure that the event is a collision candidate.
Results
=======
Forward Jets
------------
We searched for forward jets ($3 < |\eta| < 5$) in the very first $pp$ collisions data which were collected at $\sqrt{s} =$ 0.9 TeV at LHC [@EventDisplay]. In Fig. 2, an event is presented which has one forward jet and one backward jet with a corrected jet $p_{T} >$ 10 GeV.
![Display of an event with two forward jets. Data were collected by CMS in 2009 at $\sqrt{s} =$ 0.9 TeV $pp$ collisions.[]{data-label="fig:2"}](Jets_forward_forward.png){width="80.00000%"}
Energy Flow
-----------
The energy flow in the pseudorapidity region 3.15 < $|\eta|$ < 4.9 is measured for minimum bias events, and the ratio of energy flow is given by
$$R^{\sqrt{s_1},\sqrt{s_2}}_{Eflow} = \frac{\frac{1}{N_{\sqrt{s_1}}}\frac{dE_{\sqrt{s_1}}}{d\eta}}{\frac{1}{N_{\sqrt{s_2}}}\frac{dE_{\sqrt{s_2}}}{d\eta}}$$
where $dE_{\sqrt{s}}$ is energy deposition in a region $d\eta$ (integrated over azimuthal angle $\phi$), $N_{\sqrt{s}}$ corresponds to the number of selected minimum bias events, $\sqrt{s_1}$ refers to either 2.36 TeV or 7 TeV and $\sqrt{s_2}$ refers to 0.9 TeV.
![The ratio of energy flow for $\sqrt{s_1} =$ 2.36 TeV to $\sqrt{s_2} =$ 0.9 TeV (left), and for $\sqrt{s_1} =$ 7 TeV to $\sqrt{s_2} =$ 0.9 TeV (right) as a function of $\eta$. Uncorrected data are shown as points and the red line is the prediction from PYTHIA tune D6T [[@Pythia; @PythiaManual]]{}.[]{data-label="fig:3"}](236divided09.png "fig:"){width="40.00000%"}![The ratio of energy flow for $\sqrt{s_1} =$ 2.36 TeV to $\sqrt{s_2} =$ 0.9 TeV (left), and for $\sqrt{s_1} =$ 7 TeV to $\sqrt{s_2} =$ 0.9 TeV (right) as a function of $\eta$. Uncorrected data are shown as points and the red line is the prediction from PYTHIA tune D6T [[@Pythia; @PythiaManual]]{}.[]{data-label="fig:3"}](7divided09.png "fig:"){width="40.00000%"}\
In Fig. 3, the ratio of energy flows for different collision energies is shown as the average of the HF(+) and HF(-) responses [@DetPer]. The pseudorapidity region 3.15 < $|\eta|$ < 4.9 is divided into 5 bins following the transverse segmentation of the HF calorimeters. The energy deposition in a given $d\eta$ is integrated over $\phi$ and 4 GeV threshold is applied on the measured energy to be included in $dE/d\eta$ calculation. This threshold ensures that the noise is removed. Uncertainties due to systematic effects are not shown. Also there is no correction applied on energy scale.
Conclusions
===========
In this study, an analysis of the very first $pp$ collisions data collected at LHC is presented. Forward jets, at $|\eta|>3$, have been observed for the first time in hadron-hadron collisions at this forward rapidities. Ratio of energy flow for minimum bias events at different $\sqrt{s}$ has been measured at the detector level. The results indicate that the energy flow is larger at forward rapidities at higher center-of-mass energies.
At the time of writing, new measurements on minimum bias events and events having a hard scale defined by a dijet with $E_{T,jet} >$ 8 GeV ($E_{T,jet} >$ 20 GeV for $\sqrt{s} =$ 7 TeV) in $|\eta| <$ 2.5 have been studied. Systematic effects are included and the results are compared to predictions from Monte Carlo event generators that were tuned to describe the charged particle spectra seen at central rapidities. A detailed description is under preparation.
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank T. Yetkin for useful discussions throughout this work, G. Brona, H. Jung and K. Piotrzkowski for useful feedback and suggestions, Y. Onel for his support.
[99]{} T. Sjostrand and M. van Zijl, “A Multiple Interaction Model for the Event Structure in Hadron Collisions”, Phys. Rev. **D36** (1987) 2019. doi:10.1103/PhysRevD.36.2019. T. Sjostrand and M. van Zijl, “Multiple parton-parton interactions in an impact parameter picture”, Phys. Lett. **B188** (1987) 149 doi:10.1016/0370-2693(87)90722-2. David d’Enterria (for the CMS Collaboration), “Small-x QCD studies with CMS at the LHC” J. Phys. G: Nucl. Part. Phys. **34** S709 (2007). doi: 10.1088/0954-3899/34/8/S79. S. Chatrchyan et al. (CMS Collabration), “The CMS Experiment at the CERN LHC,” JINST **3**, S08004, pp. 145-149 (2008), doi:10.1088/1748-0221/3/08/S08004. S. Abdullin et al. (CMS Collaboration), “Design, Performance, and Calibration of CMS Forward Calorimeter Wedges,” Eur. Phys. J. **C53**, 139-166 (2008). A. J. Bell, “The design and construction of the beam scintillation counter for CMS”, CERN-THESIS-2009-062 (2008). CMS Collaboration, “Event displays of forward jets at 900 GeV”, CMS DP-2010/006 (2010). R. Field, “Studying the underlying event at CDF and the LHC”, arXiv:1003.4220. T. Sjostrand et al., “PYTHIA 6.4 physics and manual”, JHEP **05** (2006) 026, arXiv:hep-ph/0603175. CMS Collaboration, “Energy Flow Ratios in HF at Different Collision Energies”, CMS DP-2010/007 (2010).
[^1]: For CMS Collaboration.
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abstract: 'We show that every $k$-tree of toughness greater than $\frac{k}{3}$ is Hamilton-connected for $k \geq 3$. (In particular, chordal planar graphs of toughness greater than $1$ are Hamilton-connected.) This improves the result of Broersma et al. (2007) and generalizes the result of B[ö]{}hme et al. (1999). On the other hand, we present graphs whose longest paths are short. Namely, we construct $1$-tough chordal planar graphs and $1$-tough planar $3$-trees, and we show that the shortness exponent of the class is $0$, at most $\log_{30}{22}$, respectively. Both improve the bound of B[ö]{}hme et al. Furthermore, the construction provides $k$-trees (for $k \geq 4$) of toughness greater than $1$.'
author:
- 'Adam Kabela[^1]'
title: '**Long paths and toughness of *k*-trees and chordal planar graphs**[^2] '
---
Introduction
============
We continue the study of Hamiltonicity and toughness of $k$-trees following Broersma et al. [@ktree] and of chordal planar graphs following B[ö]{}hme et al. [@chpl]. We recall that for a positive integer $k$, a *$k$-tree* is either the graph $K_k$ (that is, the complete graph on $k$ vertices) or a graph containing a vertex whose neighbourhood induces $K_k$ and whose removal gives a $k$-tree. Clearly, $k$-trees are chordal graphs. We recall that the *toughness* of a graph $G$ is the minimum, taken over all separating sets $X$ of vertices of $G$, of the ratio of $|X|$ to the number of components of $G - X$. The toughness of a complete graph is defined as being infinite. We say that a graph is *$t$-tough* if its toughness is at least $t$.
In [@ktree], Broersma et al. showed that certain level of toughness implies that a $k$-tree has a Hamilton cycle (see also [@LZ; @SW]).
\[broersma\] Let $k \geq 2$. Every $\frac{k+1}{3}$-tough $k$-tree (except for $K_2$) is Hamiltonian.
In the same paper, they constructed $1$-tough $k$-trees which have no Hamilton cycle for every $k \geq 3$.
An older result considering toughness and Hamiltonicity in another subclass of chordal graphs is due to B[ö]{}hme et al. [@chpl] who showed the following:
\[bohme\] Every chordal planar graph (on at least $3$ vertices) of toughness greater than $1$ is Hamiltonian.
In [@gerl], Gerlach generalized Theorem \[bohme\] for planar graphs whose separating cycles of length at least four have chords. In this paper, we present a different generalization of Theorem \[bohme\] which also improves the result of Theorem \[broersma\].
The mentioned results were motivated by the following conjecture stated by Chvátal [@chva].
\[conj:ch\] There exists $t$ such that every $t$-tough graph (on at least $3$ vertices) is Hamiltonian.
Conjecture \[conj:ch\] remains open. Partial results are known for some restricted classes of graphs; for instance, for different subclasses of chordal graphs (see [@ktree; @chpl; @all; @spli; @KKS]), and for the class of chordal graphs itself (see [@18] or [@10]). The best known lower bounds regarding Conjecture \[conj:ch\] for chordal graphs and for general graphs were shown in [@74]. The study of toughness of graphs (and Conjecture \[conj:ch\] in particular) is well-documented by a series of survey papers, we refer the reader to [@surv] (for more recent results, see [@Broersma]). In addition to the result of Theorem \[bohme\], B[ö]{}hme et al. [@chpl] presented $1$-tough chordal planar graphs whose longest cycles are relatively short (compared to the number of vertices of the graph); and using the notion of shortness exponent by Gr[ü]{}nbaum and Walther [@sef], they argued the following:
\[89\] The shortness exponent of the class of $1$-tough chordal planar graphs is at most $\log_9 8$.
We recall that the *shortness exponent* of a class of graphs $\Gamma$ is the $\liminf$, taken over all infinite sequences $G_n$ of non-isomorphic graphs of $\Gamma$ (for $n$ going to infinity), of the logarithm of the length of a longest cycle in $G_n$ to base equal to the number of vertices of $G_n$. For more results considering the shortness exponent, see the survey [@owens]. To conclude this section, we mention that by the combination of results of Moser and Moon [@momo] and Chen and Yu [@3-conn], the shortness exponent of the class of $3$-connected planar graphs equals $\log_3 2$.
New results {#s2}
===========
We recall that a graph is *Hamilton-connected* if for every pair of its vertices, there is a Hamilton path between them. Clearly, every Hamilton-connected graph (on at least $3$ vertices) is Hamiltonian. Using a simple argument, we improve the result of Theorem \[broersma\] as follows. (This also improves the result of [@LZ] since Hamilton-connected chordal graphs are, in fact, panconnected.)
\[main\] Let $k \geq 3$. Every $k$-tree of toughness greater than $\frac{k}{3}$ is Hamilton-connected. Furthermore, every $1$-tough $2$-tree (except for $K_2$) is Hamiltonian.
The proof of Theorem \[main\] is given in Section \[s3\]. We also show that under this toughness restriction a graph is chordal planar if and only if it is a $3$-tree or $K_1$ or $K_2$ (see Lemma \[coin\]). In particular, Theorem \[main\] implies that chordal planar graphs of toughness greater than $1$ are Hamilton-connected (it generalizes the result of Theorem \[bohme\]). In the other direction, we present $1$-tough chordal planar graphs and $1$-tough planar $3$-trees whose longest paths and cycles are relatively short.
In particular, for every $\varepsilon > 0$, there exists a $1$-tough chordal planar graph $G$ whose longest path has less than $|V(G)|^{\varepsilon}$ vertices. In Section \[s6\], we note that such graphs can be obtained by considering the square of particular trees. Consequently, we adjust the result of Theorem \[89\] as follows:
\[0\] The shortness exponent of the class of $1$-tough chordal planar graphs is $0$.
We remark that the graphs constructed in [@chpl] are $3$-connected, so the bound $\log_9 8$ of Theorem \[89\] also applies to the shortness exponent of the class of $1$-tough planar $3$-trees (see Lemma \[3\]). In Section \[s7\], we use the standard construction for bounding the shortness exponent (for more details regarding this construction, see for instance [@owens] or [@update]), and we improve this bound by the following:
\[79\] The shortness exponent of the class of $1$-tough planar $3$-trees is at most $\log_{30}{22}$.
In Section \[s8\], we extend the used construction, and we remark that there are $k$-trees of toughness greater than $1$ whose longest paths are relatively short for every $k \geq 4$. (Meanwhile, $3$-trees of toughness greater than $1$ are Hamilton-connected by Theorem \[main\].) This remark slightly improves the lower bound on toughness of non-Hamiltonian $k$-trees presented in [@ktree], and contradicts the suggestion of [@SW].
Tough enough *k*-trees are Hamilton-connected {#s3}
=============================================
In this section, we prove Theorem \[main\]. Simply spoken, the proof is inductive; we choose a vertex on a path and we extend the path using particular neighbours of this vertex.
For a vertex $v$, we let $N(v)$ denote its *neighbourhood*, that is, the set of all vertices adjacent to $v$. We say a set $S \subseteq N(v)$ is a *squeeze* by $v$ if the following properties are satisfied for $S$ and $R = N(v) {\setminus}S$.
- $2 \geq |S| \geq 1$ and $|R| \geq 2$.
- Every vertex of $S$ is adjacent to at least $|R| - 1$ vertices of $R$, and every vertex of $R$ is adjacent to at least $|S| - 1$ vertices of $S$.
The basic ingredient for applying the induction is the following:
\[path\] Let $P$ be some set of vertices of a graph $G$ and let $x_1$, $x_2$ and $v$ be distinct vertices of $P$ and let $S$ be a squeeze by $v$. If $G - S$ has a path between $x_1$ and $x_2$ whose vertex set is $P$, then $G$ has such path whose vertex set is $P \cup S$.
We let $uv$ and $vw$ be the edges (incident with $v$) of the considered path in $G - S$. We note that the graph induced by $\{u,v,w\} \cup S$ has a Hamilton path between $u$ and $w$. Thus, we can extend the considered path into a path between $x_1$ and $x_2$ whose vertex set is $P \cup S$.
We recall that a vertex whose neighbourhood induces a complete graph is called *simplicial*. For further reference, we state the following fact (shown, for instance, in [@update]).
\[simpl\] Adding a simplicial vertex to a graph does not increase its toughness.
By definition, $k$-trees can be viewed as graphs constructed iteratively from $K_k$ by adding one new simplicial vertex of degree $k$ in each step. We recall that a vertex adjacent to all vertices of a graph is called *universal*. Considering a non-universal vertex $v$ of a $k$-tree and the set $S$ of all its neighbours of degree $k$, we say $v$ is a *twig* if $N(v) {\setminus}S$ induces $K_k$ and $|S| \geq 1$; and we say $S$ is the *bud* of this twig. We note the following two facts:
\[newTwig\] Let $k \geq 1$ and let $G$ and $G^+$ be $k$-trees such that $G$ is obtained from $G^+$ by removing a simplicial vertex. If $t$ is a twig in $G$ but not in $G^+$, then a vertex of the bud of $t$ is a twig in $G^+$.
Since $t$ is a twig in $G$ but not in $G^+$, there exists a vertex $t'$ adjacent to $t$ such that $t'$ has degree $k$ in $G$, and degree $k+1$ in $G^+$. Clearly, $t'$ is a twig in $G^+$.
\[twig\] Let $k \geq 1$ and let $G$ be a $k$-tree (on at least $k+3$ vertices) of toughness greater than $\frac{k}{3}$. Then $G$ has a twig. Furthermore, if $v$ is a twig of $G$ and $S$ is its bud, then $G-S$ is a $k$-tree of toughness greater than $\frac{k}{3}$. In addition, if $k \geq 2$, then $S$ is a squeeze by $v$.
We consider an iterative construction of $G$, and we let $T$ denote the $k$-tree on $k+3$ vertices which is obtained in the corresponding iteration of the construction. Proposition \[simpl\] implies that the toughness of $T$ is at least the toughness of $G$, and we observe that there exists only one $k$-tree on $k+3$ vertices of toughness greater than $\frac{k}{3}$ (for a fixed $k$). We note that $T$ has a twig. Thus, Lemma \[newTwig\] implies that $G$ has a twig.
We consider a twig $v$ in $G$ and its bud $S$, and we let $R = N(v) {\setminus}S$. Clearly, $G-S$ is a $k$-tree. Furthermore, the toughness of $G-S$ is at least the toughness of $G$ (by Proposition \[simpl\]).
In addition, we note that every vertex of $S$ is adjacent to precisely $|R| - 1$ vertices of $R$. Since $v$ is non-universal, the toughness of $G$ implies that no two vertices of $S$ have the same neighbourhood. In particular, for $k = 2$, we have $|S| \leq 2$. For $k \geq 3$, the same follows from the fact that $G - R - v$ has at least $|S| + 1$ components and $|R| = k$. Clearly, if $k \geq 2$ then $|R| \geq 2$; and we conclude that $S$ is a squeeze by $v$.
We note that, with Lemmas \[path\] and \[twig\] on hand, we can easily show Hamiltonicity of $k$-trees of toughness greater than $\frac{k}{3}$. (We remark that $2$-trees of toughness greater than $\frac{2}{3}$ are, in fact, $1$-tough.)
\[ham\] Let $k \geq 2$. Every $k$-tree (except for $K_2$) of toughness greater than $\frac{k}{3}$ is Hamiltonian.
We let $G$ be the considered $k$-tree, and we let $n$ denote the number of its vertices. Clearly, if $n \leq k + 2$, then $G$ is Hamiltonian. We can assume that $n \geq k + 3$. We suppose that the statement is satisfied for graphs on at most $n-1$ vertices, and we show it for $G$.
By Lemma \[twig\], $G$ has a twig $v$; and we let $S$ be the bud of $v$. Furthermore, $G-S$ is a $k$-tree of toughness greater than $\frac{k}{3}$. (Clearly, $G-S$ is distinct from $K_2$.) By the hypothesis, $G-S$ has a Hamilton cycle, and we view it as a Hamilton path containing $v$ as an interior vertex. By Lemmas \[path\] and \[twig\], we can prolong this path and obtain a Hamilton path in $G$ whose ends are adjacent, that is, a Hamilton cycle.
Aiming for the Hamilton-connectedness, we shall need two additional ingredients which are given by Lemma \[twoGood\] and Proposition \[stave\]. For $k \geq 2$, a *basic $3$-twig* is the graph obtained from $K_{k+1}$ by choosing its three different subgraphs $K_k$ and by adding one new simplicial vertex to each of them. (For instance, the basic $3$-twig for $k = 3$ is the graph $B$ depicted in Figure \[f:79\].)
\[twoGood\] Let $k \geq 1$ and let $G$ be a $k$-tree (on at least $k+4$ vertices) of toughness greater than $\frac{k}{3}$. If $G$ is distinct from the basic $3$-twig, then $G$ has two non-adjacent twigs (whose buds are disjoint).
We consider an iterative construction of $G$, and we note that all $k$-trees obtained during the construction have toughness greater than $\frac{k}{3}$ (by Proposition \[simpl\]). We consider the $k$-tree on $k+4$ vertices, and we observe that either it is the basic $3$-twig or it has two non-adjacent twigs. (Clearly, the buds of non-adjacent twigs are disjoint.) In particular, we can assume that $G$ has more than $k+4$ vertices.
Consequently, we note that the $k$-tree on $k+5$ vertices obtained during the construction has two non-adjacent twigs. Using Lemma \[newTwig\], we conclude that $G$ has two non-adjacent twigs.
In a graph $G$, we say a *$\Theta$-spanner* between vertices $x_1$ and $x_2$ is a spanning subgraph of $G$ consisting of three paths with the same ends $x_1$, $x_2$ such that (except for the ends) these paths are mutually disjoint, and each of them has at least one interior vertex. We shall use $\Theta$-spanners to address the setting in which the ends of the desired Hamilton path are the only twigs of a $k$-tree. (We note that a similar idea appeared in [@all].)
\[stave\] Let $k \geq 3$ and let $G$ be a $k$-tree (distinct from $K_4$) of toughness greater than $\frac{k}{3}$ and let $x_1$ and $x_2$ be distinct vertices of degree $k$. Then $G$ has a $\Theta$-spanner between $x_1$ and $x_2$.
Clearly, $K_k$ has no vertex of degree $k$. Furthermore, there exists only one $k$-tree on $k+1$ vertices and one on $k+2$ vertices, and only one $k$-tree on $k+3$ vertices has the required toughness (for a fixed $k$).
Considering these $k$-trees, we note that the statement is satisfied for graphs on at most $k + 3$ vertices. We let $n$ denote the number of vertices of $G$, and we assume that $n \geq k + 4$. We suppose that the statement is satisfied for graphs on at most $n-1$ vertices, and we show it for $G$.
Let us suppose that there is a twig $v$ and its bud $S$ such that neither $x_1$ nor $x_2$ belongs to $S$. By Lemma \[twig\] and by the hypothesis, we can consider a $\Theta$-spanner between $x_1$ and $x_2$ in $G - S$; and we let $P$ be the set of vertices of one of the three paths between $x_1$ and $x_2$ of this $\Theta$-spanner such that $v$ belongs to $P$. By Lemmas \[path\] and \[twig\], there is a path with the same ends whose vertex set is $P \cup S$. Thus, $G$ has a $\Theta$-spanner between $x_1$ and $x_2$.
We assume that every twig is adjacent to $x_1$ or $x_2$. By Lemma \[twoGood\], we can assume that there is a twig $x'_1$ and its bud $S'$ such that $x_1$ belongs to $S'$ and $x_2$ does not. Clearly, $x'_1$ has degree $k$ in $G - S'$. We consider a $\Theta$-spanner $Y$ between $x'_1$ and $x_2$ in $G - S'$; and we let $N$ denote the set of all vertices adjacent to $x'_1$ in $Y$. We choose a vertex $y$ of $N$ such that $y$ is adjacent to $x_1$ in $G$. Clearly, the subgraph of $Y$ induced by $N \cup \{ x'_1 \} {\setminus}\{ y \}$ is a path, and we apply Lemmas \[path\] and \[twig\] and extend this path by adding vertices of $S'$. We consider the resulting path and the edge $x_1 y$, and we extend the graph $Y - x'_1$ into a $\Theta$-spanner between $x_1$ and $x_2$ in $G$.
Finally, we use the tools introduced in this section and prove Theorem \[main\].
For $k = 2$, the statement is satisfied by Lemma \[ham\]. We assume that $k \geq 3$. We let $G$ be a $k$-tree of toughness greater than $\frac{k}{3}$, and we let $n$ denote the number of its vertices. We note that if $n \leq k + 3$, then $G$ is Hamilton-connected; so we can assume that $n \geq k + 4$. We suppose that the statement is satisfied for graphs on at most $n-1$ vertices, and we show it for $G$ (that is, we show that for an arbitrary pair of vertices $x_1$ and $x_2$, $G$ has a Hamilton path between $x_1$ and $x_2$).
Let us suppose that $G$ has a twig distinct from $x_1$ and $x_2$. By Lemma \[twoGood\], we can choose a twig $v$ such that $x_1$ does not belong to the bud $S$ of $v$. In case $x_2$ belongs to $S$, we consider a Hamilton path between $x_1$ and $v$ in $G - x_2$, and we extend it by adding the edge $v x_2$. In case neither $x_1$ nor $x_2$ belongs to $S$, we consider a Hamilton path between $x_1$ and $x_2$ in $G - S$, and we note that it can be extended into a desired path in $G$ (by Lemmas \[path\] and \[twig\]).
We assume that every twig of $G$ belongs to $\{ x_1, x_2 \}$. By Lemma \[twoGood\], we can assume that $x_1$ and $x_2$ are non-adjacent twigs and the corresponding buds $S_1$ and $S_2$ are disjoint. We consider the graph $G' = G - S_1 - S_2$. We note that $G'$ is distinct from $K_4$ and $x_1$ and $x_2$ have degree $k$ in $G'$, and that $G'$ is a $k$-tree of toughness greater than $\frac{k}{3}$ (by Lemma \[twig\]).
We consider a $\Theta$-spanner $Z$ between $x_1$ and $x_2$ in $G'$ given by Proposition \[stave\]. Clearly, $Z$ forms three paths in $G' - x_1 - x_2$. We note that we can join these paths (using the adjacency of their ends and using the vertices of $S_1$ and $S_2$) and obtain a Hamilton path from $S_1$ to $S_2$ in $G - x_1 - x_2$. Thus, we get a Hamilton path between $x_1$ and $x_2$ in $G$.
To clarify the relation between Theorem \[bohme\] and the case $k = 3$ of Theorem \[main\], we note the following:
\[coin\] A graph of toughness greater than $1$ is chordal planar if and only if it is either a $3$-tree or $K_1$ or $K_2$.
For convenience, we include a short proof of Lemma \[coin\]. We shall use the facts stated in Lemmas \[3\] and \[emb\] (shown by Patil [@patil] and by Markenzon et al. [@mjp Lemma 24], respectively). We recall that a graph is *$H$-free* if it contains no copy of the graph $H$ as an induced subgraph.
\[3\] Let $k \geq 1$. A graph (distinct from $K_k$) is a $k$-tree if and only if it is $k$-connected chordal and $K_{k+2}$-free.
\[emb\] Let $G$ be a $3$-tree. Then $G$ is planar if and only if $G - C$ consists of at most two components for every set of vertices $C$ inducing $K_3$.
The combination of Lemmas \[3\] and \[emb\] gives the desired equivalence.
We consider a chordal planar (and thus $K_5$-free) graph. By the assumption on toughness, the graph is either $3$-connected or $K_1$ or $K_2$ or $K_3$, and we apply the case $k = 3$ of Lemma \[3\].
For the other direction, we consider a $3$-tree of toughness greater than $1$. We note that a removal of three vertices creates at most two components, and we apply Lemma \[emb\].
Long paths in $\mathbf{1}$-tough chordal planar graphs {#s6}
======================================================
In this section, we shall show the following:
\[short2\] For every $n_0$, there exists a $1$-tough chordal planar graph on $n>n_0$ vertices whose longest cycle has $4\log_2 \frac{n+2}{3}$ vertices and whose longest path has $2(\log_2 \frac{n+2}{3})^2 + 2$ vertices.
In particular, the first part of Proposition \[short2\] immediately implies the result of Theorem \[0\].
We consider an infinite sequence of non-isomorphic graphs given by Proposition \[short2\]. We recall that a graph on $n$ vertices belonging to this sequence has a longest cycle on $4\log_2 \frac{n+2}{3}$ vertices. Consequently, the considered shortness exponent is at most $\displaystyle \lim_{n\to\infty} \log_n (4\log_2 \tfrac{n+2}{3}) = 0$.
We recall that a tree is *cubic* if every non-leaf vertex has degree $3$. In order to prove Proposition \[short2\], we consider the square of ‘balanced’ cubic trees, and we combine several known facts (recalled in Theorems \[square\], \[t\] and Propositions \[c\] and \[p\]). We let $G^2$ denote the *square* of a graph $G$, that is, the graph on the same vertex set as $G$ in which two vertices are adjacent if and only if their distance in $G$ is either $1$ or $2$. Studying squares of trees, Neuman [@Neuman] presented necessary and sufficient conditions for the existence of a Hamilton path between a given pair of vertices. As a corollary, the characterization of trees whose square has a Hamilton cycle (Hamilton path) follows. (Later, these results were also proven separately, see [@hs; @Gould].) We consider the trees depicted in Figure \[f:FX\], and we recall these characterizations (see Theorem \[square\]). Similarly as above, we recall that a graph is *${\mathcal H}$-free* if it contains no copy of a graph from the family ${\mathcal H}$ as an induced subgraph.
![The trees $S(K_{1,3})$, $S(K_{1,5})$ and the families of trees ${\mathcal F}$ and ${\mathcal X}$. The trees of ${\mathcal F}$ are obtained from two copies of $S(K_{1,3})$ by joining their central vertices with a path (possibly an edge) and adding one new vertex adjacent (by a pendant edge) to each interior vertex of this path. The trees of ${\mathcal X}$ are obtained from three copies of $P_5$ and from a tree containing precisely three leaves by identifying each of these leaves with the central vertex of one $P_5$.[]{data-label="f:FX"}](FX.pdf)
\[square\] Let $T$ be a tree. The following statements are satisfied:
1. $T^2$ is Hamiltonian if and only if $T$ (on at least $3$ vertices) is $S(K_{1,3})$-free.
2. $T^2$ has a Hamilton path if and only if $T$ is $S(K_{1,5})$-free, ${\mathcal F}$-free and ${\mathcal X}$-free.
In addition, we recall the following property of squares of graphs (shown by Chvátal [@chva]).
\[t\] The square of a $k$-connected graph is $k$-tough.
We recall that (as observed by Fulkerson and Gross [@elim]) a graph $G$ is chordal if and only if it has a *perfect elimination ordering*, that is, an ordering $(v_1, v_2, \dots, v_n)$ of all vertices of $G$ such that $v_i$ is a simplicial vertex of $G_i$ for every $i = 1, 2, \dots, n$, where $G_i$ is the subgraph of $G$ induced by $\{ v_1, v_2, \dots, v_i \}$. We note the following:
\[c\] The square of a tree is a chordal graph.
Clearly, a perfect elimination ordering of the tree is a perfect elimination ordering of its square.
We shall also use the following fact (which we view as a corollary of the characterization of graphs whose squares are planar by Harary et al. [@hkw]).
\[p\] Let $T$ be a tree. Then $T^2$ is planar if and only if $T$ has no vertex of degree greater than $3$.
Finally, we construct graphs which have the properties stated in Proposition \[short2\].
We let $T$ be a cubic tree (on at least $4$ vertices) having a vertex such that the distances from this vertex to every leaf are the same; and we let $r$ denote this distance. By Theorem \[t\] and Propositions \[c\] and \[p\], $T^2$ is a $1$-tough chordal planar graph.
We let $n$ denote the number of vertices of $T$. By simple counting arguments, we get that $n = 3 \cdot 2^r - 2$ (that is, $r = \log_2 \frac{n+2}{3}$) and that a largest $S(K_{1,3})$-free subtree of $T$ has $4r$ vertices.
Furthermore, $T$ is $S(K_{1,5})$-free and ${\mathcal F}$-free (since $T$ is a cubic tree). We consider a largest ${\mathcal X}$-free subtree, say $L$, and we show that it has $2r^2+2$ vertices. We let $L_0$ be the tree obtained from $L$ by removing all leaves of $L$, and we let $n_i$ be the number of vertices of degree $i$ in $L_0$ (for $i= 1,2,3$). We note that all vertices of degree $3$ in $L_0$ belong to a common path (since $L$ is ${\mathcal X}$-free). Hence, $n_3 \leq 2r-3$, and therefore $n_2 \leq (r-2)^2$ and $n_1 \leq 2r - 1$. Thus, $L$ has at most $n_3 + 2 n_2 + 3n_1 = 2r^2+2$ vertices (that is, at most $n_3 + n_2 + n_1$ vertices of $L_0$ plus the removed leaves). Lastly, we note that there is an ${\mathcal X}$-free subtree of $T$ on $2r^2+2$ vertices. We conclude that a longest cycle of $T^2$ has $4\log_2 \frac{n+2}{3}$ vertices and its longest path has $2(\log_2 \frac{n+2}{3})^2 + 2$ vertices by Theorem \[square\].
Long paths in $\mathbf{1}$-tough planar $\mathbf{3}$-trees {#s7}
==========================================================
In order to prove Theorem \[79\], we show the following:
\[short79\] Let $n$ be a non-negative integer and let $c(n) = 1 + 62(1 + 22 + \dots + 22^n)$. Then there exists a $1$-tough planar $3$-tree $H_n$ on $1 + 70(1 + 30 + \dots + 30^n)$ vertices whose longest cycle has $c(n)$ vertices and whose longest path has $c(n) + 2 + 2( c(0) + c(1) + \dots + c(n-1))$ vertices.
We note that the desired result follows as a corollary of Proposition \[short79\].
We consider the sequence of graphs $H_1, H_2, \dots$ given by Proposition \[short79\]; and for every $n \geq 0$, we let $f(n)$ denote the number of vertices of $H_n$. Clearly, $$f(n) = 1 + \tfrac{70}{29}(30^{n+1} - 1)
\quad
\textnormal{and}
\quad
c(n) = 1 + \tfrac{62}{21}(22^{n+1} - 1).$$ Thus, $$\lim_{n\to\infty} \log_{f(n)} c(n) = \log_{30}{22},$$ and therefore the considered shortness exponent is at most $\log_{30}{22}$.
In the remainder of this section, we construct the graphs $H_n$ and prove Proposition \[short79\]. We remark that, as well as in [@chpl], we shall use the standard construction for bounding the shortness exponent; the improvement of the bound comes with a choice of a more suitable starting graph $H_0$. The reasoning behind this choice is similar to the one applied in [@update].
![The graph $B$ and the construction of the graph $H_0$. The graph $H_0$ is obtained by replacing each of the highlighted triangles (of the graph depicted on the left) with a copy of $B$ in the natural way (by identifying the vertices of the highlighted triangle with the vertices of degree $5$ in $B$). The numbers represent the ordering of vertices of $H_0$.[]{data-label="f:79"}](22of30.pdf)
We consider the graph $H_0$ constructed in Figure \[f:79\]; and we let $u_1, u_2, u_3$ denote the vertices of its outer face in the present embedding. We note that $H_0$ contains $30$ vertices of degree $3$; and we call these vertices *white*. For every $n \geq 0$, we let $H_{n+1}$ be a graph obtained from $H_n$ by replacing every white vertex of $H_n$ with a copy of $H_0$ and by adding edges which connect the vertex $u_1, u_2, u_3$ of this copy to precisely $1,2,3$ neighbours of the replaced vertex, respectively. We note the following:
\[3-trees\] For every $n \geq 0$, the graph $H_n$ is a planar $3$-tree.
In accordance with the ordering suggested in Figure \[f:79\], we let $u_1, u_2, \dots, u_{71}$ denote the vertices of $H_0$.
We show that the graphs $H_n$ are $3$-trees. Clearly, $\{ u_1, u_2, u_3 \}$ induces $K_3$, and we consider adding vertices $u_4, u_5, \dots, u_{71}$ in sequence (in this order), and we observe that $H_0$ is a $3$-tree (by definition).
We view the replacement of a white vertex by a copy of $H_0$ as identifying this white vertex with the vertex $u_1$ of this copy and adding vertices $u_2, u_3, \dots, u_{71}$ of this copy in sequence, and we note that the resulting graph is a $3$-tree. Consequently, $H_n$ is a $3$-tree for every $n \geq 0$.
We consider the planar embedding of $H_0$ given by Figure \[f:79\]. When replacing a white vertex by a copy of $H_0$, we proceed in two steps. First, we remove the white vertex, and we note that its neighbourhood induces a facial cycle. Next, we embed a copy of $H_0$ inside this facial cycle, and we observe that the additional edges can be embedded as non-crossing. We conclude that $H_n$ is planar for every $n \geq 0$. (Alternatively, the planarity can be observed using Lemma \[emb\].)
To verify the toughness of the graphs $H_n$, we shall use the following lemma (shown in [@update]).
\[constr2\] For $i = 1,2$, let $G^+_i$ and $G_i$ be $t$-tough graphs such that $G_i$ is obtained by removing vertex $v_i$ from $G^+_i$. Let $U$ be a graph obtained from the disjoint union of $G_1$ and $G_2$ by adding new edges such that the minimum degree of the bipartite graph $(N(v_1), N(v_2))$ is at least $t$. Then $U$ is $t$-tough.
In order to apply Lemma \[constr2\], we determine the toughness of $H^+_0$, that is, the graph obtained from $H_0$ by adding one auxiliary vertex $x$ adjacent to $u_1, u_2$ and $u_3$.
\[1-tough\] The graphs $H^+_0$ and $H_0$ are $1$-tough.
We consider a separating set $S$ of vertices of $H^+_0$. If $u_4$ belongs to a component of $H^+_0 - S$, then every other component has precisely one vertex, and we note that $|S| > c(H^+_0 - S)$.
We assume that $u_4$ belongs to $S$. Except for $u_4$, the vertices adjacent to a white vertex are called *black*. Except for $u_4$ and $x$, the non-white and non-black vertices are called *blue*. We consider the set consisting of all white vertices and all black vertices which have no blue neighbour, and we let ${\mathcal I}$ denote the set of all components of $H^+_0 - S$ whose every vertex belongs to the considered set.
We shall use a discharging argument. We assign charge $1$ to every component of $H^+_0 - S$, and we distribute all assigned charge among the vertices of $S$ according to the following rules.
- The component containing $x$ (if there is such) gives its charge to $u_4$.
- The total charge of all components of ${\mathcal I}$ is distributed equally among black vertices of $S$.
- The total charge of all remaining components is distributed equally among blue vertices of $S$.
We observe that every vertex of $S$ receives charge at most $1$, that is, $|S| \geq c(H^+_0 - S)$. Thus, $H^+_0$ is $1$-tough. Consequently, $H_0$ is $1$-tough by Proposition \[simpl\].
\[1-toughHn\] For every $n \geq 0$, the graph $H_n$ is $1$-tough.
By Proposition \[1-tough\], $H^+_0$ and $H_0$ are $1$-tough. We consider an iterative construction of $H_n$ (replacing white vertices by copies of $H_0$ in sequence). We shall apply Lemma \[constr2\]. The graph at a current iteration plays the role of $G^+_1$ and the replaced vertex the role of $v_1$, and $H^+_0$ and $H_0$ play the role of $G^+_2$ and $G_2$. Using Lemma \[constr2\] repeatedly, we note that in each step of the construction we obtain a $1$-tough graph. We conclude that $H_n$ is $1$-tough.
We recall the standard construction for bounding the shortness exponent (this construction produces graphs whose longest cycles are relatively short). The idea of the construction is formalized in the following definition and in Lemma \[cyc\] (which was proven in [@update]).
An *arranged block* is a $5$-tuple $(G_0, j, W, O, k)$ where $G_0$ is a graph, $j$ is the number of vertices of $G_0$, and $W$ and $O$ are disjoint sets of vertices of $G_0$ such that the vertices of $W$ are simplicial and independent and $O$ induces a complete graph and such that every cycle in $G_0$ contains at most $k$ vertices of $W$.
\[cyc\] Let $(G_0, j, W, O, k)$ be an arranged block such that $k \geq 1$. For every $n \geq 1$, let $G_n$ be a graph obtained from $G_{n-1}$ by replacing every vertex of $W$ with a copy of $G_0$ (which contains $W$ and $O$), and by adding arbitrary edges which connect the neighbourhood of the replaced vertex with the set $O$ of the copy of $G_0$ replacing this vertex. Then $G_n$ has $1 + (j - 1)(1 + |W| + \dots + |W|^n)$ vertices and its longest cycle has at most $1 + (\ell - 1)(1 + k + \dots + k^n)$ vertices where $\ell = j - |W| + k$.
Finally, we show that the constructed graphs $H_n$ have all properties stated in Proposition \[short79\].
By Propositions \[3-trees\] and \[1-toughHn\], $H_n$ is a $1$-tough planar $3$-tree (for every $n \geq 0$). By a simple counting argument, we get that $H_n$ has $1 + 70(1 + 30 + \dots + 30^n)$ vertices.
We observe that a path in $H_0$ contains at most $22 + z$ white vertices where $z$ is the number of white ends of the path. In particular, every cycle in $H_0$ contains at most $22$ white vertices. By Lemma \[cyc\], a longest cycle in $H_n$ has at most $c(n)$ vertices. We let $p(n) = c(n) + 2 + 2( c(0) + c(1) + \dots + c(n-1))$ and $w(n) = 22^{n+1} + 2(1 + 22 + \dots + 22^n)$. For the sake of induction, we show that every path in $H_n$ has at most $p(n)$ vertices, and furthermore that it contains at most $w(n)$ white vertices (a similar idea was used in [@update]). We note that the claim is satisfied for $n = 0$, and we proceed by induction on $n$. We let $P$ be a path in $H_n$, and we consider suppressing vertices of $P$ as follows. For every newly added copy of $H_0$, we suppress all but one vertex of the copy and we replace the remaining vertex (if there is such) by the corresponding replaced vertex of $H_{n-1}$; and we let $P'$ be the resulting graph. Since the neighbourhood of every replaced vertex induces a complete graph, $P'$ is a path; and we view $P'$ as a path in $H_{n-1}$. By the hypothesis, $P'$ contains at most $w(n-1)$ white vertices. Thus, $P$ visits at most $w(n-1)$ of the newly added copies of $H_0$.
Similarly, we choose an arbitrary newly added copy of $H_0$, and we suppress all vertices of $P$ not belonging to this copy. Since $\{ u_1, u_2, u_3 \}$ induces a complete graph, the resulting graph is a path in $H_0$ (possibly empty or trivial). Considering such paths for all newly added copies of $H_0$, and considering the set of all their ends, we note that at most two white vertices belong to this set. Hence, in total these paths contain at most $63 \cdot w(n-1) + 2$ vertices. We note that $$p(n) = p(n-1) - w(n-1) + 63 \cdot w(n-1) + 2.$$ Thus, $P$ has at most $p(n)$ vertices. Furthermore, we note that $P$ contains at most $w(n) = 22 \cdot w(n-1) + 2$ white vertices. To conclude the proof, we extend the earlier observation as follows. In fact, there are paths in $H_0$ containing $22 + z$ white and all non-white vertices such that all non-white ends belong to $\{ u_1, u_2\}$. Using these paths, we observe that $H_n$ has a cycle on $c(n)$ vertices and a path on $p(n)$ vertices.
On *k*-trees of toughness greater than one {#s8}
==========================================
To conclude the paper, we remark that for every $k \geq 4$, there are $k$-trees of toughness greater than $1$ whose longest paths are relatively short. For brevity, we omit enumerating the exact length of these paths.
We consider the $1$-tough $3$-trees $H_n$ given by Proposition \[short79\]. Clearly, adding a universal vertex to a $k$-tree gives a $(k+1)$-tree. For every $k \geq 4$ and every $n \geq 0$, we let $H_{n,k}$ denote the graph obtained by adding $k-3$ universal vertices to $H_n$; and we note that $H_{n,k}$ is a $k$-tree of toughness greater than $1$.
We consider a path in $H_{n,k}$. We remove the universal vertices of $H_{n,k}$ from this path, and we view the resulting forest (whose components are paths) as a subgraph of $H_n$. By Proposition \[short79\], every path of this forest is relatively short. Consequently, we observe that for every $k \geq 4$, there exists $n_0$ such that if $n \geq n_0$, then a longest path in $H_{n,k}$ is relatively short. (We note that the same idea can be applied to the graphs constructed in [@chpl].)
Acknowledgement {#acknowledgement .unnumbered}
===============
The author would like to thank Jakub Teska for his mentorship and for inspiring discussions (which led to a weaker version of Theorem \[0\]) which partly motivated this study, and to thank the anonymous referees for their helpful suggestions and comments.
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[^1]: Department of Mathematics, Institute for Theoretical Computer Science, and European Centre of Excellence NTIS, University of West Bohemia, Pilsen, Czech Republic. Email: `[email protected]`.
[^2]: The research was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports and by the project 17-04611S of the Czech Science Foundation.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We introduce quantum dimer models on lattices made of corner-sharing triangles. These lattices includes the kagome lattice and can be defined in arbitrary geometry. They realize fully disordered and gapped dimer-liquid phase with topological degeneracy and deconfined fractional excitations, as well as solid phases. Using geometrical properties of the lattice, several results are obtained exactly, including the full spectrum of a dimer-liquid. These models offer a very natural - and maybe the simplest possible - framework to illustrate general concepts such as fractionalization, topological order and relation to $\mathbb{Z}_2$ gauge theories.'
author:
- 'G. Misguich'
- 'D. Serban'
- 'V. Pasquier'
title: ' Quantum Dimer Model on the Kagome Lattice: Solvable Dimer Liquid and Ising Gauge Theory'
---
Quantum dimer models (QDM) were introduced by Rokhsar and Kivelson [@rk88] in the context of resonating valence-bond (RVB) theories for the high-temperature superconductors [@anderson87]. Such models are expected to describe the dynamics of singlet bonds (dimer) in quantum disordered spin-$\frac{1}{2}$ antiferromagnets. They can describe two generic phases: spin-liquids where the system breaks no symmetry at all and dimer (or valence-bond) crystals where long-range dimer-dimer correlations develops. Recently a genuine liquid phase with a finite correlation length was found in a QDM by Moessner and Sondhi on the triangular lattice [@ms01]. Such liquid states have attracted a lot of interest because they display both fractional excitations and topological order [@wen91]. While fractionalization could play an important role in some theories of high-temperature superconductors [@anderson87; @sf01], the topological properties of these liquid states have been proposed as possible devices to implement quantum bits for quantum computations [@kitaev; @ioffe02].
In this Letter we introduce QDM which realize such a dimer-liquid phase. Due to the geometric properties of lattices made of corner-sharing triangles (the simplest two-dimensional example being the kagome lattice [^1]), these models are solved exactly. As for the solvable point of the QDM on the square [@rk88] and triangular [@ms01] lattices (see also Refs. [@ns01] and [@bfg01]), the ground-state is the equal-amplitude superposition of all dimer coverings in a given topological sector [^2]. Such state has been first considered by Sutherland [@s88] (SRK state in the following) and is the prototype of resonating valence-bond (RVB) state. However, our model has several important differences with previous analogs : 1) Not only the ground state but all excited states wave functions are known. Elementary excitations are (pairs of) non interacting and gapped vortices [@rc89k89] (called [*visons*]{} in the recent literature [@sf01]). 2) The model can be solved on any geometry: torus, discs or spheres. This allows to investigate the interplay between topology, ground-state degeneracy and elementary excitations in an very simple way. 3) Dimer-dimer correlations are strictly zero above one lattice spacing. This makes this SRK wave function the most possible disordered dimer liquid state. 4) This state is [*inside*]{} the liquid phase, it is not lying at the phase boundary with a crystalline phase. 5) It is known [@msf02] that QDM can be obtained as special limits of ${\mathbb{Z}_2}$ gauge theories, the gauge variable being the dimer number on a bond. Here we show a complete equivalence with a ${\mathbb{Z}_2}$ gauge theory. This allows to investigate the confinement transition which goes with a dimer crystallization in a simple QDM which exhibits a liquid to solid transition accompanied by a vison condensation.
For all these reasons the model we introduce is more than a toy model but the simplest possible RVB liquid. It is a “free dimer liquid” point in the short-ranged RVB phase. It shows the generic properties which characterize that state of matter but almost any quantity can be computed exaclty. It is a natural starting point for perturbative expansions toward more realistic models.
[*Medial lattice construction*]{}.— The dimer models described in this Letter can be defined on lattices made of corner-sharing triangle, constructed in the following way. Let $H$ be a trivalent lattice (each site has three neighbors). The lattice $K$, where the dimers live, is the medial lattice of $H$, [*i.e.*]{} the sites of $K$ are the midpoints of the bonds of $H$ (Fig. \[medial\]). If $H$ is the honeycomb lattice, $K$ is the kagome lattice (Fig. \[KagHexVison\]a). In the following, unless mentioned otherwise, we use kagome for simplicity, where plaquettes of $H$ are hexagons. We stress however that all results can be generalized straightforwardly to other lattices (squagome [@sg01] lattice for instance, as well as one-dimensional examples).
[*Pseudospin representation*]{}.— Let us begin with the definition of a simple dimer model. For each hexagon $h$, we define an operator $\sigma^x(h)$ as the sum of all possible kinetic energy terms involving $h$ only: $$\sigma^x(h)=\sum_{\alpha=1}^{32}
\left|d_\alpha(h) \right>\left< \bar{d}_\alpha(h)\right|
+
\left| \bar{d}_\alpha(h)\right>\left<d_\alpha(h)\right|
\label{eq:sigmax}$$ The sum runs over the $32$ loops on kagome which enclose a single hexagon and around which dimers can be moved (see Ref [@ze95] for an explicit list). The shortest loop is the hexagon itself, it involves 3 dimers. 4, 5 and 6-dimers moves are also possible by including 2, 4 and 6 additional triangles (the loop length must be even). The largest loop is the star. For each loop $\alpha$ we associate the two ways dimers can be placed along that loop: $\left|d_\alpha(h)\right>$ and $\left|\bar{d}_\alpha(h)\right>$.
For a given dimer covering $\left|D\right>$, all the kinetic operators in the sum but one annihilate $\left|D\right>$. As a consequence, $\sigma^x(h)^2=1$. One can further check that these operators flip the pseudospin variables $\sigma^z(h)$ introduced by Elser and Zeng [@ez93; @ze95] (EZ) to label dimer coverings [^3]. It is important to note that $\sigma^z$ operators depend on the choice of a reference ${\left|D_0\right>}$ and are not local, unlike $\sigma^x$. The $\sigma^x(h)$ [*commute with each other*]{}. This is not obvious from Eq. \[eq:sigmax\] and it is most easily demonstrated in terms of the arrow representation that we introduce below.
[*Arrow representation*]{}.— A correspondence between dimer coverings on the kagome lattice and [*sets or arrows*]{} as illustrated in Fig. \[KagHexVison\]a was introduced by Elser and Zeng [@ez93]. Each arrow has two possible directions: it points toward the interior of one of the two neighboring triangles. If site $i$ belongs to a dimer $(i,j)$ its arrow must point toward the triangle the site $j$ belongs to. Consider a triangle without any dimer, this arrow rule implies that it will have three outgoing arrows. Other triangles will have two incoming arrows and one outgoing arrow. In other words, the number of outgoing arrows is constrained to be odd.
The number of dimer coverings is $2^{N/3+1}$ where $N$ is the number of sites [@hw88elser89; @ez93]. $K$ has $N$ sites and $N$ arrows, $2N/3$ triangles and one constraint per triangle. However only $2N/3-1$ constraints are independent because their product for all triangles is equal to one. We are left with $N/3+1$ Ising degrees of freedom. The existence of this arrow representation is a central reason for which QDM considerably simplify on these lattices. For example $\sigma^x(h)$ translates very simply in the arrow representation: it flips the six arrows sitting around hexagon $h$, which clearly conserves the constraint for all triangles and commutes from hexagon to hexagon.
[*Rokhsar-Kivelson point*]{}.— Consider the following Hamiltonian: $$\mathcal{H}_0=-\Gamma\sum_{h} \sigma^x(h)
\label{eq:Hsigmax}$$ where the sum runs over hexagons (sites of the dual of $H$ in general). Although very simple in the pseudospin variables, this Hamiltonian is not obviously solvable when written with dimer operators. In the pseudospin variables the ground-state is a fully polarized ferromagnet in the $x$ direction, which is the sum of all pseudospin configurations in the EZ $\sigma^z$ basis. Back to dimers, this is nothing but the sum of all dimer configurations in a given topological sector, that is a SRK wave-function [@rk88]. The ground-state appears to be unique in each topological sector, which gives a global 4-fold degeneracy on the torus.
[*Correlations*]{}.— Correlations in the SRK state of the kagome lattice are particularly simple: irreducible dimer-dimer correlations are strictly zero when their corresponding triangles do not touch. The arrows on two bonds are independent provided they are not involved in a common constraint, that is a common triangle. As a result, dimer on the kagome lattice are the most possible disordered: they are independent above a finite distance.
[*Gap*]{}.— The whole spectrum is known. The $\sigma^x$ operators commute from hexagon to hexagon but physical dimer states must satisfy $\prod_h\sigma^x(h)=1$. This constraint comes from the arrow representation since $\prod_h\sigma^x(h)$ flips all the arrows [*twice*]{} and therefore keeps all dimerizations unchanged. The first excited state appears not to be a single but a [*pair*]{} of flipped hexagons with energy cost $\Delta=4\Gamma$.
[*Visons*]{}.— Despite of the simplicity of the model in the pseudospin variables, its excited states are not local when expressed with dimer degrees of freedom. A $\sigma^x(h)=-1$ hexagon is a vortex excitation (also called [*vison*]{} [@sf01]). Consider a string which goes from an hexagon $a$ to and hexagon $b$ (see Fig. \[KagHexVison\]b) and let $\Omega(a,b)$ be the operator which measures the parity $\pm1$ of the number of dimers crossing that string. $\Omega(a,b)$ commutes with all $\sigma^x(h)$, except for the ends of the string: $\sigma^x(a)\Omega(a,b)=-\Omega(a,b)\sigma^x(a)$. A dimer move changes the sign of $\Omega(a,b)$ if and only if the associated loop crosses the string an odd number of times, which can only be done by surrounding one end of the string. This shows that $\Omega(a,b)$ flips the $x$ component of the pseudospin in $a$ and $b$ [^4] and $\Omega(a,b)\left|\psi\right>$ is precisely the excited state of energy $4\Gamma$ discussed above. Visons appear to be perfectly localized in this model.
[*Spinons*]{}.— One can consider the system with two [*static*]{} unpaired sites (spinons or holes). As for others QDM at a SRK point [@rk88; @ms01], the sum of all dimerizations in a given sector remains an exact eigenstate. The energy turns out to be independent of the relative distance between spinons, which is a strong indication that spinons would be also deconfined if they had kinetic energy. Eventually, we note that taking a spinon around a vison will change the wave function by a $-1$ factor, as expected [@rc89k89].
[*Visons and topology*]{}.— On a closed surface of genus $g$, the spectrum has a degeneracy given by the number of topological sectors $2^{\rm 2g}$ and excitations are [*pairs*]{} of visons, as already mentioned. When the sample has edges, the constraint $\prod_h\sigma^x(h)=1$ is not valid anymore. To handle this case we introduce $\sigma^x(\tilde{h})$ which flips the arrows along the edge and which restores $\sigma^x(\tilde{h})\prod_h\sigma^x(h)=1$. The excitations are still pairs of visons but one vison can be located in the hole [@sf01]. In this case the gap reduces to $2\Gamma$. In an cylinder geometry, as in Ref. [@ioffe02] for instance, we have two sectors and a doubly degenerate spectrum. It is interesting to note that this dimer liquid has no low-energy edge states.
[*Liquid - solid transition*]{}.— We consider a new QDM which is a generalization of Eq. \[eq:Hsigmax\]: $$\mathcal{H}_1=\mathcal{H}_0 - J \sum_{\left<h,h'\right>}\sigma^z(h)\sigma^z(h')
\label{eq:ising}$$ where the second sum runs over pairs of neighboring hexagons. The $\sigma^z$ operators are those defined by EZ, they depend on the choice of a reference dimerization ${\left|D_0\right>}$. A term $\sigma^z(h)\sigma^z(h')$ is local, it is $=1$ if the arrow which is in between $h$ and $h'$ is in the same position as in the reference state and $-1$ otherwise. Since $\sigma^z(h)\sigma^z(h')$ is equivalent to $\Omega(h,h')$, we see that such a term allows to create, annihilate and move visons in the system. From another point of view, the $J$ term in Eq. \[eq:ising\] counts the number of arrows to be flipped to recover the reference state. If $J$ goes to $+\infty$, this Hamiltonian obviously selects the reference state as the ground-state and dimers are completely frozen. In the pseudospin language $\mathcal{H}_1$ is an Ising ferromagnet in transverse field, which displays a second order phase transition at a critical value of $\Gamma$ separating two phases: a ferromagnetic phase with $\left<\sigma^z\right>>0$ and paramagnetic phase with $\left<\sigma^z\right>=0$. From what we know of the $\Gamma=0$ and $\Gamma\to+\infty$ limits we can identify the first one with a dimer solid and the second one with the liquid. From the Ising point of view the solid phase is characterized by $\left<\sigma^z\right>>0$. The two ferromagnetic Ising states ($\left<\sigma^z\right>=\pm1$) correspond to the same dimer state on a closed surface but differ along the edge for open systems. The Ising magnetization $\left<\sigma^z(h)\right>$ provides a non-local order parameter for the dimer solid, $\left<\Omega(h,\infty)\right>$, which involves a string going to infinity.
Up to a sign, $\sigma^z(h)$ is the sum of a creation and annihilation operators of a vison. The dimer solidification can therefore be interpreted as the onset of off-diagonal order and macroscopic occupation number (of the zero-momentum state) for the visons. This QDM realizes a condensation of topological defects (“kinks” [@fs78; @kogut] or visons [@sf01]) at the confinement transition.
Notice that although we call it a solid, the large-$J$ phase does not go with a [*spatial*]{} spontaneous symmetry breaking since the $J$ part of $\mathcal{H}_1$ depends on an arbitrary reference state through the $\sigma^z$ operators; this term acts as an external potential which tends to pin the dimers along the reference state.
[*${\mathbb{Z}_2}$ gauge theory*]{}.— The model of Eq. \[eq:ising\] is the dual of a ${\mathbb{Z}_2}$ gauge theory [@wegner] (in its Hamiltonian formulation) where the gauge degrees of freedom $\tau^z(i)$ live on the bonds of $H$ ([*i.e.*]{} sites of $K$). By definition $\tau^z(i)$ is the operator which flips the arrow at site $i$ and gauge-invariant observable are made of products of $\tau^z$ around closed loops. Spatial gauge transformations require the $x$ component, which we define with respect to the reference state ${\left|D_0\right>}$: $\tau^x(i)=1$ if the arrow $i$ has the same orientation as in ${\left|D_0\right>}$ and $\tau^x(i)=-1$ otherwise. For every site of $H$ (every triangle of $K$) the constraint reads $\tau^x(i_1)\tau^x(i_2)\tau^x(i_3)=1$ where $i_1$, $i_2$ and $i_3$ are the bonds of $H$ emanating from that site. This expresses the fact that physical states must be gauge invariant. This shows a one-to-one correspondence between physical state of the gauge theory and dimer coverings of $K$ and the redundancy in the gauge theory is solved by the dimer coverings. We wish to express $\mathcal{H}_0$ with the gauge degrees of freedom. Since $\sigma^x(h)$ operator flips all the arrows around $h$, it becomes the plaquette operator: $$\sigma^x(h)=\prod_{i} \tau^z(i)
\label{eq:tauz}$$ where the product runs over the bonds of $H$ surrounding $h$. The Ising interactions can be written $$\sigma^z(h)\sigma^z(h')=\tau^x(i)
\label{eq:taux}$$ where $i$ is the common link between $h$ and $h'$ (see Fig. \[medial\]). As a result the QDM of Eq. \[eq:ising\] translates into the Hamiltonian of a ${\mathbb{Z}_2}$ gauge theory (in continuous time), which is a manifestation of the well-known duality between ${\mathbb{Z}_2}$ gauge theories and Ising models [@wegner; @kogut]. Such a ${\mathbb{Z}_2}$ gauge theory has a confined and deconfined phases. A classical result is that they can be distinguished through the expectation value of a gauge-invariant Wilson loop [@wegner; @kogut] $$W(\omega)=< \prod_{i\in \partial\omega} \tau^z(i) >
=
< \prod_{i\in \omega} \sigma^x(i) >
\label{eq:W}$$ where the first product runs over a close loop $\partial\omega$ which surrounds the area $\omega$. $W(\omega)$ changes from a [*perimeter law*]{} $\sim\exp(-|\partial\omega|)$ to an [*area law*]{} $\sim\exp(-|\omega|)$ when going from the deconfined at large $\Gamma$ to the confined phase at large $J$. The right-hand side of Eq. \[eq:W\] expresses $W(\omega)$ as a correlation function for the dimer problem. $W(\omega)$ moves dimers along the loop $\partial\omega$ and the mapping to the gauge theory tells us that its long-distance behavior characterizes the liquid and frozen phases.
[*Liquid - crystal transition*]{}.— $\mathcal{H}_1$ has the disadvantage of depending on a specific reference state. To remedy for that we introduce a QDM which is reference free and restores the full lattice symmetry: $$\mathcal{H}_2=\mathcal{H}_0
- J \sum_{h}s^z(h)
\label{eq:h2}$$ where $s^z(h)=\pm1$ is diagonal in the dimer basis and counts a factor $-1$ per anticlockwise arrow around the hexagon $h$. In the $J\to\infty$ limit the system selects $s^z(h)=1$ everywhere. This can be achieved if the system spontaneously breaks the translation symmetry and crystallizes in an ordered pattern of six-dimer “stars” described in Ref. [@sm02]. These ordered states are degenerate with others ($\sim2^{3L}$ where $L$ is the linear size) obtained by shifting the dimers along any straight line.
We have looked numerically (diagonalizations up to 54 kagome sites) at $\mathcal{H}_2$ and found evidence for a single and continuous transition from the liquid to the star crystal at $J\simeq\Gamma$. This appears as a collapse of the first excitation of the liquid, which then transforms into a degenerate ground-state of the crystal phase. We claim that the critical point is exactly at $J=\Gamma$ from a duality argument. From the arrow representation it is clear that $\sigma^x(h)$ and $s^z(h')$ commute except if $h$ and $h'$ are neighbors, in which case they anticommute. Indeed the algebraic relations of $\sigma^x$ and $s^z$ are completely symmetric. In particular, we observed numerically that the ground-state energy is [*exactly*]{} symmetric with respect to the exchange of $\Gamma$ and $J$ (this is not true for all the eigenstates) and the critical point must lie at the self-dual point $J=\Gamma$.
[*Conclusions*]{}.— The kagome lattice has the remarkable property that dimer coverings correspond to the physical states of an Ising model on the triangular lattice and, by duality, of a ${\mathbb{Z}_2}$ gauge theory on the hexagonal one. Exploiting this, we introduced QDMs for which exact results are derived. In particular, we obtained for the first time the full spectrum of a QDM realizing a dimer liquid phase. It explicitly realizes fractionalized excitations and topological order. Through several models we showed that QDM on the kagome lattice are very simple and natural tools to investigate the connexions between frustrated magnets, RVB physics and spin-charge separation.
[*Acknowledgments*]{}.— We are grateful to M. Gaudin, K. Mallick, R. Moessner, C. Lhuillier and M. Feigel’man for several fruitful discussions. Numerical diagonalizations of QDM models were done on the Compacq alpha server of the CEA under project 550.
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[^1]: The QDM discussed in this work [*do not*]{} aim at describing the physics of the spin-$\frac{1}{2}$ kagome antiferromagnet. We are currently investigating a different QDM for this later problem.
[^2]: Dimer coverings are grouped in topological sectors by considering their transition graphs, that is the set of loops obtained by superposing two coverings on top of each other. Two coverings with only topologically trivial loops are in the same sector.
[^3]: EZ pseudospins variables are different from the well-known mapping \[M. E. Fisher, J. Math. Phys. [**7**]{}, 1776 (1966)\] used to solve the Ising model on a planar lattices in terms of dimers.
[^4]: Up to a global sign (reference dependent) $\Omega(a,b)$ is equal to $\sigma^z(a)\sigma^z(b)$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A growing number of shock compression experiments, especially those involving laser compression, are taking advantage of *in situ* x-ray diffraction as a tool to interrogate structure and microstructure evolution. Although these experiments are becoming increasingly sophisticated, there has been little work on exploiting the textured nature of polycrystalline targets to gain information on sample response. Here, we describe how to generate simulated x-ray diffraction patterns from materials with an arbitrary texture function subject to a general deformation gradient. We will present simulations of Debye-Scherrer x-ray diffraction from highly textured polycrystalline targets that have been subjected to uniaxial compression, as may occur under planar shock conditions. In particular, we study samples with a fibre texture, and find that the azimuthal dependence of the diffraction patterns contains information that, in principle, affords discrimination between a number of similar shock-deformation mechanisms. For certain cases we compare our method with results obtained by taking the Fourier Transform of the atomic positions calculated by classical molecular dynamics simulations. Illustrative results are presented for the shock-induced $\alpha$–$\epsilon$ phase transition in iron, the $\alpha$–$\omega$ transition in titanium and deformation due to twinning in tantalum that is initially preferentially textured along \[001\] and \[011\]. The simulations are relevant to experiments that can now be performed using 4th generation light sources, where single-shot x-ray diffraction patterns from crystals compressed via laser-ablation can be obtained on timescales shorter than a phonon period.'
author:
- David McGonegle
- Despina Milathianaki
- Bruce A Remington
- Justin S Wark
- Andrew Higginbotham
title: 'Simulations of [*in situ*]{} x-ray diffraction from uniaxially-compressed highly-textured polycrystalline targets'
---
Introduction
============
The response of matter to shock compression has been a subject of study for more than a half a century [@Rice1958; @Duvall1963; @Davison1979; @Swegle1985; @Kalantar2005], and early in this field of research it was recognised that the high transient stresses to which materials could be subject caused them to yield and deform plastically, and, in certain circumstances, to undergo rapid polymorphic phase transitions [@Minshall1955; @Bancroft1956; @Fowles1961; @Hayes1974]. An understanding of the pertinent physics operating at the lattice level that underpins the material response has long been sought, and has been a strong motivator in the development of lattice-level and structural measurement techniques with sufficient temporal resolution to interrogate detailed material response during the deformation process itself. Of particular relevance to the work presented here are recent advances in ultra-fast x-ray diffraction. The first diffraction patterns of crystals subjected to shock compression taken during the passage of the shock itself had exposure times of ten of nanoseconds [@Johnson1970a; @Johnson1971; @Johnson1972; @Johnson1972a] . Since this pioneering work significant progress has been made by the use of x-ray sources based on diode-technology [@Germer1979], synchrotron emission[@Bilderback2005], and the use of x-rays emitted from plasmas created by irradiation with high-power lasers [@Wark1987; @Wark1989]. More recently however, the development of so-called 4th generation light sources, such as the Linac Coherent Light Source (LCLS), has allowed high-quality single-shot diffraction patterns to be obtained in just a few tens of femtoseconds – freezing motion on a timescale faster than the shortest phonon period in the system.[@Milathianaki2013]
The transient x-ray sources mentioned above have been used to further our understanding of the response of both single-crystals and polycrystalline matter to both shock and quasi-isentropic compression [@Johnson1971; @Johnson1972a; @Wark1989; @Whitlock1995; @Gupta1999; @Rigg1998; @D'Almeida2000; @Loveridge-Smith2001; @Kalantar2003; @Kalantar2005; @Hawreliak2006; @Murphy2010a; @Hawreliak2011; @Rygg2012; @Comley2013; @Milathianaki2013]. A number of x-ray diffraction techniques have been developed to monitor material response, including divergent beam geometry[@Kalantar2003a], white-light Laue diffraction[@Suggit2010], Debye-Scherrer diffraction[@Johnson1970a; @Johnson1971; @Johnson1972; @Johnson1972a] and the use of energy-dispersive single-photon counting[@Higginbotham2014a]. While studies of both single-crystals and polycrystalline matter have been undertaken, in the case of polycrystalline samples very little attention has yet been paid to the role of texture in these uniaxial compression experiments - that is to say the distribution function of grain orientations within a particular polycrystalline specimen. A number of manufacturing methods, such as rolling and epitaxial growth, result in characteristic textures due to the way in which the materials have been processed. This texture has an influence on a range of physical properties such as strength, electrical conductivity and wave propagation [@Wenk2004]. Therefore texture plays an important role in understanding material response.
Furthermore, whilst the texture of a sample will influence its response to rapid uniaxial compression, it will also have a profound influence on the way in which the sample diffracts. As certain grain orientations are more likely to occur than others, the intensity of a particular Debye-Scherrer ring (corresponding to a certain set of Miller indices) will have a strong azimuthal dependence, and this dependence can in turn be used to extract texture information. Indeed, significant static studies of the texture of polycrystalline samples have been undertaken with synchrotron sources for many years. [@Wenk1997; @Wenk2003; @Wenk2004; @Wenk2005; @Ischia2005; @Barton2012; @Vogel2012]. Wenk and co-workers provide an overview of the use of synchrotrons in such texture analysis [@Wenk2003]. While the texture of a material is often represented by a set of pole figures which can be measured directly via x-ray diffraction, prediction of anisotropic material properties requires knowledge of the full orientational distribution function (ODF). The ODF gives the probability of a crystallite having an given orientation, therefore provides a complete description about the texture of the sample. Pole figures, being a 2D projections of the ODF, result in some texture information being lost, although methods have been developed to obtain an approximate ODF from pole figures [@Wenk2003].
Given that the preferred orientation defined by texture links both the diffraction patterns observed, and the sample response, it is logical to question whether specific information can be gleaned via [*in situ*]{} diffraction studies of samples with known, well-defined texture. For example, bulk rotations of the crystal lattice, or changes in the crystal structure, such as Martensitic phase changes, will result in an altered texture that could be used to distinguish between different mechanisms of atomic rearrangement. This reorientation has been observed in previous work using both neutron sources [@Brown2005] and synchrotrons[@Wenk2003; @Vogel2012], although only at relatively modest pressures compared with those we are interested in here.
Ideally one would wish to understand the detailed response to shock compression of samples as a function of their ODF, and to then predict the resultant diffraction patterns. We outline in section \[S:Method\] the method by which this could in principle be done. However, in terms of the results of particular simulations, our present goal is more modest: we restrict ourselves to crystals that are highly fibre-textured - that is to say all of the grains within the sample initially have very similar orientations with respect to a single axis (the fibre axis), only deviating slightly in angle from that of a high-symmetry direction. As all of the grains have similar orientations along a given axis, which we also take to be the axis of compression, we might expect that we can, to a reasonable approximation, model certain aspects of their response to shock or quasi-isentropic compression using single-crystal parameters. In particular, when determining how the material will respond to compression and shear, we assume that the elastic stresses, supported by elastic strains, can be calculated by using elastic constants appropriately chosen to mimic single crystal response along the pertinent directions. In terms of x-ray diffraction, however, the finite range of orientations determined by the ODF is such that a monochromated, non-divergent incident x-ray beam can diffract from a reasonably large subset of the grains, both in the shocked and un-shocked case. A similar approach to that outlined above has recently been used to observe, via femtosecond diffraction, the ultimate compressive strength of copper subjected to shock compression at strain-rates of order 10$^9$ s$^{-1}$[@Milathianaki2013]. We shall show that breaking the symmetry of the system, such that the direction of the incident x-rays and the target normal (parallel to the compression direction) are no longer antiparallel, allows us to determine specific information about the system under study.
The layout of the paper is as follows. In section \[S:Method\] we describe the method by which we determine the analytic diffraction pattern as a function of compression, and the angle that the sample normal makes to the x-ray beam. The background to the molecular dynamics simulations are outlined in section \[S:MD\] before, in section \[S:Results\] providing the results for simulations for the $\alpha$–$\epsilon$ phase transition in iron, the $\alpha$–$\omega$ transition in titanium, and deformation due to twinning in tantalum. We then conclude with a discussion of the results, and suggestions for experiments and further work.
Method {#S:Method}
======
We have previously shown how to calculate the position of Debye-Scherrer rings from a polycrystal strained by an arbitrary deformation gradient in the Voigt limit [@Higginbotham2014], and for the sake of completeness we restate briefly the results here. We envisage a planar sample orientated such that its surface normal $\bf{n}$ lies along the $z^{\prime}$ direction. As mentioned above, we allow symmetry to be broken, such that the direction of incidence of x-rays for wavevector $\mathbf{k_0}$ can be non-parallel to $\mathbf{n}$, and along the direction $z$. The rotation matrix $\mathbf{R}$ transforms from the unprimed (x-ray) to primed (target) co-ordinate system. Debye-Scherrer diffraction occurs when the Laue condition is met: $\mathbf{G}=\mathbf{R}{^T}{\mathbf{G}}^{\prime}=\mathbf{k}-\mathbf{k_0}$, where $\mathbf{k}$ is the wave-vector of the diffracted x-ray. Deformation of the crystal in real space is represented by the arbitrary deformation gradient $\mathbf{F}$ applied in the target (primed) co-ordinate system, which corresponds to the analogue in reciprocal space of $({\mathbf{F}}^{T})^{-1}$. With these relations we see that the reciprocal lattice vector for an unstrained crystal in x-ray co-ordinates, $\mathbf{G_0}$, is transformed under deformation to a new reciprocal lattice vector $\mathbf{G}$, where $$\begin{aligned}
\mathbf{G_0} & = \mathbf{R}{^T}\mathbf{F}{^T}\mathbf{RG} = \mathbf{\boldsymbol\alpha G}, \label{eg:transform}\end{aligned}$$ where $\mathbf{\boldsymbol\alpha} = \mathbf{R}{^T}\mathbf{F}{^T}\mathbf{R}$. By noting that if a plane is to diffract once the sample has been strained, it must meet the Laue condition, and that for the unstrained case, $\left|\mathbf{G_0} \right| = \frac{2\pi}{d_0}$, it is possible to solve equation \[eg:transform\] to yield the Bragg angle as a function of azimuthal angle around the $z$-axis, $\phi$, for an arbitrary deformation gradient – that is to say we know the direction of the scattered wavevector, $\mathbf{k}$, and hence can determine via simple ray-tracing where the diffraction will impinge on a distant detector. In Higginbotham *et al.* [@Higginbotham2014], diffraction patterns for numerous deformation gradients and target geometries (with respect to the x-ray beam) are shown. However, in these cases the sample was assumed to be isotropic in texture, and thus satisfying equation \[eg:transform\] was deemed a sufficient condition for diffraction to occur.
However, a non-isotropic ODF places further constraints on the possibility of diffraction for a given Bragg angle and azimuthal position around the Debye-Scherrer ring, as the ODF provides the measure of the probability of finding a crystal with a given $\mathbf{G_0}$ existing in the original unstrained sample. The route forward for simulating diffraction for a crystal with known original ODF under a known arbitrary deformation gradient is now clear: we use the ODF to determine the probability of a given crystallite having the appropriate $\mathbf{G_0}$ existing within the sample, we then determine $\mathbf{G}$ from equation \[eg:transform\] (and hence $\mathbf{k}$ from the Laue condition), and use ray-tracing to propagate the diffracted beam to the detector assuming that the intensity is proportional to the probability of finding the original $\mathbf{G_0}$, as determined by the ODF, and taking into account factors such as multiplicity.
Although the above approach is general, within the rest of this paper we restrict our study to the particular case of a simple fibre texture where the crystallites have nearly identical orientation in the axial direction, but close to random radial orientation. Our motivations for this are due to the fact that this allows us to treat the mechanical response of the polycrystal to be well-approximated by that of a single crystal with orientation aligned with the fibre axis and that the technique of fibre diffraction under ambient conditions is well established [@Polanyi1921; @Polanyi1923]. Furthermore, recent experiments using femtosecond x-rays created with $4^{\textrm{th}}$ generation light sources to diffract from uniaxially compressed samples have employed targets with this type of texture,[@Milathianaki2013] and the thin films that have hitherto been used in these experiments often grow with such preferential orientation.
Fibre textured samples have a greatly simplified ODF. For the case of perfect fibre texture, each crystallite has a single crystallographic direction, the fibre orientation, $\mathbf{v_1}$, associated with the reciprocal lattice vector $(h_1 ,k_1 , l_1)$, which for all crystallites are aligned parallel to the sample normal, $\mathbf{n}$. Each grain is then deemed to possess a random orientation when rotated about the axis, $\mathbf{v_1}$. If we consider a particular plane within a crystallite, with miller indices $(h_2 ,k_2 , l_2)$ (which are the same set of miller indices associated with $\mathbf{G_0}$) to which the reciprocal lattice vector is $\mathbf{v_2}$, then the value of $\mathbf{\hat v_1} \cdot \mathbf{\hat v_2}$ will be a constant. However, when $z$ and $z^{\prime}$ are non-parallel, $\mathbf{\hat{v}_1} \cdot \mathbf{\hat{G}_0}$ varies as a function of azimuthal angle around $z$. Thus, for perfect fibre texture the Debye-Scherrer pattern is not a ring pattern, but an array of points defined by simultaneously satisfying equation \[eg:transform\], as well as the condition $\mathbf{\hat{v}_1} \cdot \mathbf{\hat{G}_0} = \mathbf{\hat{v}_1} \cdot \mathbf{\hat v_2}$. Note that this condition does not take into account multiplicity, and therefore one must consider this condition for each member of $\mathbf{v_2}$ in the $\{h_2 ,k_2 , l_2\}$ family, since they do not necessarily result in the same values of $\mathbf{\hat{v}_1} \cdot \mathbf{\hat v_2}$.
However, real fibre-texture samples contain crystallites that are not perfectly aligned axially, and the volume fraction of crystallites with reciprocal lattice vectors $\mathbf{v_1}$, $P(\mathbf{v_1})$, will be a rapidly decreasing function of $\mathbf{\hat v_1} \cdot \mathbf{\hat v_n}$, where now $\mathbf{\hat v_n} = < \mathbf{\hat v_1}>$. We further assume that for a given volume fraction with a certain $\mathbf{v_1}$, the directions $\mathbf{v_2}$ of the normals to the planes with miller indices $[h_2 ,k_2 , l_2]$ within the crystallites will be arranged randomly azimuthally around the axis $\mathbf{v_1}$, such that the possible orientations of $\mathbf{v_2}$ are simply constrained by $\mathbf{v_2} \cdot \mathbf{v_1}$ being equal to the value it would take for a single crystal: that is to say $P(\mathbf{v_2}) \propto P(\mathbf{v_1})$ subject to the $\mathbf{v_2} \cdot \mathbf{v_1}$ constraint. The diffraction condition is once more defined by simultaneously satisfying equation \[eg:transform\], as well as the condition $\mathbf{\hat{v}_1} \cdot \mathbf{\hat{G}_0} = \mathbf{\hat{v}_1} \cdot \mathbf{\hat v_2}$., but now the intensity of the diffraction is proportional to $P(\mathbf{v_1})$. As the possible directions of $\mathbf{v_1}$ can now vary over a constrained range, our pattern is a series of arcs, rather than points. For the sake of simplicity in our simulations here we assume that the volume fractions of crystallite orientiations are uniform over a small range of angles, i.e. $P(\mathbf{v_1}) = C$, a constant, for $| \arccos (\mathbf{\hat{v}_1} \cdot \mathbf{\hat{G}_0}) - \arccos ( \mathbf{\hat{v}_1} \cdot \mathbf{\hat v_2}) | \le \delta$, and $P(\mathbf{v_1}) = 0$ for $| \arccos (\mathbf{\hat{v}_1} \cdot \mathbf{\hat{G}_0}) - \arccos (\mathbf{\hat{v}_1} \cdot \mathbf{\hat v_2}) | > \delta$. This method effectively finds the intersection between the Ewald and Polanyi spheres [@Polanyi1921; @Polanyi1923; @Stribeck2009], however for the case of anisotropic strains, the Polanyi sphere is deformed into an ellipsoid. Note that by changing the x-ray energy or the sample orientation, which varies the size of Ewald sphere or rotates the Polanyi sphere respectively, the position of the arcs on the Debye-Scherrer ring also change, and that by varying these parameters, different parts of reciprocal space can be interrogated.
Molecular Dynamics Simulations {#S:MD}
==============================
\[MD\_sim\]
One advantage of the fibre texture discussed above is that one knows (to within the texture width) the crystallographic orientation of grains with respect to an applied planar compression front. This is particularly important as crystallite orientation can drastically alter the material response under uniaxial loading [@Murphy2010a; @Dupont2012; @Smith2012; @Zong2014]. In the case that grain size is comparable to sample thickness[@Milathianaki2013], one can approximate the response of the sample as being close to that of a suitably oriented single crystal. This is particularly pertinent if one wishes to compare results with those of molecular dynamics, where state of the art polycrystalline simulations are still generally limited to grain sizes of 5-100nm [@Bringa2005; @Kadau2007; @Jarmakani2008; @Bringa2010; @Gunkelmann2014], far below the grain sizes of typical experimental samples.\
In order to relax the requirements on computational power, we present a method of simulating the response of a fibre textured target by manipulation of a single crystal simulation. We do this by first performing a 3D Fourier Transform (FT) of the computed electron density of the single crystal [@Kimminau2008]. This provides us with a momentum space representation of the lattice which describes the allowed scattering vectors for diffraction.\
Working with the intensity of this FT, we first note that any polar dependence around the compression axis can be neglected due the random rotational distribution of grains in a fibre textured sample. Considering a cylindrical geometry, we therefore produce a 2D representation of the FT, which flattens the data into its $\left(\rho, z\right)$ components, effectively integrating around $\phi$. In the case of a perfect (zero texture width) fibre texture, this 2D representation correctly describes all scattering.\
For the case of finite texture width, one can imagine that the misorientations of the grains are simply related to a rotation about the origin of the 2D representation, and so to mimic the width we sum rotated representations for angles between $\pm\delta$. This new representation necessarily still retains the cylindrical symmetry required for fibre texture, but correctly accounts for the distribution of grain alignments. Of course, as $\delta$ becomes larger than a few degrees, the underlying assumption that all grains react in a similar manner to a well aligned single crystal will break down. However, for this paper we will assume suitably small texture widths of $\approx 5^\circ$ where this approach works well.\
One can now raytrace simulated diffraction patterns directly from this 2D representation by only considering the $\left(\rho, z\right)$ component of the scattering vectors expressed in this cylindrical target geometry.
![(Color online) A simulated ray trace of diffraction from a \[001\] fibre textured polycrystalline HCP Fe formed under shock compression. The blue overlaid lines show the positions of arcs from uniaxially compressed BCC, while the red lines show the positions of arcs from HCP with OR $\left[001\right]_{\textrm{bcc}}\left|\right| \left[2\bar{1}\bar{1}0\right]_{\textrm{hcp}}$, $\left[110\right]_{\textrm{bcc}}\left|\right| \left[0002\right]_{\textrm{hcp}}$. For clarity the x-ray energies used to trace the bcc and hcp overlays were offset by 1%. The corners of the detector are located at a scattering angle $2\mathrm{\theta} = 74.2^{\circ}$.[]{data-label="fig:Fe_HCP"}](Fe.eps){width="0.5\linewidth"}
Results {#S:Results}
=======
We now present results for three different fibre-textured polycrystals subject to uniaxial compression: iron with a \[001\] fibre texture, titanium with \[0001\] fibre-texture, and tantalum with \[001\] and \[011\] fibre-textures. For all three crystals we make assumptions about the deformation mechanisms, and use the approach of section \[S:Method\] to predict diffraction patterns. For two of the cases - iron and tantalum - we also compare our calculations with diffraction patterns simulated using the method of molecular dynamics, as outlined in section \[S:MD\].
$\alpha$–$\epsilon$ Phase Transition in \[001\] Iron
----------------------------------------------------
The $\alpha$–$\epsilon$ phase transition in iron is an example Martensitic transformation, characterised by a collective movement of atoms across distances that are typically smaller than a nearest-neighbour spacing. These type of transitions are well suited to laser compression studies, since the timescales on which they occur are comparable to the short pulses that can be attained in laser experiments. Importantly, for these non-diffusional transitions, an orientational relationship (OR) exists between the two phases. While the OR does not uniquely determine the mechanism by which the phase transition occurs, it can significantly reduce the number of candidate mechanisms. The ability to measure the OR *in situ* is therefore highly desirable.
As an example of determination of an OR, we take the case of \[001\] oriented iron, where the phase transition mechanism is well understood. Molecular dynamic simulations undertaken by Kadau *et al.* aimed to understand iron’s bcc-hcp shock induced phase transition [@Kadau2002]. The results of these investigations were later reproduced with remarkable fidelity in experimental x-ray diffraction studies[@Kalantar2005; @Hawreliak2006]. In both MD and experiment, the OR was described as $\left[001\right]_{\textrm{bcc}}\left|\right| \left[2\bar{1}\bar{1}0\right]_{\textrm{hcp}}$, $\left[110\right]_{\textrm{bcc}}\left|\right| \left[0002\right]_{\textrm{hcp}}$ [@Kadau2002] . One can therefore consider this OR as reorienting a fibre textured sample from $\left(002\right)_{\textrm{bcc}}$ to $\left(2\bar{1}\bar{1}0\right)_{\textrm{hcp}}$.
Following Kadau, we simulate a 100x100x800 cell (288x288x2301Å) iron single crystal shocked along the \[001\] direction by 0.7km $\textrm{s}^{-1}$ piston using the same Voter-Chen potential used in Kadau’s work [@Kadau2002]. For this piston velocity, the material does not reach the 18.4% uniaxial compression needed to create ideal HCP, instead reaching only 13.8% [@Hawreliak2006], resulting in an anisotropically strained HCP structure. A 3D FT was performed on a section of the material behind the shock front. The FT was modified to mimic that of a fibre textured polycrystal, using the method described in section \[MD\_sim\]. Figure \[fig:Fe\_HCP\] shows the resultant ray trace [@Kimminau2008] for a detector in transmission geometry, using a 12keV x-ray source and with the sample normal rotated at an angle $30^{\circ}$ to the incoming x-rays. The overlaid red lines show the predicted diffraction pattern (using the methods of section \[S:Method\]) of strained HCP iron described above, with a c/a ratio of 1.73, a texture direction along $\left[11\bar{2}0\right]_{\textrm{hcp}}$ and a textured width of $5^{\circ}$. The blue lines show the pattern from 13.8% uniaxially compressed BCC. As expected, there is agreement between the raytrace and predictions from the molecular dynamics simulations, supporting the validity of the approach outlined in section \[S:Method\].\
![(Color online) The predicted diffraction pattern using 7.5keV x-rays from the $\omega$ phase of a shocked $\left[0001\right]$ fibre textured Ti sample, with the sample normal rotated at an angle $25^{\circ}$ to the incoming x-rays. The green arcs labelled A and B correspond to the $\left\{10\bar{1}1\right\}$ and $\left\{11\bar{2}0\right\}$ sets of planes respectively. The corners of the diagram are located at a scattering angle $2\mathrm{\theta} = 70.5^{\circ}$.[]{data-label="fig:Ti"}](Ti.eps){width="0.5\linewidth"}
$\alpha$–$\omega$ Phase Transition in \[0001\] Titanium
-------------------------------------------------------
Variant Orientational Relationship Ref
--------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------
I $\left(0001\right)_{\alpha}\left|\right| \left(10\bar{1}1\right)_{\omega}$ , $\left[10\bar{1}0\right]_{\alpha}\left|\right| \left[\bar{1}011\right]_{\omega}$ [@Usikov1973; @Trinkle2003; @Song1995]
II $\left(0001\right)_{\alpha}\left|\right| \left(1\bar{2}10\right)_{\omega}$ , $\left[1\bar{2}10\right]_{\alpha}\left|\right| \left[0001\right]_{\omega}$ [@Usikov1973; @Silcock1958]
Zong $ \left(0001\right)_{\alpha}\left|\right| \left(10\bar{1}0\right)_{\omega}$ , $\left[10\bar{1}0\right]_{\alpha}\left|\right| \left[11\bar{2}3\right]_{\omega}$ [@Zong2014]
: The three ORs that correspond to various proposed mechanisms of the $\alpha - \omega$ phase transition in \[0001\] Titanium
\[tab:TiORs\]
The group IV transition metals titanium (Ti), zirconium (Zr) and hafnium (Hf) have a hexagonal close packed structure ($\alpha$) under ambient conditions, but exhibit a Martensitic phase transition to another hexagonal structure ($\omega$) under high pressure. Although the $\alpha$–$\omega$ transition is well established, the mechanism by which it occurs, and therefore the OR between the phases, is still not fully understood, especially under different loading conditions. A summary of the ORs for various proposed mechanisms is given in table \[tab:TiORs\]. The first two ORs were proposed by Usikov and Zilbershtein [@Usikov1973] in a TEM study of statically compressed Zr and Ti, by arguing that the transition occurs via an intermediate $\beta$ phase, and are usually referred to as Variant I and II. Earlier work by Silcock [@Silcock1958] on the $\omega$ phase in Ti alloys proposed a different direct mechanism which corresponds to the Variant II OR. Later computational work by Trinkle [@Trinkle2003] demonstrated that a new mechanism, known as TAO-1, had a lower energy barrier than that proposed by Silcock. This new mechanism produced the Variant I OR. Experimental work by Song and Gray [@Song1995] observed the OR $\left(0001\right)_{\alpha}\left|\right| \left(10\bar{1}1\right)_{\omega}$, $\left[10\bar{1}0\right]_{\alpha}\left|\right| \left[11\bar{2}3\right]_{\omega}$, although independent analysis re-examining this data led to the conclusions that is actually a subset of Variant I[@Jyoti1997]. We have therefore associated Song and Gray’s work with Variant I in table \[tab:TiORs\].
Molecular dynamics simulations performed by Zong *et al.* [@Zong2014] found that Ti shocked along the $\left[0001\right]$ direction resulted in a mechanism that gave an OR differing from both Variant I and II, as listed in the table. Note that each OR results in a different fibre texture direction for the $\omega$-phase for uniaxial compression of an initially fibre-textured $\alpha$-phase target, and thus a target with the normal tilted with respect to the incident x-ray beam is well suited to provide some discrimination between variants, therefore providing some insight into possible transformation mechanisms.
The work by Zong [@Zong2014] found that Ti shocked along the $\left[0001\right]$ direction by a piston with a velocity of 0.9km $\textrm{s}^{-1}$ resulted in transformation to the $\omega$ phase, with lattice parameters $a_{\omega} = 4.61\,\textrm{\AA}$ and $c_{\omega} = 2.82\,\textrm{\AA}$. Using these values, we calculate the diffraction pattern from the $\omega$ phase of a shock-compressed $\left[0001\right]$ sample of Ti with an angular texture width of $\pm 5^{\circ}$. The sample normal is set at angle $25^{\circ}$ to the incoming x-rays, which have an energy of at 7.5keV. The predicted diffraction patterns for each of the possible variants are shown in Figure \[fig:Ti\]. The blue, red and green lines correspond to the diffraction from $\omega$ material of Variant I, Variant II and the Zong OR respectively, while the dotted black line corresponds to the diffraction from an untextured polycrystalline sample. For clarity, the blue lines have been slightly offset outside the true Debye-Scherrer rings, while the red lines have been slightly offset inside. A clear difference can be seen in the diffraction patterns for the different variants, thus allowing the OR and hence a subset of mechanisms to be determined by the azimuthal position of the diffracting arcs around the Debye-Scherrer ring. It is important to note that the ability to discriminate clearly between all three ORs is not guaranteed, and relies on a judicious choice of both x-ray energy and tilt angle.
Figure \[fig:Ti\] also demonstrates that lines with similar $d$-spacings do not necessarily appear at similar azimuthal positions. For example in an ideal $\omega$ crystal, the $\left\{10\bar{1}1\right\}$ and $\left\{11\bar{2}0\right\}$ planes have the same spacing and are therefore completely unresolvable by powder diffraction from an untextured sample. However, since these planes form different angles to the sample normal, within a textured sample their corresponding arcs appear at different azimuthal angles around the Debye-Scherrer ring, allowing them to be resolved. This is shown in Figure \[fig:Ti\], where the green arcs labelled A correspond to the $\left\{10\bar{1}1\right\}$ set of planes, while the green arc labelled B corresponds to the $\left\{11\bar{2}0\right\}$ set of planes. By resolving these two arcs, we are able to gain information that cannot be obtained via powder diffraction from an untextured sample; in this case on any small departure from the ideal c/a ratio. We note that it is only possible to resolve lines with similar $d$-spacings if the angle between G-vectors of each of the planes and the sample normal is significantly different.
{width="100.00000%"}
Twinning in \[001\] and \[011\] Tantalum
----------------------------------------
Tantalum provides an appealing case to study, owing to its multitude of competing plasticity mechanisms (a combination of dislocation and deformation twinning mediated responses). This is of particular interest, as Debye-Scherrer diffraction in polycrystalline samples with completely random texture cannot distinguish between slip and twinning. Most experimental work into twinning under uniaxial dynamic compression has been performed using either explosive lenses or gas guns[@Murr1997; @Hsiung2000; @Florando2013; @Pang2014]. However, to date, time resolved laser diffraction experiments have failed to yield any evidence of twinning *in situ*, although residual twinning has been observed in laser driven shock recovery experiments[@Lu2012; @Lu2013].
As with phase transitions, the reorientation of the lattice caused by twinning will result in a different crystallographic direction being oriented along the fibre direction. This may, in turn, lead to a signature in the diffraction pattern. A similar technique using neutron diffraction has been used to observe twinning in magnesium, which occurs at much lower pressures than in tantalum [@Brown2005].
Molecular dynamics simulations performed by Higginbotham *et al.* have predicted a significant twinning fraction in \[001\] tantalum under shock compression[@Higginbotham2013b]. In that work, the sample was found to be almost completely twinned when compressed by a piston with a velocity of 0.9 km $\textrm{s}^{-1}$, corresponding to a uniaxial compression in the elastic wave by 18%. The authors noted that after an initial uniaxial compression of 18.4%, twinning could be achieved by shuffling alternating $\left(\bar{1}12\right)$ planes in the $\left<111\right>$ direction. They therefore proposed this to be the mechanism by which the twinning occurred, with the material reaching its final state via elastic relaxation, although they caution that, given the relatively simple nature of the potential used, they do not claim to exactly model what will occur in practice in shocked Ta. However, the observed shuffling provides a possible mechanism for how twinning of bcc materials may occur under shock compression.
{width="100.00000%"}
We repeated the simulations in this work, using a 100x100x800 cell (330x330x2644Å) Ta single crystal, modelled using the EFS potential [@Dai2006], and shocked along the $\left[001\right]$ direction by a piston travelling at 0.9km $\textrm{s}^{-1}$. The per atom structure factor (PASF) [@Higginbotham2013b] was used to distinguish between twinned and untwinned material in the plastically deformed material behind the shock front. A 3D FT was performed on a stable region within the plastically deformed material behind the shock front, as well as the twinned region separately, and both were modified in the way described in section \[MD\_sim\]. The diffraction pattern was simulated assuming a 12keV x-ray source, with the angle between the incoming x-rays and the sample normal being $25^{\circ}$. The results are shown in Figure \[fig:Ta\_001\_Raytrace\]. For the case of slip, while small rotations of the crystal lattice have been observed [@Suggit2012], no large reorientation is expected, and thus the fibre orientation will remain close to $\left[001\right]$. However, by comparing the observed and predicted positions of the diffraction arcs for a $\left[001\right]$ textured sample including the strains described above (shown in Figure \[fig:Ta\_001\_Raytrace\]a), it is clear that there has been significant reorientation of crystallites within the sample, indicative of twinning.
In Figure \[fig:Ta\_001\_Raytrace\]b we plot the predicted diffraction pattern using the methods of section \[S:Method\] assuming that the sample has undergone twinning, and is subject to the longitudinal and transverse strains noted above. In order to find the diffraction pattern resulting from the shuffling mechanism above, one must consider how this affects the lattice vectors of the crystal. The sample is first compressed uniaxially along the z axis by $18.4\%$. The shuffling then has the effect of causing the crystal to be reflected in the $\left(112\right)$ twinning plane. Note that as crystal is uniaxially compressed, this causes the $\left[001\right]$ direction to be reflected to the $\left[111\right]$ direction of the compressed crystal, rather than the $\left[221\right]$ direction expected under hydrostatic conditions (see Figure 6 of Higginbotham *et al.*[@Higginbotham2013b]). Lastly, the twinned material returns towards the hydrostat, by relaxing along the longitudinal direction and compression along the transverse directions in the new rotated coordinates. The final longitudinal and transverse strains were measured to be -0.109 and -0.052 respectively. However, in the textured sample, the effect of a finite texture width must also be considered. In this case, the initial compression occurs at slight angle to the texture direction, which results in a slightly different orientation of the twinning plane. The crystal is then reflected in this altered twinning plane, and relaxes as before. To create the predicted diffraction pattern, the lattice vectors of many crystallite orientations within the desired texture width were deformed by the method given above. These were then used to calculate deformed reciprocal lattice vectors, and thus the resultant diffraction pattern.
Excellent agreement can be seen between the analytic solution, and the MD simulation, demonstrating that twinning has occurred, although slight differences can be observed which are due to the small angle assumptions used in section \[MD\_sim\] to simulate the 3D FT of a fibre textured target. Additionally, there are some very weak arcs in the data corresponding to plastically compressed material (Figure \[fig:Ta\_001\_Raytrace\]a). The ratio of intensities of lines from twin and slip deformed material is indicative of the twin fraction.
While most theoretical work on twinning in Ta has concentrated on the $\left[001\right]$ direction, recent MD studies by Ravelo *et al.* have suggested shocking along the $\left[011\right]$ direction may be more favourable for deformation twinning, due to the lower observed shear stress threshold for twin nucleation[@Ravelo2013]. This agrees with gas gun recovery work on Ta single crystals, which found a significantly higher twin volume fraction in shocks along $\left[011\right]$, compared to the $\left[001\right]$ and $\left[111\right]$ directions[@Florando2013]. Furthermore, as the $\left[011\right]$ direction is the preferred direction for epitaxial growth of fibre textured thin films, this direction is particularly well suited for this technique. The $\left[011\right]$ orientation exhibits two types of $\left\{112\right\}\left<111\right>$ twin systems, which result in different fibre orientations. Under hydrostatic conditions, the first type causes no change in the fibre texture, while the second causes a reorientation to the $\left[\bar{4}11\right]$ direction. However, only the second type has non-zero Schmid factors, and it is therefore assumed that only twins of this type occur. It follows that diffraction arcs corresponding to a $\left[001\right]$ fibre orientation are from untwinned material, while arcs corresponding to fibre orientations close to the $\left[\bar{4}11\right]$ type directions are from the two twinned variants.
We repeated the simulations in this work, using a 100x100x800 cell (330x330x2644Å) Ta single crystal, single crystal, modelled using Ravelo’s Ta1 EAM potential [@Ravelo2013], and shocked along the $\left[011\right]$ direction by a piston travelling at 0.62km $\textrm{s}^{-1}$. Again the PASF was used to distinguish between twinned and untwinned material, and a 3D FT was performed on each region separately. In the twinned material, the $\left[\bar{9}22\right]$ direction of the compressed crystal is close to the compression axis, which is consistent with twinning after an initial uniaxial compression, similar to the \[001\] case. The diffraction pattern was simulated assuming a 10keV x-ray source, with the angle between the incoming x-rays and the sample normal being $45^{\circ}$. The results are shown in Figure \[fig:Ta\_011\_Raytrace\]. Since the twinning mechanism in \[011\] Ta is not well understood, the predicted pattern was produced for the structure found measured with the FT. Again, good agreement is seen between the analytic solution and the MD simulation, although in this case, there are strong arcs corresponding to both twinned and untwinned material, suggesting a significant amount of both are present in the sample.
Discussion
==========
The examples we have given above demonstrate that x-ray diffraction from uniaxially compressed fibre-textured targets can in principle yield information on deformation mechanisms, be they due to phase transformations or twinning. The breaking of the symmetry of the problem, by tilting the normal of the polycrystalline target with respect to the incident x-ray beam allows the encoding of such information in the azimuthal distribution of intensity in the Debye-Scherrer rings. We envisage that the methods that we have outlined here will aid in the design of experiments that have as their goal the elucidation of such mechanisms. It is worth considering, however, that the choice of initial fibre direction is important in determining what structural information can be extracted. In particular it should be noted that for \[001\] fibre oriented Fe and Ta samples, these orientation do not have the lowest surface energy, and thus are not the typical orientations in which thin polycrystalline foils of these materials grow. It may thus be that some effort is required to fabricate suitable samples. This is not an issue for the case of \[0001\] Ti or \[011\] Ta, which are usually grown with these textures. Beyond the four demonstration cases given above, it is clear that further work could concentrate on a variety of different samples and deformation mechanisms. In addition, we believe that the technique may have other advantages for the study of samples subject to shock or quasi-isentropic compression. Owing to the high strain rates present in such experiments, high dislocation densities [@Bringa2006] or small grain sizes under phase transformation may ensue [@Kadau2002; @Hawreliak2006], resulting in broad diffraction peaks that are hard to resolve simply in terms of scattering angle, and thus would not necessarily be easily amenable to study by techniques such as Rietveld refinement. However, tilting of a target and separation of diffraction peaks azimuthally offers a possible route to finding structural solutions under the extreme pressures that can be obtained via laser-ablation. We believe that the technique we have outlined could have application to laser based shock and compression experiments that make use of emergent $4^{\textrm{th}}$ generation light sources, which offer incredibly bright, narrow bandwidth x-ray sources, with unprecedented temporal resolution.
Ackowledgments
==============
DM acknowledges support from LLNL under Subcontract No. B595954. AH acknowledges financial support from AWE.
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|
{
"pile_set_name": "ArXiv"
}
|
[ **The Gauge-Higgs Legacy of the LHC Run II** ]{}
Anke Biekötter^1^, Tyler Corbett^2^, and Tilman Plehn^1^
[**1**]{} Institut für Theoretische Physik, Universität Heidelberg, Germany\
[**2**]{} Niels Bohr International Academy and Discovery Centre, Niels Bohr Institute, University of Copenhagen, Denmark\
[email protected]
Abstract {#abstract .unnumbered}
========
[**We present a global analysis of the Higgs and electroweak sector based on LHC Run II and electroweak precision observables. We show which measurements provide the leading constraints on Higgs-related operators, and how the achieved LHC precision makes it necessary to combine rate measurements with electroweak precision observables. The SFitter framework allows us to include kinematic distributions beyond pre-defined ATLAS and CMS observables, independently study correlations, and avoid Gaussian assumptions for theory uncertainties. These Run II results are a step towards a precision physics program at the LHC, interpreted in terms of effective operators.**]{}
------------------------------------------------------------------------
------------------------------------------------------------------------
[feynman]{}
Introduction {#sec:intro}
============
After the discovery of a light, likely fundamental Higgs boson largely compatible with the Standard Model [@higgs], the LHC has focused on precision studies of electroweak symmetry breaking [@review]. From a theoretical as well as from an experimental perspective, the appropriate interpretation framework for such LHC precision analyses are effective Lagrangians [@eft1; @eft2; @kaoru; @kilian; @trott]. They require us to fix the (propagating) particle content and the underlying symmetry structure. For the former, experimental observations point to the Standard Model content, possibly extended by a dark matter agent coupling to the Higgs sector. Concerning the interactions, we can assume the Higgs doublet structure of the Standard Model, which intertwines the Higgs sector and the electroweak gauge sector [@linear]. The corresponding analyses based on Run I data [@legacy1; @legacy2; @barca; @runI_th; @runI_ex; @laura] and first analyses based on Run II data [@runII_eng; @runII_concha] prove that the LHC has successfully transitioned to a precision physics experiment.
In the effective theory version of the Standard Model [@trott] we assume that departures of Higgs or gauge boson interactions from their SM predictions are characterized by a new energy scale $\Lambda$. It is crucial that this energy scale is not kinematically accessible at the LHC, which means that the corresponding new particles never appear on their mass shell. This condition defines the validity of the EFT approach [@validity]. Because the range of energies accessible in the kinematic regime of the LHC does not guarantee a strong hierarchy of scales [@englert], we can then think of an effective Lagrangian representing classes of new physics models [@eft_model; @higgsmultiplets].
One of the great advantages of the SMEFT framework is that it allows for global analyses of LHC measurements not only in the Higgs and electroweak gauge sectors, but also in the QCD sector [@qcd_eft; @krauss], the top sector [@top_eft; @topfitter], or the flavor sector [@flavor_eft]. For LHC Run I there exist analyses combining Higgs measurements with LEP data [@lep_gauge] or, even better, di-boson production at the LHC searching for anomalous triple gauge vertices [@lhc_gauge; @legacy2]. At this point we find that in the effective Lagrangian framework the LHC limits are surpassing the LEP limits, because effective operators with a momentum dependence can be tested either through high precision or through large momentum flow [@lep_lhc]. Similarly, at the level of Run II precision we should not hard-code the electroweak precision constraints into our operator basis [@fermionic]. Fermionic operators affect electroweak precision data and LHC data in different combinations with the usual bosonic operators, and this correlation generally weakens the constraints on operators contributing to Higgs physics only. This brings the number of SMEFT operators considered in our global Higgs analysis to 20, plus invisible decays. Two of these operators turn out to be successfully constrained by non-Higgs observables, so they do not have to be considered in the actual analysis.
In this paper we present an <span style="font-variant:small-caps;">SFitter</span> analysis of the Higgs and gauge sector at the LHC and electroweak precision data. As usual, we do not rely on pre-defined results from ATLAS and CMS, but evaluate event counts in total rate measurements and kinematic distributions using our in-house framework whenever available [@sfitter_orig; @sfitter_delta]. This allows us to correlate systematic uncertainties, define our own treatment of theoretical uncertainties, and account for non-Gaussian constraints. We start by defining our relevant operator basis in Sec. \[sec:higgs\_sector\] and \[sec:ewpo\_sector\]. We then compare possible Higgs-sector constraints on operators measured in other LHC analyses in Sec. \[sec:top\_sector\]. With this operator basis we then report on a global LHC analysis, starting with a comparison of Run I and Run II results, adding electroweak precision observables, and discussing the interplay of the two kinds of operators in detail in Sec. \[sec:higgs\]. Our final result brings us a significant step closer to a global <span style="font-variant:small-caps;">SFitter</span> SMEFT analysis.
Higgs and gauge sector {#sec:higgs_sector}
======================
The linear effective Lagrangian is an $SU(3)_c \otimes SU(2)_L \otimes
U(1)_Y$-symmetric extension of the renormalizable Standard model, but with the SM field content. It is ordered by inverse powers of the new physics scale [@eft1; @eft2; @linear], $$\begin{aligned}
{5}
\lag = \sum_x \frac{f_x}{\Lambda^2} \; \ope_x \;\;,
\label{eq:def_f}\end{aligned}$$ Neglecting lepton number violation at dimension five the first order of new physics effects is dimension six, with 59 baryon-number conserving operators, barring flavor structure and Hermitian conjugation [@eft2]. We follow the definition of the relevant operator basis of Ref. [@barca]: first, we restrict the initial set to $P$-even and $C$–even operators[^1]. We then use the equations of motion to rotate to a basis where there are no blind directions linked to electroweak precision data. We then neglect all operators that cannot be studied at the LHC yet or which are strongly constrained from other LHC measurements. This includes the $HHH$ vertex [@hhh], the Higgs interactions with light-generation fermions, and some operators discussed in Sec. \[sec:top\_sector\]. We are left with 18 dimension-6 operators, ten of which do not influence electroweak precision observables at tree level [@barca], $$\begin{aligned}
{9}
\ope_{GG} &= \phi^\dagger \phi \; G_{\mu\nu}^a G^{a\mu\nu} \quad
&\ope_{WW} &= \phi^{\dagger} \hat{W}_{\mu \nu} \hat{W}^{\mu \nu} \phi \quad
&\ope_{BB} &= \phi^{\dagger} \hat{B}_{\mu \nu} \hat{B}^{\mu \nu} \phi
\notag \\
\ope_W &= (D_{\mu} \phi)^{\dagger} \hat{W}^{\mu \nu} (D_{\nu} \phi)
& \ope_B &= (D_{\mu} \phi)^{\dagger} \hat{B}^{\mu \nu} (D_{\nu} \phi)
\notag \\
\ope_{\phi 2} &= \frac{1}{2} \partial^\mu ( \phi^\dagger \phi )
\partial_\mu ( \phi^\dagger \phi ) \quad
&\ope_{WWW} &= \tr \left( \hat{W}_{\mu \nu} \hat{W}^{\nu \rho}
\hat{W}_\rho^\mu \right)
\label{eq:operators} \\
\ope_{e\phi,33} &= \phi^\dagger\phi \; \bar L_3 \phi e_{R,3} \qquad
&\ope_{u\phi,33} &= \phi^\dagger\phi \; \bar Q_3 \tilde \phi u_{R,3} \qqquad
&\ope_{d\phi,33} &= \phi^\dagger\phi \; \bar Q_3 \phi d_{R,3} \notag \; .\end{aligned}$$ The covariant derivative acting on the Higgs is $D_\mu = \partial_\mu+
i g' B_\mu/2 + i g \sigma_a W^a_\mu/2$, and the field strengths are $\hat{B}_{\mu \nu} = i g' B_{\mu \nu}/2$ and $\hat{W}_{\mu\nu} = i
g\sigma^a W^a_{\mu\nu}/2$. This ad-hoc rescaling of the field strength can be motivated through our expectations from known UV-completions, but it has no effect on our analysis or its interpretation. The effective Lagrangian which we use to interpret Higgs and triple-gauge vertex (TGV) measurements at the LHC is $$\begin{aligned}
\lag_\text{eff} \supset
&- \frac{\alpha_s }{8 \pi} \frac{f_{GG}}{\Lambda^2} \ope_{GG}
+ \frac{f_{WW}}{\Lambda^2} \ope_{WW}
+ \frac{f_{BB}}{\Lambda^2} \ope_{BB} \notag \\
&+ \frac{f_W}{\Lambda^2} \ope_W
+ \frac{f_B}{\Lambda^2} \ope_B
+ \frac{f_{\phi 2}}{\Lambda^2} \ope_{\phi 2}
+ \frac{f_{WWW}}{\Lambda^2} \ope_{WWW} \notag \\
&+ \frac{f_\tau m_\tau}{v \Lambda^2} \ope_{e\phi,33}
+ \frac{f_b m_b}{v \Lambda^2} \ope_{d\phi,33}
+ \frac{f_t m_t}{v \Lambda^2} \ope_{u\phi,33}
+ \text{invisible decays}\;.
\label{eq:ourlag1}\end{aligned}$$ For invisible Higgs decays we do not include a term in the Lagrangian and consequently describe it in terms of an invisible partial width. It is best constrained through WBF Higgs production [@eboli_zeppenfeld]. All operators except for $\ope_{WWW}$ contribute to Higgs interactions. Their contributions to the several Higgs vertices, including non-SM Lorentz structures, are described in Ref. [@legacy1].
Some of the operators in Eq. contribute to the self-interactions of the electroweak gauge bosons. They can be linked to specific deviations in the Lorentz structures entering the $WWZ$ and $WW\gamma$ interactions, historically written as $\kappa_\gamma,
\kappa_Z, g_1^Z, g_1^\gamma$, $\lambda_\gamma$, and $\lambda_Z$ [@kaoru_tgv]. After using electromagnetic gauge invariance to fix $g_1^\gamma = 1$, the shifts are defined by $$\begin{aligned}
\Delta \lag_\text{TGV} =&
- i e \; (\kappa_\gamma-1) \; W^+_\mu W^-_\nu \gamma^{\mu \nu}
- \frac{i e \lambda_\gamma}{ m_W^2} \; W_{\mu \nu}^+ W^{- \nu \rho} \gamma_\rho^\mu
- \frac{i g_Z \lambda_Z}{ m_W^2} \; W_{\mu \nu}^+ W^{- \nu \rho} Z_\rho^{\;\mu}
\notag \\
&- i g_Z \; (\kappa_Z -1) \; W^+_\mu W^-_\nu Z^{\mu \nu}
- i g_Z \; ( g_1^Z -1) \; \left( W^+_{\mu \nu} W^{- \mu} Z^\nu - W^+_\mu Z_\nu W^{- \mu \nu}
\right) \notag \\
=& - i e \; \frac{g^2 v^2}{8 \Lambda^2} \left( f_W + f_B \right) \; W^+_\mu W^-_\nu \gamma^{\mu \nu}
- i e \; \frac{3 g^2 f_{WWW}}{4 \Lambda^2} \; W_{\mu \nu}^+ W^{- \nu \rho} \gamma_\rho^\mu \notag \\
&- i g_Z \; \frac{g^2 v^2}{8 c_w^2 \Lambda^2} \left( c_w^2 f_W - s_w^2 f_B \right) \; W^+_\mu W^-_\nu Z^{\mu \nu}
- i g_Z \; \frac{3 g^2 f_{WWW}}{4 \Lambda^2} \; W_{\mu \nu}^+ W^{- \nu \rho} Z_\rho^{\; \mu} \notag \\
&- i g_Z \; \frac{g^2 v^2 f_W}{8 c_w^2 \Lambda^2} \; \left( W^+_{\mu \nu} W^{- \mu} Z^\nu - W^+_\mu Z_\nu W^{- \mu \nu}
\right) \; ,
\label{eq:tgvlag}\end{aligned}$$ where $e = g s_w$ and $g_Z = g c_w$. The two notational conventions are equivalent for gauge-invariant models and linked as $$\begin{aligned}
\kappa_\gamma &= 1+
\frac{g^2 v^2}{8\Lambda^2}
\left( f_W + f_B \right) \qqqquad
\kappa_Z = 1+ \frac{g^2 v^2}{8 c_w^2\Lambda^2} \left(c_w^2 f_W - s_w^2 f_B \right) \notag \\
g_1^Z &= 1+ \frac{g^2 v^2}{8 c_w^2\Lambda^2} f_W \qqqquad
g_1^\gamma = 1 \qqqquad
\lambda_\gamma = \lambda_Z =
\frac{3 g^2 m_W^2}{2 \Lambda^2} f_{WWW}\; .
\label{eq:wwv}\end{aligned}$$ The three Wilson coefficients relevant for our analysis of di-boson production are $f_B$, $f_W$ and $f_{WWW}$, plus the operators influencing electroweak precision data discussed in Section \[sec:ewpo\_sector\]. To get a very rough idea what kind of new physics scales we can probe in the electroweak gauge and Higgs sector we quote the typical range from the global Run I analyses, $$\begin{aligned}
\frac{\Lambda}{\sqrt{|f|}} \gtrsim 300~...~500~\gev
\qquad \text{(Higgs-gauge analysis at Run~I~\cite{legacy2}).}
\label{eq:rough_higgs}\end{aligned}$$ We note that already the Run I di-boson measurements clearly outperform the corresponding LEP measurements evaluated in the effective operator basis of Eq..
If we deviate from this scenario and consider instead the more generic non-linear or chiral effective Lagrangian [@nonlinear1; @nonlinear2], the parametrization would be extended. In the most generic scenario, the TGV couplings defined above depend on a larger number of parameters and the correlations from gauge dependence are lost. Furthermore, the deviations generated by non-linear operators in the TGVs could be completely de-correlated to the deviations generated in the Higgs interactions. For the Higgs sector alone, the linear and non-linear analyses can be trivially mapped onto each other [@legacy1].
Electroweak precision sector {#sec:ewpo_sector}
============================
While the Lagrangian in Eq. does not include tree-level contributions to electroweak precision observables, we know that at the level of 13 TeV data the corresponding operators should not be neglected [@fermionic; @runII_eng; @runII_concha]. This means that we need to add two bosonic operators $$\begin{aligned}
\ope_{\phi 1} = (D_\mu \phi)^\dagger \; \phi \phi^\dagger \; (D^\mu \phi)
\qqqquad
\ope_{BW} = \phi^\dagger \hat{B}_{\mu\nu} \hat{W}^{\mu\nu} \phi \; ,
\label{eq:ope_new1}\end{aligned}$$ which affect gauge and Higgs interactions. In addition we consider the fermionic Higgs-gauge operators $$\begin{aligned}
\ope_{\phi Q,ij}^{(1)} &=\phi^\dagger (i\,{\Dfb}_{\mu} \phi) (\bar Q_{i}\gamma^\mu Q_i) \qquad & \ope_{\phi Q,ij}^{(3)} &=\phi^\dagger (i\,{\Dfba}_{\!\!\mu} \phi) (\bar \Psi_i\gamma^\mu \frac{\sigma_a}{2} \Psi_i) \notag \\
\ope_{\phi L,ij}^{(1)} &=\phi^\dagger (i\,{\Dfb}_{\mu} \phi) (\bar L_{i}\gamma^\mu L_i) \qquad & \ope_{\phi L,ij}^{(3)}&=\phi^\dagger (i\,{\Dfba}_{\!\!\mu} \phi) (\bar L_i\gamma^\mu \frac{\sigma_a}{2} L_i) \notag \\
\ope_{\phi u,ij}^{(1)} &=\phi^\dagger (i\,{\Dfb}_{\mu} \phi) (\bar u_{R,i}\gamma^\mu u_{R,i}) \qquad & \ope_{LLLL} &= (\bar{L} \gamma_\mu L) \; (\bar{L} \gamma^\mu L) \notag \\
\ope_{\phi d,ij}^{(1)} &=\phi^\dagger (i\,{\Dfb}_{\mu} \phi) (\bar d_{R,i}\gamma^\mu d_{R,i}) \notag \\
\ope_{\phi e,ij}^{(1)}&=\phi^\dagger (i\,{\Dfb}_{\mu} \phi) (\bar e_{R,i}\gamma^\mu e_{R,i}) \notag \\
\ope_{\phi ud,ij}^{(1)} &=\tilde \phi^\dagger (i\,{\Dfb}_{\mu} \phi) (\bar u_{R,i}\gamma^\mu d_{R,i})
\label{eq:ope_new2}\end{aligned}$$ The operator $\ope_{\phi ud,ij}^{(1)}$ contains the charged current $\bar{u}_R
\gamma^\mu d_R$ [@runII_concha]. Given that it does not interfere with the Standard Model and the known flavor physics constraints we will ignore it in our analysis, the same way we exclude for example dipole operators.
operator $H f \bar f$ $Z q q$ $W q q'$ $Z l \bar l$ $W l \nu$
----------------------- -------------- ---------- ---------- -------------- -----------
$\ope_{\phi 1}$ $\times$ $\times$ $\times$ $\times$ $\times$
$\ope_{BW}$ $\times$ $\times$ $\times$ $\times$
$\ope_{\phi Q}^{(3)}$ $\times$ $\times$
$\ope_{\phi Q}^{(1)}$ $\times$
$\ope_{\phi u}^{(1)}$ $\times$
$\ope_{\phi d}^{(1)}$ $\times$
$\ope_{\phi e}^{(1)}$ $\times$
: List of operators affecting electroweak precision observables and their effect on fermionic couplings testable at the LHC.[]{data-label="tab:operator_influence"}
The first eight operators generate anomalous weak boson couplings to fermions, while they do not affect the Higgs coupling to fermions, see Tab. \[tab:operator\_influence\]. They do modify the Higgs couplings to weak bosons and fermions, for instance introducing point-like $HVff$ interactions. We also include the 4-lepton operator $\ope_{LLLL}$ as it induces a shift in the Fermi constant. For our study we assume diagonal and generation independent Wilson coefficients for the fermionic operators affecting the electroweak currents. Further, we will eliminate the leptonic operators $\ope_{\phi L,ii}^{(1)}$ and $\ope_{\phi L,ii}^{(3)}$ using the equations of motion: $$\begin{aligned}
&2\ope_B+\ope_{BW}+\ope_{WW}+g^2\left(\ope_{\phi 4}-\frac{1}{2}\ope_{\phi 2}\right) =-\frac{g^2}{4}\sum_i\left(\ope_{\phi L,ii}^{(3)}+\ope_{\phi Q,ii}^{(3)}\right)
\notag \\
&2\ope_B+\ope_{BW}+\ope_{BB}+g'^2 \left(\ope_{\phi 1}-\frac{1}{2}\ope_{\phi 2}\right) = \notag \\
&\phantom{Halllllloooooooooo} -\frac{g'^2}{2}\sum_i\left(-\frac{1}{2}\ope_{\phi L,ii}^{(1)}+\frac{1}{6}\ope_{\phi Q,ii}^{(1)}-\ope_{\phi e,ii}^{(1)}+\frac{2}{3}\ope_{\phi u,ii}^{(1)}-\frac{1}{3}\ope_{\phi d,ii}^{(1)}\right) \; .\end{aligned}$$ Assuming a universal flavor structure this leaves us with the additional contributions to our effective Lagrangian, $$\begin{aligned}
\lag_\text{eff} \supset
&+ \frac{f_{\phi 1}}{\Lambda^2} \ope_{\phi 1}
+ \frac{f_{BW}}{\Lambda^2} \ope_{BW}
+ \frac{f_{LLLL}}{\Lambda^2} \ope_{LLLL} \notag \\
&+ \frac{f_{\phi Q}^{(1)}}{\Lambda^2} \ope_{\phi Q}^{(1)}
+ \frac{f_{\phi d}^{(1)}}{\Lambda^2} \ope_{\phi d}^{(1)}
+ \frac{f_{\phi u}^{(1)}}{\Lambda^2} \ope_{\phi u}^{(1)}
+ \frac{f_{\phi e}^{(1)}}{\Lambda^2} \ope_{\phi u}^{(1)}
+ \frac{f_{\phi Q}^{(3)}}{\Lambda^2} \ope_{\phi Q}^{(3)} \; .
\label{eq:ourlag2}\end{aligned}$$ Together with Eq. this defines the operator basis for our global analysis, altogether 18 operators plus the invisible Higgs branching ratio. While the additional operators affect many of our LHC measurements, they are also strongly constrained by electroweak precision observables. The challenge is that the bosonic operators in Eq. and the fermionic operators in Eq. not only contribute to electroweak precision physics, but also to di-boson or Higgs production at an observable level. Because the two data sets combine very different combinations of operators, we have to combine our Run II analysis with a set of electroweak precision observables. We follow Ref. [@unitarity] and review this approach briefly. Our $Z$-pole observables are $$\begin{aligned}
\Big\{ \Gamma_Z,\, \sigma_h^{0},\,
\mathcal{A}_l(\tau^{\rm pol}),\,
R_l^0,\, \mathcal{A}_l(\text{SLD}),\, A_\text{FB}^{0,l},\,
R_c^0,\, R_b^0,\, \mathcal{A}_c,\, \mathcal{A}_b,\, A_\text{FB}^{0,c},\, A_\text{FB}^{0,b}(\text{SLD/LEP-I}) \Big\} \; .
\label{eq:zobs}\end{aligned}$$ with measurements and correlations taken from Ref [@ewwg]. We also include the $W$-observables $$\begin{aligned}
\Big\{ m_W,\, \Gamma_W,\, \br (W\to l\nu) \Big\} \; ,
\label{eq:wobs}\end{aligned}$$ with values taken from Ref. [@pdg]. The SM predictions for these observables are taken from Ref. [@deBlas:2017wmn]. We note that for the SM prediction of the $W$-mass this includes the full one- and two-loop EW and two-loop QCD corrections of $\mathcal{O}(\alpha\alpha_s)$ as well as some 3-loop contributions. The contributions of our dimension-6 operators can be found in Ref. [@unitarity], where we limit ourselves to linear contributions from the higher-dimensional operators considered in our fit. This approximation is justified as long as the dimension-6 corrections are small, $f m_Z^2/\Lambda^2 \ll 1$ assuming that the typical energy scale of electroweak precision data is around $m_Z$. The standard analyses of electroweak precision data indeed give individual limits of the kind $$\begin{aligned}
\frac{\Lambda}{\sqrt{|f|}} \gtrsim 4~...~10~\tev
\qquad \text{(electroweak precision data~\cite{deBlas:2017wmn}).}
\label{eq:rough_ewpo}\end{aligned}$$ These limits significantly exceeds the expected sensitivity of the global LHC analysis from Eq., which naively suggests that it is not necessary to combine the two sectors. In the discussion of our global fit in Section \[sec:higgs\] we will see how the fermionic Higgs-gauge operators nevertheless lead to visible effects at the LHC.
QCD and top sectors {#sec:top_sector}
===================
[cccc]{}
(90,75)
&
(90,75)
&
(90,75)
&
(90,75)
\
$f_{tG} \, v \, p^\mu$ &$f_{tG} \, v$ &$f_{tG} \, p^\mu$ &$f_{tG}$
An operator which should be added to any basis confronted with LHC data is the anomalous triple gluon coupling $$\begin{aligned}
\ope_G &= f_{abc}
G_{a \nu}^\rho G_{b \lambda}^\nu G_{c \rho}^\lambda
\qquad \text{with} \qquad
\lag_\text{eff} \supset
\frac{g_s f_G}{\Lambda^2} \, \ope_G \; ,
\label{eq:g3}\end{aligned}$$ with $G_a^{\rho \nu} = \partial^\rho G_a^\nu - \partial^\nu G_a^\rho -
i g_s f_{abc} G^{b \rho} G^{c \nu}$. It contributes to any gluon-induced LHC process, for instance Higgs production with a hard jet. While it only affects kinematic distributions with an additional hard parton, it needs to be taken into account when we use the same distribution to separate $\ope_{u \phi,33}$ effects from $\ope_{GG}$. On the other hand, it can be constrained by ATLAS multi-jet data at 13 TeV, giving the 95% CL limits [@krauss] $$\begin{aligned}
\frac{\Lambda}{\sqrt{f_G}} > 5.2 \, (5.8)~\tev
\qquad \text{observed (expected) from multi-jets.}
\label{eq:reach_qcd}\end{aligned}$$ This limits the possible effects on Higgs production rates beyond anything a global Higgs analysis would be sensitive to in the absence of a dedicated enhancement mechanism in Higgs rates.
A critical feature of Higgs analyses is the combination of direct and indirect measurements of the top-Higgs coupling in gluon fusion and associated Higgs-top production [@eft_higgs_top]. The chromo-magnetic top operator $$\begin{aligned}
\ope_{tG}=(\bar Q\sigma^{\mu\nu}T^Au_R) \; \tilde H \; G_{\mu\nu}^A \; ,\end{aligned}$$ will, in principle, affect these observables [@runII_eng] and has been studied extensively in top-EFT analyses [@topfitter]. The interaction vertices induced by $\ope_{tG}$ are shown in Fig. \[fig:OtGdiagrams\]. The first two diagrams contribute to top pair production, the second set to $t\bar t H$ production. In each case one of the interactions is proportional to the momentum flowing through the vertex.
To constrain $f_{tG}$ in a Higgs fit we can consider gluon fusion and $t\bar{t}H$ production with additional jets. However, extra hard gluons in the final state are a typical higher-order effect and likely suppressed. Alternatively, we can use momentum-dependent distributions in $t\bar{t}H$ production. The third vertex in Fig. \[fig:OtGdiagrams\] allows for such effects, but only includes a single gluon and will therefore be suppressed by an $s$-channel propagator counteracting the dimension-6 momentum enhancement.
We can estimate the extent to which this operator can be constrained. The most promising distribution currently available is the $H_T$ distribution in the all-hadronic $$\begin{aligned}
pp \to t\bar t H \to t \bar{t} \; b \bar{b}\end{aligned}$$ signature released by CMS [@Sirunyan:2018ygk]. In Fig. \[fig:HTdistribution\] we reproduce their $H_T$ distribution as well as the distribution in the presence of two benchmark values of $f_{tG}$. We generate the relevant $ttH$ process merged with one additional jet using <span style="font-variant:small-caps;">Madgraph5</span> [@madgraph] and <span style="font-variant:small-caps;">Pythia8</span> [@pythia], combined with <span style="font-variant:small-caps;">Delphes3</span> [@delphes]. The two benchmarks each correspond to $$\begin{aligned}
\frac{\Lambda}{\sqrt{|f_{tG}|}} &\gtrsim 1~\tev
&\qquad &\text{(top sector~\cite{topfitter})}\notag \\
\frac{\Lambda}{\sqrt{|f_{tG}|}} &\gtrsim 320~\gev
&\qquad &\text{(Higgs sector~\cite{runII_eng})}
\label{eq:reach_top}\end{aligned}$$ From Fig. \[fig:HTdistribution\] we see that our expected sensitivity is comparable to the Higgs study and not competitive with the top-sector constraints. This comes as no surprise. Figure \[fig:OtGdiagrams\] indicates that $\ope_{tG}$ contributes directly to $t\bar t$ production with orders of magnitude more events than in the Higgs sector. In addition, $t\bar{t}H$ production is plagued by large backgrounds, so indeed $\ope_{tG}$ and $\ope_G$ can both be neglected in global Higgs analyses.
SFitter framework
=================
production decay ATLAS CMS
------------- ----------------------------- ------------------------------ ------------------------------------------------------
$H \to WW$ [@ATLAS13WW; @ATLAS13tthLep] [@CMS13WW; @CMS13tthLep1; @CMS13tthLep2]
$H \to ZZ$ [@ATLAS13ZZ; @ATLAS13tthLep] [@CMS13ZZ; @CMS13ZZv2; @CMS13tthLep1; @CMS13tthLep2]
$H \to \gamma\gamma$ [@ATLAS13aa] [@CMS13aa]
$H \to \tau \tau$ [@ATLAS13tthLep] [@CMS13tautau; @CMS13tthLep1; @CMS13tthLep2]
$H \to Z\gamma$ [@ATLAS13Za] [@CMS13Za]
WBF $H \to \text{inv}$ [@CMS13WBFInv]
WBF $H \to \tau \tau$ [@CMS13tautau]
$VH$ $H \to b\bar{b}$ [@ATLAS13Vhbb] [@CMS13Vhbb]
$VH$ $H \to \tau \tau$ [@CMS13Vhtautau]
$VH$ $H \to \text{inv}$ [@ATLAS13ZhInv] [@CMS13ZhInv]
$VH$ $H \to b\bar{b}$ ($m_{VH}$) [@ATLAS13VhEXO]
$t\bar{t}H$ $H \to \gamma\gamma$ [@ATLAS13tthObs] [@CMS13aa]
$t\bar{t}H$ $H \to ZZ \to 4 \ell$ [@ATLAS13tthObs] [@CMS13ZZ; @CMS13ZZv2]
$t\bar{t}H$ $H \to WW,ZZ,\tau\tau$ [@ATLAS13tthLep] [@CMS13tthLep1; @CMS13tthLep2]
$t\bar{t}H$ $H \to b\bar{b}$ [@ATLAS13tthbb] [@CMS13tthbb]
: List of Run II Higgs measurements included in our analysis. For the $m_{VH}$ distribution our highest-momentum bin with observed events starts at $m_{VH} = 990$ GeV and $1.2$ TeV for the $0 \ell$ and $1 \ell$ final states.[]{data-label="tab:higgs_data"}
In <span style="font-variant:small-caps;">SFitter</span> analyses we prefer not to rely on the pre-processed rate modifiers by ATLAS and CMS whenever possible. Instead, we extract the signal and background rates from the experimental publications and apply our own uncertainty treatment. This includes correlated and uncorrelated systematic uncertainties as well as a flat likelihood within the allowed band by theoretical uncertainties. For analyses using multivariate analysis techniques, where the number of events in each signal region is only illustrated after simple cuts rather than the full analysis, we implement the signal strength modifiers but separate for example the theory uncertainties. All signal efficiencies and higher-order effects we extract as the difference between our simulation and the numbers quoted by ATLAS and CMS.
For Higgs and di-boson signals we use <span style="font-variant:small-caps;">MadGraph5</span> [@madgraph] for the event generation, <span style="font-variant:small-caps;">Pythia6</span> [@pythia] for parton shower and hadronization, and <span style="font-variant:small-caps;">Delphes3</span> [@delphes] for the detector simulation. Branching ratios including dimension-6 effects are given by the extended version of <span style="font-variant:small-caps;">Hdecay</span> [@hdecay]. For new physics effects in the production process we use the same tool chain as for the Standard Model, combined with our <span style="font-variant:small-caps;">FeynRules</span> [@feynrules] implementation of the dimension-6 operators and assume that detector effects as well as higher-order corrections scale with the SM case in the fiducial volume of the SM-like measurement. For total rate measurements using the bulk of the phase space this approximation is obviously justified. For our kinematic distributions this is less clear, so we have checked that our approach is approximately correct [@fermionic; @sally].
As usual for our <span style="font-variant:small-caps;">SFitter</span> analysis we allow for the modification of the production amplitude through dimension-6 operators including the interference with the SM amplitude and the squared term in the Wilson coefficient. The latter becomes relevant whenever the interference with the Standard Model is suppressed. Given the estimates of Eq. and Eq. we simplify our analysis by neglecting diagrams which are modified by bosonic and fermionic operators at the same time and interfere with the SM amplitude. In our discussion of the results we will see that indeed large effects from the fermionic operators do not appear in this topology. Finally, we neglect dimension-6 squared contributions of the fermionic operators to the gauge boson branching ratios, because they will be strongly suppressed following Eq. with a typical energy scale $m_V$ in the gauge boson decays. For the same reason we neglect the effects of the fermionic operators on the decays of gauge bosons coming from Higgs decays. The hierarchy of scales combined with the well-defined external energy scale $E \lesssim m_H$ will render them numerically irrelevant.
For Higgs and di-boson we start with the set of Run I measurements discussed in Refs [@legacy1; @legacy2]. We add the Run II Higgs measurements shown in Tab. \[tab:higgs\_data\] and the Run II di-boson measurements shown in Tab. \[tab:vv\_data\]. Because the dimension-6 Lagrangian introduces new Lorentz structures and hence predicts significantly different event kinematics from the Standard Model, kinematic distributions scaling with energy are especially powerful. An attractive case is a $m_{VH}$ distribution from an ATLAS resonance search [@ATLAS13VhEXO], which we include for the zero-lepton and one-lepton final states. We re-bin the reported result such that the most relevant high bins include a statistically meaningful number of events, giving us measurements exceeding $m_{VH}
= 1$ TeV.[^2] The other side of the kinematics medal is that measurements from $H \to 4\ell$ decays can be safely neglected in a global analysis. The reason is that the momentum flow through the Higgs decay vertex is cut off by the on-shell condition, so any measurement in $VH$ or WBF production will surpass their impact on a global analysis [@info_geo].
Based on all measurements we first construct a multi-dimensional, full exclusive likelihood map. As long as we are only interested in small deviations from the Standard Model, a key assumption to be able to use an effective field theory approach, we can assume that local SM-like minima are also the global minima in this likelihood map. There exist three standard ways to explore the log-likelihood distribution around the minimum: first, we can use a naive, <span style="font-variant:small-caps;">Minuit</span>-like approach, approximating the functional form around the minimum by a quadratic function. This assumption is not appropriate once we allow for non-Gaussian errors, for example a flat shape covering the theoretical error bar. Second, we can construct a Markov chain over the parameter space. Here the problem is that different directions in the space of Wilson coefficients behave differently, which makes it hard to define a universal and efficient proposal function. Nevertheless, we check our results against such a Markov chain analysis and usually find encouraging agreement. For our numerical analysis we define 10.000 toy measurements, modeling the Poisson, Gaussian or flat input distributions. For each toy experiment we determine the best-fitting point in the space of Wilson coefficients, combine these values to a histogram, effectively profile over the remaining parameters, and determine the 68% and 95% ranges around the SM-like central value. For the error bands we require the log-likelihood values at the lower and upper ends to be identical.
Because our approch gives us full control over the log-likelihood distribution we can compare these limits with a dual Gaussian fit to the log-likelihood in one dimension. We find good agreement for all Wilson coefficients, even though Fig. \[fig:toys\] shows that for example the profile likelihood for $f_W$ does not have a symmetric Gaussian shape. Obviously, the shape for the invisible Higgs width is distorted, because it does not allow for negative branching ratios. While we quote the error bars for the non-Gaussian analysis we quote the results from the Gaussian fit whenever we give a best-fit point for a Wilson coefficient. For additional details on the <span style="font-variant:small-caps;">SFitter</span> framework we refer to Refs. [@sfitter_orig; @sfitter_delta].
![Distributions of the toy experiments for the operators $\ope_\tau$, $\ope_{\phi 2}$ and $\ope_W$ as well as the invisible Higgs branching ratio, based on the full LHC data set. The lines show the 95% CL limits from the histogram (black) and the double-Gaussian fit (red).[]{data-label="fig:toys"}](noewpo_single/chi2_update95__noewpoMINS-1_dez13_D6ftau "fig:"){width="45.00000%"} ![Distributions of the toy experiments for the operators $\ope_\tau$, $\ope_{\phi 2}$ and $\ope_W$ as well as the invisible Higgs branching ratio, based on the full LHC data set. The lines show the 95% CL limits from the histogram (black) and the double-Gaussian fit (red).[]{data-label="fig:toys"}](noewpo_single/chi2_update95__noewpoMINS-1_dez13_D6fphi2 "fig:"){width="45.00000%"}\
![Distributions of the toy experiments for the operators $\ope_\tau$, $\ope_{\phi 2}$ and $\ope_W$ as well as the invisible Higgs branching ratio, based on the full LHC data set. The lines show the 95% CL limits from the histogram (black) and the double-Gaussian fit (red).[]{data-label="fig:toys"}](noewpo_single/chi2_update95__noewpoMINS-1_dez13_D6fw "fig:"){width="45.00000%"} ![Distributions of the toy experiments for the operators $\ope_\tau$, $\ope_{\phi 2}$ and $\ope_W$ as well as the invisible Higgs branching ratio, based on the full LHC data set. The lines show the 95% CL limits from the histogram (black) and the double-Gaussian fit (red).[]{data-label="fig:toys"}](noewpo_single/chi2_update95__noewpoMINS-1_dez13_Hii.pdf "fig:"){width="45.00000%"}
channel distribution \#bins max \[GeV\]
-------- -------------------------------------------------------- --------------------------- -------- ------------- --------------------------
$WW\rightarrow \ell^+\ell^{\prime -}+\met \; (0j)$ leading $p_{T,\ell}$ 4 350 $20.3~\ifb$ [@atlas8ww]
$WW\rightarrow \ell^+\ell^{(\prime) -}+\met \; (0j)$ $m_{\ell\ell^{(\prime)}}$ 7 575 $19.4~\ifb$ [@cms8ww]
$WZ\rightarrow \ell^+\ell^{-}\ell^{(\prime)\pm}$ $m_T^{WZ}$ 6 450 $20.3~\ifb$ [@atlas8wz]
$WZ\rightarrow \ell^+\ell^{-}\ell^{(\prime)\pm}+\met $ $p_T^{Z \to \ell\ell}$ 8 350 $19.6~\ifb$ [@cms78wz]
13 TeV $WZ\rightarrow \ell^+\ell^{-}\ell^{(\prime)\pm}$ $m_T^{WZ}$ 7 675 $36.1~\ifb$ [@ATLAS13WZ]
: List of Run I and Run II di-boson measurements included in our analysis. The maximum value in GeV indicates the lower end of the highest-momentum bin we consider.[]{data-label="tab:vv_data"}
One caveat applies to all analyses based on effective Lagrangians: we consider the dimension-6 Lagrangian of Eq. and Eq. the appropriate desciption of the physics effects beyond the Standard Model. Note that this statement by no means implies that for example the dimension-6-squared contributions have to be smaller than those from the dimension-6 interference with the Standard Model [@interference]. There exist many physics reasons why this could be a valid physics effect, and the discrepancy between the generic LHC reach given by Eq. and the generic reach of electroweak precision data in Eq. will be discussed as an example for such effects in the next section. Instead, we simply need to ensure that no particle of the UV-completions which we approximate with our effective Lagrangian contributes as a propagating degree of freedom on its mass shell [@validity]. To this end, computing the effects of dimension-8 operators can give useful hints about the validity of the dimension-6 truncation [@dim8], but it does not have to.
Finally, in the spirit of the effective field theory we only consider SM-like scenarions, which means that we neglect all secondary solutions for example with switched signs of Yukawa couplings. Assuming weakly interacting new physics such effects require scales $\Lambda \sim m_H$, so we expect these models to be best tested in direct LHC searches rather than a global analysis. In any case, the observation of a sign switch for example in a Yukawa coupling as part of a global analysis would signal a breakdown of the renormalizable Standard Model and its symmetry structure and would prompt us to modify our SMEFT hypothesis. Of course, when it comes to searching for effects in kinematic distributions, these two search strategies are closely related, for example when we directly search for mass peaks in the same distributions that we indirectly test for shoulders (as an early sign of a mass peak appearing in data) [@validity].
Global analysis {#sec:higgs}
===============
![Allowed 95% CL ranges for individual Wilson coefficients $f_x/\Lambda^2$ from a one-dimensional profile likelihood. We show results from Run I (red) and using the additional Run II measurements (blue). We neglect all operators contributing to electroweak precision observables at tree level.[]{data-label="fig:global_noewpo"}](Higgsupdate_noewpoMINS-1_logo){width="70.00000%"}
Before we attempt a proper global analysis of the Higgs and electroweak gauge sector we can ask what the impact of the additional 13 TeV data given in Tabs. \[tab:higgs\_data\] and \[tab:vv\_data\] is. Aside from a generic improvement in many of the standard measurements, we expect a significant impact from the new $t\bar{t}H$ measurements, the significant observation of fermionic Higgs decays, and from the re-casted $m_{VH}$ distribution to very large energies. In Fig. \[fig:global\_noewpo\] we indeed see that the limits on $f_t$, $f_b$ and $f_\tau$ have improved by more than a factor of two. Obviously, the top Yukawa measurement directly affects the Higgs coupling to gluons, $\ope_{GG}$, because it can only be extracted after we subtract the measured top loop contribution. Because $\ope_{\phi 2}$ leads to a Higgs wave function renormalization and $\ope_b$ modifies the total Higgs width, they are strongly correlated in the global analysis. After Run II they are not only well determined, both of them also show symmetric Gaussian log-likelihood distributions. We also see a very significant improvement in the limit on $f_W$ and $f_B$, which is driven by associated $VH$ production. However, from Fig. \[fig:toys\] we know that the error bar on $f_W$ is by no means symmetric and Gaussian. The operators showing the least improvement compared to Run I are $\ope_{WW}$ and $\ope_{BB}$, reflecting the lack of high-impact kinematic WBF measurements in the Run II data set. Moreover, $\ope_{WWW}$ only affects the gauge sector, and in Tab. \[tab:vv\_data\] we see that the analysis is still dominated by a set of extremely successful kinematic measurements at Run I.
Finally, our global limit on the Higgs branching ratio to invisible particles is $$\begin{aligned}
\text{BR}_\text{inv} < 38\%
\qquad \text{at $95\%$~CL,}\end{aligned}$$ with a best-fit point of $\text{BR}_\text{inv} = 14\%$. This is significantly weaker than the limits quoted for example by CMS [@CMS13WBFInv], because our global analysis does not assume the underlying Higgs production rates to be SM-like. Indeed, we observe a strong correlation of the invisible branching ratio with $\ope_{\phi 2}$ and its universal Higgs wave function renormalization. If rather than profiling over it we fix $f_{\phi 2} =
0$, our limit becomes $\text{BR}_\text{inv} < 26\%$ in agreement with the experimental results. Altogether, we find that Run II systematically probes energy scales $\Lambda/\sqrt{f}$ between 400 GeV and 800 GeV through Higgs measurement.
![Invariant mass distribution $m_{ZH}$ normalized to the Standard Model. The shaded areas correspond to $\Lambda/\sqrt{|f_B|} < 380$ GeV and $\Lambda/\sqrt{|f^{(1)}_{\phi
u}|} < 2.6$ TeV, where the interference with the Standard Model amplitude can have either sign.[]{data-label="fig:mzh_dist"}](figures/mZh_dist_fb_fphiu1){width=".65\textwidth"}
(100,70)
(100,70)
(70,70)
![Correlations between the fermionic and bosonic operators (top row), and between the usual bosonic operators (bottom row). For the latter we show the purely LHC results (left) and the results after including the additional fermionic operators.[]{data-label="fig:corr_phiq3_w"}](corr_BRinv2MINS-1/corr_D6fw_D6fphiQ3_BRinv2MINS-1 "fig:"){width="35.00000%"} ![Correlations between the fermionic and bosonic operators (top row), and between the usual bosonic operators (bottom row). For the latter we show the purely LHC results (left) and the results after including the additional fermionic operators.[]{data-label="fig:corr_phiq3_w"}](corr_BRinv2MINS-1/corr_D6fb_D6fphiu1_BRinv2MINS-1 "fig:"){width="35.00000%"}\
![Correlations between the fermionic and bosonic operators (top row), and between the usual bosonic operators (bottom row). For the latter we show the purely LHC results (left) and the results after including the additional fermionic operators.[]{data-label="fig:corr_phiq3_w"}](corr_noewpoMINS-1/corr_D6fw_D6fb_noewpoMINS-1 "fig:"){width="35.00000%"} ![Correlations between the fermionic and bosonic operators (top row), and between the usual bosonic operators (bottom row). For the latter we show the purely LHC results (left) and the results after including the additional fermionic operators.[]{data-label="fig:corr_phiq3_w"}](corr_BRinv2MINS-1/corr_D6fw_D6fb_BRinv2MINS-1 "fig:"){width="35.00000%"}
The large improvement of the limits on $\ope_B$ at Run II forces us to consider the interplay with the fermionic operators from Eq. and their limits from electroweak precision data, Eq.. From a scale separation point of view it is seems counter-intuitive that $\ope_{\phi u}^{(1)}$ or $\ope_{\phi Q}^{(3)}$, for which $\Lambda/\sqrt{f}$ is constrained around one order of magnitude more strongly than for $\ope_W$ and much more strongly for all other operators shown in Fig. \[fig:global\_noewpo\], should have any effect on the LHC analysis [@fermionic]. In Fig. \[fig:mzh\_dist\] we see how the fermionic and bosonic operators affect for example $ZH$ production. The key observation is that the fermionic operator contributes via the 3-point $qqZ$ and the 4-point $qqHZ$ vertices, whereas the bosonic operators require the same $s$-channel $Z$-propagator we see in the Standard Model. We show the corresponding Feynman diagrams in Fig. \[fig:feyn\_vh\]. From the structure of the dimension-6 operator we can infer the scalings $$\begin{aligned}
\frac{g f_{\phi Q} \, v^2}{\Lambda^2}
\quad (qqZ)
\qqquad \text{versus} \qqquad
\frac{g f_{\phi Q} \, v}{\Lambda^2}
\quad (qqZH) \qqquad
\label{eq:energy_scaling}\end{aligned}$$ The $m_{ZH}$ distribution shown in Fig. \[fig:mzh\_dist\] is one of our most powerful observables. We have confirmed that for the fermionic operator it is entirely dominated by the 4-point interaction, even though the 3-point interaction does interfere with the Standard Model. This is due to the energy scaling in Eq., which will eventually also lead to unitarity violation [@unitarity].
It is interesting to see how two operators with an apparently very different new physics scale contribute to the $m_{ZH}$ distribution at around the same rate. This can be understood by the definitions of the operators, where the 4-point contribution from $\ope_{\phi u}^{(1)}$ lacks a second power of the coupling, while the definition of $\ope_B$ adds two powers of the coupling to a 3-point vertex. Over most of the parameter range shown in Fig. \[fig:mzh\_dist\] the dimension-6-squared contribution dominates, giving us a mis-match of four powers of the coupling just from the definitions of the Wilson coefficients.
We confirm these findings in Fig. \[fig:corr\_phiq3\_w\], where we show the resulting correlations in our global analysis, once we include the full Lagrangian of Eqs. and . We see a clear correlation between $f_B$ and $f_{\phi u}^{(1)}$ from $ZH$ production, as well as between $f_W$ and $f_{\phi Q}^{(3)}$ from $WH$ production. This correlation relates very different values of the new physics scales for the fermionic and bosonic operators. In the lower panels we see how this weakens the limits on the bosonic operators $f_B$ and $f_W$ after profiling over the fermionic Wilson coefficients, and how it re-induces a correlation between them.
All of this discussion clearly defines a new challenge for global Higgs analyses once we reach Run II levels of precision: we need to include the additional operators shown in Eq. [@fermionic; @runII_eng; @runII_concha]. As argued above, this is at least in part due to a relative enhancement of the fermionic Higgs-gauge operators through their 4-point interactions. We show the result of our global analyses in Fig. \[fig:global\_all\], both at the 68% and 95% confidence levels. As LHC observables we consider the same measurements as Fig. \[fig:global\_noewpo\], but now combined with electroweak precision observables and including an extended set of operators. While the triple-gluon operator $\ope_G$ and the chromo-magnetic operator $\ope_{tG}$ appear in a global Higgs analysis, we have shown in Sec. \[sec:top\_sector\] that their best limits come from dedicated studies and after considering these limits their effects on the Higgs observables will not be visible. We therefore include them in the SMEFT-like result shown in Fig. \[fig:global\_all\], but quote the constraints from non-Higgs analyses.
First, we see that the 68% and 95% confidence limits scale like we would expect from Gaussian uncertainties. Directly comparing the results for the bosonic operators without and with the fermionic operators we see that as expected from Fig. \[fig:corr\_phiq3\_w\] the results on $f_B$ are roughly a factor of two weaker once we profile over the fermionic Wilson coefficients. We also see weaker limits on $f_W$ and $f_{\phi 2}$, which propagate through the entire effective Lagrangian describing the global analysis.
The constraints from our global analysis on the fermionic Higgs-gauge operators are typically a factor 10 to 100 stronger than for the bosonic operators. With $f_{\phi Q}^{(3)}$ and $f_{\phi d}^{(1)}$ the global fit also constrains operators which are relatively poorly probed by electroweak precision observables alone. These limits are in the range of $\Lambda/\sqrt{f} \approx 3$ TeV at 68% CL, indicating that LHC observables can also be especially sensitive to these operators. Again, for these results it is crucial that our global Higgs analysis covers Higgs observables and di-boson observables at the LHC, combined with electroweak precision data.
![Allowed 68% and 95% CL ranges for individual Wilson coefficients $f_x/\Lambda^2$ from a one-dimensional profile likelihood. All results include the Run II measurements combined with electroweak precision data. We quote the best results for $\ope_G$ [@krauss] and $\ope_{tG}$ [@topfitter] from non-Higgs analyses.[]{data-label="fig:global_all"}](Higgsupdate_BRinv2MINS-1_all_logo){width="99.00000%"}
Summary {#sec:conclu}
=======
We have presented a global analysis of the LHC Run I and Run II measurements related to Higgs and di-boson measurements in the framework of an effective Lagrangian to dimension six. The increasingly strong constraints from Run II and especially the developing LHC sensitivity to anomalous gauge boson couplings to quarks require a combination of the LHC analysis with electroweak precision data. In our global Higgs and electroweak analysis we include 18 bosonic and fermionic dimension-6 operators. For two more operators we quote limits from other analyses, after confirming that they are more constraining than our Higgs analysis. Finally, we include invisible Higgs decays through their branching ratio. This set of operators defines a significant step towards a global SMEFT analysis in the LHC era and towards a global precision analysis of LHC data.
In the <span style="font-variant:small-caps;">SFitter</span> framework we directly analyze ATLAS and CMS measurements rather than pre-defined pseudo-observables, include correlations for systematic and theoretical uncertainties, and exploit kinematic distributions to large momentum transfer. For LHC data alone we find that all limits from Run I are consistently improved by Run II, especially in the Yukawa sector and from the kinematic measurements of $VH$ production. At 95% CL the typical Run II limits range around $\Lambda/\sqrt{f} = 400~...~800$ GeV. Through new 4-point vertices fermionic Higgs-gauge operators have an anomalously large effect on associated Higgs production. This induces strong correlations between fermionic operators and $f_{B,W}$, in spite of stringent constraints from electroweak precision data. Profiling over the fermionic Wilson coefficients weakens the limits on $f_B$ by a factor two. At the same time, LHC observables allow us to constrain fermionic operators like $f_{\phi d}^{(1)}$ far beyond the reach of electroweak precision data, indicating that the interaction between the two sectors of our global fit is mutual. For several bosonic operators our analysis probes $\Lambda/\sqrt{f}$ values up to the TeV range, while the fermionic Higgs-gauge operators are consistently constrained to $5~...~10$ TeV.
**Acknowledgments**
We are grateful to Dirk Zerwas and Michael Rauch for their continuous support of <span style="font-variant:small-caps;">SFitter</span> and for many discussions related to LHC measurements. We would like to warmly thank Tatjana Lenz and Ruth Jacobs for their help with the $m_{VH}$ distribution. We would like to thank the DAAD Australia exchange program for funding our project *Precision Higgs Physics at the LHC* (57390316). AB is funded through the Graduiertenkolleg *Particle physics beyond the Standard Model* (GRK 1940). TC acknowledges generous support from the Villum Fonden and partial support by the Danish National Research Foundation (DNRF91) through the Discovery centre. The authors acknowledge support by the state of Baden-Württemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO).
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[^1]: Before trying to prove for example $CP$-violation through a global fit we advocate dedicated $CP$ tests for the Higgs and gauge sector [@cp].
[^2]: We would happily thank ATLAS for help with this analyis result and we are grateful to the actual authors communicating with us. However, our EFT analysis was officially considered as no appropriate re-casting of a $VH$ resonance search, so there is nothing we can thank ATLAS for.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A poset is $(\chain{r}+\chain{s})$-free if it does not contain two incomparable chains of size $r$ and $s$, respectively. We prove that when $r$ and $s$ are at least $2$, the First-Fit algorithm partitions every $(\chain{r}+\chain{s})$-free poset $P$ into at most $8(r-1)(s-1)w$ chains, where $w$ is the width of $P$. This solves an open problem of Bosek, Krawczyk, and Szczypka ([*SIAM J. Discrete Math., 23(4):1992–1999, 2010*]{}).'
address:
- |
Département d’Informatique\
Université Libre de Bruxelles\
Brussels, Belgium
- |
Department of Mathematics\
University of South Carolina\
Columbia, South Carolina
author:
- Gwenaël Joret
- 'Kevin G. Milans'
bibliography:
- 'paper.bib'
title: 'First-Fit is Linear on Posets Excluding Two Long Incomparable Chains'
---
[^1]
[^2]
Introduction
============
A [chain]{} in a poset is a set of elements that are pairwise comparable, and an [antichain]{} is a set of elements that are pairwise incomparable. The [height]{} of a poset is the size of a largest chain, and the [width]{} is the size of a largest antichain. In the [on-line chain partitioning problem]{}, the elements of an unknown poset $P$ are revealed one by one in some order. Each time a new element $x$ is presented, one has to assign a color to $x$, maintaining the property that each color class is a chain. The goal is to minimize the number of chains in the resulting chain partition of $P$.
This classical problem has received increased attention in the recent years; see, for example, the survey by Bosek, Felsner, Kloch, Krawczyk, Matecki, and Micek [@BFKKMM_sub]. In this context, the quality of a solution is typically compared against the width $w$ of $P$. Since elements of an antichain must receive distinct colors, at least $w$ colors are needed. By Dilworth’s theorem, if all elements of $P$ are presented before any are colored, then $w$ colors suffice. In the on-line setting, more colors are needed.
Let ${\mathrm{val}}(w)$ be the least $k$ such that there is an on-line algorithm that partitions posets of width $w$ into at most $k$ chains. Establishing that ${\mathrm{val}}(w)$ is finite when $w\ge 2$ is challenging. In 1981, Kierstead [@K_CM] proved that ${\mathrm{val}}(w) \le (5^w - 1)/4$. For nearly three decades, Kierstead’s result was the best known upper bound on ${\mathrm{val}}(w)$. Recently, Bosek and Krawczyk [@BK_prepa] showed that ${\mathrm{val}}(w) \le w^{16 \lg w}$ (see [@BFKKMM_sub] for a proof sketch). From below, Szemerédi proved that ${\mathrm{val}}(w) \ge \binom{w+1}{2}$ (see [@K_CM; @BFKKMM_sub]), and Bosek [*et al.*]{} [@BFKKMM_sub] showed that ${\mathrm{val}}(w) \ge (2-o(1))\binom{w+1}{2}$. One of the central questions in the theory of on-line problems on partial orders is whether ${\mathrm{val}}(w)$ is bounded above by a polynomial in $w$.
In this paper, we are interested in the performance of an on-line chain partitioning algorithm called First-Fit. Using the positive integers for colors, First-Fit colors $x$ with the least $j$ such that $x$ and all elements previously assigned color $j$ form a chain. It is known that, for general posets, the number of chains used by First-Fit is not bounded by a function of $w$. In fact, Kierstead [@K_CM] showed that First-Fit uses arbitrarily many chains on posets of width $2$ (see also [@BKS_SIDMA]).
Nevertheless, First-Fit performs well on certain classes of posets, such as interval orders. An *interval order* is a poset whose elements are closed intervals on the real line, with $[a,b] < [c,d]$ if and only if $b<c$. Let ${\mathrm{FF}}(w)$ be the maximum number of chains that First-Fit uses on interval orders of width $w$. Kierstead [@K_SIDMA] proved that ${\mathrm{FF}}(w)\le 40w$. Kierstead and Qin [@KQ_DM] subsequently improved the bound, showing that ${\mathrm{FF}}(w)\le 25.8w$. Later, Pemmaraju, Raman, and Varadarajan [@PRV_SODA] (see also [@PRV_TALG]) proved that ${\mathrm{FF}}(w)\le 10w$ with an elegant argument known as the Column Construction Method. Their proof was later refined by Brightwell, Kierstead, and Trotter [@BKT_mscript] and independently by Narayanaswamy and Babu [@NS_ORDER] to show that ${\mathrm{FF}}(w)\le 8w$.
From early results of Kierstead and Trotter [@KT_SCCGTC], it follows that ${\mathrm{FF}}(w) \ge (3+\eps)w$ for some positive $\eps$. Chrobak and Ślusarek [@CS_RAIRO] showed that ${\mathrm{FF}}(w) \ge 4w-9$ when $w\ge 4$ and subsequently improved the multiplicative constant to $4.45$ at the expense of a weaker additive constant. In 2004, Kierstead and Trotter [@KT_unpub] proved that ${\mathrm{FF}}(w) \ge 4.99w - c$ for some constant $c$ with the aid of a computer. Recently, Kierstead, Smith, and Trotter [@KST_unpub] proved that for each positive $\eps$, there is a constant $c$ such that ${\mathrm{FF}}(w) \ge (5-\eps)w - c$.
If $P$ and $Q$ are posets, then $P+Q$ denotes the poset obtained from disjoint copies of $P$ and $Q$ where elements in the copy of $P$ are incomparable to elements in the copy of $Q$. A poset $P$ is [$Q$-free]{} if no induced subposet of $P$ is isomorphic to $Q$. We denote by [$\chain{r}$]{} the poset consisting of a chain of size $r$. Fishburn [@F_JMP] characterized the interval orders as the posets that are $(\chain{2}+\chain{2})$-free. When $r$ and $s$ are at least two, the family of $(\chain{r}+\chain{s})$-free posets contains the family of interval orders. Bosek, Krawczyk, and Szczypka [@BKS_SIDMA] showed that when $r\ge s$, First-Fit partitions every $(\chain{r}+\chain{s})$-free poset into at most $(3r - 2)(w - 1)w + w$ chains. They asked whether First-Fit uses only a linear number of chains, in terms of $w$, on $(\chain{r}+\chain{s})$-free posets, as it does on interval orders. This question also appears in the survey of Bosek [*et al.*]{} [@BFKKMM_sub] and in a recent paper of Felsner, Krawczyk, and Trotter [@FKT_sub].
We give a positive answer to this question by showing that First-Fit partitions every $(\chain{r}+\chain{s})$-free poset into at most $8(r-1)(s-1)w$ chains. As far as we know, this also provides the first proof that some on-line algorithm uses $o(w^{2})$ chains on $(\chain{r}+\chain{s})$-free posets. Our proof is strongly influenced by the Column Construction Method of Pemmaraju [*et al.*]{} [@PRV_TALG] and can be viewed as a generalization of that technique from interval orders to $(\chain{r}+\chain{s})$-free posets.
In Section \[sec\_evo\], we present our generalization of the Column Construction Method and establish several of its properties. In Section \[sec\_thm\], we combine these results with a structural lemma about $(\chain{r}+\chain{s})$-free posets to obtain our main result.
Evolution of Societies {#sec_evo}
======================
Let $P$ be a poset. A [[First-Fit chain partition]{}]{} is an ordered partition $C_{1}, \dots, C_{m}$ of $P$ into non-empty chains such that if $i<j$ and $x\in C_j$, then some element in $C_i$ is incomparable to $x$. Note that if $C_1, \ldots, C_m$ is a [First-Fit chain partition]{}, then First-Fit produces this partition when elements in $C_1$ are presented first, followed by elements in $C_2$, and continuing through elements in $C_m$. Conversely, every ordered partition produced by First-Fit is a [First-Fit chain partition]{}.
A [group]{} is a set of elements in $P$. A [$t$-society]{} is a pair $(S,F)$ where $S$ is a set of groups and $F$ is a [friendship function]{} from $S\times [t]$ to $S\cup\{\star\}$, where $[t]$ denotes the set $\{1,\ldots,t\}$. Each group $X\in S$ has slots for up to $t$ friends. We say that [$X$ lists $Y$ as a friend in slot $k$]{} if $F(X,k)=Y$. It is possible that $X$ does not list any friend in slot $k$, in which case $F(X,k)=\star$.
The overview of our proof is as follows. Given an poset $P$, we first exploit the structure of $P$ to define an initial $t$-society $(S_0,F_0)$ for some $t$ depending on $s$. Next, we fix a [First-Fit chain partition]{} $C_1, \ldots, C_m$, which we extend to an infinite sequence of chains by defining $C_j = \ns$ for $j>m$. We allow the initial $t$-society to evolve, generating a sequence of $t$-societies $(S_0,F_0),\ldots,(S_n,F_n)$. For $j\ge 1$, the $t$-society $(S_j,F_j)$ is obtained from $(S_{j-1},F_{j-1})$ by following certain rules that depend on $C_j$ and the previous transitions. It is helpful to view the $t$-societies as vertices of a path and to associate the edge joining $(S_{j-1},F_{j-1})$ and $(S_j,F_j)$ with the chain $C_j$.
During the evolution, we maintain that $S_0 \supseteq S_1 \supseteq \cdots \supseteq S_n$. The evolution ends when a $t$-society $(S_n,F_n)$ is generated where $S_n = \ns$. The proof proceeds in two parts. First, we show that a long evolution implies that some group in the initial $t$-society is large. Second, given an poset $P$, we show how to construct an initial $t$-society of groups inducing subposets of height at most $r-1$ that leads to a long evolution. Because large posets of bounded height contain large antichains, we obtain a lower bound on the width of $P$.
In our societies, friendship is a lifetime commitment: if $F_{j-1}(X,k)=Y$ and $\{X,Y\}\subseteq S_j$, then $F_j(X,k)=Y$. If $X$ survives the transition from $S_{j-1}$ to $S_j$ but $Y$ does not, then $X$ either chooses a new friend for its $k$th slot or leaves its $k$th slot empty according to the rules of a [replacement scheme]{}. We postpone the presentation of the details of our replacement scheme and the construction of the initial $t$-society.
A group $X$ may survive the transition from $S_{j-1}$ to $S_j$ in three ways, each of which defines a transition type. We use the first three Greek letters $\alpha$, $\beta$, and $\gamma$ to name the transition types. When $a \in \{\alpha,\beta,\gamma\}$ and $i\le j$, we define $N_{i,j}^a(X)$ to be the number of transitions of type $a$ that $X$ makes in the evolution from $(S_i,F_i)$ to $(S_j,F_j)$.
Let $\eps = 1/2t$; in Lemma \[lem:large-group\], we will find that this choice of $\eps$ is optimal. We now describe the rules that govern which groups survive the $j$th transition from $S_{j-1}$ to $S_j$. Let $X$ be a group in $S_{j-1}$.
1. If $X$ has non-empty intersection with $C_j$, then $X$ makes an $\alpha$-transition from $S_{j-1}$ to $S_j$.
2. Otherwise, if some friend of $X$ in the $t$-society $(S_{j-1},F_{j-1})$ has non-empty intersection with $C_j$, then $X$ makes a $\beta$-transition from $S_{j-1}$ to $S_j$.
3. Otherwise, if there is an $i$ such that $N_{i,j-1}^\alpha(X) > \eps(j-i)$, then $X$ makes a $\gamma$-transition from $S_{j-1}$ to $S_j$.
If none of the three rules apply, then $X\not\in S_j$, and other groups that list $X$ as a friend and survive to $S_j$ update their list of friends according to the replacement scheme.
First, we show that a long evolution implies that some group is large. We need several lemmas.
\[lem:single-col\] Fix an evolution $(S_0,F_0),\ldots,(S_n,F_n)$ of $t$-societies. Let $Y_1, Y_2, \ldots, Y_q$ be a list of groups and let $[a_1,b_1], \ldots, [a_q,b_q]$ be a sequence of disjoint intervals with integral endpoints in $[0,n]$ such that $b_j$ is the largest integer such that $Y_j \in S_{b_j}$. The sum $\sum_{j=1}^q N_{a_j,b_j}^\alpha(Y_j)$ is at most $\eps n$.
If $a_j=b_j$, then clearly $N_{a_j,b_j}^\alpha(Y_j) = 0$. Hence, we may assume that $0 \le a_1<b_1<\cdots <a_q <b_q$. Also, $b_q < n$ because $S_n = \ns$. Note that $Y_j \not\in S_{b_j+1}$. It follows that $Y_j$ did not satisfy the third condition in the transition from $S_{b_j}$ to $S_{b_j + 1}$ and therefore $N_{a_j,b_j}^\alpha (Y_j) \le \eps(b_j + 1 - a_j)$. Also, $\sum_{j=1}^q (b_j+1 - a_j) \le n$ because the intervals $[a_j,b_j]$ are disjoint subsets of $[0,n-1]$ with integral endpoints.
Our next lemma provides a bound on the number of $\beta$-transitions that a group can make if it survives to the last non-empty $t$-society.
\[lem:few-betas\] Fix an evolution $(S_0,F_0),\ldots,(S_n,F_n)$ of $t$-societies. If $X \in S_{n-1}$, then $N_{0,n-1}^\beta(X) \le t\eps n$.
Let $X \in S_{n-1}$, and for each $k\in[t]$, let $\mc{Y}_k$ be the set of groups that $X$ lists as a friend in slot $k$ at some point in the evolution. If $X$ makes a $\beta$-transition from $S_{j-1}$ to $S_j$, then there is a slot $k$ and group $Y\in \mc{Y}_k$ such that $F_{j-1}(X,k) = Y$ and $Y$ has non-empty intersection with $C_j$. Because $Y \in S_{j-1}$ and $Y$ has non-empty intersection with $C_j$, we have that $Y$ makes an $\alpha$-transition from $S_{j-1}$ to $S_j$. It follows that $$N_{0,n-1}^\beta(X) \le \sum_{k=1}^t \sum_{Y\in\mc{Y}_k} N_{I(Y)}^\alpha(Y)$$ where $I(Y)$ is denotes the interval during which $X$ lists $Y$ as a friend. (Formally, $j\in I(Y)$ if and only if $F_j(X,k)=Y$ for some $k\in[t]$.) It suffices to show that $\sum_{Y\in\mc{Y}_k} N_{I(Y)}^\alpha(Y) \le \eps n$ for each $k\in[t]$. Because $\{I(Y)\st Y\in\mc{Y}_k\}$ are disjoint intervals, the bound follows from Lemma \[lem:single-col\].
Next, we show that for each group $X$, the $\alpha$-transitions that $X$ makes constitute a large fraction of the total number of $X$’s transitions not of type $\beta$.
\[lem:many-alphas\] Fix an evolution $(S_0,F_0),\ldots,(S_n,F_n)$ of $t$-societies. If $X$ is a group, then $N_{0,j}^\alpha(X) \ge \eps(N_{0,j}^\alpha(X) + N_{0,j}^\gamma(X))$ for each $j$ with $X\in S_j$.
If $j=0$, then the inequality holds. For $j\ge 1$, the inequality holds immediately by induction unless $X$ makes a $\gamma$-transition from $S_{j-1}$ to $S_j$. In this case, there is some $i$ such that $N_{i,j-1}^\alpha(X) > \eps (j-i)$. Applying the inductive hypothesis to obtain a lower bound on $N_{0,i}^\alpha(X)$, it follows that $$\begin{aligned}
N_{0,j}^\alpha(X) &= N_{0,i}^\alpha(X) + N_{i,j-1}^\alpha(X) \\
& \ge \eps(N_{0,i}^\alpha(X)+N_{0,i}^\gamma(X)) + \eps(j-i) \\
& \ge \eps(N_{0,i}^\alpha(X)+N_{0,i}^\gamma(X)) + \eps(N_{i,j}^\alpha(X)+N_{i,j}^\gamma(X)) \\
& = \eps(N_{0,j}^\alpha(X)+N_{0,j}^\gamma(X))\end{aligned}$$ as required.
We are now able to show that a long evolution implies that some group is large.
\[lem:large-group\] Fix an evolution $(S_0,F_0),\ldots,(S_n,F_n)$ of $t$-societies. If $X\in S_{n-1}$, then $|X| \ge (n-2)/4t$.
Whenever $X$ makes an $\alpha$-transition from $S_{j-1}$ to $S_j$, it has non-empty intersection with chain $C_j$. Because the chains are disjoint, it follows that $|X|\ge N_{0,n-1}^\alpha(X)$. By Lemma \[lem:many-alphas\], we have that $N_{0,n-1}^\alpha(X) \ge \eps(N_{0,n-1}^\alpha(X) + N_{0,n-1}^\gamma(X))$. Note that $X$ makes $n-1$ transitions in total, because $X\in S_{n-1}$. Hence $N_{0,n-1}^\alpha(X) + N_{0,n-1}^\beta(X) + N_{0,n-1}^\gamma(X) = n-1$. By Lemma \[lem:few-betas\], we have that $N_{0,n-1}^\alpha(X) + t\eps n + N_{0,n-1}^\gamma(X) \ge n-1$. Consequently, $N_{0,n-1}^\alpha(X) \ge \eps(n-1 - t \eps n)$. With $\eps = 1/2t$, we obtain $N_{0,n-1}^\alpha(X) \ge (n-2)/4t$ as required.
The Initial Society and Replacement Scheme {#sec_thm}
==========================================
It remains to describe the initial $t$-society and our replacement scheme. Both depend on the following structural lemma about posets. The [height]{} of an element $x$, denoted $h(x)$, is the size of a largest chain with maximum element $x$.
\[lem:init-society\] Let $r$ and $s$ be integers with $r\ge 2$ and $s\ge 2$, and let $P$ be an poset. There is a function $I$ which assigns to each element $x\in P$ a non-empty set of consecutive integers $I(x)$ with the following properties.
1. For each integer $k$, the set $\{x\in P\st k\in I(x)\}$ induces a subposet of height at most $r-1$.
2. If $x$ and $y$ are incomparable in $P$, then either $I(x)$ and $I(y)$ have non-empty intersection, or at most $s-2$ integers are strictly between $I(x)$ and $I(y)$.
Let $q$ be the height of $P$. For each $x\in P$, let $Z(x)$ be the set of all elements $z$ such that $P$ contains a chain of size $r$ with minimum element $x$ and maximum element $z$. When $Z(x)$ is non-empty, define $b(x)$ to be the minimum height of an element in $Z(x)$; we set $b(x)=q+1$ when $Z(x)=\ns$. Let $I(x) = \{h(x), \ldots, b(x) - 1\}$.
Fix an integer $k$ and let $X = \{x\in P\st k\in I(x)\}$. Suppose for a contradiction that $X$ contains a chain $x_1 < \cdots < x_r$. Since $x_r \in X$, we have that $k \in I(x_r)$, which implies that $h(x_r) \le k$. Similarly, $k\in I(x_1)$ and therefore $k \le b(x_1) - 1$. Since $x_r \in Z(x_1)$, it follows that $b(x_1)\le h(x_r)$. Hence $h(x_r)\le k \le h(x_r) - 1$, a contradiction. It follows that (1) holds.
It remains to check (2). Suppose that $x$ and $y$ are incomparable. If $I(x)$ and $I(y)$ have non-empty intersection, then (2) holds. Hence, we may assume that every integer in $I(x)$ is less than every integer in $I(y)$. Let $i$ be the greatest integer in $I(x)$ and let $j$ be the least integer in $I(y)$, and note that $i<j\le q$. Since $i\in I(x)$ but $i+1\not\in I(x)$, it follows that $b(x)-1 = i$. Because $i<q$, it follows that $b(x)=i+1\le q$ and therefore $Z(x) \ne\ns$. Hence, there is a chain $x=x_1<\cdots <x_r$ in $P$ with $h(x_r) = i+1$. Similarly, $h(y) = j$ and there is a chain $y = y_j > \cdots > y_1$ in $P$ with $h(y_k)=k$ for each $k\in [j]$. Let $X=\{x_1,\ldots,x_r\}$ and let $Y=\{y_{i+1},\ldots,y_j\}$. We claim that every element in $X$ is incomparable to every element in $Y$. If $x_a \le y_b$, then transitivity implies that $x = x_1 \le y_j = y$, contrary to the assumption that $x$ and $y$ are incomparable. Conversely, if $y_a \le x_b$, then transitivity implies that $y_{i+1}\le x_r$. But $y_{i+1}\le x_r$ is impossible because $y_{i+1}$ and $x_r$ are distinct (since $x_{r} \not \leq y_{i+1}$) and have the same height. Hence every element in $X$ is incomparable to every element in $Y$ as claimed.
It follows that $X \cup Y$ induces a copy of $\chain{r} + \chain{j-i}$ in $P$. Because $P$ is , we have that $j-i \le s-1$ and therefore the set of integers $\{i+1,\ldots,j-1\}$ strictly between $I(x)$ and $I(y)$ has size at most $s-2$.
We now have the tools necessary to describe the initial $t$-society and our replacement scheme. While our transition rules require only that each $S_j$ is a set of groups, our replacement scheme imposes additional structure on $S_j$. In particular, our replacement scheme treats $S_j$ as a list of groups. Let $q$ be the height of $P$. With $I$ as in Lemma \[lem:init-society\], we define $X_k = \{x\in P\st k\in I(x)\}$ when $1\le k \le q$ and set $S_0 = X_1,\ldots,X_q$. This ordering is preserved throughout the evolution: if $Y$ appears before $Z$ in $S_0$ and $\{Y,Z\} \subseteq S_j$, then $Y$ also appears before $Z$ in $S_j$. When $L$ is a list of objects $a_1, \ldots, a_n$, we define $\dist_L(a_i,a_j) = |j-i|$. For convenience, when $Y$ and $Z$ are groups in $S_j$, we define $\dist_j(Y,Z) = \dist_{S_j}(Y,Z)$.
Let $t=2(s-1)$. In the initial $t$-society $(S_0, F_0)$, we define $F_0$ so that if $Y$ and $Z$ are distinct groups in $S_0$ with $\dist_0(Y,Z) \le s-1$, then $F(Y,k) = Z$ for some slot $k$. If fewer than $2(s-1)$ groups in $S_0$ are at distance at most $s-1$ from $Y$, then some slots are empty (formally, $F(Y,k) = \star$). Our replacement scheme maintains that in $t$-society $(S_j,F_j)$, a group $Y$ lists as friends all other groups $Z$ such that $\dist_j(Y,Z) \le s-1$. This is possible to maintain since $\dist_j(Y,Z) < \dist_{j-1}(Y,Z)$ only occurs when some group $Z' \in S_{j-1}$ with $\dist_{j-1}(Y,Z') < \dist_{j-1}(Y,Z)$ does not survive the transition from $(S_{j-1},F_{j-1})$ to $(S_j,F_j)$. It follows that at least as many of $Y$’s friendship slots become available as are needed to accommodate the groups $Z$ with $\dist_{j-1}(Y,Z)>s-1$ and $\dist_{j}(Y,Z)\le s-1$. Our replacement scheme places these groups in $Y$’s available friendship slots arbitrarily. As before, unused slots are assigned the value $\star$.
Our next aim is to show that our initial $t$-society and replacement scheme lead to a long evolution. We first prove an analogue of Lemma 4.2 in [@PRV_TALG].
\[lem:lem1-analogue\] Let $C_1, \ldots, C_m$ be a [First-Fit chain partition]{}, and define $C_j = \ns$ for $j>m$. Let $(S_0,F_0)$ be our initial $t$-society, and let $(S_0,F_0),\ldots,(S_n,F_n)$ be the evolution resulting from our replacement scheme. For each $i$, we have that $\bigcup_{X\in S_i} X \supseteq \bigcup_{j>i} C_j$.
By induction on $i$. By Lemma \[lem:init-society\], $I(x) \ne \ns$ for each element $x$, and therefore $\bigcup_{X\in S_0} X$ contains all elements in $P$.
Let $i\ge 1$ and consider an element $y\in C_j$ with $j>i$. Because $C_1,\ldots,C_m$ is a [First-Fit chain partition]{}, there is an element $z\in C_i$ such that $y$ and $z$ are incomparable. By induction, there are groups $Y\in S_{i-1}$ and $Z\in S_{i-1}$ with $y\in Y$ and $z\in Z$. Among all such pairs $\{Y,Z\}$, choose $Y$ and $Z$ to minimize $\dist_{i-1}(Y,Z)$. We claim that $\dist_{i-1}(Y,Z)\le s-1$. Indeed, if $\dist_{i-1}(Y,Z)\ge s$, then there are at least $s-1$ groups in $S_{i-1}$ that are strictly between $Y$ and $Z$ in the list $X_1, \ldots, X_q$. By our selection of $Y$ and $Z$, none of these groups contain $y$ or $z$. Hence, it follows that the index of each such group is strictly between $I(y)$ and $I(z)$, contradicting Lemma \[lem:init-society\].
Because $\dist_{i-1}(Y,Z) \le s-1$, our replacement scheme ensures that $Y$ lists $Z$ as a friend in some slot. Because $z \in Z \cap C_i$, some friend of $Y$ in $(S_{i-1},F_{i-1})$ has non-empty intersection with $C_i$. It follows that $Y$ either makes an $\alpha$-transition or a $\beta$-transition from $S_{i-1}$ to $S_i$. Hence $y\in Y \in S_i$ and therefore $y\in \bigcup_{X\in S_i}X$ as required.
\[lem:long-evolution\] Let $C_1, \ldots, C_m$ be a [First-Fit chain partition]{}, and define $C_j = \ns$ for $j>m$. Let $(S_0,F_0)$ be our initial $t$-society, and let $(S_0,F_0),\ldots,(S_n,F_n)$ be the evolution resulting from our replacement scheme. We have that $n\ge m+2$.
Let $y\in C_m$. By Lemma \[lem:lem1-analogue\], there is a group $Y\in S_{m-1}$ with $y\in Y$. Because $Y$ has non-empty intersection with $C_m$, we have that $Y$ makes an $\alpha$-transition from $S_{m-1}$ to $S_m$. Also, $N_{m-1,m}^\alpha(Y) = 1$ and $\eps((m+1)-(m-1)) = 2\eps = 1/t = 1/(2(s-1)) \le 1/2$, and therefore $Y$ is eligible to make a $\gamma$-transition from $S_m$ to $S_{m+1}$. Hence $Y\in S_{m+1}$. Because the evolution ends with an empty $t$-society, it follows that $n\ge m+2$.
Putting all the pieces together, we obtain our main theorem.
If $r$ and $s$ are at least $2$ and $P$ is an poset of width $w$, then First-Fit partitions $P$ into at most $8(r-1)(s-1)w$ chains.
Let $C_1,\ldots,C_m$ be a [First-Fit chain partition]{}, and define $C_j = \ns$ for $j>m$. Obtain our initial $t$-society $(S_0,F_0)$ from Lemma \[lem:init-society\], and let $(S_0,F_0),\ldots,(S_n,F_n)$ be the evolution obtained with our replacement scheme. By Lemma \[lem:long-evolution\], we have that $n\ge m+2$. By Lemma \[lem:large-group\], some group $X\in S_0$ has size at least $(n-2)/4t = (n-2)/(8(s-1)) \ge m/(8(s-1))$. By Lemma \[lem:init-society\], the height of $X$ is at most $r-1$. It follows that $X$ is the union of $r-1$ antichains, and therefore $w \ge |X|/(r-1) \ge m/(8(s-1)(r-1))$.
Concluding Remarks
==================
The following related problem is open: for which posets $Q$ of width $2$ is there a function $f_{Q}(w)$ such that First-Fit partitions every $Q$-free poset of width $w$ into at most $f_{Q}(w)$ chains? The same question applies when $f_{Q}(w)$ is restricted to be a polynomial or a linear function of $w$. We note that these problems are only interesting for posets $Q$ of width $2$. Indeed, there is a trivial linear bound when $Q$ is a chain, and the example of Kierstead [@K_CM] implies that no such function exists when the width of $Q$ is at least $3$.
Addendum {#addendum .unnumbered}
========
While this article was under review, Bosek, Krawczyk, and Matecki [@BKM_unpub] proved that for each poset $Q$ of width $2$, there is a function $f_{Q}(w)$ such that First-Fit partitions every $Q$-free poset of width $w$ into at most $f_{Q}(w)$ chains. Our second question remains open.
Acknowledgements {#acknowledgements .unnumbered}
================
This research was initiated while the authors were attending the First Montreal Spring School in Graph Theory, held in Montreal in May 2010. The authors are grateful to the organizers of the school for providing a stimulating working environment. The second author thanks William T. Trotter for engaging talks on the subject and for relaying the problem studied in this article.
[^1]: This work was supported in part by the Actions de Recherche Concertées (ARC) fund of the Communauté française de Belgique. The first author is a Postdoctoral Researcher of the Fonds National de la Recherche Scientifique (F.R.S.–FNRS)
[^2]: The second author acknowledges support of the National Science Foundation through a fellowship funded by the grant “EMSW21-MCTP: Research Experience for Graduate Students” (NSF DMS 08-38434).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
A Bianchi type-I cosmological model in the presence of a magnetic flux along a cosmic string is investigated. A nonlinear spinor field is used to simulate the cosmological cloud of strings. It is shown that the spinor field simulation offer the possibility to solve the system of Einstein’s equation without any additional assumptions. It is shown that the present model is nonsingular at the end of the evolution and does not allow the anisotropic Universe to turn into an isotropic one.
Pacs: 95.30.Sf; 98.80.Jk; 04.20.Ha
Key words: Bianchi type-I model, cosmological string, magnetic field, nonlinear spinor field
author:
- |
Bijan Saha [^1]\
[*Laboratory of Information Technologies*]{}\
[*Joint Institute for Nuclear Research, Dubna*]{}\
[*141980 Dubna, Moscow region, Russia*]{}
- |
Mihai Visinescu [^2]\
[*Department of Theoretical Physics*]{}\
[*National Institute for Physics and Nuclear Engineering*]{}\
[*Magurele, P. O. Box MG-6, RO-077125 Bucharest, Romania*]{}
title: 'Bianchi type-I model with cosmic string in the presence of a magnetic field: spinor description'
---
Introduction
============
Though the present day Universe is well described by an isotropic and homogeneous Friedmann-Robertson-Walker (FRW) model, there are serious theoretical arguments about the existence of an anisotropic phase in the evolution of the Universe [@misner]. These ideas were further supported by the observational data from COBE (Cosmic Background Explorer) and WMAP (The Wilkinson Microwave Anisotropy Probe) where small anisotropy in the microwave background radiation was found. These lead many cosmologists to consider the anisotropic model to describe the initial phase of the evolution which eventually decays into an isotropic FRW one.
After the famous paper by A. Guth [@guth], scalar field was hugely used in simulating different cosmological models. But the question occurs, if other fields can contribute to the evolution of the Universe. As an alternative the spinor field was used due to its sensitivity to the gravitational one [@henprd; @sahaprd; @greene; @SBprd04; @kremer1; @ECAA06; @sahaprd06; @BVI]. The nonlinear spinor field proved to be able to describe different cosmological models. For example it was shown that a suitable choice of nonlinearity (i) accelerates the isotropization process, (ii) gives rise to a singularity-free Universe and (iii) generates late time acceleration.
Moreover, it is shown that a nonlinear spinor field can be used to simulate a perfect fluid from ekpyrotic matter to phantom matter [@shikin; @spinpf0; @spinpf].
At the same time the string cosmological models have been used in attempts to describe the early Universe and to investigate anisotropic dark energy component including a coupling between dark energy and a perfect fluid (dark matter) [@KM; @KM1]. Cosmic strings are one dimensional topological defects associated with spontaneous symmetry breaking in gauge theories. Their presence in the early Universe can be justified in the frame of grand unified theories (GUT).
On the other hand, the magnetic field has an important role at the cosmological scale and is present in galactic and intergalactic spaces. Any theoretical study of cosmological models which contain a magnetic field must take into account that the corresponding Universes are necessarily anisotropic. Among the anisotropic spacetimes, Bianchi type-I space (BI) seems to be the most convenient for testing different cosmological models.
The object of this paper is to investigate a BI string cosmological model in the presence of a magnetic flux. For this purpose we use the nonlinear spinor field simulation as was described recently in a number of papers [@shikin; @spinpf0; @spinpf].
Basic equations {#basic}
===============
We study the evolution of the Universe in presence of a cosmic string and a magnetic flux in the framework of a BI anisotropic cosmological model. For a BI spacetime the line element is given by $$ds^2 = (dt)^2 - a_1^2(t) (dx^1)^2 - a_2^2(t)(dx^2)^2 - a_3^2(t)
(dx^3)^2\,. \label{BI}$$ There are three scale factors $a_i$ $(i=1,2,3)$ which are functions of time $t$ only and consequently three expansion rates. In principle all these scale factors could be different and it is useful to express the mean expansion rate in terms of the average Hubble rate: $$\label{Hubble}
H = \frac{1}{3}\Bigl(\frac{\dot a_1}{a_1}+\frac{\dot a_2}{a_2}+
\frac{\dot a_3}{a_3}\Bigr)\,,$$ where over-dot means differentiation with respect to $t$.
Einstein’s gravitational field equations, corresponding to the metric have the form
\[BID\] $$\begin{aligned}
\frac{\ddot a_2}{a_2} +\frac{\ddot a_3}{a_3} + \frac{\dot a_2}{a_2}\frac{\dot
a_3}{a_3}&=& \kappa T_{1}^{1}\,,\label{11}\\
\frac{\ddot a_3}{a_3} +\frac{\ddot a_1}{a_1} + \frac{\dot a_3}{a_3}\frac{\dot
a_1}{a_1}&=& \kappa T_{2}^{2}\,,\label{22}\\
\frac{\ddot a_1}{a_1} +\frac{\ddot a_2}{a_2} + \frac{\dot a_1}{a_1}\frac{\dot
a_2}{a_2}&=& \kappa T_{3}^{3}\,,\label{33}\\
\frac{\dot a_1}{a_1}\frac{\dot a_2}{a_2} +\frac{\dot
a_2}{a_2}\frac{\dot a_3}{a_3}+\frac{\dot a_3}{a_3}\frac{\dot
a_1}{a_1}&=& \kappa T_{0}^{0}\,, \label{00}\end{aligned}$$
where $\kappa$ is the gravitational constant. The energy momentum tensor for a system of cosmic string and magnetic field in a comoving coordinate is given by $$T_{\mu}^{\nu} = \rho u_\mu u^\nu - \lambda x_\mu x^\nu
+ E_\mu^\nu\,, \label{imperfl}$$ where $\rho$ is the rest energy density of strings with massive particles attached to them and can be expressed as $\rho = \rho_{p} +
\lambda$, where $\rho_{p}$ is the rest energy density of the particles attached to the strings and $\lambda$ is the tension density of the system of strings [@letelier; @pradhan; @tade] which may be positive or negative. Here $u_i$ is the four velocity and $x_i$ is the direction of the string, obeying the relations $$u_iu^i = -x_ix^i = 1, \quad u_i x^i = 0\,. \label{velocity}$$
In $E_{\mu\nu}$ is the electromagnetic field [@lich] and in what follows we shall choose the string and also the magnetic field along $x^1$ direction. In our model the electromagnetic field tensor $F^{\alpha \beta}$ has only one non-vanishing component, namely $$F_{23} = h\,,
\label{f23}$$ where $h$ is presumed to be constant. For the electromagnetic field $E_\mu^\nu$ one gets the following non-trivial components $$E_0^0 = E_1^1 = - E_2^2 = - E_3^3 = \frac{h^2}
{2 {\bar \mu} a_2^2 a_3^2}
\equiv \frac{1}{2}\frac{\beta^2}{(a_2 a_3)^2}\,.
\label{E}$$ where $\bar \mu$ is a constant characteristic of the medium and called the magnetic permeability. Typically $\bar \mu$ differs from unity only by a few parts in $10^{-5}$ ($\bar \mu > 1$ for paramagnetic substances and $\bar \mu < 1$ for diamagnetic). To simplify the notation, we include in a constant $\beta$ the value of the electromagnetic field, $h$, and the magnetic permeability, $\bar \mu$.
Using comoving coordinates we have the following components of energy momentum tensor [@ass]: $$\label{total}
T_{0}^{0} - \rho = T_{1}^{1} - \lambda = - T_{2}^{2} =
- T_{3}^{3} = \frac{\beta^2}{2}\frac{a_1^2}{\tau^2}\,,$$ where we introduce the volume scale of the BI space-time [@sahaprd] $$\tau = \sqrt{-g} = a_1 a_2 a_3 \,, \label{taudef}$$ which is connected with the Hubble rate , namely $
\frac{\dot \tau}{\tau} = 3 H\,.
$
In view of $T_{2}^{2} = T_{3}^{3}$ from one finds $$a_2 = a_3 D \exp \biggl(X \int \frac{dt'}{\tau}
\biggr)\,, \label{a2a3}$$ with $D$ and $X$ some integration constants.
On the other hand, summation of Einstein equations , , and three times gives: $$\frac{\ddot \tau}{\tau}= \frac{1}{2}\kappa \Bigl(3\rho + \lambda +
\beta ^2 \frac{a_1^2}{\tau^2} \Bigr)\,.
\label{dtau1}$$
Taking into account the conservation of the energy-momentum tensor, i.e., $T_{\mu;\nu}^{\nu} = 0$, after a little manipulation of one obtains [@SV; @SRV]: $$\label{rholambda}
\dot \rho + \frac{\dot \tau}{\tau}\rho - \frac{\dot a_1}{a_1}\lambda
= 0\,.$$
It is customary to assume a relation between $\rho$ and $\lambda$ in accordance with the state equations for strings. The simplest one is a proportionality relation [@letelier]: $$\label{rhoalphalambda}
\rho = \alpha \lambda \,.$$ Among the most usual choices of the constant $\alpha$ we mention the following: $$\label{alpha}
\alpha =\left \{
\begin{array}{ll}
1,& \quad {\rm geometric\,\,\,string}\\
1 + \omega, & \quad \omega \ge 0, \quad p \,\,{\rm
string\,\,\,or\,\,\,
Takabayasi\,\,\,string}\\
-1, & \quad {\rm Reddy\,\,\,string}
\end{array}
\right.$$
It is possible to consider a more general barotropic relation $$\label{rholambdagen}
\rho = \rho ( \lambda) \,.$$ than the linear relation , subject to the restrictions imposed by the energy conditions. The weak energy condition as well as the strong one require $\rho \geq \lambda$ with $\lambda
\geq 0$ or $\rho \geq 0$ with $\lambda < 0$ and the dominant energy condition implies $\rho \geq 0$ and $\rho^2 \geq \lambda^2$ [@letelier].
Spinor field approach
=====================
Recently it was shown that a nonlinear spinor can be used to simulate different types of perfect fluids including those called ekpyrotic, phantom matter and dark energy [@shikin; @spinpf0; @spinpf]. Here we show that it is possible to describe cosmic strings in terms of spinor fields as well.
Spinor field simulation
-----------------------
We shall simulate the cloud formed by massive cosmic strings with particles attached along their extensions with a nonlinear spinor field described by the Lagrangian: $$L_{\rm sp} = \frac{i}{2} \biggl[{\bar \psi}\gamma^{\mu} \nabla_{\mu} \psi-
\nabla_{\mu} \bar \psi \gamma^{\mu} \psi \biggr] - m{\bar \psi}\psi + F\,,
\label{lspin}$$ with $F$ being some arbitrary function of the scalar $S = \bar \psi\psi$. For the simulation of the present cosmological model in which the anisotropic scale factors $a_i$ are functions solely of time it is adequate to assume that the spinor field depends on $t$ only. Then the corresponding components of energy-momentum tensor take the form
$$\begin{aligned}
T_0^0 &=& mS - F\,, \label{t00s}\\
T_i^i &=& S \frac{dF}{dS} - F\,\,\,, i=1,2,3\,. \label{t11s}\end{aligned}$$
In what follows we describe the energy density $\rho$ of the string by $ T_0^0$ and the tension density $\lambda$ by $ T_1^1$ in agreement with . Inserting and into we find $$S \frac{dF}{dS} - (1 -\frac{1}{\alpha})F - \frac{m}{\alpha} S= 0\,,
\label{eos1s}$$ with the solution $$F = C_1 S^{(\alpha - 1)/\alpha} + mS\,. \label{sol1}$$ Here $C_1$ is an integration constant. The positivity of $T_0^0$ imposes some restriction on $C_1$, namely $C_1 \le 0$. Setting $C_1
= - \nu$ we find the cosmic string can be described by the spinor field Lagrangian $$L_{\rm sp} = \frac{i}{2} \biggl[\bar \psi \gamma^{\mu} \nabla_{\mu}
\psi- \nabla_{\mu} \bar \psi \gamma^{\mu} \psi \biggr] - \nu
S^{(\alpha - 1)/\alpha}\,, \label{lspincs}$$ remarking the disappearance of the mass term [@shikin].
With these preparatives we get
$$\begin{aligned}
\rho = T_0^0 &=& \nu S^{(\alpha - 1)/\alpha}\,, \label{t00sf}\\
\lambda = T_1^1 &=& \frac{\nu}{\alpha} S^{(\alpha - 1)/\alpha}\,.
\label{t11sf}\end{aligned}$$
Variation on with respect to $\psi$ and $\bar \psi$ gives spinor field equations with nonlinear terms. On the other hand from the spinor field equations for $S$ one finds [@sahaprd] $$\dot S + \frac{\dot \tau}{\tau} \, S = 0\,,\label{S}$$ with the solution $$S = \frac{C_0}{\tau}\,,$$ $C_0$ being a constant.
Taking into account this simple behavior of $S$ we have finally
$$\begin{aligned}
\rho &=& R \tau^{-\frac{\alpha -1}{\alpha}}\,, \label{rho}\\
\lambda &=& \frac{R}{\alpha}\tau^{-\frac{\alpha - 1}{\alpha}}\,,
\label{lambda}\end{aligned}$$
with the constant $ R = \nu C_0^{\frac{\alpha - 1}{\alpha}}$.
Using the above formulas for $\rho$ and $\lambda$, from we can determine the anisotropic factor $a_1$: $$\label{a_1}
a_1 = A_1\tau\,,$$ $A_1$ being a constant of integration. On the other hand, from equations and we obtain $$\label{a_2}
a_2 = \sqrt{\frac{D}{A_1}} \exp \biggl(\frac{X}{2} \int
\frac{dt'}{\tau}\biggr)\,,$$ and $$\label{a_3}
a_3 = \frac{1}{\sqrt{A_1 D}} \exp \biggl(-\frac{X}{2} \int
\frac{dt'}{\tau}\biggr)\,.$$
Asymptotic behavior {#asymptotic}
-------------------
In what follows, we study the equation for $\tau$ in details. Using the above expressions for $\rho, \lambda , a_1$ we get from : $$\ddot \tau= \frac{\kappa}{2}\frac {3 \alpha + 1}{\alpha}R
\tau^{\frac{1}{\alpha}} + \frac{\kappa \beta^2 A_1^2 }{2}
\tau\,.\label{dtaunu}$$ We can evaluate the derivative of $\tau$ with respect to $t$ which leads finally to a solution in quadrature $$\int \frac{d\tau}{\sqrt{[\kappa R(3\alpha+1)/(\alpha +
1)]\tau^{(\alpha + 1)/\alpha} + [\kappa \beta^2 A_1^2/2] \tau^2 +
C}} = t + t_0\,, \label{1st}$$ $C$ and $t_0$ being some integration constants.
In spite of the fact that this equation cannot be explicitly solved, the asymptotic behavior of the solutions for $t\rightarrow \infty$ could be found.
### Case I {#caseI}
In most cases, provides a standing expansion of the volume scale of the BI spacetime for $t$ growing.
Indeed, for $$\label{1-1}
\frac{\alpha + 1}{\alpha} \leq 2 \,,$$ i.e. $\alpha \geq 1$ or $\alpha < 0$, the term with $\tau^2$ at the denominator of the l. h. s. of is dominant and we get an exponential behavior $$\label{tauexp}
\tau \propto \exp t \,,$$ for large $t$. In this case, only the anisotropic factor $a_1$ presents an exponential increase for $t \rightarrow \infty$, while the scale factors $a_2$ and $a_3$ tend to constants.
Concerning the asymptotic behavior of $\rho$ and $\lambda$ we infer from , that they tend to zero as $$\label{rholambdainfty}
\rho\,, \lambda \propto \frac{1}{\exp \left (\frac{\alpha -1}{\alpha} t
\right ) } \,.$$
### Case II
For $0 < \alpha < 1$, there is no solution with a $\tau \rightarrow
\infty$ behavior for $t \rightarrow \infty$. In this case from one finds: $$\tau^{\frac{\alpha -1}{2 \alpha}} \propto t. \nonumber$$
Since $\alpha -1 < 0$, $\tau$ cannot tend to infinity as $t \to \infty$. Therefore the model does not admit a consistent solution for $0 < \alpha < 1$ in agreement with the discussion of the general barotropic equation from Section \[basic\].
Taking into account the absence of consistent solutions for $0 < \alpha < 1$ as it was shown above, for the present spinor simulation, in what follows we shall refer only to the situations described in subsection \[caseI\] (Case I). It is worth also mentioning that the present model does not support in any case a vanishing of $\tau$ for $t \rightarrow \infty$.
Numerical simulations
---------------------
In this subsection we graphically illustrate the evolution of energy density $\rho$, volume scale $\tau$ and metric functions $a_1$, $a_2$ and $a_3$. In doing so we rewrite the equation for $\tau$ in terms of the Hubble parameter $H$ and introduce a new function $T$:
\[system\] $$\begin{aligned}
\dot \tau &=& 3H\tau, \\
\dot H &=& - 3H^2 + \frac{\kappa}{6}\frac {3 \alpha + 1}{\alpha}R
\tau^{\frac{1 - \alpha}{\alpha}} + \frac{\kappa \beta^2 A_1^2 }{6},\\
\dot T &=& \frac{T}{\tau}.\end{aligned}$$
We also rewrite the metric functions in terms of $T$, which now read $$a_1 = A_1\tau, \quad a_2 = \sqrt{\frac{D}{A_1}} T^{X/2}, \quad a_3 =
\sqrt{\frac{1}{A_1 D}} T^{-X/2}. \label{mfnew}$$
Let us now numerically solve the system . In doing so for simplicity we set $\kappa = 1$, $A_1 = 1$, $D = 1$ and $X =1$. In Fig. \[rhosppf0\] we plot the evolution of energy density $\rho$ for different values of $\alpha$, namely for $\alpha = -1$ and $\alpha = 1.5$. Evolution of $\tau$ corresponding to these values of parameters in shown in Fig. \[tausppf\]. For simplicity we also set the initial values of $\tau$, $H$ and $T$ to be unity.
The numerical simulations support the behavior described in , . From Fig. \[tausppf\] one finds at first, in the case of a negative $\alpha$ the Universe expands slower than it does as for a positive $\alpha$, though in both cases we have an exponential growth for large time.
In Figs. \[abc+105\] and \[abc-1\] we plot the behavior of metric functions for positive and negative values of $\alpha$. As is seen from the figures, with the expansion of the Universe $a_1$ increases exponentially, while $a_2, a_3$ tend to constants. Thus we see that introduction of cosmic string does not allow isotropization of initially anisotropic space-time. In the following subsections we discuss the singularity problem and isotropization process in detail.
Singularities
-------------
The next task is to investigate the space-time singularities in the present model. Let us rewrite the metric functions in the following form: $$a_i = C_i \tau^{N_i} \exp \biggl(Y_i \int
\frac{dt'}{\tau}\biggr)\,,\label{ai}$$ where $C_1 = A_1$, $C_2 = \sqrt{D/A_1}$, $C_3 =
1/\sqrt{DA_1}$, $N_1 = +1$,$N_2 = N_3 = 0$,$Y_1 = 0$, $Y_2 = X/2$, and $Y_3 = -X/2$. Then the first and the second derivatives of the metric functions take the form:
\[singvis\] $$\begin{aligned}
\frac{\dot a_i}{a_i} &=& N_i \frac{\dot{\tau}}{\tau} +
\frac{Y_i}{\tau}\,, \label{singvis1}\\
\frac{\ddot a_i}{a_i} &=& N_i \frac{\ddot{\tau}}{\tau} + (N_i^2 -
N_i) \Bigl(\frac{\dot{\tau}}{\tau}\Bigr)^2 + (2N_i - 1) Y_i \frac{
\dot{\tau}}{\tau^2} + \frac{Y_i^2}{\tau^2}\,.\end{aligned}$$
We study the singularities analyzing the regularity properties of the Kretschmann scalar which for the metric reads $$\mathcal{K} = 4\Bigl[\Bigl(\frac{\ddot a_1}{a_1}\Bigr)^2 + \Bigl(\frac{\ddot
a_2}{a_2}\Bigr)^2 + \Bigl(\frac{\ddot a_3}{a_3}\Bigr)^2 +
\Bigl(\frac{\dot a_1}{a_1}\frac{\dot a_2}{a_2}\Bigr)^2 +
\Bigl(\frac{\dot a_2}{a_2}\frac{\dot a_3}{a_3}\Bigr)^2 +
\Bigl(\frac{\dot a_3}{a_3}\frac{\dot a_1}{a_1}\Bigr)^2\Bigr]\,.
\label{Kretsch}$$ Evidently, a singularity can occur when some or all scale factors $a_i$ tend to zero or infinity. We shall follow the criteria given in [@Bronshik] which states:
\(i) $t$ finite, some $a_i \to \infty$. [*If at least one scale factor becomes infinite at finite $t$, it is a curvature singularity.*]{}
\(ii) $t$ finite, some $a_i \to 0$. [*If more than one scale factor turns to zero at finite $t$, it is a singularity. If only one scale factor is zero at finite $t$, the spacetime can be nonsingular.*]{}
\(iii) $t \to \infty$, some $a_i \to \infty$. [*Such an asymptotic can only be singular if at least one scale factor grows faster than exponentially:*]{}\
$a_i(t) \gg \exp{(k|t|)},\quad k = {\rm const.} > 0.$
(iv)$t \to \infty$, some $a_i \to 0$. [*Such an asymptotic can only be singular if at least one scale factor vanishes faster than exponentially:*]{}\
$a_i(t) \ll \mathcal{O}(\exp{[-k|t|]}),\quad k = {\rm const.} > 0.$
Looking at the asymptotic behaviors described above (Section \[asymptotic\]) we conclude that in the spinor field approach of the cosmic strings the present model is nonsingular at the end of the evolution. No scale factor of the BI metric presents a growth faster than exponentially or vanishing faster than exponentially for $t \to \infty$.
Isotropization
--------------
Since the present-day Universe is surprisingly isotropic, it is important to see whether our anisotropic BI model evolves into an isotropic FRW model. Isotropization means that at large physical times $t$, when the volume factor $\tau$ tends to infinity, the three scale factors $a_i(t)$ grow at the same rate. Two wide-spread criteria of isotropization read
\[aniso\] $$\begin{aligned}
{\cal A} &=& \frac{1}{3} \sum\limits_{i=1}^{3} \frac{H_i^2}{H^2} -
1 \to 0\,,\\
\Sigma^2 &=& \frac{1}{2} {\cal A} H^2 \to 0\,.\end{aligned}$$
Here ${\cal A}$ and $\Sigma^2$ are the average anisotropy and shear, respectively, while $H_i = \dot{a_i}/a_i$ is the directional Hubble parameter evaluated in . We also investigate the isotropization condition proposed in [@Bronshik] $$\frac{a_i}{a}\Bigl|_{t \to \infty} \to {\rm const}\,, \label{isocon}$$ where $a(t) = \tau^{1/3}$ is the average scale factor. Provided that condition is fulfilled, by rescaling some of the coordinates, we can make $a_i/a \to 1$, and the metric will become manifestly isotropic at large $t$.
In order to study the isotropization process in the present model, we evaluate from $$\frac{a_i}{a} = \frac{a_i}{\tau^{1/3}} = C_i \tau^{N_i-1/3} \exp
\biggl(Y_i \int \frac{dt'}{\tau}\biggr)\,,\label{isocon1}$$ Taking into account the value of $N_i$ and $Y_i$ we see that $a_i/a$ do not tend to constants as $t \to \infty$. For example, in the generic case with the exponential expansion of the volume of the Universe , $a_1/a \to \infty$, $a_2/a \to 0$ and $a_3/a \to 0$. So in the case considered, no isotropization process takes place. Indeed, at the early stage of evolution, where $\tau \to 0$ we assume $a_1 \to \infty$, $a_2 \to
0$ and $a_3 \to 0$, that is at this stage the Universe looks like a one-dimensional string. In the asymptotic region where $t \to
\infty$ and $\tau \to \infty$ we have $a_1 \to \infty$, while $a_2$ and $a_3$ evaluating to finite values.
Let us also examine the possibility of an isotropization process using conditions . In view of from in the generic case one finds:
\[aniso1\] $$\begin{aligned}
{\cal A} &=& 2 \,,\\
\Sigma^2 &=& \frac{1}{9} \,.\end{aligned}$$
As one sees, both ${\cal A}$ and $\Sigma^2$ are some positive defined, non vanishing quantities. Thus, in general, the present model does not undergo an isotropization process, the anisotropic feature of the Universe from the early stages being retained.
Conclusions
===========
We have studied the evolution of a anisotropic universe given by a BI cosmological model in presence of cosmic strings and magnetic field. In doing so we exploit the spinor simulation of cosmic strings that allows us to solve the system of Einstein’s equations. Exact solutions have been supplemented with some numerical evaluations. It is shown that the presence of cosmic strings does not allow the anisotropic Universe to evolve into an isotropic one.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The authors gratefully acknowledge the support from the joint Romanian-LIT, JINR, Dubna Research Project, theme no. 05-6-1060-2005/2010. M.V. is partially supported by the CNCSIS Program IDEI - 571/2008, Romania.
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T. Koivisto and D.F. Mota, [*Phys. Lett. B*]{} [**644**]{}, 104 (2007). T. Koivisto and D.F. Mota, [*Phys. Rev. D*]{} [**75**]{}, 023518 (2007). P.S. Letelier, [*Phys. Rev. D*]{} [**28**]{}, 2414 (1983). A. Pradhan, A. K. Yadav, R. P. Singh and V. K. Singh, [*Astrophys. Space Sci.*]{} [**312**]{}, 145 (2007). G.S. Khadekar and S.D. Tade, [*Astrophys. Space Sci.*]{} [**310**]{}, 47 (2007). A. Lichnerowicz, [*Relativistic Hydrodynamics and Magnetohydrodynamics*]{}, (Benjamin, New York, 1967). B. Saha, [*Astrophys. Space Sci.*]{} [**299**]{}, 149 (2005). B. Saha and M. Visinescu, [*Astrophys. Space Sci.*]{} [**315**]{}, 99 (2008). B. Saha, V. Rikhvitsky and M. Visinescu, [*Cent. Eur. J. Phys.*]{} [**8**]{}, 113 (2010). K.A. Bronnikov, E.N. Chudaeva and G.N. Shikin, Class. Quantum Grav. [**21**]{}, 3389 (2004).
[^1]: E-mail: [email protected]; URL: http://wwwinfo.jinr.ru/\~bijan/
[^2]: E-mail: [email protected]; URL: http://www.theory.nipne.ro/\~mvisin/
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Effects of difference in the spin and parity distributions for the surrogate and neutron-induced reactions are investigated. Without assuming specific (schematic) spin-parity distributions, it was found that the surrogate ratio method can be employed to determine neutron fission and capture cross sections if 1) weak Weisskopf-Ewing condition (defined in this paper) is satisfied, 2) there exist two surrogate reactions whose spin-parity distributions of the decaying nuclei are almost equivalent, and 3) difference of the representative spin values between the neutron-induced and surrogate reactions is no much larger than 10 $\hbar$. If these conditions are satisfied, we need not to know the spin-parity distributions populated by the surrogate method. Instead, we should just select a pair of surrogate reactions which will populate the similar spin-parity distributions, using targets having similar structure and reactions having the similar reaction mechanisms. Achievable accuracy is estimated to be around 5 and 10 % for fission and capture channels, respectively, for nuclei of the Uranium region. The surrogate absolute method, on the contrary, can be marginally applicable to determination of fission cross sections. However, there will be little hope to apply this method for capture cross section measurements unless the spin-parity distributions in the neutron-induced and surrogate reactions are fairly close to each other or the difference can be corrected theoretically. The surrogate ratio method was shown also to be a robust method in the presence of breakup reactions, again, without assuming specific breakup reaction mechanisms.'
author:
- Satoshi CHIBA
- Osamu IWAMOTO
title: Verification of the Surrogate Ratio Method
---
Introduction
============
With the advance of nuclear science and technology, neutron cross sections of unstable nuclei, such as minor actinides (MAs) and long-lived fission products (LLFPs), are becoming more and more necessitated. Neutron cross sections of radioactive nuclei also play important roles in astrophysical nucleosynthesis. In spite of the importance, however, measurement of neutron cross sections are extremely difficult for these nuclei since preparation of enough amount of sample is difficult or practically impossible. At the same time, theoretical determination of the fission and capture cross sections still suffers from a large uncertainty if there exists no experimental data; an error of factor of 2, namely the uncertainty of 100 %, will be a reasonable estimate. These fundamental problems prevent us from accurate determination of neutron cross sections of unstable nuclei including MAs and LLFPs.
Recently, a new method, called surrogate method, has come to be used actively to determine neutron cross sections of unstable nuclei (see, e.g., Refs. [@cramer70; @britt79; @petit04; @boyer06; @younes03a; @younes03b; @escher06; @lyles07; @forssen07; @lesher09; @allmond09] and references therein). This is a method which uses (multi) nucleon transfer reactions (both stripping and pick-up) or inelastic scattering on available target nuclei and produce the same compound nuclei as those of the desired neutron-induced reactions, and measure the decay branching ratios leading to capture and/or fission channel. Identification of the produced compound nuclei and their excitation energies can be done by detection of the ejectile species and their energies.
At a first glance, it seems to be a simple and effective method to simulate the neutron-induced reactions. However, the thing is not that easy. Even if we produce the same compound nuclei at the same excitation energy as produced in the desired neutron-induced reactions, the spin-parity distributions are plausibly different between them. Since we are interested in low-energy neutron cross sections relevant to reactor applications and astrophysics, the produced compound nuclei decay statistically, and the branching ratio is strongly influenced by the spin and parity. Therefore, difference of the spin-parity distributions between the surrogate and neutron-induced reactions must be properly taken into account in converting the branching ratio determined by the surrogate method to the one for neutron-induced reactions. Up to now, however, it has not been able to deduce the spin-parity distribution in the surrogate reactions, since they are normally multi-nucleon transfer reactions, the reaction mechanisms of which are not understood well. What have been done so far is to assume that the decay branching ratio does not depend on the spin-parity and ignore the difference; the so-called Weisskopf-Ewing condition, or to assume schematic (rather arbitrary) spin-parity distributions for the surrogate reaction and argue that they do not affect the decay branching ratio sensitively. Both of these approaches, however, are based on arbitrary assumptions which have not been justified theoretically nor experimentally. On the other hand, it is also true that the surrogate method has yielded a rather accurate cross sections, verified when the corresponding neutron data are available. Therefore, it is natural to expect that there is a certain condition to equate the results from the surrogate method and the neutron-induced reactions. However, the condition under which the surrogate method works is not clearly understood yet.
In this paper, we investigate the spin-parity dependence of the branching ratios of Uranium isotopes to the fission and capture channels and clarify the condition for the surrogate (ratio) method to work, and estimate the accuracy achievable by it.
Surrogate “absolute" and “ratio" methods
========================================
In the surrogate method, we measure a branching ratio to a specific decay channel, normally the fission or capture channel by populating the same kind of compound nucleus as the desired neutron-induced reactions . We denote the decay channel by a subscript $i$ ($i$ = fission or capture), and then the surrogate method hopefully gives a ratio of the neutron cross section $\sigma^n_i$ to the total neutron reaction cross section $\sigma^n_{R}$ of the compound system, namely, $$\label{eq:a}
R^S_{i}\stackrel{?}{=}\frac{\sigma^n_{i}}{\sigma^n_{R}},$$ The symbol $R^S_i$ denotes the branching ratio of the nucleus decaying to channel $i$ populated by the surrogate reaction, and is defined later by Eq. (\[eq:2a\]). By multiplying it the total reaction cross section $\sigma^n_{R}$ calculated by the optical or coupled-channel model, we can determine the neutron cross section $\sigma^n_{i}$. Here, a question mark is explicitly shown since it is not obvious if this equality holds or not. It is due to the reason that the spin-parity distributions populated in the surrogate (left-hand-side) and neutron-induced (right-hand-side) reactions are different, and the branching ratio is dependent on them in general. This is the very fundamental problem to be resolved for the surrogate method to yield correct neutron-indeed cross sections. This method is referred to as the surrogate absolute method. On the contrary, these ratios can be measured for two nearby nuclei 1 and 2 by using the same kind of surrogate reactions, $S_{1}$ and $S_{2}$, e.g., ($t,p$) reactions on different targets. If we know the neutron cross section $\sigma_{i}^{n_{2}}$ for the reaction leading to the same compound nucleus as the $S_{2}$ reaction, we can determine the neutron cross section ($\sigma_{i}^{n_{1}}$) which leads to the same compound nucleus as $S_{1}$ reaction via the equality (with a question mark) $$\begin{aligned}
\label{eq:b}
\frac{R_{i}^{S_{1}}}{R_{i}^{S_{2}}} &\stackrel{?}{=}& \frac{\frac{\sigma_{i}^{n_{1}}}
{\sigma_{R}^{n_1}}}{\frac{\sigma_{i}^{n_{2}}}{\sigma_{R}^{n_2}}}, \\
\rightarrow \sigma_{i}^{n_{1}} &\stackrel{?}{=}&
\sigma_{i}^{n_{2}}\cdot
\frac{\sigma_{R}^{n_{1}}}{\sigma_{R}^{n_{2}}}
\cdot\frac{R_{i}^{S_1}}{R_{i}^{S_2}
},~~(i=\mathrm{fission~or~capture}).\end{aligned}$$ Here, $\sigma_{i}^{n_{j}}$ denotes the neutron fission ($i$=fission) or capture ($i$=capture) reaction cross section, and $\sigma_{R}^{n_{j}}$ the total neutron reaction cross section for the reaction $n_{j}$ ($j$ = 1 or 2). Provided that the above equations hold, we can determine the neutron cross section $\sigma_{i}^{n_{1}}$ from this formula, since we know $\sigma_{i}^{n_{2}}$, we measure the ratio $R_{i}^{S_1}/R_{i}^{S_2}$ and we can calculate the ratio of the reaction cross sections $\sigma
_{R}^{n_{1}}/\sigma_{R}^{n_{2}}$ by the coupled-channel theory rather accurately[@capote08; @kunieda09]. This method is referred to as the surrogate ratio method or relative surrogate method. It is naively expected to give a result better than the surrogate absolute method, since we do not need to know in the relative method all the experimental artifacts such as the detector efficiency and geometrical factor required to deduce the ratio in the absolute method. However, all these methods require a fact that the branching ratios are equal for the surrogate and the neutron-induced reactions. This is true only when 1) the ratios are independent of the spin-parity of the decaying nuclei (Weisskopf-Ewing condition[@we]), or 2) the spin-parity distributions are equivalent for the surrogate and neutron reactions, or 3) the ratio is not sensitive to the difference of the spin-parity distributions between the neutron-induced and surrogate reactions. Below, we will investigate if these assumptions are justified or not, and when justified, what accuracy will be.
Computational method and results
================================
We use the Hauser-Feshbach theory[@hf] to calculate the decay branching ratios of various spin-parity ($J^{\pi}$) states of $^{239}$U by using CCONE code system[@ccone]. It represents a nucleus produced by n+$^{238}$U reactions and corresponding surrogate reactions such as $^{237}$U($t,p$)$^{239}$U. This nucleus was chosen just as an example. In the calculation, the same parameter values for discrete level structures, transmission coefficients, level density, fission barrier and GDR as used in the evaluation of JENDL Actinoid File 2008[@jac08] were used. Therefore, the present calculation contains realistic information of the characteristics of participating nuclei adjusted to reproduce neutron cross sections.
Figures 1 and 2 shows the branching ratios (decay probabilities) to the fission (Fig. 1) and capture (Fig. 2) channels for various $J^{\pi}$ states of $^{239}$U up to $J^{\pi}$ = (21/2)$^{\pm}$ and neutron energy of 5 MeV. The upper panels in Figs. 1 and 2 show branching ratios from positive parity states, while the lower ones denote those from negative parity states. If the Weisskopf-Ewing condition is fulfilled, the various lines in these figures must coincide (at least approximately); if it is the case both of the surrogate absolute and ratio methods can be justified. However, Fig. 1 shows that the fission decay ratio varies depending on $J^{\pi}$ by about 15 % at 5 MeV but variation is about 50 % at 1.5 MeV. The convergence is much worse for the capture channel as shown in Fig. 2; the branching ratios scatter by a factor of about 10 at 5 MeV, and the variation is much larger at lower energies. Therefore, we have to conclude that there is only little hope to use surrogate method to determine neutron capture cross sections at these energies, since the low-energy neutron-induced reactions bring only small angular momentum to the compound system in general, while the surrogate method will bring much more. The absolute surrogate method, therefore, will never work to measure capture cross sections unless the spin-parity distribution between the neutron-induced and surrogate reactions are fairly close to each other or the difference is corrected theoretically. It will be also only marginally applicable to measure the fission cross sections.
However, the $J^{\pi}$ dependence of the branching ratios to the fission and capture reactions show rather systematic behaviors. Above 2.5 MeV, the fission probability shown in Fig. 1 increases monotonically as *J* increases. Same trend is true for the capture reaction. Since it was found also to be true for other compound nuclei in this mass region, $^{236}$U and $^{237}$U (not shown here), we may expect that there is a possibility to cancel out the large $J^{\pi}$ dependence by taking ratios of the branching ratios for each $J^{\pi}$. We have done such calculations and the results are shown in Figs. 3 and 4. Figure 3 shows the ratios of fission probabilities (branching ratios) for $^{239}$U and $^{237}$U for various values of $J^{\pi}$. We can notice an astonishingly good convergence. The thick black line denotes the ratios of the neutron fission probabilities ($\sigma^n_f/\sigma^n_R$) for the corresponding neutron-induced reactions. All the curves converges to the ratio of the neutron fission probabilities very well. The deviation is only a level of 3 % at 5 MeV. The largest scatter lies at about 1.6 MeV, but the scatter around the neutron curve is only a level of several % nominally, while that was about 50 % in Fig. 1. This means that we can determine the unknown fission cross sections by taking this kind of ratio if we know one of the other neutron cross section. The convergence seems to be valid also for somewhat higher value of spins. Similar convergence, although less dramatic, can be seen in Fig. 4 for capture probabilities. At 5 MeV, the ratios of the capture branching ratios for the 2 nuclei scatter only by about $\pm$ 5 % around those for the neutron capture reaction. At energies from 2.5 to 4 MeV, the surrogate ratios are all larger than the neutron ratio, but the deviation is still only 10 %. The same ratios were compared for various $J^+$ states produced in the neutron-induced reactions on $^{197}$Au and $^{193}$Ir in Fig. 5. We can notice that very good mutual convergence up to 8$^+$ and equivalence to the neutron ratio are obtained in this mass region as well. Therefore, these data can be used to determine the GDR parameters at an energy region of, e.g., 2 to 5 MeV to normalize the calculated neutron capture cross section, and these parameters can be used to calculate the capture cross sections at lower energies since the Hauser-Feshbach theory can predict the shape of the energy dependent cross section rather accurately if normalization is given correctly at certain energies. Therefore, there is a fair possibility that we can determine the neutron capture cross section with accuracy of several % by the surrogate ratio method in combination with a theoretical calculation. The convergence of the ratios of fission and capture probabilities are very important to validate the surrogate technique and can be a base of the validity of surrogate ratio method.
Formal verification of the surrogate ratio method
=================================================
In the previous section, we have seen that the ratios of fission and capture probabilities at various values of $J^\pi$ between 2 nuclei have a dramatic convergence to the ratios of the neutron reactions. This can be utilized to verify the surrogate ratio method as follows. Let 2 surrogate reactions used for the ratio method be denoted as *S*$_{1}$ and *S*$_{2}$, and corresponding neutron reactions as *n*$_{1}$ and *n*$_{2}$. The reactions *S*$_{j}$ and *n*$_{j}$ (*j*=1,2) are chosen to lead to the same compound nucleus. Let us assume that we know the neutron cross section $\sigma
_{i}^{n_{2}}$ for the $n_{2}$ reaction . The branching ratio of the surrogate reaction for channel $i$ ($i$ = fission or capture) may be written as $B_{i}^{S_{j}}(U,J^{\pi})$, where $U$ denotes the equivalent neutron energy ($U$ can be the excitation energy as well). Then, the identity of the branching ratios shown in Figs. 3 and 4 can be expressed as $$\label{eq:1}
\frac{B_{i}^{S_{1}}(U,J^{\pi})}{B_{i}^{S_{2}}(U,J^{\pi})}
%=\frac{B_{i}^{S_{1}}(U)}{B_{i}^{S_{2}}(U)}
=\frac{R_{i}^{n_{1}}(U)}{R_{i}^{n_{2}}(U)}$$ to the accuracy mentioned above, where $$\label{eq:2}
R_{i}^{n_{j}} \equiv \frac{\sigma_{i}^{n_{j}}}{\sigma_{R}^{n_{j}}}.$$ Relation of the $B^x_i$ and $R^x_i$ $(x=S_j~{\rm or}~n_j)$ are expressed as follows: $$\label{eq:2a}
R_{i}^{x_{j}}(U) \equiv \frac{\sum_{J^{\pi}}\sigma^{x_{j}}(U,J^{\pi})\cdot
B_{i}^{x_{j}}(U,J^{\pi})}{\sum_{J^{\pi}}\sigma^{x_{j}}(U,J^{\pi})},$$ where $\sigma^{x_{j}}(U,J^{\pi})$ denotes the formation cross section of $J^\pi$ states in reaction $x_j$ including the factor of $(2J+1)$. Equation (\[eq:1\]) can be rewritten as $$\label{eq:3}
B_{i}^{S_{1}}(U,J^{\pi})=B_{i}^{S_{2}}(U,J^{\pi})\cdot\frac{R_{i}^{n_{1}}%
(U)}{R_{i}^{n_{2}}(U)}.$$ Then, the decay probability for reaction $i$ in surrogate $S_{1}$ measurement, $R_{i}^{S_{1}}$, can be written as $$\begin{aligned}
\label{eq:4}
R_{i}^{S_{1}}(U) & = &\frac{\sum_{J^{\pi}}\sigma^{S_{1}}(U,J^{\pi})\cdot
B_{i}^{S_{1}}(U,J^{\pi})}{\sum_{J^{\pi}}\sigma^{S_{1}}(U,J^{\pi})} \nonumber \\
&=&\frac
{\sum_{J^{\pi}}\sigma^{S_{1}}(U,J^{\pi})\cdot B_{i}^{S_{2}}(U,J^{\pi}
)\cdot\frac{R_{i}^{n_{1}}(U)}{R_{i}^{n_{2}}(U)}}{\sum_{J^{\pi}}\sigma^{S_{1}
}(U,J^{\pi})}\nonumber\\
& = &\frac{R_{i}^{n_{1}}(U)}{R_{i}^{n_{2}}(U)}\cdot\frac{\sum_{J^{\pi}}
\sigma^{S_{1}}(U,J^{\pi})\cdot B_{i}^{S_{2}}(U,J^{\pi})}{\sum_{J^{\pi}}
\sigma^{S_{1}}(U,J^{\pi})}.\end{aligned}$$ Since the 2 surrogate reactions $S_{1}$ and $S_{2}$ are assumed to be carried out for a pair of nuclei having similar mass and structure, the distribution of the formation cross section $\sigma^{S_{1}}(U,J^{\pi})$ will be fairly close to that of $\sigma^{S_{2}}(U,J^{\pi})$ if the nuclear structure and reaction mechanisms are similar to each other. We can write this similarity as $\sigma^{S_{1}}(U,J^{\pi}) = \alpha \sigma^{S_{2}}(U,J^{\pi})$, where the symbol $\alpha$ denotes a constant such as the kinematical factor. If the dependence of $\alpha$ on $J^\pi$ is ignorable, Eq. (\[eq:4\]) reads $$\begin{aligned}
\label{eq:5}
R_{i}^{S_{1}}(U)&=&\frac{R_{i}^{n_{1}}(U)}{R_{i}^{n_{2}}(U)}\cdot\frac{\sum
_{J^{\pi}} \alpha \sigma^{S_{2}}(U,J^{\pi})\cdot B_{i}^{S_{2}}(U,J^{\pi})}
{\sum_{J^{\pi}} \alpha \sigma^{S_{2}}(U,J^{\pi})} \nonumber \\
&=&\frac{R_{i}^{n_{1}}(U)}{R_{i}
^{n_{2}}(U)}\cdot R_{i}^{S_{2}}(U)\end{aligned}$$ by definition. This equation is equivalent to Eq. (\[eq:b\]). Since we know $R_{i}^{n_{2}}(U)$, and we measure $R_{i}^{S_{1}}/R_{i}^{S_{2}}$ in surrogate ratio method, we can obtain $R_{i}^{n_{1}}$ to the accuracy mentioned above. This gives an explanation of the reason why the surrogate ratio method works.
The essential point in the verification is the equality given in Eq. (\[eq:1\]) and equality of the $J^\pi$ spectra of the 2 surrogate reactions. The latter implies that the $J^\pi$ distributions in the surrogate reactions can be different from those of the neutron-induced reactions. What is important is that 2 surrogate reactions should yield equivalent $J^{\pi}$ distributions. It can be easily achieved in experiments by selecting targets having similar structure and using the same reaction for the both surrogate reactions. However, the difference of the representative spin between the neutron- induced and surrogate reactions should not be much larger than about 10 $\hbar$. We define the equality given in Eq. (\[eq:1\]) as “[**weak Weisskopf-Ewing condition**]{}". This condition is different from the standard Weisskopf-Ewing condition, which is written as $$\label{eq:}
B_{i}^{S_{j}}(U,J^{\pi})=B_{i}^{S_{j}}(U)=R_{i}^{n_{j}}(U).$$ If this standard condition is satisfied, we can determine the branching ratios by the surrogate absolute method. Unfortunately, it is not the case for the reactions investigated in this paper, especially it is a poor assumption for the capture reaction as shown in Fig. 2.
The surrogate ratio method has another advantage over the absolute method. Since the surrogate method uses multi-nucleon transfer reactions very often, there is a possibility, when the corresponding neutron energy increases, that the nucleons expected to be transferred to bound states of the target is actually transferred to an unbound state, eventually leading to the breakup (or preequilibrium) reactions such as $^{238}$U($t,np$)$^{239}$U instead of expected reaction $^{238}$U($t,p$)$^{240}$U. This effect can be also canceled out by the surrogate ratio method as follows.
Let us denote the bound states as “Q", and unbound ones as “P". Since we measure the ejectile (e.g., $p$), the production cross section of it contains transitions to both the Q- and P-states of the residual nuclei. On the contrary, the true decay occurs only via the Q-states. Therefore, the decay probabilities measured in the surrogate method in the presence of breakup reaction, $R_{i}^{S_{1}}(P+Q)$, can be written as $$\label{eq:8}
R_{i}^{S_{1}}(P+Q)=
\frac{\sum_{J^{\pi}}\hat{Q}\sigma^{S_{1}}(U,J^{\pi})\cdot
B_{i}^{S_{1}}(U,J^{\pi})}{\sum_{J^{\pi}}(\hat{P}+\hat{Q})\sigma^{S_{1}}(U,J^{\pi})}
\leq R_{i}^{S_{1}}(U)
=
\frac{\sum_{J^{\pi}}\hat{Q}\sigma^{S_{1}}(U,J^{\pi})\cdot
B_{i}^{S_{1}}(U,J^{\pi})}{\sum_{J^{\pi}}\hat{Q}\sigma^{S_{1}}(U,J^{\pi})},$$ where the $\hat{Q}$ and $\hat{P}$ denote fractions of transitions to the Q- and P-states, respectively, and $\hat{P}+\hat{Q}$=1. The same is true for the $S_{2}$ reaction Therefore, the ratio of the measured surrogate reaction ratios reads $$\begin{aligned}
\label{eq:9}
\frac{R_{i}^{S_{1}}(P+Q)}{R_{i}^{S_{2}}(P+Q)} &=&
\frac{
\frac{\sum_{J^{\pi}}\hat{Q}\sigma^{S_{1}}(U,J^{\pi})\cdot B_{i}^{S_{1}}(U,J^{\pi})}{\sum_{J^{\pi}}(\hat{P}+\hat{Q})\sigma^{S_{1}}(U,J^{\pi})}}
{\frac{\sum_{J^{\pi}}\hat{Q}\sigma^{S_{2}}(U,J^{\pi})\cdot B_{i}^{S_{2}}(U,J^{\pi})}{\sum_{J^{\pi}}(\hat{P}+\hat{Q})\sigma^{S_{2}}(U,J^{\pi})}} \nonumber \\
&=&\frac{R_{i}^{n_{1}}(U)}{R_{i}^{n_{2}}(U)}\cdot\frac{\sum_{J^{\pi}}\hat
{Q}\sigma^{S_{2}}(U,J^{\pi})\cdot B_{i}^{S_{2}}(U,J^{\pi})}{\sum_{J^{\pi}}
\hat{Q}\sigma^{S_{2}}(U,J^{\pi})\cdot B_{i}^{S_{2}}(U,J^{\pi})} \nonumber \\
&=&\frac{R_{i}^{n_{1}}(U)}{R_{i}^{n_{2}}(U)},\end{aligned}$$ where the weak Weisskopf-Ewing condition (Eq. (\[eq:3\])) and proportionality of $\sigma^{S_{1}}(U,J^{\pi})$ and $\sigma^{S_{2}}(U,J^{\pi})$ were employed. Therefore, the surrogate ratio method has a capability to work even when breakup (or preequilibrium) reaction occurs.
Even though the derivation here is qualitative, it was enough to assume that the ratios of $\hat{P}$ and $\hat{Q}$ to be the same for the 2 surrogate reactions used in the ratio method. This can be satisfied if the breakup mechanisms are the same, which is a reasonable assumption. Again, it must be noted that we do not need to understand the breakup reaction mechanism itself, which is a formidable task, but just require them to be the same for the 2 reactions employed in the ratio method. It can be easily verified experimentally by observing the spectra of emitted particles. This may explain the reason why the ratio method worked to measure the $^{236}$U($n,f$) cross section for energies above several MeV as reported by Lyles *et al.*[@lyles07] where the 2nd and 3rd chance fission occur, which corresponds to the condition that the breakup reaction can occur in the surrogate method.
Concluding remarks
==================
We have investigated the condition that the surrogate reaction should work. It was found that the surrogate absolute method will give a marginal result for fission cross sections but it seems to be hopeless to apply it for the capture cross section measurements. On the contrary, it was shown that, without assuming any specific (schematic) spin parity distributions, the surrogate ratio method has a high potential to determine neutron fission and capture cross sections. The achievable accuracy would be around 3$\sim$5 % for the fission and 10 % for the capture cross sections under the condition investigated in this work (up to difference of spin values of between neutron-induced and surrogate reactions of around 10 $\hbar$) for nuclei in Uranium region at around 2.5 to 5 MeV. The success is brought by the weak Weisskopf-Ewing condition, namely, $J^{\pi}$ by $J^{\pi}$ convergence of the branching ratios and their coincidence to the neutron reaction ratio, defined in this work. Furthermore, it is important to select a pair of nuclei, one of which is the reference nucleus, having similar properties so that the excitation spectra of various $J^\pi$ states can be considered almost equivalent. These conditions are the basis for the surrogate ratio method to work. Furthermore, it was shown to be rather robust even breakup reaction occurs. This was shown again without assuming any breakup reaction mechanisms. Altogether, the surrogate ratio method was proved to be a useful method to determine neutron cross sections for which the direct measurements using neutrons are not possible. Generally speaking, however, application of the surrogate method must be done with a caution. It will be very sensitive to the spin and parity of the decaying nucleus at low energies since transitions to discrete levels, which differ nucleus to nucleus, occupy a dominant part of the decay branch there. This is the reason why the weak Weisskopf-Ewing condition tends to be violated at lower energies.
It must be also noted that we use a standard Hauser-Feshbach calculation using models and parameters adjusted to reproduce neutron cross sections, but the results may have some dependence on them. Such a dependence, however, is expected also to be small in the surrogate ratio method, since many factors in models and parameters can cancel out in the ratio quantities.
The authors are grateful to Drs. A. Iwamoto, K. Nishio, S. Hashimoto and Y. Aritomo for fruitful discussions. Present study is the result of “Development of a Novel Technique for Measurement of Nuclear Data Influencing the Design of Advanced Fast Reactors" entrusted to Japan Atomic Energy Agency (JAEA) by the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT).
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 Decay probabilities (branching ratios) to the fission channel from various $J^{\pi}$ states of $^{239}$U. (a) : positive parity states, (b): negative parity states](u238+n-fis1.eps)
 Decay probabilities (branching ratios) to the capture channel from various $J^{\pi}$ states of $^{239}$U. (a): positive parity states, (b): negative parity states](u238+n-cap1.eps)
 Ratios of decay probabilities (branching ratios) to the fission channel from various $J^{\pi}$ states of $^{239}$U and $^{237}$U. (a): positive parity states, (b): negative parity states](u238-236-fis1.eps)
 Ratios of decay probabilities (branching ratios) to the capture channel from various $J^{\pi}$ states of $^{239}$U and $^{237}$U. (a): positive parity states, (b): negative parity states](u238-236-cap1.eps)
 Ratios of decay probabilities (branching ratios) to the capture channel from various $J^{+}$ states of $^{198}$Au and $^{194}$Ir as a function of corresponding neutron energy in the case they are produced by neutron-induced reactions. Similar results were obtained for negative parity states.](au197-ir193-a1-cap-pos.eps)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Many computer vision problems can be posed as learning a low-dimensional subspace from high dimensional data. The low rank matrix factorization (LRMF) represents a commonly utilized subspace learning strategy. Most of the current LRMF techniques are constructed on the optimization problems using $L_1$-norm and $L_2$-norm losses, which mainly deal with Laplacian and Gaussian noises, respectively. To make LRMF capable of adapting more complex noise, this paper proposes a new LRMF model by assuming noise as Mixture of Exponential Power (MoEP) distributions and proposes a penalized MoEP (PMoEP) model by combining the penalized likelihood method with MoEP distributions. Such setting facilitates the learned LRMF model capable of automatically fitting the real noise through MoEP distributions. Each component in this mixture is adapted from a series of preliminary super- or sub-Gaussian candidates. Moreover, by facilitating the local continuity of noise components, we embed Markov random field into the PMoEP model and further propose the advanced PMoEP-MRF model. An Expectation Maximization (EM) algorithm and a variational EM (VEM) algorithm are also designed to infer the parameters involved in the proposed PMoEP and the PMoEP-MRF model, respectively. The superseniority of our methods is demonstrated by extensive experiments on synthetic data, face modeling, hyperspectral image restoration and background subtraction.'
author:
- 'Xiangyong Cao, Qian Zhao, Deyu Meng$^\ast$, Yang Chen, Zongben Xu[^1]. [^2].'
bibliography:
- 'mybibfile.bib'
title: 'Low-rank Matrix Factorization under General Mixture Noise Distributions'
---
Low-rank matrix factorization, mixture of exponential power distributions, Expectation Maximization algorithm, face modeling, hyperspectral image restoration, background subtraction.
Introduction
============
Many computer vision, machine learning, data mining and statistical problems can be formulated as the problem of extracting the intrinsic low dimensional subspace from input high-dimensional data. The extracted subspace tends to deliver the refined latent knowledge underlying data and thus has a wide range of applications including structure from motion [@tomasi1992shape], face recognition [@wright2009robust], collaborative filtering [@koren2008factorization], information retrieval [@deerwester1990indexing], social networks [@cheng2012fused], object recognition [@turk1991eigenfaces], layer extraction [@ke2001subspace] and plane-based pose estimation [@sturm2000algorithms].
![From left to right: Original hyperspectral image (HSI), reconstructed image, two extracted noise images with their histograms by the proposed methods. (Top: $EP_{0.2}$ noise image and histogram. Bottom: $EP_{1.8}$ noise image and histogram).[]{data-label="intro_fig"}](Intro_fig1){width="1\linewidth"}
Low rank matrix factorization (LRMF) is one of the most commonly utilized techniques for subspace learning. Given a data matrix $\mathbf{Y}\in\mathcal{R}^{m\times n}$ with entries $y_{ij}s$, the LRMF problem can be mathematically formulated as $$\begin{aligned}
\label{LRMF}
\min_{\mathbf{U},\mathbf{V}}||\mathbf{W}\odot(\mathbf{Y}-\mathbf{U}\mathbf{V}^{T})||,\end{aligned}$$ where $\mathbf{W}$ is the indicator matrix with $w_{ij} = 0$ if $y_{ij}$ is missing and 1 otherwise, and $\mathbf{U}\in \mathcal{R}^{m\times r}$ and $\mathbf{V}\in \mathcal{R}^{n\times r}$ are low-rank matrices ($r<\min(m,n)$). The operator $\odot$ denotes the Hadamard product (the component-wise multiplication) and $||\cdot||$ corresponds to a certain noise measure.
Under the assumption of Gaussian noise, it is natural to utilize the $L_2$-norm (Frobenius norm) as the noise measure, which has been extensively studied in LRMF literatures [@srebro2003weighted; @buchanan2005damped; @okatani2007wiberg; @aguiar2008spectrally; @zhao2010successively; @okatani2011efficient; @wen2012solving; @mitra2010large]. However, it has been recognized in many real applications that these methods constructed on $L_2$ norm are sensitive to outliers and non-Gaussian noise. In order to introduce robustness, the $L_1$-norm based models [@ke2005robust; @eriksson2010efficient; @zheng2012practical; @kwak2008principal; @shu2014robust; @ji2010robust] have attracted much attention recently. However, the $L_1$-norm is only optimal for Laplace-like noise and still very limited for handling various types of noise encountered in real problems. Taking the hyper-spectral image (HSI) as an example, it has been investigated in [@zhang2014hyperspectral] that there are mainly two kinds of noise embedded in such type of data, i.e., sparse noise (stripe and deadline) and Gaussian-like noise, as depicted in Fig. \[intro\_fig\]. The stripe noise is produced by the non-uniform sensor response which conducts the deviation of gray values of the original image continuously towards one direction. This noise always very sparsely located on edges and in texture areas of an image. The deadline noise, which is induced by some damaged sensor, results in zero or very small pixel values of entire columns of images along some HSI bands. The Gaussian-like noise is induced by some random disturbation during the transmission process of hyper-spectral signals. It is easy to see that such kind of complex noise cannot be well fit by either Laplace or Gaussian, which means that neither $L_1$-norm nor $L_2$-norm LRMF models are proper for this type of data.
Very recently, some novel models were presented to expand the availability of LRMF under more complex noise. The key idea is assuming that the noise follows a more complicated mixture of Gaussians (MoG) [@meng2013robust], which is expected to better fit real noise, since the MoG constructs a universal approximator to any continuous density function in theory [@maz1996approximate]. However, this method still cannot finely adapt real data noise. On one hand, MoG can approximate a complex distribution, e.g. Laplace, only under the assumption that the number of components goes to infinity, while in applications only a finite number of components can be specified. On the other hand, it also lacks a theoretically sound manner to properly select the number of Gaussian mixture components based on the practical noise extent mixed in data. Thus, it is crucial to construct a better strategy with more adaptive distribution modeling capability on data noises beyond MoG.
In this paper, we propose a new LRMF method with a more general noise model to address the aforementioned issues. Specifically, we encode the noise as a mixture distribution of a series of sub- and super-Gaussians (i.e., general exponential power (EP) distribution), and formulate LRMF as a penalized MLE model, called PMoEP model [@cao2015PMoEP]. Moreover, by facilitating the local continuity of noise components, we embed Markov random field into the PMoEP model and propose the PMoEP-MRF model. Then we design an Expectation Maximization (EM) algorithm and a variational EM (VEM) algorithm to estimate the parameters involved in the proposed PMoEP model and PMoEP-MRF model, respectively, and prove their convergence. The two new methods are not only capable of adaptively fitting complex real noise by EP noise components with proper parameters, but also able to automatically learn the proper number of noise components from data, and thus can better recover the true low-rank matrix from corrupted data as verified by extensive experiments.
The rest of the paper is organized as follows. In Section II, the related work regarding LRMF is discussed. In Section III, we first present the PMoEP model and the corresponding EM algorithm, and then conduct the convergence analysis of the proposed algorithm. The PMoEP-MRF model and the corresponding variational EM algorithm are proposed in Section IV. In Section V, extensive experiments are conducted to substantiate the superiority of the proposed models over previous methods. Finally, conclusions are drawn in Section VI. Throughout the paper, we denote scalars, vectors, and matrices as the non-bold letters, bold lower case letters, and bold upper case letters, respectively.
Related work
============
The $L_2$ norm LRMF with missing data has been studied for decades. Gabriel and Zamir [@gabriel1979lower] proposed a weighted SVD method as the early attempt for this task. They used alternated minimization to find the principal subspace underlying the data. Srebro and Jaakkola [@srebro2003weighted] proposed the Weighted Low-rank Approximation (WLRA) algorithm to enhance efficiency of LRMF calculation. Buchanan and Fitzgibbon [@buchanan2005damped] further proposed a regularized model that adds a regularization term and then adopts the damped newton algorithm to estimate the subspaces. However, it cannot handle large-scale problems due to the infeasibility of computing the Hessian matrix over a large number of variables. Okatani and Deguchi [@okatani2007wiberg] showed that a Wiberg marginalization strategy on $\mathbf{U}$ and $\mathbf{V}$ can provide a better and robust initialization and proposed the Wiberg algorithm that updates $\mathbf{U}$ via least squares while updates $\mathbf{V}$ by a Gauss-Newton step in each iteration. Later, the Wiberg algorithm was extended to a damped version to achieve better convergence by Okatani et al. [@okatani2011efficient]. Aguiar et al. [@aguiar2008spectrally] deduced a globally optimal solution to $L_2$-LRMF with missing data under the assumption that the missing data has a special Young diagram structure. Zhao and Zhang [@zhao2010successively] formulated the $L_2$- norm LRMF as a constrained model to improve its stability in real applications. Wen et al. [@wen2012solving] adopted the alternating strategy to solve the $L_2$-norm LRMF problem. Mitra et al. [@mitra2010large] proposed an augmented Lagrangian method to solve the $L_2$-norm LRMF problem for higher accuracy. However, all of these methods minimize the $L_2$-norm or its variations and is only optimal for Gaussian-like noise.
To make subspace learning method less sensitive to outliers, some robust loss functions have been investigated. For example, De la Torre and Black [@de2003framework] adopted the Geman-McClure function and then used the iterative reweighted least square (IRLS) method to solve the induced optimization problem. In the last decade, the $L_1$-norm has become the most popular robust loss function along this research line. Ke and Kanade [@ke2005robust] initially replaced the $L_2$-norm with the $L_1$-norm for LRMF, and then solved the optimization by alternated convex programming (ACP) method. Kwak [@kwak2008principal] later proposed to maximize the $L_1$-norm of the projection of data points onto the unknown principal directions instead of minimizing the residue. Eriksson and Hengel [@eriksson2010efficient] experimentally showed that the ACP approach does not converge to the desired point with high probability, and thus introduced the $L_1$-Wiberg approach to address this issue. Zheng et al. [@zheng2012practical] added more constraints to the factors $\mathbf{U}$ and $\mathbf{V}$ for $L_1$-norm LRMF, and solved the optimization by ALM, which improved the performance in structure from motion application. Within the probabilistic framework, Wang et al. [@wang2012probabilistic] proposed probabilistic robust matrix factorization (PRMF) that modeled the noise as a Laplace distribution, which has been later extended to fully Bayesian settings by Wang and Yeung [@wang2013bayesian]. However, these methods optimize the $L_1$-norm and thus are only optimal for Laplace-like noise.
Beyond Gaussian or Laplace, other types of noise assumptions have also been attempted recently to make the model adaptable to more complex noise scenarios. Lakshminarayanan et al. [@lakshminarayanan2011robust] assumed that the noise is drawn from a student-t distribution. Babacan et al. [@babacan2012sparse] proposed a Bayesian methods for low-rank matrix estimation modeling the noise as a combination of sparse and Gaussian. To handle more complex noise, Meng and De la Torre [@meng2013robust] modeled the noise as a MoG distribution for LRMF, and later was extended to the Bayesian framework by Chen et al. [@chen2015bayesian] and to RPCA by Zhao et al. [@zhao2014robust]. Although better than traditional methods, these methods are still very limited in dealing with complex noise in real scenarios.
LRMF with MoEP noise
====================
In this section, we first present the new LRMF model with MoEP noise, called PMoEP model, and then design an EM algorithm to solve it. Finally, we give the convergence analysis of the proposed EM algorithm and the implementation issues.
PMoEP model
-----------
In LRMF, from a generative perspective, each element $y_{ij}(i=1,2,\dots,m,j=1,2,\dots,n)$ of the data matrix $\mathbf{Y}$ can be modeled as $$\begin{aligned}
y_{ij}=\mathbf{u}_{i}\mathbf{v}_{j}^{T} + e_{ij},\end{aligned}$$ where $\mathbf{u}_{i}$ and $\mathbf{v}_{i}$ represent the $i^{th}$ row vectors of $\mathbf{U}$ and $\mathbf{V}$, respectively, and $e_{ij}$ is the noise embedded in $y_{ij}$. Instead of assuming that the noise obeys Gaussian [@srebro2003weighted], Laplace [@ke2005robust] or MoG [@meng2013robust] distributions as previous methods, we assume that the noise $e_{ij}$ follows more flexible mixture of Exponential Power (EP) distributions: $$\begin{aligned}
\label{MoEP}
\mathbb{P}(e_{ij}) = \sum_{k=1}^K \pi_{k} f_{p_{k}}(e_{ij};0,\eta_k),\end{aligned}$$ where $\pi_{k}$ is the mixing proportion with $\pi_{k}\geq 0$ and $\sum_{k=1}^{K}\pi_{k}=1$, $K$ is the number of the mixture components and $f_{p_{k}}(e_{ij};0,\eta_k)$ denotes the $k^{th}$ EP distribution with parameter $\eta_{k}$ and $p_{k} (p_{k}>0)$. Let $\mathbf{p}=[p_1,p_2,\dots,p_{K}]$, in which each $p_{k}$ can be variously specified. As defined in [@mineo2005software], the density function of the EP distribution ($p>0$) with zero mean is $$\begin{aligned}
\label{EP_pdf}
f_{p}(e;0,\eta) &=& \frac{p\eta^{\frac{1}{p}}}{2\Gamma(\frac{1}{p})}\exp\{-\eta|e|^{p}\},\end{aligned}$$ where $\eta$ is the precision parameter, $p$ is the shape parameter and $\Gamma(\cdot)$ is the Gamma function. By changing the shape parameter $p$, the EP distribution describes both leptokurtic ($0<p<2$) and platykurtic ($p>2$) distributions. In particular, we obtain the Laplace distribution with $p=1$, the Gaussian distribution with $p=2$ and the Uniform distribution with $p\rightarrow \infty$ (see Fig. \[EPpdf\]). Therefore, all previous cases including $L_2$, $L_1$, MoG and any combinations of them are just special cases of MoEP. By setting $\eta=1/(p\sigma^{p})$, the EP distribution (\[EP\_pdf\]) can be equivalently written as $EP_{p}(e;0,p\sigma^{p})$.
![The probability density function of EP distributions.[]{data-label="EPpdf"}](EPpdf){width="0.85\linewidth"}
In our model, we assume that each noise $e_{ij}$ is equipped with an indicator variable $\mathbf{z}_{ij}=[z_{ij1},z_{ij2},\dots,z_{ijK}]^{T}$, where $z_{ijk}\in \{0,1\}$ and $\sum_{k=1}^{K}z_{ijk}=1$. $z_{ijk}=1$ implies that the noise $e_{ij}$ is drawn from the $k^{th}$ EP distribution. $\mathbf{z}_{ij}$ obeys a multinomial distribution $\mathbf{z}_{ij}\sim \mathcal{M}(\boldsymbol{\pi})$, where $\boldsymbol{\pi}=[\pi_1,\pi_2,\dots,\pi_{K}]^{T}$. Then we have: $$\begin{aligned}
\mathbb{P}(e_{ij}|\mathbf{z}_{ij}) &=& \prod_{k=1}^{K} f_{p_{k}}(e_{ij};0,\eta_{k})^{z_{ijk}},\\
\mathbb{P}(\mathbf{z}_{ij};\boldsymbol{\pi}) &=& \prod_{k=1}^{K}\pi_{k}^{z_{ijk}}.\end{aligned}$$ Denoting $\mathbf{E}=(e_{ij})_{m\times n}$, $\mathbf{Z}=(\mathbf{z}_{ij})_{m\times n}$ and $\mathbf{\Theta}=\{\boldsymbol{\pi},\boldsymbol{\eta},\mathbf{U},\mathbf{V}\}$ with $\boldsymbol{\eta}=[\eta_1,\eta_2,\dots,\eta_K]^{T}$, the *complete likelihood function* can then be written as $$\begin{aligned}
\mathbb{P}(\mathbf{E},\mathbf{Z};\mathbf{\Theta})&=& \prod_{i,j\in \Omega}\prod_{k=1}^{K}[\pi_{k}f_{p_{k}}(e_{ij};0,\eta_{k})]^{z_{ijk}},\end{aligned}$$ where $\Omega$ is the index set of the non-missing entries in $\mathbf{Y}$. Then the *log-likelihood function* is $$\label{loglikelihood}
l(\mathbf{\Theta})=\log{\mathbb{P}(\mathbf{E};\mathbf{\Theta})}=\log{\sum_{\mathbf{Z}}\mathbb{P}(\mathbf{E},\mathbf{Z};\mathbf{\Theta})},$$ and the *complete log-likelihood function* is $$\begin{aligned}
l^{C}(\mathbf{\Theta})&=&\log{\mathbb{P}(\mathbf{E},\mathbf{Z};\mathbf{\Theta})}\nonumber \\
&=&\sum_{i,j\in \Omega}\sum_{k=1}^{K}z_{ijk}[\log{\pi_{k}}+\log{f_{p_{k}}(e_{ij};0,\eta_{k})}].\end{aligned}$$
As aforementioned in introduction, determining the number of components $K$ is an important problem for the mixture model. Thus, various model selection techniques can be readily employed to resolve this issue. Most conventional methods are based on the likelihood function and some information theoretic criteria, such as AIC and BIC. However, Leroux [@leroux1992consistent] showed that these criteria may overestimate the true number of components. On the other hand, Bayesian approaches [@ormoneit1998averaging; @zivkovic2004recursive] have also been used to find a suitable number of components of the finite mixture model. But the computation burden and statistical properties of the Bayesian method limit its use to a certain extent. Here we adopt a recently proposed method by Huang et al. [@huang2013model] for this aim of selecting EP mixture number, and construct the following penalized MoEP (PMoEP) model: $$\begin{aligned}
\label{PMoEPmodel}
\max_{\mathbf{\Theta}}\left\{l_{P}^{C}(\mathbf{\Theta})=l^{C}(\mathbf{\Theta})-P(\boldsymbol{\pi};\lambda)\right\},\end{aligned}$$ where $$\begin{aligned}
P(\boldsymbol{\pi};\lambda)=n\lambda \sum_{k=1}^{K}D_{k}\log{\frac{\epsilon+\pi_{k}}{\epsilon}},\end{aligned}$$ with $\epsilon$ being a very small positive number, $\lambda$ being a tuning parameter ($\lambda>0$), and $D_{k}$ being the number of free parameters for the $k^{th}$ component. In the proposed PMoEP model, $D_{k}$ equals 2 (for $\pi_{k}$ and $\eta_{k}$).
EM algorithm for PMoEP model
----------------------------
In this subsection, we propose an EM algorithm to solve the proposed PMoEP model (\[PMoEPmodel\]). The EM algorithm is an iterative procedure and thus we assume that $\mathbf{\Theta}^{(t)}=\{\{\boldsymbol{\pi}^{(t)}\},\{\boldsymbol{\eta}^{(t)}\},\mathbf{U}^{(t)},\mathbf{V}^{(t)}\}$ is the estimation at the $t^{th}$ iteration. In the following, we will introduce the two steps of the proposed EM algorithm.
In the E step, we compute the conditional expectation of $z_{ijk}$ given $e_{ij}$ by the Bayes’ rule: $$\begin{aligned}
\label{updategamma}
\gamma_{ijk}^{(t+1)} = \frac{\pi_{k}^{(t)}f_{p_k}(y_{ij}-\mathbf{u}_{i}^{(t)}(\mathbf{v}_{j}^{(t)})^{T})|0,\eta_{k}^{(t)})}
{\sum_{l=1}^{K}\pi_{l}^{(t)}f_{p_l}(y_{ij}-\mathbf{u}_{i}^{(t)}(\mathbf{v}_{j}^{(t)})^{T})|0,\eta_{l}^{(t)}))}.\end{aligned}$$ Then, it is easy to construct the so-called $Q$ function: $$\begin{aligned}
\begin{split}
Q(\mathbf{\Theta},\mathbf{\Theta}^{(t)})\!&=\!\sum_{i,j\in \Omega,k}\gamma_{ijk}^{(t+1)}[\log{f_{p_k}(y_{ij}\!-\!\mathbf{u}_{i}\mathbf{v}_{j}^{T};\eta_{k})}\!+\!\log{\pi_{k}}]\nonumber\\
& - n\lambda \sum_{k=1}^{K}D_{k}\log{\frac{\epsilon+\pi_{k}}{\epsilon}}.
\end{split}\end{aligned}$$
In the M-step, we update $\mathbf{\Theta}$ by maximizing the $Q$ function. For $\boldsymbol{\pi}$ and $\boldsymbol{\eta}$, it is easy to obtain the update equations by taking the first derivative of $Q$ with respect to them respectively, and finding the zero points through: $$\label{updatepi}
\pi_{k}^{(t+1)}\!=\!\max\left\{0,\frac{1}{1\!-\!\lambda \hat{D}}\left[\frac{\sum_{i,j\in \Omega}\gamma_{ijk}^{(t+1)}}{|\Omega|}\!-\!\lambda D_{k}\right]\right\},$$ $$\label{updatetheta}
\eta_{k}^{(t+1)}\!=\!\frac{N_{k}}{p_{k}\sum_{i,j\in \Omega}\gamma_{ijk}^{(t+1)}|y_{ij}\!-\!\mathbf{u}_{i}^{(t)}(\mathbf{v}_{j}^{(t)})^{T}|^{p_{k}}},$$ where $\hat{D} = \sum_{k=1}^{K}D_{k} = 2K$, $N_{k} = \sum_{i,j\in \Omega}\gamma_{ijk}^{(t+1)}$ and $|\Omega|$ is the number of non-missing elements. To update $\mathbf{U}, \mathbf{V}$, we need to maximize the following function: $$\begin{aligned}
\label{update_s}
-\sum_{i,j\in \Omega}\sum_{k=1}^{K}\gamma_{ijk}^{(t+1)}
\eta_{k}^{(t+1)}|y_{ij}-\mathbf{u}_{i}^{(t)}(\mathbf{v}_{j}^{(t)})^{T}|^{p_k},\end{aligned}$$ which is equivalent to solving[^3] $$\label{subproblem_uv}
\min_{\mathbf{U}, \mathbf{V}}\sum_{k=1}^{K}||\mathbf{W}_{(k)}\odot (\mathbf{Y}-\mathbf{U}\mathbf{V}^{T})||_{p_{k}}^{p_{k}},$$ where the element $w_{(k)ij}$ of $\mathbf{W}_{(k)}\in \mathcal{R}^{m\times n}(k=1,\dots,K)$ is\
$$w_{(k)ij}=\begin{cases}(\eta_{k}^{(t+1)}\gamma_{ijk}^{(t+1)})^{\frac{1}{p_{k}}}, \quad i,j\in\Omega\\~~~~~~~~~~0, ~~~~~~~~~~~ i,j\notin\Omega \end{cases}.$$ To solve (\[subproblem\_uv\]), we resort to augmented Lagrange multipliers (ALM) method. By introducing auxiliary variable $\mathbf{L}=\mathbf{U}\mathbf{V}^{T}$, (\[subproblem\_uv\]) can be equivalently rewritten as $$\label{subproblem_uv2}
\begin{split}
&\min_{\mathbf{U},\mathbf{V}}\sum_{k=1}^{K}||\mathbf{W}_{(k)}\odot (\mathbf{Y}-\mathbf{L})||_{p_{k}}^{p_{k}},
\quad s.t~\mathbf{L} = \mathbf{U}\mathbf{V}^{T}.
\end{split}$$ The augmented Lagrangian function can be written as: $$\label{lagrangefunc}
\begin{split}
L(\mathbf{U},\mathbf{V},\mathbf{L},\mathbf{Y},\rho)&=\sum_{k=1}^{K}||\mathbf{W}_{(k)}\odot (\mathbf{Y}\!-\!\mathbf{L})||_{p_{k}}^{p_{k}}\\
&+\langle\mathbf{\Lambda},\mathbf{L}\!-\!\mathbf{U}\mathbf{V}^{T}\rangle+\frac{\rho}{2}||\mathbf{L}\!-\!\mathbf{U}\mathbf{V}^{T}||_{F}^{2},
\end{split}$$ where $\mathbf{\Lambda}\in \mathcal{R}^{m\times n}$ is the Lagrange multiplier and $\rho$ is a positive scalar. Then the optimization (\[subproblem\_uv2\]) can be solved by alternatively updating all involved variables and multipliers as follows $$\begin{aligned}
\label{optProcess}
\begin{cases}
&\!\left(\mathbf{U}^{(s+1)},\mathbf{V}^{(s+1)}\right)\!=\! \underset{\mathbf{U},\mathbf{V}}{\arg\min} L(\mathbf{U},\mathbf{V},\mathbf{L}^{(s)},\mathbf{\Lambda}^{(s)},\rho^{(s)}),\\
&\!\mathbf{L}^{(s+1)}\!=\! \underset{\mathbf{L}}{\arg\min} L(\mathbf{U}^{(s+1)},\mathbf{V}^{(s+1)},\mathbf{L},\mathbf{\Lambda}^{(s)},\rho^{(s)}),\\
&\!\mathbf{\Lambda}^{(s+1)}\!=\!\mathbf{\Lambda}^{(s)} + \rho^{(s)}(\mathbf{L}^{(s+1)}\!-\!\mathbf{U}^{(s+1)}(\mathbf{V}^{(s+1)})^{T}),\label{alg1:eq3}\\
&\!\rho^{(s+1)} \!=\! \alpha\rho^{(s)}\label{alg1:eq4},
\end{cases}\end{aligned}$$ where $\alpha$ is a preset constant which is slightly larger than 1, guaranteeing the gradually increasing value for $\rho$ in each iteration. Now we discuss how to solve the subproblems involved in the above procedure.
\(1) *Update* $\mathbf{U},\mathbf{V}$. The following subproblem needs to be solved: $$\label{uvupdate}
\min_{\mathbf{U},\mathbf{V}}||\mathbf{L}^{(s)}+\frac{1}{\rho^{(s)}}\mathbf{\Lambda}^{(s)}-\mathbf{U}\mathbf{V}^{T}||_{F}^{2},$$ which can be accurately and efficiently solved by the SVD method.
\(2) *Update* $\mathbf{L}$. We need to solve the following problem: $$\begin{aligned}
\label{subproblemL}
\begin{split}
&\min_{\mathbf{L}}\sum_{k=1}^{K}||\mathbf{W}_{(k)}\!\odot\! (\mathbf{Y}\!-\!\mathbf{L})||_{p_{k}}^{p_{k}}\!+\!\langle\mathbf{\Lambda}^{(s)},\mathbf{L}\rangle\\
&+\frac{\rho^{(s)}}{2}||\mathbf{L}-\mathbf{U}^{(s+1)}(\mathbf{V}^{(s+1)})^{T}||_{F}^{2}.
\end{split}\end{aligned}$$ This problem seems to be more difficult due to its non-convexity and non-smoothness. However, we can divide it into $mn$ independent scalar optimization problems as follows: $$\begin{aligned}
\label{updateE}
\begin{cases}
\begin{split}
&\min_{l_{ij}}\sum_{k}\eta_{k}\gamma_{ijk}|y_{ij}-l_{ij}|^{p_{k}}
+\frac{\rho^{(s)}}{2}l_{ij}^{2}\\
&~~~~~~~~~+((\mathbf{\Lambda}_{ij}^{(s)})-\rho^{(s)}\mathbf{u}_{i}\mathbf{v}_{j}^{T})l_{ij},~~~~~~~~(i,j)\in\Omega\\
&\underset{l_{ij}}{\min}\frac{\rho^{(s)}}{2}l_{ij}^{2}+ ((\mathbf{\Lambda}^{(s)})_{ij}-\rho^{(s)}\mathbf{u}_{i}\mathbf{v}_{j}^{T})l_{ij}.~~~(i,j)\notin\Omega
\end{split}
\end{cases}\end{aligned}$$ Letting $s_{ij} = y_{ij}-l_{ij}$, (\[updateE\]) is equivalent to $$\begin{aligned}
\label{sij}
\begin{cases}
&\underset{s_{ij}}{\min}\frac{1}{2}(t_{ij}-s_{ij})^{2}+ \frac{1}{\rho^{(s)}}\sum_{l}\eta_{l}\gamma_{ijl}|s_{ij}|^{p_{l}},~(i,j)\in\Omega\\
&\underset{s_{ij}}{\min}\frac{1}{2}(t_{ij}-s_{ij})^{2},~(i,j)\notin\Omega
\end{cases}\end{aligned}$$ where $t_{ij}=-\mathbf{u}_{i}\mathbf{v}_{j}^{T}+y_{ij}+\frac{1}{\rho^{(s)}}(\mathbf{\Lambda}_{ij}^{(s)})$. Then, for each $(i,j)\in\Omega$, (\[sij\]) is equivalent to the following subproblem: $$\begin{aligned}
\label{subproblem_e}
\min_{s_{ij}} \frac{1}{2}(t_{ij}-s_{ij})^{2} + \frac{1}{\rho}\sum_{l=1}^{K}\eta_{l}\gamma_{ijl}|s_{ij}|^{p_{l}}.\end{aligned}$$ This problem requires to optimize a scalar variable, and we take its first derivative with respect to $s_{ij}$ and then adopt the well-known Newton method to easily approach a local minimum of it. The procedure of updating $\mathbf{L}$ by ALM method can then be listed in Algorithm \[alg1\].
The initialization of $\mathbf{L}^{(0)}$, $\mathbf{\Lambda}^{(0)}$ and $s=0$. $\mathbf{U}$ and $\mathbf{V}$. Updating $\mathbf{U}^{(s+1)}$ and $\mathbf{V}^{(s+1)}$ via Eq. (\[uvupdate\]); Updating $\mathbf{L}^{(s+1)}$ via Eqs. (\[sij\]) and (\[subproblem\_e\]). Updating $\mathbf{\Lambda}^{(s+1)}$ via Eq. (\[alg1:eq3\]). Updating $\alpha^{(s+1)}$ via Eq. (\[alg1:eq4\]).
[**[Remark:]{}**]{} If $f_{k}$ is specified as the density of a Gaussian distribution, the PMoEP model degenerates to the penalized MoG (PMoG) model. The optimization process of the PMoG model is almost the same as the PMoEP except the minimization form of (\[subproblem\_uv\]). In this case, the optimization problem (18) has the following form $$\begin{aligned}
\label{PMoG_dif}
\min_{\mathbf{U}, \mathbf{V}}||\mathbf{\tilde{W}}\odot (\mathbf{Y}-\mathbf{U}\mathbf{V}^{T})||_{2}^{2},\end{aligned}$$ and then any off-the-shelf weighted $L_2$ norm LRMF method can be adopted to solve it. It should be noted that the PMoG method so conducted is different from the previous MoG method [@meng2013robust] due to its augmented automatic mixture-component-number learning capability.
The proposed EM algorithm for PMoEP model can now be summarized in Algorithm \[alg2\].
Data $\mathbf{Y}$; The algorithm parameters: rank $r$ and $\lambda$.
Parameter $\mathbf{\Theta}$, the number of mixture components $K_{final}$ and posterior probability $\boldsymbol{\gamma}=(\gamma_{ijk})_{m\times n\times K_{final}}$. $\mathbf{\Theta}^{(t)}\!=\!\{\boldsymbol{\pi}^{(t)},\boldsymbol{\eta}^{(t)}, \mathbf{U}^{(t)}, \mathbf{V}^{(t)}\}$, the number of initial mixture components $K_{start}$, preset candidates $\mathbf{p}=[p_1,\dots,p_{K_{start}}]$, tolerance $\epsilon$ and $t=0$. Updating $\boldsymbol{\gamma}^{(t)}$ via Eq. ;\
Updating $\boldsymbol{\pi}^{(t)}$ via Eq. , and removing the component with $\pi_{k}^{(t)}=0$;\
Updating $\boldsymbol{\eta}^{(t)}$ via Eq. ;\
Updating $\mathbf{U}^{(t)}, \mathbf{V}^{(t)}$ via Algorithm \[alg1\].\
$t = t + 1;$
Convergence Analysis of EM algorithm
------------------------------------
In this subsection, we show the convergence property of the proposed EM algorithm for PMoEP model.
\[theorem1\] Let $l_{P}^{C}(\mathbf{\Theta}) = l(\mathbf{\Theta})-P(\boldsymbol{\pi};\lambda)$, where $l(\Theta)$ is defined in (\[loglikelihood\]). If we assume that $\{\mathbf{\Theta}^{(t)}\}$ is the sequence generated by Algorithm \[alg2\] and the sequence of likelihood values $\{l_{P}^{C}(\mathbf{\Theta}^{(t)})\}$ is bounded above, then there exits a constant $l^{\star}$ such that $$\lim_{t\rightarrow \infty}l_{P}^{C}(\mathbf{\Theta}^{(t)}) = l^{\star},$$ where $$\mathbf{\Theta}^{(t)}\!=\!\underset{\mathbf{\Theta}}{\arg\max}\left\{\Omega(\mathbf{\Theta}|\mathbf{\Theta}^{(t-1)})\!+\!P(\boldsymbol{\pi}^{(t-1)};\lambda)\!-\!P(\boldsymbol{\pi};\lambda)\right\},$$ and $$\Omega(\mathbf{\Theta}|\mathbf{\Theta}^{(t-1)})\!=\!\sum_{\mathbf{Z}}\mathbb{P}(\mathbf{Z}|\mathbf{E};\mathbf{\Theta}^{(t-1)})
\log{\frac{\mathbb{P}(\mathbf{E},\mathbf{Z};\mathbf{\Theta})}{\mathbb{P}(\mathbf{E},\mathbf{Z};\mathbf{\Theta}^{(t-1)})}}.$$
The proof is listed in Appendix A.
Implementation Issues
---------------------
In the proposed PMoEP algorithm, there are three involved preset parameters, $K_{start}$, $p$ and $\lambda$. Throughout all our experiments, we just simply set $K_{start}$ as a not large number as $4-10$ based on a coarse empirical estimate on the noise complexity inside data. Once $K_{start}$ is initialized, the length of vector $\mathbf{p}=[p_{1},p_{2},\dots,p_{K_{start}}]$ in PMoEP is determined. In all our experiments, the elements in $\mathbf{p}$ are selected ranging over the interval between 0.1 and 2. For the setting of parameter $\lambda$ , we first provide a series of candidates $\lambda$ and then adopt the modified BIC to select a good $\lambda$ among these candidates based on the modified BIC criterion. This criterion has been proven to be able to yield consistent component number estimation of the finite Gaussian mixture model [@huang2013model]. Specifically, the modified BIC criterion is defined as $$\label{bic}
\mbox{BIC}(\lambda)\!=\! \sum_{i,j\in\Omega}\log{\{\sum_{k=1}^{\hat{K}}\hat{\pi}_{k}f_{k}(e_{ij};\hat{\eta}_{k})\}}\!-\! \frac{1}{2}(\sum_{k=1}^{\hat{K}}D_{k})\log{|\Omega|}.$$ Then we can select the proper $\hat{\lambda}$ by $$\hat{\lambda} = \arg\max_{\lambda}\mbox{BIC}(\lambda),$$ where $|\Omega|$ is the number of non-missing elements, $\hat{K}$ is the estimate of the number of components, $\hat{\pi}_{k}$ is the estimate of parameter $\pi_{k}$, and $\hat{\eta}_{k}$ is the estimate of parameter $\eta_{k}$ for maximizing (\[PMoEPmodel\]) for a given $\lambda$.
PMoEP with Markov Random Field
==============================
In this section, we first propose an advanced PMoEP-MRF model. Then, we introduce a variational EM (VEM) algorithm to solve it. Finally, we also show the convergence analysis for the proposed algorithm.
PMoEP-MRF Model
---------------
In some practical applications, we often have certain noise prior knowledge. By introducing the prior into modeling, noise can be more appropriately modeled and thus the performance of the model is expected to be further improved. In video data, we can utilize the spatial and temporal smoothness prior. Specifically, for a certain pixel in one video frame, the pixels located near it both spatially and temporally tend to have similar distribution to it. Therefore, by facilitating the local continuity of noise components, we can embed Markov Random Field (MRF) into the PMoEP model. Note that the random variable $\mathbf{z}_{ij}$ determines the cluster label of noise $e_{ij}$ in PMoEP model, and the aforementioned spatial and temporal relationships among adjacent pixels imply that they incline to possess similar $\mathbf{z}_{ij}$ values. Therefore, we integrate into the distribution of $\mathbf{z}_{ij}$ with such prior smoothness knowledge as: $$\mathbf{z}_{ij}\sim \mathcal{M}(\mathbf{z}_{ij};\boldsymbol{\pi})\prod_{(p,q)\in\mathcal{N}(i,j)}\psi(\mathbf{z}_{ij},\mathbf{z}_{pq}),$$ where $$\psi(\mathbf{z}_{ij},\mathbf{z}_{pq})=
\frac{1}{C}\prod_{k}\exp\left[\tau(2z_{ijk}\!-\!1)(2z_{pqk}\!-\!1)\right],$$ where $\tau$ is a positive scalar parameter (we set $\tau=10$ in experiments), $C$ is a normalization constant of $\psi(\mathbf{z}_{ij},\mathbf{z}_{pq})$ and $\mathcal{N}(i,j)$ is the neighborhood of the $(i,j)$ entry. Specifically, when $z_{ijk}$ and $z_{pqk}$ achieve the same value (0 or 1), $\psi(\mathbf{z}_{ij},\mathbf{z}_{pq})$ will have higher value, and thus this term readily encode the expected prior information. After defining the new distribution of $\mathbf{z}_{ij}$, the distribution of $\mathbf{Z}$ can be written as $$\begin{aligned}
\begin{split}
\mathbb{P}(\mathbf{Z};\boldsymbol{\pi})&=\frac{1}{C}\prod_{i,j\in\Omega,k}\pi_{k}^{z_{ijk}}\\
&\prod_{i,j\in\Omega,k}\prod_{(p,q)\in \mathcal{N}(i,j)}\exp\left[\tau(2z_{ijk}\!-\!1)(2z_{pqk}\!-\!1)\right].
\end{split}\end{aligned}$$ Then, the *complete likelihood function* can be written as $$\begin{aligned}
\begin{split}
\mathbb{P}(\mathbf{E},\mathbf{Z};\mathbf{\Theta})&=\mathbb{P}(\mathbf{E}|\mathbf{Z};\boldsymbol{\eta})\mathbb{P}(\mathbf{Z};\boldsymbol{\pi})\\
&= \frac{1}{C}\prod_{i,j\in \Omega,k}[\pi_{k}f_{p_k}(y_{ij}\!-\!\mathbf{u}_{i}\mathbf{v}_{j}^{T};0,\eta_{k})]^{z_{ijk}}\\
&\prod_{i,j\in\Omega,k}\prod_{(p,q)\in \mathcal{N}(i,j)}\!\!\!\!\!\exp\!\left[\tau(2z_{ijk}\!-\!1)(2z_{pqk}\!-\!1)\right],
\end{split}\end{aligned}$$ and the *complete log-likelihood function* is $$\begin{aligned}
\begin{split}
l^{C}(\mathbf{\Theta})&\!=\!\log{\mathbb{P}(\mathbf{E},\mathbf{Z};\mathbf{\Theta})}\\
&\!=\!\sum_{i,j\in \Omega,k}z_{ijk}[\log{\pi_{k}}\!+\!\log{f_{p_k}(y_{ij}\!-\!\mathbf{u}_{i}\mathbf{v}_{j}^{T};0,\eta_{k})}]\\
\quad &\!+\!\tau\sum_{i,j\in\Omega,k}\sum_{(p,q)\in \mathcal{N}(i,j)}\!\!\!(2z_{ijk}\!-\!1)(2z_{pqk}\!-\!1) \!+\!const.
\end{split}\end{aligned}$$ In the next section, we will introduce a variational EM algorithm to solve this PMoEP-MRF model in detail.
Variational EM algorithm for PMoEP-MRF model
--------------------------------------------
Since EM requires the computation of conditional distribution $\mathbb{P}(\mathbf{Z}|\mathbf{E})$ which is not tractable. In such PMoEP-MRF model, we resort to the variational method that aims at optimizing a lower bound of $\log{\mathcal{L}(\mathbf{E})}$, denoted by $$\begin{aligned}
\mathcal{J}(R_{\mathbf{E}}) = \log{\mathcal{L}(\mathbf{E})} - KL[R_{\mathbf{E}}(\mathbf{Z}),\mathbb{P}(\mathbf{Z}|\mathbf{E})],\end{aligned}$$ where $KL$ denotes the Kullback$-$Leibler divergence, $\mathbb{P}(\mathbf{Z}|\mathbf{E})$ is the true conditional distribution of the indicator variables $\mathbf{Z}$ given $\mathbf{E}$, and $R_{\mathbf{E}}(\mathbf{Z})$ is an approximation of the conditional distribution. $\mathcal{J}(R_{\mathbf{E}})$ equals to $\log{\mathcal{L}(\mathbf{E})}$ if and only if $R_{\mathbf{E}}(\mathbf{Z})=\mathbb{P}(\mathbf{Z}|\mathbf{E})$.
As shown above, we are not able to calculate $\mathbb{P}(\mathbf{Z}|\mathbf{E})$, so we will look for the best (in terms of $KL$ divergence) $R_{\mathbf{E}}(\mathbf{Z})$ in a certain class of distributions. Specifically, we constrain the variational distribution $R_{\mathbf{E}}(\mathbf{Z})$ to have the following form: $$\begin{aligned}
R_{\mathbf{E}}(\mathbf{Z})&=&\prod_{i,j}R(\mathbf{z}_{ij};\boldsymbol{\gamma}_{ij}),\end{aligned}$$ where $R(\mathbf{z}_{ij};\boldsymbol{\gamma}_{ij})=\prod_{ij}\prod_{k}\gamma_{ijk}^{z_{ijk}},\sum_k \gamma_{ijk} = 1$, and $\boldsymbol{\gamma}$ is the variational parameter. Then, the lower bound $\mathcal{J}(R_{\mathbf{E}})$ to be maximized can be written as $$\begin{aligned}
\label{lowbound}
\begin{split}
\mathcal{J}(R_{\mathbf{E}}) &= E_{R_{\mathbf{E}}(\mathbf{Z})}\{\log{\mathbb{P}(\mathbf{E},\mathbf{Z})}\} - E_{R_{\mathbf{E}}(\mathbf{Z})}\{R_{\mathbf{E}}(\mathbf{Z})\},\\
&= \sum_{i,j\in\Omega,k}\left[\log{\pi_{k}+\log{f_{p_k}(e_{ij};0,\eta_{k})}}\right]\\
&+\tau\sum_{i,j\in\Omega,k}\sum_{(p,q)\in \mathcal{N}(i,j)}(2\gamma_{ijk}-1)(2\gamma_{pqk}-1)\\
&-\sum_{i,j\in\Omega,k}\gamma_{ijk}\log{\gamma_{ijk}}+const.
\end{split}\end{aligned}$$ We can easily adopt alternative search strategy for the maximization problem on $\mathcal{J}(R_{\mathbf{E}})$ by alternatively solving the sub-problems: (i) with respect to $R_{\mathbf{E}}$ and (ii) with respect to parameters $\mathbf{U},\mathbf{V},\boldsymbol{\pi},\boldsymbol{\eta}$. The following Proposition \[pro1\] and \[pro2\] provide the solutions of optimization problem (i) and (ii), respectively.
\[pro1\] *(Variational E-step)* Given parameters $\mathbf{\Theta}=\{\mathbf{U},\mathbf{V},\boldsymbol{\pi},\boldsymbol{\eta}\}$, the optimal variational parameters $\hat{\gamma}_{ij} = \underset{\boldsymbol{\gamma}}{\arg\max}~\mathcal{J}(R_{\mathbf{E}})$ satisfy the following fixed point relation: $$\label{updategamma_fixpoint}
\gamma_{ijk}\propto\pi_{k}f_{p_k}(e_{ij};0,\eta_{k})\exp\{\tau\sum_{(p,q)\in \mathcal{N}(i,j)}\gamma_{pqk}\}.$$
Based on (\[lowbound\]), we maximize $\mathcal{J}(R_{\mathbf{E}})$ with respect to $\boldsymbol{\gamma}_{ij}s$, subject to $\sum_k\gamma_{ijk}=1$, for all $i,j$, i.e. to maximize $\mathcal{J}(R_{\mathbf{E}})+\sum_{ij}[\lambda_{ij}(\sum_{k}\gamma_{ijk}-1)]$ where $\lambda_{ij}$ is the Lagrangian multiplier. The derivative with respect to $\gamma_{ijk}$ is $$\log{\pi_{k}\!+\!\log{f_{p_k}(e_{ij};0,\eta_{k})}}\!+\!\tau\!\!\!\!\sum_{(p,q)\in \mathcal{N}(i,j)}\gamma_{pqk}\!-\!\log{\gamma_{ijk}}\!-\!1\!+\!\lambda_{ij}.$$ This derivative is null iff $\gamma_{ijk}$ satisfy the relation given in the proposition, and $\exp(-1+\lambda_{ij})$ is the the normalizing constant.
\[pro2\] *(Variational M-step)* Given the variational parameters $\boldsymbol{\gamma}_{ij}s$, the values of parameters $\mathbf{U},\mathbf{V},\boldsymbol{\pi},\boldsymbol{\eta}$ that maximize $\mathcal{J}(R_{\mathbf{E}})$ can be calculated in the same way as the M step in the EM algorithm of PMoEP model.
The proposed variational EM algorithm for PMoEP-MRF model can then be summarized in Algorithm \[alg3\].
Data $\mathbf{Y}$, rank $r$, $\tau$ and $\lambda$. Parameter $\mathbf{\Theta}$, mixture components number $K_{final}$ and $\boldsymbol{\gamma}=(\gamma_{ijk})_{m\times n\times K_{final}}$.
$\mathbf{\Theta}^{(0)}\!=\!\{\boldsymbol{\pi}^{(0)},\boldsymbol{\eta}^{(0)}, \mathbf{U}^{(0)}, \mathbf{V}^{(0)}\}$, the initial mixture components number $K_{start}$, preset candidates $\mathbf{p}=[p_1,\dots,p_{K_{start}}]$, tolerance $\epsilon$ and $t=0$. Updating $\boldsymbol{\gamma}^{(t)}$ via the fixed-point Eq. ;\
Updating $\boldsymbol{\pi}^{(t)}$ via Eq. , and removing the component with $\pi_{k}^{(t)}=0$;\
Updating $\boldsymbol{\eta}^{(t)}$ via Eq. ;\
Updating $\mathbf{U}^{(t)}, \mathbf{V}^{(t)}$ via Algorithm \[alg1\];\
$t = t + 1.$
Convergence Analysis of Variational EM algorithm
------------------------------------------------
In this subsection, we show the convergence property of the proposed EM algorithm for PMoEP model.
\[theorem2\] Given $\lambda$, Algorithm \[alg3\] generates a sequence $\{\{\boldsymbol{\gamma}_{ij}^{(t)}\}, \mathbf{\Theta}^{(t)}\}\}_{t=1}^{\infty}$ which increases $\mathcal{J}(R_{\mathbf{E}})$ such that $$\mathcal{J}(R_{\mathbf{E}};\{\boldsymbol{\gamma}_{ij}^{(t+1)}\}, \mathbf{\Theta}^{(t+1)}\})\geq \mathcal{J}(R_{\mathbf{E}};\{\boldsymbol{\gamma}_{ij}^{(t)}\},\mathbf{\Theta}^{(t)}\}).$$
This is a direct consequence of Propositions \[pro1\] and \[pro2\], which both guarantee that $\mathcal{J}(R_{\mathbf{E}})$ monotonically increases in iteration.
It is easy to see that $\mathcal{J}(R_{\mathbf{E}};\{\boldsymbol{\gamma}_{ij}^{(t)}\},\mathbf{\Theta}^{(t)}\})$ is upper bounded, and thus the convergence of Algorithm 3 can be guaranteed.
Experimental Results
====================
To evaluate the performance of the proposed PMoEP method, its special case PMoG and the PMoEP-MRF method, we conducted a series of experiments on both synthetic and real data. Five state-of-the-art LRMF methods were considered for comparison, including Mixture of Gaussion method (MoG [@meng2013robust]), Laplace noise methods (CWM [@meng2013cyclic], RegL1ALM [@zheng2012practical]) and Gaussian noise methods (Damped Wiberg (DW) [@okatani2011efficient] and SVD). All experiments were implemented in Matlab R2014a on a PC with 3.60GHz CPU and 12GB RAM.
\[1\]
[c|c|c]{} &\
& PMoG & PMoEP\
Gaussian &
-------------------------
$K_{final}=1$,
$\lambda_{select}=0.01$
-------------------------
: \[table1\]The Parameter Selection for PMoG and PMoEP.
&
--------------------------------
$K_{final}=1, p_{select} = 2$,
$\lambda_{select}=0.15$
--------------------------------
: \[table1\]The Parameter Selection for PMoG and PMoEP.
\
Exponential Power &
--------------------------
$K_{final}=3$,
$\lambda_{select}=0.001$
--------------------------
: \[table1\]The Parameter Selection for PMoG and PMoEP.
&
-------------------------------
$K_{final}=1,p_{select}=0.2$,
$\lambda_{select}=0.3$
-------------------------------
: \[table1\]The Parameter Selection for PMoG and PMoEP.
\
Laplace &
--------------------------
$K_{final}=3$,
$\lambda_{select}=0.001$
--------------------------
: \[table1\]The Parameter Selection for PMoG and PMoEP.
&
-----------------------------
$K_{final}=1,p_{select}=1$,
$\lambda_{select}=0.1$
-----------------------------
: \[table1\]The Parameter Selection for PMoG and PMoEP.
\
Sparse &
--------------------------
$K_{final}=2$,
$\lambda_{select}=0.005$
--------------------------
: \[table1\]The Parameter Selection for PMoG and PMoEP.
&
--------------------------------
$K_{final}=2,p_{select}=[2,2]$
, $\lambda_{select}=0.005$
--------------------------------
: \[table1\]The Parameter Selection for PMoG and PMoEP.
\
Mixuture 1 &
-------------------------
$K_{final}=2$,
$\lambda_{select}=0.01$
-------------------------
: \[table1\]The Parameter Selection for PMoG and PMoEP.
&
-----------------------------------
$K_{final}=2,p_{select}=[1.5,2]$,
$\lambda_{select}=0.005$
-----------------------------------
: \[table1\]The Parameter Selection for PMoG and PMoEP.
\
Mixuture 2 &
--------------------------
$K_{final}=1$,
$\lambda_{select}=0.001$
--------------------------
: \[table1\]The Parameter Selection for PMoG and PMoEP.
&
-----------------------------------
$K_{final}=2,p_{select}=[0.5,2]$,
$\lambda_{select}=0.005$
-----------------------------------
: \[table1\]The Parameter Selection for PMoG and PMoEP.
\
Synthetic simulations
---------------------
Several synthetic experiments with different noise settings were designed to compare the performance of the proposed methods and other competing methods. We first randomly generated $30$ low rank matrices with size $40\times20$ and rank 4. Each of these matrices was generated by the multiplication of two low-rank matrices $\mathbf{U}_{gt}\in \mathcal{R}^{40\times 4}$ and $\mathbf{V}_{gt}\in \mathcal{R}^{20\times 4}$, and $\mathbf{Y}_{gt}=\mathbf{U}_{gt}\mathbf{V}_{gt}^{T}$ is the ground truth matrix. Then, we randomly specified 20% elements of $\mathbf{Y}_{gt}$ as missing entries. Next, we added different types of noise to the non-missing entries as follows: (1) *Gaussian noise*: $\mathcal{N}(0,0.04)$. (2) *Exponential power noise*:[^4] $EP_{0.2}(0,0.2^{p}p), p=0.2$. (3) *Laplace noise*: $\mathcal{L}(0,0.2)$. (4) *Sparse noise*: 12.5% of the non-missing entries were corrupted with uniformly distributed noise on \[-20,20\]. (5) *Mixture noise 1*: 25% of the entries were corrupted with uniformly distributed noise on \[-5,5\], 25% were contaminated with Gaussian noise $\mathcal{N}(0,0.04)$ and the remaining 50% are corrupted with Gaussian noise $\mathcal{N}(0,0.01)$. (6) *Mixture noise 2*: 37.5% of the entries were corrupted with $EP(0,0.1^{p}p), p=0.5$, 50% were contaminated with Laplace noise $\mathcal{L}(0,0.3)$ and the remaining 50% were corrupted with Gaussian noise $\mathcal{N}(0,0.01)$. Then we get the noisy matrix $\mathbf{Y}_{no}$. Six measures were utilized for performance assessment: $$\begin{split}
&C1 \!=\! ||\mathbf{W}\!\odot\!(\mathbf{Y}_{no}\!-\!\tilde{\mathbf{U}}\tilde{\mathbf{V}}^{T})||_{1},\nonumber~C2 \!=\! ||\mathbf{W}\!\odot\! (\mathbf{Y}_{no}\!-\!\tilde{\mathbf{U}}\tilde{\mathbf{V}}^{T})||_{2}, \nonumber\\
&C3 = ||\mathbf{Y}_{gt}-\tilde{\mathbf{U}}\tilde{\mathbf{V}}^{T}||_{1}, \nonumber~~C4 = ||\mathbf{Y}_{gt}-\tilde{\mathbf{U}}\tilde{\mathbf{V}}^{T}||_{2}, \nonumber\\
&C5 = subspace(\mathbf{U}_{gt},\tilde{\mathbf{U}}), \nonumber~~C6 = subspace(\mathbf{V}_{gt},\tilde{\mathbf{V}}),
\end{split}$$ where $\tilde{\mathbf{U}},\tilde{\mathbf{V}}$ are the outputs of the corresponding competing method, and $subspace(\mathbf{U}_1$,$\mathbf{U}_2$) denotes the angle between subspaces spanned by the columns of $\mathbf{U}_1$ and $\mathbf{U}_2$. Note that $C1$ and $C2$ are the optimization objective function for $L_1$ and $L_2$ norm LRMF problems, while the latter four measures ($C3-C6$) are more faithful to evaluate whether a method recovers the correct subspaces.
\[0.9\]
[ccccccc]{} & PMoEP & PMoG & MoG & DW & CWM & RegL1ALM\
\
C1 & 40.97 & 41.00 & 41.00 & 41.00 & 39.23 & [**[36.60]{}**]{}\
C2 & [**[4.16]{}**]{} & [**[4.16]{}**]{} & [**[4.16]{}**]{} & [**[4.16]{}**]{} & 5.67 & 5.27\
C3 & [**[3.27]{}**]{} & [**[3.27]{}**]{} & [**[3.27]{}**]{} & [**[3.27]{}**]{} & 6.01 & 4.94\
C4 & [**[3.90e+1]{}**]{} & 3.91e+1 & 3.91e+1 & 3.91e+1 & 5.09e+1 & 5.09e+1\
C5 & [**[4.22e-2]{}**]{} & [**[4.22e-2]{}**]{} & [**[4.22e-2]{}**]{} & [**[4.22e-2]{}**]{} & 5.71e-2 & 5.33e-2\
C6 & [**[3.01e-2]{}**]{} & [**[3.01e-2]{}**]{} & [**[3.01e-2]{}**]{} & [**[3.01e-2]{}**]{} & 4.55e-2 & 3.79e-2\
\
C1 & 3.60e+2 & 3.42e+2 & 3.23e+2 & 4.30e+2 & [**[3.21e+2]{}**]{} & 3.65e+2\
C2 & 1.30e+3 & 1.04e+3 & 1.18e+3 & [**[6.27e+2]{}**]{} & 1.17e+3 & 8.51e+2\
C3 & [**[1.72e+2]{}**]{} & 4.49e+4 & 2.17e+3 & 5.06e+3 & 1.73e+2 & 7.77e+4\
C4 & [**[2.32e+2]{}**]{} & 4.67e+2 & 2.60e+2 & 6.29e+2 & 2.40e+2 & 9.68e+2\
C5 & [**[3.31e-1]{}**]{} & 5.67e-1 & 4.11e-1 & 9.19e-1 & 3.39e-1 & 1.16\
C6 & [**[2.19e-1]{}**]{} & 4.97e-1 & 2.31e-1 & 8.94e-1 & 2.61e-1 & 1.11\
\
C1 & 7.63e+1 & 7.29e+1 & 7.13e+1 & 7.76e+1 & 7.24e+1 & [**[6.80e+1]{}**]{}\
C2 & 1.72e+1 & 2.57e+1 & 2.44e+1 & **[1.68e+1]{} & 2.16e+1 & 2.10e+1\
C3 & [**[1.27e+1]{}**]{} & 2.02e+1 & 1.84e+1 & 1.31e+1 & 1.69e+1 & 1.42e+1\
C4 & [**[7.54e+1]{}**]{} & 9.37e+1 & 8.99e+1 & 7.69e+1 & 8.33e+1 & 7.85e+1\
C5 & [**[9.17e-2]{}**]{} & 1.15e-1 & 1.07e-1 & 9.22e-2 & 1.07e-1 & 9.80e-2\
C6 & [**[6.30e-2]{}**]{} & 8.25e-2 & 7.84e-2 & 6.49e-2 & 8.24e-2 & 6.56e-2\
\
C1 & [**[8.12e+2]{}**]{} & [**[8.12e+2]{}**]{} & [**[8.12e+2]{}**]{} & 1.17e+3 & 8.20e+2 & 8.73e+2\
C2 & 1.08e+4 & 1.08e+4 & 1.08e+4 & [**[5.12e+3]{}**]{} & 1.06e+4 & 5.95e+3\
C3 & [**[2.37e-12]{}**]{} & [**[2.37e-12]{}**]{} & 7.94e-12 & 3.09e+4 & 9.75e+1 & 1.59e+6\
C4 & [**[2.54e-5]{}**]{} & 2.55e-5 & 3.48e-5 & 2.12e+3 & 6.03e+1 & 4.89e+3\
C5 & [**[3.87e-8]{}**]{} & 3.87e-8 & 6.63e-8 & 1.48 & 2.83e-1 & 1.47\
C6 & [**[2.28e-8]{}**]{} & 2.29e-8 & 4.44e-8 & 1.39 & 6.25e-2 & 1.54\
\
C1 & 4.49e+2 & 4.55e+2 & 5.25e+2 & 5.25e+2 & [**[4.33e+2]{}**]{} & 4.35e+2\
C2 & 1.36e+3 & 1.25e+3 & [**[8.49e+2]{}**]{} & 8.51e+2 & 1.12e+3 & 1.16e+3\
C3 & [**[1.53e+2]{}**]{} & 6.52e+4 & 8.98e+2 & 8.93e+2 & 3.01e+2 & 1.56e+4\
C4 & [**[1.66e+2]{}**]{} & 4.38e+2 & 6.02e+2 & 6.00e+2 & 2.87e+2 & 5.15e+2\
C5 & [**[3.28e-1]{}**]{} & 5.79e-1 & 6.47e-1 & 6.60e-1 & 4.30e-1 & 7.88e-1\
C6 & [**[1.18e-1]{}**]{} & 3.78e-1 & 5.01e-1 & 5.01e-1 & 2.93e-1 & 6.84e-1\
\
C1 & 9.01e+1 & 8.93e+1 & 8.76e+1 & 9.60e+1 & 8.83e+1 & [**[8.32e+1]{}**]{}\
C2 & 3.37e+1 & 4.04e+1 & 3.99e+1 & [**[2.72e+1]{}**]{} & 3.53e+1 & 3.42e+1\
C3 & [**[1.72e+01]{}**]{} & 2.62e+1 & 2.49e+1 & 2.13e+1 & 2.40e+1 & 1.87e+1\
C4 & [**[8.57e+01]{}**]{} & 1.04e+2 & 1.01e+2 & 9.71e+1 & 9.77e+1 & 8.87e+1\
C5 & [**[1.02e-01]{}**]{} & 1.23e-1 & 1.24e-1 & 1.09e-1 & 1.21e-1 & 1.07e-1\
C6 & [**[6.39e-02]{}**]{} & 8.41e-2 & 8.14e-2 & 7.02e-2 & 8.96e-2 & 6.62e-2\
**
![Visual comparison of the ground truth (denote by True) noise probability density functions and those estimated (denote by Est) by the PMoEP method in the synthetic experiments. The embedded sub-figures depict the zoom-in of the indicated portions.[]{data-label="pdfdraw"}](PDF_draw1){width="1\linewidth"}
![From left to right: original face images, reconstructed faces by PMoEP, PMoG, MoG, RegL1ALM, DW, CWM and SVD.[]{data-label="FaceRecover"}](Face_Recover1){width="1\linewidth"}
We set the rank of all the competing methods to $4$ and adopt the random initialization strategy for all the methods. For each method, we first run with 20 random initializations and then select the best result with respect to the corresponding objective value of the method. The performance of each method was evaluated as the average results over the 30 random matrices in terms of the six measures, and the results are summarized in Table \[simuTab1\]. We also report the final selections of the mixture number $K_{final}$ and the corresponding parameter $\lambda_{select}$ for PMoG and PMoEP in Table \[table1\].
From Table \[simuTab1\], we can observe that $L_2$-norm methods DW, MoG, PMoG and our proposed PMoEP methods achieve the best performance than others in Gaussian noise case. In Laplace noise case, our PMoEP method performs best and $L_1$ method RegL1ALM achieves similar results. When the noise is Exponential Power, PMoEP evidently outperforms other competing methods in term of criteria C3$-$C6. In sparse noise case, PMoEP and PMoG perfom the best and MoG achieves comparable good results with PMoEP. Moreover, when the noise gets more complex, PMoEP achieves the best performance, which attributes to the high flexibility of PMoEP to model unknown complex noise. These results then substantiate that our proposed PMoEP method can estimate a better subspace from the noisy data than other competing methods.
The promising performance of PMoEP method in these cases can be easily explained by Fig. \[pdfdraw\], which compares the ground truth noise distributions and the estimated ones by the PMoEP method. It can be easily observed that the estimated noise distributions well match the true ones, which naturally conducts its good reconstruction capability to the true low-rank matrix.
Face modeling
-------------
This experiment aims to test the effectiveness of PMoG and PMoEP methods in face modeling application. We choose the first and the second subset of the Extended Yale B database[^5], and each subset consists of 64 faces of one person with size $192\times168$ and then generate two data matrices, each of which is with size $32256\times 64$. Typical images are shown in the first column of Fig. \[FaceRecover\]. We set the rank as $4$ [@basri2003lambertian] and adopt two initialization strategies, namely random and SVD for all competing methods. Then we report the best result among the results in terms of the object value of the corresponding model utilized by each method. Some reconstructed faces of different methods are visually compared in Fig. \[FaceRecover\].
From Fig. \[FaceRecover\], it is easy to observe that, the proposed PMoEP and PMoG methods, as well as the other competing ones, can remove the cast shadows and saturations in faces. However, our PMoEP and PMoG methods perform better than other ones on faces containing a large dark region. Such face images contain both significant cast shadow and saturation noises, which correspond to the highly dark and bright areas in face, and camera noise [@nakamura2005image], which is much amplified in the dark areas. Compared with other competing methods, PMoEP method is capable of better extracting such complex noise configurations, and thus leads to its better face reconstruction performance.
![ Restoration results of band 103 in Urban data set: (a) original bands. (b)-(g) reconstructed bands by PMoEP, PMoG, MoG, RegL1ALM, CWM and SVD.[]{data-label="103bandRec"}](103_test_cp){width="1.0\linewidth"}
![ Restoration results of band 206 in Urban data set: (a) original bands. (b)-(g) reconstructed bands by PMoEP, PMoG, MoG, RegL1ALM, CWM and SVD.[]{data-label="206bandRec"}](206_test_cp){width="1.0\linewidth"}
![ Restoration results of band 207 in Urban data set: (a) original bands. (b)-(g) reconstructed bands by PMoEP, PMoG, MoG, RegL1ALM, CWM and SVD.[]{data-label="207bandRec"}](207_test_cp){width="1.0\linewidth"}
![Restoration results of band 107 in Urban data set: (a) original bands. (b)-(g) reconstructed bands by PMoEP, PMoG, MoG, RegL1ALM, CWM and SVD.[]{data-label="107bandRec"}](107_test_cp){width="1.0\linewidth"}
![ Restoration results of band 152 in Terrain data set: (a) original bands. (b)-(g) reconstructed bands by PMoEP, PMoG, MoG, RegL1ALM, CWM and SVD.[]{data-label="152Rec"}](Rec_band_152_cp){width="1.0\linewidth"}
Hyperspectral Image Restoration
-------------------------------
In this section, we evaluate the performance of our proposed PMoEP method on hyperspectral image restoration problem. Two real hyperspectral image (HSI) data sets[^6] were used.
The first dataset is Urban HSI data. This dataset contains $210$ bands, each of which is $307\times307$, and some bands are seriously polluted by atmosphere and water and corrupted by noises with complex structures, as shown in Fig. \[intro\_fig\]. We reshape each band as a vector, and stack all the vectors into a matrix, resulting in the final data matrix with size $94249\times210$. The second one is the Terrain dataset. The original images are of size $500\times307\times210$. We use all the bands in our experiments and thus generate a $153500\times210$ data matrix. Therefore, we get two data matrices used to test our methods. All the competing methods were implemented, except DW method which encounters the ‘out of memory’ problem.
![ Restoration results of band 206 in Terrain data set: (a) original bands. (b)-(g) reconstructed bands by PMoEP, PMoG, MoG, RegL1ALM, CWM and SVD.[]{data-label="206Rec"}](Rec_band_206_cp){width="1.0\linewidth"}
![ Restoration results of band 139 in Terrain data set: (a) original bands. (b)-(g) reconstructed bands by PMoEP, PMoG, MoG, RegL1ALM, CWM and SVD.[]{data-label="139Rec"}](Rec_band_139_cp){width="1.0\linewidth"}
{width="0.8\linewidth"}
{width="0.8\linewidth"}
{width="0.8\linewidth"}
The reconstructed hyperspectral images of bands $103$, $206$, $207$ and $107$ in Urban dataset and bands $152$, $206$ and $139$ in Terrain dataset are shown in Fig. \[103bandRec\]$-$\[107bandRec\] and Fig. \[152Rec\]$-$\[139Rec\], respectively. For easy observation, an area of interest is amplified in the restored images obtained by all the competing methods. It can be easily seen from the figures that for some bands containing evident stripes and deadlines, the image restored by the proposed PMoEP method is clean and smooth, while the results obtained by the other competing ones contain evident stripe area. In addition, as is demonstrated in Fig. \[107bandRec\] and Fig. \[139Rec\], the PMoEP method can effectively recover the seriously polluted bands, while the other methods failed on them. These results show that our proposed PMoEP method can not only remove complicated noises embedded in HSI, but also can perform robust in the presence of extreme outlier cases like in Fig. \[107bandRec\] and Fig. \[139Rec\].
Then we give more quantitative comparison by showing the vertical mean profiles and horizontal mean profiles of band 207 in Urban dataset and band 152 in Terrain dataset before and after reconstruction in Fig. \[M207\] and Fig. \[M152\]. The horizontal axis of Fig. \[M207\] represents the column (left) and row (right) number, and the vertical axis represents the mean DN value of each column (left) and row (right). It is easy to observe that the curves in Fig. \[M207\](a) and \[M152\](a) (right) have drastic fluctuations for the original image. This is deviated from the prior knowledge that the adjacent bands should possess similar shapes since they are captured under relatively similar sensor settings. After the reconstruction, the fluctuations in vertical direction have been reduced by most of the methods. While in the horizontal direction (see Fig. \[M207\] (right) and Fig. \[M152\] (right)), the PMoEP method provides evidently smoother curves, which indicates that the stripes in the horizontal direction have been removed more effectively by our method. The results are consistent with the recovered HSIs in Fig. \[207bandRec\] and Fig. \[152Rec\].
The better performance of PMoEP over other methods is due to its more powerful ability in noise modeling. Specifically, as depicted in Fig. \[UrbanNoise\], PMoEP can more properly extract noise information from the corrupted images with physical meanings, such as sparse strips, sparse deadlines, and dense Gaussian noise, while other competing methods fail to do so.
\[ForDet\] {width="0.85\linewidth"}
Background Subtraction
----------------------
In this section, we evaluate the performance of our proposed methods on background subtraction problem. The background subtraction from a video sequence captured by a static camera can be modeled as a low-rank matrix analysis problem [@wright2009robust]. All the nine standard video sequences[^7] provided by Li et.al [@li2004statistical] were adopted in our evaluation, including simple and complex background. Ground truth foreground regions of 20 frames were provided for each sequence.
We compared our PMoEP and PMoEP-MRF methods with the state-of-the-art LRMF methods: SVD, RegL1ALM, CWM and MoG methods. To conduct the experiments, we first ran each method on each video sequence to estimate the background. Then we obtained the recovered foreground by calculating the absolute values of the difference between the original frame and the estimated background. For MoG, PMoEP and PMoEP-MRF methods, we obtained the foreground by selecting the noise component with largest variance.
For quantitative evaluation, we first introduce some evaluation indices. We measure the recovery accuracy of the support in the foreground by comparing the true support $S$ with the detected support $\tilde{S}$. We regard it as a classification problem and thus can evaluate the results using precision and recall, which are defined as: $$precision = \frac{TP}{TP+FP},~~~ recall = \frac{TP}{TP+FN},$$ where $TP$, $FP$, $TN$ and $FN$ represent the numbers of true positive, false positive, true negative and false negative, respectively. For simplicity, we adopt $F\textrm{-}measure$ that combines precision and recall together: $$F\textrm{-}measure = 2\times \frac{precision\times recall}{precision+recall}.$$ The higher Fmeasure value means the better recovery accuracy of the support. Additionally, the recovered support $\tilde{S}$ is obtained by thresholding the recovered foreground $E$ with a threshold value that gives the maximal Fmeasure. For all competing methods, we adopt two initialization strategies, namely, random and SVD. Then we report the best result among the two initializations. The results are summarized in Table \[simuTab2\].
\[0.75\]
[cccccccc]{} Video & SVD & RegL1ALM & CWM & MoG & PMoEP & PMoEP-MRF\
\
Campus & 0.4716 & [**[0.5308]{}**]{} & ***0.5301*** & 0.4633 & 0.5065 & 0.5115\
Lobby & 0.7623 & 0.7679 & ***0.7681*** & [**[0.7724]{}**]{} & 0.7650 & 0.7444\
ShoppingMall & 0.6990 & ***0.7138*** & [**[0.7173]{}**]{} & 0.6387 & 0.7037 & 0.7015\
Bootstrap & 0.6234 & [**[0.6749]{}**]{} & 0.6533 & 0.4234 & 0.6404 & ***0.6635***\
Hall & 0.4104 & 0.4659 & 0.4624 & 0.4523 & ***0.5372*** & [**[0.5438]{}**]{}\
Curtain & 0.5273 & 0.5342 & 0.5316 & 0.7869 & [**[0.7895]{}**]{} & ***0.7888***\
Fountain & 0.4989 & 0.5298 & 0.5262 & 0.5782 & ***0.6843*** & [**[0.7295]{}**]{}\
WaterSurface & 0.3416 & 0.2840 & 0.2920 & 0.5979 & ***0.8515*** & [**[0.8651]{}**]{}\
Escalator & 0.2675 & 0.2998 & 0.2972 & 0.2675 & ***0.3255*** & [**[0.3408]{}**]{}\
Average & 0.5113 & 0.5334 & 0.5309 & 0.5534 & ***0.6448*** & [**[0.6543]{}**]{}\
From Table \[simuTab2\], it can be easily seen that our proposed PMoEP and PMoEP-MRF methods outperform other methods in the sequences of Hall, Curtain, Fountain, WaterSurface and Escalator, of which the background is with complex shapes. For the sequences with simple background, including Bootstrap, ShoppingMall, Campus and Lobby, the performances of all the methods are almost the same. On average, the PMoEP method achieves the second best performance. Compared with the PMoEP method, the PMoEP-MRF method slightly improves the average performance due to the modeling of spatial and temporal smoothness prior knowledge under foreground using Markov random field.
The better performance of PMoEP and PMoEP-MRF methods can be visually shown in Fig. 15. It can be easily seen from the figure that the proposed PMoEP and PMoEP-MRF can perform comparably well as other methods in simple foreground cases, while evidently better in much complicated scenarios, e.g., videos with dynamic background.
Conclusions
===========
In this paper, we model the noise of the LRMF problem as a Mixture of Exponential Power (MoEP) distributions and proposes a penalized MoEP (PMoEP) model by combining the penalized likelihood method with the MoEP distributions. Moreover, by facilitating the local continuity of noise components along both space and time of a video, we embed Markov random field into PMoEP and then propose the PMoEP-MRF model. Compared with the current LRMF methods, our PMoEP method performs better in a wide variety of synthetic and real complex noise scenarios including face modeling, hyperspectral image restoration, and background subtraction applications. Additionally, our methods are capable of automatically learning the number of components from data, and thus can be used to deal with more complex applications. In the future, we’ll attempt to extend the noise modeling methodology under PMoEP to more computer vision and machine learning tasks, e.g., the high-order low rank tensor factorization problems.
Acknowledgements {#acknowledgements .unnumbered}
================
This research was supported by 973 Program of China with No.3202013CB329404, the NSFC projects with No.11131006, 91330204 and 61373114.
Proof of Theorem 1
==================
\(i) First, we calculate that $$\begin{aligned}
\begin{split}
l_{P}^{\textsc{C}}(\mathbf{\Theta})-l_{P}^{C}(\mathbf{\Theta}^{(t)})
&=l(\mathbf{\Theta})-l(\mathbf{\Theta}^{(t)})+P(\pi^{(t)};\lambda)-P(\pi;\lambda)\nonumber \\
&=\log{\sum_{\mathbf{Z}}\mathbb{P}(\mathbf{Z}|\mathbf{E};\mathbf{\Theta}^{(t)})\frac{\mathbb{P}(\mathbf{E}|\mathbf{Z};\mathbf{\Theta})\mathbb{P}(\mathbf{Z};\mathbf{\Theta})}{\mathbb{P}(\mathbf{Z}|\mathbf{E};\mathbf{\Theta}^{(t)})}}\\
&~~~-\log{\mathbb{P}(\mathbf{E};\mathbf{\Theta}^{(t)})}+P(\pi^{(t)};\lambda)-P(\pi;\lambda)\nonumber \\
&\geq \sum_{\mathbf{Z}}\mathbb{P}(\mathbf{Z}|\mathbf{E};\mathbf{\Theta}^{(t)})\log{\frac{\mathbb{P}(\mathbf{E}|\mathbf{Z};\mathbf{\Theta})\mathbb{P}(\mathbf{Z};\mathbf{\Theta})}{\mathbb{P}(\mathbf{Z}|\mathbf{E};\mathbf{\Theta}^{(t)})}}\\
&~~~-\log{\mathbb{P}(\mathbf{E};\mathbf{\Theta}^{(t)})}+P(\pi^{(t)};\lambda)-P(\pi;\lambda)\nonumber \\
&=\sum_{\mathbf{Z}}\mathbb{P}(\mathbf{Z}|\mathbf{E};\mathbf{\Theta}^{(t)})\log{\frac{\mathbb{P}(\mathbf{E}|\mathbf{Z};\mathbf{\Theta})\mathbb{P}(\mathbf{Z};\mathbf{\Theta})}{\mathbb{P}(\mathbf{Z}|\mathbf{E};\mathbf{\Theta}^{(t)})\mathbb{P}(\mathbf{E};\mathbf{\Theta}^{(t)})}}\\
&~~~+P(\pi^{(t)};\lambda)-P(\pi;\lambda).\nonumber
\end{split}
\end{aligned}$$ Let $\Omega(\mathbf{\Theta}|\mathbf{\Theta}^{(t)})=\sum_{\mathbf{Z}}\mathbb{P}(\mathbf{Z}|\mathbf{E};\mathbf{\Theta}^{(t)})\log{\frac{\mathbb{P}(\mathbf{E}|\mathbf{Z};\mathbf{\Theta})\mathbb{P}(\mathbf{Z};\mathbf{\Theta})}
{\mathbb{P}(\mathbf{Z}|\mathbf{E};\mathbf{\Theta}^{(t)})\mathbb{P}(\mathbf{E};\mathbf{\Theta}^{(t)})}}$, then $$\label{lemma1}
l_{P}^{C}(\mathbf{\Theta})\geq l_{P}^{C}(\mathbf{\Theta}^{(t)}) + \Omega(\mathbf{\Theta}|\mathbf{\Theta}^{(t)})+P(\pi^{(t)};\lambda)-P(\pi;\lambda)\nonumber.$$\
(ii) In the M step of Algorithm 1, it is obvious that $$\begin{aligned}
\mathbf{\Theta}^{(t+1)}\!&=&\! \underset{\mathbf{\Theta}}{\!\arg\max}\left\{\sum_{\mathbf{Z}}\mathbb{P}(\mathbf{Z}|\mathbf{E};\mathbf{\Theta}^{(t)})\log{\mathbb{P}(\mathbf{E},\mathbf{Z};\mathbf{\Theta})}\!-\!P(\pi;\lambda)\right\}\nonumber\\
\!&=&\!\underset{\mathbf{\Theta}}{\!\arg\max}\left\{\sum_{\mathbf{Z}}\mathbb{P}(\mathbf{Z}|\mathbf{E};\mathbf{\Theta}^{(t)})\frac{\log{\mathbb{P}(\mathbf{E},\mathbf{Z};\mathbf{\Theta})}}
{\log{\mathbb{P}(\mathbf{E},\mathbf{Z};\mathbf{\Theta}^{(t)})}}\!-\!P(\pi;\lambda)\right\}\nonumber\\
\!&=&\!\underset{\mathbf{\Theta}}{\!\arg\max}\left\{\Omega(\mathbf{\Theta}|\mathbf{\Theta}^{(t)})+P(\pi^{(t)};\lambda)-P(\pi;\lambda)\right\}.\nonumber
\end{aligned}$$ Thus, we have $$\begin{aligned}
\begin{split}
&\Omega(\mathbf{\Theta}^{(t+1)}|\mathbf{\Theta}^{(t)})+P(\pi^{(t)};\lambda)-P(\pi^{(t+1)};\lambda)\\
&\geq
\Omega(\mathbf{\Theta}^{(t)}|\mathbf{\Theta}^{(t)})+P(\pi^{(t)};\lambda)-P(\pi^{(t)};\lambda)=0
\end{split}
\end{aligned}$$ Then, we can easily derive that $$\label{inequ}
l_{P}^{C}(\mathbf{\Theta}^{(t+1)})\geq l_{P}^{C}(\mathbf{\Theta}^{(t)}).\nonumber$$ Based on (\[inequ\]), the sequence $\{l_{P}^{G}(\mathbf{\Theta}^{(t)})\}_{t=1}^{\infty}$ is nondecreasing and bounded above. Therefore, there exits a constant $l^{\star}$ such that $$\lim_{t\rightarrow \infty}l_{P}^{C}(\mathbf{\Theta}^{(t)}) = l^{\star}.\nonumber$$
Exponential Power Distribution
==============================
Three different forms of Exponential Power Distribution
-------------------------------------------------------
The Exponential Power Distribution ($\mu=0$) has the following three equivalent forms: $$\label{form1}
f_{p}(x;0,\sigma)=\frac{1}{2\sigma p^{\frac{1}{p}}\Gamma(1+\frac{1}{p})}\exp\left\{-\frac{|x|^{p}}{p\sigma^{p}}\right\}.\nonumber$$ Let $\tau = (p\sigma^{p})^{\frac{1}{p}}$, then $$\label{form2}
f_{p}(x;0,\tau)=\frac{1}{2\tau\Gamma(1+\frac{1}{p})}\exp\left\{-|\frac{x}{\tau}|^{p}\right\}.\nonumber$$ Let $\eta = \frac{1}{\tau^{p}}$, then $$\label{form3}
f_{p}(x;0,\eta)=\frac{\eta^{\frac{1}{p}}}{2\Gamma(1+\frac{1}{p})}\exp\left\{-\eta|x|^{p}\right\}.\nonumber$$ Noting that $\Gamma(1+\frac{1}{p})=\frac{1}{p}\Gamma(\frac{1}{p})$, then we can represent the above three forms in equivalent forms.
Draw Samples from Exponential Power Distribution
------------------------------------------------
The second form of exponential power distribution is $$\label{form2}
f_{p}(x;0,\tau)=\frac{1}{2\tau\Gamma(1+\frac{1}{p})}\exp\left\{-|\frac{x}{\tau}|^{p}\right\}.\nonumber$$ Sampling from the exponential power distribution contains two cases: $p\geq 1$ and $0<p<1$.
### case 1: $p\geq 1$
We adopt the method proposed in [@chiodi1995generation; @marsaglia1964convenient; @mineo2005software].
### case 2: $0<p<1$
When $0<p<1$, the method proposed in [@polson2014bayesian] is used. We sample the distribution in two steps: $$\label{samplew}
(w|p) \sim \frac{1+p}{2}Ga(2+\frac{1}{p},1) + \frac{1-p}{2}Ga(1+\frac{1}{p},1),$$ $$\label{samplebeta}
(\beta|\tau,w,p) \sim \frac{1}{\tau w^{\frac{1}{p}}}\left\{1-|\frac{\beta}{\tau w^{\frac{1}{p}}}|\right\}_{+},$$ where $w$ is a intermediate variable. (\[samplew\]) can be sampled directly but (\[samplebeta\]) is difficult. Therefore, we adopt the slice sampling strategy in [@bishop2006pattern].
[^1]: Xiangyong Cao, Qian Zhao, Deyu Meng, Yang Chen, and Zongben Xu are with the School of Mathematics and Statistics and Ministry of Education Key Lab of Intelligent Networks and Network Security, Xi’an Jiaotong University, Xi’an 710049, China ([email protected], [email protected], [email protected], [email protected], [email protected])
[^2]: $^\ast$Deyu Meng is the corresponding author
[^3]: The $p$-norm of a matrix is defined as $||\mathbf{X}||_{p}=(\sum_{i,j}|x_{ij}|^{p})^{\frac{1}{p}}$.
[^4]: The method of drawing samples from a general exponential power distribution is introduced in Appendix B.
[^5]: http://vision.ucsd.edu/ leekc/ExtYaleDatabase/ExtYaleB.html
[^6]: http://www.tec.army.mil/hypercube.
[^7]: http://perception.i2r.a-star.edu.sg/bk\_model/bk\_index.html
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Functional data analysis, FDA, is now a well established discipline of statistics, with its core concepts and perspectives in place. Despite this, there are still fundamental statistical questions which have received relatively little attention. One of these is the systematic construction of confidence regions for functional parameters. This work is concerned with developing, understanding, and visualizing such regions. We provide a general strategy for constructing confidence regions in a real separable Hilbert space using hyper-ellipsoids and hyper-rectangles. We then propose specific implementations which work especially well in practice. They provide powerful hypothesis tests and useful visualization tools without using any simulation. We also demonstrate the negative result that nearly all regions, including our own, have *zero-coverage* when working with empirical covariances. To overcome this challenge we propose a new paradigm for evaluating confidence regions by showing that the distance between an estimated region and the desired region (with proper coverage) tends to zero faster than the regions shrink to a point. We call this phenomena *ghosting* and refer to the empirical regions as *ghost* regions. We illustrate the proposed methods in a simulation study and an application to fractional anisotropy tract profile data.'
address:
- 'The Pennsylvania State University, University Park, PA, USA.'
- 'The Pennsylvania State University, University Park, PA, USA.'
author:
- Hyunphil Choi
- Matthew Reimherr
bibliography:
- 'jrss-b.bib'
- 'biom1960.bib'
- 'nb.bib'
- 'fda.bib'
title: |
A Geometric Approach to Confidence Regions\
and Bands for Functional Parameters
---
Introduction
============
Functional data analysis, FDA, is a branch of statistics whose foundational work goes back at least two decades. Its development and application has seen a precipitous increase in recent years due to the emergence of new data gathering technologies which incorporate high frequency sampling. Fundamentally, FDA is concerned with data which can be viewed as samples of curves, images, shapes, or surfaces. While FDA is now a well established discipline with its core tools and concepts in place, there are still fundamental questions that have received relatively little attention. This work is concerned with developing, understanding, and visualizing confidence regions for functional data, a fundamental statistical concept which has received little attention in the FDA literature. Our approach is geometric in that we start with general hyper-ellipsoids and hyper-rectangles and show how they can be tailored to become proper confidence regions. A distinguishing feature of functional confidence regions is that, when the covariance of the estimator is estimated, nearly all confidence regions turn out to have *zero-coverage* for the parameter; this is primarily due to the infinite dimensional nature of the parameter. However, we demonstrate how most of these regions are very close to the proper regions with respect to Hausdorff distance. Such an issue does not occur in multivariate statistics and is a distinct feature of FDA. We refer to this phenomenon as *ghosting*, namely, that while one uses a confidence regions with zero-coverage, they can be shown to be arbitrarily close to a proper confidence region with the desired coverage. Of course, for these *ghost* regions to be useful, these distances must decrease faster than the rate at which the regions shrink down to a point.
Forming a confidence region for a functional parameter can equivalently be thought of as forming a confidence region for an infinite dimensional parameter. To see why this is a challenge, consider a classic multivariate confidence region. Suppose that $\theta \in {{\mathbb R}}^p$ and we have an estimator, $\hat \theta$, which is multivariate normal, $\hat \theta \sim {{\mathcal N}}_p(\theta, \Sigma)$. The classic approach to forming a $1-\alpha$ confidence region, $G_\alpha$, is to take the following ellipse $$\begin{aligned}
G_\alpha = \{ x \in {{\mathbb R}}^p : (\hat \theta - x)^\top \Sigma^{-1} (\hat \theta - x) \leq \xi_\alpha \}.
\label{e:mult_conf}\end{aligned}$$ The constant $\xi_\alpha$ is chosen so that the region achieves the proper coverage; when $\Sigma$ is known it is taken as the quantile of a $\chi^2$, while when $\Sigma$ is estimated it is taken from an $F$. For $p$ very large, at least two things happen: (1) the inversion of $\Sigma$ becomes very unstable due to small eigenvalues and (2) the constant $\xi_\alpha$ becomes very large. In fact, a naive functional analog would require $\xi_\alpha = \infty$ and the sample covariance operator would not even be invertible.
To address this problem in the functional case, there have been at least two main approaches. The first approach is to develop confidence bands via simulation techniques [@degras:2011; @cao:2012; @zheng:2014; @cao:2014]. These methods work quite well, but they shift the focus from Hilbert spaces, usually $L^2[0,1]$, to Banach spaces, such as $C[0,1]$. Given that Hilbert spaces are the foundation of the large majority of theory and methods for FDA, it is important to have a procedure which is based on Hilbert spaces. A more minor issue is that such bands, after taking into account point-wise variability, are usually built upon using a constant threshold across all time points. For most practical purposes, this works well, but for objects with highly complex intra-curve dependencies, it could be useful to adjust the bands. For example, in areas with high positive within curve correlation, the bands can be made narrower, and in areas with very weak correlation they should be made wider. Finally, simulation based approaches are computationally intensive, especially if one wants to invert the procedure to find very small p-values, which is very common in genetic studies, or increase evaluation points on the domain, i.e. work on a finer grid.
The second approach is based on functional principal component analysis, FPCA [@ramsay:silverman:2005; @yao:muller:wang:2005JASA; @goldsmith:2013]. There one uses FPCA for dimension reduction and builds multivariate confidence ellipses, which can be turned into bands using Scheffé’s method. As we will show, this procedure produces ellipses which have *zero-coverage*. The dimension reduction inherently clips part of the parameter, meaning that the true parameter will never lie in the region. As a simple illustration, imagine trying to capture a two dimensional parameter with an ellipse versus a line segment. The probability of capturing the parameter with a random ellipse can usually be well controlled, but any random line segment will fail to capture the parameter with probability one. Additionally, the bands formed from these ellipses, depend heavily on the number of FPCs used.
The paper and its contributions are organized as follows. In Section \[section:ConfRegion\] we present a new geometric approach to constructing confidence regions in real separable Hilbert spaces using hyper-ellipsoids (Section \[s:ellipse\]) and hyper-rectangles (Section \[s:rectangle\]). We show how to transform confidence hyper-ellipses into confidence bands and propose a specific ellipse which gives the smallest average squared width when turned into a band (Section \[s:bands\]). Simulations in Section \[section:Simulation\] suggest that this ellipse is an excellent starting point for practitioners. We also propose a visualization technique using rectangular regions (Section \[s:rectangle-visual\]). In Section \[section:EstConfRegion\], we detail issues involved in using estimated/empirical versions based on estimated covariances. As a negative result, we will show that nearly all empirical regions have zero–coverage. However, we justify using these regions in practice by introducing the concept of *‘ghosting’*: using regions with deficient coverage as estimates for regions with proper coverage. Lastly, in Sections \[section:Simulation\] and \[s:dti\], we provide a simulation study and an application to `DTI` data in the `R` package `refund` [@goldsmith:2012a; @goldsmith:2012b].
Constructing Functional Confidence Regions {#section:ConfRegion}
==========================================
Throughout this paper we consider a general functional parameter $\theta \in {{\mathcal H}},$ where ${{\mathcal H}}$ is a real separable Hilbert space with inner product $\langle \cdot, \cdot \rangle$. We assume that we have an estimator $\hat{\theta} \in {{\mathcal H}}$ which is asymptotically Gaussian in ${{\mathcal H}}$ in the sense that $\sqrt N (\hat \theta - \theta) \overset{d}{\to} {{\mathcal N}}(0, C_\theta)$, where $N$ is the sample size and $C_\theta$ is a covariance operator that can be estimated. Although multivariate confidence regions are ellipsoids, this geometric shape is a by–product of using quadratic forms. Here, however, we take the opposite approach. We first define the desired geometric shape and then demonstrate how to adjust the region to achieve the desired confidence level. Recall that $1-\alpha$ (asymptotic) confidence region $G_{\hat{\theta}}$ for $\theta \in {{\mathcal H}}$ is a random subset of ${{\mathcal H}}$ which satisfies ${{\mathbb P}}(\theta \in G_{\hat{\theta}}) \to 1-\alpha$. We make the following assumption to simplify arguments.
\[a:normal\] Assume that $\sqrt{N}(\hat \theta - \theta) \overset{d}{\to} {{\mathcal N}}(0, C_\theta)$, that is, is asymptotically Gaussian in ${{\mathcal H}}$ with mean zero and covariance operator $C_\theta$.
Assumption \[a:normal\] is fairly weak and satisfied by many methods for dense functional data including mean estimation [@degras:2011], covariance estimation [@zhang:wang:2016], eigenfunction/value estimation [@kokoszka:reimherr:2013b], and function-on-scalar regression [@ReNi:2014]. To achieve such a property one needs that (i) the bias of the estimate is asymptotically negligible and that (ii) the estimate is *tight* so that convergence in distribution occurs in the strong topology. While these two conditions are often satisfied, there are still many FDA settings where they are not. The bias can usually be shown to be asymptotically negligible when the number of points sampled per curve is greater than $N^{1/4}$ [@li:hsing:2010; @cai:2011; @zhang:wang:2016]. Thus our approach will not work for sparse FDA settings. The tightness assumption is often violated when estimates stem from ill-posed inverse problems. For example, in scalar-on-function regression, typical slope estimates are not tight and not asymptotically normal in the strong topology [@cardot:2007].
The backbone of our construction, and many other FDA methods, is the Karhunen-Loève, KL, expansion which gives $$\begin{aligned}
\label{e:KL}
\sqrt{N} (\hat{\theta} - \theta) = \sum_{j=1}^\infty \sqrt{\lambda_j}Z_jv_j,\end{aligned}$$ where $\{\lambda_j\}$ and $\{v_j\}$ are eigenvalues and eigenfunctions, respectively, of $C_\theta$, and $\{Z_j\}$ are uncorrelated with mean zero and unit variance. We note that this expansion holds for any random element in ${{\mathcal H}}$ with a finite second moment and that the infinite sum converges in ${{\mathcal H}}$. In the next subsections we discuss two types of regions which exploit this expansion. The first is a hyper-ellipse which is, as in the multivariate case, much easier to construct. The second is a hyper-rectangle which is not mentioned as often in the multivariate literature due to the complexity of its form. However, in Section \[section:Simulation\] we will show that in some settings the hyper-rectangle can outperform the ellipse and is usually much more interpretable.
Hyper-Ellipsoid Form {#s:ellipse}
--------------------
A hyper-ellipse in any Hilbert space can be defined as follows. One needs a center, $m \in {{\mathcal H}}$, axes, $e_1, e_2, \cdots $, which are an orthonormal basis for ${{\mathcal H}}$, and a radius for each axis, $r_1, r_2, \cdots$. The ellipse is then given by $$\left\{ h \in \mathcal{H} : \sum_{j=1}^{\infty} \frac{\langle h - m, e_j \rangle^2}{r_j^2} \leq 1 \right\}.$$ We note that this definition makes sense even when $r_j = 0$ or $\infty$. In the former one is saying that the radius in that direction is zero or ‘closed’, while in the latter one is saying that it is infinite or ‘opened’. Since our aim is to construct a confidence region for $\theta$, we will replace the arbitrary axes above with the eigenfunctions $\{v_j\}$ and the center with $\hat \theta$, to get $$E_{\hat{\theta}} := \left\{ h \in \mathcal{H} : \sum_{j=1}^{\infty} \frac{\langle h - \hat{\theta}, v_j \rangle^2}{r_j^2} \leq 1 \right\}.$$ This hyper-ellipsoid will be a $1-\alpha$ confidence region for $\theta$ if we find $\{r_j\}$ which give $${{\mathbb P}}(\theta \in E_{\hat{\theta}})
= {{\mathbb P}}\left( \sum_{j=1}^{\infty} \frac{\langle \theta - \hat{\theta}, v_j \rangle^2}{r_j^2} \leq 1 \right)
\to 1-\alpha.$$ Note that there are actually infinitely many options for $\{r_j\}$ but not all of them lead to ‘nice’ regions. We decompose $r_j^2 = N^{-1} \xi c_j^2$, where $\{c_j\}$ are predefined weights (based on $\{\lambda_j\}$) for each direction, and $\xi$ is adjusted to achieve proper coverage. We then have $$\begin{aligned}
\label{e:EllipsoidBound}
E_{\hat{\theta}} = \left\{ h \in \mathcal{H} : \sum_{j=1}^{\infty} \frac{\langle \sqrt{N}(\hat{\theta} - h), v_j \rangle^2}{c_j^2} \leq \xi \right\}.\end{aligned}$$ From it follows that the coverage is given by $$\begin{aligned}
\label{e:EllipsoidProbRule}
{{\mathbb P}}\left(\theta \in E_{\hat{\theta}} \right)
= {{\mathbb P}}\left( W_\theta \leq \xi \right)
\qquad
\text{where}
\qquad
W_{\theta} = \sum_{j=1}^{\infty} \frac{\lambda_j}{c_j^2}Z_j^2.\end{aligned}$$ Therefore, to achieve the desired asymptotic confidence level for a given $\{c_j\}$, one can take $\xi$ to be the $1-\alpha$ quantile of a weighted sum of chi-squared random variables. Though the distribution of the weighted sum of chi-squares does not have a closed form expression, fast and efficient numerical approximations exist such as the [imhof]{} function in [R]{} [@Imhof:1961:CDQ].
In choosing $\{c_j\}$ we suggest two important considerations. The first is that one wants $c_j \to 0$ so as to eliminate the effect of later dimensions. In doing so, one is also producing compact regions [@laha:roghatgi:1979]. Since probability measures over Hilbert spaces are necessarily tight [@billingsley:1995], meaning they concentrate on compact sets, a region which is not compact is overly large. Conversely, the faster that $c_j \to 0$, the larger the mean of $W_\theta$, which increases all of the radii. Therefore, it seems desirable to balance these two concerns, choosing $c_j$ which go to zero, but not overly fast.
Two popular hypothesis testing frameworks in FDA, the [*norm approach*]{} and [*PC approach*]{}, can be understood as two extreme cases in this framework. The norm approach to test $H_0: \theta = \theta_0$ uses $N\|\hat{\theta} - \theta_0 \|^2$ as the test statistic. If $H_0$ is true, this test statistic, asymptotically, is a weighted sum of $\chi^2_1$ random variables with weights $\{\lambda_j\}$. This corresponds to taking $c_j^2 = 1$ for all $j$. The resulting confidence region, which we denote as $E_{norm}$, is a ball in ${{\mathcal H}}$ and provides proper coverage for $\theta$. However, this region is not compact and therefore too large as illustrated in Figure \[fig:ShortfallOfNoncompactRegion\]. The PC approach to hypothesis testing uses $\sum_{j=1}^{J} { N \langle \hat{\theta} - \theta_0, v_j \rangle^2}{\lambda_j^{-1}}$ as the test statistic, for some finite $J$. If $H_0$ is true, this test statistic follows a $\chi^2_J$ distribution; therefore, $J$ must be a finite value even when the covariance is known. There are two possible confidence regions induced by this approach. Both regions take $c_j^2=\lambda_j$ for $j \leq J$, but for $j > J$, one could either *close them off*, $c_j = 0 $, or *open them up*, $c_j = \infty$. The former results in a compact confidence region, but we have ${{\mathbb P}}\left(\theta \in E_{\hat{\theta}} \right) = 0$, i.e. *zero-coverage* even if we make the very artificial assumption that $\theta \in \operatorname{span}\{v_1,\dots, v_J\}$ since the center of the region $\hat{\theta}$ sits outside $\operatorname{span}\{v_1,\dots, v_J\}$ *almost surely*. On the other hand, the *opened-up* region would achieve proper coverage, but the region is not even bounded, let alone compact.
There exists infinitely many options for proper $\{c_j\}$ and how to best choose them is an open question deserving further exploration. In preparing this work, a number of options were initially considered, however, we propose using the following due to 1) its ability to achieve *the narrowest average squared width band* using tools from Section \[s:bands\] 2) excellent empirical performance, and 3) its simplicity: $$c_j^2 = \lambda_j^{1/2}
\qquad \text{and} \qquad E_c := \left\{ h \in \mathcal{H} : \sum_{j=1}^{\infty} \frac{\langle \sqrt{N}(\hat{\theta} - h), v_j \rangle^2}{\sqrt{\lambda_j}} \leq \xi \right\},$$ i.e. the square root of the corresponding eigenvalues. Although $\sum_j {\lambda_j}{c^{-2}_j} \equiv \sum_j {\lambda_j^{1/2}} < \infty$ is not always guaranteed, this holds for most processes that are smoother than Brownian motion ($\lambda_j \approx j^{-2}$), and therefore would hold in most applications. If the process is rough enough such that $\sum_j {\lambda_j^{1/2}} < \infty$ is not guaranteed, one may use another criteria suggested in the Appendices, namely $c_j^2 = \left(\sum_{i\geq j} \lambda_i \right)^{1/2}$, which guarantees both $c_j \to 0$ and $\sum_j {\lambda_j}{c^{-2}_j} < \infty$ [@rudin:1976 p. 80].
Hyper-Rectangular Form {#s:rectangle}
----------------------
Our second form is a slight modification of the previous form, switching from an ellipse to a rectangle. In multivariate statistics a rectangular confidence region is often easier to interpret than an ellipse since it gives clear confidence intervals for each (principal component) coordinate. However, it is often much easier to compute an ellipse since the distributions of quadratic forms are well understood. Regardless, we will show that it can still be easily computed using a nearly closed form expression, up to a function involving the standard normal quantile function.
A hyper-rectangular region can be similarly constructed as: $$R_{\hat{\theta}}
= \left\{ h \in \mathcal{H} : \frac{| \langle h - \hat{\theta}, v_j \rangle |}{r_j} \leq 1, \forall \ j =1,2 \dots \ \right\} = \left\{ h \in \mathcal{H} : | \langle \sqrt{N}(h - \hat{\theta}), v_j \rangle | \leq c_j\sqrt{\xi} , \ \forall j \right\}$$ using the same decomposition $r_j^2 = N^{-1} \xi c_j^2$. From the KL expansion , we want $$\begin{aligned}
\label{e:RectangleProbRule}
{{\mathbb P}}\left(\theta \in R_{\hat{\theta}} \right)
= {{\mathbb P}}\left( |\sqrt{\lambda_j}Z_j| \leq c_j \sqrt{\xi} , \ \forall j \right)
\to 1-\alpha.\end{aligned}$$ When we define $z_j := \frac{c_j}{\sqrt{\lambda_j} } \sqrt{\xi}$, the remaining problem is to find proper $\{z_j\}$, or selecting the $\{c_j\}$ and finding the proper $\xi$. One may first determine $\{c_j\}$ and find the proper $\xi$, or find the proper $\{z_j\}$ directly. Again, there exists infinitely many criteria and some examples can be found in the Appendices. Among those, we propose using the following: $$z_j = \Phi_{sym}^{-1}\left[ \exp \left( {\frac{\lambda_j}{\sum_{k=1}^\infty \lambda_k} \log(1-\alpha)} \right) \right] \text{ for each } j,$$ where $\Phi^{-1}_{sym}(\cdot)$ is defined as the inverse of $\Phi_{sym}(z):={{\mathbb P}}(|Z| \leq z)$, $Z \stackrel{d}{=} {{\mathcal N}}(0,1)$. We denote this rectangular region as $R_{z}$. This criterion produces a region that is close to the one that minimizes $\sup \{ \| h - \hat{\theta} \|^2 : h \in R_{\hat{\theta}} \}$, i.e. the distance between the farthest point of the region from the center, but in a much faster way. It is simple, easy to compute, and shows an excellent empirical performance.
Visualizing Ellipses via Bands {#s:bands}
------------------------------
Visualizing a confidence ellipse is challenging even in the finite dimensional setting; it is very difficult once one goes beyond two or three dimensions. In this sense, the rectangular regions are much easier to visualize since one can simply translate them into marginal intervals and examine each coordinate separately (while still achieving simultaneous coverage). It is therefore useful to develop visualization techniques for elliptical regions. One option is to construct bands in the form of an infinite collection of point-wise intervals over the domain of the functions. To make our discussion more concrete, in this section only we assume that ${{\mathcal H}}= L^2({{\mathcal D}})$, where ${{\mathcal D}}$ is some compact subset of ${{\mathbb R}}^d$. For example, $d = 1$ for temporal curves and $d=2$ for spatial surfaces.
A symmetric confidence band in ${{\mathcal H}}$ around $\hat{\theta}$ can be understood as: $$\begin{aligned}
\label{e:bandform}
B_{\hat{\theta}}
&= \left\{ h \in \mathcal{H} : | h(x) - \hat{\theta}(x) | \leq r(x), \ \text{for } x \in {{\mathcal D}}\text{\textit{ almost everywhere}} \right\}.\end{aligned}$$ The caveat “almost everywhere" here (i.e. except on a set of Lebesgue measure zero) cannot be dropped since we are working with $L^2$ functions. The downside of using the above band, however, is that an analytic expression for $r \in {{\mathcal H}}$ usually does not exist. One therefore typically resorts to simulation based methods as in @degras:2011. The band suggested by @degras:2011 takes $c_\alpha \hat{\sigma}(t)/\sqrt{N}$ as $r(t)$ where $\hat{\sigma}(t)$ is the estimated standard deviation of $\sqrt{N}\hat{\theta}(t)$. The proper scaling factor $c_\alpha$ is then found via parametric bootstrap. We denote this band as $\hat{B}_{s}$, while denoting the one using the true covariance as $B_{s}$.
In traditional multivariate statistics and linear regression, ellipses can be transformed into point-wise intervals and bands using Scheffé’s method which, at its heart, is an application of the Cauchy-Schwarz inequality. This approach cannot be applied *as is* to our ellipses because they are infinite dimensional. However, a careful modification of Scheffé’s method can be used to generate bands. We now show a $1-\alpha$ ellipsoid confidence region $E_{\hat{\theta}}$ can be transformed into the confidence band $B_{\hat{\theta}}$ such that $E_{\hat{\theta}} \subset B_{\hat{\theta}}$ based on a modification of Scheffé’s method. Defining $$\begin{aligned}
\label{e:BandZ1}
r(x) = \sqrt{ \frac{\xi}{N}\sum_{j=1}^\infty c_j^2 v_j^2(x)}\ ,\end{aligned}$$ then we have the following theorem.
\[t:band\] If Assumption \[a:normal\] holds, $\sum c_j^2 < \infty$, and $\sum \lambda_j c_j^{-2}<\infty$, then $r(x) \in {{\mathcal H}}$ and $E_{\hat{\theta}} \subset B_{\hat{\theta}}$. Therefore, ${{\mathbb P}}(\theta \in B_{\hat{\theta}}) \geq 1-\alpha + o(1).$
These bands also lead to a convenient metric for choosing an “optimal" sequence $c_j$. In particular, we choose the $c_j$ which lead to a band with the *narrowest average squared width*. This is in general a difficult metric to quantify due to $\xi$. However, we can replace $\xi$, which is a quantile of a random variable, by its mean to obtain the following: $$\begin{aligned}
ASW(\{ c_j \} ) = \sum_{j=1}^\infty \frac{\lambda_j}{c_j^2} \sum_{i=1}^\infty {c_i^2}.\end{aligned}$$ Clearly the $\{c_j\}$ are unique only up to a constant multiple, however, it is a straightforward calculus exercise to show that one option is to take $c_j^2 = \lambda_j^{1/2}$, which is also conceptually very simple. It is also worth noting that this choice does not change with the smoothness of the underlying parameters or the covariance of the estimator; these quantities are implicitly captured by the eigenvalues themselves and thus already built into the $c_j$ with this choice. In practice, the coverage of this band will be larger than $1-\alpha$, since $E_{\hat{\theta}} \subset B_{\hat{\theta}}$ and the coverage of $E_{\hat{\theta}}$ is $1-\alpha$. Our simulation studies show that this gap is non-trivial for rougher processes, but narrows substantially for smoother ones. The band formed this way from $E_c$ will be denoted as $B_{E_c}$.
Our suggested band takes into account the covariance structure of the estimator via the eigenvalues (though the $c_j$) and the eigenfunctions. Thus, our band differs from those described in [@degras:2011] in that we do not use a constant threshold after taking into account the point-wise variance; our band adjusts locally to the within curve dependence of the estimator. We will illustrate this point further in Section \[section:Simulation\] as one of our simulation scenarios will have a dependence structure which changes across the domain. Our band will adjust to this dependence, widening in areas with low within curve dependence and narrowing when this dependence is high.
Lastly, one practical issue arises in finding proper $\xi$ since finding the quantile of weighted sum of $\chi^2$ random variables is not straightforward. One may try to invert the approximate CDF, like `imhof` in $\texttt{R}$. Alternatively, one can use a gamma approximation by matching the first two moments [@feiveson:delaney:1968]. Our simulations showed that for typical choices of $\alpha$, such as $0.1, 0.05,$ or $0.01$, a gamma approximation works well.
Estimating Confidence Regions and Ghosting {#section:EstConfRegion}
==========================================
We have, until now, treated $C_\theta$ as known for ease of exposition and to explore the infinite dimensional nature of the regions. In this section we consider the fully estimated versions. Issues arise here that do not in the multivariate setting. In particular, one typically has *zero-coverage* when working with estimated regions, but we will show that these regions are still in fact useful since they are very close to regions with proper coverage. In this sense, we call them *Ghost Regions* since they ‘ghost’ the regions with proper coverage. Here we view the empirical regions as estimators of the desired regions which have proper coverage, and then show that the distance between the two quickly converges to zero. Our purpose in doing so is to provide a theoretical justification for using the regions in practice. In Section \[section:Simulation\] we will also validate these regions through simulations.
We assume that we have an estimator $\hat{C}_\theta$ of $C_\theta$ which achieves root-$N$ consistency. Consistency of $\hat{C}_\theta$ enables us to replace $\{( v_j , \lambda_j) \}_{j=1}^{\infty}$ with the empirical versions $\{( \hat{v}_j , \hat{\lambda}_j) \}_{j=1}^{N}$[^1]. When we replace $\{v_j\}_{j=1}^\infty$ with $\{\hat{v}_j\}_{j=1}^{N}$, however, we nearly always end up with a finite number of estimated eigenfunctions (with nonzero eigenvalues). We present asymptotic theory for the hyper-ellipsoid form although similar arguments can be applied to the hyper-rectangular form.
Define ${{\mathcal H}}_J := \operatorname{span}( \{\hat{v}_j\}_{j=1}^{J} ) \subset {{\mathcal H}}$ where $J \leq N$. We construct two versions of the estimated confidence regions $$\begin{aligned}
\label{e:estimatedregion-open}
\hat{E}^{\circ}_{\hat{\theta}} = \left\{ h \in {{\mathcal H}}: \sum_{j=1}^{J} \frac{\langle h - \hat{\theta}, \hat{v}_j \rangle^2}{N^{-1}c_j^2} \leq \xi \right\} \qquad \text{and}\end{aligned}$$ $$\begin{aligned}
\begin{split}
\label{e:estimatedregion}
\hat{E}_{\hat{\theta}}
= \left\{ h \in {{\mathcal H}}_J : \sum_{j=1}^{J} \frac{\langle h - \hat{\theta}, \hat{v}_j \rangle^2}{N^{-1} c_j^2} \leq \xi \right\}
= \left\{ h \in {{\mathcal H}}: \sum_{j=1}^{\infty} \frac{\langle h - \hat{\theta}, \hat{v}_j \rangle^2}{N^{-1} c_j^2 \mathbf{1}_{j \leq J}} \leq \xi \right\},
\end{split}\end{aligned}$$ though in our theoretical results we will let $J \to \infty $ with $N$. The empirical eigenfunctions $\{\hat{v}_j\}_{j=1}^J$ can be extended to give a full orthonormal basis of ${{\mathcal H}}$. Note that $\hat{E}_{\hat{\theta}}$ is ‘closed off’ while $\hat{E}^{\circ}_{\hat{\theta}}$ is ‘opened up’ for those dimensions not captured by the first $J$ components. We take $\xi$ to be the $1-\alpha$ quantile of a weighted sum of $\chi^2$ random variables with weights $\{{\hat \lambda_j}{c^{-2}_j} \}_{j=1}^J$. Observe that $\hat{E}_{\hat{\theta}}^{\circ}$ achieves the proper coverage ${{\mathbb P}}( \theta \in \hat{E}_{\hat{\theta}}^{\circ} ) \to 1-\alpha$. However, $\hat{E}_{\hat{\theta}}^{\circ}$ cannot be compact regardless of how $\{c_j\}$ is chosen unless ${{\mathcal H}}$ is finite dimensional. If we quantify the distance between sets using Hausdorff distance, $\hat{E}_{\hat{\theta}}^{\circ}$ does not converge to $E_{\hat{\theta}}$ since it is unbounded. On the other hand, $\hat{E}_{\hat{\theta}}$ is always compact but has *zero-coverage*; we almost always have ${{\mathbb P}}( \theta \in \hat{E}_{\hat{\theta}} ) = 0$ regardless of the sample size. Therefore, neither empirical confidence regions maintains the nice properties of the ones using a known covariance – compactness and proper coverage – at the same time. However, as we will show, $\hat{E}_{\hat{\theta}}$ is close to $E_{\hat{\theta}}$ in Hausdorff distance, meaning we can use $\hat{E}_{\hat{\theta}}$ as an estimate of the desired region ${E}_{\hat{\theta}}$. With this convergence result at hand, one may prefer the closed version $\hat{E}_{\hat{\theta}}$ over $\hat{E}_{\hat{\theta}}^{\circ}$ as a confidence region. Because $\hat{E}_{\hat{\theta}}$ does not have proper coverage we call it a *ghost* region.
Convergence in the Hausdorff Metric
-----------------------------------
In this subsection we show that the Hausdorff distance, denoted $d_H$, between $\hat{E}_{\hat{\theta}}$ and ${E}_{\hat{\theta}}$ can be well controlled. In particular, we will show that this distance converges to zero faster than $N^{-1/2}$. Since this is the rate at which ${E}_{\hat{\theta}}$ shrinks to a point, this is necessary to ensure that $\hat{E}_{\hat{\theta}}$ is actually useful as a proxy for ${E}_{\hat{\theta}}$. We begin by introducing a fairly weak assumption on the distribution of $\hat{C}_\theta$. Recall that $C_\theta$ is a Hilbert-Schmidt operator (all covariance operators are) in the sense that $
\|C_{\theta}\|^2_{{{\mathcal S}}} := \sum_{j=1}^\infty \| C_{\theta}(e_j) \|^2_{{\mathcal H}}< \infty
$ where $\{e_j\}$ is any orthonormal basis of ${{\mathcal H}}$. We denote the vector space of Hilbert-Schmidt operators by ${{\mathcal S}}$, which is also a real separable Hilbert space with inner product $
\langle \Psi, \Phi \rangle_{{{\mathcal S}}} := \sum_{j=1}^\infty \langle \Psi(e_j), \Phi(e_j) \rangle_{{{\mathcal H}}}.
$ A larger space, ${{\mathcal L}}$, consists of all bounded linear operators with norm $
\| \Psi\|_{{{\mathcal L}}} = \sup_{h \in {{\mathcal H}}} \| \Psi (h)\| / \|h\|,
$ which is strictly smaller than the ${{\mathcal S}}$ norm, implying ${{\mathcal S}}\subset {{\mathcal L}}$. We now assume that we have a consistent estimate of $C_\theta$.
\[assumption:1\] Assume that we have an estimator $\hat C_\theta$ of $C_\theta$ which is root-$N$ consistent in the sense that $N {{\mathbb E}}\| \hat C_\theta - C_\theta\|_{{{\mathcal S}}}^2 = O(1)$.
The Hausdorff distance between two subsets $S_1$ and $S_2$ of ${{\mathcal H}}$ is defined as $$d_H(S_1,S_2) = \max\{\rho(S_1,S_2), \rho(S_2,S_1) \}, \qquad \text{where} \qquad \rho(S_1,S_2) = \sup_{x \in S_1} \inf_{y \in S_2} \|x - y \|_{{\mathcal H}}.$$ We say two regions $S_1$ and $S_2$ converge to each other if $d_H(S_1,S_2)$ converges to $0$. Therefore, to achieve convergence of $\hat{E}_{\hat{\theta}}$ to $E_{\hat{\theta}}$ in probability, we need $d_H(\hat{E}_{\hat{\theta}},E_{\hat{\theta}}) \xrightarrow{P} 0 $ as $N \to \infty$. To accomplish this, we separate the results for $\rho(\hat{E}_{\hat{\theta}},E_{\hat{\theta}})$ and $\rho({E}_{\hat{\theta}},\hat E_{\hat{\theta}})$. Since $\hat{E}_{\hat{\theta}}$ is the “smaller" set, the former is primarily controlled by the distance between the empirical and population level eigenfunctions. The latter is additionally influenced by how large the remaining dimension of $E_{\hat{\theta}}$ is. We define $\{\alpha_j\}$ as $$\alpha_1 := \lambda_1 - \lambda_2 \quad \text{and} \quad \alpha_j := \min\{ \lambda_j - \lambda_{j+1}, \lambda_{j-1} - \lambda_{j} \} \text{ for } j = 2, \dots.$$ Our primary convergence results are given in the following two theorems.
\[thm:Convergence1\] If Assumptions \[a:normal\] and \[assumption:1\] hold and $c_1 \geq c_2 \geq \dots$ then with probability one $$\begin{aligned}
\label{e:order1}
\rho(\hat{E}_{\hat{\theta}},E_{\hat{\theta}}) \leq \left[ \sum_{j=1}^{J} \frac{8 \xi c_1^2 \|\hat C_\theta - C_\theta\|_{{\mathcal L}}^2}{N \alpha_j^{2}} \right]^{\frac{1}{2}}.
\end{aligned}$$
\[thm:Convergence2\] If Assumptions \[a:normal\] and \[assumption:1\] hold and $c_1 \geq c_2 \geq \dots$ then with probability one $$\begin{aligned}
\rho(E_{\hat{\theta}}, \hat{E}_{\hat{\theta}})
\leq \left[c_J^2 N^{-1} \xi \right]^{\frac{1}{2}} + \left[ \sum_{j=1}^{J} \frac{8 \xi c_1^2 \|\hat C_\theta - C_\theta\|_{{\mathcal L}}^2}{N \alpha_j^{2}} \right]^{\frac{1}{2}}.
\label{e:order2}
\end{aligned}$$
With Theorems \[thm:Convergence1\] and \[thm:Convergence2\] in hand, we can characterize the overall convergence rate for $d_H(\hat{E}_{\hat{\theta}},E_{\hat{\theta}}) $, but we first need more explicit assumptions on the rates for the eigenvalues, $\lambda_j$, and weights, $c_j^2$.
\[a:rate1\] Assume that there exist constants $K > 1 $, $\delta > 1$, and $\gamma > 0$ such that $$\frac{1}{Kj^{\delta}} \leq \lambda_j \leq \frac{K}{ j^{\delta}},
\qquad
\frac{1}{Kj^{\delta+1}} \leq \lambda_j - \lambda_{j+1} \leq \frac{K}{ j^{\delta+1}},
\quad \text{ and } \quad
\frac{1}{K j^{ 2 \gamma}} \leq c_j^2 \leq \frac{K}{j^{ 2 \gamma}},$$ for all $j = 1, \dots$, where we have $0 < 2 \gamma < \delta -1$.
The first two assumptions are quite common in FDA. One needs to control the rate at which the eigenvalues go to zero as well as the spread of the eigenvalues which influences how well one can estimate the corresponding eigenfunctions, though this can likely be slightly relaxed [@reimherr:2015]. The rate at which $c_j^2$ decreases to zero also needs to be well controlled, and, in particular, it cannot go to zero much faster than $\lambda_j$.
\[thm:ConvergenceOverall\] Assume that Assumptions \[a:normal\], \[assumption:1\] and \[a:rate1\] hold, then
1. The $J$ which balances and is $J = N^{\frac{1}{2 \delta + 3 + 2 \gamma}}$.
2. The overall convergence rate is then ${{\mathbb E}}[d_H(\hat{E}_{\hat{\theta}},E_{\hat{\theta}})^2 ] \leq
O \left( N^{-\left( 2 - \frac{2\delta + 3}{2 \delta + 3 + 2\gamma} \right)} \right)$.
Theorem \[thm:ConvergenceOverall\] shows that the squared distance between the ghost region $\hat{E}_{\hat{\theta}}$ and the desired region $E_{\hat{\theta}}$ goes to zero faster than $N^{-1}$, which is the rate at which $E_{\hat{\theta}}$ shrinks to a point. This suggests that $\hat{E}_{\hat{\theta}}$ is a viable proxy for ${E}_{\hat{\theta}}$, even though it has zero coverage.
Interestingly, the rate is better the faster that $c_j$ tends to zero, i.e. for larger values of $\gamma$. At first glance, this may suggest that one should actively try and find $c_j$ which tend to zero as fast as possible. However, by changing the $c_j$ one is changing the confidence region into a potentially less desirable one. In particular, as we will soon see the choice of $c_j^2 = \lambda_j^{1/2}$ leads to a suboptimal convergence rate in Theorem \[thm:ConvergenceOverall\], but, in some sense, leads to an optimal confidence band and excellent empirical performance. Thus, this may be one of the few instances in statistics where it is not necessarily desirable to have the “fastest" rate of convergence.
We finish this section by stating a Corollary for when $c_j^2 = \hat \lambda_j^{1/2}$. In this case, we also take into account that the $c_j$ are estimated from the data. Note that in Theorem \[thm:ConvergenceOverall\] it is assumed that the $c_j$ are not random, while in Theorems \[thm:Convergence1\] and \[thm:Convergence2\] the $c_j$ can be random or deterministic as long as they are nonincreasing.
\[corol:EchXConvRate1\] Let $c_j^2 = \hat \lambda_j^{1/2}.$ Then under Assumptions \[assumption:1\] and \[a:rate1\], we have $$d(\hat E_{\hat \theta}, E_{\hat \theta})^2 = O_p\left(N^{ - \frac{6\delta + 6}{5 \delta + 6}} \right).$$
Simulation {#section:Simulation}
==========
In this section, we present a simulation study to evaluate and illustrate the proposed confidence regions and bands. Throughout this section, we only consider dense FDA designs. Section \[subsection:HT\] first compares different regions for hypothesis testing. Note that comparing these regions presents a nontrivial challenge as we cannot just choose the “smallest" one as we are working in infinite dimensional Hilbert spaces. We therefore turn to using the regions for hypothesis testing, evaluating each’s ability to detect different types of changes from some prespecified patterns. In Section \[subsection:ComparisonOfBands\], we visually compare bands and examine their local coverages. Lastly, in Section \[subsection:SimulationOnData\], we consider more complicated mean and covariance structures borrowed from the DTI data in Section \[s:dti\] and examine the effects of smoothing.
Hypothesis Testing {#subsection:HT}
------------------
We consider the hypothesis testing $H_0 : \theta = \theta_0$ vs. $H_1 : \theta \neq \theta_0 $. For a given confidence region $G_{\hat{\theta}}$, the natural testing rule is to reject $H_0$ if $\theta_0 \notin G_{\hat{\theta}}$. For ellipses and rectangles, however, we compare $\theta_0$ only in the directions included in the construction of the confidence regions to alleviate the ghosting issue and mimic how the methods would be used in practice. We calculate p-values (detailed in the Appendices) and compare them to $\alpha$. For bands like $B_s$ and $B_{E_c}$, $H_0$ will be rejected if $\theta_0$ sits outside the bands at least one evaluation point over the domain.
In this section and in Section \[subsection:ComparisonOfBands\], we take ${{\mathcal H}}= L^2[0,1]$ and consider an $iid$ sample $\{X_i(t)\}_{i=1}^N$, $t\in[0,1]$ from a Gaussian process ${{\mathcal N}}(\theta, C_\theta)$. To estimate $\theta$ and $C_\theta$ (when unknown), we use the standard estimates [@hkbook] $\hat{\theta}(t) = N^{-1}\sum_{i=1}^{N}X_i(t)$ and $\hat{C}_\theta(t,s) = (N-1)^{-1} \sum_{i=1}^N ( X_i(t) - \hat{\theta}(t) ) ( X_i(s) - \hat{\theta}(s) )$. To emulate functions on the continuous domain $[0,1]$, functions are evaluated at 100 equally spaced points over $[0,1]$.
### Verifying Type I Error {#subsection:TypeIError}
#### Regions with Known Covariance:
We first verify Type I error rates assuming the true covariance operator is known. Each setting was repeated 50,000 times according to the following procedure:
1. Generate a sample $\{X_i\}_{i=1}^N {\stackrel{iid}{\sim}}{{\mathcal N}}(\theta,C_\theta)$. For the mean function we take $\theta(t) := 10t^3-15t^4+6t^5$, which was used in @degras:2011 and @Hart:1986:KRE. For the covariance operator, we use a Matérn covariance $C_\theta(t,s) := \frac{.25^2}{\Gamma(\nu) 2^{\nu-1}} \left( \sqrt{2\nu}|t-s|\right)^\nu K_\nu \left(\sqrt{2\nu}|t-s| \right)$, where $K_\nu(\cdot)$ is the modified Bessel function of the second kind, and $\nu$ is the smoothness parameter.
2. Find $\hat{\theta}$ from the sample, and perform hypothesis testings on $\theta_0 = 10t^3-15t^4+6t^5$, which is the same as $\theta$, based on the confidence regions using $C_\theta$, the true covariance.
To represent small/large sample size and rough/smooth processes, the four combinations of $N=25$, $N=100$ and $\nu=1/2,$ $\nu=3/2$ were used. Table \[tbl:ActualAlphaTrueCov\] summarizes proportion of the rejections.
$N$ $\nu$ $E_{norm}$ $E_{PC}$ $B_{s}$ $E_{c}$ $R_{z}$ $R_{zs}$ $B_{E_c}$
----- --------------------------- ------------ ---------- --------- --------- --------- ---------- -----------
25 $\sfrac{1}{2} \ (rough)$ .048 .049 .049 .049 .048 .049 .000
25 $\sfrac{3}{2} \ (smooth)$ .051 .051 .053 .050 .051 .049 .025
100 $\sfrac{1}{2} \ (rough)$ .051 .049 .052 .051 .050 .050 .000
100 $\sfrac{3}{2} \ (smooth)$ .050 .049 .047 .049 .049 .049 .023
: Type I error rates with known covariance. $E_{norm}$, $E_{PC}$, and $E_{c}$ represent ellipsoid regions from norm approach, FPCA approach, and the proposed one, respectively. $B_{s}$ is the simulation based band while $B_{E_c}$ is the band based on $E_{c}$. $R_{z}$ is the proposed rectangular region and $R_{zs}$ is the small sample version of $R_{z}$, which uses only eigenfunctions (but not eigenvalues) of $C_\theta$.[]{data-label="tbl:ActualAlphaTrueCov"}
All the methods are satisfactory except for the transformed band $B_{E_c}$, which generates a conservative band as expected. For the ellipsoid and rectangular regions, up to the very last PCs were used – trimming out only $\lambda_j < 10^{-18}$ – and the results were still stable. Although not presented here, the results were robust against the number of PCs used.
#### Regions with Unknown Covariance:
We now use $\hat C_\theta$ instead of $C_\theta$ in the step 2 above, and use PCs to capture at least $99.9\%$ of estimated variance, i.e. took $J$ such that $J = \min_j ( \sum_{i=1}^j \hat{\lambda}_i / \sum_{i=1}^{N-1}\hat{\lambda}_i \geq .999)$, for all ellipsoid and rectangular regions. For the FPCA based region, we additionally took $J=3$, which explained approximately $90\%$ of the variability. Table \[tbl:ActualAlpha\] summarizes the proportions of the rejections and the following can be observed.
1. Coverage of $\hat{E}_{PC}$ is very sensitive to the number of PCs used and works well only when the number is relatively small. This reenforces the common concern of how to best choose $J$ in practice. In contrast, $\hat{E}_{norm}$ does not have this question. Our proposed methods $\hat{E}_{c}$ and $\hat{R}_{z}$ lie somewhere between the two and choosing $J$ is not a concern as long as the very late PCs are dropped.
2. When $N$ is small, the small sample modification of the rectangular region ($\hat{R}_{zs}$) achieves slightly conservative but seemingly the best result. $\hat{E}_{norm}$ follows closely, possibly due to its lower dependency on later PCs. The details on $\hat{R}_{zs}$ can be found in the Appendices.
$N$ $\nu$ $\hat{E}_{norm}$ $\hat{E}_{PC}$ $\hat{E}_{PC(3)}$ $\hat{B}_{s}$ $\hat{E}_{c}$ $\hat{R}_{z}$ $\hat{R}_{zs}$ $\hat{B}_{E_c}$ $PC^*$
----- ---------------- ------------------ ---------------- ------------------- --------------- --------------- --------------- ---------------- ----------------- -------- -- -- --
25 $\sfrac{1}{2}$ .057 .162 .069 .087 .071 .069 .041 .013 21
25 $\sfrac{3}{2}$ .061 .132 .090 .071 .068 .068 .047 .039 5
100 $\sfrac{1}{2}$ .052 .255 .056 .058 .060 .059 .050 .001 53
100 $\sfrac{3}{2}$ .052 .066 .057 .054 .053 .053 .049 .026 5
: Type I error rates with an estimated covariance. $E_{norm}$, $E_{PC}$, and $E_{c}$ represent ellipsoid regions from norm approach, FPCA approach, and the proposed one, respectively. $B_{s}$ is the simulation based band while $B_{E_c}$ is the band based on $E_{c}$. $R_{z}$ is the proposed rectangular region and $R_{zs}$ is the small sample version of $R_{z}$, which uses only eigenfunctions (but not eigenvalues) of $C_\theta$.[]{data-label="tbl:ActualAlpha"}
*\* Median number of PCs required to capture $\geq 99.9\%$ of estimated variance.*
We emphasize here the dependence on choosing $J$ for both the FPCA and our new approach. As is well known, FPCA based methods are very sensitive to the choice of $J$ as it places all eigenfunctions on an “equal footing”. However, later eigenfunctions are often estimated very poorly, which can result in very bad type 1 error rates when $J$ is taken too large. In contrast, our approach is not as sensitive to the choice of $J$ since later eigenfunctions are down weighted. In our simulations, they remained well calibrated as long as the very late FPCs are dropped, e.g. after capturing $99\%$ of the variance.
### Comparing Power
To compare the power of the hypothesis tests, we gradually perturb $\theta$ – the actual sample generating mean function – from $\theta_0$ by an amount $\Delta \in {{\mathbb R}}$. To emulate what one might encounter in practice, three scenarios are considered:
1. shift: $\theta_0(t) = 10t^3-15t^4+6t^5$, $\theta(t) = \theta_0(t) + \Delta$,
2. scale: $\theta_0(t) = 10t^3-15t^4+6t^5$, $\theta(t) = \theta_0(t)(1+\Delta)$,
3. local shift: $\theta_0(t) = \max\left\{0, -10|t-0.5| + 1 \right\}$, $\theta(t) = \max\left\{0, -10|t-0.5| + 1+\Delta \right\}$.
A visual representation of the three scenarios can be found in the left column of Figure \[fig:PowerComparison\]. We estimate $C_\theta$ throughout, reduce the number of repetitions to 10,000, and use the same combinations of the sample size ($N=25, \ 100$) and smoothness ($\nu=$$, \ $$ $). For the $\hat{E}_{PC}$ method, the first 3 PCs were again used to ensure an acceptable Type I error. For other ellipsoid and rectangular regions, $J$ was taken to explain approximately 99.9% of the variance as in the previous section. Power plots for $N=100$ and $\nu=\sfrac{1}{2}$ can be found in the right column of Figure \[fig:PowerComparison\], and a summary is given in Table \[tbl:AveragePower\]\]. The result for other combinations of sample size and smoothness can be found in the Appendices, but they all lead to the same conclusions:
1. In scenario 1, $\hat{E}_{PC(3)}$ has the lowest power while the other regions performs similarly.
2. In scenario 2, $\hat{E}_{PC(3)}$ has the highest power while $E_{norm}$ has the lowest. Our hyper-ellipse method $\hat E_c$ has only slightly less power than the FPCA method. Our rectangular method, $\hat R_z$, and the band of @degras:2011 have about the same power, but both are lower than the ellipse.
3. In Scenario 3, our proposed regions $\hat{E}_{c}$ and $\hat{R}_{z}$ far outperform existing ones. Note that $\theta$ differs from $\theta_0$ only on a fraction of the domain and the size of the departure is also small. Due to the small $\| \theta - \theta_0\|$, therefore, $\hat{E}_{norm}$ performs the worst. The FPCA method, $\hat{E}_{PC(3)}$ and Degras’s band fall quite a bit behind our proposed methods, but still better than the norm approach. The $\hat{E}_{PC(3)}$ performs much better when the process is smooth and therefore the ‘signal’ is captured in earlier dimensions – although it still falls short from the proposed ones.
As a conclusion, we recommend using $\hat E_c$ in practice for hypothesis testing purposes. We base this recommendation on 1) its power is at the top or near the top in every scenario; 2) its type I error is well-maintained as long as very late PCs are dropped; 3) it is less sensitive to the number of PCs used as long as the number is reasonably large; 4) it is easy to compute; and 5) it can be used to construct a band. Being able to make this recommendation is quite substantial as previous work has focused on the norm versus PC approach, where clearly one does not always outperform the other [@ReNi:2014].
Scenario $\hat{E}_{norm}$ $\hat{E}_{PC(3)}$ $\hat{B}_{s}$ $\hat{E}_{c}$ $\hat{R}_{z}$ $\hat{R}_{zs}$
----------------- ------------------ ------------------- --------------- --------------- --------------- ----------------
1\. Shift .623 .560 .617 .625 .607 .598
2\. Scale .411 .549 .503 .522 .496 .480
3\. Local Shift .234 .568 .504 .759 .770 .749
: Average Power over $\Delta$ for each Scenario[]{data-label="tbl:AveragePower]"}
Comparison of Bands {#subsection:ComparisonOfBands}
-------------------
In this section we compare the shape of $\hat{B}_{E_c}$ with $\hat{B}_{s}$, the two band forms of confidence regions, along with point-wise 95% confidence intervals denoted as ‘naive-t’. For this purpose, we consider three different scenarios regarding the smoothness of $\hat{\theta}$. The procedure can be summarized as follows:
1. Generate a sample $\{X_i\}_{i=1}^N {\stackrel{iid}{\sim}}{{\mathcal N}}(\theta,C_\theta)$, using the same mean function $\theta(t)$ as in Section \[subsection:TypeIError\]. For the covariance operator, three scenarios are considered:
1. The same Matérn covariance in Subsection \[subsection:TypeIError\] with $\nu = \sfrac{1}{2}$ (rough).
2. The same Matérn covariance in Subsection \[subsection:TypeIError\] with $\nu = \sfrac{3}{2}$ (smooth).
3. $C_\theta(t,s) := \frac{.25^2}{\Gamma(\nu) 2^{\nu-1}} \left( \sqrt{2\nu}|t^{10}-s^{10}|\right)^\nu K_\nu \left(\sqrt{2\nu}|t^{10}-s^{10}| \right)$ with $\nu=\sfrac{1}{2}$. This generate processes that transition from smooth to rough by ‘warping’ the domain.
2. Find $\hat{\theta}$ and $\hat{C}_\theta$, and generate symmetric bands around $\hat{\theta}$ using $\hat{C}_\theta$.
Figure \[figure:ConfidenceBands\] shows sample paths ($N=25$) from the three different covariance operators on the first row, their 95% simultaneous confidence bands on the second row, and local coverage rates on the third row. The findings can be summarized as follows :
1. The proposed band $\hat{B}_{E_c}$ is wider than $\hat{B}_{s}$ for rougher processes, but almost identical to $\hat{B}_{s}$ for smoother ones, except for the far ends of the domain.
2. In the third case, $\hat{B}_{E_c}$ is narrower than $\hat{B}_{s}$ in the smoother areas, while $\hat{B}_{s}$ maintains the same width. $\hat{B}_{E_c}$ adjusts its width such that it gets narrow in the smooth areas (higher within curve dependence) and wider in the rough areas.
3. Due to its construction, $\hat{B}_{E_c}$ does not bear any local under-coverage issue, and therefore any pattern in the third row in Figure \[figure:ConfidenceBands\] can be rather related to its over-coverage.
We conclude that the confidence band $\hat{B}_{E_c}$ is an effective visualization tool to use in practice especially when the estimate $\hat{\theta}$ is relatively smooth. For smoother estimates, it is nearly identical to the parametric bootstrap but is much faster to compute since it requires no simulation. This is important as our band is conservative, utilizing a Scheffé-type inequality. It suggests that not much is lost in using such an approach as long as the parameter estimate is sufficiently smooth. If the hypothesis tests and our confidence bands are in disagreement, say due to the conservative nature of $\hat{B}_{E_c}$, then it is recommended to follow up with a parametric bootstrap to get tighter bands.
Simulation based on DTI data {#subsection:SimulationOnData}
----------------------------
Although the mean and the covariances in Subsection \[subsection:HT\] and \[subsection:ComparisonOfBands\] are chosen to mimic common functional objects, actual data in practice may show much more complex structures. In this subsection we use a mean and a covariance structure from the `DTI` dataset in the `R` package `refund`. *This DTI data were collected at Johns Hopkins University and the Kennedy-Krieger Institute*. More details on this data set can be found in @goldsmith:2012a and @goldsmith:2012b. This dataset contains fractional anisotropy tract profiles of the corpus callosum for two groups – healthy control group and multiple sclerosis case group, observed over 93 locations. In this subsection, we took only the first visit scans of the case group in which the sample size is 99. We will denote this sample as original sample.
First, we estimated the sample mean and covariance from the original sample and considered them as parameters. While the mean was estimated by penalizing $2^{nd}$ derivative with leave-one-out cross-validation to achieve a smooth mean function, the covariance was estimated using the standard method (but using the smoothed mean) to mimic the roughness of the original sample. Using these mean and covariance, we generated multiple (10,000) Gaussian simulation samples of the sample size $99$. The use of Gaussian sample could be justified by the distribution of coefficients on each principal components in the original sample.
For each generated sample, two different estimation procedures were taken to look at the effect of smoothing. First approach is to simply smooth the sample using quardic bspline basis (with equally spaced knots) and use the standard estimates, and the second approach is to directly smooth the mean function by penalizing $2^{nd}$ derivative (or curvature), in which the covariance was estimated accordingly as shown in the Appendices. For both approaches, leave-one-out cross validation was used to find the number of basis functions and the penalty size, respectively. To reduce the computation time, those values were pre-determined from the original sample and applied to the simulation samples.
Other than the explicit differences in approaches for smoothing – data first vs directly on the estimate, and bspline vs penalty on curvature –, the first smoothing would introduce bias because the 15 bspline functions would not fully recover the assumed mean function. The empirical bias from the first smoothing was $6.1$ times larger than the second one.
Figure \[figure:PointwiseCoverage\] compares confidence bands from the two smoothing schemes. Although they do not snow any material difference in the shapes of the bands, we get slightly narrower bands from bspline smoothing (left column). This may cause under-coverage for $\hat{B}_s$, but does not work adversely for the proposed band $\hat{B}_{E_c}$ which generally provides over-coverage. While the narrow band for $\hat{B}_{E_c}$ is mainly caused by more explicit dimension reduction or more smoothing, but for $\hat{B}_s$ and naive-t, bias seems to be the main source of it – This can be supported by the local coverage patterns in the figure.
Table \[tbl:ActualAlpha\_Sim2\] compares coverage rates of non-band form regions using two different smoothing schemes and cutting points for $J$. Note that we see only minor difference between the two smoothing approaches, and the effect of $J$ is essentially the same as in Section \[subsection:TypeIError\]; the coverage of FPC based method $\hat{E}_{PC}$ deteriorate fast as $J$ increases while $\hat{E}_{norm}$ it not affected, and $J$ that explains about $99\%$ of variance does not raise major concern in the proposed regions $\hat{E}_{c}$, $\hat{R}_{z}$, and $\hat{R}_{zs}$.
Smoothing
------------- ------------------ ---------------- --------------- --------------- ---------------- -------- ------------------ ---------------- --------------- --------------- ---------------- --------
var. $\geq$ $\hat{E}_{norm}$ $\hat{E}_{PC}$ $\hat{E}_{c}$ $\hat{R}_{z}$ $\hat{R}_{zs}$ $PC^*$ $\hat{E}_{norm}$ $\hat{E}_{PC}$ $\hat{E}_{c}$ $\hat{R}_{z}$ $\hat{R}_{zs}$ $PC^*$
0.90 .949 .942 .948 .947 .952 5 .949 .942 .946 .948 .954 5
0.95 .949 .937 .946 .945 .951 7 .949 .933 .944 .946 .953 8
0.99 .949 .904 .940 .939 .947 11 .949 .887 .937 .940 .949 15
0.999 .949 .849 .935 .933 .943 15 .948 .572 .922 .904 .926 24
: Coverage rates of non-band form regions using two different smoothing approaches[]{data-label="tbl:ActualAlpha_Sim2"}
*\* Median number of PCs required to capture desired (estimated) variance.*
Data Example {#s:dti}
============
In this section, we further illustrate the usage of suggested methods using the same `DTI` dataset. We now take both control and case group of the first visit scans to look at their differences in mean, in which the sample sizes are 42 and 99, respectively.
Visualization via Bands {#subsection:data_visualization}
-----------------------
The first step is to visually compare the two sample mean functions, and make confidence bands for the mean difference. Figure \[figure:DTIBand\] shows the two sample means, followed by 95% confidence bands for the mean difference using $\hat{B}_{E_c}$ and $\hat{B}_{s}$, assuming unequal variances. Although the proposed band $\hat{B}_{E_c}$ is wider than $\hat{B}_{s}$ when the standard estimates from the raw data are used (middle), it gets narrower when the data are smoothed (right). For smoothing, we used quadric bsplines with two-fold cross-validations on the mean difference to choose the number of basis functions. In this case 11 basis functions were chosen and we used equally spaced knots. We observe that the bands do not cover zero($0$) for most of the domain except for the beginning and the very end part.
Hypothesis Testing {#hypothesis-testing}
------------------
The result of hypothesis testing $H_0 : \mu_{\text{ctrl}} = \mu_{\text{case}}$ versus $H_0 : \mu_{\text{ctrl}} \neq \mu_{\text{case}}$ using different regions is summarized in Table \[tbl:DTITesting\]. The proposed regions $\hat{E}_{c}$ and $\hat{R}_{z}$ yield at least comparable p-values with existing ones like $\hat{E}_{norm}$ and $\hat{E}_{PC(3)}$. Since there exists an overall shift in the difference of the mean functions, little room could be found for the proposed regions to outperform $\hat{E}_{norm}$. Small sample version $\hat R_{zs}$ achieves a bit larger p-value as expected, but not materially.
Data Var. $\geq$ $\hat{E}_{norm}$ $\hat{E}_{PC}$ $\hat{E}_{PC(3)}$ $\hat{E}_{c}$ $\hat{R}_{z}$ $\hat{R}_{zs}$ $PC^*$
---------- ------------- ------------------ ---------------- ------------------- --------------- --------------- ---------------- --------
Raw 0.99 $6.6E^{-14}$ $2.6E^{-10}$ $2.1E^{-13}$ $2.3E^{-14}$ $2.5E^{-13}$ $2.1E^{-11}$ 22
Smoothed 0.99 $1.6E^{-13}$ $4.0E^{-14}$ $8.7E^{-14}$ $1.1E^{-14}$ $2.2E^{-13}$ $1.9E^{-11}$ 11
: P-values from hypothesis testings based on different regions.[]{data-label="tbl:DTITesting"}
*\* Number of PCs used to capture desired variance except for $\hat{E}_{PC(3)}$ which uses only $3$ PCs*
In @pomann:2016 two sample tests were developed and illustrated using the same data. There they use a bootstrap approach to calculate p–values. A p–value of approximately zero is reported based on 5000 repetitions, which means that the p-value $< 2 \times 10^{-4}$. Since our approach is based on asymptotic distributions, not simulations, we are able to give more precise p–values which are of the order $10^{-14}$ for the lowest and $10^{-11}$ for the highest.
Visual Decomposition using Rectangular Region {#s:rectangle-visual}
---------------------------------------------
One merit of a rectangular region is that it can be expressed as intersection of marginal intervals. Note that since eigenfunctions are uniquely determined up to signs, it does not help to look at the signs of coefficients. Figure \[figure:DTI\_Marginals\] shows confidence intervals for the absolute values of coefficients for each PC using $\hat{R}_{z}$. We observe that only the confidence interval for the first PC does not cover zero. Based on this, we can infer that there exists a significant difference between the two mean functions along the $1^{st}$ PC, but the two means are not significantly different in any other features. In this sense, this visual decomposition serves as hypothesis testings on PCs while maintaining family-wise level at $\alpha$. Although we made intervals for absolute coefficients to visually represent the importance of each PCs, one may choose to make intervals for absolute $z$-scores to make later intervals more visible.
Once the overall shapes of confidence intervals are obtained, one may choose to examine specific PCs. Figure \[figure:DTICoefPC1\] shows the interval for the $1^{st}$ PC as a band along the $1^{st}$ eigenfunction. This now reveals that the departure is caused by the ‘downward’ shift of the case group, and confirms that this is the main source of the mean departure in Figure \[figure:DTIBand\]. Lastly, we mention that smoothing here also makes little difference in the ‘shapes’ of the intervals in Figure \[figure:DTI\_Marginals\] and \[figure:DTICoefPC1\] except for the effect of smoothing itself – smoother ($1^{st}$) eigenfunction and more variance captured in early PCs.
Discussion
==========
Each of the proposed and existing regions (and the corresponding hypothesis tests) has pros and cons, and therefore the decision on which region to use in practice would depend on many factors including the nature of the data, the purpose of the research, etc. However, we believe that we have clearly demonstrated that the proposed hyper-ellipses, $\hat E_c$, or hyper-rectangles, $\hat R_z$, make excellent candidates as the “default" of choice. In our simulations, they were at the top or near the top, in terms of power, in every setting. Deciding between ellipses versus rectangles comes down to how the regions will be used. If the focus is on the principal components and interpreting their shapes, then the rectangular regions make an excellent choice. If the FPCs are of little to no interest, then the hyper–ellipses combined with their corresponding band make an excellent choice, especially if the parameter estimate is relatively smooth. However, for rougher estimates, we recommend sticking with the simulation based bands like @degras:2011 as opposed to the bands generated from the ellipses.
We also believe that the discussed perspectives on coverage and ghosting will be useful for developing and evaluating new methodologies. From a theoretical point of view, working with infinite dimensional parameters presents difficulties which are not found in scalar or multivariate settings. In particular, it is common for methods to “clip" the infinite dimensional parameters. In practice the clipping may or may not have much of an impact – for example the FPCA methods are very sensitive to this clipping while our ellipses and rectangles are not – but in all cases it introduces an interesting theoretical challenge. Our ghosting framework will be useful as it provides a sound basis for using regions with deficient coverage.
For the first time, the construction of confidence regions and bands has been placed into a Hilbert space based framework together which has become a primary model for many FDA methodologies. However, we believe there is a great deal of additional work to be done in this area and that it presents some exciting opportunities. For example, are there other metrics for determining which confidence region to use? Do these metrics lead to different choices of $c_j$? How can we choose $J$, the number of PCs to use in practice without undermining proper coverage considering poor estimation of later PCs? Are there other shapes beyond ellipses and rectangles which are useful? Can we use better metrics than Hausdorff for evaluating convergence? Many open questions remain which we hope other researchers will find interesting.
Proofs
======
In this section we gather all of the proofs and necessary lemmas.
The ${{\mathcal H}}$ norm of $r(x)$ is given by $$\| r\|^2 = \frac{\xi}{N}\sum_{j=1}^\infty c_j^2.$$ This will be finite if $\sum_{j=1}^\infty c_j^2 < \infty$ and $|\xi| < \infty$, the latter of which is guaranteed when $\sum \lambda_j c_j^{-2}<\infty $. Therefore $r(x)$ is in ${{\mathcal H}}$.
To show $E_{\hat{\theta}} \subset B_{\hat{\theta}}$, take $h \in E_{\hat{\theta}}$. Using the Cauchy-Schwartz inequality and , we get $$\begin{aligned}
\left( h(x) - \hat{\theta}(x) \right)^2 &= \left(\sum_{j=1}^\infty \langle h-\hat{\theta}, v_j \rangle v_j(x) \right)^2
= \left(\sum_{j=1}^\infty \frac{\langle \sqrt{N}(h-\hat{\theta}), v_j \rangle}{c_j} \frac{1}{\sqrt{N}}c_jv_j(x) \right)^2 \\
& \leq \sum_{j=1}^\infty \frac{\langle \sqrt{N}(h-\hat{\theta}), v_j \rangle^2}{c_j^2} \sum_{j=1}^{\infty} \frac{1}{N}c_j^2 v_j^2(x) \\
& \leq \sum_{j=1}^{\infty} \frac{\xi}{N}c_j^2 v_j^2(x) \equiv r^2(x),
\end{aligned}$$ for $x$ *almost everywhere*, which then implies $h \in B_{\hat{\theta}}$ and thus $E_{\hat{\theta}} \subset B_{\hat{\theta}}$ as desired.
\[lemma:DiffinNormExpansion\] Define $\alpha_1 := \lambda_1 - \lambda_2$ and $\alpha_j := \min\{ \lambda_j - \lambda_{j+1}, \lambda_{j-1} - \lambda_{j} \}$ for $j = 2, \dots$. Then with probability 1 $$\begin{aligned}
\label{e:DiffinNormExpansion}
\| \hat v_j - v_j \| \leq \frac{ 2 \sqrt{2} \| \hat C_\theta - C_\theta\|_{{{\mathcal L}}}}{ \alpha_j}
\qquad \text{and} \qquad
| \hat \lambda_j - \lambda_j| \leq \| \hat C_\theta - C_\theta\|_{{{\mathcal L}}}.
\end{aligned}$$
See @hkbook.
Our aim is to show that for any $x \in \hat{E}_{\hat{\theta}}$ there exists $y \in E_{\hat{\theta}}$, s.t. $\| y - x\|$ is bounded by the RHS of . For any $x \in \hat{E}_{\hat{\theta}}$, take $y \in {{\mathcal H}}$ s.t., $y = \hat{\theta} + \sum_{j=1}^{J} \langle x - \hat{\theta}, \hat{v}_j \rangle v_j$. We then have that $$\sum_{j=1}^\infty \frac{\langle y - \hat{\theta} , v_j\rangle^2}{N^{-1} c_j^2}
= \sum_{j=1}^{J} \frac{\langle x - \hat{\theta} , \hat{v}_j\rangle^2}{N^{-1}c_j^2}
\leq \xi,$$ which implies that $y \in E_{\hat{\theta}}$ follows from $x \in \hat{E}_{\hat{\theta}}.$ Turning to the difference between $x$ and $y$ we have that $$\begin{aligned}
\| y - x \|^2
&= \| (y - \hat{\theta}) - (x - \hat{\theta}) \|^2 \\
&= \left\| \sum_{j=1}^{J} \langle x - \hat{\theta}, \hat{v}_j \rangle v_j - \sum_{j=1}^{J} \langle x - \hat{\theta} , \hat{v}_j \rangle \hat{v}_j \right\|^2
= \left\| \sum_{j=1}^{J} \langle x- \hat{\theta} , \hat{v}_j \rangle (v_j - \hat{v}_j) \right\|^2.
\end{aligned}$$ From Cauchy-Schwarz inequality, the above is bounded by $$\sum_{j=1}^J \langle x- \hat{\theta} , \hat{v}_j \rangle ^2 \sum_{j=1}^J \| v_j - \hat v_j\|^2
\leq N^{-1} \xi c_1^2 \sum_{j=1}^J \| v_j - \hat v_j\|^2,$$ which holds uniformly in $x$. Using Lemma \[lemma:DiffinNormExpansion\] we get $$\begin{aligned}
\sum_{j=1}^{J} \left\| v_j - \hat{v}_j \right\|^2
& \leq \sum_{j=1}^{J} \frac{8 \|\hat C_\theta - C_\theta\|_{{\mathcal L}}^2 }{\alpha_j^{2}}.
\end{aligned}$$ Therefore, $$\begin{aligned}
\rho(\hat{E}_{\hat{\theta}},E_{\hat{\theta}}) ^2 \leq \sum_{j=1}^{J} \frac{8 \xi c_1^2 \|\hat C_\theta - C_\theta\|_{{\mathcal L}}^2 }{N \alpha_j^{2}}
\end{aligned}$$ as claimed.
Again, we aim to show that for any $y \in E_{\hat{\theta}}$ there exists $x \in \hat{E}_{\hat{\theta}}$, such that $\| y - x\|$ achieves the claimed bound. Take $y_J := \hat{\theta} + \sum_{j=1}^J \langle y - \hat{\theta}, v_j \rangle v_j$, and $x := \hat{\theta} + \sum_{j=1}^J \langle y - \hat{\theta}, v_j \rangle \hat{v}_j$. As before, $x \in \hat{E}_{\hat{\theta}}$ follows from $y \in E_{\hat{\theta}}$. We then use a triangle inequality to obtain $$\begin{aligned}
\label{e:local1}
\|y - x\| \leq \|y - y_J \| + \|y_J - x \|.
\end{aligned}$$ The first term is bounded by $$\begin{aligned}
\|y - y_J\|^2
& = \sum_{j=J+1}^\infty \langle y - \hat{\theta}, v_j\rangle^2
= \sum_{j=J+1}^\infty c_j^2 \frac{\langle y - \hat{\theta}, v_j\rangle^2}{c_j^2}
\leq c_{J}^2 \sum_{j=J+1}^\infty \frac{\langle y - \hat{\theta}, v_j\rangle^2}{c_j^2}
\leq c_J^2 N^{-1} \xi,
\end{aligned}$$ uniformly in $y$. Using the same arguments as in Theorem \[thm:Convergence1\] we obtain $$\begin{aligned}
\|y_J - x\|^2 & = \left\| \sum_{j=1}^J \langle y - \hat{\theta} , v_j \rangle (v_j - \hat{v}_j) \right\|^2
\leq \sum_{j=1}^{J} \frac{8 \xi c_1^2 \|\hat C_\theta - C_\theta\|_{{\mathcal L}}^2 }{N \alpha_j^{2}}
\end{aligned}$$ uniformly in $y$ as well. Therefore, $$\begin{aligned}
\rho(E_{\hat{\theta}}, \hat{E}_{\hat{\theta}}) = & \left[c_J^2 N^{-1} \xi \right]^{\frac{1}{2}}+
\left[ \sum_{j=1}^{J} \frac{8 \xi c_1^2 \|\hat C_\theta - C_\theta\|_{{\mathcal L}}^2}{N \alpha_j^{2}} \right]^{\frac{1}{2}}
\end{aligned}$$ as claimed.
\[l:sum\] Let $\alpha > 0$, then as $J \to \infty$ $$\sum_{j = 1}^J j^{\alpha} \approx \frac{J^{\alpha + 1}}{\alpha+1}
\quad \text{ i.e. } \quad
\frac{\sum_{j = 1}^J j^{\alpha}}{J^{\alpha + 1}/(\alpha+1) } \to 1.$$
We can rewrite $$\sum_{j = 1}^J j^{\alpha} = J^{\alpha + 1} \sum_{j = 1}^J \frac{1}{J } \left(\frac{j}{J}\right)^{\alpha}.$$ Using the definition of the Riemann integral we have that $$J^{\alpha + 1} \sum_{j = 1}^J \frac{1}{J } \left(\frac{j}{J}\right)^{\alpha} \approx J^{\alpha + 1} \int_{0}^1 x^{\alpha} \ dx =
\frac{J^{\alpha + 1}}{\alpha + 1},$$ which is the desired result.
We begin by analyzing the sum of the $\alpha_j^{-2}$. Applying Assumption \[a:rate1\] and Lemma \[l:sum\] we have that $$\begin{aligned}
\sum_{j=1}^J \alpha_j^{-2} & \leq K \sum_{j=1}^J j^{2\delta+2}
\approx \frac{K}{2\delta+3} J^{2\delta+3}.
\end{aligned}$$ This implies that is bounded by $$\begin{aligned}
\label{e:rate1.1}
{{\mathbb E}}\rho(\hat E_{\hat \theta}, E_{\hat \theta})^2
\leq \sum_{j=1}^J \frac{8 \xi c_1^2 {{\mathbb E}}\| \hat C_\theta - C_\theta \|_{{{\mathcal L}}}^2}{N \alpha_j^2}
\leq J^{{2 \delta + 3}} N^{-2} O(1).
\end{aligned}$$ The second distance can be bounded using the simple scalar relationship $(a + b)^2 \leq 2a^2 + 2 b^2$, which gives $$\rho(E_{\hat \theta}, \hat E_{\hat \theta})^2
\leq 2 c_J^2 N^{-1} \xi + \sum_{j=1}^J \frac{16 \xi c_1^2 \| \hat C_\theta - C_\theta\|_{{{\mathcal L}}}^2}{N \alpha_j^2}.$$ Using Assumption \[a:rate1\] we then have that $${{\mathbb E}}[\rho(E_{\hat \theta}, \hat E_{\hat \theta})^2]
= J^{-2\gamma} N^{-1} O(1) + J^{{2 \delta + 3}} N^{-2} O(1).$$ Setting the two errors equal to each other yields $$J^{-2\gamma} N^{-1} = J^{{2 \delta + 3}} N^{-2}
\Longrightarrow J = N^{\frac{1}{2\delta + 3 + 2 \gamma}}.$$ This yields an overall error of $${{\mathbb E}}d_H(E_{\hat \theta}, \hat E_{\hat \theta})^2
\leq N^{- \left(2 - \frac{2\delta + 3 }{2\delta + 3 + 2\gamma} \right)} O(1) .$$ as claimed.
Recall that $c_j^2 = \hat \lambda_j^{1/2}$. Denote $\tilde c_j^2 =\lambda_j^{1/2}$ and the resulting $J$ dimensional confidence region as $\tilde E_{\hat \theta}$. Here $\tilde E_{\hat \theta}$ acts an intermediate step between $\hat E_{\hat \theta}$ and $E_{\hat \theta}$. When using $\tilde E_{\hat \theta}$ we have that $ 2 \gamma = \delta /2 $. The rate in Theorem \[thm:ConvergenceOverall\] then has an exponent of $$- \left( 2 - \frac{2\delta + 3}{2\delta + 3 + \delta/2}\right)
= - \frac{6\delta + 6}{5 \delta + 6},$$ which means that $$d_H(\tilde E_{\hat \theta}, E_{\hat \theta})^2 = O_P( N^{ - \frac{6\delta + 6}{5 \delta + 6}}).$$ We will now show that the distance $ d_H(\hat E_{\hat \theta}, \tilde E_{\hat \theta})^2$ is of a smaller order, which implies that $d_H(\hat E_{\hat \theta}, E_{\hat \theta})^2$ has the same rate as $ d_H(\tilde E_{\hat \theta}, E_{\hat \theta})^2$, as desired.
Recall that if $x \in \hat E_{\hat \theta}$ then it satifies $$\sum_{j=1}^J \frac{\langle x - \hat \theta, \hat v_j \rangle^2 }{N^{-1} c_j^2 } < \xi.$$ We can create a scaled $x$, call it $\tilde x$ such that it is also in $\tilde E_{\hat \theta}$ by noticing that $$\sum_{j=1}^J \frac{\langle x - \hat \theta, \hat v_j \rangle^2 }{N^{-1} \tilde c_j^2 }
= \sum_{j=1}^J \frac{\langle x - \hat \theta, \hat v_j \rangle^2 }{N^{-1} c_j^2 } \frac{c_j^2}{\tilde c_j^2}
\leq \max_{j=1,\dots, J} \frac{c_j^2}{\tilde c_j^2} \sum_{j=1}^J \frac{\langle x - \hat \theta, \hat v_j \rangle^2 }{N^{-1} c_j^2 }
\leq \xi \max_{j=1,\dots, J} \frac{c_j^2}{\tilde c_j^2}.$$ So $\tilde x \in \tilde E_{\hat \theta}$ if $ \tilde x = \hat \theta + \left(\max_{j=1,\dots, J} c_j^2 / \tilde c_j^2 \right)^{-\frac{1}{2}} (x - \hat \theta)$. Now the difference between $x$ and $\tilde x$ is given by, using the simple scalar relationship $(1-\sqrt{a})^2 \leq (1-a)^2$ for $a > 0$, $$\begin{aligned}
\|x - \tilde x\|^2 = \sum_{j=1}^J \langle x - \tilde x, \hat v_j \rangle^2 & = \left(1 - \left(\max_{j=1,\dots, J} c_j^2 / \tilde c_j^2 \right)^{-\frac{1}{2}}\right)^2
\sum_{j=1}^J \langle x - \hat \theta, \hat v_j \rangle^2 \\
& \leq \left(1 - \left(\max_{j=1,\dots, J} c_j^2 / \tilde c_j^2 \right)^{-1}\right)^2
\sum_{j=1}^J \langle x - \hat \theta, \hat v_j \rangle^2 \\
& \leq \xi N^{-1} c_1^2 \left(1 - \left(\max_{j=1,\dots, J} c_j^2 / \tilde c_j^2 \right)^{-1}\right)^2 \\
& \leq \xi N^{-1} c_1^2
\max_{j=1,\dots, J} \frac{(c_j^2 - \tilde c_j^2)^2}{c_j^4} \\
& \leq \frac{\xi N^{-1} c_1^2}{c_J^{4}}
\max_{j=1,\dots, J} (c_j^2 - \tilde c_j^2)^2.
\end{aligned}$$ Using a Taylor expansion and Lemma \[lemma:DiffinNormExpansion\] one has that $ \max_{j=1,\dots, J} (c_j^2 - \tilde c_j^2)^2 = O_P(N^{-1})$ and $c_J^{-4} = O_P(\lambda_J^{-1})$. We therefore have that $$\rho(\hat E_{\hat \theta}, \tilde E_{\hat \theta})^2
= \frac{1}{N^2 \lambda_J}O_P(1).$$ Plugging in the optimal $J$ we get that $$\rho(\hat E_{\hat \theta}, \tilde E_{\hat \theta})^2 =
J^{\delta}N^{-2} O_P(1)
= N^{- \left(2 - \frac{\delta}{2 \delta + 3 + \delta/2} \right)} O_P(1)
= N^{- \frac{8 \delta +12 }{5 \delta + 6} } O_P(1)
= N^{- \frac{6 \delta +6 }{5 \delta + 6} } o_P(1).$$ Nearly identical arguments will yield the same result for the reverse $\rho(\tilde E_{\hat \theta}, \hat E_{\hat \theta})^2$. Thus $d(\hat E_{\hat \theta}, \tilde E_{\hat \theta})^2$ is of a lower order than $d(\tilde E_{\hat \theta}, E_{\hat \theta})^2$ and the claim holds.
Other Criteria {#section:OtherCriteria}
==============
Both in hyper-ellipsoid and hyper-rectangular regions, there exists infinitely many options to find $\{c_j\}$ that determines their shapes. The following introduces a few more criteria that may be found to be interesting.
Hyper-Ellipsoid
---------------
In hyper-ellipsoid, one may take $$c_j^2 = \left( \sum_{i=j}^\infty \lambda_i \right) ^ {1/2},$$ i.e. the square root of the tail sum of the eigenvalues. Recall that $\sum_{j=1}^\infty \lambda_j < \infty$ since it is equal to the trace of the covariance operator. It is therefore clear that $c_j \to 0$ and the region is compact. What is not as obvious is that one also has $\sum_{j=1}^\infty {\lambda_j}{c^{-2}_j} < \infty$, which means that the resulting $W_\theta$ is a random variable with finite mean and the resulting region can obtain the proper coverage. Showing this is actually an interesting real analysis exercise and we refer the reader to @rudin:1976 [p. 80] for more details. We denote this region as $E_{c1}$
Hyper-Rectangle
---------------
For hyper-rectangular regions, the first possibility is to use the same $\{c_j\}$ used in hyper-ellipsoid and find $\{z_j\}$ numerically. Since $\{z_j\}$ is uniquely determined by $\xi$ once $\{c_j\}$ is given, one can easily search for $\xi$ that satisfies . We will denote the region achieved in this way using $c_j^2 = \left( \sum_{i=j}^\infty \lambda_i \right) ^ {1/2}$ of $E_{c1}$ as $R_{c1}$, and the one uses $c_j^2 = \lambda_j^{1/2}$ of $R_{c}$ as $E_{c}$
Next approach we considered is to find $\{z_j\}$ that minimizes $\sup \{ \| h - \hat{\theta} \|^2 : h \in R_{\hat{\theta}} \}$, i.e. the distance between the farthest point of the region from the center. It is equivalent to finding a rectangular region that has smallest ${{\mathcal H}}$ norm. The problem reduces to minimizing $\frac{1}{N} \sum_{j=1}^{\infty} \lambda_j z_j^2$ under the constraint $\sum_{j=1}^{\infty}\log \Phi_{sym}(z_j) = \log(1-\alpha)$. The solution for this problem can be found in the Appendices Section \[Appendix:FindZ1\], and the following summarizes the steps to follow :
1. Define a function $f(z) := \exp(z^2 / 2) \Phi_{sym}(z)$
2. Define $f^{-1}(\cdot)$, the inverse of $f(\cdot)$. This can be achieved numerically by univariate optimization in practice.
3. Find $M^* = \arg\min_{M}
\left| \prod_{j} \Phi_{sym}\left\{ f^{-1} \left( \frac{M}{\sqrt{2\pi} \lambda_j}\right) \right\} - (1-\alpha) \right|$. This also can be achieved numerically by univariate optimization.
4. Take $z_j = f^{-1}\left( \frac{M^*}{\sqrt{2\pi}\lambda_j}\right)$ for each $j$.
We will denote this region as $R_{z1}$
Yet another approach, which is much simpler, is to start from $\sum_{j=1}^{\infty}\log \Phi_{sym}(z_j) = \log(1-\alpha) $ and distribute $\log(1-\alpha)$ among $\Phi_{sym}(z_j)$’s, possibly assigning more weight to early $j$’s to narrow down length along the early eigenfunctions. For example, $\{z_j\}$ that satisfies $\log \Phi_{sym}(z_j) = \frac{\lambda_j^\rho}{\sum_{k=1}^\infty \lambda_k^\rho} \log(1-\alpha)$ is an intuitive option as long as $\sum_j \lambda_j^\rho < \infty$ is satisfied for some positive $\rho$. This then has a closed form solution $z_j = \Phi_{sym}^{-1}\left[ \exp \left( {\frac{\lambda_j^\rho}{\sum_{k=1}^\infty \lambda_k^\rho} \log(1-\alpha)} \right) \right]$ for each $j$. Therefore, any $\rho \geq 1$ can be used regardless the smoothness of the process, and $\rho = 1$ leads to what we denoted as $R_{z}$ in the main text. We empirically observed that $\{z_j\}$ found this way is similar to $\{z_j\}$ found from $R_{z1}$ above.
Small Sample Version {#Appendix:SmallSample}
====================
When $C_\theta$ is unknown, we relied on the consistency of $\hat{C}_\theta$ and therefore replaced $\{( v_j , \lambda_j) \}_{j=1}^{\infty}$ with $\{( \hat{v}_j , \hat{\lambda}_j) \}_{j=1}^{J}$ to construct empirical versions of confidence regions. Although this approach is still valid, one might want to look at an alternative if $N$ is too small. We suggest here a simple technique to respond to this concern. To utilize an explicit form of $\hat{C}_\theta$, we will turn our attention to a special case of $\hat{\theta}$ in this section only. Note, however, that same idea can be applied in other situations too. Consider $\hat{\theta} = \bar{X}$, where $\bar{X} = N^{-1}\sum_{i=1}^N X_i$, and $X_i \stackrel{iid}{\sim} {{\mathcal N}}(\theta, C_\theta)$. The standard covariance estimator $\hat{C}_\theta$ is $$\begin{aligned}
\hat{C}_\theta = \frac{1}{N-1} \sum_{i=1}^{N} (X_i - \bar{X}) \otimes (X_i - \bar{X}).\end{aligned}$$ By KL expansion , we can write $X_i = \theta + \sum_{j=1}^\infty \sqrt{\lambda_j}Z_{ij}v_j$ where $Z_{ij} \stackrel{iid}{\sim}{{\mathcal N}}(0,1)$. Denoting $\bar{Z}_{\cdot j} := N^{-1}\sum_{i=1}^N Z_{ij}$, $$\begin{aligned}
\hat{C}_\theta &= \frac{1}{N-1} \sum_{i=1}^N \left( \sum_{j=1}^\infty \sqrt{\lambda_j}(Z_{ij} -\bar{Z}_{\cdot j}) v_j \right) \otimes
\left( \sum_{k=1}^\infty \sqrt{\lambda_{k}}(Z_{ik} -\bar{Z}_{\cdot k}) v_{k} \right) \\
&= \sum_{j=1}^\infty \sum_{k=1}^\infty \sqrt{\lambda_j \lambda_{k}} \frac{1}{N-1} \sum_{i=1}^N (Z_{ij} -\bar{Z}_{\cdot j})(Z_{ik} -\bar{Z}_{\cdot k}) \left( v_j \otimes v_{k} \right) \end{aligned}$$ gives basis expansion of $\hat{C}_\theta$ using $\{(v_j \otimes v_k)\}_{j,k}$, an orthonormal basis of ${{\mathcal S}}$, other than the common expansion of $\hat{C}_\theta = \sum_{j=1}^{N-1} \hat{\lambda}_j (\hat{v}_j \otimes \hat{v}_j)$.
Therefore, the coefficient of $\hat{C}_\theta$ with respect to $(v_j \otimes v_j)$ becomes $$\begin{aligned}
\label{e:tildeLambda}
\tilde{\lambda}_j := \langle \hat{C}_\theta, (v_{j} \otimes v_{j}) \rangle_{{{\mathcal S}}} = \frac{\lambda_j}{N-1} \sum_{i=1}^{N} (Z_{ij} - \bar{Z}_{\cdot j})^2\end{aligned}$$ Observer that $V_j := \sum_{i=1}^{N} (Z_{ij} - \bar{Z}_{\cdot j})^2 \stackrel{d}{=} \chi^2_{N-1}$ and is independent from $Z_j := \sqrt{N }\bar{Z}_{\cdot j} \stackrel{d}{=} {{\mathcal N}}(0,1)$. Therefore, $T_j := Z_j / \sqrt{V_j/(N-1)}$ follows $t$ distribution with $N-1$ degree of freedom, which are also mutually independent among $j$’s. Finally, using $\sqrt{\lambda_j} = \sqrt{\tilde{\lambda}_j}/\sqrt{V_j / (N-1)}$ in , we achieve expansion $$\begin{aligned}
\begin{split}
\label{e:ExpansionT}
\sqrt{N} (\hat{\theta} - \theta)
= \sum_j \sqrt{\lambda_j}Z_j v_j
= \sum_j \sqrt{\tilde{\lambda}_j} T_j v_j.
\end{split}\end{aligned}$$
Expansion implies that we can work with exact distribution with the knowledge of eigenfunctions, not both eigenfunctions and eigenvalues. In practice where eigenfunctions are not known, we will still replace $\{(\tilde{\lambda}_j, v_j)\}$ with $\{(\hat{\lambda}_j, \hat{v}_j)\}$. Note, however, that this approach now depend only on the consistency of $\{\hat{v}_j\}$, not those of both $\{\hat{v}_j\}$ and $\{\hat{\lambda}_j\}$. Simulation study in Section \[section:Simulation\] confirms that this approach is actually appealing.
The rest of this section discusses how to obtain confidence regions utilizing expansion $\eqref{e:ExpansionT}$. All the following implementations need replacement of $\{\lambda_j\}$ with $\{\tilde{\lambda}_j\}$ when $\{v_j\}$ is known, although $\{\hat{\lambda}_j\}$ will be used for both $\{\lambda_j\}$ and $\{\tilde{\lambda}_j\}$ when $\{v_j\}$ or covariance operator is unknown.
#### Hyper-Rectangular Regions
For hyper-rectangular regions, one can consider (at least) two options. First option is to maintain the same ratio of radii. One replace $z_j$ with $t_j := c z_j$, where $c > 1$ can be numerically found to satisfy $\prod_{j} {{\mathbb P}}(|T_j| \leq t_j) = 1-\alpha$. Another option is to take $\{t_j\}$ such that $\mathbb{P} (|Z_j| \leq z_j) = \mathbb{P} (|T_j| \leq t_j)$ for each $j$. Favoring the simplicity of implementation, we used the second option in our simulation study, and was denoted with suffix ‘$s$’, for example, as $R_{zs}$. Note, however, that the first option is not costly either.
#### Hyper-Ellipsoid Regions
For a hyper ellipsoid region, we observe that $$\begin{aligned}
W_{\theta} = \sum_j \frac{\lambda_j Z_j^2}{c_j^2}
= \sum_j \frac{\tilde{\lambda}_j }{c_j^2} Z_j^2 \frac{\lambda_j}{\tilde{\lambda_j}}
= \sum_j \frac{\tilde{\lambda}_j }{c_j^2} \frac{Z_j^2}{V_j / (N-1)}. \end{aligned}$$ By defining $F_j := \frac{Z_j^2}{V_j / (N-1)}$, $W_\theta = \sum_j \frac{\tilde{\lambda}_j }{c_j^2} F_j$ follows weighted sum of independent $\mathcal{F}_{1,N-1}$ distributions, with weights being $\{\tilde{\lambda}_j c_j^{-2}\}$. This region was not implemented since no (numerical) tool for the distribution of sum of weighted $\mathcal{F}$ seemed to be available yet.
Finding $\{z_j\}$ for $R_{z1}$ {#Appendix:FindZ1}
==============================
Under the constraint of $\sum_{j=1}^{\infty}\log \Phi_{sym}(z_j) = \log(1-\alpha)$, we want to find $$\begin{aligned}
\left\{z_{j}\right\}
&= \arg \min_{\left\{z_{j}\right\}} \sup \{ \| h - \hat{\theta} \|^2 : h \in S_{\hat{\theta}} \}
= \arg \min_{\left\{z_{j}\right\}} \frac{1}{N} \sum_{j=1}^{\infty} \lambda_j z_j^2
= \arg \min_{\left\{z_{j}\right\}} \sum_{j=1}^{\infty} \lambda_j z_j^2. \end{aligned}$$ Take $
L(M, z_1, \cdots) := \sum_{j} \lambda_j z_j^2 - M \left( \sum_{j} \log \Phi_{sym}(z_j) - \log (1-\alpha) \right)
$ where $M$ is Lagrange multiplier. To minimize $L$, set $$\begin{aligned}
\frac{\partial}{\partial{z_j}} L &= 2\lambda_j z_j -
M \{\Phi_{sym}(z_j)\}^{-1} \frac{\partial}{\partial{z_j}}\int_{-z_j}^{z_j}(2\pi)^{-1/2}e^{-x^2/2}dx \\
&= 2\lambda_j z_j - M \{\Phi_{sym}(z_j)\}^{-1} \sqrt{\frac{2}{\pi}}e^{-z_j^2/2} = 0.\end{aligned}$$ We then achieve $
M = \sqrt{2\pi} \lambda_j e^{z_j^2/2}z_j \Phi_{sym}(z_j)
$ for each $j$. By defining a function $f(z) := e^{z^2/2}z\Phi_{sym}(z)$ which is increasing in $z > 0$, we can write $z_j$ as $z_j = f^{-1}\left(M / \sqrt{2\pi}\lambda_j \right)$. Therefore, each $z_j$ is can be uniquely determined once $M$ is found. We can numerically search for such $M$ that satisfies $$\begin{aligned}
-\frac{\partial}{\partial{M}} L &= \sum_{j} \log \Phi_{sym}(z_j) - \log (1-\alpha)= 0\end{aligned}$$ This is not computationally difficult nor expensive since $\frac{\partial}{\partial{M}} L$ is monotone in $M$ and we can easily find upper and lower bounds for $M$ from those for $z_1$. For the range of $z_1$, we require $z_1 \geq \Phi_{sym}^{-1}(1-\alpha) \equiv z_{1,lb}$ where $\Phi_{sym}^{-1}$ is inverse of $\Phi_{sym}$. We also want $z_1$ to be smaller than at least average of $z_i$’s, so we can take $z_{1,ub}$, the upper bound of $z_1$, to satisfy $\prod_{j=1}^p \Phi_{sym}(z_{1,ub}) = (\Phi_{sym}(z_{1,ub}))^p = 1-\alpha$, i.e., $z_{1,ub} = \Phi_{sym}^{-1}\left\{(1-\alpha)^{1/p}\right\}$. This gives a practical range for $M$ as $
\sqrt{2\pi} \lambda_1 f(z_{1,lb}) \leq M \leq \sqrt{2\pi} \lambda_1 f(z_{1,ub})
$.
#### Finding p-value
Observe that $M = \sqrt{2\pi} \lambda_j f(z_j)$ hold for all $j$. We can then take $M^\star := \sup_j \{\sqrt{2\pi} \lambda_j f(z_j)\}$ as our statistic to get p-value from, namely $1 - \prod_{j=1}^p \Phi_{sym}(z_j^\star)$ where $z_j^\star := f^{-1}\left(M^\star / \sqrt{2\pi}\lambda_j \right)$.
Calculating $P$-values for HT from Confidence Regions {#supp:PvalueRectangle}
=====================================================
This section describes steps to find $p$-values from the hypothesis testing baseds on proposed hyper-ellipsoid and hyper-rectangular regions. Note that $p$-value can be interpreted as the smallest $\alpha$ that makes the confidence region to cover $\theta_0$. For estimated regions, we replace $\{(v_j, \lambda_j)\}$ or $\{(v_j, \tilde{\lambda}_j)\}$ with $\{(\hat{v}_j, \hat{\lambda}_j)\}$.
Hyper-ellipsoid Regions
-----------------------
We find the observed test statistic $W^* := \sum_{j} N \langle \hat{\theta} - \theta_0 , v_j \rangle^2 / c_j^2$ and get $p$-value as ${{\mathbb P}}\left(W_\theta \geq W^* \right)$ where $W_\theta$ is a weighted sum of $\chi^2$ random variables with weights $\{\lambda_j/c_j^2\}$.
Hyper-rectangular Regions
-------------------------
For hyper-rectangular regions, we first need to find z-score for each $j$ as $z_j^* = \langle \sqrt{N}(\hat{\theta} - \theta_0), v_j \rangle / \sqrt{\lambda_j}$. The next step differs by each criterion.
1. When $\{c_j\}$ was determined first ($R_c$, $R_{c1}$) : We find $\sqrt{\xi}^\star = \sup_j z^*_j \sqrt{\lambda_j}c_j^{-1}$, which serves as the test statistic, and get $p$-value as $1 - \prod_j \Phi_{sym}(z_j^\star)$ where $z_j^{\star} := \frac{c_j}{\sqrt{\lambda_j} } \sqrt{\xi}^{\star}$.
2. $R_{z}$ : We utilize $1- \alpha = \exp\left( \frac{\sum_k \lambda_k}{\lambda_j} \log \Phi_{sym}(z_j)\right)$ for each $j$ and therefore use\
$\inf_j \left[ 1 - \exp\left( \frac{\sum_k \lambda_k}{\lambda_j} \log \Phi_{sym}(z^*_j)\right) \right]$ as our p-value.
3. $R_{z1}$ : We find $M^\star = \sup_j \sqrt{2\pi}\lambda_jf(z_j^*)$, which serves as the test statistic, and get $p$-value as $1 - \prod_j \Phi_{sym}(z_j^\star)$ where $z_j^\star = f^{-1}(\frac{M^\star}{\sqrt{2\pi}\lambda_j})$.
For the small sample versions of hyper-rectangular regions, we find $t$-score for each $j$ as $t_j^* := \langle \sqrt{N}(\hat{\theta} - \theta_0), v_j \rangle / \sqrt{\tilde{\lambda}_j}$ and convert them into $z$-scores such that $\Phi_{sym}(z_j^*)={{\mathbb P}}(|T_j| \leq t_j^*)$ for each $j$. We can then follow the same step for each region as described above.
Smoothing by penalty on $2^{nd}$ derivative and it’s covariance
===============================================================
Let $\{X_i\}_{i=1}^{N} \subset {{\mathcal H}}= L^2[0,1]$ be an independent sample with mean $\mu$. Using a basis expansion we can write $X_i \approx \sum_{j=1}^J x_{ij}e_j$ and $\mu = \sum_{j=1}^J m_{j}e_j$ where $\{e_j\}_{j=1}^J$ is a basis of ${{\mathcal H}}$, and satisfy $\|e_j\| = 1$ and the second derivative $e_j^{(2)}$ exists for all $j$. We assume that $J$ is large so that the degree of smoothing on $\mu$ is controlled primarily through the penalty. A standard smoothed estimate for $\mu$ can be found by penalizing the second derivative $$\begin{aligned}
\hat \mu = \arg \min_\mu N^{-1} \sum_{i=1}^N \| X_i - \mu \|^2 + \lambda \| \mu^{(2)} \| \end{aligned}$$ where $\lambda$ is the smoothing parameter. Note $$\begin{aligned}
\| X_i - \mu \|^2 &= \left\| \sum_{j} (x_{ij} - m_j)e_j \right \|^2 =
\left\langle \sum_{j} (x_{ij} - m_j)e_j , \sum_{j'} (x_{ij'} - m_{j'})e_{j'} \right\rangle \\
&= \sum_j \sum_{j'} ( x_{ij}x_{ij'} - x_{ij}m_{j'} - x_{ij'}m_j + m_jm_{j'}) \langle e_j, e_{j'} \rangle
= {{\bf x}}_i'B{{\bf x}}_i - 2 {{\bf x}}_i' B {{\bf m}}+ {{\bf m}}' B {{\bf m}}\end{aligned}$$ where ${{\bf x}}_i = (x_{i1}, \dots , x_{iJ})'$, ${{\bf m}}= (m_{i}, \dots , m_{J})'$, and $B_{jj'} = \langle e_j, e_{j'} \rangle $. Likewise, $$\begin{aligned}
\| \mu^{(2)} \|^2 &= \left\| \sum_{j} m_j e_j^{(2)} \right \|^2 =
\left\langle \sum_{j} m_j e_j^{(2)}, \sum_{j'} m_{j'} e_{j'}^{(2)} \right\rangle
= \sum_j \sum_{j'} m_jm_{j'} \langle e_j^{(2)} , e_{j'}^{(2)} \rangle
= {{\bf m}}' D {{\bf m}}\end{aligned}$$ where $D_{jj'} = \langle e_j^{(2)}, e_{j'}^{(2)} \rangle $. Therefore, the target function can be expressed as $$N^{-1} \sum_{i=1}^N \left( {{\bf x}}_i'B{{\bf x}}_i - 2 {{\bf x}}_i' B {{\bf m}}\right) + {{\bf m}}' B {{\bf m}}+ \lambda {{\bf m}}' D {{\bf m}}$$ using familiar matrix notations. By taking $\partial Q / \partial {{\bf m}}= 0$ and defining $\bar{{{\bf x}}} := N^{-1}\sum_{i=1}^N {{\bf x}}_i$, we get $$\hat{{{\bf m}}} = (B + \lambda D)^{-1} B \bar{{{\bf x}}}$$ which then gives estimate of $\mu$ as $\hat{\mu} = \sum_{j=1}^J \hat{m}_j e_j$. By defining $A := (B + \lambda D)^{-1} B$, we get the covariance of $\hat{{{\bf m}}}$ as $\Sigma_{\hat{{{\bf m}}}} := A \Sigma_{\bar{{{\bf x}}}} A'$ where $\Sigma_{\bar{{{\bf x}}}} := \operatorname{Cov}(\bar{{{\bf x}}})$. Finally, the covariance operator of $\hat{\mu}$, $C_{\hat{\mu}}$, takes a bivariate function form of $
C_{\hat{\mu}}(t,s) = \sum_j \sum_{j'} \operatorname{Cov}(\hat{m}_j,\hat{m}_{j'})e_j(t)e_{j'}(s)
$, or $
C_{\hat{\mu}} = \sum_j \sum_{j'} \operatorname{Cov}(\hat{m}_j,\hat{m}_{j'})(e_j \otimes e_{j'})
$ as an operator.
Power Tables
============
The following four tables come from Subsection 4.1.2
\[supplement:PowerTable\]
**1. Shift** 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Avg.
-------------------- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
$\hat{E}_{norm}$ 0.09 0.16 0.29 0.47 0.63 0.78 0.88 0.96 0.98 1.00 0.62
$\hat{E}_{PC(3)}$ 0.09 0.14 0.24 0.37 0.52 0.67 0.81 0.91 0.95 0.99 0.57
$\hat{E}_{c1}$ 0.09 0.16 0.30 0.47 0.64 0.79 0.89 0.96 0.98 1.00 0.63
$\hat{E}_{c}$ 0.10 0.17 0.31 0.48 0.64 0.79 0.89 0.96 0.98 1.00 0.63
$\hat{R}_{z1}$ 0.09 0.15 0.28 0.44 0.60 0.75 0.86 0.94 0.98 0.99 0.61
$\hat{R}_{z}$ 0.09 0.15 0.28 0.44 0.60 0.75 0.87 0.95 0.98 0.99 0.61
$\hat{R}_{z1s}$ 0.06 0.11 0.23 0.38 0.54 0.70 0.83 0.93 0.97 0.99 0.57
$\hat{R}_{zs}$ 0.06 0.11 0.23 0.38 0.55 0.71 0.83 0.93 0.97 0.99 0.58
$\hat{B}_{s}$ 0.12 0.20 0.34 0.51 0.66 0.80 0.90 0.96 0.98 1.00 0.65
**2. Scale** 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Avg.
$\hat{E}_{norm}$ 0.06 0.09 0.13 0.20 0.30 0.41 0.54 0.68 0.79 0.88 0.41
$\hat{E}_{PC(3)}$ 0.08 0.13 0.22 0.35 0.51 0.65 0.80 0.89 0.95 0.98 0.56
$\hat{E}_{c1}$ 0.07 0.11 0.18 0.28 0.42 0.56 0.71 0.84 0.92 0.96 0.51
$\hat{E}_{c}$ 0.08 0.13 0.20 0.32 0.47 0.61 0.76 0.87 0.94 0.97 0.53
$\hat{R}_{z1}$ 0.08 0.12 0.19 0.29 0.41 0.56 0.71 0.83 0.91 0.95 0.50
$\hat{R}_{z}$ 0.08 0.11 0.19 0.28 0.41 0.56 0.70 0.82 0.91 0.95 0.50
$\hat{R}_{z1s}$ 0.05 0.08 0.12 0.20 0.32 0.46 0.61 0.76 0.86 0.92 0.44
$\hat{R}_{zs}$ 0.05 0.07 0.12 0.20 0.32 0.46 0.61 0.75 0.85 0.92 0.44
$\hat{B}_{s}$ 0.10 0.14 0.22 0.33 0.47 0.60 0.74 0.85 0.92 0.96 0.53
**3. Local Shift** 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Avg.
$\hat{E}_{norm}$ 0.06 0.07 0.08 0.10 0.14 0.19 0.26 0.36 0.49 0.62 0.23
$\hat{E}_{PC(3)}$ 0.08 0.12 0.20 0.33 0.48 0.65 0.78 0.87 0.93 0.96 0.54
$\hat{E}_{c1}$ 0.07 0.10 0.18 0.35 0.63 0.89 0.98 1.00 1.00 1.00 0.62
$\hat{E}_{c}$ 0.09 0.14 0.31 0.61 0.88 0.99 1.00 1.00 1.00 1.00 0.70
$\hat{R}_{z1}$ 0.09 0.17 0.40 0.69 0.91 0.99 1.00 1.00 1.00 1.00 0.72
$\hat{R}_{z}$ 0.09 0.17 0.39 0.68 0.91 0.98 1.00 1.00 1.00 1.00 0.72
$\hat{R}_{z1s}$ 0.04 0.07 0.19 0.42 0.72 0.92 0.99 1.00 1.00 1.00 0.64
$\hat{R}_{zs}$ 0.04 0.07 0.18 0.41 0.71 0.91 0.98 1.00 1.00 1.00 0.63
$\hat{B}_{s}$ 0.10 0.13 0.21 0.31 0.45 0.61 0.75 0.86 0.94 0.97 0.53
: $N=25, \ \nu=1/2$
**1. Shift** 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Avg.
-------------------- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
$\hat{E}_{norm}$ 0.08 0.15 0.27 0.41 0.57 0.73 0.84 0.92 0.97 0.99 0.59
$\hat{E}_{PC(3)}$ 0.10 0.16 0.25 0.36 0.51 0.67 0.80 0.89 0.95 0.98 0.57
$\hat{E}_{c1}$ 0.09 0.16 0.27 0.42 0.58 0.74 0.85 0.93 0.97 0.99 0.60
$\hat{E}_{c}$ 0.09 0.16 0.28 0.42 0.58 0.74 0.85 0.93 0.97 0.99 0.60
$\hat{R}_{z1}$ 0.09 0.15 0.26 0.41 0.56 0.72 0.84 0.92 0.96 0.99 0.59
$\hat{R}_{z}$ 0.09 0.15 0.26 0.41 0.56 0.72 0.84 0.92 0.97 0.99 0.59
$\hat{R}_{z1s}$ 0.06 0.12 0.22 0.35 0.50 0.67 0.81 0.89 0.95 0.98 0.56
$\hat{R}_{zs}$ 0.06 0.12 0.22 0.36 0.51 0.67 0.81 0.90 0.95 0.98 0.56
$\hat{B}_{s}$ 0.09 0.17 0.28 0.43 0.59 0.75 0.86 0.93 0.97 0.99 0.61
**2. Scale** 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Avg.
$\hat{E}_{norm}$ 0.07 0.09 0.13 0.18 0.27 0.37 0.49 0.61 0.73 0.83 0.38
$\hat{E}_{PC(3)}$ 0.11 0.18 0.29 0.43 0.60 0.75 0.87 0.94 0.98 0.99 0.61
$\hat{E}_{c1}$ 0.08 0.12 0.19 0.30 0.45 0.62 0.77 0.89 0.95 0.98 0.53
$\hat{E}_{c}$ 0.08 0.12 0.20 0.31 0.46 0.62 0.77 0.89 0.95 0.98 0.54
$\hat{R}_{z1}$ 0.08 0.13 0.21 0.33 0.48 0.64 0.79 0.89 0.95 0.98 0.55
$\hat{R}_{z}$ 0.08 0.13 0.21 0.32 0.48 0.64 0.78 0.89 0.95 0.98 0.55
$\hat{R}_{z1s}$ 0.06 0.09 0.15 0.24 0.39 0.54 0.70 0.83 0.92 0.97 0.49
$\hat{R}_{zs}$ 0.06 0.09 0.15 0.24 0.38 0.53 0.70 0.82 0.92 0.96 0.49
$\hat{B}_{s}$ 0.08 0.12 0.20 0.30 0.43 0.59 0.72 0.83 0.91 0.96 0.51
**3. Local Shift** 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Avg.
$\hat{E}_{norm}$ 0.07 0.06 0.08 0.09 0.11 0.15 0.19 0.24 0.33 0.43 0.17
$\hat{E}_{PC(3)}$ 0.13 0.28 0.52 0.77 0.93 0.99 1.00 1.00 1.00 1.00 0.76
$\hat{E}_{c1}$ 0.09 0.16 0.49 0.88 0.96 0.98 0.98 0.99 1.00 1.00 0.75
$\hat{E}_{c}$ 0.09 0.17 0.48 0.87 0.96 0.98 0.98 0.99 1.00 1.00 0.75
$\hat{R}_{z1}$ 0.21 0.81 0.96 0.98 0.99 1.00 1.00 1.00 1.00 1.00 0.89
$\hat{R}_{z}$ 0.20 0.81 0.96 0.98 0.99 1.00 1.00 1.00 1.00 1.00 0.89
$\hat{R}_{z1s}$ 0.10 0.63 0.94 0.97 0.98 0.99 1.00 1.00 1.00 1.00 0.86
$\hat{R}_{zs}$ 0.09 0.62 0.93 0.97 0.98 0.99 1.00 1.00 1.00 1.00 0.86
$\hat{B}_{s}$ 0.09 0.13 0.21 0.31 0.46 0.62 0.75 0.87 0.94 0.98 0.54
: $N=25, \ \nu=3/2$
**1. Shift** 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Avg.
-------------------- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
$\hat{E}_{norm}$ 0.08 0.15 0.28 0.46 0.63 0.79 0.90 0.96 0.98 1.00 0.62
$\hat{E}_{PC(3)}$ 0.08 0.12 0.20 0.34 0.51 0.67 0.82 0.91 0.96 0.99 0.56
$\hat{E}_{c1}$ 0.08 0.16 0.28 0.46 0.63 0.79 0.90 0.96 0.99 1.00 0.62
$\hat{E}_{c}$ 0.09 0.16 0.28 0.46 0.63 0.79 0.90 0.96 0.99 1.00 0.62
$\hat{R}_{z1}$ 0.08 0.14 0.25 0.42 0.60 0.76 0.88 0.95 0.98 0.99 0.61
$\hat{R}_{z}$ 0.08 0.14 0.26 0.42 0.60 0.76 0.88 0.95 0.98 0.99 0.61
$\hat{R}_{z1s}$ 0.07 0.13 0.24 0.41 0.58 0.75 0.87 0.94 0.98 0.99 0.60
$\hat{R}_{zs}$ 0.07 0.13 0.24 0.41 0.58 0.75 0.88 0.94 0.98 0.99 0.60
$\hat{B}_{s}$ 0.09 0.15 0.28 0.45 0.61 0.77 0.89 0.95 0.98 0.99 0.62
**2. Scale** 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Avg.
$\hat{E}_{norm}$ 0.06 0.08 0.13 0.19 0.29 0.41 0.55 0.69 0.81 0.89 0.41
$\hat{E}_{PC(3)}$ 0.07 0.11 0.20 0.33 0.50 0.65 0.79 0.90 0.96 0.98 0.55
$\hat{E}_{c1}$ 0.07 0.10 0.16 0.26 0.41 0.55 0.71 0.84 0.92 0.97 0.50
$\hat{E}_{c}$ 0.07 0.11 0.18 0.29 0.45 0.60 0.75 0.86 0.94 0.98 0.52
$\hat{R}_{z1}$ 0.07 0.10 0.17 0.27 0.41 0.55 0.70 0.83 0.92 0.96 0.50
$\hat{R}_{z}$ 0.07 0.10 0.17 0.26 0.41 0.54 0.70 0.83 0.92 0.96 0.50
$\hat{R}_{z1s}$ 0.06 0.09 0.15 0.25 0.38 0.52 0.68 0.81 0.91 0.96 0.48
$\hat{R}_{zs}$ 0.06 0.09 0.15 0.24 0.38 0.52 0.68 0.81 0.91 0.96 0.48
$\hat{B}_{s}$ 0.07 0.10 0.18 0.28 0.43 0.56 0.71 0.83 0.91 0.96 0.50
**3. Local Shift** 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Avg.
$\hat{E}_{norm}$ 0.05 0.06 0.08 0.10 0.12 0.17 0.24 0.36 0.49 0.68 0.23
$\hat{E}_{PC(3)}$ 0.07 0.12 0.21 0.34 0.51 0.69 0.84 0.93 0.97 0.99 0.57
$\hat{E}_{c1}$ 0.06 0.09 0.19 0.39 0.76 0.96 1.00 1.00 1.00 1.00 0.64
$\hat{E}_{c}$ 0.07 0.17 0.47 0.89 1.00 1.00 1.00 1.00 1.00 1.00 0.76
$\hat{R}_{z1}$ 0.07 0.20 0.56 0.90 0.99 1.00 1.00 1.00 1.00 1.00 0.77
$\hat{R}_{z}$ 0.07 0.19 0.55 0.90 0.99 1.00 1.00 1.00 1.00 1.00 0.77
$\hat{R}_{z1s}$ 0.06 0.15 0.46 0.84 0.99 1.00 1.00 1.00 1.00 1.00 0.75
$\hat{R}_{zs}$ 0.06 0.14 0.45 0.83 0.99 1.00 1.00 1.00 1.00 1.00 0.75
$\hat{B}_{s}$ 0.06 0.10 0.17 0.28 0.41 0.57 0.72 0.85 0.92 0.97 0.50
: $N=100, \ \nu=1/2$
**1. Shift** 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Avg.
-------------------- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
$\hat{E}_{norm}$ 0.08 0.13 0.26 0.41 0.58 0.73 0.85 0.93 0.97 0.99 0.59
$\hat{E}_{PC(3)}$ 0.07 0.11 0.20 0.32 0.48 0.63 0.78 0.89 0.95 0.98 0.54
$\hat{E}_{c1}$ 0.08 0.14 0.26 0.40 0.58 0.73 0.85 0.93 0.97 0.99 0.59
$\hat{E}_{c}$ 0.08 0.14 0.26 0.41 0.58 0.73 0.85 0.93 0.97 0.99 0.59
$\hat{R}_{z1}$ 0.07 0.13 0.25 0.39 0.56 0.71 0.84 0.92 0.97 0.99 0.58
$\hat{R}_{z}$ 0.07 0.13 0.25 0.39 0.56 0.72 0.84 0.92 0.97 0.99 0.58
$\hat{R}_{z1s}$ 0.07 0.12 0.24 0.38 0.55 0.70 0.83 0.92 0.96 0.99 0.58
$\hat{R}_{zs}$ 0.07 0.13 0.24 0.38 0.55 0.70 0.83 0.92 0.96 0.99 0.58
$\hat{B}_{s}$ 0.08 0.14 0.26 0.41 0.57 0.73 0.85 0.93 0.97 0.99 0.59
**2. Scale** 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Avg.
$\hat{E}_{norm}$ 0.06 0.08 0.12 0.18 0.25 0.35 0.47 0.61 0.73 0.84 0.37
$\hat{E}_{PC(3)}$ 0.07 0.13 0.24 0.40 0.56 0.73 0.87 0.95 0.98 0.99 0.59
$\hat{E}_{c1}$ 0.06 0.10 0.17 0.28 0.42 0.60 0.75 0.88 0.95 0.98 0.52
$\hat{E}_{c}$ 0.06 0.10 0.17 0.28 0.43 0.60 0.76 0.89 0.95 0.98 0.52
$\hat{R}_{z1}$ 0.06 0.10 0.17 0.29 0.44 0.62 0.78 0.89 0.95 0.98 0.53
$\hat{R}_{z}$ 0.06 0.10 0.17 0.29 0.44 0.61 0.77 0.89 0.95 0.98 0.53
$\hat{R}_{z1s}$ 0.06 0.09 0.16 0.28 0.42 0.59 0.75 0.88 0.95 0.98 0.52
$\hat{R}_{zs}$ 0.06 0.09 0.16 0.27 0.42 0.58 0.75 0.87 0.94 0.98 0.51
$\hat{B}_{s}$ 0.07 0.10 0.18 0.29 0.42 0.57 0.71 0.84 0.91 0.96 0.50
**3. Local Shift** 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Avg.
$\hat{E}_{norm}$ 0.05 0.06 0.07 0.08 0.10 0.13 0.16 0.22 0.29 0.37 0.15
$\hat{E}_{PC(3)}$ 0.10 0.24 0.48 0.74 0.91 0.98 1.00 1.00 1.00 1.00 0.74
$\hat{E}_{c1}$ 0.07 0.13 0.44 0.93 1.00 1.00 1.00 1.00 1.00 1.00 0.76
$\hat{E}_{c}$ 0.07 0.14 0.43 0.92 1.00 1.00 1.00 1.00 1.00 1.00 0.76
$\hat{R}_{z1}$ 0.16 0.90 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.91
$\hat{R}_{z}$ 0.16 0.90 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.91
$\hat{R}_{z1s}$ 0.13 0.87 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.90
$\hat{R}_{zs}$ 0.13 0.86 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.90
$\hat{B}_{s}$ 0.07 0.11 0.18 0.30 0.44 0.60 0.74 0.86 0.94 0.98 0.52
: $N=100, \ \nu=3/2$
[^1]: In practice we usually have less than $N$ empirical eigenfunctions due to the estimation of other parameters.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper we consider Hopf’s Lemma and the Strong Maximum Principle for supersolutions to $$\sum_{i=1}^N g_i(u_{x_i}^2)u_{x_ix_i}=0$$ under suitable hypotheses that allow $g_i$ to assume value zero at zero.'
address:
- 'Dipartimento di Matematica e Applicazioni, Universitá degli Studi di Milano-Bicocca, Via R. Cozzi 53, 20125 Milano'
- 'Dipartimento di Matematica e Applicazioni, Universitá degli Studi di Milano-Bicocca, Via R. Cozzi 53, 20125 Milano'
- 'Dipartimento di Matematica e Applicazioni, Universitá degli Studi di Milano-Bicocca, Via R. Cozzi 53, 20125 Milano'
author:
- 'S. Bertone'
- 'A. Cellina'
- 'E. M. Marchini'
title: 'On Hopf’s Lemma and the Strong Maximum Principle'
---
Introduction
============
Let $\Omega\subset\mathbb R^N$ be a connected, open and bounded set; we call $\Omega$ regular if for every $z\in\partial\Omega$, there exists a tangent plane, continously depending on $z$. We say that $\Omega $ satisfies the interior ball condition at $z$ if there exists an open ball $B\subset\Omega$ with $z\in\partial B$. On $\Omega$, consider the operator $$\label{elJ}
F(u)=\sum_{i=1}^N g_i(u_{x_i}^2)u_{x_ix_i},$$ where $g_i:[0,+\infty)\to[0,+\infty)$ are continuous functions.
When $F$ is elliptic, two classical results hold.\
[*Hopf’s Lemma:\
Let $\Omega$ be regular, let $u$ be such that $F(u)\leq0$ on $\Omega$. Suppose that there exists $z\in\partial\Omega$ such that $$u(z)<u(x),\quad\mbox{ for all $x$ in $\Omega$}.$$ If, in addition, $\Omega$ satisfies the interior ball condition at $z$, we have $$\frac{\partial u}{\partial\nu}(z)<0,$$ where $\nu$ is the outer unit normal to $B$ at $z$.\
*]{}
[*The Strong Maximum Principle:\
Let $u$ be such that $F(u)\leq0$ on $\Omega$, then if $u$ it attains minimum in $\Omega$, it is a constant.*]{}\
In 1927 Hopf proved the Strong Maximum Principle in the case of second order elliptic partial differential equations, by applying a comparison technique, see [@h]. For the class of quasilinear elliptic problems, many contributions have been given, to extend the validity of the previous results, as in [@bnv; @dpr; @d; @fq; @gt; @g; @ps; @ps2; @psz; @se; @se2; @v].
In the case in equation (\[elJ\]) we have $g_i\equiv1$, for every $i$, then $F(u)=\Delta u$, and we find the classical problem of the Laplacian, see [@e; @gt].
On the other hand, when there exists $i\in\{1,\dots,N\}$ such that $g_i\equiv0$ on an interval $I=[0,T]\subset\mathbb R$, the Strong Maximum Principle does not hold. Indeed, in this case, it is always possible to define a function $u$ assuming minimum in $\Omega$ and such that $\sum_{i=1}^N g_i(u_{x_i}^2)u_{x_ix_i}=0$. For instance, let $g_N(t)=0$, for every $t\in[0,2]$. The function $$u(x_1,\dots,x_N)=\left\{\begin{array}{ll}
-(x_N^2-1)^4 & \mbox{ if }-1\leq x_N\leq1\\
0 & \mbox{ otherwise}
\end{array}
\right.,$$ satisfies (\[elJ\]) in $\mathbb R^N$.
We are interested in the case when $0\leq g_i(t)\leq1$ and it does not exist $i$ such that $g_i\equiv0$ on an interval. Since $g_i$ could assume value zero, the equation (\[elJ\]) is non elliptic.
The results known so far, for the validity of Hopf’s Lemma and of the Strong Maximum Principle, suggest that, for possibly non elliptic equations, but arising from a functional having rotational symmetry, this validity shall depend only on the behaviour of the functions $g_i$ near zero, see [@cel].
In this paper, we prove, in section 3, a sufficient condition for the validity of the Hopf’s Lemma and of the Strong Maximum Principle; a remarkable feature of this condition is that it concerns only the behaviour of the function $g_i(t)$ that goes fastest to zero, as $t$ goes to zero. Hopf’s lemma and the Strong Maximum Principle are essentially the same result as long as we can build subsolutions whose level lines can have arbitrarily large curvature. This need not be always possible for problems not possessing rotational symmetry. This difficulty will be evident in sections 4 and 5. In these sections, a more restricted class of equations is considered, namely when all the functions $g_i$, for $i=1,\dots,N-1$, are $1$ and only $g_N$ is allowed to go to zero. In this simpler class of equations we are able to show that the condition $$\lim_{t\rightarrow0^+}\frac{\left(g_N(t)\right)^{3/2}}{tg_N^\prime(t)}>0$$ is at once necessary for the validity of Hopf’s Lemma and sufficient for the validity of the Strong Maximum Principle.
Preliminary results
===================
We impose the following local assumptions.\
*Assumptions (L):\
\
There exists $\overline t>0$ such that:\
i) on $[0,\overline t\:]$, for every $i=1,\dots,N-1$, $$0\leq g_N(t)\leq g_i(t)\leq1;$$*
ii\) $g_N$ is continuous on $[0,\overline t\,]$; positive and differentiable on $(0,\overline t\,]$;\
iii) on $(0,\overline t\,]$, the function $t\to g_N(t)+g_N^\prime(t)t$ is non decreasing.
\
Notice that, in case [*ii)*]{} above is violated, the Strong Maximum Principle does not hold; and that condition [*iii)*]{} above includes the case of the Laplacian, $g_i(t)\equiv1$; and, finally, that under these assumptions, $g_i$ could assume value zero at most for $t=0$.
Moreover, we can consider the equation $$\sum_{i=1}^N g_i(u_{x_i}^2)u_{x_ix_i}=0$$ as the Euler-Lagrange equation associated to the functional $$J(u)=\int_\Omega{L(\nabla u)d\Omega}=\int_\Omega{\frac{1}{2}\left(\sum_{i=1}^N f_i(u_{x_i}^2)u_{x_i}^2\right)d\Omega},$$ where $L(\nabla u)$ is strictly convex in $\{(u_{x_1},\dots,u_{x_N}):u_{x_i}^2\leq\overline t, \mbox{ for every }i=1,\dots,N\}$. Indeed, fix $i$. Let $f_i$ be a solution to the differential equation $$\label{eqg}
g_i(t)=f_i(t)+5tf_i^\prime(t)+2t^2f_i^{\prime\prime}(t),$$ for $t\in[0,\overline t\,]$. Since $$\frac{\partial^2 L}{\partial u_{x_i}^2}(u_{x_i}^2)=f_i(u_{x_i}^2)+5u_{x_i}^2f_i^\prime(u_{x_i}^2)+2u_{x_i}^4f_i^{\prime\prime}(u_{x_i}^2)=g_i(u_{x_i}^2),$$ we have that $$\mbox{div}\nabla_{\nabla u}L(\nabla u)=\sum_{i=1}^N\frac{\partial^2 L}{\partial u_{x_i}^2}u_{x_ix_i}=\sum_{i=1}^N g_i(u_{x_i}^2)u_{x_ix_i}.$$ Moreover, the strict convexity of $L(\nabla u)$ in $\{(u_{x_1},\dots,u_{x_N}):u_{x_i}^2\leq\overline t,\mbox{ for every }i=1,\dots,N\}$ follows by the fact that $g_i$ is positive in $(0,\overline t\,]$.\
Since we will need general comparison theorems that depend on the global properties of the solutions, i.e. on their belonging to a Sobolev space, we will need also a growth assumption on $g_i$ (assumption [*(G)*]{}) to insure these properties of the solutions.\
[*Assumption (G):\
Each function $f_i$ as defined in (\[eqg\]), is bounded and $f_i(u_{x_i}^2)u_{x_i}^2$ is strictly convex.* ]{}\
Any function $g_i$ satisfying assumptions [*(L)*]{} on $[0,\overline t\,]$ can be extended so as to satisfy assumption [*(G)*]{} on $[0,+\infty)$. In fact, it is enough to extend $g_i$ to $(\,\overline t,+\infty)$ by setting $g_i(t)=f_i(\,\overline t\,)$, for $t>\overline t$.\
Let $\Omega$ be open, and let $u\in W^{1,2}(\Omega)$. The map $u$ is a weak solution to the equation $F(u)=0$ if, for every $\eta\in C^{\infty}_0(\Omega)$, $$\int_\Omega{\langle\nabla L(\nabla u(x)),\nabla\eta(x)\rangle dx}=0.$$ $u$ is a weak subsolution ($F(u)\geq0$) if, for every $\eta\in C^{\infty}_0(\Omega)$, $\eta\geq0$, $$\int_\Omega{\langle\nabla L(\nabla u(x)),\nabla\eta(x)\rangle dx}\leq0.$$ $u$ is a weak supersolution ($F(u)\leq0$) if, for every $\eta\in C^{\infty}_0(\Omega)$, $\eta\geq0$, $$\int_\Omega{\langle\nabla L(\nabla u(x)),\nabla\eta(x)\rangle dx}\geq0.$$ We say that a function $w\in W^{1,2}(\Omega)$ is such that $w_{|\partial \Omega}\leq0$ if $w^{+}\in W^{1,2}_0(\Omega)$.
The growth assumption [*(G)*]{} assures that, if $u\in W^{1,2}(\Omega)$, then $\nabla L(\nabla u(x))\in L^2(\Omega)$. The strict convexity of $L$ implies the following comparison lemma.
\[comp\]
Let $\Omega$ be a open and bounded set, let $v\in W^{1,2}(\Omega)$ be a subsolution and let $u\in W^{1,2}(\Omega)$ be a supersolution to the equation $F(u)=0$. If $v_{|\partial \Omega}\leq u_{|\partial \Omega}$, then $v\leq u$ a.e. in $\Omega$.
We wish to express the operator $$F(v)=\sum_{i=1}^N g_i(v_{x_i}^2)v_{x_ix_i}$$ in polar coordinates. Set $$\left\{\begin{array}{l}
x_1=\rho\cos\theta_{N-1}\dots\cos\theta_2\cos\theta_1\\
x_2=\rho\cos\theta_{N-1}\dots\cos\theta_2\sin\theta_1\\
\dots\\
x_N=\rho\sin\theta_{N-1}
\end{array}
\right.$$ so that $$v_{x_i}=v_{\rho}\frac{x_i}{\rho}\quad\mbox{ and }\quad v_{x_ix_i}=v_{\rho\rho}\left(\frac{x_i}{\rho}\right)^2+\frac{v_{\rho}}{\rho}\left[1-\left(\frac{x_i}{\rho}\right)^2\right].$$ When $v$ is a radial function, $F$ reduces to $$F(v)=\sum_{i=1}^Ng_i\left(v_{\rho}^2\left(\frac{x_i}{\rho}\right)^2\right)\left[v_{\rho\rho}\left(\frac{x_{i}}{\rho}\right)^{2}+\frac{v_{\rho}}{\rho}\left(1-\left(\frac{x_{i}}{\rho}\right)^{2}\right)\right]=$$ $$v_{\rho\rho}\sum_{i=1}^Ng_i\left(v_{\rho}^2\left(\frac{x_i}{\rho}\right)^2\right)\left(\frac{x_i}{\rho}\right)^2+\frac{v_{\rho}}{\rho}\sum_{i=1}^Ng_i\left(v_{\rho}^2\left(\frac{x_i}{\rho}\right)^2\right)\left(1-\left(\frac{x_i}{\rho}\right)^2\right).$$ In general, we do not expect that the equation $F(v)=0$ admits radial solutions. However we will use the expression of $F$ valid for radial functions in order to reach our results.\
The following technical lemmas will be used later.
\[ln\]
Let $n=2,\dots,N$ and set $$h_n(a)=g_N\left(\frac{t(1-a)}{n-1}\right)(1-a)+g_N(ta)a.$$ For every $0<t\leq\overline t$ ($\overline t$ defined in assumptions (L)), $h_n(a)\geq h_n(1/n)$, for every $a$ in $[0,1]$.
Since, on $(0,\overline t\,]$, the function $t\to g_N(t)+g_N^\prime(t)t$ is non decreasing, we have that $$h_n^{\prime}(a)=-g_N\left(\frac{t(1-a)}{n-1}\right)-g^\prime_N\left(\frac{t(1-a)}{n-1}\right)\frac{t(1-a)}{n-1}+g_N(ta)+g_N^\prime(ta)ta\geq0$$ if and only if $a\geq1/n$, so that $h_n(a)\geq h_n(1/n)$, for every $a\in[0,1]$.
\[lind\]
For every $0<t\leq\overline t$ ($\overline t$ defined in assumptions (L)), we have that $$\sum_{i=1}^N g_N\left(t\left(\frac{x_i}{\rho}\right)^2\right)\left(\frac{x_i}{\rho}\right)^2\geq g_N\left(\frac{t}{N}\right).$$
We prove the claim by induction on $N$.
Let $N=2$. Set $a=\sin^2\theta_1$. Applying Lemma \[ln\] we obtain that $$g_N\left(t\left(\frac{x_1}{\rho}\right)^2\right)\left(\frac{x_1}{\rho}\right)^2+g_N\left(t\left(\frac{x_2}{\rho}\right)^2\right)\left(\frac{x_2}{\rho}\right)^2=$$ $$g_N(t(1-a))(1-a)+g_N(ta)a\geq g_N\left(\frac{t}{2}\right).$$
Suppose that the claim is true for $N-1$, i.e. $$\sum_{i=1}^{N-1}g_N\left(t\left(\frac{x_i}{\rho}\right)^2\right)\left(\frac{x_i}{\rho}\right)^2\geq g_N\left(\frac{t}{N-1}\right).$$ Let us prove it for $N$. Set $$\left\{\begin{array}{l}
y_1=\rho\cos\theta_{N-2}\dots\cos\theta_2\cos\theta_1\\
y_2=\rho\cos\theta_{N-2}\dots\cos\theta_2\sin\theta_1\\
\dots\\
y_{N-1}=\rho\sin\theta_{N-2}
\end{array}
\right.$$ and set $a=\sin^2\theta_{N-1}$. Applying Lemma \[ln\] we obtain that $$\sum_{i=1}^{N}g_N\left(t\left(\frac{x_i}{\rho}\right)^2\right)\left(\frac{x_i}{\rho}\right)^2=$$ $$\sum_{i=1}^{N-1}g_N\left(t\left(\frac{y_i}{\rho}\right)^2(1-a)\right)\left(\frac{y_i}{\rho}\right)^2(1-a)+g_N(ta)a\geq$$ $$g_N\left(\frac{t(1-a)}{N-1}\right)(1-a)+g_N(ta)a\geq g_N\left(\frac{t}{N}\right),$$ and the claim is proved.
A sufficient condition for the validity of Hopf’s Lemma and of the Strong Maximum Principle
===========================================================================================
Consider the improper Riemann integral $$\int^\xi_0{\frac{g_N(\zeta^2/N)}{\zeta}d\zeta}=\lim_{\widehat{\xi}\to 0}\int^\xi_{\widehat{\xi}}{\frac{g_N(\zeta^2/N)}{\zeta}d\zeta}$$ as an extended valued function $G$, $$G(\xi)=\int^\xi_0{\frac{g_N(\zeta^2/N)}{\zeta}d\zeta},$$ where we mean that $G(\xi)\equiv+\infty$ whenever the integral diverges.\
We wish to prove the following lemma.
\[hopfr\]
Let $\Omega\subset\mathbb R^N$ be a connected, open and bounded set. Let $u\in W^{1,2}(\Omega)\cap C\left(\overline\Omega\right)$ be a weak solution to $$\sum_{i=1}^N g_i(u_{x_i}^2)u_{x_ix_i}\leq0.$$ In addition to the assumptions (L) and (G) on $g_i$, assume that $G(\xi)\equiv+\infty$. Suppose that there exists $z\in\partial\Omega$ such that $$u(z)<u(x),\quad\mbox{ for all $x$ in $\Omega$}$$ and that $\Omega$ satisfies the interior ball condition at $z$. Then $$\frac{\partial u}{\partial\nu}(z)<0,$$ where $\nu$ is the outer unit normal to $B$ at $z$.
As an example of an equation satisfying the assumptions of the theorem above, consider the Laplace equation $\Delta u=0$. The functions $g_i\equiv1$ satisfy the assumptions [*(L)*]{} and [*(G)*]{}, and $$G(\xi)=\int^\xi_0{\frac{1}{\zeta}d\zeta}=+\infty.$$
Another example is obtained setting $$g_N(t)=\dfrac{1}{|\ln(t)|}$$ for $0\leq t\leq1/e$. The assumptions [*(L)*]{} and [*(G)*]{} are satisfied; moreover, for $0\leq\xi^2/N\leq1/e$, $$G(\xi)=\int^{\xi}_{0}{\frac{d\zeta}{\zeta|\ln(\zeta^2/N)|}}=+\infty.$$\
a\) Assume that $u(z)=0$ and that $B=B(O,r)$. We prove the claim by contradiction. Suppose that $$\frac{\partial u}{\partial\nu}(z)\geq0,$$ where $\nu$ is the outer unit normal to $B$ at $z$. Let $\epsilon=\min\left\{u(x):x\in\overline{B(O,r/2)}\right\}$; we have that $\epsilon>0$. Set $$\omega=B(O,r)\setminus\overline{B(O,r/2)}.$$
b\) We seek a radial function $v\in W^{1,2}(\omega)\cap C(\overline{\omega})$ satisfying $$\label{eqnrad}
\left\{\begin{array}{ll}
v \mbox{ is a weak solution to }F(v)\geq0 & \mbox{ in } \omega\\
v>0 & \mbox{ in }\omega\\
v=0 & \mbox{ in }\partial B(O,r)\\
v\leq\epsilon & \mbox{ in }\partial B(O,r/2)\\
v_\rho(z)<0.\\
\end{array}
\right.$$ Consider the Cauchy problem $$\label{cr}
\left\{\begin{array}{l}
\zeta^\prime=-\dfrac{N-1}{\rho}\dfrac{\zeta}{g_N(\zeta^2/N)}\\
\\
\zeta(r/2)=-\dfrac{\epsilon}{r}.\\
\end{array}
\right.$$ There exists a unique local solution $\zeta$ of (\[cr\]), such that $$\int^{\zeta(\rho)}_{-\frac{\epsilon}{r}}{\frac{g_N(\zeta^2/N)}{\zeta}d\zeta}=\int^{\rho}_{r/2}{-\frac{N-1}{s}ds}=-(N-1)\ln\left(\frac{2\rho}{r}\right).$$ We claim that $\zeta$ is defined in $[r/2,+\infty)$. Indeed, suppose that $\zeta$ is defined in $[r/2,\tau)$, with $\tau<+\infty$. Since $\zeta^\prime>0$, $\zeta$ is an increasing function, so that $\tau<+\infty$ if and only if $\lim_{\rho\to\tau}\zeta(\rho)=0$. But $$-\infty=\lim_{\rho\to\tau}\int^{\zeta(\rho)}_{-\frac{\epsilon}{r}}{\frac{g_N(\zeta^2/N)}{\zeta}d\zeta}=\lim_{\rho\to\tau}-(N-1)\ln\left(\frac{2\rho}{r}\right),$$ a contradiction. Hence, the solution $\zeta$ of (\[cr\]) is defined in $[r/2,+\infty)$.
Setting $v_\rho=\zeta$, since, for every $\rho\in(r/2,r)$, $$-\frac{\epsilon}{r}<v_\rho(\rho)<0,$$ we have that the function $$v(\rho)=\int_{r}^\rho{v_{\rho}(s)ds}$$ solves the problem $$v_{\rho\rho}g_N\left(\frac{v_{\rho}^2}{N}\right)+\frac{v_{\rho}}{\rho}(N-1)=0,$$ in particular, $v(\rho)>0$ and $v_\rho(\rho)<0$, for every $\rho\in(r/2,r)$, $v(r)=0$ and $v(r/2)\leq\epsilon$.
Since $v_{\rho\rho}\geq0$ and $-\sqrt{\overline{t}}\leq v_{\rho}\leq0$, for every $\rho\in(r/2,r)$, by the hypotheses on $g_i$ and by Lemma \[lind\], we have that $$F(v)=v_{\rho\rho}\sum_{i=1}^Ng_i\left(v_{\rho}^2\left(\frac{x_i}{\rho}\right)^2\right)\left(\frac{x_i}{\rho}\right)^2+\frac{v_{\rho}}{\rho}\sum_{i=1}^Ng_i\left(v_{\rho}^2\left(\frac{x_i}{\rho}\right)^2\right)\left(1-\left(\frac{x_i}{\rho}\right)^2\right)\geq$$ $$v_{\rho\rho}\sum_{i=1}^Ng_N\left(v_{\rho}^2\left(\frac{x_i}{\rho}\right)^2\right)\left(\frac{x_i}{\rho}\right)^2+\frac{v_{\rho}}{\rho}\sum_{i=1}^N\left(1-\left(\frac{x_i}{\rho}\right)^2\right)=$$ $$v_{\rho\rho}\sum_{i=1}^Ng_N\left(v_{\rho}^2\left(\frac{x_i}{\rho}\right)^2\right)\left(\frac{x_i}{\rho}\right)^2+\frac{v_{\rho}}{\rho}(N-1)\geq v_{\rho\rho}g_N\left(\frac{v_{\rho}^2}{N}\right)+\frac{v_{\rho}}{\rho}(N-1).$$ The function $v$ solves (\[eqnrad\]), indeed, $v$ is in $C^2(\overline\omega)$ and it is such that $F(v)\geq0$ and $v>0$ in $\omega$, $v(r)=0$, $v(r/2)\leq\epsilon$ and $v_\rho(z)<0$.\
c) Since $u,v\in W^{1,2}(\omega)\cap C(\overline{\omega})$, $v$ is a weak subsolution and $u$ is a weak solution to $F(u)=0$, and $v_{|\partial \omega}\leq u_{|\partial \omega}$, applying Lemma \[comp\], we obtain that $v\leq u$ in $\omega$. From $$v_\rho(z)=\frac{\partial v}{\partial\nu}(z)<\frac{\partial u}{\partial\nu}(z),$$ it follows that there exists $x^0\in\omega$ such that $v(x^0)>u(x^0)$, a contradiction.
From Hopf’s Lemma we derive:
\[thsmpr\]
Let $\Omega\subset\mathbb R^N$ be a connected, open and bounded set. Let $u\in W^{1,2}(\Omega)\cap C\left(\overline\Omega\right)$ be a weak supersolution to $$\sum_{i=1}^N g_i(u_{x_i}^2)u_{x_ix_i}=0.$$ In addition to the assumptions (L) and (G) on $g_i$, assume that $G(\xi)\equiv+\infty$. Then, if $u$ attains its minimum in $\Omega$, it is a constant.
a\) Assume $\min_\Omega u=0$ and set $\mathcal{C}=\{x\in\Omega:u(x)=0\}$. By contradiction, suppose that the open set $\Omega\setminus\mathcal{C}\neq\emptyset$.\
b) Since $\Omega$ is a connected set, there exist $s\in\mathcal{C}$ and $R>0$ such that $B(s,R)\subset\Omega$ and $B(s,R)\cap(\Omega\setminus\mathcal{C})\neq\emptyset$. Let $p\in B(s,R)\cap(\Omega\setminus\mathcal{C})$. Consider the line $\overline{ps}$. Moving $p$ along this line, we can assume that $B(p,d(p,\mathcal{C}))\subset(\Omega\setminus\mathcal{C})$ and that there exists one point $z\in\mathcal{C}$ such that $d(p,\mathcal{C})=d(p,z)$. Set $r=d(p,\mathcal{C})$. W.l.o.g. suppose that $p=O$.\
c) The set $\Omega\setminus\mathcal{C}$ satisfies the interior ball condition at $z$, hence Hopf’s Lemma implies $$\frac{\partial u}{\partial\nu}(z)<0.$$ But this is a contradiction: since $u$ attains minimum at $z\in\Omega$, we have that $Du(z)=0$.\
A necessary condition for the validity of Hopf’s Lemma
======================================================
In this and the following section we consider the operator $$\label{gN}
F(u)=\sum_{i=1}^{N-1}u_{x_ix_i}+g(u_{x_N}^2)u_{x_Nx_N},$$ We wish to provide a necessary condition for the validity of Hopf’s Lemma in a class of non elliptic equations.
Consider the case $$G(\xi)=\int^\xi_0{\frac{g(\zeta^2/N)}{\zeta}d\zeta}<+\infty.$$
\[hopfn\]
Consider the operator (\[gN\]), where $g$ satisfies assumptions (L) and (G), and on $(0,\overline t\,]$ ($\overline{t}$ defined in assumptions (L)), $$g^\prime(t)>0 \quad\mbox{and}\quad g(t)+g^\prime(t)t\leq1.$$ If $$\lim_{t\rightarrow0^+}\frac{\left(g(t)\right)^{3/2}}{tg^\prime(t)}=0,$$ then there exist: an open regular region $\Omega\subset\mathbb R^N$; a radial function $u\in C^2(\Omega)$ such that $F(u)\leq0$ in $\Omega$ and a point $z\in\partial\Omega$ such that $u(z)=0$, $u(z)\leq u(x)$ for every $x\in\Omega$ and $$\frac{\partial u}{\partial\nu}(z)=0$$ where $\nu$ is the outer unit normal to $\Omega$ at $z$.\
If, in addition, we assume that $$\label{C2}
\frac{g(t)}{tg^\prime(t)} \quad \mbox{is bounded in} \quad (0,\overline t\,]$$ then $\Omega$ satisfies the interior ball condition at $z$.
When $\lim_{t\rightarrow0^+}{g^\prime(t)t}$ exists, it follows that $$\lim_{t\rightarrow0^+}\frac{\left(g(t)\right)^{3/2}}{tg^\prime(t)}$$ exists, and that $$g(t)+g^\prime(t)t\leq1\,,\mbox{ on }(0,\overline t\,].$$
Indeed, we have that $$\lim_{t\rightarrow0^+}{\left(g(t)+g^\prime(t)t\right)}=0.$$ Otherwise, there exists $K>0$ such that, when $0<t\leq\overline t$, $g^\prime(t)t\geq K$, so that $$g(t)t=\int^t_0{\left(g(s)+g^\prime(s)s\right)ds}\geq Kt,$$ and $g(t)\geq K$. From $$\int^{\xi^2}_0{\frac{g(t)}{t}dt}<+\infty,$$ it follows that $\lim_{t\rightarrow0^+}{g(t)}=0$, a contradiction.\
The map $$g(t)=\frac{1}{|\ln(t)|^k},$$ with $k>2$, for $0\leq t\leq1/e$, satisfies the assumption $$\lim_{t\rightarrow0^+}\frac{\left(g(t)\right)^{3/2}}{tg^\prime(t)}=0.$$\
The following lemma is instrumental to the proofs of the main results.
\[l1\]
Let $g$ satisfies assumptions (L) and (G). Suppose that for every $0<t\leq\overline t$, $$g^\prime(t)\geq0\quad\mbox{ and }\quad g(t)+g^\prime(t)t\leq1.$$ Set $$k_1(a)=(1-a)+ag(ta)\quad\mbox{ and }\quad k_2(a)=-a-(1-a)g(ta)$$ For every $0<t\leq\overline t$ ($\overline t$ defined in assumptions (L)), $k_1$ and $k_2$ are non increasing in $[0,1]$.
Since, for every $0<t\leq\overline t$, $$g^\prime(t)\geq0\quad\mbox{ and }\quad g(t)+g^\prime(t)t\leq1,$$ we have that, for every $0\leq a\leq1$, $$k_1^{\prime}(a)=-1+g(ta)+g^\prime(ta)ta\leq0$$ and $$k_2^{\prime}(a)=-1+g(ta)-(1-a)g^\prime(ta)t=-1+g(ta)+g^\prime(ta)ta-g^\prime(ta)t\leq0.$$
a\) Let $v$ be a radial function. Setting $a=\sin^2\theta_{N-1}$, (\[gN\]) reduces to $$F(v)=v_{\rho\rho}\left(1-a+ag(v_{\rho}^2a)\right)+\frac{v_{\rho}}{\rho}\left(N-2+a+(1-a)g(v_{\rho}^2a)\right).$$ Let $a=1$, we seek a solution to $$\label{supersol}
v_{\rho\rho}g(v_\rho^2)+(N-1)\frac{v_{\rho}}{\rho}=0$$ such that $v_\rho(R(1)+1)=0$ and $v_\rho(\rho)<0$, for every $\rho\in[2,R(1)+1)$. Consider the Cauchy problem $$\label{csup}
\left\{\begin{array}{l}
\zeta^\prime=-\dfrac{N-1}{\rho}\dfrac{\zeta}{g(\zeta^2)}\\
\\
\zeta(2)=-1.\\
\end{array}
\right.$$
We are interested in a negative solution $\zeta$. Define $R(1)$ to be the unique positive real solution to $$G(-1)-(N-1)\ln\left(\frac{R(1)+1}{2}\right)=0,$$ i.e. $$R(1)=2e^{\frac{G(-1)}{N-1}}-1.$$ The solution $\zeta$ of (\[csup\]), satisfies $$G(\zeta(\rho))-G(-1)=\int^{\zeta(\rho)}_{\zeta(2)}{\frac{g(t^2)}{t}dt}=\int^{\rho}_{2}{-\frac{N-1}{s}ds}=-(N-1)\ln\left(\frac{\rho}{2}\right).$$ Then, for every $\rho\in(2,R(1)+1)$, $G(\zeta(\rho))>0$ and $\zeta(\rho)<0$, while $\zeta(R(1)+1)=0$. Setting $v_{\rho}(\rho)=\zeta(\rho)$ and $$v(\rho)=\int_{R(1)+1}^{\rho}{v_{\rho}(s)ds},$$ we obtain that $v$ solves (\[supersol\]) and, for every $\rho\in(2,R(1)+1)$, $$v_{\rho}(\rho)<v_{\rho}(R(1)+1)=0\quad\mbox{ and }\quad v(\rho)>v(R(1)+1)=0.$$
b\) Set, for $\rho\in(1,R(1)]$, $u(\rho)=v(\rho+1).$ Since, for the function $v$, we have $$v_{\rho\rho}(\rho)g(v_\rho^2(\rho))+(N-1)\frac{v_{\rho}(\rho)}{\rho}=0,$$ at $\rho+1$ we obtain $$\label{B}
u_{\rho\rho}(\rho)g(u_\rho^2(\rho))+(N-1)\frac{u_{\rho}(\rho)}{\rho+1}=0.$$ This equality yields, for $\rho\in(1,R(1))$, $$u_{\rho\rho}g(u_\rho^2)+(N-1)\frac{u_{\rho}}{\rho}=-(N-1)u_{\rho}\left(\frac{1}{\rho+1}-\frac{1}{\rho}\right)<0.$$
c\) Let $1/2<a<1$. We wish to find $R(a)\leq R(1)$ such that $u$ is a solution to $$F(u)=u_{\rho\rho}\left(1-a+ag(u_{\rho}^2a)\right)+\frac{u_{\rho}}{\rho}\left(N-2+a+(1-a)g(u_{\rho}^2a)\right)\leq0,$$ for $\rho\in(1,R(a))$. Since $$F(u)=\frac{-u_{\rho}}{\rho(\rho+1)g((u_\rho)^2)}\left[\rho(N-1)\left(1-a+ag((u_{\rho})^2a)\right)-\right.$$ $$\left.(\rho+1)g((u_{\rho})^2)\left(N-2+a+(1-a)g((u_{\rho})^2a)\right)\right],$$ setting $$k(\rho)=$$ $$\rho(N-1)\left(1-a+ag((u_{\rho})^2a)\right)-(\rho+1)g((u_{\rho})^2)\left(N-2+a+(1-a)g((u_{\rho})^2a)\right),$$ we have that $F(u)\leq0$ if and only if $k(\rho)\leq0$. From $$\frac{d}{d\rho}g((u_{\rho})^2)\leq\frac{d}{d\rho}g((u_{\rho})^2a)\leq0,$$ and $N-2+a\geq(N-1)a$ for $N\geq2$, applying Lemma \[l1\], we have that $$k^\prime(\rho)=(N-1)(1-a+ag((u_{\rho})^2a)-g((u_{\rho})^2)(N-2+a+(1-a)g((u_{\rho})^2a))+$$ $$\rho(N-1)a\frac{d}{d\rho}g((u_{\rho})^2a)-(\rho+1)\frac{d}{d\rho}g((u_{\rho})^2)(N-2+a+(1-a)g((u_{\rho})^2a))-$$ $$(\rho+1)g((u_{\rho})^2)(1-a)\frac{d}{d\rho}g((u_{\rho})^2a)\geq$$ $$\rho\left[(N-1)a\frac{d}{d\rho}g((u_{\rho})^2a)-\frac{d}{d\rho}g((u_{\rho})^2)(N-2+a+(1-a)g((u_{\rho})^2a))-\right.$$ $$\left.g((u_{\rho})^2)(1-a)\frac{d}{d\rho}g((u_{\rho})^2a)\right]\geq$$ $$\rho(N-1)a\left(\frac{d}{d\rho}g((u_{\rho})^2a)-\frac{d}{d\rho}g((u_{\rho})^2)\right)\geq0.$$ Since the function $k(\rho)$ is non decreasing, it follows that $F(u)\leq0$, for every $\rho\in(1,R(a))$, if and only if $$k(R(a))\leq0.$$ We have that $$k(R(a))=R(a)(N-1)\left(1-a+ag((u_{\rho}(R(a)))^2a)\right)-$$ $$(R(a)+1)g((u_{\rho}(R(a)))^2)\left(N-2+a+(1-a)g((u_{\rho}(R(a)))^2a)\right)\leq$$ $$(N-1)\left[R(a)(1-a)-g((u_{\rho}(R(a)))^2)a\right]\leq$$ $$(N-1)\left[R(1)(1-a)-g((u_{\rho}(R(a)))^2)a\right]\leq$$ $$(N-1)\left[R(1)(1-a)-\frac{g((u_{\rho}(R(a)))^2)}{2}\right].$$ We define $R(a)$ to be a solution to $$\label{A}
R(1)(1-a)-\frac{g((u_{\rho}(R(a)))^2)}{2}=0.$$\
d) In order to solve (\[A\]) for the unknown $R(a)$, recalling that $1-a=\cos^2\theta_{N-1}=c^2$, let $$h(r)=\sqrt{\frac{g((u_{\rho}(r))^2)}{2R(1)}}.$$ The function $h$ is decreasing, differentiable and with inverse differentiable. We have that $|c|=h(R(1-c^2))$, so that $R(1-c^2)=h^{-1}(|c|)$, $R(1-c^2)$ is increasing in $|c|$ and $$\lim_{c\rightarrow0}R(1-c^2)=R(1).$$ Let $0<|\bar{c}|<1/2$ be such that $R(1-\bar{c}^2)\geq1$, so that, for $c^2\leq\bar{c}^2$, we have $R(1-c^2)\geq R(1-\bar{c}^2)\geq1$. We have obtained that, for every $1\geq a\geq 1-|c|^2$, there exists $R(a)$ such that (\[A\]) holds. It follows that $$k(R(a))\leq\left[R(1)(1-a)-\frac{g((u_{\rho}(R(a)))^2)}{2}\right]=0,$$ so that the function $u$ solves $F(u)\leq0$ for every $\rho\in(1,R(a))$.\
e) Set $\Omega=\{(x_1,\dots,x_N)\in\mathbb R^N:\rho\in(1,R(1-c^2))\mbox{ and }|c|<|\bar c|\}$. $\Omega\subset\mathbb R^N$ is a connected, open and bounded set and $u\in W^{1,2}(\Omega)\cap C\left(\overline\Omega\right)$ is a weak solution to $F(u)\leq0.$ The point $z=(R,0,\dots,0)\in\partial\Omega$ is such that $u(z)<u(x)$, for all $x\in\Omega$. We wish to show that $\Omega$ is regular in a neighborhood of $z=(R,0,\dots,0)$. Since $\frac{d}{dc}R(1-c^2)$ exists, in $(0,|\bar c|)$, to prove our claim it is sufficient to show that $$\lim_{c\rightarrow0}\frac{d}{dc}R(1-c^2)=0.$$ Recalling (\[B\]), we have that $$\frac{d}{dc}R(1-c^2)=\left(h^{-1}(c)\right)^\prime=\frac{1}{h^\prime(R(1-c^2))}=$$ $$\label{dR}
-\frac{\sqrt{2}R(1)}{N-1}\left(R(1-c^2)+1\right)\frac{\left(g((u_{\rho}(R(1-c^2)))^2)\right)^{3/2}}{(u_{\rho}(R(1-c^2)))^2g^\prime((u_{\rho}(R(1-c^2)))^2)}.$$ Since $$\lim_{c\rightarrow0}u_{\rho}(R(1-c^2))=0$$ and $$\lim_{t\rightarrow0^+}\frac{\left(g(t)\right)^{3/2}}{tg^\prime(t)}=0,$$ it follows that $$\lim_{c\rightarrow0}\frac{d}{dc}R(1-c^2)=0.$$ Since ${\frac{d}{d\theta}\cos\theta_{N-1}}_{|\theta_{N-1}=\frac{\pi}{2}}=1$, this shows that $\frac{d}{d\theta}R(1-\cos^2\theta_{N-1})=0$, and $\Omega$ is regular.\
f) To prove the validity of the interior ball condition at $z=(R,0,\dots,0)$, it is enough to show that the second derivative of $R(1-c^2)$ is bounded at $c=0$, i.e. that $$\left|\frac{1}{c}\frac{d}{dc}R(1-c^2)\right|$$ is bounded. Set $$t(c)=(u_{\rho}(R(1-c^2)))^2,$$ from (\[dR\]) we obtain $$\frac{d}{dc}R(1-c^2)=-\frac{\sqrt{2}R(1)}{N-1}\left(R(1-c^2)+1\right)\frac{(g(t(c)))^{3/2}}{t(c)g^\prime(t(c))},$$ and from (\[B\]) $$\frac{dt(c)}{dc}=2u_{\rho}(R(1-c^2))u_{\rho\rho}(R(1-c^2))\frac{d}{dc}(R(1-c^2))=$$ $$2\sqrt{2}R(1)\frac{(g(t(c)))^{1/2}}{g^\prime(t(c))}$$ and $$\frac{d}{dc}(g(t(c)))^{1/2}=\frac{g^\prime(t(c))}{2(g(t(c)))^{1/2}}\frac{dt(c)}{dc}=\sqrt{2}R(1).$$ From $g(t(0))=0$, we obtain that $$(g(t(c)))^{1/2}=\sqrt{2}R(1)c$$ and $$\frac{(g(t(c)))^{3/2}}{ct(c)g^\prime(t(c))}=\sqrt{2}R(1)\frac{g(t(c))}{t(c)g^\prime(t(c))}.$$ From condition (\[C2\]) we obtain $$\left|\frac{1}{c}\frac{d}{dc}R(1-c^2)\right|\leq M.$$
units < 1cm, 1cm> x from -7 to 7, y from -1 to 9 at 0 0 ([.]{}) 2 3.464102 3.5 6.062178 / -2 3.464102 -3.5 6.062178 / -60 degrees from -2 3.464102 center at 0 0 -15 degrees from 2.546488 7.304819 center at -1.696153 3.062178 -14.99999 degrees from -3.5 6.062178 center at 1.696151 3.062177 -89.99998 degrees from -2.546491 7.304819 center at -0.00000143 4.75833 ([.]{}) 0 0 -0.00000143 9.35961 / 2 3.464102 0 0 / 0 0 -2 3.464102 / 3.5 6.062178 4.679805 8.105659 / -3.5 6.062178 -4.679805 8.105659 / -80 degrees from -5.373454 6.403832 center at 0 0 -10.00001 degrees from 2 3.464102 center at 0 0 -9.99999 degrees from -2.571151 3.064178 center at 0 0 at .5 0 at .49 8.8 at 2.3 3 at 5 7 at 4.6 5.9 at 1.3 6
A sufficient condition for the validity of the Strong Maximum Principle
=======================================================================
Consider the case $$G(\xi)=\int^\xi_0{\frac{g(\zeta^2/N)}{\zeta}d\zeta}<+\infty.$$
We wish to prove the following theorem.
\[thsmp\]
Let $\Omega\subset\mathbb R^N$ be a connected, open and bounded set. Let $u\in W^{1,2}(\Omega)\cap C\left(\overline\Omega\right)$ be a weak supersolution to $$\sum_{i=1}^{N-1}u_{x_ix_i}+g(u_{x_N}^2)u_{x_Nx_N}=0,$$ where $g$ satisfies assumptions (L) and (G) and, on $(0,\overline t\,]$ ($\overline{t}$ defined in assumptions (L)), $$g^\prime(t)\geq0\quad\mbox{ and }\quad g(t)+g^\prime(t)t\leq1.$$ Moreover, suppose that there exists $K>0$ such that, for every $0<\xi^2/N\leq\overline{t}$ , we have $$\sqrt{g(\xi^2/N)}\leq K\left(e^{G(\xi)}-1\right).$$ Then, if $u$ attains its minimum in $\Omega$, it is a constant.
When the function $g$ satisfies the condition $$g^\prime(t)t\leq2K\left(g(t)\right)^{3/2},$$ for every $0<t\leq\overline t$, then it satisfies $$g(t)+g^\prime(t)t\leq1$$ and $$\sqrt{g(\xi^2/N)}\leq K\left(e^{G(\xi)}-1\right),$$ for every $0<\xi^2/N\leq\overline t$.
Indeed, since $G(\xi)<+\infty$, we have that $\lim_{t\rightarrow0}g(t)=0$. Hence, we can suppose that $g(t)\leq1/(2K+1)$, for $0<t\leq\overline t$, so that $$g(t)+g^\prime(t)t\leq g(t)+2K\left(g(t)\right)^{3/2}\leq1.$$ Moreover, since $g(0)=0$, $G(0)=0$ and $$\left(\sqrt{g(\xi^2/N)}\right)^\prime\leq2K\frac{g(\xi^2/N)}{\xi}\leq K\left(e^{G(\xi)}-1\right)^\prime,$$ we obtain that $$\sqrt{g(\xi^2/N)}\leq K\left(e^{G(\xi)}-1\right).$$
Among the functions $g$ such that $$\lim_{t\rightarrow0^+}\frac{\left(g(t)\right)^{3/2}}{tg^\prime(t)}$$ exists, there exists $K>0$ such that, for every $0<\xi^2/N\leq\overline{t}$, we have $$\sqrt{g(\xi^2/N)}\leq K\left(e^{G(\xi)}-1\right)$$ if and only if $$\lim_{t\rightarrow0^+}\frac{\left(g(t)\right)^{3/2}}{tg^\prime(t)}>0.$$
An example of a map satisfying the assumptions of the theorem above, is given by $$g(t)=\frac{1}{(\ln(t))^2},$$ for $0\leq t\leq1/e^4$. For $0\leq\xi^2/N\leq1/e^4$, we have $$G(\xi)=\int^{\xi}_{0}{\frac{1}{\zeta(\ln(\zeta^2/N))^2}d\zeta}=-\frac{1}{2\ln(\xi^2/N)},$$ and $$\sqrt{g(\xi^2/N)}=\frac{1}{|\ln(\xi^2/N)|}\leq2\left(e^{\frac{1}{2|\ln(\xi^2/N)|}}-1\right)=e^{G(\xi)}-1.$$
Set $$\mathcal R(\lambda,\lambda_N)=\{(x_1,\dots,x_N):|x_i|\leq\lambda,\mbox{ for }i=1,\dots,N-1, |x_N|\leq\lambda_N\}.$$
To the opposite of the proof of Lemma \[hopfr\] and Theorem \[thsmpr\], we will build a subsolution that [*is not*]{} radially symmetric. This construction is provided by next theorem.
\[subsol\]
Under the same assumptions on $g$ as on Theorem \[thsmp\], for every $r>0$ and every $\epsilon$, there exist: $l,l_N$; an open convex region $\mathcal{A}\subset\mathcal R(l,l_N)$; a function $v\in W^{1,2}(\omega)\cap C^1(\omega)\cap C(\overline{\omega})$, where $\omega=B(\mathcal{A},r)\setminus\overline{\mathcal{A}}$, such that
i\) $0\leq l\leq2Kr$, and $0\leq l_N\leq r/4$;
ii)$$\label{eqnN}
\left\{\begin{array}{ll}
v \mbox{ is a weak solution to }F(v)\geq0 & \mbox{ in } \omega\\
v>0 & \mbox{ in }\omega\\
v=0 & \mbox{ in }\partial B(\mathcal{A},r)\\
v\leq\epsilon & \mbox{ in }\partial\mathcal{A}.\\
\end{array}
\right.$$
Fix $r$; we can assume that $\epsilon$ is such that $0<\epsilon^2/r^2\leq\overline t$ and that $$2\sqrt{g\left(\frac{\epsilon^2}{r^2}\right)}+\frac{1}{2}\sqrt{g\left(\frac{\epsilon^2}{r^2}\right)}\left|\ln g\left(\frac{\epsilon^2}{r^2}\right)\right|\leq\frac{1}{4K}.$$ Fix the origin $O^0=(0,\dots,0)$, and set polar coordinates as $$\left\{\begin{array}{l}
x_1=\rho\cos\theta_{N-1}\dots\cos\theta_2\cos\theta_1\\
x_2=\rho\cos\theta_{N-1}\dots\cos\theta_2\sin\theta_1\\
\dots\\
x_N=\rho\sin\theta_{N-1}.
\end{array}
\right.$$\
1) When $w$ is a radial function, setting $a=\sin^2\theta_1$ , $F$ reduces to $$F(w)=w_{\rho\rho}\left(1-a+ag(w_{\rho}^2a)\right)+\frac{w_{\rho}}{\rho}\left(N-2+a+(1-a)g(w_{\rho}^2a)\right).$$ For $a=1$, we seek a solution to $$\label{pN}
(N-1)\frac{w_{\rho}}{\rho}+w_{\rho\rho}g(w_\rho^2)=0.$$ such that $w_\rho(R(1))=-\epsilon/r$ and $w_\rho(\rho)<0$, for every $\rho\in[R(1),R(1)+r)$. Consider the Cauchy problem $$\label{c2}
\left\{\begin{array}{l}
\zeta^\prime=-\dfrac{N-1}{\rho}\dfrac{\zeta}{g(\zeta^2)}\\
\\
\zeta(R(1))=-\dfrac{\epsilon}{r}.\\
\end{array}\right.$$ We are interested in a negative solution $\zeta$. Define $R(1)$ to be the unique positive real solution to $$G(-\epsilon/r)-(N-1)\ln\left(\frac{R(1)+r}{R(1)}\right)=0,$$ i.e. $$R(1)=\frac{r}{e^{\frac{G(-\epsilon/r)}{N-1}}-1}.$$ Consider the unique solution $\zeta$ of (\[c2\]), such that $\zeta(R(1))=-\epsilon/r$, i.e., such that $$G(\zeta(\rho))-G(-\epsilon/r)=\int^{\zeta(\rho)}_{\zeta(R(1))}{\frac{g(t^2)}{t}dt}=\int^{\rho}_{R(1)}{-\frac{N-1}{s}ds}=-(N-1)\ln\left(\frac{\rho}{R(1)}\right).$$ Then, for every $\rho\in[R(1),R(1)+r)$, $G(\zeta(\rho))>0$ and $\zeta(\rho)<0$, while $\zeta(R(1)+r)=0$. Setting $w_{\rho}(\rho)=\zeta(\rho)$ and $$w(\rho)=\int_{R(1)+r}^{\rho} {w_{\rho}(s)ds},$$ we obtain that $w$ solves (\[pN\]) and, for every $\rho\in(R(1),R(1)+r)$, $$-\epsilon/r=w_{\rho}(R(1))<w_{\rho}(\rho)<w_{\rho}(R(1)+r)=0$$ and $$0=w(R(1)+r)<w(\rho)<w(R(1))\leq\epsilon.$$
2\) Applying Lemma \[l1\], we infer that the function $w$ defined in 1) is actually a solution to $$F(w)=w_{\rho\rho}(1-a)+\frac{w_{\rho}}{\rho}(N-2+a)+g(w_\rho^2a)\left(w_{\rho\rho}a+\frac{w_{\rho}}{\rho}(1-a)\right)\geq0,$$ for every $0\leq a\leq1$ and every $\rho\in(R(1),R(1)+r)$.\
3) Let $\bar{a}<1$. We wish to find the smallest $R(\bar{a})>0$ such that, setting $$w^{\bar{a}}(\rho)=w(\rho-R(\bar{a})+R(1)),$$ the function $w^{\bar{a}}$ is a solution to $$\label{eqa}
F(w^{\bar{a}})=w^{\bar{a}}_{\rho\rho}(1-\bar{a})+\frac{w^{\bar{a}}_{\rho}}{\rho}(N-2+\bar{a})+g((w^{\bar{a}}_\rho)^2\bar{a})\left(w^{\bar{a}}_{\rho\rho}\bar{a}+\frac{w^{\bar{a}}_{\rho}}{\rho}(1-\bar{a})\right)\geq0,$$ for every $\rho\in(R(\bar{a}),R(\bar{a})+r)$.
Since, for the function $w$, we have $$(N-1)\frac{w_{\rho}(\rho)}{\rho}+g(w_\rho^2(\rho))w_{\rho\rho}(\rho)=0,$$ at $\rho-R(\bar{a})+R(1)$ we obtain $$(N-1)\frac{w^{\bar{a}}_{\rho}(\rho)}{\rho-R(\bar{a})+R(1)}+g((w^{\bar{a}}_\rho(\rho))^2) w^{\bar{a}}_{\rho\rho}(\rho)=0.$$ This equality yields $$(N-2+\bar{a})\frac{w^{\bar{a}}_{\rho}}{\rho}+\bar{a}g((w^{\bar{a}}_{\rho})^2\bar{a})w^{\bar{a}}_{\rho\rho}=$$ $$\frac{w^{\bar{a}}_{\rho}}{\rho(\rho-R(\bar{a})+R(1))}\left((N+2-\bar{a})(\rho-R(\bar{a})+R(1))
-\bar{a}\rho(N-1)\frac{g((w^{\bar{a}}_{\rho})^2\bar{a})}{g((w^{\bar{a}}_{\rho})^2)}\right)$$ and $$(1-\bar{a})\left(w^{\bar{a}}_{\rho\rho}+\frac{w^{\bar{a}}_{\rho}}{\rho}g((w^{\bar{a}}_{\rho})^2\bar{a})\right)=$$ $$\frac{(1-\bar{a})w^{\bar{a}}_{\rho}}{\rho(\rho-R(\bar{a})+R(1))}\left((\rho-R(\bar{a})+R(1))
g((w^{\bar{a}}_{\rho})^2\bar{a})-\frac{(N-1)\rho}{g((w^{\bar{a}}_{\rho})^2)}\right).$$
Since $$F(w^{\bar{a}})=(N-2+\bar{a})\frac{w^{\bar{a}}_{\rho}}{\rho}+\bar{a}g((w^{\bar{a}}_{\rho})^2\bar{a})w^{\bar{a}}_{\rho\rho}+(1-\bar{a})\left(w^{\bar{a}}_{\rho\rho}+\frac{w^{\bar{a}}_{\rho}}{\rho}g((w^{\bar{a}}_{\rho})^2)\right),$$ we obtain that $F(w^{\bar{a}})\geq0$ if and only if $$(\rho-R(\bar{a})+R(1))\left(N-2+\bar{a}+(1-\bar{a})g((w^{\bar{a}}_{\rho})^2\bar{a})\right)-$$ $$\frac{(N-1)\rho}{g((w^{\bar{a}}_{\rho})^2)}\left(1-\bar{a}+\bar{a}g((w^{\bar{a}}_{\rho})^2\bar{a})\right)\leq0.$$ Set $$k(\rho)=(\rho-R(\bar{a})+R(1))\left(N-2+\bar{a}+(1-\bar{a})g((w^{\bar{a}}_{\rho})^2\bar{a})\right)-$$ $$\frac{(N-1)\rho}{g((w^{\bar{a}}_{\rho})^2)}\left(1-\bar{a}+\bar{a}g((w^{\bar{a}}_{\rho})^2\bar{a})\right)$$ Since $$\frac{d}{d\rho}g((w^{\bar{a}}_{\rho})^2)\leq\frac{d}{d\rho}g((w^{\bar{a}}_{\rho})^2\bar a)\leq0,$$ applying Lemma \[l1\], we have that $$k^\prime(\rho)=$$ $$\left(N-2+\bar{a}+(1-\bar{a})g((w^{\bar{a}}_{\rho})^2\bar{a})\right)+
(\rho-R(\bar{a})+R(1))(1-\bar{a})\frac{d}{d\rho}g((w^{\bar{a}}_{\rho})^2\bar a)-$$ $$(N-1)\left(\frac{1}{g((w^{\bar{a}}_{\rho})^2)}-\frac{\rho}{(g((w^{\bar{a}}_{\rho})^2))^2}\frac{d}{d\rho}g((w^{\bar{a}}_{\rho})^2)\right)\left(1-\bar{a}+\bar{a}g((w^{\bar{a}}_{\rho})^2\bar{a})\right)-$$ $$(N-1)\frac{\rho}{g((w^{\bar{a}}_{\rho})^2)}\frac{d}{d\rho}g((w^{\bar{a}}_{\rho}(\rho))^2\bar a)\bar{a}\leq$$ $$\frac{(N-1)\rho}{(g((w^{\bar{a}}_{\rho})^2))^2}\frac{d}{d\rho}g((w^{\bar{a}}_{\rho})^2)\left(1-\bar{a}+\bar{a}g((w^{\bar{a}}_{\rho})^2\bar{a})\right)-\frac{(N-1)\rho}{g((w^{\bar{a}}_{\rho})^2)}\frac{d}{d\rho}g((w^{\bar{a}}_{\rho}(\rho))^2\bar a)\bar{a}\leq$$ $$\frac{(N-1)\rho}{(g((w^{\bar{a}}_{\rho})^2))^2}\frac{d}{d\rho}g((w^{\bar{a}}_{\rho})^2)\left(1-\bar{a}+\bar{a}g((w^{\bar{a}}_{\rho})^2\bar{a})-\bar{a}g((w^{\bar{a}}_{\rho})^2)\right)\leq$$ $$\frac{(N-1)\rho}{(g((w^{\bar{a}}_{\rho})^2))^2}(1-\bar{a})g((w^{\bar{a}}_{\rho})^2)\frac{d}{d\rho}g((w^{\bar{a}}_{\rho})^2)\leq0.$$ Since the function $k(\rho)$ is non increasing, it follows that $F(w^{\bar{a}})\geq0$, for every $\rho\in(R(\bar{a}),R(\bar{a})+r)$, if and only if $$R(1)\left(N-2+\bar{a}+(1-\bar{a})g((w^{\bar{a}}_{\rho}(R(\bar{a})))^2\bar{a})\right)-$$ $$\frac{(N-1)R(\bar{a})}{g((w^{\bar{a}}_{\rho}(R(\bar{a}))^2)}\left(1-\bar{a}+\bar{a}g((w^{\bar{a}}_{\rho}(R(\bar{a})))^2\bar{a})\right)=$$ $$R(1)\left(N-2+\bar{a}+(1-\bar{a})g\left(\frac{\epsilon^2}{r^2}\bar{a}\right)\right)-
\frac{(N-1)R(\bar{a})}{g\left(\frac{\epsilon^2}{r^2}\right)}\left(1-\bar{a}+\bar{a}g\left(\frac{\epsilon^2}{r^2}\bar{a}\right)\right)\leq0,$$ i.e. if and only if $$R(\bar{a})\geq g\left(\frac{\epsilon^2}{r^2}\right)\frac{R(1)}{N-1}\frac{N-2+\bar{a}+g\left(\frac{\epsilon^2}{r^2}\bar{a}\right)(1-\bar{a})}{1-\bar{a}+g\left(\frac{\epsilon^2}{r^2}\bar{a}\right)\bar{a}}.$$ Hence, we define $$R(\bar{a})=g\left(\frac{\epsilon^2}{r^2}\right)\frac{R(1)}{N-1}\frac{N-2+\bar{a}+g\left(\frac{\epsilon^2}{r^2}\bar{a}\right)(1-\bar{a})}{1-\bar{a}+g\left(\frac{\epsilon^2}{r^2}\bar{a}\right)\bar{a}}.$$\
4) The function $w^{\bar{a}}$ defined in point 3) is a solution to $$F(w^{\bar{a}})=w^{\bar{a}}_{\rho\rho}(1-a)+\frac{w^{\bar{a}}_{\rho}}{\rho}(N-2+a)+
g((w^{\bar{a}}_\rho)^2a)\left(w^{\bar{a}}_{\rho\rho}a+\frac{w^{\bar{a}}_{\rho}}{\rho}(1-a)\right)\geq0,$$ for every $a<\bar{a}$. Indeed, applying Lemma \[l1\] we obtain that, for every $\rho\in(R(\bar{a}),R(\bar{a})+r)$, $$(\rho-R(\bar{a})+R(1))\left(N-2+a+(1-a)g((w^{\bar{a}}_{\rho})^2a)\right)-$$ $$\frac{(N-1)\rho}{g((w^{\bar{a}}_{\rho})^2)}\left(1-a+ag((w^{\bar{a}}_{\rho})^2a)\right)\leq$$ $$(\rho-R(\bar{a})+R(1))\left(N-2+\bar{a}+(1-\bar{a})g((w^{\bar{a}}_{\rho})^2\bar{a})\right)-$$ $$\frac{(N-1)\rho}{g((w^{\bar{a}}_{\rho})^2)}\left(1-\bar{a}+\bar{a}g((w^{\bar{a}}_{\rho})^2\bar{a})\right)\leq0.$$\
5) Assume we have a partition $\alpha$ of $[0,\pi/2]$, $\alpha=\{0=\alpha_n<\dots<\alpha_1<\alpha_0=\pi/2\}$. This partition defines two partitions of $[0,1]$, given by $c_i=\cos\alpha_i$ and $s_i=\sin\alpha_i$.
Consider the sums $$S_1(\alpha)=\sum_{i=1}^{n}\left(R(1-c_{i-1}^2)-R(1-c_i^2)\right)c_i=R(1)c_1+\sum_{i=1}^{n-1}R(1-c_i^2)\left(c_{i+1}-c_i\right)$$ and $$S_2(\alpha)=\sum_{i=1}^{n}R(s_{i-1}^2)(s_{i-1}-s_i)=R(1)(1-s_1)+\sum_{i=1}^{n-1}R(s_i^2)(s_i-s_{i+1}),$$ where, in the previous equalities, we have taken into account that $R(1-c_n^2)=R(0)=0$. Our purpose is to provide a partition $\alpha$ and corresponding estimates for $S_1(\alpha)$ and $S_2(\alpha)$ that are independent of $\epsilon$.
The sums $$\sum_{i=1}^{n-1}R(1-c^2_i)(c_{i+1}-c_i)\quad\mbox{ and }\quad\sum_{i=1}^{n-1}R(s_i^2)(s_i-s_{i+1})$$ are Riemann sums for the integrals $$\int^1_{c_1}R(1-c^2)dc\quad\mbox{ and }\quad\int^{s_1}_0R(s^2)ds.$$ Consider the first integral. From $$\frac{N-1-c^2+g\left(\frac{\epsilon^2}{r^2}(1-c^2)\right)c^2}{c^2+g\left(\frac{\epsilon^2}{r^2}(1-c^2)\right)(1-c^2)}\leq\frac{N-1}{c^2}$$ we obtain that $$\int^1_{c_1}R(1-c^2)dc=g\left(\frac{\epsilon^2}{r^2}\right)\frac{R(1)}{N-1}\int^1_{c_1}\frac{N-1-c^2+g\left(\frac{\epsilon^2}{r^2}(1-c^2)\right)c^2}{c^2+g\left(\frac{\epsilon^2}{r^2}(1-c^2)\right)(1-c^2)}dc\leq$$ $$R(1)g\left(\frac{\epsilon^2}{r^2}\right)\int^1_{c_1}\frac{dc}{c^2}.$$ Set $$S_x^*(c)=R(1)c+R(1)g\left(\frac{\epsilon^2}{r^2}\right)\int^1_c{\frac{db}{b^2}}=R(1)\left(c+g\left(\frac{\epsilon^2}{r^2}\right)\left(\frac{1}{c}-1\right)\right)=$$ $$R(1)g\left(\frac{\epsilon^2}{r^2}\right)\left(\frac{c}{g\left(\frac{\epsilon^2}{r^2}\right)}+\frac{1}{c}-1\right).$$ Evaluating the last term at the minimum point $c=\sqrt{g\left(\frac{\epsilon^2}{r^2}\right)},$ we obtain $$S_x^*\left(\sqrt{g\left(\frac{\epsilon^2}{r^2}\right)}\right)=R(1)g\left(\frac{\epsilon^2}{r^2}\right)\left(\frac{2}{\sqrt{g\left(\frac{\epsilon^2}{r^2}\right)}}-1\right)=$$ $$\frac{2r\sqrt{g\left(\frac{\epsilon^2}{r^2}\right)}}{e^{G(\epsilon/r)}-1}-R(1)g\left(\frac{\epsilon^2}{r^2}\right).$$ We fix $c_1=\sqrt{g\left(\frac{\epsilon^2}{r^2}\right)}$, so that $\alpha_1=\arccos{\sqrt{g\left(\frac{\epsilon^2}{r^2}\right)}}$.
Consider the second integral $$\int^{s_1}_0R(s^2)ds.$$ From $$\frac{N-2+s^2+g\left(\frac{\epsilon^2}{r^2}s^2\right)(1-s^2)}{1-s^2+g\left(\frac{\epsilon^2}{r^2}\right)s^2}\leq\frac{N-2+s^2+g\left(\frac{\epsilon^2}{r^2}s^2\right)(1-s^2)}{1-s^2}$$ we obtain that $$\int^{s_1}_0R(s^2)ds=g\left(\frac{\epsilon^2}{r^2}\right)\frac{R(1)}{N-1}\int^{s_1}_0
\frac{N-2+s^2+g\left(\frac{\epsilon^2}{r^2}s^2\right)(1-s^2)}{1-s^2+g\left(\frac{\epsilon^2}{r^2}s^2\right)s^2}ds\leq$$ $$g\left(\frac{\epsilon^2}{r^2}\right)\frac{R(1)}{N-1}\int^{s_1}_0\frac{N-2+s^2+g\left(\frac{\epsilon^2}{r^2}\right)(1-s^2)}{1-s^2}ds.$$ Set $$S_y^*(s)=R(1)(1-s)+g\left(\frac{\epsilon^2}{r^2}\right)\frac{R(1)}{N-1}\int^s_0\frac{N-2+b^2+g\left(\frac{\epsilon^2}{r^2}\right)(1-b^2)}{1-b^2}db=$$ $$R(1)(1-s)+g\left(\frac{\epsilon^2}{r^2}\right)\frac{R(1)}{N-1}\left[\left(g\left(\frac{\epsilon^2}{r^2}\right)-1\right)s+\frac{N-1}{2}\ln\left(\frac{1+s}{1-s}\right)\right].$$ Since $$1-\sqrt{1-g\left(\frac{\epsilon^2}{r^2}\right)}\leq g\left(\frac{\epsilon^2}{r^2}\right),$$ evaluating the last term at the point $s_1=\sin\alpha_1=\sqrt{1-g\left(\frac{\epsilon^2}{r^2}\right)}$, we obtain $$S_x^*\left(\sqrt{1-g\left(\frac{\epsilon^2}{r^2}\right)}\right)=R(1)\left(1-\sqrt{1-g\left(\frac{\epsilon^2}{r^2}\right)}\right)+$$ $$g\left(\frac{\epsilon^2}{r^2}\right)\frac{R(1)}{N-1}\left[\left(g\left(\frac{\epsilon^2}{r^2}\right)-1\right)\sqrt{1-g\left(\frac{\epsilon^2}{r^2}\right)}-\frac{1}{2}\ln\left(\frac{1-\sqrt{1-g\left(\frac{\epsilon^2}{r^2}\right)}}{1+\sqrt{1-g\left(\frac{\epsilon^2}{r^2}\right)}}\right)\right]\leq$$ $$R(1)\sqrt{g\left(\frac{\epsilon^2}{r^2}\right)}\left(\sqrt{g\left(\frac{\epsilon^2}{r^2}\right)}+\frac{1}{2}\sqrt{g\left(\frac{\epsilon^2}{r^2}\right)}\left|\ln g\left(\frac{\epsilon^2}{r^2}\right)\right|\right).$$ To define the other points of the required partition $\alpha$, consider the integrals $$\int^1_{c_1}{R(1-c^2)dc}\quad\mbox{ and }\quad\int^{s_1}_0R(s^2)ds.$$ Set $$\sigma=R(1)g\left(\frac{\epsilon^2}{r^2}\right).$$ By the basic theorem of Riemann integration, taking a partition $\alpha$ with mesh size small enough, the value of the Riemann sums $$\sum_{i=1}^{n-1}R(1-c^2_i)(c_{i+1}-c_i)\quad\mbox{ and }\quad\sum_{i=1}^{n-1}R(s^2_i)(s_i-s_{i+1})$$ differs from $$\int^1_{c_1}{R(1-c^2)dc}\quad\mbox{ and }\quad\int^{s_1}_0R(s^2)ds$$ by less than $\sigma$. In particular we obtain $$S_1(\alpha)=R(1)c_1+\sum_{i=1}^{n-1}R(1-c^2_i)\left(c_{i+1}-c_i\right)\leq R(1)c_1+\int^1_{c_1}{R(1-c^2)dc}+\sigma\leq$$ $$\frac{2r\sqrt{g\left(\frac{\epsilon^2}{r^2}\right)}}{e^{\frac{G(\epsilon/r)}{N-1}}-1}\leq2Kr$$ and $$S_2(\alpha)=R(1)(1-s_1)+\sum_{i=1}^{n-1}R(s^2_i)(s_i-s_{i+1})\leq$$ $$R(1)(1-s_1)+\int^{s_1}_0R(s^2)ds+\sigma\leq$$ $$\frac{r\sqrt{g\left(\frac{\epsilon^2}{r^2}\right)}}{e^{\frac{G(\epsilon/r)}{N-1}}-1}\left(2\sqrt{g\left(\frac{\epsilon^2}{r^2}\right)}+\frac{1}{2}\sqrt{g\left(\frac{\epsilon^2}{r^2}\right)}\left|\ln g\left(\frac{\epsilon^2}{r^2}\right)\right|\right)\leq\frac{r}{4}.$$\
6) With respect to the coordinates fixed at the beginning of the proof, consider $x_i\geq0$. Set $$\mathcal{D}_0=\{(x_1,\dots,x_N):R(1)<\rho<R(1)+r,\sqrt{a_1}\leq\sin\theta_{N-1}\leq\sqrt{a_0}=1\}$$ and on $\mathcal{D}_0$ define the function $$v_0(x_1,\dots,x_N)=w(\rho(x_1,\dots,x_N)).$$ By point 1), the function $v_0$ is of class $C^2(int(\mathcal{D}_0))$ and satisfies, pointwise, the inequality $F(v_0)\geq0$. Having defined $v_0$, define $v_1$ as follows. Set: $$O_1(\theta_1,\dots,\theta_{N-2})=(O^1_{x_1},\dots,O^1_{x_N})=$$ $$(R(1)-R(a_1))\left(\sqrt{1-a_1}\cos\theta_{N-2}\dots\cos\theta_1,\sqrt{1-a_1}\cos\theta_{N-2}\dots\sin\theta_1,\dots,\sqrt{a_1}\right),$$ $$\rho_1(x_1,\dots,x_N)=\sqrt{(x_1-O^1_{x_1})^2+\dots+(x_N-O^1_{x_N})^2}$$ and $$\sin\theta_{N-1}^1(x_1,\dots,x_N)=\frac{x_N-O^1_{x_N}}{\rho_1(x_1,\dots,x_N)}.$$ Recalling the definition of $w^{a_1}$ in 3), consider $$\mathcal{D}_1=\left\{(x_1,\dots,x_N):R(a_1)<\rho_1(x_1,\dots,x_N)<R(a_1)+r,\right.$$ $$\left.\sqrt{a_2}\leq\sin\theta_{N-1}^1(x_1,\dots,x_N)\leq\sqrt{a_1}\right\}$$ and, on $\mathcal{D}_1$, set $$v_1(x_1,\dots,x_N)=w^{a_1}(\rho_1(x_1,\dots,x_N)).$$ The function $v_1$ is of class $C^2(int(\mathcal{D}_1))$. We claim that $v_1$ still satisfies $F(v_1)\geq0$. Remark that the set of the points $O^1$ is equal to $$\mathcal O^1=\left\{(x_1,\dots,x_N):\rho=R(1)-R(a_1),\quad\sin\theta_{N-1}=\sqrt{a_1}\right\}$$ and that for every point $p\in\mathcal{D}_1$, the corresponding point $O^1(p)$ is the projection of $p$ on $\mathcal O^1$, while $\rho_1(p)=d(p,\mathcal O^1)$. Then we obtain $$\frac{\partial\rho_1}{\partial\theta_i}=0 \quad\mbox{for every}\quad i=1,\dots,N-2,$$ $$\frac{\partial O^1_{x_i}}{\partial x_i}\geq0 \quad\mbox{for every}\quad i=1,\dots,N-1,$$ $$\frac{\partial O^1_{x_N}}{\partial x_N}=0.$$ and $$\nabla v_1=\frac{w^{a_1}_{\rho}(\rho_1)}{\rho_1} \left(x_1-O^1_{x_1},\dots,x_N-O^1_{x_N}\right)=$$ $$w^{a_1}_{\rho}(\rho_1)\left(\cos\theta_{N-1}^1\dots\cos\theta_1,\cos\theta_{N-1}^1\dots\sin\theta_1,\dots,\sin\theta_{N-1}^1\right),$$ $$(v_1)_{x_ix_i}=w^{a_1}_{\rho\rho}(\rho_1)\left(\frac{x_i-O^1_{x_i}}{\rho_1}\right)^2
+\frac{w^{a_1}_{\rho}(\rho_1)}{\rho_1}\left(1-\left(\frac{x_i-O^1_{x_i}}{\rho_1}\right)^2
-\frac{\partial O^1_{x_i}}{\partial x_i}\right)=$$ $$w^{a_1}_{\rho\rho}(\rho_1)\cos^2\theta_{N-1}\dots\sin^2\theta_{i-1}
+\frac{w^{a_1}_{\rho}(\rho_1)}{\rho_1}\left(1-\cos^2\theta_{N-1}\dots\sin^2\theta_{i-1}
-\frac{\partial O^1_{x_i}}{\partial x_i}\right).$$ Then $$F(v_1)=\sum_{i=1}^{N-1}(v_1)_{x_ix_i}+(v_1)_{x_Nx_N}g((v_1)_{x_N}^2)=$$ $$w^{a_1}_{\rho\rho}(\rho_1)\left(\cos^2\theta_{N-1}^1+\sin^2\theta_{N-1}^1g\left((w^{a_1}_{\rho}(\rho_1))^2\sin^2\theta_{N-1}^1\right)\right)+$$ $$\frac{w^{a_1}_{\rho}(\rho_1)}{\rho_1}\left(N-2+\sin^2\theta_{N-1}^1
+\cos^2\theta_{N-1}^1g\left((w^{a_1}_{\rho}(\rho_1))^2\sin^2\theta_{N-1}^1\right)\right)-$$ $$\frac{w^{a_1}_{\rho}(\rho_1)}{\rho_1}\sum_{i=1}^{N-1}\frac{\partial O^1_{x_i}}{\partial x_i}\geq0$$ since $w^{a_1}(\rho_1)$ verifies equation (\[eqa\]).
The sets $\mathcal{D}_0$ and $\mathcal{D}_1$ intersect on $\sin\theta_2(x_1,\dots,x_N)=\sin\theta_2^1(x_1,\dots,x_N)=\sqrt{a_1}$. For a point $(x_1,\dots,x_N)$ in this intersection we have $$\rho_1(x_1,\dots,x_N)=\rho(x_1,\dots,x_N)-(R(1)-R(a_1)).$$ Hence, on $\mathcal{D}_0\cap\mathcal{D}_1$ $$R(a_1)\leq\rho_1(x_1,\dots,x_N)\leq R(a_1)+r$$ if and only if $$R(1)\leq\rho(x_1,\dots,x_N)\leq R(1)+r,$$ and the functions $v_0$ and $v_1$ coincide: $$v_1(x_1,\dots,x_N)=w^{a_1}(\rho_1(x_1,\dots,x_N))=w(\rho_1(x_1,\dots,x_N)+R(1)-R(a_1))=$$ $$w(\rho(x_1,\dots,x_N))=v_0(x_1,\dots,x_N).$$ The formula $$\bar{v}(x_1,\dots,x_N)=\left\{\begin{array}{ll}
v_0(x_1,\dots,x_N) & \mbox{ on }\mathcal{D}_0\\
v_1(x_1,\dots,x_N) & \mbox{ on }\mathcal{D}_1\\
\end{array}
\right.$$ defines a function $\bar{v}$ in $C^0(int(\mathcal{D}_0\cup\mathcal{D}_1))$. We claim that it is also in $C^1(int(\mathcal{D}_0\cup\mathcal{D}_1))$.
In fact, we have $$\nabla v_0(x_1,\dots,x_N)=\frac{w_{\rho}(\rho(x_1,\dots,x_N))}{\rho(x_1,\dots,x_N)}(x_1,\dots,x_N),$$ $$\nabla v_1(x_1,\dots,x_N)=
\frac{w^{a_1}_{\rho}(\rho_1(x_1,\dots,x_N))}{\rho_1(x_1,\dots,x_N)}\left(x_1-O^1_{x_1},\dots,x_N-O^1_{x_N}\right)$$ and, on $\mathcal{D}_0\cap\mathcal{D}_1$, $$\frac{1}{\rho_1(x_1,\dots,x_N)}\left(x_1-O^1_{x_1},\dots,x_N-O^1_{x_N}\right)=$$ $$\frac{1}{\rho(x_1,\dots,x_N)}(x_1,\dots,x_N).$$ On $int(\mathcal{D}_0)$ and $int(\mathcal{D}_1)$, the function $\bar{v}$ is of class $C^2$ and satisfies, pointwise, the inequality $F(\bar{v})\geq0$. We claim that $\bar{v}$ is also in $W^{1,2}(int(\mathcal{D}_0\cup\mathcal{D}_1))$ and that it is a weak solution to $F(\bar{v})\geq0$ on $int(\mathcal{D}_0\cup\mathcal{D}_1)$. In fact, for every $\eta\in C^\infty_0(int(\mathcal{D}_0\cup\mathcal{D}_1))$, applying the divergence theorem separately to $int(\mathcal{D}_0)$ and to $int(\mathcal{D}_1)$, we obtain $$\int_{int(\mathcal{D}_0\cup\mathcal{D}_1)}
{\left[\mbox{div}\nabla_{\nabla v} L(\nabla\bar{v}(x))\eta(x)+\langle\nabla L(\nabla\bar{v}(x)),\nabla\eta(x)\rangle\right]dx}=$$ $$\int_{int(\mathcal{D}_0)\cup int(\mathcal{D}_1)}
{\left[\mbox{div}\nabla_{\nabla v} L(\nabla\bar{v}(x))\eta(x)+\langle\nabla L(\nabla\bar{v}(x)),\nabla\eta(x)\rangle\right]dx}=$$ $$\int_{\partial(int(\mathcal{D}_0))}{\eta(x)\langle\nabla L(\nabla\bar{v}(x)),\mbox{\bf n}(x)\rangle dl}+\int_{\partial(int(\mathcal{D}_1))}{\eta(x)\langle\nabla L(\nabla\bar{v}(x)),\mbox{\bf{n}}(x)\rangle dl}=$$ $$\int_{\mathcal{D}_0\cap\{\sin\theta_{N-1}=\sqrt{a_1}\}}{\eta(x)\langle\nabla L(\nabla\bar{v}(x)),\mbox{\bf n}(x)\rangle dl}+$$ $$\int_{\mathcal{D}_1\cap\{\sin\theta^1_{N-1}=\sqrt{a_1}\}}{\eta(x)\langle\nabla L(\nabla\bar{v}(x)),\mbox{\bf n}(x)\rangle dl}.$$ The last term equals zero, since $\bar{v}\in C^1(int(\mathcal{D}_0\cup\mathcal{D}_1))$. Hence, when if $\eta\geq0$, we have that $$\int_{int(\mathcal{D}_0\cup\mathcal{D}_1)}{\langle\nabla L(\nabla\bar{v}(x)),\nabla\eta(x)\rangle dx}\leq0,$$ as we wanted to show.
Assuming defined $O^{n-2}(\theta_1,\dots,\theta_{N-2})$ and a function $v\in C^1(int(\mathcal{D}_0\cup\dots\cup\mathcal{D}_{n-2}))$, consider $$O^{n-1}(\theta_1,\dots,\theta_{N-2})=(O^{n-1}_{x_1},\dots,O^{n-1}_{x_N})=$$ $$O^{n-2}(\theta_1,\dots,\theta_{N-2})+(R(a_{n-2})-R(a_{n-1}))$$ $$\left(\sqrt{1-a_{n-1}}\cos\theta_{N-2}\dots\cos\theta_1,\sqrt{1-a_{n-1}}\cos\theta_{N-2}\dots\sin\theta_1,\dots,\sqrt{a_{n-1}}\right).$$ Set $$\rho_{n-1}(x_1,\dots,x_N)=\sqrt{(x_1-O^{n-1}_{x_1})^2+\dots+(x_N-O^{n-1}_{x_N})^2},$$ $$\sin\theta_{N-1}^{n-1}(x_1,\dots,x_N)=\frac{x_N-O^{n-1}_{x_N}}{\rho_{n-1}(x_1,\dots,x_N)}$$ $$\mathcal{D}_{n-1}=\left\{(x_1,\dots,x_N):R(a_n)<\rho_{n-1}(x_1,\dots,x_N)<R(a_{n-1})+r,\right.$$ $$\left.0=\sqrt{a_n}\leq\sin\theta_{N-1}^{n-1}(x_1,\dots,x_N)\leq\sqrt{a_{n-1}}\right\}$$ and define on $\mathcal{D}_{n-1}$ the function $$v_{n-1}(x_1,\dots,x_N)=w^{a_{n-1}}(\rho_{n-1}(x_1,\dots,x_N)).$$ Set $\mathcal{D}=int(\mathcal{D}_0\cup\dots\cup\mathcal{D}_{n-1})$, the same considerations as before imply that the function $$\bar{v}(x_1,\dots,x_N)=\left\{\begin{array}{ll}
v_0(x_1,\dots,x_N) & \mbox{ on }\mathcal{D}_0\\
\dots & \dots \\
v_{n-1}(x_1,\dots,x_N) & \mbox{ on }\mathcal{D}_{n-1}\\
\end{array}
\right.$$ is such that $\bar{v}\in W^{1,2}(\mathcal{D})\cap C^1(\mathcal{D})\cap C(\overline{\mathcal{D}})$ and it is a weak solution to $F(\bar{v})\geq0$ on $\mathcal{D}$. This completes the construction of $\bar{v}$ as a weak solution to $F(\bar{v})\geq0$ on $\mathcal{D}_0\cup\dots\cup\mathcal{D}_{N-1}$.
Set $O^*=(O^*_{x_1},\dots,O^*_{x_N})=(0,\dots,0,R(1)-l_N)$. We have that $$\mathcal{D}_0\cup\dots\cup\mathcal{D}_{n-1}\subset\left\{(x_1,\dots,x_N):0\leq x_i\leq l+r
\mbox{ for }i=1,\dots,N-1,\right.$$ $$\left.O^*_{x_N}\leq x_N\leq O^*_{x_N}+l_N+r)\right\}.$$
Define the full domain $\omega$ and the solution by symmetry with respect to the point $O^*$. Figure \[fA\] shows this construction in dimension $N=2$ and for $n-1=2$. Hence the solution will be in $W^{1,2}(\omega)\cap C^1(\omega)\cap C(\overline{\omega})$ and a weak solution of $F(v)\geq0$ on $\omega$.
units < .35cm, .35cm> x from -15 to 15, y from -16 to 10 at 0 0 from 0 8 to 0 -15 0 -15 5.952838 7.216293 / at 0 -15 2.070553 -7.272593 10.6742 5.014687 / at 2.070553 -7.272593 from 0 0 to 13.28468 0 at 7.162878 0 5.4352 5.284443 0 5.284443 / from 2.070553 -7.272593 to 2.070553 6.727407 from 7.162878 0 to 7.162878 6 9.527046 3.376383 2.070553 3.376383 / from 0 -15 to 2 -15 from 2.070553 -7.272593 to 4.070553 -7.272593 -74.99997 degrees from .258819 -14.03407 center at 0 -15 -54.99998 degrees from 2.644129 -6.453442 center at 2.070553 -7.272593 ([.]{}) -30 degrees from -5.4352 5.284443 center at 0 -15 -30 degrees from -5.952838 7.216293 center at 0 -15 30 degrees from -5.435199 -5.284443 center at 0 15 30 degrees from -5.952837 -7.216293 center at 0 15 -20 degrees from 5.4352 5.284442 center at 2.070553 -7.272593 -20 degrees from 5.952839 7.216294 center at 2.070553 -7.272593 20.00001 degrees from -5.4352 5.284442 center at -2.070553 -7.272593 20.00001 degrees from -5.952838 7.216294 center at -2.070553 -7.272593 -20 degrees from -5.435199 -5.284443 center at -2.070553 7.272593 -20 degrees from -5.952837 -7.216294 center at -2.070553 7.272593 20 degrees from 5.4352 -5.284442 center at 2.070553 7.272593 19.99999 degrees from 5.952838 -7.216294 center at 2.070553 7.272593 -110 degrees from 9.527046 3.376383 center at 7.162878 0 -110 degrees from 10.6742 5.014687 center at 7.162878 0 -110 degrees from -9.527045 -3.376384 center at -7.162878 0 -110 degrees from -10.6742 -5.014688 center at -7.162878 0 ([.]{}) at -1 0 at -1 -15 at 1.070553 -7.272593 at 7.962878 -.9 at 1.5 -14 at 3.570553 -6.472593 at 5.2352 6.284443 at -5.4352 6.084442 at -5.4352 1.284442 at 1.5 6.8 at 8.2352 5.284443 at 11.52705 2.676383
7\) The previous construction yields a region $\mathcal{A}$ centered in $O^*$, a corresponding region $\omega$ and a function $v$ that solves (\[eqnN\]). The change of coordinates $\hat x_1=x_1,\dots,\hat x_{N-1}=x_{N-1}$, $\hat x_N=x_N-O^*_N$, centers $\mathcal{A}$ at the origin and proves the theorem.\
In order to prove Theorem \[thsmp\], we need this further lemma.
\[dist\]
Consider the sets $\mathcal A$ and $\mathcal{R}(O^*,l,l_N)$, where $\mathcal A$, $O^*$, $l$, $l_N$ have been defined in Theorem \[thsmp\]. Then, for every $p\in\partial\mathcal R$, $$d(p,\overline{\mathcal A})<l_N.$$
Set $q=O^*+(l,\dots,l,l_N)$. We prove that $$d(q,\overline{\mathcal A})<l_N.$$ Set $p_i=O^*+(0,\dots,0,l,0,\dots,0)$ and let $\Pi^{N-1}$ the hyperplane passing through $p_1,\dots,p_N$. Since $\overline{\mathcal A}$ is convex and $p_i\in\overline{\mathcal A}$, we obtain that $$d(q,\overline{\mathcal A})<d(q,\Pi^{N-1})<l_N.$$ See Figure \[fdist\].
units < .4cm, .4cm> x from -16 to 15, y from -8 to 8 from 14 0 to -14 0 from 0 8 to 0 -8 ([.]{}) -30 degrees from -5.4352 5.284443 center at 0 -15 30 degrees from -5.435199 -5.284443 center at 0 15 -20 degrees from 5.4352 5.284442 center at 2.070553 -7.272593 20.00001 degrees from -5.4352 5.284442 center at -2.070553 -7.272593 -20 degrees from -5.435199 -5.284443 center at -2.070553 7.272593 20 degrees from 5.4352 -5.284442 center at 2.070553 7.272593 -110 degrees from 9.527046 3.376383 center at 7.162878 0 -110 degrees from -9.527045 -3.376384 center at -7.162878 0 ([.]{}) from 11.28468 6 to -11.28468 6 from -11.28468 6 to -11.28468 -6 from -11.28468 -6 to 11.28468 -6 from 11.28468 -6 to 11.28468 6 11.28468 0 0 6 / 11.28468 6 8.797606 1.322362 / at -3 3 at -5.6 6.8 at -.8 .8 at 12.08468 6.6 at .8 6.8 at 12.08468 .6 at 5.68468 6.6 at 12.08468 3
a\) Suppose that $u$ attains its minimum in $\Omega$, and assume $\min_\Omega u=0$ and set $\mathcal{C}=\{x\in\Omega:u(x)=0\}$. By contradiction, suppose that the open set $\Omega\setminus\mathcal{C}\neq\emptyset$.\
b) Since $\Omega$ is a connected set, there exist $s\in\mathcal{C}$ and $R>0$ such that $B(s,R)\subset\Omega$ and $B(s,R)\cap(\Omega\setminus\mathcal{C})\neq\emptyset$. Let $p\in B(s,R)\cap(\Omega\setminus\mathcal{C})$. Consider the line $\overline{ps}$. Moving $p$ along this line, we can assume that $B(p,d(p,\mathcal{C}))\subset(\Omega\setminus\mathcal{C})$, and that there exists one point $z\in\mathcal{C}$ such that $d(p,\mathcal{C})=d(p,z)$.\
c) Fix $r$: $$0<r<\frac{d(p,\mathcal{C})}{32(N-1)K^2+\frac{7}{8}}.$$ Set $$\epsilon(r)=\min\left\lbrace u(z):z\in\overline{B\left(p,d(p,\mathcal{C})-\dfrac{r}{4}\right)}\right\rbrace,$$ we have that $\epsilon(r)>0$, and we set $\epsilon=\min\{\epsilon(r),r\bar{\xi}\}$.\
d) For $r$ and $\epsilon$ as defined in c), consider: $l$, $l_N$, $\mathcal{A}$ and $v$ as defined in Theorem \[subsol\]. Without loss of generality, since the set $\mathcal{A}$ is symmetric with respect to both coordinate axis, we can suppose that $\overline{pz}$ belongs to the first quadrant, i.e. that, for every $i=1,\dots,N$, $z_i\geq p_i$, where $z=(z_1,\dots,z_N)$ and $p=(p_1,\dots,p_N)$.
Define the point $q$ on the segment $\overline{pz}$ such that $d(q,p)=d(p,z)-\dfrac{r}{2}$. Set $q^*=q-(l,\dots,l,l_N)$, $\mathcal{R}(q^*,l,l_N)=q^*+\mathcal{R}(l,l_N)$, $\mathcal{A}^*=q^*+\mathcal{A}$ and $v^*(x+q^*)=v(x)$. We first claim that $$\mathcal{R}(q^*,l,l_N)\subset B\left(p,d(p,\mathcal{C})-\dfrac{r}{4}\right).$$ Let $t\in\mathcal{R}(q^*,l,l_N)$, then $t$ can be written as $(q_1-2\alpha_1l,\dots,q_{N-1}-2\alpha_{N-1}l,q_N-2\alpha_Nl_N)$, with $0\leq\alpha_i\leq1$, for $i=1,\dots,N$. Since $r<\frac{d(p,\mathcal{C})}{32(N-1)K^2+\frac{7}{8}}$, we have that $$d(t,p)^2=\sum_{i=1}^N(q_i-2\alpha_il_i-p_i)^2=d(q,p)^2+\sum_{i=1}^N4\alpha_i^2l_i^2-\sum_{i=1}^N4\alpha_il_i(q_i-p_i)\leq$$ $$\left(d(p,\mathcal{C})-\dfrac{r}{2}\right)^2+\sum_{i=1}^N4l_i^2\leq\left(d(p,\mathcal{C})-\dfrac{r}{2}\right)^2+16(N-1)K^2r^2+\frac{r^2}{4}<\left(d(p,\mathcal{C})-\dfrac{r}{4}\right)^2.$$ See Figure \[fmax\].
units < .35cm, .35cm> x from -12 to 21, y from -4 to 21 from 0 20.5 to 0 0 from 0 0 to 20.5 0 -120 degrees from -6.180338 19.02113 center at 0 0 -116 degrees from -5.262691 17.21349 center at 0 0 -111 degrees from -4.410198 15.38019 center at 0 0 ([.]{}) from 13.10643 9.177223 to 7.106433 9.177223 from 7.106433 9.177223 to 7.106433 7.177223 from 7.106433 7.177223 to 13.10643 7.177223 from 13.10643 7.177223 to 13.10643 9.177223 0 0 18.02135 12.61868 / from 10.10643 9.177223 to 10.10643 8.177223 from 10.10643 8.177223 to 13.10643 8.177223 at 0 0 at 16.38304 11.47153 at 13.10643 9.177223 at -.8 -.8 at 17.38304 11.47153 at -10.5 14.5 at -10.5 16.5 at -11 18.5 at 13.2 10 at 11.6 9.8 at 9.5 8.5 at 5.5 8.2
Since $\mathcal{A}^*\subset\mathcal{R}(q^*,l,l_N)$, we have obtained that $$\mathcal{A}^*\subset B\left(p,d(p,\mathcal{C})-\dfrac{r}{4}\right),$$ so that $u\geq\epsilon$ in $\partial\mathcal A^*$.
By Lemma \[dist\], $$d(q,\overline{\mathcal A^*})<l_N\leq\frac{r}{4},$$ we have that $$d(z,\overline{\mathcal A^*})\leq d(z,q)+d(q,\overline{\mathcal A^*})<\frac{3}{4}r,$$ so that $$z\in\omega^*=B(\mathcal A^*,r)\setminus\overline{\mathcal A^*}.$$
e\) The function $v^*$ satisfies $$\left\{\begin{array}{ll}
v^*\mbox{ is a weak solution to }F(v)\geq0 & \mbox{ in }\omega^*\\
v^*>0 & \mbox{ in }\omega^*\\
v^*=0 & \mbox{ in }\partial B(\mathcal{A}^*,r)\\
v^*\leq\epsilon & \mbox{ in }\partial\mathcal{A}^*.\\
\end{array}
\right.$$ Since $u,v^*\in W^{1,2}(\omega^*)\cap C(\overline{\omega^*})$, $v^*$ is a weak subsolution and $u$ is a weak solution to $F(u)=0$, and $v^*_{|\partial \omega^*}\leq u_{|\partial\omega^*}$, applying Lemma \[comp\], we obtain that $u\geq v^*$ in $\omega^*$. But $u(z)=0<v^*(z)$, a contradiction.\
[DDDDD]{}
, [*The principal eigenvalue and maximum principle for second-order elliptic operators in general domains*]{}, Communications on Pure and Applied Mathematics [**47**]{} (1994), 1, 47–92.
, [*Analyse fonctionnelle, théorie et applications*]{}, Masson, Paris, 1983.
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, [*Nonlinear partial differential equations and free boundaries*]{}, Pitman Research Notes in Mathematics, Vol. 106, 1985.
, [*Partial Differential Equations*]{}, Graduate Studies in Mathematics, Volume 19, American Mathematical Society, Providence, Rhode Island.
, [*On the strong maximum principle for quasilinear elliptic equations and systems*]{}, Adv. Differential Equations [**7**]{} (2002), 1, 25–46.
, [*Elliptic Partial Differential Equations of Second Order*]{}, Springer-Verlag, Berlin Heidelberg New York Tokyo, 1983.
, [*Strong maximum principle for a quasi-linear equation with applications*]{}, Ann. Acad. Sci. Fenn. Ser. A VI Phys. [**21**]{} (1978), 25 pp.
, [*Elementare Bemerkungen über die LÂŽösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus*]{}, Sitzungsberichte Preussiche Akademie Wissenschaften, Berlin, 1927, 147–152.
, [*The strong maximum principle revisited*]{}, J. Differential Equations [**196**]{} (2004), 1, 1–66.
, [*A note on the strong maximum principle for elliptic differential inequalities*]{}, J. Math. Pures Appl. (9) [**79**]{} (2000), 1, 57–71.
, [*A strong maximum principle and a compact support principle for singular elliptic inequalities*]{}, J. Math. Pures Appl. (9) [**78**]{} (1999), 8, 769–789.
, [*On the strong maximum principle for quasilinear second order differential inequalities*]{}, J. Functional Anal. 1970, 184–193.
, [*Commentary on Hopf strong maximum principle*]{}, C.S. Morawetz, J.B. Serrin, Y.G. Sinai (Eds.), Selected Works of Eberhard Hopf with commentaries, Amer. Math. Soc., Providence, RI, 2002.
, [*A strong maximum principle for degenerate elliptic operators*]{}, Comm. Partial Differential Equations [**4**]{} (1979), 11, 1201–1212.
, [*A strong maximum principle for some quasilinear elliptic equations*]{}, Appl. Math. Optim. [**12**]{} (1984), 3, 191–202.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'S. T. Banks'
- 'S. T. Bramwell'
title: 'Magnetic frustration in the context of pseudo-dipolar ionic disorder.'
---
Introduction
============
Disordered phases of matter may be categorized in terms of the relative importance of frustrated interparticle interactions and quenched positional disorder (‘frustration plus disorder’). For example, if the frustrated interactions are magnetic and the positional disorder is uncorrelated then one arrives at a recipe for a spin glass, a concept which underpins much thinking about the nature of disordered states [@fisher_and_hertz]. However, it seems reasonable to ask if the combination of frustration with [*correlated*]{} positional disorder can lead to distinct behaviour that is not encompassed in the usual spin glass paradigm.
Pseudo-dipolar positional disorder is characterised by a two-particle correlation function in reciprocal space, $g({\bf q})$, that decays like a dipole-dipole interaction. It contrasts strongly with conventional disorder, in which $g({\bf q})$ decays like a screened Coulomb interaction (an exponentially cut-off power law). Spontaneously generated pseudo-dipolar disorder occurs in ice rule ferroelectrics [@YoungbloodAxe] and a magnetic equivalent has been observed in spin ice [@Fennell]. However, to our knowledge, there has been no comprehensive theoretical study of the effect of pseudo-dipolar positional disorder on magnetic properties (although Villain has previously discussed the topic in the context of insulating spin glasses [@Villain]). In particular, the effect of varying the relative strengths of competing exchange interactions within a system possessing pseudo-dipolar positional disorder has remained an unsolved problem. Such a study could be relevant, either ideally, or approximately, to many real systems. For example, a combination of ice-rules positional disorder with magnetic frustration could be realised experimentally in certain inverse spinel ferrites [@anderson_1956_1008] or by the family of fluoride pyrochlores, exemplified by CsNiCrF$_6$. In these systems two ionic species (e.g. Ni$^{2+}$, Cr$^{3+}$) are distributed over the pyrochlore lattice [and Anderson showed that by minimizing the Coulomb interaction the distribution of ions should obey the ice rules (although the perfection of the ice rules in these systems has not been experimentally determined).]{} However it is not clear that spin glass states are a generic property of this experimental class [@Steiner; @Alba]. Hence it is relevant to ask the questions, what kind of magnetic states should one ideally expect, and what are their experimental signatures? Here we consider the case of ideal ice-rules disorder. Such a study is not only of interest to illuminate an alternative to the traditional spin glass paradigm but is also relevant to the concept of the Coulomb phase [@henley_2010_179], a general consequence of pseudo-dipolar correlations. Furthermore, there is currently much interest in ‘charge ice’ [@Fulde; @Udagawa; @Ishizuka], in which mobile electrons can adopt ice-rule configurations, leading to fractional excitations [@Fulde]. Our system provides a counterpoint to the electron system, in that has quenched or static charge disorder, which influences dynamical spin degrees of freedom.
The Model
=========
The model we study consists of equal numbers of two species of classical Heisenberg spins ${\ensuremath{\mathbf{s}}}_a$ and ${\ensuremath{\mathbf{s}}}_b$, randomly distributed across the pyrochlore lattice (a cubic array of corner-linked tetrahedra) and subject to the ‘ice rule’ constraint that there are two spins of each type on every tetrahedron. The spins are coupled by Heisenberg exchange parameters that take one of three possible values, $J_a$, $J_b$ or $J_{ab}$ depending on whether the neighbouring spins are both of type $a$, both of type $b$ or one of each type respectively. In the case that all spin [lengths]{} and exchange parameters are identical, the model reduces to either a pyrochlore lattice ferromagnet or antiferromagnet. While the former orders conventionally, the latter remains in a cooperative paramagnetic state down to $T=0$ [@moessner_1998_2929] and gives rise to a distinctive pinch-point magnetic structure factor indicative of pseudo-dipolar spin-spin correlations . The case where spin and exchange parameters differ is addressed by expressing the spin Hamiltonian as a sum over contributions from individual tetrahedra, $$H = -\frac{1}{N_T}\sum_{\alpha=1}^{N_T}\left(\sum_{\langle i,j
\rangle_{\alpha}}J_{ij}^{\alpha}{\ensuremath{\mathbf{s}}}_i^{\alpha}.{\ensuremath{\mathbf{s}}}_j^{\alpha}\right).
\label{eq:Hplaq}$$ Here $N_T$ is the number of tetrahedra and $\langle
i,j\rangle_{\alpha}$ indicates all pairs of spins $i$ and $j$ on plaquette (tetrahedron) $\alpha$ with exchange interaction $J_{ij}^{\alpha}$.
A single tetrahedron
--------------------
We begin by focusing on a single tetrahedron and, with no loss of generality, we assign spins ${\ensuremath{\mathbf{s}}}_1$ and ${\ensuremath{\mathbf{s}}}_2$ to be of type $a$ and spins ${\ensuremath{\mathbf{s}}}_3$ and ${\ensuremath{\mathbf{s}}}_4$ to be of type $b$. We then define three angular variables: $\phi_a$ as the angle between spins ${\ensuremath{\mathbf{s}}}_1$ and ${\ensuremath{\mathbf{s}}}_2$; $\phi_b$ as the angle between spins ${\ensuremath{\mathbf{s}}}_3$ and ${\ensuremath{\mathbf{s}}}_4$; $\theta$ as the angle between the resultants $\mathbf{S}_a=({\ensuremath{\mathbf{s}}}_1+{\ensuremath{\mathbf{s}}}_2)$ and $\mathbf{S}_b=({\ensuremath{\mathbf{s}}}_3+{\ensuremath{\mathbf{s}}}_4)$. In these coordinates the Hamiltonian for the single tetrahedron becomes $$\begin{aligned}
H & = & -J_as_a^2\cos\phi_a-J_bs_b^2\cos\phi_b\nonumber\\
& & -2J_{ab}s_as_b
(1+\cos\phi_a)^{\frac{1}{2}}(1+\cos\phi_b)^{\frac{1}{2}}\cos\theta.
\label{eq:ham_3}\end{aligned}$$ The ground state magnetic configurations then separate into two classes, characterized by the sign of $J_{ab}$, allowing an effective phase diagram to be mapped out in the space spanned by the reduced variables $J_a/|J_{ab}|$ and $J_b/|J_{ab}|$ (Fig. \[fig:phase\_diag\]). The topology of the resulting phase diagram is independent of the sign of $J_{ab}$, as are the equations of the boundaries between regions. In region I, spins of the same species are aligned parallel. Spins of different species are either parallel or anti-parallel depending on the sign of $J_{ab}$. This behaviour extends even into the quadrants of $J$-space where one or both intra-species interactions are antiferromagnetic. The extent of this domination by $J_{ab}$ is governed by the ratio $s_a/s_b$ as indicated by the equations of the boundaries between region I and regions II and III (Figure \[fig:phase\_diag\]).
Regions II and III are identical on interchanging the labels $a$ and $b$. In the ground state, spins of one species are perfectly parallel whilst the antiferromagnetic coupling of the other species is frustrated by the coupling $J_{ab}$. These spins cant away from the collinear axis defined by the first species through an angle [equal to half that between spins of type $x$,]{} $$\cos\phi_x = \frac{2J_{ab}^2s_y^2}{|J_x|^2s_x^2}-1,
\label{eq:cant}$$ where $\{x,y\}=\{a,b\}\,\,\, (\{b,a\})$ in region II (III) and $|J_x|>s_y/s_x$. The single species sublattice order parameter is defined in these regions as [ $$m^{(x)} =
\frac{1}{N_xs_x}\sqrt{\left(\sum_{i\in\{x\}}{\ensuremath{\mathbf{s}}}_i\right).\left(\sum_{i\in\{x\}}{\ensuremath{\mathbf{s}}}_i\right)}
\label{eq:mgs0}$$ where $N_x$ is the number of spins of type $x$ and]{} the sum is over all spins of type $x$ (our definition anticipates extending the theory to a macroscopic lattice). Combined with (\[eq:cant\]) we see that the ground state order parameter is inversely proportional to $J_x$: $${\ensuremath{m_{\mathrm{gs}}}}^{(x)} = \frac{s_y{|J_{ab}|}}{s_x|J_x|}.
\label{eq:mgs1}$$ This order parameter is defined only in terms of the $x$ spins as the $y$ spins are perfectly ordered with respect to each other (parallel or anti-parallel depending on the sign of $J_{ab}$) throughout regions II and III. Within regions I and IV we simply define ${\ensuremath{m_{\mathrm{gs}}}}^{(x)}=1,0$ respectively.
In region IV the intra-species antiferromagnetic interactions dominate, leading to configurations with $\phi_a=\phi_b=\pi$. Thus the third term of (\[eq:ham\_3\]) is zero and the Hamiltonian is independent of $\theta$. In this region, spins of different species are effectively decoupled. For $J_{ab}>0$ the confluence of the phase boundaries is particularly interesting as at this point the ordered ferromagnet becomes degenerate with the antiferromagnetic spin liquid.
Extension to the macroscopic lattice
------------------------------------
Consider now a macroscopic pyrochlore lattice having two A and two B ions per tetrahedron. A walker, starting at some ion of type A and following a path only through sites populated by A ions (without retracing its steps) will always return to its starting point. Furthermore, the path traced out will have no branches but will form a continuous closed loop containing an even number of lattice sites. The whole lattice is tiled with such loops – every lattice site belongs to one (and only one) of these closed, even membered, loops of spins of a single species (referred to simply as ‘loops’ from now on). The statistics of such loops have recently been discussed in the context of the magnetic Coulomb phase [@ludovic]. In regions I, II and III, A type loops and B type loops interact with each other via $J_{ab}$ which has the effect of imposing long range order. By contrast, in region IV there is no coupling between loops, although spins within a given loop are perfectly antiferromagnetically ordered with respect to each other. We consider this soup of uncoupled, closed, antiferromagnetic loops to be a novel spin-liquid like phase; frozen interactions within loops exist within a framework of two mobile degrees of freedom per loop. There are no energy barriers to facilitate global spin freezing and so, in the absence of free energy barriers or dynamical constraints, there can be no spin glass transition in region IV. This is in contrast to the Heisenberg pyrochlore antiferromagnet (HPAFM) with weak random and uncorrelated bond disorder [@bellier_castella_2001_1365; @saunders_2007_157201] to which a number of points in region IV are closely related.
In all regions of $J$-space, the ground states of the single tetrahedron are robust to stacking, with no extra frustration incurred. The phase diagram in Fig. \[fig:phase\_diag\] should therefore be equally valid for the macroscopic lattice.
The decoupling of loops in region IV requires either $\phi_a^{\alpha}=\pi$ or $\phi_b^{\alpha}=\pi$ on every tetrahedron $\alpha$, both of which are true in the ground state. The individual loops may then be viewed as independent one-dimensional [Heisenberg]{} chains, which may be arbitrarily long in the thermodynamic limit. For $T>0$, $\phi_a^{\alpha}$ and $\phi_b^{\alpha}$ may differ from their ground state values due to the excitation of low energy spin waves. The Hamiltonian then regains its dependence on the variable $\theta$ which is likely to dramatically slow the dynamics. Our numerical simulations suggest, however, that this slowing does not amount to truly broken ergodicity. Relaxing the ice rules constraint on the ion placement prevents this dynamical slowing down by providing extra unconstrained [magnetic]{} degrees of freedom in plaquettes with all ions of the same species or ions in a 1:3 ratio. [It is readily shown that 1:3 and 3:1 tetrahedra place fewer constraints on the magnetic degrees of freedom than do the 2:2 tetrahedra. This result may be understood intuitively given that tetrahedra with all spins of the same species are the least magnetically constrained of all.]{}
Numerical Simulations
=====================
Numerical evidence from Monte Carlo simulations is in agreement with our theoretical predictions. Simulations were performed on lattices with $L^3$ cubic unit cells ($L=4,7$, corresponding to 1024 spins and 5488 spins respectively) for which a short loop algorithm [@rahman_1972_4009] was used to generate ion configurations obeying the ice rules constraint. In all simulations we observed loops on all scales, from the smallest possible (six membered rings) up to loops spanning the system. With one exception (discussed below) the energy scale was defined by $|J_{ab}|$ and a single spin flip Metropolis algorithm was employed, with spin updates confined to a small solid angle.
To investigate spin freezing, we recorded the Edwards-Anderson order parameter $$q_{\mathrm{EA}} =
\frac{1}{N}\sum_{i=1}^{N}\left\langle\mathbf{s}_i\right\rangle^2,$$ in the region $0.01\le T/|J_{ab}|\le 0.1$ with $L=7$. We chose [$s_a=3/2$, $s_b=1$]{}, (corresponding to the magnitudes of the magnetic ions in CsNiCrF$_6$) and focused on the point $(J_a/|J_{ab}|,J_b/|J_{ab}|)=(-1.1,-1.1)$, ensuring three different bond contributions to the energy. We simulated both the completely random and ice rules constrained models as described above. At each temperature the systems were annealed in five steps from $T/|J_{ab}|=1$ with $10^6$ MCS/s (Monte Carlo Steps per spin) ($10^7$ MCS/s were used at and below $T/|J_{ab}|=0.03$) for equilibration at each step. Data was recorded over $10^6$ ($10^7$) MCS/s and averaged over ten disorder configurations. In the absence of the ice rules constraint $q_{EA}$ is essentially zero at all the temperatures studied. Imposing the constraint leads to a significant slowing of the dynamics, but with $10^7$ MCS/s $q_{EA}$ remains below 0.05 even at $T/|J_{ab}|=0.01$. As already noted, this behaviour is in strong contrast with that of HPAFM with weak random bond disorder [@bellier_castella_2001_1365; @saunders_2007_157201].
Figure \[fig:op\] shows the variation of the order parameter $m^{(x)}$ along the indicated lines in $J$-space. The agreement between [the theoretical and numerical results]{} is striking and this level of accuracy has been achieved with relatively small scale simulations ($L=4$, [$s_a=3/2$, $s_b=1$]{} and MCS/s=$10^6$). These results validate the extension of our analytical solution to the macroscopic lattice.
For the macroscopic lattice, differences in the ground state behaviour between the regions should manifest themselves in the magnetic structure factor, examples of which are shown in the lower panels of Figure \[fig:phase\_diag\]. For region I, long range collinear order produces sharp Bragg peaks. In regions II and III, sharp peaks arise from the components of all spins along the pseudo-collinear axis (shown for region II of Figure \[fig:phase\_diag\]), however scattering from just the perpendicular component of the canted spins (shown for region III of Figure \[fig:phase\_diag\]) reveals interesting diffuse scattering that suggests spin liquid like correlations transverse to the ordered component. We suggest that this might be called a ‘semi-spin liquid’, in analogy with a semi-spin glass [@Villain]. In region IV $S(\mathrm{Q})$ has the characteristic structure factor of an algebraic spin liquid, with pinch points indicative of the pseudo-dipolar correlations, although without the same clarity observed previously in studies of the HPAFM [@canals_2001_1323; @henley_2005_014424; @isakov_2004_167204]. The origins of such correlations in our model are not trivial. Unlike the pure HPAFM, the ground state in region IV has spins which interact only within a single loop. For such a ground state to exhibit pinch-point scattering would indicate that dipolar correlations emerge purely as a consequence of the geometric distribution of loops, as governed by the ion configuration. The spin-spin correlation function is then nothing more than the probability that the two spins are on the same loop. To confirm this assertion we examined a toy model representative of the ideal ground state of the system: $J_{ab}$ was set to zero and perfect Néel order was enforced within each loop. We assigned a randomly selected easy axis to each loop and measured $S(Q)$ for the resulting configuration. This process was repeated for a number of sets of randomly chosen easy axes and the resulting structure factors averaged. The resolution was improved by averaging again over a number of disorder configurations. The results [(Figure \[fig:ideal-IV\])]{} clearly show signs of dipolar spin-spin correlations emerging from this system of magnetically independent 1$d$ chains. To obtain the data in this figure we averaged over 50 disorder configurations with $L=7$. $S(Q)$ was averaged over 10 easy-axis configurations per lattice.
A direct consequence of the above observations is that the magnetic scattering is acting as a probe of the structural disorder. There is however an inherent limit on the resolution of the pinch-point scattering pattern which can result from a single realisation of the quenched disorder on a finite lattice. Unlike the pure HPAFM, for which the bow-tie pattern is well resolved even for relatively small lattices, the ice rules constrained binary pyrochlore described here has the lengths and spatial arrangement of its loops predetermined by a particular ion configuration. The pure HPAFM however allows for a dynamic interchange of spins between loops, in effect sampling a large number of loop configurations with a corresponding increase in resolution. [In practical terms, this should have a similar effect to averaging the quenched disorder over many equivalent ion configurations. ]{}
Conclusions
===========
In conclusion, for the ideal model considered we have demonstrated the suppression of spin glass behaviour and the emergence of novel spin liquid and semi-spin liquid phases. It will be of interest to re-examine the magnetic behaviour of the fluoride pyrochlores in the light of this result. In particular, certain fluoride pyrochlores have been reported to show spin glass transitions [@Steiner] however there is a growing body of evidence that these compounds do not form traditional spin glasses below the supposed freezing temperature [@Alba; @Ramirez_2; @Harris]. We conclude that if these are true spin glass transitions, they must be a consequence of disorder or interactions beyond those considered here. At a more general level we have illustrated a counter example to the idea that geometric frustration and positional disorder must combine to generate a spin glass, although we have not ruled out the possibility that the spin liquid states we have identified may be highly sensitive to further quenched disorder of a different character. [Finally, our results have shed light on an issue of rather general importance that is pertinent to the interpretation of neutron scattering patterns of disordered magnets. Thus, quenched atomic or ionic disorder is generally characterised by an energy scale much higher than that of the magnetic interactions, so the magnetic structure factor should generally be affected by the structural disorder correlations, but the question is to what degree, and what does this signify? We have identified a limiting case where the magnetic and structural disorder correlations are closely connected, and this connection is easily comprehended in terms of the model of ‘unfrustrated’ spin loops we describe. The opposite limiting case, which one would expect to apply to an ideal spin glass, is where the frustration is purely magnetic in origin and there is no dependence of magnetic on structural correlations. Real systems are likely to lie between these two limiting cases, so our results may be of some general relevance to the interpretation of neutron scattering patterns, including those of canonical spin glasses [@murani_1999_131] and weakly disordered spin glass systems such as ‘SCGO’ [@Ramirez] and Y$_2$Mo$_2$O$_7$ [@Gingras; @Wiebe]. ]{}
It is a pleasure to thank Mark Harris, Tom Fennell, Chris Henley, Peter Holdsworth and John Chalker for very stimulating discussions. S. T. Banks thanks the Ramsay Memorial Fellowship Trust for funding through a Ramsay Memorial Fellowship.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay algebras over a field, for codimension two algebras and for Gorenstein algebras of codimension three. In fact, we prove stronger bounds than the conjectured ones allowing us to characterize the extremal cases. This may be seen as a converse to the multiplicity formula of Huneke and Miller that inspired the conjectural bounds.'
address:
- 'Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA'
- 'Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, KY 40506-0027, USA'
- 'FB Mathematik/Informatik, Universität Osnabrück, 49069 Osnabrück, Germany'
author:
- Juan Migliore
- Uwe Nagel
- Tim Römer
title: The multiplicitiy conjecture in low codimensions
---
[^1]
Introduction
============
Let $R = K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$ with its standard grading where $\deg x_i=1$ for $i=1,\dots,n$. Let $R/I$ be a standard graded $K$-algebra, where $I$ is a graded ideal of codimension $c$. We denote by $e(R/I)$ the multiplicity of $R/I$. When $I$ is a saturated ideal defining a closed subscheme $V \subset {\mathbb P}^{n-1}$, $e(R/I)$ is just the degree $\deg V$ of $V$. Consider the minimal graded free resolution of $R/I$ $$0 \rightarrow
\bigoplus_{j \in {\mathbb{Z}}} R(-j)^{\beta_{p,j}^R(R/I)}
\rightarrow \dots \rightarrow
\bigoplus_{j \in {\mathbb{Z}}} R(-j)^{\beta_{1,j}^R(R/I)}
\rightarrow R
\rightarrow 0$$ where we denote by $\beta_{i,j}^R(R/I)=\operatorname{Tor}_i^R(R/I,K)_j$ the graded Betti numbers of $R/I$ and $p$ is the projective dimension of $R/I$. Let $c$ denote the codimension of $R/I$. Then $c \leq p$ and equality holds if and only if $R$ is Cohen-Macaulay. We define $m_i = \min \{j \in {\mathbb{Z}}: \beta_{i,j}^R(R/I)\neq 0 \}$ and $M_i = \max \{j \in {\mathbb{Z}}: \beta_{i,j}^R(R/I)\neq 0 \}$. When there is any danger of ambiguity, we write $m_i(I)$ and $M_i(I)$. The algebra $R/I$ has a [*pure resolution*]{} if $m_i = M_i$ for all $i$. In this case we write $d_i$ for the unique $j$ such that $\beta_{i,j}^R(R/I)\neq 0$. It was shown by Huneke and Miller [@HM] that if $R/I$ is Cohen-Macaulay with a pure resolution then $$e(R/I) = \left ( \prod_{i=1}^p d_i \right )/ p! .$$ Extending this, Herzog, Huneke and Srinivasan made the following multiplicity conjecture:
\[conj\] If $R/I$ is Cohen-Macaulay then $$\left ( \prod_{i=1}^p m_i \right )/ p! \leq e(R/I) \leq \left (
\prod_{i=1}^p M_i \right )/p!.$$
Conjecture \[conj\] has been studied extensively, and partial results have been obtained. In [@HS], Herzog and Srinivasan proved it in the following cases: $R/I$ is a complete intersection; $I$ is a perfect ideal with quasi-pure resolution (i.e. $m_i(R/I)\geq M_{i-1}(R/I)$); $I$ is a perfect ideal of codimension 2; $I$ is a codimension 3 Gorenstein ideal with five minimal generators; $I$ is a Gorenstein monomial ideal of codimension 3 with at least one generator of smallest possible degree (relative to the number of generators); $I$ is a perfect stable ideal; $I$ is a perfect squarefree strongly stable ideal. Furthermore, Herzog and Srinivasan proved that the [*upper bound*]{} of Conjecture \[conj\] holds for all codimension 3 Gorenstein ideals. In addition, Guardo and Van Tuyl [@GV] proved that the conjecture holds for powers of complete intersections, and Gold, Schenck and Srinivasan [@GSS] proved it in certain cases where $I$ is linked to something “simple." In addition, Srinivasan [@srinivasan] proved a stronger bound for Gorenstein algebras with quasi-pure resolutions. (cf. Remark \[rem-hema\].)
The non Cohen-Macaulay case has also been studied. Here it is necessary to replace the projective dimension by the codimension in Conjecture \[conj\]. It was observed in [@HS] that the lower bound is false. However, Herzog and Srinivasan proved the upper bound in these cases: $I$ is a stable ideal; $I$ is a squarefree strongly stable ideal; $I$ is an ideal with $d$-linear resolution. In addition, Gold [@gold] proved it for codimension two lattice ideals; this was generalized by Römer [@roemer] for all codimension two ideals. It was also proved by Gasharov, Hibi and Peeva [@GHP] for [**a**]{}-stable ideals and more generally by Römer [@roemer] for componentwise linear ideals.
In this paper we begin with a new, stronger version of Conjecture \[conj\] in the codimension two case:
\[thm-intro-2\] Let $R/I$ be a graded Cohen-Macaulay algebra of codimension two. Then the following lower and upper bounds hold:
1. $
e(R/I)
\geq
\frac{1}{2} m_1 m_2 + \frac{1}{2} (M_2-M_1) (M_2-m_2 + M_1-m_1)
$;
2. $
e(R/I) \leq \frac{1}{2} M_1 M_2 - \frac{1}{2} (m_2-m_1)
( M_2-m_2 + M_1-m_1 ).
$
As an immediate consequence of this result, we obtain the following characterization for the sharpness of Conjecture \[conj\] in the codimension two Cohen-Macaulay case. This can be viewed as a converse to the Huneke-Miller result [@HM] mentioned above.
\[cor-intro-2\] Let $R/I$ be a graded Cohen-Macaulay algebra of codimension two. Then the following conditions are equivalent:
1. $e(R/I) = \frac{1}{2} m_1 m_2 $;
2. $e(R/I) = \frac{1}{2} M_1 M_2 $;
3. $R/I$ has a pure minimal graded free resolution.
In Section \[sec-2\] we give two proofs of these results. The first is based on some formulas in [@HS]. The second one is more self-contained and uses the methods that allow us obtain the results discussed below. We also discuss the non Cohen-Macaulay case. In particular, we show that even a natural weakening of the lower bound in Conjecture \[conj\] could only be true for reduced ideals (cf. Remark \[rem-naCM\]). In Section \[sec-3\] we prove a stronger version of Conjecture \[conj\] for Gorenstein ideals of codimension three:
\[ht 3 gor bd\] Let $R/I$ be a graded Gorenstein algebra of codimension three. Then the following lower and upper bounds hold:
1. $e(R/I)\geq \frac{1}{6} m_1 m_2 m_3 + \frac{1}{6}(M_3-M_2)2(M_2-m_2+M_1-m_1)$;
2. $e(R/I) \leq \frac{1}{6} M_1 M_2 M_3 - \frac{1}{12} M_3(M_2-m_2+M_1-m_1)$.
As an immediate consequence we get that the lower bound of Conjecture \[conj\] holds for codimension three Gorenstein ideals. This was the last open case of the conjecture in low codimensions where structure theorems of minimal graded free resolutions are available. As in the case of codimension two perfect ideals we can characterize when Conjecture \[conj\] is sharp providing again a converse to the Huneke-Miller formula in [@HM].
\[cor-sharp-gor\] Let $R/I$ be a graded Gorenstein algebra of codimension three. Then the following conditions are equivalent.
1. $e(R/I) = \frac{1}{6} m_1 m_2 m_3$;
2. $e(R/I) = \frac{1}{6} M_1 M_2 M_3 $;
3. $R/I$ has a pure minimal free resolution.
Our method of proof consists in exhibiting a specific example for each possible set of Betti numbers and a reduction procedure that allows us to proceed by induction. While the first idea seems difficult to extend we expect that the reductions via basic double links will be useful in other cases, too.
We conclude this note with an explicit formula for the multiplicity of a Gorenstein ideal in terms of the degrees of the entries of its Buchsbaum-Eisenbud matrix (Proposition \[prop-def-formula\]). It is a by-product of the proof of Theorem \[ht 3 gor bd\].
Codimension two Cohen-Macaulay algebras {#sec-2}
=======================================
Let $K$ be a field, $R=K[x_1,\dots,x_n]$ be the polynomial ring and $I \subset R$ a graded ideal of height two such that $R/I$ is a Cohen-Macaulay ring. It follows from the Hilbert-Burch theorem (e.g. see [@BRHE] for details) that $I$ has a minimal graded free resolution of the form $$\label{eq-2-res}
0 \to
\bigoplus_{i=1}^{m-1}R(-f_i)
\stackrel{{\varphi}}{\longrightarrow}
\bigoplus_{i=1}^{m}R(-e_i)
\to
J
\to
0$$ Let $u_i=f_i-e_i$ and $v_i=f_i-e_{i+1}$. The following was observed, for example, in [@HTV]:
1. $u_i \geq v_i \geq 0$ for $i=1,\dots,m-1$;
2. $u_{i+1} \geq v_i$ for $i=1,\dots,m-2$;
3. $e_1=v_1+\dots+v_{m-1}$ and $e_m=u_1+\dots+u_{m-1}$;
4. $f_1=v_1+\dots+v_{m-1}+u_1$ and $f_{m-1}=u_1+\dots+u_{m-1}+v_{m-1}$;
5. $
e(R/I)=
\sum_{i=1}^{m-1} u_i(v_i+\dots + v_{m-1})
=
\sum_{i=1}^{m-1} v_i(u_1+\dots + u_{i}).
$
Herzog and Srinivasan proved in [@HS] the formulas (see proof of Theorem 2.1):
1. $\sum_{i=2}^{m-1}(v_{i-1} + v_i)(v_i+\dots + v_{m-1})
=
(v_1+\dots+v_{m-1})(v_2+\dots+v_{m-1})
$;
2. $\sum_{i=1}^{m-2}(u_{i} + u_{i+1})(u_1+\dots + u_{i})
=
(u_1+\dots+u_{m-1})
(u_1+\dots+u_{m-2})
$.
Note that $e_1=m_1$, $e_{m}=M_1$, $f_1=m_2$ and $f_{m-1}=M_2$. Following the proof of Theorem 2.1 in [@HS], we can show Theorem \[thm-intro-2\] of the introduction.
\(a) Using $$\begin{aligned}
2u_1
=
(u_1-v_1) + u_1+v_1
=
e_2-e_1 + u_1+ v_1
\\
2u_i
=
(2u_i-v_{i-1}-v_i) + v_{i-1}+v_i
=
(e_{i+1}-e_i + f_i - f_{i-1}) + v_{i-1}+v_i\end{aligned}$$ and the first formula of Herzog and Srinivasan above we compute $$\begin{aligned}
2e(R/i)
& = &
\sum_{i=1}^{m-1} 2u_i(v_i+\dots + v_{m-1})\\
& = &
e_1f_1
+
(u_1-v_1)(v_1+\dots + v_{m-1})
+
\sum_{i=2}^{m-1} (2u_i-v_i-v_{i-1})(v_i+\dots + v_{m-1})
\\
& = &
m_1m_2
+
(e_2-e_1)(v_1+\dots + v_{m-1})
+
\sum_{i=2}^{m-1} (e_{i+1}-e_i + f_i - f_{i-1})(v_i+\dots + v_{m-1})
\\
& = &
m_1m_2
+
\sum_{i=1}^{m-1}(e_{i+1}-e_i)(v_i+\dots + v_{m-1})
+
\sum_{i=2}^{m-1} (f_i - f_{i-1})(v_i+\dots + v_{m-1})
\\
& \geq &
m_1m_2
+
v_{m-1}
(
\sum_{i=1}^{m-1}(e_{i+1}-e_i)
+
\sum_{i=2}^{m-1} (f_i - f_{i-1})
)
\\
& = &
m_1m_2
+
(M_2-M_1)
(
M_1-m_1
+
M_2-m_2
).\end{aligned}$$ Dividing by 2, the desired formula follows.
\(b) Similarly, using the second formula of Herzog and Srinivasan we compute $$\begin{aligned}
2e(R/I)
& = &
\sum_{i=1}^{m-1} 2v_i(u_1+\dots + u_{i})
\\
& = &
e_m f_{m-1}
+
(v_{m-1}-u_{m-1})(u_1+\dots + u_{m-1})
+
\sum_{i=1}^{m-2} (2v_i-u_i-u_{i+1})(u_1+\dots + u_{i})
\\
& = &
M_1M_2
-
(e_{m}-e_{m-1})(u_1+\dots + u_{m-1})
-
\sum_{i=1}^{m-2} (e_{i+1}-e_i + f_{i+1} - f_{i})(u_1+\dots + u_{i})
\\
& = &
M_1M_2
-
\sum_{i=1}^{m-1}
(e_{i+1}-e_i)(u_1+\dots + u_{i})
-
\sum_{i=1}^{m-2} (f_{i+1} - f_{i})(u_1+\dots + u_{i})
\\
& \leq &
M_1M_2
-
u_1
(
\sum_{i=1}^{m-1}
(e_{i+1}-e_i)
+
\sum_{i=1}^{m-2} (f_{i+1} - f_{i})
)
\\
& = &
M_1M_2
-
(m_2-m_1)
(
M_1-m_1
+
M_2-m_2
)\end{aligned}$$
Now we present alternative, more self-contained proofs for the bounds in Theorem \[thm-intro-2\] that use some methods of liaison theory. Its purpose is twofold. They illustrate the principles we will use in the following section to discuss Gorenstein ideals of codimension three and they allow us to provide some of the relations we will use there.
The map ${\varphi}$ in the minimal free resolution (\[eq-2-res\]) is represented by the Hilbert-Burch matrix of $I$. By reordering we can arrange that the degree matrix is $$\label{deg mat}
A = \begin{bmatrix}
a_{1,1} & a_{1,2} & \dots & a_{1,t+1} \\
\vdots & \vdots & & \vdots \\
a_{t,1} & a_{t,2} & \dots & a_{t,t+1}
\end{bmatrix}$$ where the entries are increasing from bottom to top and from left to right, so $a_{t,1}$ is the smallest and $a_{1,t+1}$ is the largest. Notice that in order to be a degree matrix of a height two Cohen-Macaulay ideal, the main diagonal has to be strictly positive ([@GM3], page 3142 and see [@sauer], page 84).
\[rem-deg-matrix\] Let us rewrite the degree matrix $A$ as follows $$A = \begin{bmatrix}
a_1 & b_1 & & *\\
& \ddots & \ddots & \\
* & & a_t & b_t \\
\end{bmatrix}$$ Note that $A$ is completely determined by $a_1,\ldots,a_{t},b_1,\ldots,b_{t}$. By our ordering of degrees we have in particular $$b_t \geq a_t \quad \mbox{and} \quad b_{t-1} \geq a_t \; \mbox{provided} \; t
\geq 2.$$ The minimal generators of $I$ have degrees $a_1 + \ldots + a_j + b_{j+1} + \ldots + b_t$, $j = 0,\ldots,t$ and the syzygies of $I$ have degrees $a_1 + \ldots + a_j + b_{j} + \ldots + b_t$, $j = 1,\ldots,t$. Thus, we obtain $$\label{formulas}
\begin{array}{rcl}
m_1 & = & a_1 + \ldots + a_t \\
M_1 & = & b_{1} + \ldots + b_t \\
m_2 & = & a_1 + \ldots + a_t + b_t = m_1 + b_t \\
M_2 & = & a_1 + b_{1} + \ldots + b_t.
\end{array}$$
Now we will re-prove the lower bound for the multiplicity in Theorem \[thm-intro-2\]:
We induct on $t \geq 1$. It $t = 1$ then $I$ is a complete intersection. Its resolution is $$\label{koszul}
0 \rightarrow R(-m_1-M_1) \rightarrow
\begin{array}{c}
R(-M_1) \\
\oplus \\ R(-m_1)
\end{array}
\rightarrow I \rightarrow 0.$$ Then we get $$\begin{array}{rcl}
\frac{1}{2} m_1 m_2 + \frac{1}{2} (M_2-M_1) (M_2-m_2 + M_1-m_1) &
= & \frac{1}{2} m_1 (m_1+M_1) + \frac{1}{2} m_1(M_1-m_1) \\[1ex]
& = & m_1 M_1 = e(R/I).
\end{array}$$
Now assume $t \geq 1$ and let $I'$ be an ideal whose degree matrix of the Hilbert-Burch matrix is $$A' = \begin{bmatrix}
a_1 & b_1 & & & *\\
& \ddots & \ddots & \\
& & a_t & b_t \\
* & & & a_{t+1} & b_{t+1}
\end{bmatrix}.$$ Note that the multiplicity of $R/I'$ is completely determined by the degree matrix, so it suffices to consider an example of an ideal for any degree matrix. Basic double linkage is then used in order to apply the induction hypothesis. The idea will be to show that we can reduce to an ideal with degree matrix $A$ (see Remark \[rem-deg-matrix\]), i.e. we remove the last row and the last column.
It is easy to see that the following monomial ideal has $A'$ as its degree matrix $$I' = (y^{b_1 + \ldots + b_{t+1}}, x^{a_1} y^{b_2 + \ldots + b_{t+1}},
\ldots, x^{a_1 + \ldots + a_{t+1}}).$$ Write it as $$\label{eq-J'-J}
I' = x^{a_1 + \ldots + a_{t+1}}R + y^{b_{t+1}} I.$$ Then the monomial ideal $I$ has $A$ as its degree matrix where $A$ is obtained by deleting the last row and column of $B$. Thus, we may apply induction to $I$. Let $m_1,m_2,M_1,M_2$ be the corresponding invariants for $I$, and let $m_1',m_2',M_1',M_2'$ be those of $I'$. Moreover, in order to simplify notation we set $$a := a_{t+1}, \; b:= b_{t+1}, \; \mbox{and} \; c := b_t.$$ Using Remark \[rem-deg-matrix\], we see that $$\label{comp-deg}
\begin{array}{rcl}
m_1' & = & m_1 + a \\
M_1' & = & M_1 + b \\
m_2' & = & m_2 + a + b -c \\
M_2' & = & M_2 + b.
\end{array}$$ Moreover, we have $e(R/I') = e(R/I) + m_1' b$. Using the formulas above we get by induction $$\begin{aligned}
\lefteqn{ m_1'm_2' + (M_2'-M_1') (M_2'-m_2' + M_1'-m_1')} \\[1ex]
& = & (m_1+a)(m_2+a+b-c) \\
& & + [(M_2+b) - (M_1+b)] [(M_2+b) - (m_2+a+b-c) + (M_1 + b) - (m_1+a)] \\[1ex]
& = & m_1 m_2 + (m_1 + a) (a+b-c) + m_2 a \\
& & + (M_2 - M_1) [M_2 - m_2+ M_1 - m_1 + b + c - 2a] \\[1ex]
& \leq & 2 e(R/I) + (m_1 + a) (a+b-c) + m_2 a +
(M_2 - M_1) (b+c-2a) \\[1ex]
& = & 2 e(R/I') - 2 (m_1 + a) b + (m_1 + a) (a+b-c) + (m_1 + c) a + \\
& & (M_2 - M_1) (b+c-2a)
\\[1ex]
& = & 2 e(R/I') + [m_1 + M_1 - M_2] (2a-b-c) + a(a-b).\end{aligned}$$ Note that we used the relation $m_2 = m_1 + c$. But $b \geq a$ and $c \geq a$ and $M_1 + m_1 \geq M_2$ by \[rem-deg-matrix\], so $$\begin{array}{rcll}
\frac{1}{2} m_1'm_2' + \frac{1}{2} (M_2'-M_1') (M_2'-m_2' +
M_1'-m_1') & \leq & e(R/I')
\end{array}$$ as desired. (Note the strict inequality unless $a = b = c$ or $t = 2$ and $a=b$.)
In a similar way we can re-prove the upper bound in Theorem \[thm-intro-2\].
We use again induction on $t \geq 1$. First suppose that $I$ is a complete intersection. Then we have the resolution (\[koszul\]), and we obtain $$\begin{array}{rcl}
\frac{1}{2} M_1 M_2 -
\frac{1}{2}
(m_2-m_1)
(
M_2-m_2
+
M_1-m_1
) & = &
\frac{1}{2} M_1 (M_1 + m_1) -
\frac{1}{2} M_1 (M_1-m_1) \\[1ex]
& = & m_1 M_1 = e(R/I).
\end{array}$$
Now for the general case, we again use induction with the set-up of the previous proof. We have $$\begin{aligned}
\lefteqn{ M_1'M_2' - (m_2'-m_1')
(
M_2'-m_2'
+
M_1'-m_1'
)}
\\[1ex]
& = & (M_1+b)(M_2+b)
\\
& & - [(m_2 + a + b - c) - (m_1+a)] [M_2 -
m_2+ M_1 - m_1 + b + c - 2a]
\\[1ex] & = & M_1M_2 + b (M_1 + M_2) + b2 \\
& & - [m_2 + b - c - m_1] [M_2 - m_2+ M_1 - m_1 + b + c - 2a]
\\[1ex]
& \geq & 2 e(R/I) + b (M_1 + M_2 + b) - b
(b+c -2a) - (b-c) [M_2 - m_2+ M_1 - m_1]
\\[1ex]
& = & 2 e(R/I') - 2 (m_1 + a) b + b (M_1 + M_2 + b) - b (b+c -2a) \\
& & - (b-c) [M_2 - m_2+ M_1 - m_1] \\[1ex]
& = & 2 e(R/I') + b (M_1 + M_2 - 2m_1 -c) - (b-c) [M_2 - m_2+ M_1 - m_1] \\[1ex]
& = & 2 e(R/I') + c (M_1 + M_2 - 2m_1 -c)\\[1ex]
& \geq & 2 e(R/I').\end{aligned}$$ (Note the strict inequality unless $I$ has a pure resolution.)
In a similar way we can prove another upper bound that extends the upper bound of Herzog and Srinivasan. The following proposition has a hypothesis that is a bit technical, but it has a more satisfying conclusion than the bound of Theorem \[thm-intro-2\](b) in one case.
\[upper bd prop\] Let $I$ be a height two Cohen-Macaulay ideal with degree matrix $$A = \begin{bmatrix}
a_1 & b_1 & && *\\
d & a_2 & b_2 \\
&& \ddots & \ddots & \\
* & & & a_t & b_t
\end{bmatrix}.$$ as in Remark \[rem-deg-matrix\] (but note the new variable $d$ in the $(1,2)$ spot if $t \geq 2$). Then either one of the following conditions is sufficient to conclude that $$e(R/I) \leq \frac{1}{2} M_1M_2 - (M_1-m_1) - (M_2 - m_2).$$
- all of the entries of $A$ are $\geq 2$;
- $a_1-2d+1 \geq 0$, provided $t \geq 2$.
We omit the details and leave the proof to the reader.
Consider the degree matrix $$B =
\left [
\begin{array}{ccccc}
2 & 2 & 2 & 2 \\
2 & 2 & 2 & 2 \\
1 & 1 & 1 & 1\end{array}
\right ]$$ This comes, for example, from the ideal $I' = (x5, x^4y, x^2y3,y5)$. It is easy to check that $M_1' = m_1' = 5, M_2' = 7, m_2' = 6$, and $e(R/I') = 17$. If we consider the $2 \times 3$ submatrix $A$, as in the proof, we have $m_1 = M_1 = 4, m_2
= M_2 = 6$. Then $$e(R/I')>\frac{1}{2} M_1'M_2' - (M_1'-m_1') - (M_2'-m_2') = 16.5.$$ So we see that the conclusion of Proposition \[upper bd prop\] does not hold here, and indeed $a_1-2d+1 = -1$.
In fact, it is not too difficult to show that a $t \times (t+1)$ degree matrix consisting of 2’s in the first $(t-1)$ rows and 1’s in the last row satisfies $m_1 = M_1 = 2t-1$, $m_2 = 2t$, $M_2 = 2t+1$, and $e(R/I')
= 2t2-1$. However, one checks that $$\frac{1}{2} M_2M_2 - (M_1-m_1) - (M_2-m_2) = 2t2 - \frac{3}{2}$$ so this gives an example of any size that violates the bound of Proposition \[upper bd prop\].
As mentioned in the introduction, it was shown by Huneke and Miller [@HM] that if $R/I$ is Cohen-Macaulay of codimension $c$ with a pure resolution then $$e(R/I) = \left ( \prod_{i=1}^c d_i \right )/ c! ,$$ where of course $d_i = m_i = M_i$ is the shift in the $i$-th free module of the pure resolution. Corollary \[cor-intro-2\] in the introduction may be seen as a converse to this result.
The claim follows from Theorem \[thm-intro-2\] because $m_2 > m_1$ and $M_2 > M_1$.
We end this section with a remark about the non Cohen-Macaulay case in codimension two.
\[rem-naCM\]
It is known that the lower bound of Conjecture \[conj\] is false in the non Cohen-Macaulay case. Even the weaker statement $$e(R/I) \geq \frac{1}{c!} \prod_{i=1}^c m_i$$ (where $c$ is the codimension of $I$) is false. An easy example is the case of two skew lines in ${\mathbb P}^3$. In codimension two, however, the analogous upper bound is true ([@roemer]). Is there a different lower bound that is true? One natural guess is that one might be able to replace $c!$ by some suitable integer $k$. That is, perhaps $$e(R/I) \geq \frac{1}{k} \prod_{i=1}^c m_i$$ for suitable $k$. One immediately sees that at the very least, we should assume that our ideals are unmixed. For instance, starting with any curve, adding points does not change the multiplicity but makes the $m_i$ arbitrarily large.
The next observation is that we must assume that $I$ is reduced in order to hope for a lower bound of the type $\frac{1}{k} \prod_{i=1}^c m_i$. Indeed, consider ideals in $k[x_0,x_1,x_2,x_3]$ of the form $$I = (x_0,x_1)^t + (F)$$ where $F$ is a polynomial that is smooth along the line defined by $(x_0,x_1)$, and $\deg
F \geq t+1$. Then $I$ defines an unmixed curve of multiplicity $t$ (cf. [@MPP]), and one quickly sees that $m_1 = t$ and $m_2 = t+1$. Hence we would need $k \geq t+1$, which can be made arbitrarily large by choosing large $t$.
However, if we do assume that $I$ is reduced, there may be such a bound. Indeed, experiments with [Macaulay]{} ([@macaulay]) have not yet yielded a counterexample to the guess $$e(R/I) \geq \frac{1}{5} m_1 m_2$$ at least among unmixed height two [*reduced*]{} non arithmetically Cohen-Macaulay curves in ${\mathbb P}^3$.
Codimension three Gorenstein algebras {#sec-3}
=====================================
We now turn to height three Gorenstein ideals. Our approach here is similar to that of the alternative proofs given in the previous section. In [@HS] the upper bound stated in Conjecture \[conj\] was proved for such ideals. The lower bound was proved only when the number of generators is five (or of course three). In this section we will prove an improved version of Conjecture \[conj\], and as a consequence we will again (as in the codimension two case) immediately obtain as a corollary that sharpness occurs (necessarily for both bounds) if and only if the resolution is pure.
Let $I \subset R$ be a height three Gorenstein ideal. The possible graded Betti numbers of such ideals were described in [@BE] and in [@D], and it was shown in [@GM5] that any such set of graded Betti numbers in fact occurs for some reduced arithmetically Gorenstein set of points in $\mathbb P3$. In fact more was shown in [@GM5].
\[rem-Gor-const\] Suppose that $I$ has a minimal graded free resolution $$\label{desired resol}
0 \rightarrow R(-m_3) \rightarrow \bigoplus_{i=1}^{2t+1} R(-{\beta}_i)
\rightarrow
\bigoplus_{i=1}^{2t+1} R(-{\alpha}_i) \rightarrow I \rightarrow 0$$ where ${\alpha}_1 \leq \dots \leq {\alpha}_{2t+1}$, ${\beta}_1 \leq \dots \leq {\beta}_{2t+1}$ (this is slightly different from the notation of [@GM5]), and $m_3 = M_3$. It was shown in [@GM5] that
- \[fact1\] there exists a Cohen-Macaulay ideal $J \subset R$ with minimal graded free resolution $$\label{resol of C}
0 \rightarrow \bigoplus_{i=1}^t R(-{\beta}_i) \rightarrow \bigoplus_{i=1}^{t+1}
R(-{\alpha}_i) \rightarrow J \rightarrow 0.$$
- \[fact2\] there are homogeneous polynomials $f, g \in J$ of degree $m_1(I) =
{\alpha}_1$ and $M_2(I) = {\beta}_{2t+1} = m_3-{\alpha}_1$, respectively, such that $\tilde{J} := (f, g) : J$ does not have any components in common with $J$ (i.e. $J$ and $\tilde{J}$ are geometrically linked by $(f,
g)$), thus $(f, g) = J \cap \tilde{J}$.
- \[fact4\] the ideal $I := J + \tilde{J}$ has the desired minimal free resolution (\[desired resol\]). (Note that not all Gorenstein ideals arise in this way; this only says that numerically for any set of graded Betti numbers this construction produces a Gorenstein ideal with the given Betti numbers, but this is enough for our purposes.)
A famous result of Buchsbaum and Eisenbud in [@BE] says that without loss of generality we may choose bases so that the middle map of the resolution (\[desired resol\]) is represented by a skew symmetric matrix, $M$, and that the minimal generators of $I$ are then given by the maximal Pfaffians of that matrix. However, we may represent the degree matrix $B$ corresponding to $M$, much as we did in (\[deg mat\]), so that the entries are increasing from bottom to top and from left to right. Then the resulting degree matrix $B$ is [*symmetric*]{} about the [*non-main*]{} diagonal. In particular, we have $$\label{eq-3-deg-mat}
B = \begin{bmatrix}
b_t & & & & & & *\\
a_t & \ddots \\
& \ddots & b_1 \\
& & a_1 & d \\
& & & a_1 & b_1 & & \\
& & && \ddots & \ddots & \\
* & & & & & a_t & b_t \\
\end{bmatrix}$$ where $$A := \begin{bmatrix}
a_1 & b_1 & & *\\
& \ddots & \ddots & \\
* & & a_t & b_t \\
\end{bmatrix}$$ is the degree matrix of the ideal $J$ (that has been used to produce $I$).
Furthermore, comparing the resolutions (\[desired resol\]) and (\[resol of C\]) we obtain in conjunction with the formulas (\[formulas\]) that $$\label{3-formulas}
\begin{array}{rcccl}
m_1 & = & m_1 (J) & = & a_1 + \ldots + a_t \\
m_2 & = & m_2 (J)& = & a_1 + \ldots + a_t + b_t = m_1 + b_t \\
m_3 & = & & & d + 2 (b_1 + \ldots + b_t).
\end{array}$$
We now are ready to show our improvement of Conjecture \[conj\] for Gorenstein ideals of codimension three.
Our proof will be by induction on the size of the degree matrix, $A$, of the Buchsbaum-Eisenbud matrix of $I$. First, let $t = 1$, i.e. $I$ is a complete intersection. Let $m_1, y, M_1$ be the degrees of the minimal generators of $I$. Then $m_1 \leq y \leq M_1$ and we get
$$\begin{aligned}
\lefteqn{ m_1 m_2 m_3 + (M_3-M_2)2(M_2-m_2+ M_1-m_1) } \\[1ex]
& = & m_1 (m_1 + y) (m_1 + y + M_1) + 2 m_12 (M_1 - m_1) \\[1ex]
& = & m_1 (2 y + m_1 - y) (3 M_1 +m_1+ y - 2 M_1) + 2 m_12 (M_1 - m_1) \\[1ex]
& = & 6 m_1 y M_1 + m_1 \left [ (m_1 + y) (m_1 + y - 2M_1) + (m_1 - y) 3 M_1 + 2 m_1 (M_1 - m_1) \right ] \\[1ex]
& = & 6 e(R/I) + m_1 \left [ (m_1 + y) (y - M_1) + 3 (m_1 - y) M_1 + (m_1 -y) (M_1 - m_1) \right ] \\[1ex]
& \leq & 6 e(R/I)\end{aligned}$$
proving the lower bound.
The upper bound is shown similarly. We have $$\begin{aligned}
\lefteqn{M_1 M_2 M_3 - \frac{1}{2} M_3(M_2-m_2+M_1-m_1) } \\[1ex]
& = & M_1 (y + M_1) (m_1 + y + M_1) - (m_1 + y + M_1) (M_1 - m_1) \\[1ex]
& = & M_1 (2 y + M_1 - y) (3 m_1 + y + M_1 - 2m_1) - (m_1 + y + M_1)
(M_1 - m_1) \\[1ex]
& = & 6 m_1 y M_1 + M_1 \left [ (M_1 - y) (m_1 + y + M_1) + 2 y (y +
M_1 - 2m_1) \right ] \\
& & - (m_1 + y + M_1) (M_1 - m_1) \\[1ex]
& = & 6 e(R/I) + 2 y M_1(y - m_1) + (m_1 + y + M_1) M_1
(M_1 - y) \\[1ex]
& &+ \left [2 y M_1 - (m_1 + y + M_1) \right ] (M_1 - m_1) \\[1ex]
& \geq & 6 e(R/I)\end{aligned}$$ because if $M_1 = 1$ then we must have $m_1 = y = M_1$ and the last estimate becomes an equality. But if $M_1 \geq 2$ then we get $2 y M_1 - (m_1 + y + M_1) \geq 2 y M_1 - (2y + M_1) \geq 0$ because for any two integers $k, l \geq 2$ we have $k l \geq k+l$. The upper bound follows.
Now assume $t \geq 1$ and let $I'$ be the Gorenstein ideals whose degree matrix is $$\label{eq-3-ind-deg-mat}
B' = \begin{bmatrix}
b_{t+1} \\
a_{t+1} & b_t & & & & & & *\\
& a_t & \ddots \\
& & \ddots & b_1 \\
& & & a_1 & d \\
& & & & a_1 & b_1 & & \\
& & & && \ddots & \ddots & \\
& & & & & & a_t & b_t \\
& * & & & & & & a_{t+1} & b_{t+1}
\end{bmatrix}$$ and that has been produced using the Cohen-Macaulay ideal $J'$ (cf. Remark \[rem-Gor-const\]) with degree matrix $$A' = \begin{bmatrix}
a_1 & b_1 & & & *\\
& \ddots & \ddots & \\
& & a_t & b_t \\
* & & & a_{t+1} & b_{t+1}
\end{bmatrix}.$$ To simplify notation, we set as in the codimension two case $$a := a_{t+1}, \quad b := b_{t+1}, \quad \mbox{and} \; c := b_t.$$ Let $I$ be the Gorenstein ideal whose Buchsbaum-Eisenbud matrix is obtained from the Buchsbaum-Eisenbud matrix of $I'$ by stripping the top and bottom rows, and the rightmost and leftmost columns such that the degree matrix is $B$ as in (\[eq-3-deg-mat\]). Let $m_1, m_2, m_3,$ $M_1, M_2, M_3$ be the invariants of $I$ and let $m'_1, m'_2, m'_3, M'_1, M'_2, M'_3$ be those of $I'$. Self-duality of the resolution of $R/I$ provides $$M_1 = m_3 - m_2 \quad \mbox{and} \quad M_2 = m_3 - m_1.$$ It follows that $$M_2 - m_2 = M_1 - m_1 = m_3 - m_1 - m_2 = M_1 + M_2 - M_3.$$ Thus, the formulas (\[3-formulas\]) provide $$\label{eq-rel-I-I'}
\begin{array}{rclcrcl}
m'_1 & = & m_1 + a & \mbox{ } & M'_1 & = & M_1 + b + c -a \\
m'_2 & = & m_2 + a + b - c & \mbox{ } & M'_2 & = & M_2 + 2 b - a \\
m'_3 & = & m_3 + 2 b & \mbox{ } & M'_3 & = & M_3 + 2b.
\end{array}$$ We also need the relation between the multiplicities of $R/I$ and $R/I'$.
\[rel bet G and G’\] $
\begin{array}{rcl}
e(R/I') & = & e(R/I) + b (m_1 + a) (M_2 + b -a).
\end{array} $
To see this, we may assume temporarily that $R/I$ and $R/I'$ have dimension one. Thus, the ideals $J$ and $J'$ used to produced $I$ and $I'$ (as in Remark \[rem-Gor-const\]) define curves. Denote their arithmetic genera by $g$ and $g'$, respectively. As preparation, we first relate the multiplicities of $R/I$ and $R/J$ and then the genera of $J$ and $J'$.
Using the notation of Remark \[rem-Gor-const\], we have that $I = J + \tilde{J}$ and that ${\mathfrak c}:= J \cap \tilde{J}$ is a complete intersection of type $(m_1, M_2)$. Hence, we have graded isomorphisms (cf., e.g., [@N-gorliaison], Lemma 3.5) $$K_{R/J} (4- m_1 - M_2) \cong \tilde{J}/(J \cap \tilde{J}) \cong (J + \tilde{J})/J = I/J$$ where $K_{R/J}$ denotes the canonical module of $R/J$. It follows that $$\label{rel bet I and J}
e(R/I) = (m_1 +M_2-4) \cdot e(R/J) - (2g-2).$$ Numerically, we may assume that $J'$ is a basic double link of $J$ (cf. (\[eq-J’-J\])), i.e.there is a complete intersection $(f, g)$ of type $(m_1+a, b)$ such that $J' = f R + g J$. Hence using the formula for the genus of a complete intersection (see for instance [@book], page 36), Proposition 4.1(b) in [@N-gorliaison] provides $$g' = g + \frac{1}{2} b (m_1 + a) (m_1 + a + b -4) + b \cdot e(R/J).$$ Therefore, using formula (\[rel bet I and J\]) for $I'$ as well as (\[eq-rel-I-I’\]) we obtain $$\begin{aligned}
e(R/I') & = & (m'_1 + M_2' - 4) \cdot e(R/J') - (2 g' - 2) \\
& = & (m_1 + M_2 + 2 b - 4) \left [ e(R/J) + b (m_1 + a) \right ] \\
& & - \left [ 2 g + b (m_1 + a) (m_1 + a + b -4) + 2 b \cdot e(R/J) - 2 \right ] \\
& = & (m_1 +M_2-4) \cdot e(R/J) - (2g-2) + b (m_1 + a) (M_2 + b -a) \\
& = & e(R/I) + b (m_1 + a) (M_2 + b -a),\end{aligned}$$ as claimed.
Now we are ready for the induction step. We assume that the bounds hold for $I$, and we prove them for $I'$. We will use the above numbered equations without comment.
We begin with the lower bound. We have to show that $$\label{eq-low-b}
e(R/I') \geq \frac{1}{6} m'_1 m'_2 m'_3 + \frac{1}{3} (m'_1)2 (m'_3 - m'_1 - m'_2).$$ Unfortunately, we need some rather lengthy computation. We have $$\begin{aligned}
\lefteqn{ m'_1 m'_2 m'_3 + 2 (m'_1)2 (m'_3 - m'_1 - m'_2) } \\[1ex]
& = & (m_1 + a) (m_2 + a + b - c) (m_3 + 2b) + 2 (m_1 + a)2 (m_3 -
m_1 - m_2 - (2a - b -c)) \\[1ex]
& = & m_1 m_2 m_3 + 2 m_12 (m_3 - m_1 - m_2) \\
& & + \left [a (m_2 + a + b - c)(m_3 + 2b) + m_1 (m_2 + a + b -c) 2 b
+ m_1 (a+ b -c) m_3 \right ] \\
& & + 2 \left [ a (2 m_1 + a)(m_3 - 2 m_1 - c) - (m_1 + a)2 (2a - b -
c) \right ] \\[1ex]
& \leq & 6 e(R/I) \\
& & + \left [a (m_1 + a + b)(m_3 + 2b) + m_1 (m_1 + a + b) 2 b + m_1
(a+ b -c) m_3 \right ] \\
& & + 2 \left [ a (2 m_1 + a)(m_3 - m_1 - m_2) - (m_1 + a)2 (2a - b -
c) \right ]\end{aligned}$$ by the induction hypothesis and because of $m_2 - c = m_1$ (by \[3-formulas\]).
Using Claim \[rel bet G and G’\] and essentially ordering for $m_3$ we obtain $$\begin{aligned}
\lefteqn{ m'_1 m'_2 m'_3 + 2 (m'_1)2 (m'_3 - m'_1 - m'_2) } \\[1ex]
& \leq & 6 e(R/I') - 6 b (m_1 + a) (m_3 - m_1 + b -a) \\
& & + \left [a (m_1 + a + b)(m_3 + 2b) + m_1 (m_1 + a + b) 2 b + m_1
(a+ b -c) m_3 \right ] \\
& & + 2 \left [ a (2 m_1 + a)(m_3 - m_1 - m_2) - (m_1 + a)2 (2a - b -
c) \right ] \\[1ex]
& = & 6 e(R/I') \\
& & + m_3 \left [ -6b (m_1 + a) + a (m_1 + a + b) + (a+b-c) m_1 \right ]
+ 2a (2m_1 + a) (m_3 - 2m_1 - c) \\
& & - 6b (m_1 + a) (-m_1 + b -a) + a (m_1 +a + b) 2b + m_1 (m_1 + a + b) 2b
- 2 (m_1 + a)2 (2a - b -c) \\[1ex]
& = & 6 e(R/I') \\
& & + m_3 \left [ m_1 (2a -5b -c) + a (a-5b) \right ] + 2a (2m_1 + a) (m_3 - 2m_1 - c) \\
& & - 6b (m_1 + a) (-m_1 + b -a) + 2b (m_1 + a) (m_1 +a + b) - 2 (m_1 + a)2 (2a - b -c) \\[1ex]
& = & 6 e(R/I') \\
& & + m_3 \left [ m_1 (2a -b -c) + a (a-b) - 4b (m_1 + a) \right ] + 2a (2m_1 + a) (m_3 - 2m_1 - c) \\
& & + 2b (m_1 + a) (4 m_1 + 4 a - 2b) - 2 (m_1 + a)2 (2a - b -c) \\[1ex]
& = & 6 e(R/I') \\
& & + m_3 \left [ m_1 (2a -b -c) + a (a-b) \right ] + 2a (2m_1 + a) (m_3 - 2m_1 - c) \\
& & + 4b (m_1 + a) (- m_3 + 2 m_1 + 2 a - b) - 2 (m_1 + a)2 (2a - b -c) \\[1ex]
& = & 6 e(R/I') + a (a-b) m_3 + (2a-b-c) \left [ m_1 m_3 + 4b (m_1 + a) - 2 (m_1 + a)2 \right ] \\
& & + (m_3 - 2m_1 -c) \left [ 2a (2m_1 + a) - 4b (m_1 + a) \right ]\end{aligned}$$ where we used $-m_3 + 2m_1 + 2a - b = -m_3 + 2m_1 + c + 2a - b -c$. Observing that $a \leq b,\; a \leq c$, and $m_3 \geq m_1 + m_2 = 2 m_1
+ c$ we get $$\begin{aligned}
\lefteqn{ m'_1 m'_2 m'_3 + 2 (m'_1)2 (m'_3 - m'_1 - m'_2) } \\[1ex]
& \leq & 6 e(R/I') + (2a-b-c) \left [ m_1 m_3 + 4b (m_1 + a) - 2 (m_1 + a)2 \right ] \\
& & + (m_3 - 2m_1 -c) \left [ 4 m_1 (a-b) + 2a (a -2b) \right ] \\[1ex]
& \leq & 6 e(R/I') + (2a-b-c) \left [ m_1 m_3 + 4b (m_1 + a) - 2 (m_1 + a)2 \right ] \\[1ex]
& \leq & 6 e(R/I')\end{aligned}$$ because $$m_1 m_3 + 4b (m_1 + a) - 2 (m_1 + a)2 = m_1 (m_3 - 2 m_1 + 4 (b-a)) + 2a (2b - a) \geq 0.$$ This completes the proof of the lower bound.
Turning to the upper bound, we have to show that: $$e(R/I') \leq \frac{1}{6} M'_1 M'_2 M'_3 - \frac{1}{6}
M'_3 (M'_1 + M'_2 - M'_3).$$ To start with, we have: $$\begin{aligned}
\lefteqn{ M'_1 M'_2 M'_3 - M'_3 (M'_1 + M'_2 - M'_3) } \\[1ex]
& = & (M_1 + b+c-a) (M_2 + 2b - a) ( M_3+ 2b) \\
& & - (M_3 + 2b) (M_1 + M_2 - M_3 + b + c -2a) \\[1ex]
& = & M_1 M_2 M_3 - M_3 (M_1 + M_2 - M_3) \\
& & + (M_1 +b+c-a) (2b-a) (M_3 + 2b) + (M_1 + b+c-a) M_2 2b +
(b+c-a) M_2 M_3 \\
& & - M_3 (b+c-2a) - 2b (M_1 + M_2 - M_3 + b+c-2a) \\[1ex]
& \geq & 6 e(R/I) \\
& & + (M_2 + b-a) (2b-a) (M_3 + 2b) + (M_2 + b-a) M_2 2b + (b+c-a) M_2 M_3 \\
& & - M_3 (b+c-2a) - 2b (2 M_2 - M_3 + b-2a) \\[1ex]\end{aligned}$$ by the induction hypothesis and because of $M_2 = M_1 + c$.
Using Claim \[rel bet G and G’\] and essentially ordering for $M_3$ we obtain $$\begin{aligned}
\lefteqn{ M'_1 M'_2 M'_3 - M'_3 (M'_1 + M'_2 - M'_3) } \\[1ex]
& \geq & 6 e(R/I') - 6 b (M_3 - M_2 + a) (M_2 + b - a) \\
& & + (M_2 + b-a) (2b-a) (M_3 + 2b) + (M_2 + b-a) M_2 2b + (b+c-2a) M_2 M_3 \\
& & - M_3 (b+c-2a) - 2b (2 M_2 - M_3 + b-2a) \\[1ex]
& = & 6 e(R/I') \\
& & + M_3 \left [ (M_2 + b-a) (2b -a - 6b) + (b+c-a) M_2 \right ] \\
& & + (M_2 + b-a) \left [ (2b-a) 2b + 2b M_2 + 6b (M_2 -a) \right ] \\
& & - M_3 (b+c-2a) - 2b (2 M_2 - M_3 + b-2a) \\[1ex]
& = & 6 e(R/I') \\
& & + M_3 \left [ (M_2 + b-a) (- 4b) + (b+c-2a) M_2 - a (b-a) \right ] \\
& & + (M_2 + b-a) \left [ 8b M_2 + (2b-4a) 2b \right ] \\
& & - M_3 (b+c-2a) - 2b (2 M_2 - M_3 + b-2a) \\[1ex]
& = & 6 e(R/I') \\
& & + M_3 \left [ (b+c-2a) M_2 - a (b-a) \right ] \\
& & + (M_2 + b-a) 4 b \left [ 2 M_2 - M_3 + b-2a \right ] \\
& & - M_3 (b+c-2a) - 2b (2 M_2 - M_3 + b-2a) \\[1ex]
& = & 6 e(R/I') \\
& & + M_3 \left [ (b+c-2a) (M_2 - 1) - a (b-a) \right ] \\
& & + (M_2 + b-a- \frac{1}{2}) 4 b \left [ 2 M_2 - M_3 + b-2a \right ]. \\\end{aligned}$$ Observe that $b \geq a$, $c \geq a$, and $M_2 \geq m_2 \geq c + 1 \geq
a + 1$. It follows that $$\begin{aligned}
b+c - 2a & \geq & b - a, \\
M_2 - 1 & \geq & a,\end{aligned}$$ thus $$\begin{aligned}
(b+c-2a) (M_2 -1) & \geq & a (b - a).\end{aligned}$$ Since we also have $$2 M_2 - M_3 + b-2a = (M_1 + M_2 - M_3) + (b+ c-2a) \geq 0$$ we obtain $$M'_1 M'_2 M'_3 - M'_3 (M'_1 + M'_2 - M'_3) \geq 6 e(R/I')$$ and the upper bound follows.
\[rem-hema\] In [@srinivasan], Srinivasan proved, compared to Conjecture \[conj\], stronger bounds for Gorenstein ideals of arbitrary codimension, but with quasi-pure resolutions. A resolution is quasi-pure if $m_i \geq
M_{i-1}$ for all $i$. In case of a codimension three Gorenstein ideal $I$ her bounds are $$\frac{1}{6} m_1 M_2 M_3 \leq e(R/I) \leq \frac{1}{6} M_1 m_2 m_3.$$ Note that the lower bound is not even true for arbitrary complete intersections. For example, a complete intersection of type $(2, 2,
5)$ gives a counterexample. On the other hand, the upper bound is true for all complete intersections and we wonder if it is true for all Gorenstein ideals of codimension three.
The method of proof of Theorem \[ht 3 gor bd\] also provides an explicit formula for the multiplicity of a Gorenstein ideal in terms of the degrees of the entries of its Buchsbaum-Eisenbud matrix.
\[prop-def-formula\] Let $I$ be a homogeneous Gorenstein ideal of codimension three with $2 t +1$ minimal generators. Order its Buchsbaum-Eisenbud matrix such that its degree matrix is $B$ as in (\[eq-3-deg-mat\]). Then we have $$e(R/I) = \sum_{j=1}^t b_j \cdot (a_1 + \ldots + a_j) \cdot (d +
\sum_{i=1}^{j-1}
(2 b_i - a_i) + b_j - a_j).$$
This follows easily from Claim \[rel bet G and G’\]. Indeed, let $t
= 1$. Then $I$ is a complete intersection with minimal generators of degree $a_1, b_1, d +
b_1 - a_1$ and the claim follows. Let $t \geq 2$. Then Claim \[rel bet G and G’\] provides the assertion using the formulas (\[3-formulas\]) and $M_2 = m_3 - m_1$.
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[^1]: Part of the work for this paper was done while the first author was sponsored by the National Security Agency under Grant Number MDA904-03-1-0071 and the second author was supported by a Special Faculty Research Fellowship from the University of Kentucky.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'As contemporary software-intensive systems reach increasingly large scale, it is imperative that failure detection schemes be developed to help prevent costly system downtimes. A promising direction towards the construction of such schemes is the exploitation of easily available measurements of system performance characteristics such as average number of processed requests and queue size per unit of time. In this work, we investigate a holistic methodology for detection of abrupt changes in time series data in the presence of quasi-seasonal trends and long-range dependence with a focus on failure detection in computer systems. We propose a trend estimation method enjoying optimality properties in the presence of long-range dependent noise to estimate what is considered “normal” system behaviour. To detect change-points and anomalies, we develop an approach based on the ensembles of “weak” detectors. We demonstrate the performance of the proposed change-point detection scheme using an artificial dataset, the publicly available Abilene dataset as well as the proprietary geoinformation system dataset.'
author:
- '\'
-
bibliography:
- 'references\_v4.bib'
title: |
Detecting Performance Degradation\
of Software-Intensive Systems in the Presence\
of Trends and Long-Range Dependence
---
Introduction
============
The last decade has witnessed the emergence of a novel type of high-tech systems: the software-intensive systems [@iso42010]. The latter[^1] include digital communication systems, internet systems (including devices, data transfer networks and services), call centers, automated power grids, intellectual transport systems, electronic trading platforms and many others. The obvious requirement for such systems is the effective, reliable and uninterrupted operation. However, recent studies of large-scale software-intensive systems indicate quite the opposite state of affairs: due to their sheer scale[^2] “software and hardware failures will be the norm rather than the exception” [@Yigitbasi2010]. According to the research, the dominant cause of costly and dangerous system failures are the software failures which makes software “the most problematic element of large-scale systems” [@northrop2006ultra].
Among the efforts undertaken in order to improve system reliability a major role is played by failure detection which aims to identify failures based on the analysis of data collected during the system operation. Examples of such data include the average number of processed requests and the queue size per time unit, the volume of transferred traffic, the number of rejected queries, etc. During both unexpected events (such as network equipment failures and network attacks) and scheduled occasions (e.g. data center maintenance and system software upgrades) the data experience abrupt deviations from the target state. The goal then is to detect sudden changes (referred to as anomalies or disorders) in the flow of the observed data. The detection is to be performed online; within the online (sequential) setting, as long as the behavior of the observations is consistent with the target state, one is content to let the process continue. If the state changes, then one is interested in detecting the change as rapidly as possible. Problems concerned with constructing efficient procedures for detecting changes in observed stochastic processes are known in the literature as change-point detection problems [@Polunchenko2012].
-0.2in
In the present work, we investigate the change-point detection problem for localization and diagnosis of anomalies in large-scale software-intensive systems in the presence of quasi-periodic trends and long-range dependence. The key step in the change-point detection approach is the specification of what is “normal” and “abnormal” state. This problem represents a challenge due to a number of reasons. First, the systems we consider here experience anthropogenic “nearly periodic” load variations that are difficult to model due to a complex load shape and its random variations over time. An example of quasi-periodic time series we investigate in this work is shown in Fig. \[fig:data\_example\]; they reflect weekly and daily load profiles for several internet services.
The second essential property of data flows in large-scale computer systems is that long-term correlations are present in these quantities, i.e., they are statistically self-similar [@Leland1994]. As self-similarity (also referred to as *long-range dependence* or *LRD*) has significant impact on queueing performance and represents the dominant cause of load “bursts”, the model should be able to efficiently capture it.
A natural approach to change-point detection involves utilization of statistical detection procedures such as the CUSUM procedure [@Page1954], the control charts procedure [@Shewhart1931], etc., as they possess certain efficiency properties. In turn, for these procedures to be implemented, the change-point model (the model of the “normal” and the “abnormal” signals) must be specified. The latter often cannot be specified accurately; as a consequence, even theoretically optimal procedures suffer significant degradation in change-point performance.
Finally, a considerable difficulty is caused by the large scale of contemporary software-intensive systems. For instance, volume of the dataset measured at Yandex[^3] reaches hundreds of thousands of characteristics, while other authors report software systems consisting of up to tens of thousand nodes [@Yigitbasi2010]. As the cost of manual model selection for each individual observed signal might be unacceptable, one should consider an automatic approach to model learning.
In this paper, we present an optimal method for signal estimation and an efficient procedure for change-point and anomaly detection in the presence of quasi-periodic trends and long-range dependence. We use our theoretical results regarding the structure of the optimal filter to construct a practical trend estimation algorithm. Using the estimate, we develop the change-point detection algorithm based on the ensemble of “weak” detectors to improve change-point detection performance when the standard assumptions regarding the change-point model are violated. We briefly describe existing change-point detection approaches as well as some of the conventional filtering techniques in Sect. \[related\_work\]. In Sect. \[model\_spec\], we specify our time series model and propose the model estimation algorithm. In Sect. \[changepoint\_detection\], we consider the particular change-point detection problem for our model and develop the ensemble-based change-point detection method. Sect. \[applications\] presents the evaluation results for a simulated and two real-world datasets: a publicly available Abilene network dataset and a proprietary Yandex dataset.
Related Work {#related_work}
============
A vast body of research covers the problem of failure detection in computer systems, and efficient detection algorithms have been developed for anomaly detection in computer networks, data stream networks, etc, see, e.g., [@Pham2014; @Casas2010] and references therein. In these applications, the change-point detection problem is investigated for the case of stationary random series, which is a well-studied setting (See [@Polunchenko2012] for a bird’s eye review).
Process stationarity assumption is rather restrictive for practice since in many applications the observed process is non-stationary. While no specific assumptions about the structure of the observed process are made, purely data-driven approaches such as principal component analysis (PCA) and its modifications are often taken under consideration [@Casas2010]. PCA and the subspace methods classify the observed data into “normal” and “abnormal” subspaces and have proven themselves efficient in anomaly detection applications [@Pham2014; @Lakhina2004; @Casas2010].
Change-point detection approaches mentioned above are difficult to apply directly to our problem. On the one hand, the observed data in our system are non-stationary; on the other hand, these data are characterized by trends and LRD noise that make PCA and the subspace methods ineffective.
A body of research covers a vast number of trend modeling and estimation approaches, such as multiple exponential smoothing [@winters1960forecasting], autoregressive models [@findley1998new], decomposition methods [@hodrick1997postwar], parametric and nonparametric regression [@artemov2015nonparametric]. Neither of the approaches incorporates an explicit model of LRD; consequently, efficient trend estimation in the presence of LRD cannot be achieved. On the contrary, our trend extraction approach relies on an explicit model of LRD signal and yields theoretically efficient estimates.
Trend Estimation in the Presence\
of Long-Range Dependence {#model_spec}
=================================
LRD and the Fractional Brownian Motion {#fbm_theory}
--------------------------------------
Long-range dependence is a phenomenon shared by many natural and technical systems. It relates to the rate of decay of statistical dependence of points with increasing time interval. In relation to software-intensive systems, LRD may be qualified as the presence of “burstiness” across an extremely wide range of time scales [@Leland1994]. During the last decades, the fractional Brownian motion has been established as the standard model for LRD signals.
The fractional Brownian motion (fBm) was introduced by Kolmogorov in connection with his works on the theory of turbulence [@kolmogorov1940wiener] and later was constructively defined by Mandelbrot [@Mandelbrot1968]. In what follows, we adopt the notation from [@artemov2015optimal]. A standard fBm ${\ensuremath{{B^H}= ({B^H}_t)_{0 \leqslant t \leqslant T}}}$ with Hurst exponent $H\in\left(0,1\right)$ on $\left[0,T\right]$ is a Gaussian process with continuous trajectories, ${B^H}_0 = 0,$ $\mathbf{E}{B^H}_t = 0,$ $\mathbf{E}{B^H}_s {B^H}_t =
\frac{1}{2}\left(t^{2H} + s^{2H} - \left|t-s\right|^{2H}\right).$ When $H=\frac{1}{2}$, the process ${B^H}$ is a standard Brownian motion but in the case $H\neq\frac{1}{2}$ the process ${B^H}$ is not a semimartingale. In many applications, process ${B^H}$ is used for modeling of time series with very chaotic movements (the case $H < 1/2$) and with a relatively smooth behavior (the case $H \geqslant 1/2$).
The Specification of the Theoretical Filter {#filter_theory}
-------------------------------------------
Let the observed continuous-time process ${\ensuremath{X = (X _t)_{0 \leqslant t \leqslant T}}}$ satisfy the relation $$X_{t}=\sum\limits _{i=0}^{n}\theta_{i}\varphi_i(t) + \sigma B_{t}^{H},
\label{eq:observation_explicit_form}$$ where $\{\varphi_i(t)\}_{i=0}^{n}$ is a dictionary of differentiable functions on $[0, T]$, ${\ensuremath{{B^H}= ({B^H}_t)_{0 \leqslant t \leqslant T}}}$ is the standard fBm on $[0, T]$ with a known Hurst index $H$, and the variance $\sigma > 0$ is assumed to be known. The unknown parameters $\{\theta_i\}_{i=0}^n$ must be estimated using the observations $\left\{X_{s},0\leqslant s\leqslant t\right\}$ available up to time $t$. In [@artemov2015optimal], theoretical results regarding the structure of the optimal filter in for the general dictionary of functions $\{\varphi_i(t)\}_{i=0}^{n}$ were obtained for the case of (a) the maximum likelihood estimate and (b) the Bayesian estimate. For the purpose of the current work, we use the maximum likelihood (ML) filter to estimate a smooth trend against the LRD noise. We assume that:
- the dictionary consists of power functions: $\varphi_i(t) = t^i, i=0, \ldots, 3$, allowing to estimate the polynomial trend $f(t)$;
- the value of the Hurst exponent $H$ is known (in practice, $H$ can be estimated from the observations using such approaches as introduced in [@kirichenko2011comparative; @hardstone2012detrended]);
- the value of the variance $\sigma$ is known (in fact, the filter from [@artemov2015optimal] does not depend on the variance, see below).
The ML estimate ${\widehat{{\bm{\theta}}}_{\mathrm{ML}}}$ for the drift parameter ${\bm{\theta}}=(\theta_0, \ldots, \theta_3)$ is given by $$\label{eq:ml_estimate_generic}
{\widehat{{\bm{\theta}}}_{\mathrm{ML}}}= {\bm{R}_H}^{-1}(t) {\bm{\psi}^H}_t,$$ where ${\bm{R}_H}(t) = ({\bm{R}_H}(t))_{ij}$ and ${\bm{\psi}^H}_t = (({\bm{\psi}^H}_t)_0, \ldots, ({\bm{\psi}^H}_t)_3)$ are defined by $({\bm{R}_H}(t))_{ij} = \alpha_H(i,j) t^{i+j-2H}$ and $({\bm{\psi}^H}_t)_i = \beta_{H}(i) \int\limits _{0}^{t}s^{i-1}d{M^H}_s,$ where $\lambda_{H} = 2H\frac{\Gamma(3-2H) \Gamma(1/2 + H)} {\Gamma(3/2-H)},$ $\alpha_H(i,j) = \lambda^{-1}_{H}\beta_{H}(i)\beta_{H}(j)\frac{2-2H}{i+j-2H},$ $\beta_{H}(i) = i \tfrac{2-2H+i-1}{2-2H}\tfrac{\Gamma\left(3-2H\right)}{\Gamma\left(3-2H+i-1\right)} \tfrac{\Gamma\left(3/2-H+i-1\right)}{\Gamma\left(3/2-H\right)},$ $ {i,j=0,\,\ldots,n},$ and ${\ensuremath{{M^H}= ({M^H}_t)_{0 \leqslant t \leqslant T}}}$ is a martingale defined by ${M^H}_t \equiv \kappa_{H}^{-1} \int\limits _{0}^{t} s^{1/2-H}(t-s)^{1/2-H}d X_s,$ $\kappa_{H} = 2H\Gamma(3/2 - H)\Gamma(1/2 + H) \enspace.$
The Trend Estimation Algorithm with LRD Correction {#ml_estimation_algorithm}
--------------------------------------------------
The algorithm assumes the observations are taken according to the model $$\label{eq:general_observations_model}
X_t = f(t) + \eta^H(t), \qquad t \geqslant 0,$$ where the trend $f(t)$ is some smooth function observed in the LRD noise $\eta^H(t)$. Taking advantage of the smoothness of the trend $f(t)$, we approximate it using some finite-order polynomial $\sum_{i = 0}^n \theta_i (t - t_0)^i$ in the neighbourhood of any $t_0 > 0$. We model $\eta^H(t)$ using the fractional Gaussian noise (fGn) $Z_t^H$ with some (unknown but nonrandom) variance $\sigma(t)$ and Hurst exponent $H$: $\eta^H(t) = \sigma(t) Z_t^H$. Given the noisy observations $\{(X_k, t_k)\}_{k = 1}^{\ell}$, the goal is to estimate the expected value $f(t) = \mathbf{E} X_t$ for any $t \geqslant 0$. The following algorithm provides a solution to this problem.
1. Consider an interval $[a,b]$ and select observations window $W(a, b) = \{(X_k, t_k): a \leqslant t_k \leqslant b\}$.
2. Compute the estimate $\widehat{f}_{[a,b]}(t)$ of the trend $f(t)$ for $a \leqslant t \leqslant b$:
1. Assuming a cubic polynomial model for the observations \[fbm\_algorithm:theta\_estimate\] $$\label{eq:locpoly_observations_model}
X_k = \sum \limits_{i = 0} ^3 \theta_i (t_k - t_0)^i + \sigma Z_k^H,$$ where $(X_k, t_k) \in W(a, b)$, $t_0 = (a + b) / 2$, $\sigma$ is assumed to be constant, and $H = \frac{1}{2}$, estimate the value of ${\bm{\theta}}= (\theta_0, \ldots, \theta_3)$ using the maximum likelihood estimate ${\widehat{{\bm{\theta}}}_{\mathrm{ML}}}$ described in Sect. \[filter\_theory\].
2. \[fbm\_algorithm:trend\_estimate\] Compute the trend estimate on $[a,b]$ using the relation $\widehat{f}_{[a,b]}(t) = \sum _{i = 0} ^3 ({\widehat{{\bm{\theta}}}_{\mathrm{ML}}})_i (t - t_0)^i$ for each $t \in [a, b]$.
3. \[fbm\_algorithm:var\_estimate\] Compute the variance estimate $\widehat{\sigma}$ as the sample variance of residuals $\{X_k - \widehat{f}_{[a,b]}(t_k) \mid t_k \in [a, b] \}$.
4. Compute the estimate of the Hurst exponent $\widehat{H}$ using an approach from [@hardstone2012detrended] and the standardized residuals $\{(X_k - \widehat{f}_{[a,b]}(t_k)) / \widehat{\sigma} \mid t_k \in [a, b] \}$.
5. Using the Hurst exponent estimate $\widehat{H}$, compute corrected trend and variance estimates in a)–c).
3. We use the sliding window $[a,b]=[a, a + \Delta]$ with sufficiently large $\Delta$ and obtain $n_{[a,b]}(t) = \left\vert{A(t)}\right\vert$ local corrected estimates $\widehat{f}_{[a,b]}(t)$ for each $t \geqslant 0$, where $A(t) = \{(a, b) \mid t \in [a, b]\}$. To obtain the final estimate $\widehat{f}(t)$ we average the corrected estimates using the relation $\widehat{f}(t) = \frac{1}{n_{[a,b]}(t)} \sum\limits_{(a,b)\in A(t)} \widehat{f}_{[a,b]}(t) \enspace.$
The two-step procedure for computing the estimate $\widehat{f}(t)$ is necessary since in practice the Hurst exponent is unknown but important constant strongly influencing an estimation performance, see Fig. \[fig:mc\_trend\_filter\_vs\_hurst\]. By applying the correction in the algorithm steps \[fbm\_algorithm:theta\_estimate\]–\[fbm\_algorithm:var\_estimate\] we achieve better trend estimation accuracy compared to a generic approach with $H = \frac{1}{2}$, see Fig. \[fig:mc\_correction\_effect\].
\
\[ width=0.6, \][iterations\_cubic\_trendhurst\_new\_v2]{}
Change-point Detection in the Presence\
of Trends and Long-Range Dependence {#changepoint_detection}
=======================================
The Change-point Model
----------------------
We consider the following change-point model for the noise $\eta^H(t)$ in : $$\label{eq:short_term_change_process}
\eta^H(t) = \mu \mathds{1}_{[\theta, \theta + \Delta t]}(t) + \sigma Z^H_t,
\qquad t \geqslant 0,$$ where $\theta$ is an unknown time of a change, $\mu$ is an unknown change magnitude, $\sigma$ is an unknown (non-random) variance, and $Z^H_t$ is the fGn. The characteristic duration $\Delta t$ of the considered change is short; hence the change represents a local deviation in the values of the observed series, see Fig. \[fig:trajectory\_real\].
To detect the change, we introduce a residual process ${\ensuremath{R = (R _t)_{t \geqslant 0}}}$ $$\label{eq:residual_change_process}
R_t = \sigma^{-1} (X_t - \widehat{X}_t),
\qquad t \geqslant 0,$$ where $X_t$ is the signal with a known variance $\sigma$ observed in and $\widehat{X}_t$ is an estimate of $X_t$ obtained via filtering algorithm described in Sect. \[ml\_estimation\_algorithm\]. In absence of a change, $R_t$ is an approximately zero-mean process with unit variance, however, in presence of a change, neither of these properties holds. Note that $R_t$ is a natural estimate for $Z^H_t$ and $\sigma R_t$ is a natural estimate for the noise component $\eta^H(t)$. We use the process $R_t$ in Sect. \[change\_point\_detection\] to detect the change.
The Ensemble-based Change-point Detection Procedure {#change_point_detection}
---------------------------------------------------
The standard assumptions regarding the change-point model state that pre- and post-change distributions are Gaussian i.i.d. with different (yet known) parameters [@Shewhart1931; @Page1954; @Polunchenko2012]. These assumptions are heavily violated in our case due to (a) the approximation error introduced by substitution of the real trend $f(t)$ with a locally cubic trend, (b) the estimation error introduced by the estimation algorithm in Sect. \[ml\_estimation\_algorithm\], (c) the unknown change signature, and (d) the modeling errors due to interpreting noise in the real signal as the fBm. Moreover, the absence of accurate detection procedures for LRD signals makes the change-point detection performance low when “classical” change-point detection methods are used.
Let $\Pi_1, \ldots, \Pi_n$ denote $n$ change-point detection procedures, such as the cumulative sum (CUSUM) procedure [@Page1954] based on the process ${\ensuremath{T = (T _t)_{t \geqslant 0}}}$: $$\label{eq:cusum_definition}
T_t =
\max(0,T_{t - 1} + \zeta_t),
\quad T_0 = 0, \quad {t \geqslant 0},$$ where $\zeta_t = \log (f_0(X_t) / f_{\infty}(X_t))$ is the log-likelihood ratio, and $f_{\infty}(\cdot)$ and $f_0(\cdot)$ are one-dimentional pre- and post-change distributions, respectively. Each procedure $\Pi_k$ prescribes to stop observations at time $\tau_k$ which is the first hitting time of some process ${\ensuremath{S^k = (S^k _t)_{t \geqslant 0}}}$ to a level $h_k > 0$: $\tau_k = {\ensuremath{\inf \{{t \geqslant 0}: S^k_t \geqslant h_k\}}}$. We further consider a set of *signals* $\big\{{\ensuremath{s^k = (s^k _t)_{t \geqslant 0}}}\big\}_{k = 1}^{n}$ defined by $s^k_t = S^k_t / h_k, {t \geqslant 0}$. We call the procedure $\mathrm{A}$ an *ensemble* if its stopping time $\tau_{\mathrm{A}}$ is defined as the first hitting time of some process ${\ensuremath{a = (a _t)_{t \geqslant 0}}}$ to a specified level $h_{\mathrm{A}} > 0$: $\tau_{\mathrm{A}} = {\ensuremath{\inf \{{t \geqslant 0}: a_t \geqslant h_{\mathrm{A}}\}}}$, where $$\label{eq:general_ensemble}
a_t = \psi(\bm{\lambda}; \mathbf{S}^1_t, \ldots, \mathbf{S}^n_t),$$ $\bm{\lambda} \in {{\rm I\!R}}^d$ ($d\geqslant n$) and $\mathbf{S}^k_t = \{s^k_s, 0 \leqslant s \leqslant t\}$ is the history of the signal ${\ensuremath{s^k = (s^k _t)_{t \geqslant 0}}}$ up to the time $t$, $k = 1, \ldots, n$. Each ensemble is completely defined by the choice of the “aggregation function” $\psi(\cdot)$. In this work, we consider a *logistic regression-based* classifier for which the aggregation function could be written as $$\label{eq:logistic_weighted_p_ensemble}
a_t = \psi_{\textsc{Log}-p}(\bm{\lambda}; \mathbf{S}^1_t, \ldots, \mathbf{S}^n_t) =
\sigma\Big(\sum\limits _{j = 0} ^{p} \sum\limits _{k = 1} ^{n} \lambda_{kj} s^k_{t - j} - \lambda_0\Big),$$ where $\sigma(x) = 1 / (1 + e^{-x})$ is the logistic function. The value $a_t$ can be interpreted as a posterior probability of a change-point given the observations history $\mathbf{X}_t = \{X_s, 0 \leqslant s \leqslant t\}$ up to the moment $t$. Note that for this ensemble the threshold $h_{\mathrm{A}}$ must be chosen to belong to the interval $(0, 1)$ [@artemov2015ensembles].
Learning Ensemble Parameters
----------------------------
Ensemble parameters $\bm{\lambda} \in {{\rm I\!R}}^d$ can be *learned* to optimize a certain performance measure. Let $\mathcal{X}^{\ell} = \{(X^i, Y^i)\}_{i=1}^{\ell}$ be the labeled data where each point $(X^i, Y^i) \in \mathcal{X}^{\ell}$ is a pair, its first component ${\ensuremath{X^i = (X^i _t)_{0 \leqslant t \leqslant T}}}$ being a sample path of the observations, and its label ${\ensuremath{Y^i = (Y^i _t)_{0 \leqslant t \leqslant T}}}$ being an “abnormal” state indicator: $Y^i_t = \mathds{1}_{\mathcal{T}^{i}_{0}}(t)$. Let $T_{\infty}^i$ and $T_0^i$ be the durations of “normal” and “abnormal” states $\mathcal{T}^{i}_{\infty}$ and $\mathcal{T}^{i}_{0}$ for each point $(X^i, Y^i), i=1, \ldots, \ell$, respectively. We formulate the problem of learning the parameters $\bm{\lambda} \in {{\rm I\!R}}^d$ of an ensemble as an optimization problem $\mathbf{F} (\mathrm{A}) \to \inf \limits_{\bm{\lambda} \in {{\rm I\!R}}^d}$ for the Average Relative Error Rate measure $$\begin{gathered}
\label{eq:ara_measures}
\mathbf{F} (\mathrm{A}) =
c_{\infty}
\mathbf{E}_{\infty}
\Bigg[
\frac {
\int \mathds{1}_{\{a_t \geqslant h_{\mathrm{A}}\}}(t) \mathds{1}_{\mathcal{T}_{\infty}}(t) dt
}
{
\int \mathds{1}_{\mathcal{T}_{\infty}}(t) dt
}
\Bigg] + \\
c_{0}
\mathbf{E}_{0}
\Bigg[
\frac {
\int \mathds{1}_{\{a_t < h_{\mathrm{A}}\}}(t) \mathds{1}_{\mathcal{T}_{0}}(t) dt
}
{
\int \mathds{1}_{\mathcal{T}_{0}}(t) dt
}
\Bigg],\end{gathered}$$ where $c_{\infty}$ and $c_{0}$ are the costs of false alarm and false silence, respectively. As $\mathbf{F} (\mathrm{A})$ is a non-differentiable function and cannot be optimized using standard approaches, we introduce its empirical approximation $\widehat{\mathbf{F}}_{\mathrm{D}}(\mathrm{A})$ defined by $$\begin{gathered}
\label{eq:empirical_ara_upper_bound}
\widehat{\mathbf{F}}_{\mathrm{D}}(\mathrm{A}) =
\frac{1}{\ell} \sum\limits_{i=1} ^{\ell}
\Bigg\{
\frac {c_{\infty}} {T^i_{\infty}}
\sum\limits_{t \in \mathcal{T}^i_{\infty}} \sigma(a_t - h_{\mathrm{A}})
+ \\
\frac {c_{0}} {T^i_{0}}
\sum\limits_{t \in \mathcal{T}^i_{0}} \sigma(h_{\mathrm{A}} - a_t),
\Bigg\}\end{gathered}$$ where $\sigma(x) = 1 / (1 + e^{-x})$ is the logistic function. Note now that the function $\widehat{\mathbf{F}}_{\mathrm{D}}(\mathrm{A})$ is differentiable w.r.t. the ensemble parameters $\bm{\lambda} \in {{\rm I\!R}}^d$ and can therefore be optimized using standard methods.
Performance Evaluation {#applications}
======================
Evaluation Datasets {#sec:evaluation_datasets}
-------------------
We study the performance of filtering and change-point detection algorithms on two artificial datasets <span style="font-variant:small-caps;">Artificial-Easy</span> and <span style="font-variant:small-caps;">Artificial-Hard</span> and on two real-world datasets: the publicly available Abilene network dataset and on the proprietary Yandex dataset.
Artificial datasets consist of one-week samples of artificial data $\{(X_k, t_k)\}_{k = 1}^{K}$, $K = 2016,$ measured at consecutive 5-minute intervals according to the model $X_k = f(t_k) + \eta^H(t_k)$, where $f(t_k) = A \sin(2\pi t_k / T)$ with $A = 1.5, T = 288$, and $\eta^H(t)$ is the LRD noise process. To model the change-point in the artificial data, for each replication of the sample we generate the LRD noise $\eta^H(t)$ according to the model in with $\sigma = 1$, a random change-point time: $\theta \sim U(T, 6T)$, a random change-point duration: $\Delta t \sim U(5, 100)$, and ${\ensuremath{Z^H = (Z^H _t)_{t \geqslant 0}}}$ formed as a discrete approximation of the fGn process with $H = 0.95$. For <span style="font-variant:small-caps;">Artificial-Easy</span>, we set the change-point magnitude $\mu = 5$, and for <span style="font-variant:small-caps;">Artificial-Hard</span>, change-point magnitude is set to $\mu = 3$. Despite this seemingly large magnitude, as we show below, the change-points we generated are remarkably hard to detect, due to the presence of seasonal trends and LRD noise, see Fig. \[fig:season\_approx\] (left). We generated 1000 independent replications of the sample for training the ensemble and another 1000 for testing. We denote these dataset $\mathcal{X}^{\ell}_{\mathrm{TRAIN}}$ and $\mathcal{X}^{\ell}_{\mathrm{TEST}}$, where $\ell = 1000$, respectively.
The Abilene dataset[^4] describes network load in the Abilene network in terms of the amount of traffic transmitted between network endpoints during consecutive 5-minute intervals. The data is available for the period of March 1, 2004 to September 10, 2004, and consists of 132 different time series describing traffic transmitted between 12 different network nodes located in 12 different locations across the USA. An example of Abilene data is shown in Fig. \[fig:data\_example\], bottom-left, for 4 different pairs of endpoints for a particular measurement period. The Abilene dataset is frequently used for evaluation of anomaly detection methods due to its complex structure and presence of both short-lived and long-lived anomalies [@Lakhina2004; @Casas2010].
The Yandex dataset consists of time series describing the performance of a geoinformation system at Yandex. Each time series is sampled at consecutive 5-minute intervals and it represents the total number of requests processed by the system. An example of Yandex time series is shown in Fig. \[fig:data\_example\] (top-left) and in Fig. \[fig:trajectory\_real\] (right) along with labels displaying the anomalies subject to detection.
Evaluated Procedures {#sec:evaluated_procedures}
--------------------
We train the ensemble using five “weak” detectors: the cumulative sum detector, the Shiryaev-Roberts detector, the Shewhart detector, the changepoint detector, and the posterior probability process detector (for details, refer to [@artemov2015ensembles], Sect. 2).
We empirically compare the performance of our ensemble-based procedure to that of several well-studied approaches, specifically, threshold-based procedure, CUSUM procedure, and the subspace method. The threshold-based procedure <span style="font-variant:small-caps;">EWMA-Threshold</span> uses EWMA to estimate the mean $\widehat{\mu}_t$ and variance $\widehat{\sigma}^2_t$ of the time series $X_t$, obtains the residuals $R_t = (X_t - \widehat{\mu}_t) / \widehat{\sigma}_t$, and calculates the fraction of the residual points within the time window $[t - \Delta, t]$ located above the threshold $h$. The stopping time for raising the alarm is defined as $\tau_{\mathrm{THR}} = \inf \{k \geqslant 1:
S_k \geqslant h_{\mathrm{THR}}\}$ where $S_k =
\sum_{i = k - \Delta}^k \mathds{1}_{ \{R_i \geqslant h\} } (i)$. The threshold $h_{\mathrm{THR}}$, the per-point threshold $h$ and the window size $\Delta$ are algorithm parameters; we only report results regarding the calibrated values of these parameters which result in best performance of the procedure. The <span style="font-variant:small-caps;">EWMA-CUSUM</span> procedure replaces the statistic ${\ensuremath{S = (S _t)_{t \geqslant 0}}}$ defined above with the CUSUM statistic $T_t$ defined in . The densities $f_{\infty}(\cdot)$ and $f_0(\cdot)$ are assumed to be normal with unit variances and means $\mu_{\infty} = 0$ and $\mu_0 = \mu_{\infty} + \delta$, respectively. The parameter $\delta$ is selected to obtain the best performance on training set in terms of the area under the precision-recall curve. The subspace method <span style="font-variant:small-caps;">PCA</span> is closely related to the singular spectrum analysis (SSA) approach and subspace methods from the literature [@Lakhina2004; @Casas2010; @vautard1992singular]. In the <span style="font-variant:small-caps;">PCA</span> procedure, a decomposition of the time series ${\ensuremath{X = (X _t)_{t \geqslant 0}}}$ is obtained using the SSA procedure, and the component $\mathbf{X}^{\mathrm{RES}}_t$ living in the residual subspace is considered. The statistic ${\ensuremath{P = (P _t)_{t \geqslant 0}}}$ of the procedure is the norm of the residual component: $P_t = \Vert\mathbf{X}^{\mathrm{RES}}_t\Vert$. We note that the subspace method benefits greatly from pretraining on historic data. To exploit this advantage, we supplied the SSA procedure with a week of historic data to obtain a better decomposition. We call this procedure <span style="font-variant:small-caps;">PCA-Pretraining</span>. Note that no other procedure receives any additional input when trained.
Trend Approximation Accuracy {#sec:trend_extraction_accuracy}
----------------------------
We first compare the trend extraction accuracy on the dataset <span style="font-variant:small-caps;">Artificial-Easy</span>. We use the relative root mean squared forecast error $\mbox{RRMSE}(X_t, \widehat{X}_t) = \sqrt{\frac{1}{K} \sum_{t=1}^K (X_t - \widehat{X}_t)^2/X_t^2},$ to evaluate forecasting performance. Table \[table:trend\_approximation\_rmse\] presents trend extraction accuracy on two tasks: trend approximation and one-point-ahead forecasting. Trend approximation accuracy $\mbox{RRMSE}(f(t), \widehat{f}(t))$ measures how closely the extracted trend follows the true trend $f(t)$. One-point-ahead forecasting accuracy estimates how well an algorithm predicts incoming new data $X_t$ given the observed values $\{X_k, k < t\}$. Our study shows that our approach produces significantly more accurate estimates than EWMA. An example of trend approximation is presented in Fig. \[fig:season\_approx\] for the artificial dataset and for the Abilene dataset. We note that our approach yields a smooth approximation and allows for more robust anomaly isolation, while EWMA follows the data more closely.
0.15in
[lcc]{} Method &
---------------
Trend
approximation
---------------
: Trend extraction accuracy for the artificial dataset in terms of RRMSE (%) for EWMA, PCA and our approach.[]{data-label="table:trend_approximation_rmse"}
&
-----------------
One-point-ahead
forecasting
-----------------
: Trend extraction accuracy for the artificial dataset in terms of RRMSE (%) for EWMA, PCA and our approach.[]{data-label="table:trend_approximation_rmse"}
\
EWMA & 7.84 & 7.34\
PCA & 8.96 & 5.65\
PCA-Pretraining & 5.58 & 3.80\
Ours & 5.72 & 3.06\
-0.1in
[0.52]{} {width="\linewidth"}
[0.52]{} {width="\linewidth"}
Change-point Detection Performance Measures {#sec:change_point_detection_results}
-------------------------------------------
To evaluate the change-point detection performance, we use two performance measures. The first measure is the Precision-Recall Curve, which is a standard performance measure in the area of machine learning. The second measure is the Average Relative Error Rate curve proposed in –. Before discussing the obtained results, we briefly explain how these performance measures are computed. Suppose that a procedure $\Pi$ is defined by a statistic ${\ensuremath{S = (S _t)_{t \geqslant 0}}}$. When computed on a test instance $(X^i, Y^i) \in \mathcal{X}^{\ell}_{\mathrm{TEST}}$, procedure $\Pi$ generates a trajectory $\{S^i_1, \ldots, S^i_l\}$ and for some specified threshold $h_{\mathrm{\Pi}} > 0$ produces $M_{\mathrm{\Pi}}$ segments $\big\{[t_{a_m}, t_{b_m}]\big\}_{m=1}^{M_{\mathrm{\Pi}}}$ such that $\forall t \in [t_{a_m}, t_{b_m}] \quad S^i_t \geqslant h_{\mathrm{\Pi}}$. We declare the detection $[t_{a_m}, t_{b_m}]$ true positive if it intersects with the “abnormal” segment, i.e. if $[\theta, \theta + \Delta t] \cap [t_{a_m}, t_{b_m}] \neq \emptyset$. If, on the other hand, this intersection is empty (the statistic signals outside the interval $[\theta, \theta + \Delta t]$), then the detection is declared false positive. The Precision-Recall Curve is plotted by varying the threshold $h_{\mathrm{\Pi}}$. The Average Relative Error Rate curve is a plot of Average False Positive Rate $\frac{1}{\ell} \sum_{i=1} ^{\ell}
\frac {c_{\infty}} {T^i_{\infty}}
\sum_{t \in \mathcal{T}^i_{\infty}} \mathds{1}_{\{S^i_t \geqslant h_{\mathrm{\Pi}}\}}(t)$ versus Average False Negative Rate $\frac{1}{\ell} \sum_{i=1} ^{\ell}
\frac {c_{0}} {T^i_{0}}
\sum_{t \in \mathcal{T}^i_{0}} \mathds{1}_{\{S^i_t < h_{\mathrm{\Pi}}\}}(t)$. Average Relative Error Rate can be thought of as a segmentation rather than classification measure.
Results {#sec:experiments_results}
-------
For the <span style="font-variant:small-caps;">Artificial-Easy</span> data, our approach is outperformed only by the optimal CUSUM procedure by a little margin when measured in terms of AUC, see Fig. \[fig:detection\_artificial\_easy\], left. On <span style="font-variant:small-caps;">Artificial-Hard</span>, our approach outperforms all other methods in equal conditions. However, adding more data to <span style="font-variant:small-caps;">PCA</span> to improve decomposition accuracy makes it the best on this task, see Fig. \[fig:detection\_artificial\_hard\], left. Our approach also yields the most accurate segmentations, as can be seen on both Fig. \[fig:detection\_artificial\_easy\], right, and Fig. \[fig:detection\_artificial\_hard\], left, meaning both lower average false silence and lower average false alarm durations.
We conclude that our approach significantly outperforms the rival algorithms in terms of the precision-recall characteristic. The reason for this increase in change-point detection performance is the high correlation between the true change-points and the proposed detections, as can be seen in Fig. \[fig:trajectory\_artificial\], right. We note, however, that due to the complex nature of both artificial datasets, many change-points are difficult to detect.
Trend extraction results for the two real-world datasets are presented in Fig. \[fig:trajectory\_real\] for EWMA and our approach, and in Fig. \[fig:trajectory\_real\_pca\] for PCA and our approach. As can be seen from these figures, our filtering approach would result in residuals which violate the change-point model in to a lesser extent; the ensemble then further should improve detection performance because it optimizes on the residual data. PCA-based approach performs generally comparable to our approach (and even outperforms it in case of pretraining); however, PCA requires retraining which is computationally very expensive when performed online on a large number of time series. Our filtering approach is advantageous in that it may be implemented online via a simple linear filter. More results are in Fig. \[fig:cusum\_data\_example\], where change-point detection results using the logistic regression-based ensemble are presented for both Yandex and Abilene data. We conclude that our approach is effective for both artificial and real data and can readily be applied for anomaly detection in a multitude of environments.
-0.2in
\
-0.2in
Conclusion
==========
We investigated change-point detection in the presence of quasi-seasonal trends and long-range dependent noise with an application to fault detection in software-intensive systems. We proposed an effective trend estimation algorithm based on the theoretically optimal filter and a practical change-point detection procedure based on the ensemble of “weak” detectors. An empirical study of the change-point detection procedure shows that it significantly ourperforms the standard EWMA and PCA-based algorithms when the conventional assumptions about the change-point model are violated.
Acknowledgements {#acknowledgement}
================
The research, presented in Section \[applications\] of this paper, was supported by the RFBR grants 16-01-00576 A and 16-29-09649 ofi\_m; the research, presented in other sections, was conducted in IITP RAS and supported solely by the Russian Science Foundation grant (project 14-50-00150).
[^1]: defined in ISO/IEC/IEEE 42010:2011 as systems where “software contributes essential infuences to the design, construction, deployment, and evolution of the system as a whole”
[^2]: Expressed in “number of lines of code; number of people employing the system for different purposes; amount of data stored, accessed, manipulated, and refined; number of connections and interdependencies among software components; and number of hardware elements” [@northrop2006ultra].
[^3]: Yandex is one of the largest internet companies in Europe, operating Russia’s most popular search engine and its most visited website, see <http://company.yandex.com>.
[^4]: See <http://www.cs.utexas.edu/~yzhang/research/AbileneTM>.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: ' We discuss compactness of the ${\overline\partial}$-Neumann operator in the setting of weighted $L^2$-spaces on $\mathbb C^n.$ In addition we describe an approach to obtain the compactness estimates for the ${\overline\partial}$-Neumann operator. For this purpose we have to define appropriate weighted Sobolev spaces and prove an appropriate Rellich - Kondrachov lemma.'
address: ' F. Haslinger: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria'
author:
- Friedrich Haslinger
bibliography:
- 'mybibliography.bib'
title: ' Sobolev spaces for the weighted ${\overline{\partial}}$-Neumann operator'
---
Introduction
============
Let $\Omega $ be a bounded open set in $\mathbb{R}^n,$ and $k$ a nonnegative integer. We denote by $W^k(\Omega)$ the Sobolev space $$W^k(\Omega) = \{ f\in L^2(\Omega ) \, : \, \partial^\alpha f \in L^2(\Omega ) , \, |\alpha |\le k \},$$ where the derivatives are taken in the sense of distributions and endow the space with the norm $$\|f\|_{k,\Omega} = \left ( \sum_{|\alpha |\le k} \int_\Omega |\partial^\alpha f |^2 \,d\lambda \right )^{1/2},$$ where $\alpha =(\alpha_1, \dots ,\alpha_n)$ is a multiindex , $|\alpha |=\sum_{j=1}^n \alpha_j$ and $$\partial^\alpha f =\frac{\partial^{|\alpha |}f}{\partial x_1^{\alpha_1}\dots \partial x_n^{\alpha_n}}.$$ $W^k(\Omega)$ is a Hilbert space. If $\Omega \subset \mathbb{R}^n \, , \, n\ge 2,$ is a bounded domain with a $\mathcal{C}^1$ boundary, the Rellich-Kondrachov lemma says that for $n>2$ one has $$W^1(\Omega ) \subset L^r(\Omega) \ , \ r\in [1, 2n/(n-2))$$ and that the imbedding is also compact; for $n=2$ one can take $r\in [1,\infty)$ (see for instance [@AF]), in particular, there exists a constant $C_r$ such that $$\label{eq: sob1}
\|f\|_r \le C_r \|f\|_{1,\Omega},$$ for each $f\in W^1(\Omega),$ where $$\|f\|_r = \left ( \int_\Omega |f|^r \, d\lambda \right )^{1/r}.$$ Now let $\Omega \subseteq \mathbb{C}^n ( \cong \mathbb{R}^{2n} )$ be a smoothly bounded pseudoconvex domain. We consider the ${\overline{\partial}}$-complex $$\label{eq: dbarcomplex1}
L^2(\Omega )\overset{{\overline{\partial}}}\longrightarrow L^2_{(0,1)}(\Omega)
\overset{{\overline{\partial}}}\longrightarrow \dots \overset{{\overline{\partial}}}\longrightarrow
L^2_{(0,n)}(\Omega)\overset{{\overline{\partial}}}\longrightarrow 0\, ,$$ where $L^2_{(0,q)}(\Omega)$ denotes the space of $(0,q)$-forms on $\Omega$ with coefficients in $L^2(\Omega).$ The ${\overline{\partial}}$-operator on $(0,q)$-forms is given by $$\label{eq: deriv1}
{\overline{\partial}}\left ( \sum_J\,^{'} a_J \, d{\overline}z_J \right )=
\sum_{j=1}^n \sum_J\,^{'}\ \frac{\partial a_J}{\partial {\overline}z_j}d{\overline}z_j\wedge
d{\overline}z_J,$$ where $\sum ^{'} $ means that the sum is only taken over strictly increasing multi-indices $J.$
The derivatives are taken in the sense of distributions, and the domain of ${\overline{\partial}}$ consists of those $(0,q)$-forms for which the right hand side belongs to $L^2_{(0,q+1)}(\Omega).$ So ${\overline{\partial}}$ is a densely defined closed operator, and therefore has an adjoint operator from $L^2_{(0,q+1)}(\Omega)$ into $L^2_{(0,q)}(\Omega)$ denoted by ${\overline{\partial}}^* .$
We consider the ${\overline{\partial}}$-complex $$\label{eq: complex0}
L^2_{(0,q-1)}(\Omega )\underset{\underset{{\overline{\partial}}^* }
\longleftarrow}{\overset{{\overline{\partial}}}
{\longrightarrow}} L^2_{(0,q)}(\Omega )
\underset{\underset{{\overline{\partial}}^* }
\longleftarrow}{\overset{{\overline{\partial}}}
{\longrightarrow}} L^2_{(0,q+1)}(\Omega ),$$ for $1\le q \le n-1.$
We remark that a $(0,q+1)$-form $u=\sum_{J}^{'} u_J\,d{\overline}z_J$ belongs to $\mathcal{C}^\infty_{(0,q+1)}({\overline}\Omega ) \cap {\text{dom}}({\overline{\partial}}^*)$ if and only if $$\label{eq: dom2}
\sum_{k=1}^n u_{kK} \, \frac{\partial r}{\partial z_k} =0$$ on $b\Omega$ for all $K$ with $|K|=q,$ where $r$ is a defining function of $\Omega$ with $|\nabla r(z)|=1$ on the boundary $b\Omega.$ (see for instance [@Str])
The complex Laplacian $\Box = {\overline{\partial}}\, {\overline{\partial}}^* + {\overline{\partial}}^*\, {\overline{\partial}}$, defined on the domain $${\text{dom}}(\Box) = \{ u\in L^2_{(0,q)}(\Omega ) : u\in {\text{dom}}({\overline{\partial}}) \cap {\text{dom}}({\overline{\partial}}^*) , {\overline{\partial}}u\in {\text{dom}}({\overline{\partial}}^*) , {\overline{\partial}}^* u \in {\text{dom}}({\overline{\partial}}) \}$$ acts as an unbounded, densely defined, closed and self-adjoint operator on $L^2_{(0,q)}(\Omega ),$ for $ 1\le q \le n,$ which means that $\Box = \Box^*$ and ${\text{dom}}(\Box ) = {\text{dom}}(\Box^*).$
Note that $$\label{eq: diri1}
(\Box u,u)=( {\overline{\partial}}\, {\overline{\partial}}^* u+ {\overline{\partial}}^* \, {\overline{\partial}}u,u)=\| {\overline{\partial}}u \|^2 + \| {\overline{\partial}}^* u \|^2,$$ for $u\in {\text{dom}}(\Box ).$
If $\Omega $ is a smoothly bounded pseudoconvex domain in $\mathbb{C}^n,$ the so-called basic estimate says that
$$\label{eq: diri2}
\| {\overline{\partial}}u \|^2 + \| {\overline{\partial}}^* u \|^2 \ge c \, \|u\|^2,$$
for each $u\in {\text{dom}}({\overline{\partial}}) \cap {\text{dom}}({\overline{\partial}}^* ) , \ c>0.$
This estimate implies that $ \Box : {\text{dom}}(\Box) \longrightarrow L^2_{(0,q)}(\Omega )$ is bijective and has a bounded inverse $$N_{(0,q)}: L^2_{(0,q)}(\Omega ) \longrightarrow {\text{dom}}(\Box).$$ $N_{(0,q)}$ is called ${\overline{\partial}}$-Neumann operator. In addition $$\label{eq: cont5}
\|N_{(0,q)} u\| \le \frac{1}{c} \, \|u\|.$$ Hence the ${\overline{\partial}}$-Neumann operator $N_{(0,q)}$ is continuous from $L^2_{(0,q)}(\Omega )$ into itself. Compactness of the ${\overline\partial}$-Neumann operator is relevant for a number of circumstances ([@Str]). From the point of view of the $L^2$-Sobolev theory of the ${\overline\partial}$-Neumann operator, an important application of compactness is that it implies global regularity. Kohn and Nirenberg ([@KN]) proved that compactness of $N_{(0,q)}$ on $L^2_{(0,q)}(\Omega )$ implies compactness (in particular, continuity) of $N_{(0,q)}$ from the Sobolev spaces $W^s_{(0,q)}(\Omega )$ into itself for all $s\ge 0,$ see also [@Str]. For this result the Rellich - Kondrachov lemma is important, it holds as $\Omega$ is a bounded domain.
0.3 cm
The aim of this paper is to study similar properties for the weighted ${\overline\partial}$-Neumann operator on $\mathbb C^n.$
Let $\varphi : \mathbb{C}^n \longrightarrow \mathbb{R}$ be a plurisubharmonic $\mathcal{C}^2$-function and let $$L^2( \mathbb{C}^n, e^{-\varphi}) = \{ g: \mathbb{C}^n \longrightarrow \mathbb{C} \ {\text{measurable}} \,:
\|g\|^2_\varphi =(g,g)_\varphi = \int_{\mathbb{C}^n} |g|^2 e^{-\varphi}\, d\lambda < \infty \}.$$ Let $1\le q \le n$ and $$f= \sum_{|J|=q}\, ' \, f_J\,d\overline z_J ,$$ where the sum is taken only over increasing multiindices $J=(j_1, \dots , j_q)$ and $d\overline z_J = d\overline z_{j_1} \wedge \dots \wedge d\overline z_{j_q}$ and $f_J\in L^2(\mathbb C^n, e^{-\varphi}).$
We write $f\in L^2_{(0,q)}(\mathbb C^n, e^{-\varphi})$ and define $${\overline{\partial}}f = \sum_{|J|=q}\, ' \, \sum_{j=1}^n \frac{\partial f_J}{\partial \overline z_j}\, d\overline z_j \wedge d\overline z_J$$ for $1\le q \le n-1$ and $${\text{dom}}({\overline{\partial}}) = \{ f \in L^2_{(0,q)}(\mathbb C^n, e^{-\varphi}) \, : \, {\overline{\partial}}f \in L^2_{(0,q+1)}(\mathbb C^n, e^{-\varphi}) \} .$$ In this way ${\overline{\partial}}$ becomes a densely defined closed operator and its adjoint ${\overline{\partial}}^*_\varphi$ depends on the weight $\varphi.$
We consider the weighted ${\overline{\partial}}$-complex $$L^2_{(0,q-1)}(\mathbb{C}^n , e^{-\varphi} )\underset{\underset{{\overline{\partial}}_\varphi^* }
\longleftarrow}{\overset{{\overline{\partial}}}
{\longrightarrow}} L^2_{(0,q)}(\mathbb{C}^n , e^{-\varphi} )
\underset{\underset{{\overline{\partial}}_\varphi^* }
\longleftarrow}{\overset{{\overline{\partial}}}
{\longrightarrow}} L^2_{(0,q+1)}(\mathbb{C}^n , e^{-\varphi} )$$ and we set $$\Box_{\varphi}^{(0,q)}= {\overline{\partial}}\, {\overline{\partial}}_\varphi^* + {\overline{\partial}}_\varphi^* {\overline{\partial}},$$ where $${\text{dom}}(\Box_\varphi^{(0,q)}) = \{ u \in {\text{dom}}({\overline{\partial}}) \cap {\text{dom}}({\overline{\partial}}^*_\varphi): {\overline{\partial}}u \in {\text{dom}}({\overline{\partial}}^*_\varphi), {\overline{\partial}}^*_\varphi u \in {\text{dom}}({\overline{\partial}})\}.$$
It turns out that $\Box_{\varphi}^{(0,q)}$ is a densely defined, non-negative self-adjoint operator, which has a uniquely determined self-adjoint square root $(\Box_{\varphi}^{(0,q)})^{1/2}.$ The domain of $(\Box_{\varphi}^{(0,q)})^{1/2})$ coincides with ${\text{dom}}({\overline{\partial}}) \cap {\text{dom}}({\overline{\partial}}^*_\varphi),$ which is also the domain of the corresponding quadratic form $$Q_\varphi (u,v):=({\overline{\partial}}u, {\overline{\partial}}v)_\varphi + ({\overline{\partial}}^*_\varphi u, {\overline{\partial}}^*_\varphi v)_\varphi,$$ see for instance [@Dav].
Next we consider the Levi matrix $$M_\varphi = \left ( \frac{\partial^2 \varphi}{\partial z_j \partial \overline z_k} \right )_{j,k=1}^n$$
and suppose that the lowest eigenvalue $\mu_\varphi$ of $M_\varphi$ satisfies $$\label{perss}
\liminf_{|z| \to \infty} \mu_\varphi (z)>0.$$
implies that $\Box_\varphi ^{(0,1)}$ is injective and that the bottom of the essential spectrum $\sigma_e(\Box_\varphi ^{(0,1)})$ is positive (Persson’s Theorem), see [@HaHe]. Now it follows that $\Box_\varphi ^{(0,1)}$ has a bounded inverse, which we denote by $$N_\varphi ^{(0,1)}: L^2_{(0,1)}(\mathbb{C}^n , e^{-\varphi} ) \longrightarrow L^2_{(0,1)}(\mathbb{C}^n , e^{-\varphi} ).$$ Using the square root of $N_\varphi ^{(0,1)}$ we get the basic estimates $$\label{coerc}
\|u\|^2_\varphi \le C ( \|{\overline{\partial}}u \|^2_\varphi + \|{\overline{\partial}}^*_\varphi u\|^2_\varphi ),$$ for all $u\in {\text{dom}}({\overline{\partial}}) \cap {\text{dom}}({\overline{\partial}}^*_\varphi).$
Now we will study compactness of the weighted ${\overline\partial}$-Neumann operator $N_\varphi ^{(0,1)}.$ For this purpose we will use the description of compact subsets in $L^2$-spaces, as it is done in [@Has10] Chapter 11, to derive a sufficient condition for compactness in terms of the weight function. It turns out that compactness of the ${\overline\partial}$-Neumann operator $N_\varphi ^{(0,1)}$ is equivalent to compactness of the embedding of a certain complex Sobolev space into $L^2_{(0,1)}(\mathbb{C}^n , e^{-\varphi}).$
Let
$$\mathcal{W}^{Q_\varphi}= \{ u\in L^2_{(0,1)}(\mathbb{C}^n , e^{-\varphi}) \ : \ u\in {\text{dom}}({\overline{\partial}}) \cap {\text{dom}}({\overline{\partial}}^*_\varphi) \}$$ with norm $$\label{com9}
\| u\|_{Q_\varphi} = (\| {\overline{\partial}}u \|^2_\varphi + \| {\overline{\partial}}_\varphi ^* u\|^2_\varphi )^{1/2}.$$
So $\mathcal{W}^{Q_\varphi}$ is the form domain of $Q_\varphi .$
\[sec: embed\]Suppose that the weight function $\varphi$ is plurisubharmonic and that the lowest eigenvalue $\mu_{\varphi}$ of the Levi - matrix $M_{\varphi }$ satisfies $$\label{eq: com10}
\lim_{|z|\rightarrow \infty}\mu_\varphi(z) = +\infty\, .$$ Then the embedding $$\label{eq: com11}
j_\varphi : \mathcal{W}^{Q_\varphi } \hookrightarrow L^2_{(0,1)}(\mathbb{C}^n, e^{-\varphi} )$$ is compact. Consequently, the ${\overline{\partial}}$-Neumann operator $N_\varphi^{(0,1)}$ is compact.
This result can be seen as a Rellich Kondrachov lemma for Sobolev spaces defined by complex derivatives. Notice that $$N_\varphi^{(0,1)} : L^2_{(0,1)}(\mathbb{C}^n , e^{-\varphi}) \longrightarrow L^2_{(0,1)}(\mathbb{C}^n , e^{-\varphi})$$ can be written in the form $$N_\varphi^{(0,1)} = j_\varphi \circ j_\varphi ^* \ ,$$ where $$j_\varphi ^* : L^2_{(0,1)}(\mathbb{C}^n , e^{-\varphi}) \longrightarrow \mathcal{W}^{Q_\varphi}$$ is the adjoint operator to $j_\varphi,$ see [@Has10] Section 6.2, or [@Str] Section 2.8.
0.3 cm
It is now clear that $N_\varphi^{(0,1)} $ is compact if and only if $j_\varphi$ is compact.
We have to show that the unit ball in $\mathcal{W}^{Q_\varphi }$ is relatively compact in $ L^2_{(0,1)}(\mathbb{C}^n, e^{-\varphi} )$. For this purpose we use the characterization of compact subsets in $L^2$-spaces (see [@Has10] Chapter 11).
For $u\in \mathcal{W}^{Q_\varphi}$ we have $$\| {\overline{\partial}}u \|^2_\varphi + \| {\overline{\partial}}_\varphi ^* u\|^2_\varphi \ge ( M_{\varphi } u, u)_\varphi .$$
This implies $$\label{eq: compac}
\| {\overline{\partial}}u \|^2_\varphi + \| {\overline{\partial}}_\varphi ^* u\|^2_\varphi \ge \int_{\mathbb{C}^n}\mu_\varphi(z) \, |u(z)|^2\, e^{-\varphi(z)}\,d\lambda (z) \ge \int_{\mathbb{C}^n \setminus \mathbb{B}_R} \mu_\varphi(z) |u(z)|^2 e^{-\varphi(z)} d\lambda (z),$$ where $\mathbb{B}_R$ is the ball with center $0$ and radius $R>0.$
Consequently, assumption implies that for each $\epsilon >0$ there is $R>0$ such that
$$\label{eq: com12}
\int_{\mathbb{C}^n \setminus \mathbb{B}_R} |u(z)|^2 e^{-\varphi (z)}\,d\lambda (z) < \epsilon,$$
for all $u$ in the unit ball of $\mathcal{W}^{Q_\varphi}.$ Also, the map $u \mapsto u|_{\mathbb B_R}$ is compact from $\mathcal{W}^{Q_\varphi }$ to $L^2_{(0,1)}(\mathbb B_R, e^{-\varphi} ),$ in view of the ellipticity of ${\overline{\partial}}\oplus {\overline{\partial}}^*_\varphi.$ Together with , this latter fact shows that the image of a bounded set in $\mathcal{W}^{Q_\varphi }$ is pre-compact in $L^2_{(0,1)} (\mathbb C^n, e^{-\varphi}).$ 0.5 cm In the following we will describe an approach to obtain the so-called compactness estimates for the ${\overline\partial}$-Neumann operator $N_\varphi^{(0,1)} ,$ where we follow [@Str], Propostion 4.2. For this purpose we have to define appropriate weighted Sobolev spaces and we need an appropriate Rellich - Kondrachov lemma. 2 cm
Weighted $L^2$-Sobolev spaces
=============================
2 cm
Let $z=(z_1,\dots, z_n)=(x_1+iy_1, \dots , x_n+iy_n)\in \mathbb C^n$ and write for a multiindex $$\gamma = (\gamma_1, \gamma_2, \dots , \gamma_{2n-1}, \gamma_{2n})$$ and an appropriate function $$\partial^\gamma f= \frac{\partial^{|\gamma|}f}{\partial x_1^{\gamma_1} \partial y_1^{\gamma_2} \dots
\partial x_n^{\gamma_{2n-1}} \partial y_n^{\gamma_{2n} }}.$$
\[sobo1\] We denote by $W^k(\mathbb C^n)$ the Sobolev space $$W^k(\mathbb C^n) = \{ f\in L^2(\mathbb C^n ) \, : \, \partial^\gamma f \in L^2(\mathbb C^n ) , \, |\gamma |\le k \},$$ where the derivatives are taken in the sense of distributions and endow the space with the norm $$\|f\|_{k} = \left ( \sum_{|\gamma |\le k} \int_{\mathbb C^n} |\partial^\gamma f |^2 \,d\lambda \right )^{1/2}.$$
$W^k(\mathbb C^n)$ is a Hilbert space. It is well-known that the embedding $\iota: W^1(\mathbb C^n) \hookrightarrow L^2(\mathbb C^n) $ fails to be compact. In sake of completeness we recall the easy proof: let $\psi \in \mathcal C^\infty_0(\mathbb C^n)$ be a smooth function with compact support such that ${\text{Tr}} \psi \subset B_{1/2}(0)$ and $\int_{\mathbb C^n}|\psi (z)|^2\,d\lambda (z)=1.$ For $k\in \mathbb{N}$ let $\psi_k(z)= \psi (z-\overrightarrow{k}),$ where $\overrightarrow{k} = (k, 0, \dots ,0) \in \mathbb C^n.$ Then ${\text{Tr}}\psi_k \subset
B_{1}(\overrightarrow{k})$ and $(\psi_k)_k$ is a bounded sequence in $W^1(\mathbb C^n).$ Now let $k,m\in \mathbb N$ with $k\neq m.$ Due to the fact that $\psi_k$ and $\psi_m$ have non-overlapping supports we have $$\| \psi_k - \psi_m \|^2 = \| \psi_k \|^2 + \| \psi_m \|^2 = 2,$$ and the sequence $(\psi_k)_k$ has no convergent subsequence in $L^2(\mathbb C^n).$
Let $U_\varphi : L^2(\mathbb C^n ) \longrightarrow L^2(\mathbb C^n, e^{-\varphi} )$ denote the isometry given by $U_\varphi (f)= f e^{\varphi/2},$ for $f\in L^2(\mathbb C^n ).$ The inverse is given by $U_{-\varphi} (g) = ge^{-\varphi/2},$ for $ g\in L^2(\mathbb C^n, e^{-\varphi} ).$ The appropriate weighted Sobolev spaces are determined as the images of $W^k(\mathbb C^n)$ under the isometry $U_\varphi.$ In the following we consider only Sobolev spaces of order $1.$ Let $f\in W^1(\mathbb C^n).$ Then $fe^{\varphi/2} , (\partial_jf)e^{\varphi/2} \in L^2(\mathbb C^n, e^{-\varphi}), $ where $\partial_j f$ denotes all first order derivatives of $f$ with respect to $x_j$ and $y_j$ for $j=1,\dots,n.$ Set $h=fe^{\varphi/2}.$ Then $$\begin{aligned}
\partial_j h& = (\partial_jf)e^{\varphi/2}+\frac{1}{2} f (\partial_j\varphi) e^{\varphi/2} \\
& = (\partial_jf)e^{\varphi/2} +\frac{1}{2} (\partial_j\varphi) h,\end{aligned}$$ which implies $(\partial_jf)e^{\varphi/2} = \partial_j h - \frac{1}{2} (\partial_j\varphi) h$ and 0.3 cm $U_\varphi (W^1(\mathbb C^n))= \{ h\in L^2(\mathbb C^n, e^{-\varphi}): \partial_j h - \frac{1}{2} (\partial_j\varphi) h \in L^2(\mathbb C^n, e^{-\varphi}), j=1, \dots ,2n \}.$ 0.3 cm For reasons which will become clear later, we denote $$W^1_0(\mathbb C^n, e^{-\varphi}):= U_\varphi (W^1(\mathbb C^n)),$$ and we endow the space $W^1_0(\mathbb C^n, e^{-\varphi})$ with the norm $h \mapsto ( \|h\|^2_\varphi + \sum_j \| \partial_j h - \frac{1}{2} (\partial_j\varphi) h \|^2_\varphi )^{1/2}.$ 0.3 cm in this way $U_\varphi : W^1(\mathbb C^n) \longrightarrow W^1_0(\mathbb C^n, e^{-\varphi})$ is again isometric and we have the following commutative diagram $$\begin{CD}
W^1(\mathbb C^n) @> \iota >> L^2(\mathbb C^n) \\
@V{U_\varphi} VV @VV{U_\varphi}V \\
W^1_{0}(\mathbb C^n, e^{-\varphi}) @>>\iota_\varphi > L^2(\mathbb C^n, e^{-\varphi})
\end{CD}$$ where $\iota_\varphi : W^1_0(\mathbb C^n, e^{-\varphi}) \hookrightarrow L^2(\mathbb C^n, e^{-\varphi})$ is the canonical embeddings. As $U_\varphi \, \iota = \iota_\varphi \, U_\varphi $ and $\iota $ fails to be compact, $\iota_\varphi $ is also not compact.
\[sobo2\] Let $\eta \in \mathbb R.$ We denote by $W^1_{\eta}(\mathbb C^n, e^{-\varphi})$ the Sobolev space 0.3 cm $W^1_{\eta}(\mathbb C^n, e^{-\varphi}) = \{ h\in L^2(\mathbb C^n, e^{-\varphi} ) : \partial_j h - \frac{1+\eta}{2} (\partial_j\varphi) h \in L^2(\mathbb C^n, e^{-\varphi}), j=1, \dots ,2n \},$ 0.3 cm endowed with the norm $h \mapsto ( \|h\|^2_\varphi + \sum_j \| \partial_j h - \frac{1+\eta}{2} (\partial_j\varphi) h \|^2_\varphi )^{1/2}.$
We use the notation $$X_j=\frac{\partial }{\partial x_j} - \frac{1+\eta}{2}\frac{\partial \varphi}{\partial x_j} \ {\text {and}} \
Y_j=\frac{\partial }{\partial y_j} - \frac{1+\eta}{2}\frac{\partial \varphi}{\partial y_j},$$ for $j=1,\dots ,n.$ Then $$W^1_\eta(\mathbb C^n, e^{-\varphi} )=\{ f\in L^2(\mathbb{C}^n, e^{-\varphi}) \ : X_jf,\ Y_jf \in L^2(\mathbb{C}^n, e^{-\varphi}) , j=1,\dots ,n \},$$ with norm $$\|f\|^2_{ \varphi , \eta}= \|f\|^2_{ \varphi }+\sum_{j=1}^n( \|X_jf\|^2_\varphi
+ \|Y_jf\|^2_\varphi) .$$
For suitable weight functions $\varphi,$ we can prove an analogous result to the Rellich Kondrachov lemma.
\[rellich\] Suppose that $\varphi $ is a $\mathcal{C}^2$-function satisfying
$$\label{eq: nowei}
\lim_{|z|\to \infty}(\eta^2 |\nabla \varphi (z)|^2+(1+\epsilon) \eta \, \triangle \varphi (z))= +\infty ,$$
for some $\epsilon >0,$ where $$|\nabla \varphi (z)|^2= \sum_{k=1}^n \left ( \left | \frac{\partial \varphi}{\partial x_k}\right |^2+ \left | \frac{\partial \varphi}{\partial y_k}\right |^2 \right ).$$ Then the canonical embedding $\iota_{\varphi, \eta} :W^1_\eta (\mathbb C^n,e^{- \varphi})\hookrightarrow L^2(\mathbb{C}^n, e^{-\varphi}) $ is compact.
We adapt methods from [@BDH] , [@Jo] and [@KM] and use the general result that an operator between Hilbert spaces is compact if and only if the image of a weakly convergent sequence is strongly convergent.
In addition we remark that $\mathcal C^\infty_0(\mathbb C^n)$ is dense in all spaces which are involved. For the vector fields $X_j$ and their adjoints $X_j^*$ in the weighted space $L^2(\mathbb C^n,e^{-\varphi})$ we have $X_j^*=-\frac{\partial}{\partial x_j} + \frac{1-\eta}{2}\frac{\partial \varphi}{\partial x_j}$ and $$\label{eq: expl}
(X_j+X^*_j)f=-\eta \frac{\partial \varphi}{\partial x_j}\, f \ {\text{and}} \
[X_j,X^*_j]f= -\eta \frac{\partial^2\varphi}{\partial x_j^2}\, f,$$ for $f\in \mathcal{C}^\infty_0(\mathbb{C}^n),$ and $$\label{eq: comsob1}
( [X_j,X^*_j]f,f )_\varphi=\|X^*_jf\|^2_\varphi - \|X_jf\|^2_\varphi ,$$ $$\label{eq: comsob2}
\|(X_j+X^*_j)f\|^2_\varphi \le (1+1/\epsilon)\|X_jf\|^2_\varphi + (1+\epsilon)\|X^*_jf\|^2_\varphi$$ for each $\epsilon>0,$ where we used the inequality $$|a+b|^2 \le |a|^2 + |b|^2 + 1/\epsilon \, |a|^2 + \epsilon \, |b|^2.$$ Similar relations hold for the vector fields $Y_j.$ Now we set $$\Psi (z)=\eta^2|\nabla \varphi (z)|^2+(1+\epsilon)\eta \triangle \varphi (z).$$
By , and , it follows that $$( \Psi f,f )_\varphi \le
(2+\epsilon +1/\epsilon)\sum_{j=1}^n( \|X_jf\|^2_\varphi
+ \|Y_jf\|^2_\varphi) .$$ Since $\mathcal{C}^\infty_0(\mathbb{C}^n)$ is dense in $W^1_\eta(\mathbb C^n, e^{-\varphi})$ by definition, this inequality holds for all $f\in W^1_\eta(\mathbb C^n, e^{-\varphi}).$
If $(f_k)_k$ is a sequence in $W^1_\eta(\mathbb C^n, e^{-\varphi})$ converging weakly to $0,$ then $(f_k)_k$ is a bounded sequence in $W^1_\eta(\mathbb C^n, e^{-\varphi})$ and our assumption implies that $$\Psi (z)=\eta^2 |\nabla \varphi (z)|^2+(1+\epsilon) \eta \triangle \varphi (z)$$ is positive in a neighborhood of $\infty $. So we obtain $$\begin{aligned}
\int\limits_{\mathbb{C}^n}|f_k(z)|^2e^{-\varphi (z)}\,d\lambda (z) & \le &
\int\limits_{|z|< R}|f_k(z)|^2e^{-\varphi (z)}\,d\lambda (z)\\
& + &
\int\limits_{|z|\ge R} \frac{\Psi (z) |f_k(z)|^2}{\inf \{\Psi (z) \, : \, |z|\ge R\}} \, e^{-\varphi (z)}\,d\lambda (z)\\
&\le & C_{\varphi , R}\, \|f_k\|^2_{L^2(B(0,R))}+ \frac{C_\epsilon \, \|f_k\|^2_{\varphi, \eta}}{\inf \{\Psi (z) \, : \, |z|\ge R\}}.\end{aligned}$$ Notice that in the last estimate the expression $\Psi (z)$ plays a similar role as $\mu_{\varphi} (z)$ in . It is now easily seen that the sequence $(f_k)_k$ converges also weakly to zero in $W^1(B(0,R)).$ Hence the assumption and the fact that the embedding $$W^1(B(0,R)) \hookrightarrow L^2(B(0,R))$$ is compact (classical Rellich Kondrachov Lemma, see for instance [@AF]) show that $(f_k)_k$ tends to $0$ in $L^2(\mathbb{C}^n, e^{-\varphi}).$
If $\eta =0,$ we get the case corresponding to $W^1(\mathbb C^n), $ whereas $\eta =-1$ corresponds to the Sobolev space of all functions $h\in L^2(\mathbb C^n, e^{-\varphi})$ such that all derivatives of order $1$ satisfy $\partial_j h \in L^2(\mathbb C^n, e^{-\varphi});$ in this case the higher order Sobolev spaces are defined as the spaces of all functions $h\in L^2(\mathbb C^n, e^{-\varphi})$ such that all derivatives of order $k\ge 1$ belong to $L^2(\mathbb C^n, e^{-\varphi}).$
From Theorem \[rellich\] we can also derive compactness for embeddings in Sobolev spaces without weights. For this purpose we define
\[sobo7\] Let $\eta \in \mathbb R.$ We define
$ W^1_\eta (\mathbb C^n, \nabla \varphi) := \{ f\in L^2(\mathbb C^n) : \partial_j f - \frac{\eta}{2}(\partial_j \varphi)f \in L^2(\mathbb C^n), j=1, \dots, 2n\}.$
Then $U_\varphi : W^1_\eta (\mathbb C^n, \nabla \varphi) \longrightarrow W^1_\eta (\mathbb C^n,e^{- \varphi})$ is an isometry. We consider the canonical embedding $\iota_\eta : W^1_\eta (\mathbb C^n, \nabla \varphi) \hookrightarrow L^2(\mathbb C^n)$ and we have the following commutative diagram $$\begin{CD}
W^1_\eta (\mathbb C^n, \nabla \varphi) @> \iota_\eta >> L^2(\mathbb C^n) \\
@V{U_\varphi} VV @VV{U_\varphi}V \\
W^1_{\eta}(\mathbb C^n, e^{-\varphi}) @>>\iota_{\varphi, \eta} > L^2(\mathbb C^n, e^{-\varphi})
\end{CD}$$ Hence the condition implies that the canonical embedding $\iota_\eta: W^1_\eta (\mathbb C^n, \nabla \varphi) \hookrightarrow L^2(\mathbb C^n)$ is compact.
0.4 cm Now we return to compactness of the ${\overline\partial}$-Neumann operator $N_\varphi^{(0,1)}.$ We consider the weighted Sobolev space
$W^1_{1}(\mathbb C^n, e^{-\varphi}) = \{ h\in L^2(\mathbb C^n, e^{-\varphi} ) : \partial_j h - (\partial_j\varphi) h \in L^2(\mathbb C^n, e^{-\varphi}), j=1, \dots ,2n \},$
and use $$X_j=\frac{\partial }{\partial x_j} - \frac{\partial \varphi}{\partial x_j} \ {\text {and}} \
Y_j=\frac{\partial }{\partial y_j} - \frac{\partial \varphi}{\partial y_j},$$ for $j=1,\dots ,n.$ Then $$W^1_1(\mathbb C^n, e^{-\varphi} )=\{ f\in L^2(\mathbb{C}^n, e^{-\varphi}) \ : X_jf,\ Y_jf \in L^2(\mathbb{C}^n, e^{-\varphi}) , j=1,\dots ,n \},$$ with norm $$\|f\|^2_{ \varphi , 1}= \|f\|^2_{ \varphi }+\sum_{j=1}^n( \|X_jf\|^2_\varphi
+ \|Y_jf\|^2_\varphi) .$$
We point out that each continuous linear functional $L$ on $W^1_1(\mathbb C^n, e^{-\varphi} )$ is represented by $$L(f) = \int_{\mathbb{C}^n} fg_0e^{-\varphi}\,d\lambda +
\sum_{j=1}^n \int_{\mathbb{C}^n}((X_jf)g_j+(Y_jf)h_j)e^{-\varphi }\,d\lambda,$$ for $f\in W^1_1(\mathbb C^n, e^{-\varphi} )$ and for some $g_0,g_j,h_j\in L^2(\mathbb{C}^n, e^{-\varphi}), \, j=1,\dots,n.$ In particular, each function in $L^2(\mathbb C^n, e^{-\varphi})$ can be identified with an element of the dual space $(W^1_1(\mathbb C^n, e^{-\varphi} ))' =: W^{-1}_1(\mathbb C^n, e^{-\varphi} ).$ We denote the norm in $ W^{-1}_1(\mathbb C^n, e^{-\varphi} )$ by $\| \, . \, \|_{\varphi, -1}.$ See [@Has10] Chapter 11, for more details.
If we suppose that $\varphi $ is a $\mathcal{C}^2$-function satisfying
$$\label{eq: nowei1}
\lim_{|z|\to \infty}( |\nabla \varphi (z)|^2+(1+\epsilon) \, \triangle \varphi (z))= +\infty ,$$
for some $\epsilon >0,$ then the embedding $$L^2_{(0,1)}(\mathbb C^n, e^{-\varphi}) \hookrightarrow W^{-1}_{1,(0,1)}(\mathbb C^n, e^{-\varphi} )$$ is compact by Theorem \[rellich\] and duality. So, as in [@Str], Proposition 4.2 or [@Has10], Proposition 11.20, we get the compactness estimates
\[sec: compact\] \
Suppose that the weight function $\varphi$ satisfies and $$\lim_{|z|\to \infty}( |\nabla \varphi (z)|^2+(1+\epsilon) \, \triangle \varphi (z))= +\infty ,$$ for some $\epsilon >0,$ then the following statements are equivalent.
1. The ${\overline{\partial}}$-Neumann operator $N_{\varphi}^{(0,1)}$ is a compact operator from $L_{(0,1)}^2(\mathbb{C}^n, e^{-\varphi})$ into itself.
2. The embedding of the space dom$({\overline{\partial}})\,\cap$ dom$({\overline{\partial}}_\varphi^*),$ provided with the graph norm $u\mapsto (\|u\|^2_\varphi + \|{\overline{\partial}}u\|^2_\varphi +
\|{\overline{\partial}}_\varphi ^*u\|^2_\varphi)^{1/2},$ into $L^2_{(0,1)}(\mathbb{C}^n, e^{-\varphi})$ is compact.
3. For every positive $\epsilon' $ there exists a constant $C_{\epsilon'}$ such that $$\|u\|_\varphi ^2 \le \epsilon' (\| {\overline{\partial}}u \|_\varphi ^2 + \| {\overline{\partial}}_\varphi ^*u\|_\varphi ^2) + C_{\epsilon'} \|u\|_{\varphi , -1} ^2,$$ for all $u\in$ dom$({\overline{\partial}})\,\cap\,$dom$({\overline{\partial}}_\varphi^*).$
4. For every positive $\epsilon'$ there exists $R>0$ such that $$\int_{\mathbb{C}^n \setminus \mathbb{B}_R} |u(z)|^2\, e^{-\varphi(z)}\,d\lambda(z) \le \epsilon' ( \|{\overline{\partial}}u\|_\varphi ^2 + \| {\overline{\partial}}_\varphi^* u\|_\varphi ^2)$$ for all $u\in dom\,({\overline{\partial}}) \,\cap dom\,({\overline{\partial}}_\varphi^*). $
5. The operators $${\overline{\partial}}_\varphi ^* N_{\varphi}^{(0,1)} : L_{(0,1)}^2(\mathbb{C}^n, e^{-\varphi})\cap {\text {ker}}({\overline{\partial}}) \longrightarrow L^2(\mathbb{C}^n, e^{-\varphi}) \ \ {\text {and}}$$ $${\overline{\partial}}_\varphi ^* N_{\varphi}^{(0,2)} : L_{(0,2)}^2(\mathbb{C}^n, e^{-\varphi})\cap {\text {ker}}({\overline{\partial}}) \longrightarrow L_{(0,1)}^2(\mathbb{C}^n, e^{-\varphi})$$ are both compact.
If $$\lim_{|z|\rightarrow \infty}\mu_\varphi(z) = +\infty,$$ then the condition of the Rellich-Kondrachov lemma is satisfied.
This follows from the fact that we have for the trace ${\text {tr}}(M_\varphi ) $ of the Levi - matrix $${\text {tr}}(M_\varphi )=\frac{1}{4}\triangle \varphi,$$ and since for any invertible $(n\times n)$-matrix $T$ $${\text {tr}}(M_\varphi )={\text {tr}}(TM_\varphi T^{-1}),$$ it follows that ${\text {tr}}(M_\varphi )$ equals the sum of all eigenvalues of $M_\varphi .$
We mention that for the weight $\varphi (z)=|z|^2$ the ${\overline\partial}$-Neumann operator fails to be compact (see [@Has10] Chapter 15), but condition is satisfied.
In view of Theorem \[rellich\] it is clear that for any weight satisfying and for $\eta \in \mathbb R, \eta \neq 0,$ and for some $\epsilon >0,$ the restriction of the ${\overline\partial}$-Neumann operator $N_{\varphi}^{(0,1)} $ to $W^1_{\eta, (0,1)} (\mathbb C^n,e^{- \varphi})$ is compact as an operator from $W^1_{\eta, (0,1)} (\mathbb C^n,e^{- \varphi})$ to $L^2_{(0,1)}(\mathbb C^n, e^{-\varphi}).$
1 cm ACKNOWLEDGMENT: The author wishes to thank the referee for the valuable comments.
1 cm
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Evolutionary algorithm research and applications began over 50 years ago. Like other artificial intelligence techniques, evolutionary algorithms will likely see increased use and development due to the increased availability of computation, more robust and available open source software libraries, and the increasing demand for artificial intelligence techniques. As these techniques become more adopted and capable, it is the right time to take a perspective of their ability to integrate into society and the human processes they intend to augment. In this review, we explore a new taxonomy of evolutionary algorithms and resulting classifications that look at five main areas: the ability to manage the control of the environment with limiters, the ability to explain and repeat the search process, the ability to understand input and output causality within a solution, the ability to manage algorithm bias due to data or user design, and lastly, the ability to add corrective measures. These areas are motivated by today’s pressures on industry to conform to both societies concerns and new government regulatory rules. As many reviews of evolutionary algorithms exist, after motivating this new taxonomy, we briefly classify a broad range of algorithms and identify areas of future research.'
author:
- 'Andrew N. Sloss'
- Steven Gustafson
title: '**2019 Evolutionary Algorithms Review**'
---
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Erik Woldhuis
- 'Brian P. Tighe'
- Wim van Saarloos
title: 'Wide shear zones and the spot model: Implications from the split-bottom geometry'
---
Introduction
============
It is well known that a relatively general theory for flow of granular media is still beyond reach. In part, this is due to the richness of granular flow phenomena [@nedderman; @gdr; @beverloo; @jenike] — ranging from avalanche type behavior down inclined planes or granular surfaces to hopper discharges, granular gases, sheared flows in Couette cells or glassy type rheology. Other impediments to the development of a general framework to describe granular flow are the competition between static regions and flowing regions, and the fact that flow zones are typically not much wider than a few particle diameters, so that continuum theories are questionable. Even though a general theory is still lacking, some more limited approaches aiming at describing the phenomena in a particular limit or in some special case have been relatively successful [@aranson01; @boquet02; @pouliquen06]. One particular recent proposal that appears to be quite successful for describing hopper flow and similar quasi-static flow problems is the so-called spot model introduced by Bazant and co-workers [@bazant06a; @bazant06b; @bazant07a; @bazant07b]. The central idea of this approach, illustrated in Fig. \[figspot\], is that the slow flow of dense random packings can be understood in terms of the drift and diffusion of relatively lower density regions of about 3-5 grains in diameter, called spots. While this idea is not unlike earlier approaches incorporating cooperative rearrangement effects, including the Soft Glassy Rheology phenomenology [@sollich97; @sollich98] and the Shear Transformation Zone Theory [@falk98; @lemaitre02; @lemaitre04; @bouchbinder07], an attractive feature of the spot model is that it is more specific, permitting calculation of flow properties within a number of different experimental geometries and hence comparison to experiments or simulations. A second appealing aspect of the spot model is that it aims to merge the behavior at the scale of a few grains to the more “engineering” continuum-type approaches relating slow plastic flows to the stress fields (e.g. Mohr-Coulomb theory).
![Caricature [@bazant06a; @bazant06b] of a spot, a region where the density of grains is less than in the surrounding. When the spot moves upwards, the individual grains move downwards. []{data-label="figspot"}](spot.eps)
In this paper we will explore to what extent the spot model can be applied to the wide shear zones found in the so-called split-bottom geometries, i.e., cylindrical Couette cells in which the bottom consists of two rings, the inner of which rotates with the inner cylinder, the outer with the outer cylinder [@vanhecke03; @vanhecke04; @vanhecke06; @chicago]. The shear zones found in such cells are exceptionally wide, much wider than the shear zones of order 5-10 particle diameters found in many other flow geometries [@gdr]. In addition, the shear zone bends inwards towards the inner cylinder as a function of height in a way which appears to be determined largely by the balance of torques [@Unger].
A feature of the experimental data that made us initially optimistic that the spot model in its simplest form could be applied to these wide shear zones is an empirical finding, namely that the shear rate as a function of radius follows an error function profile with an effective width that depends on height [@vanhecke03]. As the spot model leads to a diffusion-type equation for the density of spots, fundamental solutions of the diffusion equation like Gaussians or error functions can be expected to emerge quite naturally for the spot density and the associated flow field. This intuitive observation motivated us to investigate the applicability of the spot model to these wide shear zone flows.
Contrary to our expectations, our main finding is that, without significant modifications or extensions, it is hard to reconcile the spot model with the main experimental observations on the wide shear zones. To guide our discussion with a concrete example, we will first briefly discuss in section 3 the results of a simple-minded layer approximation, which permits straightforward extension of the main features of the spot model in two dimensions to the three dimensional split-bottom geometry. This crude approximation always predicts quite narrow shear zones which essentially go straight up, rather than bend inwards as is found in experiments and according to the torque balance argument [@Unger]. As we shall see, these shortcomings are, however, not due to the inadequacy of the layer approximation underlying the implementation, but instead are intimately related to the basic structure of the drift-diffusion equation for spots. Indeed, in sections \[sectionflowrule\] and \[sfssection\] we explore in generality which features of the spot model may be responsible for the incompatibility of the model with the wide shear zones.
There are essentially two different conceptual ingredients of the spot model which can be investigated separately: the more fundamental postulate that the flow can be analyzed in terms of a diffusion-type equation for the spot density, and the additional postulate that the drift vector in this diffusion equation can be calculated approximately from a Mohr-Coulomb type theory that predict stresses in materials at incipient yield. As we shall discuss, without additional modifications to the model, both features appear to be problematic for these wide shear flows. On the one hand, the drift-diffusion equation can be shown to be incompatible with the main experimental features of the wide shear zones. At the same time the co-axiality of the principal axes of the quasistatic granular stress and strain rate tensors, which according to theoretical arguments [@depken06] and recent simulations [@depken07] holds quite well in these wide quasi-static shear zones, is violated by Mohr-Coulomb stresses in the two dimensional Couette setup where the principal directions of the strain rate tensor are fixed by symmetry considerations. In the outlook we will briefly mention some of the possible modifications and extensions that may be required to describe wide shear zones within a picture of drift and diffusion of spots.
Essentials of the spot model
============================
As stated above, the essential idea of the spot model is that flow in granular matter is mediated by the opposing movement of spots, regions of excess free volume, on the order of 3-5 grain diameters in size. The excess free volume associated with a single spot is less than the volume of one grain. When a spot moves in one direction, a net particle flow is caused in the opposite direction. The movement of the spots is postulated to be the result of a combination of drift and diffusion; this is called the “stochastic flow rule". Together with the incompressibility of the flow this leads to the following central equation $$\label{eqnflowfield}
\vec{u}= \frac{L}{\Delta t} \left( - {\hat{\vec d}}\rho+ \frac{L}{2} \vec{\nabla}\rho \right) ,$$ relating the granular velocity field $\vec{u}$ to the dimensionless spot density $\rho$. Here $L$ is the spot size, $\Delta t$ the time it takes a spot to move a distance $L$, and ${\hat{\vec d}}$ the normalized drift direction vector. The “bare velocity” $\vec{u}$ is smeared out by spatial convolution with a “spot influence function” to give the final flow field. Nevertheless, the essential features of the flow are captured by $\vec{u}$ and the drift vector $\hat{d}$, on which we therefore focus. Note that the time scale $\Delta t$ sets the velocity scale, so that only one parameter, the spot size $L$, governs the balance between the drift and diffusion terms. As we shall see, this simple feature is intimately tied to the fact that the spot model tends to give rise to narrow shear bands of width $L \approx$ 3-5 grain diameters.
The above expression can be thought of as the constitutive equation of the spot model: it postulates that granular flow results from the combined effect of a systematic drive (the drift term) and a random diffusion. The equation also illustrates that there are two sides to the spot model. On the one hand, Eq. (\[eqnflowfield\]) captures the essential idea that quasistatic granular flow can be captured completely in terms of the drift and diffusion of spots of reduced local density. This idea can already be compared with known flow profiles if we know the drift direction vector ${\hat{\vec d}}$ from simple symmetry arguments. E.g., in a two-dimensional hopper discharge it is argued [@bazant06a; @bazant06b] that the drift direction vector ${\hat{\vec d}}$ points straight up, as a result of gravity. In this case, the equation for the spot density reduces to a simple diffusion equation, with the vertical direction playing a role analogous to time. We will use similar symmetry arguments below in section \[sectionflowrule\] to investigate the compatibility of the spot model with wide shear zones.
The second side of the model is evident in nontrivial geometries, in which it is necessary to calculate the drift direction vector ${\hat{\vec d}}$ explicitly. Calculation of ${\hat{\vec d}}$ requires a theory of stresses in the material; it is here that the spot model in its simple form makes contact with traditional continuum methods from engineering. In order to deal with general 2d or quasi-2d cases, Kamrin and Bazant [@bazant07a; @bazant07b] have proposed that ${\hat{\vec d}}$ can be determined from the Mohr-Coulomb plasticity theory (MCP theory). MCP theory posits that, immediately prior to plastic failure, granular materials are at incipient yield everywhere, i.e. the material is just about to fail, collapse or exhibit plastic flow along at least two lines through each point of the material, called slip lines. The direction of these slip lines is determined by the stress fields and the value of the Coulomb friction coefficient $\mu$. Kamrin and Bazant assume that the stress tensor obtained from MCP remains valid in dense, quasistatic flows. They assert that spots perform a biased random walk along the slip lines, the bias being provided by the net force on a locally fluidized material region. This then determines the effective drift vector ${\hat{\vec d}}$ from the stresses in the granular medium. Note that this formalism is essentially two-dimensional, a property that it inherits from the MCP theory. As an example of the type of result that can be achieved with the current, two-dimensional, spot-model we present both ${\hat{\vec d}}$ and the azimuthal velocity in a two-dimensional Taylor-Couette disk in Fig. \[figcouetteresults\]; both results can be found in Ref. [@bazant07a].
![(left) Vector field for the drift direction vector ${\hat{\vec d}}$ in one quarter of a cylindrical Couette cell cross section (inner and outer radii $r_{in}$ and $r_{out}$, respectively) driven with azimuthal velocity $v_{\theta, drive}$ at the inner boundary. For a Couette geometry in the absence of gravity, the drift direction vector is identical in every cross section. Note that while the drift has a large inwards radial component, this will be compensated by an equal and opposite diffusion, resulting in only azimuthal flow in the steady state. (right) The azimuthal flow velocity as a function of radius on a logarithmic scale, illustrating the almost purely exponential fall-off of the velocity, consistent with for example Ref. [@gdr].[]{data-label="figcouetteresults"}](Couetteresults4.eps)
Example: crude layer approximation for the split-bottom geometry
================================================================
In order to give an idea of the difficulties of reproducing wide shear zones within the spot model, we briefly sketch the main result of a crude and minimal extension to three dimensions that we have developed to study the steady state shear profiles in the split-bottom Couette geometry of Fig. \[figsetup2\] [@vanhecke03; @vanhecke04; @vanhecke06; @chicago]. The shear bands observed in this system can exceed 50 grain diameters, much wider than what one typically finds. The location of the center of the shear band in this system is relatively accurately determined by a simple torque minimization argument applied to an infinitely thin shear zone [@Unger]. One would like the three-dimensional spot model to reproduce this and at the same time smear out these discontinuities to the relatively wide shear zones in the velocity field that are found experimentally to be well fitted by an error function profile.
There are a number of ways to expand the two-dimensional spot model to a three-dimensional one. One option is to replace the essentially 2d MCP theory with a 3d plasticity theory. Although possible, 3d plasticity theories come at the expense of a cumbersome mathematical framework. Below we will identify difficulties with the spot model inherent to both the stochastic flow rule and MCP. Though some plasticity theories may avoid the difficulties we identify with MCP, none can correct the issues associated with applying the stochastic flow rule to wide shear zones. Thus, for simplicity, we illustrate a generalization to 3d that builds on the 2d spot model incorporating MCP. We stress once more that we choose this simple and crude implementation here only for illustrative purposes — the model can certainly be made more realistic but the arguments of sections 4 and 5 show that this will not change the basic tendency of the model to give straight narrow shear zones, as long as one sticks to Eq. (\[eqnflowfield\]).
![(a) A schematic drawing of the split-bottom Couette geometry or Leiden geometry. The central part of the bottom rotates; the walls and the outer part of the bottom are stationary. (b) A cross-section of the flow. The center and width of the shear zone are indicated. Note that it is relatively wide and moves inwards and widens with increasing height. Figure taken from Ref. [@vanhecke03] []{data-label="figsetup2"}](splitbottom.eps)
For the Couette geometry with symmetry along the axial direction, Kamrin and Bazant [@bazant07a] have already applied MCP. The corresponding ${\hat{\vec d}}$-vectors are indicated in Fig. \[figcouetteresults\]. We build on these results by using a weak gravity approximation in which we consider the granular material to be composed of thin stacked slices. In each of these slices we assume that the two-dimensional picture holds, i.e. we assume plane-strain boundary conditions. This means we can solve the two-dimensional spot model, based on MCP theory, in each slice to find the two-dimensional drift vector that would be there if this slice was the entire system [@footnote]. To this two-dimensional drift vector we then add a small vertical component due to gravity, which is constant both within each slice and between slices. The drift vector is thus $$\hat{\vec{d} }={\hat{\vec d}}_\parallel (r,\theta,z) + {\hat{\vec d}}_\perp \hat{z}.$$ Each slice is divided into two rings: an inner one that tends to rotate with the inner cylinder, and an outer one that tends to remain stationary in the lab frame. The boundary between the two rings is pinned at the split bottom but free to move with increasing height. Its position is determined self-consistently as we iterate upwards slice-by-slice.
Using this approach, one is able to calculate the azimuthal velocity in a split-bottom Taylor-Couette setup as a function of both height and radius. As can be seen from Fig. \[figresults\], these results do not match the experimental observations of Fig. \[figsetup2\]: the shear band does not move inwards for larger heights and also shows no widening — its width remains of order $L$, i.e. of the order of a few particle diameters. In the weak gravity limit, the fact that the shear bands remain of order $L$ can actually easily be understood from the result of Fig. \[figcouetteresults\]a: we already argued that as long as the ${\hat{\vec d}}$-vector points mostly in the radial direction, the spot model constitutive equation (\[eqnflowfield\]) predicts that the shear band width will be of order $L$ in the radial direction. We have explored various parameter regimes and other approximations, including a large gravity limit, but they always lead to essentially the same result: we invariably find relatively narrow shear bands that shoot straight up. Since it was shown by Unger [*et al.*]{} [@Unger] that the position of the shear band as a function of height can be obtained from a torque minimization argument, we consider it likely that it [*is*]{} possible to get the position of the shear band from a more accurate continuum stress calculation than we have done here. However, we focus here on the width of the shear band and now proceed to show that this is a generic feature of using spot model expression (\[eqnflowfield\]) also in three dimensions, independent of which theory of stresses, MCP or otherwise, is used.
![Results obtained in a three dimensional weak gravity approximation for the flow in a split-bottom Couette geometry. Plotted are the azimuthal velocities as a function of the rescaled radius for three different heights. The shear band remains narrow and centered above the split in the bottom of the cell. []{data-label="figresults"}](results.eps)
Stress independent analysis {#sectionflowrule}
===========================
For simplicity, let us consider a linear split-bottom shear cell. As sketched in Fig. \[figSFS\], this is an infinitely long rectangular container filled with grains. The bottom of the container is split lengthwise, so that the two halves can be moved relative to each other. Simulations [@depken07] have confirmed that in this rectangular cell one obtains wide shear zones just as in the cylindrical Couette cell, but the geometry is somewhat easier to analyze due to the higher symmetry. If we denote the long dimension of the setup by $y$, the vertical dimension by $z$ and the cross-channel horizontal dimension by $x$, symmetry dictates the flow must be in the $y$-direction and that physical quantities like the velocity or spot density cannot have any $y$-dependence.
In the central vertical $y$-$z$-plane along which the two halves of the setup are sheared, both ${\hat{\vec d}}_x$ and ${\hat{\vec d}}_y$ must be zero due to symmetry, and since ${\hat{\vec d}}$ is normalized ${\hat{\vec d}}_z=1$ there. Neither in the experiments nor in the simulations is there any sign that there is a nonzero velocity $u_z$ at this center line; if we therefore impose $u_z=0$ at this center plane, we obtain from Eq. (\[eqnflowfield\]): $$-\rho+\frac{L}{2} \frac{\partial \rho}{\partial z} = 0,$$ which trivially leads to the exponential height dependence $$\rho = A e^{\frac{2}{L} z}, \label{exponential}$$ with $A$ an arbitrary constant. This equation illustrates clearly the tendency of the naive extension of the spot model to 3d to lead to exponential profiles of width of order $L$. For the split-bottom geometry, such an exponential dependence can obviously not be correct: in the experiments [@vanhecke03; @vanhecke04; @vanhecke06; @chicago] as well as the simulations [@depken07], wide shear zones at filling heights of $h_{max}=50$ grain diameters are studied. Even with $L=5$ Eq. (\[exponential\]) leads to an overall height variation of the spot density by an unrealistically large factor of order $e^{20}\approx 5\cdot 10^8$ — to put this in perspective, note that one spot is thought to contain a “free” volume of about a fifth of that of a grain [@bazant06b]. Moreover, since the shear velocity in the $y$ direction away from the center line is proportional to $\rho$, as there are no gradients in the $y$ direction, such an exponential dependence would imply an exponential variation of the $u_y$ velocity with height $z$, unless ${\hat{\vec d}}_y$ is small and has a counterbalancing exponential height variation. Similar arguments apply if we analyze the extension of the profiles in the lateral (cross-channel) direction: for the profiles to be wide in the cross-channel direction, we can assume ${\hat{\vec d}}_x \simeq - \alpha x$ with $\alpha \ll 1$ (this [*Ansatz*]{} would lead to Gaussian variation of $\rho $ in the lateral direction) but the height dependence would remain exponential and close to that of (\[exponential\]). Hence we conclude that independent of the precise “flow rule" that relates ${\hat{\vec d}}$ to the local stresses in the granular medium, the extension of the spot model, based on treating all components of ${\hat{\vec d}}$ on an equal footing (as in Eq. \[eqnflowfield\]), is incompatible with the existence of wide shear zones. The reason for this is that the balance of the drift and diffusion term in Eq. (\[eqnflowfield\]) is governed by the single spot diameter length scale $L$ — given that the spot size is hypothesized to be of order $3-5$ grain diameters this almost inescapably leads to narrow shear bands of only up to ten grain diameters wide.
![A number of shear free sheets in a linear shear cell, the shear free sheet basis is indicated with $e_{1,2,3}$. Figure taken from Ref. [@depken07].[]{data-label="figSFS"}](SFS.eps)
Co-axiality and Shear Free Sheets {#sfssection}
=================================
The above discussion is independent of which particular flow rule is adopted for the connection between the stresses and the drift vector ${\hat{\vec d}}$. It is nevertheless of interest to go back and discuss the relation between the principal stress and shear directions on the basis of what we know from recent theory and simulations [@depken06; @depken07; @bazantprivate], and to explore the implications for theories on granular flow.
In the original formulation of the spot model, the so-called co-axiality flow rule is explicitly rejected and replaced by a flow rule that builds on MCP theoy. Co-axiality means that the principal axes of the stress tensor and the strain rate tensor are co-axial, i.e., aligned. However, according to the the analysis of wide shear flows of Depken [*et al.*]{} [@depken06], there are various reasons to believe that co-axiality is actually a crucial feature of these wide shear zones; later simulations confirmed this picture surprisingly well [@depken07].
In order to explore what this implies, we need to explain briefly the Shear Free Sheet (SFS) basis introduced by Depken and co-workers. A SFS is a surface of constant velocity; for the case of the rectangular split-bottom geometry the SFS’s are sketched in Fig. \[figSFS\]. These sheets naturally define the orthogonal basis spanned by the unit vectors $\hat{e}_1$ and $\hat{e}_2$ in the SFS (with $\hat{e}_2$ is taken in the direction of flow) and $\hat{e}_3$ perpendicular to the sheet. Since by construction the velocity is constant within each sheet, the strain rate within each SFS is also zero. An easy calculation then shows us that two of the principal directions of strain are at angles of $\pi/4$ relative to $\hat{e}_2$ and $\hat{e_3}$, while the third principal direction is the same as $\hat{e}_1$. In line with the theoretical arguments [@depken06], recent simulations [@depken07; @bazantprivate] confirm that co-axiality holds in these wide shear zones, and hence that the orientation of the axes of principal stress are fixed by the SFS’s.
![(solid curve) The value of $\psi$, the angle between the radial axis and the axis of minor principal stress, as a function of radius in a planar Couette setup. According to the results of Ref. [@depken06] this should always be close to $\frac{1}{4}\pi$ (dashed line) or $\frac{3}{4} \pi$ in a wide shear zone.[]{data-label="figpsi"}](psi.eps)
To illustrate how different the Mohr-Coulomb picture is from co-axiality in wide shear zones, let us return briefly to the approximation in which we think of the cylindrical Couette cell as being built up from slices of the two-dimensional (disk-like) Couette setup. Since the flow is azimuthal, the SFS’s in each slice are just concentric circles. Since the principal directions of strain rate are at $\pi/4$ angles to these sheets, and since the shear zones bend only slightly inwards towards the inner radius with increasing height, they make an angle close to $\pm \pi/4$ with the azimuthal direction. Due to co-axiality we know that this must also be true for the principal stress directions. This means that the angle between the radial axis and the minor principal stress axis is in reality always close to $\frac{1}{4}\pi$ or $\frac{3}{4}\pi$ depending on which of the two is the major and which is the minor principal stress direction. In the two-dimensional Mohr-Coulomb theory, however, the principal stress axes point in very different directions throughout most of the layer, as Fig. \[figpsi\] illustrates. In other words, any plasticity theory that presumes a violation of co-axiality, does not appear to be a viable starting point for the description of wide shear zones. Any theory based on drift and diffusion of spots will have to build in, or self-consistently lead to, co-axiality in the wide shear zones.
Outlook
=======
In putting our results into perspective we would like to stress that the spot model was not formulated specifically to apply to wide shear zones. In spite of this, the error function shear profiles found experimentally and the success of the model in capturing Gaussian hopper flow profiles led us to become optimistic that the model could capture the wide shear zones as well. Nevertheless, our analysis shows that present formulations of the spot model cannot capture the physics of the wide shear zones observed in split-bottom experiments and simulations. The main reason is that the model in its current form is based on the interplay and balance of diffusion and drift. As Eq. (\[eqnflowfield\]) clearly shows, this balance is governed by the single length scale $L$ of order 3-5 grain diameters. As a result, in its present form the structure of the model is such that it leads to localization of the shear bands on this same scale $L$.
More generally, our analysis brings up the question whether static stress considerations can be used to determine the properties of wide shear zones: In Mohr-Coulomb theory and in the present spot model, one attempts to calculate the flow fields from static stress fields in conjunction with an incipient yield postulate, but the co-axiality of the stress and strain rate tensor suggests instead a picture in which the flow “self-organizes” to a co-axial state.
To be more specific: we have based our discussion on the central result, Eq. (\[eqnflowfield\]), for the connection between the drift vector ${\hat{\vec d}}$ and the velocity field. In their discussion of the applicability of the spot model to other flow geometries, Kamrin and Bazant [@bazant07a] have independently advanced the idea that spot drift must align with special surfaces (related to “slip line admissability”) [@kensremarks]. If indeed we postulate that the spot drift is orthogonal to the $z$-direction in the central $yz$ plane of Fig. \[figSFS\], then clearly the main problem — an unrealistic exponential variation of the spot density $\rho$ with height — dissolves. If ${\hat{\vec d}}$ is not along the $z$-direction in the central $yz$ plane, then by symmetry it must be zero. Kamrin and Bazant in fact already interpreted a region with a null drift vector to be an indication that the spot mechanism is weak (a similar situation occurs in plane shear and inclined plane flow [@bazant07a]). So from that perspective too, one unfortunately has to conclude that wide shear zones include physics that go beyond the spot approach. Other possible routes to extending the spot idea to accommodate wide shear zones might be to introduce spot generation and annihilation terms [@martin] (i.e. regions of high plastic strain may act like a source of spots); to allow for evolution in spot size, thereby implicitly introducing another length scale; or to use an approach as suggested in Ref. [@bazant07a] that reconciles Bagnold type rheology with the physics of the stochastic flow rule.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Martin Bazant and Ken Kamrin for openly sharing their thoughts on the strengths and weaknesses of the spot model with us and for extensive comments on an earlier version of this manuscript. We also thank Martin Depken and Martin van Hecke for discussions, encouragement and critique.
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In the approximation, the stress boundary conditions are not really properly satisfied, most obviously at the bottom slice. Our approximation is motivated by the results of Unger et al. [@Unger], who find that loading at the boundaries of the rigid regions, rather than compressions or tractions appearing as body forces within a slice, is the dominant effect in determining the position of the shear zone. As explained in the beginning of this section, we do not attempt to correct these shortcomings here as a proper treatment will not alter our conclusions concerning the qualitative features of the shear zones implied by Eq. (\[eqnflowfield\]).
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Recent simulations by Bazant and co-workers have also confirmed co-axiality to hold in many cases (M. Bazant, private communication).
This possibility was brought to our attention in particular by K. Kamrin (private communication).
M. Bazant (private communication).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
*Objective*: The recent emergence of a new coronavirus, COVID-19, has gained extensive coverage in public media and global news. As of 24 March 2020, the virus has caused viral pneumonia in tens of thousands of people in Wuhan, China, and ten of thousands of cases in 184 other countries and territories. This study explores the potential use of Google Trends (GT) to monitor worldwide interest in this COVID-19 epidemic. GT was chosen as a source of reverse engineering data, given the interest in the topic.
*Methods*: Current data on COVID-19 is retrieved from (GT) using one main search topic: Coronavirus. Geographical settings for GT are worldwide, China, South Korea, Italy and Iran. The reported period is 15 January 2020 to 24 March 2020.
*Results*: The results show that the highest worldwide peak in the first wave of demand for information was on 31 January 2020. After the first peak, the number of new cases reported daily rose for 6 days. A second wave started on 21 February 2020 after the outbreaks were reported in Italy, with the highest peak on 16 March 2020. The second wave is six times as big as the first wave. The number of new cases reported daily is rising day by day.
*Conclusion*: This short communication gives a brief introduction to how the demand for information on coronavirus epidemic is reported through GT.
author:
- |
Artur Strzelecki\
Department of Informatics\
University of Economics in Katowice\
40-287 Katowice, Poland\
`[email protected]`\
title: 'The Second Worldwide Wave of Interest in Coronavirus since the COVID-19 Outbreaks in South Korea, Italy and Iran: A Google Trends Study'
---
Introduction
============
As of 7 pm Central European Time on 24 March, 2020, 407,485 cases of pneumonia had been reported globally [@Chen2020; @Cheng2020; @Hui2020] that were caused by the novel coronavirus that is now known as COVID-19 [@WorldHealthOrganization2020c]. 18,227 cases have resulted in death [@Parry2020a]. There have been 325,894 reported cases in other 184 countries and territories, including Italy (n=69,176), Iran (n=24,811) and South Korea (n=9,037) [@Nishiura2020; @Dong2020]. It took over three months to reach the first 100 00 confirmed cases, 12 days to reach the next 100 000, 4 days to reach the next 100 000 and only 3 days to reach the next 100 000 [@WorldHealthOrganization2020a].
On 11 February 2020, the official names were announced for the virus responsible for COVID-19 (previously known as “2019 novel coronavirus” or “2019-nCoV”) and the disease it causes. The official name of the disease is coronavirus disease (COVID-19); the virus itself is called severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) [@WorldHealthOrganization2020c]. Thus, the aim of the study is to show how GT data can be used to forecast the trends of reporting new cases.
This article can be a starting point for further analysis of information demand across search engine. Currently, Google offers its Trends service, which acts as reverse data engineering and allows data on users’ searches to be collected, which in this case is interest in the COVID-19 epidemic.
Materials and methods
=====================
The methodology for this communication follows the principles presented by [@Mavragani2019] that describe how to select the appropriate keyword(s), region(s), period, and category. Data is collected from GT and is normalized. High interest in a search query is expressed by 100, whereas a lack of interest or insufficient data is expressed by 0. GT contains data from different geographical locations that is segmented into countries, territories and cities; it also allows a custom time range to be set.
Data for the study was retrieved for the period starting 15 January 2020, as this was when relevant data started appearing on GT. The data comes from a textual search with five geographical settings: 1) worldwide, to see the global interest in coronaviruses; 2) China, where there is currently the highest number of cases; 3) South Korea, where interest has increased since 19 February because hundreds of new cases were reported; 4) Italy; and 5) Iran, where since 22 February hundreds of new cases have been reported. Data from GT related to interest in coronavirus was compared with confirmed reports of new cases provided by WHO [@WorldHealthOrganization2020a]. The collected data relates to the search topic: Coronavirus. This topic allows the popularity of all related keywords across all available languages and regions to be compared [@Kaminski2019].
Results
=======
On Google Trends, the first wave of interest in coronavirus peaked on 31 January 2020. This is measured globally for all GT data for the coronavirus search topic. Since 1 February, global interest has decreased even though the number of new cases reported daily is increasing. In the first wave, the highest number of confirmed new cases was on 5 February. In the second wave, so far the highest number of confirmed new cases in a single day was on 24 March.
Figure \[fig:fig1\] presents the global, Chinese, South Korean, Italian and Iranian results compared to the number of new COVID-19 cases. Since GT has a two-day data delay, GT data at the time of writing ends on 21 March. The left axis shows normalized GT search volume. The right axis shows new COVID-19 cases [@WorldHealthOrganization2020c]. The data interval is one day.
![Coronavirus reported cases and Google Trends data from 15 January to 24 March 2020 in China, South Korea, Italy, Iran and Worldwide.[]{data-label="fig:fig1"}](second-3-times.pdf){width="\textwidth"}
The situation has changed since a rapid increase in cases was reported in South Korea, Italy and Iran. GT data reveals the rapid growth of the second wave of interest in coronavirus since 21 February 2020. This rising interest trend is observed worldwide and in the presented countries, where a rapid increase in cases of laboratory-confirmed COVID-19 has been reported since 21 February 2020 [@WorldHealthOrganization2020b].
[llllll]{} & China & South Korea & Italy & Iran & Worldwide\
China & 1 & & & &\
South Korea & 0.762\*\*\* & 1 & & &\
Italy & 0.348\*\*\* & 0.647\*\*\* & 1 & &\
Iran & 0.575\*\*\* & 0.780\*\*\* & 0.817\*\*\* & 1 &\
Worldwide & 0.779\*\*\* & 0.890\*\*\* & 0.781\*\*\* & 0.670 & 1\
\[tab:table1\]
[llllll]{} & China & South Korea & Italy & Iran & Worldwide\
China & 1 & & & &\
South Korea & 0.072\*\*\* & 1 & & &\
Italy & 0.171\*\*\* & 0.433 & 1 & &\
Iran & 0.309\*\*\* & 0.714\*\*\* & 0.596\*\*\* & 1 &\
Worldwide & 0.777\*\*\* & 0.047\*\* & 0.297\* & 0.246\*\*\* & 1\
\[tab:table2\]
Table \[tab:table1\] presents Pearson’s correlation matrix of GT trends between Worldwide and in reported countries in the time of first wave, from 15 January to 18 February 2020. Results show that Worldwide trend is significantly, positively and strongly correlated with trends in China, South Korea and Italy.
Table \[tab:table2\] presents Pearson’s correlation matrix of GT trends between Worldwide and in reported countries in the time of second wave, from 19 February to 24 March 2020. Results show that Worldwide trend is significantly, positively correlated with trends in China, Italy, South Korea and Iran.
Discussion
==========
The key finding is that GT forecasted the rise of new cases. In first wave, new cases increased day-by-day for 6 days after the highest peak of GT worldwide interest. In the second wave, interest in coronavirus on GT is still rising, which predicts the increasing number of new cases reported daily. This implies that national health services should implement additional health measures against countries other than China.
The limitation of the study is GT data about China because of the general unavailability of Google in China. Another limitation is that data about South Korea, Italy and Iran is changing rapidly every day, thus results are only relevant to the reported date.
[10]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
Y. Chen, Q. Liu, and D. Guo, “[Emerging coronaviruses: Genome structure, replication, and pathogenesis]{},” *Journal of Medical Virology*, vol. 92, no. 4, pp. 418–423, apr 2020. \[Online\]. Available: <http://www.ncbi.nlm.nih.gov/pubmed/31967327>
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, “[Coronavirus disease (COVID-2019) situation reports]{},” 2020. \[Online\]. Available: <https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports>
A. Mavragani and G. Ochoa, “[Google trends in infodemiology and infoveillance: Methodology framework]{},” *Journal of Medical Internet Research*, vol. 21, no. 5, 2019.
M. Kami[ń]{}ski, I. [Ł]{}oniewski, and W. Marlicz, “[Global Internet Data on the Interest in Antibiotics and Probiotics Generated by Google Trends]{},” *Antibiotics*, vol. 8, no. 3, p. 147, sep 2019. \[Online\]. Available: <https://www.mdpi.com/2079-6382/8/3/147>
, “[Joint WHO and ECDC mission in Italy to support COVID-19 control and prevention efforts]{},” 2020. \[Online\]. Available: <http://www.euro.who.int/en/health-topics/health-emergencies/coronavirus-covid-19/news/news/2020/2/joint-who-and-ecdc-mission-in-italy-to-support-covid-19-control-and-prevention-efforts>
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We use scale-free networks to study properties of the infected mass $M$ of the network during a spreading process as a function of the infection probability $q$ and the structural scaling exponent $\gamma$. We use the standard SIR model and investigate in detail the distribution of $M$, We find that for dense networks this function is bimodal, while for sparse networks it is a smoothly decreasing function, with the distinction between the two being a function of $q$. We thus recover the full crossover transition from one case to the other. This has a result that on the same network a disease may die out immediately or persist for a considerable time, depending on the initial point where it was originated. Thus, we show that the disease evolution is significantly influenced by the structure of the underlying population.'
address: 'Department of Physics, University of Thessaloniki, 54124 Thessaloniki, Greece'
author:
- 'Lazaros K. Gallos and Panos Argyrakis'
title: 'Distribution of infected mass in disease spreading in scale-free networks'
---
Scale-free networks ,spreading ,SIR model ,infected mass 89.75.-Hc ,87.23.Ge
There has been a growing interest recently in the network structure [@1; @2; @DM; @3; @4; @5; @6; @7] and dynamics [@8; @9] of real-life organized systems. Many such systems, covering an extremely wide range of applications, have been recently shown [@1; @2; @DM; @3; @7] to exhibit scale-free character in their connectivity distribution, meaning that they obey a simple power law. Thus, the distribution of the connectivity of nodes, follows a law of the form $$P(k) \sim k^{-\gamma} ,$$ where $k$ is the number of connections that a node has and $\gamma$ is a network parameter which determines the degree of its connectivity. These networks have some unusual properties, thus justifying the heavy interest in recent years. For example, spreading processes in scale-free networks show a dynamic behavior which is different than in other classes of networks. These processes are plausible models for the spreading of diseases, epidemics, etc. Several models for spreading exist in the literature built on different algorithms, such as the SIR model [@Newman], the SIS model [@8], the SIRS model [@Kuperman], etc.
In a recent paper Newman [@Newman] studied analytically the SIR model in scale-free networks. The connectivity distribution had an exponential cutoff of the form $$P(k) \sim k^{-\gamma} e^{-k/\kappa} \;,$$ where $\kappa$ is an arbitrary cutoff value for $k$. This work gave a closed form solution for the epidemic size and the average outbreak size as a function of the infection probability. It showed that there is a critical infection threshold ($q_c$) only for small finite values of $\kappa$, but as $\kappa$ increases $q_c$ decreases, apparently resulting in $q_c\rightarrow 0$ (no critical threshold at all) in the limit of $\kappa \rightarrow \infty$. The same result was also shown for the SIS model by Pastor-Satorras and Vespignani [@8].
The absence of a critical threshold is not a universal network property. Actually, in the well studied cases of lattice networks and small-world networks the opposite is true [@Warren]. Such a threshold [@11; @12] is always present, which separates the infected from the uninfected regions. This has as a result that these models do not offer a very realistic picture. Recently, Warren [*et al.*]{} [@Warren] have used a heterogeneous distribution for the infection probabilities both for lattices and small-world networks. They model the variability in a population which results in a broadening of the transition regime; however, a threshold still exists and the behavior of the transition is qualitatively similar to the case of the simpler SIR model on a lattice. Because of this, scale-free networks are distinctly different regarding the predictions on the rate and efficiency of spreading. This is clearly much closer to what it is intuitively expected, and can provide useful estimates for the properties of epidemics of any kind. The same type of model could also describe a diverse set of networks, such as social networks, virus spreading on the Web, rumor spreading, signal transmission etc.
In the present study we calculate the distribution of the epidemic size, i.e. the distribution of the infected mass, for several different $\gamma$ values. This property helps us to better understand the importance of the starting point (origin) of the disease. It turns out that this distribution is not a smooth function for all networks, but depends strongly on the network density, i.e. the value of $\gamma$.
We use a simulation algorithm to construct a scale-free network comprised of $N$ nodes. We follow a network generation method which enables us to freely vary the connectivity distribution of the network. We assign a number of edges $k$ for each node by using a power-law distribution $P(k)\sim k^{-\gamma}$. Starting from the highest connectivity nodes we create links by randomly choosing $k$ other nodes. Care is taken that no duplicate links are established between the same two nodes and once a node has reached the number of edges initially assigned to it, it no longer accepts any new connections. The cutoff value for the maximum possible connectivity of a node was fixed to $N/2$.
The spreading of a disease follows the standard SIR (Susceptible, Infective, Recovered) model. Initially, all nodes are in the susceptible (S) status, and a random node is infected (I). During the first time step it tries to infect with probability $q$ the nodes linked to it, and when the attempt is successful the status of the linked node switches from S to I. The process is repeated with all infected nodes trying to influence their susceptible (S) neighbors during each time step. After trying to infect its neighbors the status of an infected site changes to recovered (R) and can no longer be infected. The simulation stops when there are no infected nodes in the system or when all nodes have been infected. Small-world networks were constructed as described in reference [@10].
We monitor the percentage $M$ of nodes infected and the duration of a disease (i.e. the time needed for the disease to either disappear or cover the entire network). In figure 1 we show for comparison purposes the percentage $M$ of infected sites as a function of $q$, for three different network types. The form of the curves for the lattice and the small world networks is similar, i.e. in all cases there is a sharp transition and a critical point. However, this behavior is unrealistic, as it does not follow the majority of actual situations, such as disease spreading in real-life networks [@Warren]. If this were to happen, e.g. the Web would be in a state of either no virus present or the entire Web (all computers in the world) would be infected, with a very small probability of having an intermediate situation with only a certain fraction of computers infected, which is the realistic picture. Scale-free networks with a high degree of connectivity ($\gamma=2.0$) follow a much smoother spreading evolution, as the mass of the infected population increases almost linearly with the infection probability q and there is no transition regime. This is in agreement with the recent formalism of Newman [@Newman]. The linear behavior can be understood as follows: On a scale-free network there exist nodes with a wide range of connectivity. For fixed infection probability $q$, the average probability for an infected node with $k$ links to spread the disease is $kq$. If this number is greater than 1, it is statistically certain that a neighbor node will be infected. If it is significantly less than 1 the disease will die out. On the other hand, for a small-world network and for a lattice network there is a characteristic mean number of links $\langle k \rangle$ assigned to each node.
Here, for a scale-free network, we look at the distribution of the quantity $M$. This is shown in figure 2, where we see that for different $\gamma$ values we have two opposite situations. First, for $\gamma=2.0$ we see that for fixed $q$ we can have both cases coexisting, depending on the connectivity of the initially infected node. For this case the distribution $\Phi(M)$ of the infected population for fixed $q$ is bimodal and comprises of two distinct parts, a strong peak at $M=1/N$ (only one node is infected) attributed to initially infected nodes with $k$=1, or more accurately $kq\ll 1$, and a Gaussian-shaped part at some higher value of $M$. The almost linear increase of $M$ with $q$ in figure 1 is due to the fact that for higher $q$ the peak of $1/N$ decreases ($kq$ increases), and the average value of the Gaussian distribution increases accordingly. We also notice in figure 2 that the gaussian part of the distribution has an average value which is larger than $q$. For example, for $q=0.1$ the peak of the distribution is close to 0.25, and when $q=0.8$ it is closer to 0.9. This means that if the peak in $1/N$ did not exist, the behavior of the curve in figure 1 would be superlinear, i.e. $M$ would always be larger than $q$ as a result of the complex connectivity of the network, but it would not reach the value of $M=1$ for infection probabilities less than 1. The interplay between the peak at $1/N$ and the gaussian part yields the final curve which follows roughly a linear increase.
This result is a superposition of the two possible states present in the case of a simple lattice, where below the threshold the distribution is a simple peak at $M=1/N$, while above the threshold the distribution peaks around $M=1$. On a scale-free network there exists a finite probability for the disease to be either eliminated immediately or cover a considerable portion of the network. Thus, for small $\gamma$ values, the prediction on the future of a disease largely depends on the place where it originates, because of the existence of the gap. This bimodality implies that every disease which survives the initial step(s) spreads over a non-zero portion of the network population. This is not the case for large $\gamma$ values, where recently [@15], results for the SIR model on a scale-free network (with $\gamma=3.0$) were presented. The distribution of $M$ did not show any bimodality, similarly to our solid curve on the left of figure 2, which corresponds to $\gamma=2.9$ and $q=0.5$. We see that the bimodality is now lost, and it is replaced by a smoothly decreasing function with increasing $M$. This result is in agreement with the work of Moreno [*et al.*]{} [@15]. Thus, the distribution changes from bimodal to unimodal as we go from a dense network to a sparse network. The crossover from one to the other takes place as it is given in the inset in figure 2, where we plot $\gamma$ vs $q$, i.e. each ($\gamma$, $q$) pair is exactly at the corresponding transition point.
These observations show that in scale-free networks nodes of high connectivity act as “boosters” to the disease spreading; even if very few nodes remain infected, by the time a high connectivity node is infected it spreads the disease over a significant number of its neighbors, even for small $q$. This fact stresses the importance of the ‘hubs’, as it has also been observed in the past in studies of the static properties for such networks [@5; @13; @14]. Similarly, the low-connectivity nodes may serve to isolate large clusters of the network. These clusters are effectively screened by the disease via the presence of the low-connectivity node, especially in the case of high $\gamma$, where the network is loosely connected.
Similar conclusions can be drawn for the duration of a disease. For small-world networks and regular lattices the duration of a disease when epidemics takes place is practically constant for infection probabilities greater than the threshold (figure 3). For scale-free networks, on the contrary, there is a slow increase of the duration as a function of $q$. However, the duration is much smaller now, which implies that even for considerable infection probabilities a disease cannot last for a long time, and a considerable portion of the network can be infected in a practically small and constant time. The duration of the disease at $q=1$ is also a measure of the network ‘diameter’, since it represents the average number of links needed to cross, before reaching all the system nodes.
Upon monitoring the distribution of uninfected sites as a function of time we observed that it followed a power law at all times. The exponent of this power law was always the same as the one used for the initial connectivity distribution, with the curve scaled down by a constant factor. Thus, sites of different connectivity are infected with the same relative rate.
Summarizing, we have investigated spreading properties on scale-free networks. For the SIR model we studied we find that the starting point of the disease is very important, because it can either stop the disease or facilitate its spreading. This phenomenon for dense networks yields a bimodal distribution for the infected mass, with a peak close to 0 and a gaussian part around a finite value of $M$. Despite the smooth increase of the infected mass with $q$ the disease spreads rapidly on the network in a practically constant time, almost independently of $q$. This rapid spreading manifests the compactness of the network (as compared to lattice and small-world networks) and its small diameter which is related to the short path length from any site of the network to another. For sparse networks this is not the case, as it also has been previously observed.
[00]{} S.H. Strogatz, Nature 410 (2001) 268. R. Albert, A.-L. Barabasi, Rev. Mod. Phys. 74 (2002) 47. S.N. Dorogovtsev, J.F.F. Mendes, Adv. Phys. 51 (2002) 1079. A.-L. Barabasi, R. Albert, Science 286 (1999) 509. R. Albert, A.-L. Barabasi, Phys. Rev. Lett. 85 (1999) 5234. R. Albert, H. Jeong, A.-L. Barabasi, Nature 406 (2000) 378. P.L. Krapivsky, S. Redner, F. Leyvraz, Phys. Rev. Lett. 85 (2000) 4629. F. Liljeros *et al.* Nature 411 (2001) 907. R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 86 (2001) 3200. R. Pastor-Satorras, A. Vazquez, A. Vespignani, Phys. Rev. Lett. 87 (2001) 258701. M.E.J. Newman, Phys. Rev. E 66 (2002) 016128. M. Kuperman, G. Abramson, Phys. Rev. Lett. 86 (2001) 2909. C.P. Warren *et al.* Math. Biosci. 180 (2002) 293. C. Moore, M.E.J. Newman, Phys. Rev. E 61 (2000) 5678; 62 (2001) 7059. D.H. Zanette, Phys. Rev. E 64 (2001) 050901(R). D.J. Watts, S.H. Strogatz, Nature 393 (1998) 440. Y.Moreno, R.Pastor-Satorras, A.Vespignani, Eur. Phys. J. B 26 (2002) 521. D.S. Callaway, M.E.J. Newman, S.H. Strogatz, D.J. Watts, Phys. Rev. Lett. 85 (2000) 5468. R. Cohen, K. Erez, D. ben-Avraham, S. Havlin, Phys. Rev. Lett. 85 (2000) 4626.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We present a design for a tunneling-current-assisted scanning near-field microwave microscope. For stable operation at cryogenic temperatures, making a small and rigid microwave probe is important. Our coaxial resonator probe has a length of approxomately 30 mm and can fit inside the 2-inch bore of a superconducting magnet. The probe design includes an insulating joint, which separates DC and microwave signals without degrading the quality factor. By applying the SMM to the imaging of an electrically inhomogeneous superconductor, we obtain the spatial distribution of the microwave response with a spatial resolution of approximately $200\ \mathrm{nm}$. Furthermore, we present an analysis of our SMM probe based on a simple lumped-element circuit model along with the near-field microwave measurements of silicon wafers having different conductivities.'
author:
- 'Hideyuki Takahashi$^{1,3}$, Yoshinori Imai$^{2,3}$, Atsutaka Maeda$^{3}$'
title: 'Low-temperature-compatible tunneling-current-assisted scanning microwave microscope utilizing a rigid coaxial resonator'
---
Introduction
============
A scanning near-field microwave microscope (SMM) enables local electrical characterization with a spatial resolution beyond the diffraction limit. Since the first SMM was produced by Ash and Nicholls [@Ash1972], various such instruments have been reported. They are divided into two categories; apertured [@Ash1972; @TabibAzar1999; @Park2001] and apertureless probes [@Wei1996]. Recent progress has mainly been on the latter type. The spatial resolution of the apertureless-type SMM has significantly improved by precise control of the tip-sample distance. Using the feedback control circuits of a scanning tunneling microscope (STM-SMM [@Imtiaz2003; @Machida2009; @Lee2010]) and an atomic force microscope (AFM-SMM [@Kim2003; @Gao2005; @TabibAzar2004; @Lai2007; @Zhang2010]), a nanometer-scale contrast of material properties can be obtained. In addition, it has been suggested that STM-SMM allows us to obtain an atomic-resolution microwave image using tunneling impedance [@Lee2010; @Reznik2014].
Such SMMs have already been used for condensed matter research to evaluate the sheet resistances of metal oxide thin films [@Park2001; @Wang2005], superconductors [@Takeuchi1997; @Steinheuer1997Dec] and combinatorial materials [@Yoo2001; @Okazaki2008]. The value of the SMM will be further enhanced by cryogenic applications, and provide us with insights regarding the origin of various physical phenomena. In fact, cryogenic SMM has revealed that intrinsic inhomogeneity is related to the metal-insulator transition [@Hyun2002; @Lai2010] and phase separation [@HT2015pC; @HT2015]. Additionally, SMM is expected to serve as a tool for locally studying inhomogeneous superconductors such as high-$T_c$ cuprate in the pseudogap phase [@Lang2002; @Kohsaka2007] and vortex matter [@HT2012; @Okada2012] because the microwave response directly reflects the low-energy quasiparticle dynamics [@Maeda2005; @HT2011].
In this paper, we present the design of a low-temperature-compatible STM-SMM utilizing a coaxial resonator probe. We achieve stable operation at cryogenic temperature by designing a small and rigid microwave probe. We also present the analysis of our SMM probe based on the simple lumped-element circuit model. Although we need a simulation method such as finite element method [@Okazaki2007] for rigorous analysis, the lumped parameter model is useful for semi-quantitatively understanding the behavior of resonator probes.
Apparatus
=========
![(a) Side (left) and cross-sectional (right) views of the SMM head. (b) Schematic of the coaxial resonator probe. (c) Simulation of the oscillating electric and magnetic fields inside the coaxial resonator probe. The red and blue color show the sign of the field direction. The dashed line in the right figure corresponds to the position of the insulating joint.[]{data-label="apparatus"}](fig1_revised.eps){width="0.9\hsize"}
Figure \[apparatus\](a) shows our SMM head, which mainly comprise a resonator probe, a piezo scanner, and body parts. The outer diameter at the thickest part is set to 50 mm to fit inside a superconducting magnet for the future experiment under magnetic field. The body parts machined from oxygen-free copper (OFC) or brass form a coarse approaching mechanism, which is in direct contact with liquid helium at the upper flange to strongly cool the resonator probe.
The resonator probe (fig. \[apparatus\](b)) was also made of OFC. It is similar to the often-used coaxial resonator [@Wei1996], except that the central conductor is divided into two parts. There is a small hole at the tapered end of the upper part. A sharpened metal tip is connected to this hole and protrudes out of the aperture of the outer conductor to simultaneously detect the tunneling current and microwave response. We use a mechanically cut or electrochemically etched Pt-Ir wire as a tip, whose curvature at the end ($r_{\mathrm{tip}}$) is approximately 100 nm. The upper part is glued to the bottom part with insulating varnish. The shielded cable for the tunneling current detection is inserted through the hole of the bottom part, and then is soldered to the post of the upper part. While this design separates the tunneling current and microwave circuits, it is not suitable for operation in the lowest transverse electromagnetic (TEM) $\lambda /4$ mode (where $\lambda$ is the wavelength) because of the considerable energy loss at the joint part. Instead, we use the second-lowest TEM $3\lambda /4$ mode at $f_0=2\pi/\omega_0=10.7\ \mathrm{GHz}$. Figure \[apparatus\](c) shows the distribution of the electromagnetic field inside the resonator. In this mode, the electric and magnetic fields are almost confined to the radial and the angular direction, respectively. The oscillating magnetic field has a node. The energy loss is significantly reduced by adjusting the joint position at this node. Consequently, we can maintain a high quality factor $Q$. The unloaded quality factor of the resonator is $Q_0=$1200-1300 at room temperature and $Q_0>$2000 below liquid nitrogen temperature.
Semi-rigid coaxial cables (0.085 inches in diameter) are used for transmission and coupling to the resonator. These cables are thermally anchored to the body part. Sample and tip interchange is performed by disconnecting the SMA connectors. After a sample is set, the vacuum can is sealed with indium wire; then, it is evacuated and helium is used as exchange gas.
![The schematic of the control circuit for SMM using FFC. Either the tunneling current or $Q$ is used as a feedback signal. []{data-label="FFC"}](fig2.eps){width="0.9\hsize"}
The measurement system comprises STM and microwave circuits. A commercial STM controller (RC4 and SC4 by SPECS Surface Nano Analysis GmbH) is used to control tunnel current feedback. The quality factor, $Q$, and resonant frequency shift, $\Delta f$, are measured either by acquiring the transmission spectrum using a PNA network analyzer or by the frequency feedback circuit (FFC) [@Steinheuer1997July]. In the latter method, a frequency-modulated (FM) microwave with a frequency expressed as $f_c+D\cos (2\pi f_m t)$ (where $f_c$, $f_m$, and $D$ are the carrier frequency, modulation rate, and frequency deviation, respectively) is used (fig. \[FFC\]). The modulation is controlled by the DC-coupled external bias circuit. When $f_c$ coincides with the resonant frequency of the coaxial probe, the transmitted power oscillates with a frequency of $2f_m$, which is converted into a voltage signal, $V_{2f}(\propto Q)$ by the diode detector (Agilent 8473C). Otherwise, the component oscillating with $f_m$ ($V_f$) becomes large. By feedback control minimizing $V_f$, $f_c$ is locked at $f_0$. The advantage of the FFC method over the method that uses a PNA network analyzer is its high speed. In addition, the FFC method allows us a tip-sample distance control not only by STM feedback (constant current mode) but also by keeping $Q$ constant (constant Q mode) [@HT2015]. However, since the phase-locked loop inside the microwave source (HP83630A) should be open when the modulation is controlled by an external circuit, a drift of the source frequency arises. A typical drift rate is approximately $10\ \mathrm{kHz/min}$, and its influence is corrected after data acquisition. In this work, SMM is always operated at $f_0$. It should be mentioned that there are other studies that use slight off-resonant excitation to optimizing the measurement condition [@Sardi2015; @Gregory2016].
Figure. \[Bi2Se3\](a)-(c) show example images taken by our SMM. The sample is cleaved Bi$_2$Se$_3$ and a temperature is 77 K. The $Q$ and $\Delta f$ images are raw images without any geometrical corrections, whereas a tilt correction is applied for topography. These are obtained using the PNA network analyzer; hence, these are free from the frequency drift of the microwave source. Therefore, these images directly reflect the performance of our SMM probe. Since this sample is homogeneous, $Q$ and $\Delta f$ are constant in most regions. The resonant characteristics change only at the edges of the terraces. These changes are related to the change in the capacitance between the tip and sample, $C_x$, which will be discussed later. Despite the long measurement time of approximately an hour, we obtain images without any drift of the microwave properties, indicating the high long-term stability of our SMM probe.
![Simultaneously acquired (a) topography; (b) quality factor; and (c) frequency shift image of the cleaved surface of Bi$_2$Se$_3$ single crystal.[]{data-label="Bi2Se3"}](fig3.eps){width="1\hsize"}
Analysis
========
To discuss the change in the resonant characteristics caused by samples having different conductivities, we first model the resonator probe by a distributed constant circuit; then, we simplify it using a lumped parameter circuit. The coaxial resonator is equivalent to a transmission line resonator described in fig. \[equivcircuit\](a). The left side corresponds to the tip end of the resonator. $\gamma(=\alpha+j\beta)$ is a propagation constant, with $\alpha$ being an attenuation constant related to the loss in the transmission line and $\beta$ being a phase constant equal to $2\pi/\lambda$ for air.
In transmission line theory, the input impedance at the position $l$ away from the load impedance, $Z_{\mathrm{load}}$, is $$Z_{\mathrm{in}}=Z_{0}\frac{Z_{\mathrm{load}} + Z_0 \tanh \gamma l}{Z_{0}+Z_{\mathrm{load}} \tanh \gamma l}.$$ Where $Z_0$ is the characteristic impedance of the transmission line, which is expressed as $$Z_0=\sqrt{\frac{\mu}{\epsilon}}\frac{\ln b/a}{2\pi}.$$ In our case, $Z_0 =70\ \mathrm{\Omega}$ using a ratio of the outer diameter to the inner one of $b/a\ =\ 3.3$. Since the resonator is closed at the right end, the impedance on the right-hand side of the dashed line in fig. \[equivcircuit\](a) is $$Z_{R}=Z_{0}\frac{\tanh\alpha l+j\tan \beta l}{1+j\tanh\alpha l\tan \beta l}.$$ When the transmission line is lossless ($\alpha=0$), we obtain $$Z_{R}=jZ_0\tan \beta l.$$ When a sample is absent, the impedance of the tip end, $Z_L$, is $Z_L=\infty$. From the resonant condition met when $Z_R+Z_L=0$, we obtain $$l=\frac{\lambda}{4}(2n-1)\ \ \ \ (n=1,2,3\cdots)$$ The TEM $3\lambda /4$ mode corresponds to the case of $n\ =\ 2$.
![(a) The equivalent trasmission line and (b) the lumped element circuit of the coaxial resonator. []{data-label="equivcircuit"}](fig4.eps){width="0.9\hsize"}
Next, we discuss the quality factor of the resonator by considering a finite transmission-line loss. When the loss is small ($\alpha l\ll 1$), $Z_R$ is expressed as $$Z_{R}=Z_{0}\frac{\alpha l\cot \beta l +j}{\cot\beta l +j\alpha l}.\nonumber\\$$ Near the resonant frequency $\omega_0$, $Z_R$ of a TEM $3\lambda /4$ resonator is $$Z_{R}=Z_{0}\frac{1+3j\alpha l \pi\Delta \omega/2\omega_0 }{\alpha l+3j\pi\Delta \omega/2\omega_0}
\simeq\frac{Z_0}{\alpha l+3j\pi\Delta \omega/2\omega_0},$$ where $\Delta\omega$ ($\Delta\omega/\omega_0 \ll 1$) is the deviation from the resonant frequency. This formation is equivalent to the input impedance of the lumped $RLC$ parallel circuit, $Z_{\mathrm{in}}=(R^{-1}+2j\Delta\omega C)^{-1}$, whose quality factor and resonant frequency are $Q=\omega_0 RC$ and $\omega_0=1/\sqrt{LC}$, respectively. Therefore, our resonator probe can be simplified by the lumped element circuit shown in fig. \[equivcircuit\](b). The corresponding $R$ and $C$ components can be determined as $$R=\frac{Z_0}{\alpha l}=\frac{2Z_0 Q}{\beta l}=\frac{2\times 70\times 2000}{3\pi/2}=6\times 10^4\ \Omega,$$ $$C=\frac{3\pi}{4\omega_0 Z_0}=0.5\ \mathrm{pF}.$$ Here we use $Q$ in the cryogenic environment, $Q_0=2000$.
![(a) $\sigma$-dependences of $Q$ and $\Delta f$ for different $C_x$ values. (b) $\sigma_2$-dependences of $Q$ and $\Delta f$. $C_x=0.1\ \mathrm{fF}$ was used for this calculation.[]{data-label="calculation"}](fig5.eps){width="1\hsize"}
The tip-sample interaction is modeled by the series of the near-field impedance of a sample and the coupling capacitance, $Z_{\mathrm{load}}=R_x +1/j\omega C_x$. For conducting samples, $R_x$ is roughly approximated as $R_x=1/\sigma r_{\mathrm{tip}}$, where $\sigma$ is the DC conductivity. $C_x$ is calculated to be on the order of 0.01 fF by applying a parallel-plate approximation ($C_x \approx \epsilon_0 \pi r_{\mathrm{tip}}^2/h$, where $\epsilon_0$ is the dielectric permeability in vacuum and $h$ is the tip height from the surface). By a straightforward calculation, we obtain the changes in $Q$ and the resonant frequency as follows: $$\frac{\Delta Q}{Q_0}\simeq-\frac{(\omega C_x R)(\omega C_x R_x)}{1+(\omega C_x R)(\omega C_x R_x)+(\omega C_x R_x)^2},$$ $$\label{fvsCx}
\frac{\Delta f}{f_0}\simeq-\frac{C_x}{2C}.$$ $Q$ exhibits a minimal value when $\omega C_x R_x =1$ (fig. \[calculation\](a)). On the other hand, $\Delta f$ exhibits a monotonous $R_x$ dependence.
In applications to superconductors, we need to consider the contribution of the imaginary part of complex conductivity, $\tilde{\sigma}=\sigma_1+j\sigma_2$. Since $\sigma_1$ and $\sigma_2$ are respectively proportional to the normal fluid and superfluid density, $\sigma_1\ll\sigma_2$ at a temperature sufficiently lower than $T_c$. Figure \[calculation\](b) shows the $\sigma_2$-dependence of the resonant characteristics at fixed $\sigma_1$. $Q$ shows monotonous change against $\sigma_2$ with a maximum slope ($dQ/d\sigma_2$) at $\sigma_1=\sigma_2$ and approaches the value at the unloaded condition when $\sigma_1\ll\sigma_2$. On the contrary, $\Delta f$ shows nonmonotonous change against $\sigma_2$. The characteristic behavior is also observed at $\sigma_1=\sigma_2$ as a peak. It is noteworthy that one cannot distinguish superconducting and non-superconducting phases only by a frequency measurement because a difference of less than $<1\ \mathrm{kHz}$ in $\Delta f$ is beyond the frequency resolution of our setup. On the contrary, a 0.1 %-difference in $Q$ between the superconducting and metallic limits is detectable.
To examine the model’s usefulness, we measure the near-field response of bulk silicon wafers with different conductivities at room temperature. Figure \[Siwafers\] shows the $h$-dependencies of the resonant characteristics. Different tips (A and B) are used in the left and right figures. $Q$ is normalized by the values at $h=100\ \mathrm{nm}$. Both $Q$ and $\Delta f$ monotonically decrease as the tip approaches the sample because of the increase of $C_x$. When we compare the data with the value at $h=0$, we find a remarkable difference between the semiconducting samples ($\sigma=10$-$20\ \mathrm{S/m}$) and the metallic ones ($\sigma=10^3\ \mathrm{S/m}$, $10^4\ \mathrm{S/m}$) ; the change in $Q$ is larger for the semiconducting samples, while the change in $\Delta f$ is larger for the metallic samples. For the insulating sample ($\sigma=0.1\ \mathrm{S/m}$), both $Q$ and $\Delta f$ exhibit their smallest $h$-dependencies. The lumped element model predicts that the change in $C_x$ will be the dominant factor affecting $\Delta f$ (Eq. \[fvsCx\]). $\Delta f$ below $h=100\ \mathrm{nm}$ corresponds to the 0.01-0.1 fF change in $C_x$, which is of the same order as the calculated value with a simple parallel-plate approximation.
The lumped element model semi-qualitatively explains the behavior of the SMM probe. However, it is too simplified for quantitative discussion. We assume that the electromagnetic field is tightly localized around the tip with a decaying length comparable to $r_{\mathrm{tip}}$. In reality, the field decays in a power-law manner [@Imtiaz2006]. A more rigorous model requires modifying the expression for the load impedance, for which the tip geometry and field distribution are considered [@Okazaki2007].
![The tip height-dependences of $Q$ and $\Delta f$ measured for silicon wafer samples with different conductivities. []{data-label="Siwafers"}](fig6.eps){width="1\hsize"}
Imaging
=======
To evaluate the performance of SMM at cryogenic temperature, we measure the spatial distribution of the microwave response of a single crystal of the iron chalcogenide superconductor Fe(Se,Te) with a PbO structure [@Fang2008]. The sample is grown by the Bridgeman method from starting materials with a nominal Se:Te ratio of 0.4:0.6. Although the annealed crystal exhibits perfect shielding below $T_c=14\ \mathrm{K}$ [@HT2011], the unannealed crystal separates into superconducting and non-superconducting phases having different chemical composition [@Speller2011; @Speller2012; @Prokes2015]. The origin of the phase separation is considered to be a large difference in the Fe-Se and Fe-Te bonding lengths [@Joseph2010].
Figure \[FeSeTe\_CC\] shows the surface topography and spatial dependence of $Q$ for Fe(Se,Te), acquired in the constant current mode. The most typical images are figs. \[FeSeTe\_CC\](a) and \[FeSeTe\_CC\](b). The surface roughness is less than 5 nm. The changes in the areas where topographical changes are observed are attributed to the abrupt change of $C_x$. As discussed above, near-field microwave measurement is sensitive not only to $R_x$ but also to $C_x$. In fact, the sharp changes at the edges of the terraces in figs. \[Bi2Se3\](b) and \[Bi2Se3\](c) are also attributed to the change in $C_x$. As $C_x$ strongly depends upon the tip-sample distance and geometry, even a 1-nm step affects the microwave images. Although this is often problematic, one can distinguish whether the contrast in images is related to sample properties or is only a geometrical artifact by carefully examining the topography and microwave image. If we find that the contrast does not correlate with topographic change in the microwave images, we can conclude that it is caused by electric inhomogeneity. Figures. \[FeSeTe\_CC\](c) and \[FeSeTe\_CC\](d) are images of the other region. In addition to the $C_x$-induced contrast, we can see the changes in the microwave response that are not correlated with topography; the contrast gradually changes from the upper left to lower right. This change is considered to be related to the inhomogeneous $R_x$.
The more effective method for separating the topographical and electrical contrast requires slight modification of the measurement system from the constant current mode. In this method, we select $Q$ as a feedback signal instead of the tunneling current. The tunneling current is monitored only so that the tip does not contact the sample. $h$ changes depending upon the local material property while $Q$ is kept constant. Since $\Delta f$ largely depends on $h$, it exhibits significant changes only when the tip crosses a boundary between regions with different conductivities. As a result, a qualitative image is obtained. The advantage of this scanning mode is that the topographical information is largely eliminated in the obtained image. We can avoid the influence of the fluctuation of $C_x$ by setting $h$ higher than the constant current mode (typically $h=10$-$20\ \mathrm{nm}$).
![(a)-(d) Topographies and $Q$ images of Fe(Se,Te) acquired in a homogeneous area (area 1) and an inhomogeneous area (area 2). []{data-label="FeSeTe_CC"}](fig7.eps){width="1\hsize"}
Figure \[FeSeTe\_CQ\](a) is the frequency image acquired in the constant $Q$ mode for the same region as area 2 in fig. \[FeSeTe\_CC\]. The topographic contrast that was present in the constant current image (fig. \[FeSeTe\_CC\](d)) has disappeared. As a result, we can find the boundary between two different phases. Fig. \[FeSeTe\_CQ\](c) shows the $h$ dependencies of the microwave response at positions corresponding to A-D in fig. \[FeSeTe\_CQ\](a). The sharp change below $h=5\ \mathrm{nm}$ is caused by a polluted layer on the sample surface. What is important is the behavior above $h=5\ \mathrm{nm}$. $Q$ in the blue region (positions A and B) is higher than that in the red region (positions C and D), and its difference is observed even at $h=100\ \mathrm{nm}$. Since the length scale of the near-field microwave is approximately $r_{\mathrm{tip}}+h$ when the tip is at height $h$ [@Imtiaz2006], these data indicate that the length scale of electric inhomogeneity is much larger than 100 nm. On the contrary, the $h$ dependencies of the frequency shift does not exhibit a significant difference between the two regions. The change solely observed in $Q$ suggests that the contrast is related to the difference in $\sigma_2$. As shown in fig. \[calculation\](b), high $Q$ is observed in the superconductive region, while the superconductivity hardly affects $\Delta f$. Therefore, in fig. \[FeSeTe\_CQ\](a), the upper-left and lower-right regions correspond to the superconducting and non-superconducting regions, respectively.
As the tip crosses the boundary between different phases, the change in the microwave response occurs within a width of 200 nm (fig. \[FeSeTe\_CQ\](b)). If we assume that the sample has a well-defined boundary and that the electrical property is homogeneous in each region, this indicates that the spatial resolution is no worse than 200 nm. This value is consistent with the curvature of the tip.
The use of sharper tips is indispensable for further improvement of the spatial resolution. However, it is expected that spatial resolution does not improve linearly with $r_{\mathrm{tip}}$ decreases. At small $r_{\mathrm{tip}}$ values, the near-field response is very small because of the weak coupling between the tip and sample. In such a situation, we have to consider the proximity effect, i.e., the contribution from an additional capacitive component that arises from a finite aspect ratio of the tip.
The combination of CC and CQ scanning modes is applicable also for distinguishing semiconducting phase from metallic phase. Our previous studies have revealed web-like mesoscopic phase separation in KFe$_x$Se$_2$ at room temperature [@HT2015pC; @HT2015].
![(a) The frequency image acquired in the constant $Q$ mode for the same region as the area 2 in fig. \[FeSeTe\_CC\]. (b) The linecut between the positions A and D in (a). (c) The tip height-depedences of $V_{\mathrm{2f}}(\propto Q)$ and the resonant frequency at the positions A-D. The offset frequency, $f_{\mathrm{offset}}=10.720974\ \mathrm{GHz}$, is subtracted from the data for clarity.[]{data-label="FeSeTe_CQ"}](fig8.eps){width="1\hsize"}
Summary
=======
We have developed the low-temperature compatible STM-SMM. The modified coaxial resonator probe allows STM operation without sacrificing the high $Q$ factor. The behavior of the SMM probe was described by the lumped element circuit, which was confirmed by the near-field response to silicon wafers having different conductivities. We also demonstrated that STM-SMM can be used for the study of inhomogeneous superconductors using two scanning modes. The spatial resolution is approximately 200nm at this time, which is as high as that in previous reports [@Imtiaz2007; @Machida2009].
A challenge for the future is to combine microwave measurement with local tunneling spectroscopy [@Machida2009]. Since both the local density of states and microwave conductivity are important for understanding the nature of quasiparticles in the superconductor, STM-SMM will be a useful tool for studying nanoscale inhomogeneity and the vortex state in superconductors.
The authors thank for Yusuke Yasutake and Tetsuo Hanaguri for technical assistance, and Susumu Fukatsu for providing us silicon wafers. This work has been supported by a Grant-in-Aid for Scientific Research(A) (Grants. No. 23244070) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. H. Takahashi also thanks the Japan Society for the Promotion of Science for financial support.
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{
"pile_set_name": "ArXiv"
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abstract: '[We summarize results from flux density monitoring campaigns performed with the 100m radio-telescope at Effelsberg and the VLA during the past 15yrs. We briefly discuss some of the statistical properties from now more than 40 high declination sources ($\delta \geq 30^\circ$), which show Intraday Variability (IDV). In general, IDV is more pronounced for sources with flat radio spectra and compact VLBI structures. For 0917+62, we present new VLBI images, which suggest that the variability pattern is modified by the occurrence of new jet components. For 0716+71, we show the first detection of IDV at millimeter wavelengths (32GHz). For the physical interpretation of the IDV phenomenon, a complex source and frequency dependent superposition of interstellar scintillation and source intrinsic variability should be considered. ]{}'
author:
- 'T.P. Krichbaum $^{1}$, A. Kraus $^{1}$, L. Fuhrmann $^{1}$, G. Cimò $^{1}$, A. Witzel $^{1}$'
title: Intraday Variability in Northern Hemisphere Radio Sources
---
=2em =15.5cm =22.6 cm =-1.0 cm =0.5cm =0.5cm
[$^1$ Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany\
e-mail: [email protected]\
]{}
[**Keywords: quasars: general, quasars: individual (0716+71, 0917+62,\
0954+65), radio continuum: ISM, scattering**]{}
Statistical Properties of IDV
=============================
A large fraction (about 20-30%) of compact flat spectrum radio sources, mostly identified as Blazars (Quasars, BLLac objects, etc.), show flux density and polarization variations on timescales of a few hours to a few days. These so called Intraday Variable (IDV) radio sources (see review of Wagner & Witzel, 1995 (WW95)) are defined via their structure function (SF), exhibiting pronounced maxima within time lags of $\sim 0.5-2$days (‘type II’ sources; cf. Heeschen et al. 1987). The Fourier analysis of the IDV timescales shows in nearly all type II sources a quasi-periodic variability pattern, i.e. a concentration of the power (in a cleaned power spectrum) to a few ($\lsim 3-5$) discrete timescales. In contrast, the so called ‘type I’ IDV sources vary slower, showing a quite monotonically increasing SF within the time of the observations, suggesting variability timescales of $\geq 2$ days. Although a $\sim 1$day timescale seems to be common for the classical ‘type II’ IDV sources, ‘sudden’ changes of the variability timescales are observed eg. in 0716+71, which changed from quasi-periodic daily to less periodic weekly oscillations (cf. Qian 1995, WW95, Kraus et al. 2001). Extremely rapid ($\lsim 0.1$days) and pronounced variations like the ones recently observed in PKS0405-38 (Kedziora-Chudczer et al. 1997) or J1819+38 (Dennett-Thorpe & de Bruyn 2000) are not observed in our high declination sample. The fastest variation observed so far by us, also show variations on a 1hr timescale, however with an amplitude of only 2% (in 0716+71 at 5 GHz in April 1993). At the moment it is not clear, if J1819+38 or PKS0405-38 should be classified as ‘type II’ IDV’s, or if it would be better to define another class of more extreme IDV sources, which would imply that it is at present unclear whether the physical process causing such extreme variability is the same as for the ‘classical’ IDV sources.
In the radio bands and for classical IDV’s, the peak-to-peak variations of the flux densities can reach up to $20-30$% (eg. for 0804+49, 0917+62), more typically, however, are a few percent. In polarization, the variations are more pronounced ($\sim 20-100$%) and are about a factor of two faster than in total intensity. The variations in total (I) and polarized (P) flux density appear often either correlated (eg. in 0716+71) or anti-correlated (eg. in 0917+62). However transitions from correlated to anti-correlated variability are observed occasionally. Correlated with the variations in P, the polarization angle (PA) varies on the same timescale by typically a few to a few ten degrees (Kraus et al. 2001). Sometimes, polarization angle swings by $180^\circ$ are observed within hours (0917+62: Quirrenbach et al. 1989; 1150+81: Kochenov & Gabuzda, 1999). A direct conclusion drawn from the polarization IDV is a multiple component structure of the radio sources, with their sub-components being of different compactness and polarization. Although this is in good agreement with the observational findings from polarization VLBI (cf. Gabuzda et al., this conference), the short variability timescales indicate component sizes much smaller than the angular resolution of present day ground and space-based VLBI ($< 0.2$mas).
Intrinsic or Extrinsic ?
========================
In Figure 1 (top), we plot the frequency dependence of the variability index $m$ (defined by $m_I=\Delta I/I$ (left) and $m_P=\Delta P/P$ (right), $\Delta I$ and $\Delta P$ are the rms amplitudes) for a small sample of sources observed with the VLA and the 100m telescope at 5 frequencies (1.5 – 15GHz). The observing interval and data sampling restrict the detection of characteristic variability timescales to $\sim 0.2 \leq t_{\rm var} \leq \sim 2.5$ days (5 days of data). Still, we use the rms-fluctuation index $m$ to illustrate the strength of variability at each frequency for [*these*]{} timescales. We note that for the given (limited) observing interval, $m$ might be only a coarse estimate of the actual variability pattern, particularly in view of the likely mixture of stochastical variations and variations on discrete times scales. A more detailed structure function analysis is beyond the scope of this paper and should be done in the future. Nevertheless, it is obvious that in polarization, the variability index $m_P$ for most sources peaks somewhere between 1.5 and 5 GHz, in contrast to the variations in total intensity, where the situation is more complicated. When looking at the change of the frequency dependence of the variability index with time in Figure 1 (bottom), it is seen that at least two types of variability exist: in 0716+71 the variability index peaks for most observing dates between 3-5GHz, whereas in 0954+65 a clear minimum of $m$ appears in this frequency range. Above 8GHz and in both sources the variability index increases with frequency!
The theory of interstellar scintillation (ISS) predicts most pronounced variations close to a critical frequency $\nu_{\rm crit}$, where strong scattering changes into weak scattering. From Figure 1 it appears, as if this transition frequency is different for each source and also changes for a given source with time (since the data base is still small, this effect should be investigated in more detail). Also the strength of variability at $\nu_{\rm crit}$ seems to be time dependent. An increase of $m$ towards higher frequencies or a ‘double peaked’ appearance of $m(\nu)$ is inconsistent with simple models of ISS, unless a source intrinsic contribution, with increasing dominance towards higher frequency, is assumed.
A stratified (multiple layer) or otherwise more inhomogeneous interstellar medium may also help to explain the complexity seen in Figure 1. As we will show in a moment, this has the potential to explain changes of $m$ with viewing direction (through our galaxy) and the occurrence of more than one characteristic variability timescale observed in a few sources (eg. 0716+71, 0917+62), but not the observed ‘sudden’ transitions between them[^1]. For refractive ISS (RISS), the variability timescale is $t \propto \theta_{\rm scat} \cdot D / v$, where $\theta_{\rm scat}=$ scattering size, $D=$ distance to scatterer, $v=$ relative velocity of scatterer. Rapid changes of $t$ therefore imply similar rapid changes of at least one of the three parameters in this equation. This could perhaps indicate a very clumpy ISM, eg. with sharp edges of quite fast moving clouds at different distances. We however think, that this is not very likely. If on the other hand, the source size $\theta_{\rm src}$ is of order of $\theta_{\rm scat}$ (‘quenched’ scattering), a ‘sudden’ ($\leq$ 1 day) change of the characteristic variability timescale $t$ implies a similar ‘sudden’ change of the source size ($t \propto t_{\rm riss} \theta_{\rm src}/ \theta_{\rm scat}$). This, however, is nothing else but rapid intrinsic variability.
Thus, variability amplitudes increasing monotonically with frequency (from radio, infrared to optical (cf. 2007+77, Peng et al. 2000) and even to X- and Gamma-rays (0716+71, Wagner et al. 1996)), and the sudden transitions of IDV timescales, could also be regarded as sign for source intrinsic variations, which at least in the case of 0716+71 and 0954+65 appear not unlikely in view of the observed broad-band radio-optical correlations (Quirrenbach et al., 1991, Wagner et al. 1993, WW95). The recent finding of IDV in 0716+71 at 9mm wavelengths (Fig. 2) strongly supports this view. At such short wavelength and high galactic latitude ($b_{\rm II}=28^\circ$), the interpretation of IDV as due to scintillation would appear more questionable. More evidence for the idea that IDV at cm-wavelengths is most likely a mixture of propagation and source intrinsic effects, also comes from a multi-frequency analysis of correlated I & P variations in 0917+62, in which correlated and anti-correlated I & P variability peaks were found at 20 and 6cm (Qian et al. 2001). Owing to the frequency dependence of scattering and the fact that for a point source I & P should vary simultaneously, the correlated peaks were interpreted by ISS, resulting in reasonable parameters for the ISM and the source. However, also an anti-correlated I & P peak was seen in the same light curve, which – despite a multiple polarized component structure assumed for the source – did not at all fit into the standard scattering model used by Qian et al.. Since also the polarization angle variations could not be fitted within this scattering model (adding even more structure components or changing the ISM doesn’t help), the conclusion must be that (i) either the present ISS model isn’t yet developed enough to explain the polarization angle variations, or (ii) the IDV variations are a mixture of extrinsic and intrinsic effects, which only mutually can explain the observed radio light-curves.
VLBI observations of IDV’s
==========================
The extreme apparent brightness temperatures ($T_B \geq 10^{16}$K) deduced from the light travel time argument require, in this source-intrinsic interpretation of IDV, relativistic Doppler-boosting factors $\delta \geq 20$. Recently, several authors claimed to have seen faster than previously known superluminal motion with speeds of up to $30 h$c (Jorstad et al. 2001, adopting ). In 0235+164, a Lorentz-factor of $\gamma= (30-50) h$ is likely (Qian et al. 2000, Romero et al. 2000). Therefore apparent brightness temperatures of up to a few times $10^{16} h^3$K could in principle be reached with source intrinsic relativistic jet physics. Much higher brightness temperatures probably indicate the presence of scattering or, if intrinsic mechanisms are at work, an exponent of the Lorentz-transformation for the intrinsic brightness temperature $T_{B,i}$ larger than 3 ($T_{B,app} \propto \delta^x T_{B,i}$ with $x \geq 3$), which could appear in non-spherical geometries (Qian et al. 1991 & 1996, Spada et al. 1999). Also other (coherent) radiation mechanisms are not yet ruled out (cf. Benford & Tzach, 2000, but see also D. Melrose, this proceeding).
Even if the basic process causing IDV is extrinsic, the typical properties of IDV are affected by source intrinsic variations. A dramatic change in the variability behaviour of 0917+62, which slowed down and nearly stopped to vary in September 1998, but resumed to be rapidly variable in February 1999 (Kraus et al. 1999), was recently interpreted as due to orbital motion of the Earth relative to the ISM (Rickett et al. 2001, Jauncey & Macquart, 2001). In order to test the suggested annual modulation of the IDV pattern, we continued to monitor the source with the 100m RT at Effelsberg. The results are shown in L. Fuhrmann’s paper (this conference), who could not confirm the annual modulation scenario.
In Figure 3, we show some new VLBI maps of the bent milli-arcsecond jet of 0917+62 obtained at half year time sampling in 1999–2000. The flux density of the most compact VLBI component C1, the ‘VLBI core’ [^2], is $\sim 200-300$mJy (total flux: $\sim 1.5$Jy). In order to explain the observed variability index of $m \simeq 5$%, the core or its associated shock component has to vary with $m= 25 - 40$%. Such large amplitudes are only possible, if strong rather than weak scattering occurs.
The 2cm maps show component motion with $\beta_{app} = (5-6) h$. Back-extrapolation of the motion of the 3 inner jet components (C2, C3, C4, Fig. 3) to zero-separation from the stationary assumed VLBI core gives their ejection times (see Fig. 4). Component C2 was ejected in 1998.1, at a time when 0917+62 showed pronounced IDV. The variation of $m$ with time after ejection of C2 is indicated by thick bars in Figure 4. The low value of $m$ in September 1998 can be interpreted as due to ‘quenched’ scattering and source expansion: as long as the size of the core region (a blend of core C1 and moving secondary component C2) is larger than the scattering size, no IDV is observed. After a while, the secondary component (C2) moves further down the jet, expands and fades. The compact VLBI core becomes dominant again and the scintillation resumes.
In order to further test this idea, we searched for additional changes in the variability index of 0917+62, using IDV observations since 1988. One other particularly low value of $m=1.4$% in April 1993 coincides nicely with an enhanced 2.8cm flux density ($\sim 0.5$Jy excess) and a slightly increased size of the VLBI core at 1.3cm at this time (Standke et al. 1996). This suggests that also in early 1993 a new jet component was born. Unfortunately we have no other high resolution VLBI data for the period 1993–1995 to test this idea.
Conclusion
==========
We conclude that the physical interpretation of IntraDay Variability still poses problems, both to the source extrinsic (ISM) and source intrinsic (AGN related) scenarios. Most likely, the IDV observed in the radio regime is a quite complex mixture of both effects. The relative contribution of these two effects seem to depend on the properties of the ISM, the viewing direction, and the source size and internal sub-structure, which itself is frequency and time dependent. There is, however, hope that with multifrequency variability studies (in intensity and polarization, cf. Fig. 1) and simultaneous VLBI monitoring (cf. Fig. 4), the extrinsic and intrinsic contribution to IDV can be separated.
For 0917+62, we conclude that temporal variations of the source size, which are correlated with total flux density variations and the ejection of new VLBI components, are likely to modify the observed scintillation pattern of compact radio sources, depending on the ratio of actual source size and scattering size.
In many compact radio sources, new jet components (and flux density outbursts observed at high frequencies) appear quite regularly at one half to a few years time intervals. It is therefore possible to observe an annual modulation in the variability pattern (caused by orbital motion) only in those objects, which are quite inactive and whose nuclear sizes remain smaller than the scattering size for periods much longer than one year.
References {#references .unnumbered}
==========
Benford, G., & Tzach, D., MN, 317, 497. Dennett-Thorpe, J., & de Bruyn, A.G., 2000, ApJ, 525, 65. Kochenov, P.Y., & Gabuzda, D.C., 1999, in: [*‘BLLac Phenomenon’*]{}, eds. L.O. Takalo & A. Sillanpaä, ASP Conf. Ser. Vol. 159, p. 460. Heeschen, D.S., et al. 1987, AJ, 94, 1493. Jorstad, S.G., et al. 2001, ApJS, 134, 181. Kedziora-Chudczer, L.L., et al. 1997, ApJ, 490, L9. Kraus, A. et al. 1999, AA, 352, 107. Kraus, A., et al. 2001, AA, submitted. Jauncey, D.L., Macquart, J.P., 2001, AA, 370, 9. Peng, B., et al. 2000, AA, 353, 937. Qian S.J., et al., 1996, in: [*‘Energy Transport in Radio Galaxies and Quasars’*]{}, eds: P. Hardee et al., ASP Conf. Ser. Vol. 100, p. 55. Qian, S.J., 1995, Chin. Astron. Astrophys. 19/1, p. 69. Qian, S.J., et al. 2000, AA, 357, 84. Qian, S.J., et al. 2001, AA, 367, 770. Quirrenbach, A., et al. 1989, AA, 226, 1. Quirrenbach, A., et al. 1991, AA, 372, 71. Rickett, B.J., et al., 2001, ApJ, 550, 11. Romero, G.E., Cellone, S.A., & Combi, J.A., 2000, AA, 360, 47. Spada, M., Salvati, M., Pacini, F., 1999, ApJ, 511, 126. Standke, K., et al., 1996, AA, 306, 27. Wagner, S.J. et al., 1993, AA, 271, 344. Wagner, S.J. et al., 1996, AJ, 111, 2187. Wagner, S.J., & Witzel, A., 1995, ARA&A, 33, 163 (WW95).
[^1]: This should be not confused with the annual modulation of $m$ due to the Earth’s orbital motion (see Fuhrmann et al., this conference; and Rickett et al. 2001).
[^2]: ‘VLBI core’ is used in this context synonymous for the transition region at the jet base from optically thick to thin emission ($\tau=1$–surface), or for the first visible shock near this base.
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{
"pile_set_name": "ArXiv"
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abstract: 'A detailed proof is given of the well-known facts that greatest common divisors exist in rings of non-Archimedean entire functions of several variables and that these rings of entire functions are almost factorial, in the sense that an entire function can be uniquely written as a countable product of irreducible entire functions.'
address: |
Department of Mathematics\
University of North Texas\
1155 Union Circle \#311430, Denton, TX 76203\
USA
author:
- William Cherry
date: 'March 16, 2011'
title: 'Existence of GCD’s and Factorization in Rings of non-Archimedean Entire Functions'
---
[^1]
In [@CY], Ye and I needed the fact that greatest common divisors exist in rings of non-Archimedean entire functions of several variables. In that paper, we wrote:
> “by standard arguments (see any book on several complex variables that discusses the Second Cousin Problem and the Poincaré Problem), we need only consider ….”
We then gave an argument of Lütkebohmert [@Lu] that the essential property held. We left it to the reader to fill in the details that this really did imply the existence of gcd’s. Cristina Toropu, now a Ph.D. student at the University of New Mexico, asked me to write up a detailed discussion of the details Ye and I omitted from the appendix to [@CY]. The result is this short note, intended primarily for students and others new to the subject of non-Archimedean analysis. The arguments presented are standard, but not, as far as I know, available in the literature in the context of non-Archimedean entire functions. Part of what I present closely parallels section 6.4 in Krantz [@Krantz], where he discusses algebraic properties of rings of analytic functions in several complex variables. Thus, this note also serves to illustrate a useful principle that someone new to the subject should keep in mind: *if a theorem in complex analysis makes use of the local ring of germs of analytic functions at a point, the appropriate substitute in non-Archimedean analysis is the ring of analytic functions on a closed ball.*
The purpose of this note is to illustrate how one transfers a local algebraic property, in this case the existence of greatest common divisors in the ring of analytic functions on a closed ball, to the global ring of entire functions. The algebraic properties of rings of analytic functions on closed balls, or more generally affinoid domains, is broadly treated in books, and so I refer, for instance, to [@BGR] for the fact that the ring of analytic functions on a closed ball is factorial and for the proof of the Weierstrass Preparation theorem.
I would like to emphasize that this note concerns functions of *several* variables. In one variable, it is not hard to see that a non-Archimedean entire function factors into an infinite product of the form $$cz^e\prod_{i\in I}\left(1-\frac{z}{a_i}\right)^{e_i},$$ where $c\in\mathbf{F},$ $e$ is a non-negative integer, $I$ is a countable index set, the $e_i$ are positive integers, and the $a_i$ are non-zero elements of $\mathbf{F}$ with at most finitely many $a_i$ in any bounded subset of $\mathbf{F};$ compare with Theorem \[almostufd\]. See [@Lazard] for a detailed treatment of the one variable case.
Given that this note resulted from discussions with a student and is intended primarily to be read by students, I am pleased to dedicate this note to the memory of Nicole De Grande-De Kimpe, to Chung-Chun Yang, and to Alain Escassut. Over the courses of their careers, each of these individuals has been encouraging and supportive of students and young mathematicians throughout the world.
I would like to thank Alain Escassut for suggesting I cite the work of Lazard and Salmon. I would also like to thank the anonymous referee for suggesting some improvements to this manuscript, and in particular for pointing out that a somewhat lengthy ad-hoc proof of one of the implications of Proposition \[unit\] that I had in an early draft was not needed.
Let $\mathbf{F}$ be an algebraically closed field complete with respect to a non-trivial non-Archimedean absolute value, which we denote by $|~|.$ Denote by $|\mathbf{F}^\times|$ the value group of $\mathbf{F},$ or in other words $$|\mathbf{F}^\times|=\{|a| : a\in\mathbf{F}^\times=\mathbf{F}\setminus\{0\}\}.$$ Because $|~|$ is non-trivial and $\mathbf{F}$ is algebraically closed, $|\mathbf{F}^\times|$ is dense in the positive real numbers. Let $\mathbf{B}^m(r)$ denote the “closed” ball of radius $r$ in $\mathbf{F}^m,$ *i.e.,* $$\mathbf{B}^m(r)=\{(z_1,\dots,z_m)\in\mathbf{F}^m :
\max |z_i| \le r\}.$$ *Henceforth, we will only consider $r\in|\mathbf{F}^\times|.$* Denote by $\mathcal{A}^m(r)$ the ring of analytic functions on $\mathbf{B}^m(r),$ or in other words the sub-ring of formal power series in the multi-variable $z=(z_1,\dots,z_m)$ with coefficients in $\mathbf{F}$ converging on $\mathbf{B}^m(r),$ *i.e.,* $$\mathcal{A}^m(r) = \left\{\sum_\gamma a_\gamma z^\gamma :
\lim_{|\gamma|\to\infty}|a_\gamma|r^{|\gamma|}=0\right\}.$$ Note that we use multi-variable and multi-index notation throughout, and that for a multi-index $\gamma=(\gamma_1,\dots,\gamma_m),$ we use $|\gamma|$ to mean $$|\gamma|=\gamma_1+\dots+\gamma_m.$$ We recall that a multi-index is said to be greater than a multi-index in the **graded lexicographical order** if or if and $\alpha$ is greater than $\beta$ in the (ungraded) lexicographical ordering, which means that for the smallest subscript $i$ such that $\alpha_i\ne\beta_i,$ we have that $\alpha_i>\beta_i.$ Comparing multi-indices or monomials based on the graded lexicographical order simply means to first compare the total degree and then to break ties between monomials of the same total degree by using the lexicographical order.
Denote the quotient field of $\mathcal{A}^m(r),$ *i.e.,* the field of meromorphic functions on $\mathbf{B}^m(r),$ by $\mathcal{M}^m(r).$ We will also want to consider analytic and meromorphic functions that do not depend on the final variable $z_m,$ and for convenience, we denote these by $\mathcal{A}^{m-1}(r)$ and $\mathcal{M}^{m-1}(r).$
Recall that the residue class field $\widetilde{\mathbf{F}}$ is defined by $$\widetilde{\mathbf{F}}=\{a \in \mathbf{F} : |a|\le1\}/
\{a \in \mathbf{F} : |a| < 1\}.$$ A property will said to be true for an $m$-tuple **over a generic residue class** if $|u_j|\le 1$ for $1\le j \le m$ and if the property holds for all such $u$ such that the reduction lies outside the zero locus in $\widetilde{\mathbf{F}}^m$ of some non-zero polynomial in $m$ variables with coefficients in $\widetilde{\mathbf{F}};$ note that $\widetilde{\mathbf{F}}$ is algebraically closed.
If $$f(z)=\sum a_\gamma z^\gamma$$ is an element of $\mathcal{A}^m(r),$ then denote by $$|f|_r=\sup_\gamma |a_\gamma|r^{|\gamma|}.$$ We begin with the non-Archimedean maximum modulus principle.
\[maxmod\] Let $f$ be an analytic function in $\mathcal{A}^m(r).$ Then, $|f(z)|\le|f|_r$ for all $z$ in $\mathbf{B}^m(r).$ Moreover, let $c$ be an element of $\mathbf{F}$ with $|c|=r.$ Then for over a generic residue class, $$|f(cu_1,\dots,cu_m)|=|f|_r.$$
See [@BGR Prop. 5.1.4/3]. I give the argument here because a solid understanding of $|f|_r$ is fundamental to most of what I do in this note. Write $$f(z)=\sum_\gamma a_\gamma z^\gamma.$$ Then, we immediately have, $$|f(z)| = \left|\sum_\gamma a_\gamma z^\gamma\right|
\le \sup |a_\gamma||z^\gamma|
\le \sup |a_\gamma|r^{|\gamma|}=|f|_r.$$ To see that equality holds for $u$ above a generic residue class, let $\Gamma$ be the set of multi-indices $\gamma$ such that Let $\gamma_0$ be a multi-index in $\Gamma,$ and let $b=a_{\gamma_0}c^{|\gamma_0|}$ so that $|b|=|f|_r.$ If $$|f(cu_1,\dots,cu_m)| < |f|_r,$$ then $$\left|\sum_{\gamma\in\Gamma}a_\gamma c^{|\gamma|}u^\gamma\right|
< |f|_r,$$ and hence $$\left|\sum_{\gamma\in\Gamma}e_\gamma u^\gamma\right|<1,
\qquad\textnormal{where~}e_\gamma=\frac{a_\gamma c^{|\gamma|}}{b}.$$ Note that $|e_\gamma|\le1$ and that $e_{\gamma_0}=1.$ In terms of residue classes, the previous inequality precisely means $$\sum_{\gamma\in\Gamma} \tilde e_\gamma {\tilde u}^\gamma=0.$$ This is a non-trivial polynomial relation, and hence we must have equality over a generic residue class.
\[mult\] The real-valued function $|~|_r$ on $\mathcal{A}^m(r)$ is a non-Archimedean absolute value on $\mathcal{A}^m(r).$
Let $f$ and $g$ be analytic functions in $\mathcal{A}^m(r).$ That $$|f+g|_r\le\max\{|f|_r,|g|_r\}$$ follows directly from the fact that $|~|$ is a non-Archimedean absolute value on $\mathbf{F}.$ To check multiplicativity, note that by Proposition \[maxmod\], there exists a point $(a_1,\dots,a_m)$ in $\mathbf{B}^m(r)$ such that $$\begin{aligned}
|f|_r &= |f(a_1,\dots,a_m)|,\\
|g|_r &= |g(a_1,\dots,a_m)|, \textnormal{~and}\\
|fg|_r &= |f(a_1,\dots,a_m)g(a_1,\dots,a_m)|,\end{aligned}$$ and so the multiplicativity of $|~|_r$ also follows from the multiplicitivity of $|~|.$
Note that we may extend $|~|_r$ to a non-Archimedean absolute value on $\mathcal{M}^m(r)$ by multiplicativity.
\[unit\] An analytic function of the form $$u(z)=1+\sum_{|\gamma|\ge1}a_\gamma z^\gamma$$ is a unit in $\mathcal{A}^m(r)$ if and only if $$\sup_{|\gamma|\ge1}|a_\gamma|r^{|\gamma|}<1.$$
If $u=1-f,$ where $f$ is in $\mathcal{A}^m(r)$ with $|f|_r<1,$ then $$1+f+f^2+f^3+\dots$$ converges to a function $v$ such that $uv=1,$ and so $u$ is a unit.
We will postpone the proof of the converse until later.
Following [@BGR], but working with $\mathcal{A}^m(r)$ instead of just $\mathcal{A}^m(1),$ we say that an analytic function $$f(z)=f(z_1,\dots,z_m)=\sum_\gamma a_\gamma z^\gamma
= \sum_{j=0}^\infty A_j(z_1,\dots,z_{m-1})z_m^j$$in $\mathcal{A}^m(r)$ thought of as a power series in $z_m$ alone with coefficients in $\mathcal{A}^{m-1}(r)$ is **of degree $n$**
- if $A_n(z_1,\dots,z_{m-1})$ is a unit in $\mathcal{A}^{m-1}(r),$
- if
- and if for all $j>n.$
The function $f$ is called simply $z_m$-distinguished if it is $z_m$-distinguished of degree $n$ for some $n\ge0.$ Note that if $f$ is $z_m$-distinguished of degree $0,$ then $f$ is a unit in $\mathcal{A}^m(r)$ by Proposition \[unit\]. An element $W$ of $\mathcal{A}^{m-1}(r)[z_m],$ *i.e.,* a polynomial in the last variable $z_m$ with coefficients analytic, but not necessarily polynomial, in the first $m-1$ variables, of degree $n$ in $z_m$ is called a **Weierstrass polynomial** if $W$ is monic and if $|W|_r=r^n.$
\[Wnotunit\] A Weierstrass polynomial of positive degree is not a unit.
Let $$W(z_1,\dots,z_m)=A_0(z_1,\dots,z_{m-1})+\dots+A_{d-1}(z_1,\dots,z_{m-1})z_m^{d-1}+z_m^d$$ be a Weierstrass polynomial of degree $d>0$ in $\mathcal{A}^{m-1}(r)[z_m].$ Factor the one-variable monic polynomial $$W(0,\dots,0,z_m)= (z_m-b_1)\cdots(z_m-b_d).$$ I claim that for some $j$ from $1$ to $d,$ we must have that $|b_j|\le r.$ For if not, then $$|W(0,\dots,0,z_m)|_r = |z_m-b_1|_r\cdots|z_m-b_d|_r > r^d,$$ which, by Proposition \[maxmod\], contradicts the hypothesis that $|W|_r=r^d.$ Hence, there is some $b$ with $|b|\le r$ such that $W(0,\dots,0,b)=0,$ and hence $W$ is not a unit, as was to be shown.
I now state the important
\[wprep\] If an analytic function $f$ in $\mathcal{A}^m(r)$ is $z_m$-distinguished of degree $n,$ then there is a unique Weierstrass polynomial of degree $n$ and a unique unit $u$ in $\mathcal{A}^m(r)$ such that $f=uW.$
\[wpfactor\] Let $f_1,f_2 ,W\in\mathcal{A}^{m-1}(r)[z_m]$ such that $W=f_1f_2$ and such that $W$ is a Weierstrass polynomial. Then, there exist units $u_1$ and $u_2$ in $\mathcal{A}^{m-1}(r)$ such that $f_j/u_j$ are also Weierstrass polynomials.
This proof is similar to [@Krantz Lemma 6.4.8].
Let $d,$ $d_1$ and $d_2$ be the degrees of $W,$ $f_1,$ and $f_2$ respectively thought of as polynomials in $z_m.$ For $j=1,2,$ write $$f_j=A_{j,d_j}(z_1,\dots,z_{m-1})z_m^{d_j}+\dots+
A_{j,0}(z_1,\dots,z_{m-1}).$$ Then, because $W$ is monic, $$z_m^d+\dots=\left(A_{1,d_1}z_m^{d_1}+\dots+
A_{1,0}\right)\cdot
\left(A_{2,d_2}z_m^{d_1}+\dots+
A_{2,0}\right),$$ and hence $A_{1,d_1}$ and $A_{2,d_2}$ are units in $\mathcal{A}^{m-1}(r),$ $$|A_{1,d_1}|_r\cdot|A_{2,d_2}|_r=1,$$ and $$\max_{\begin{array}{c}\scriptstyle 0\le i_1 \le d_1\\ \scriptstyle
0\le i_2 \le d_2\end{array}}
|A_{1,i_1}|_r\cdot|A_{2,i_2}|_r r^{i_1+i_2} \le r^d.$$ For $j=1,2,$ let $W_j=f_j/A_{j,d_j}.$ Then, $W_1$ and $W_2$ are monic and if $\{j,k\}=\{1,2\},$ we have $$\left|\frac{A_{j,i}}{A_{j,d_j}}\right|_r r^i =
|A_{j,i}|_r\cdot|A_{k,d_k}|_r \cdot r^i
\cdot\frac{r^{d_k}}{r^{d_k}} \le \frac{r^d}{r^{d_k}}=r^{d_j},$$ which precisely means that $W_1$ and $W_2$ are Weierstrass polynomials.
As the Weierstrass Preparation Theorem only applies to $z_m$-distinguished functions, we need to know that every function can be made to be $z_m$-distinguished after a simple change of variables. The standard reference [@BGR Prop. 5.2.4/2] uses a non-linear coordinate change, but a linear coordinate change will be more useful for our purposes here. Let be an $m-1$-tuple of elements $u_j$ in $\mathbf{F}$ with $|u_j|\le1.$ We consider the $\mathbf{F}$-algebra automorphism $\sigma_u$ of $\mathcal{A}^m(r)$ defined by $$\sigma_u(z_1,\dots,z_m)=
(z_1+u_1z_m,\dots,z_j+u_jz_m,\dots,z_{m-1}+u_{m-1}z_m,z_m).$$ The homomorphism $\sigma_u$ is easily seen to be an automorphism by observing that its inverse is given by $$\sigma_u^{-1}(z_1,\dots,z_m)=
(z_1-u_1z_m,\dots,z_j-u_jz_m,\dots,z_{m-1}-u_{m-1}z_m,z_m).$$
\[changevars\] Let $r,R\in|\mathbf{F^\times}|,$ with $r\le R,$ and let $$f(z)=\sum_\gamma a_\gamma z^\gamma \in \mathcal{A}^m(R)$$ be such that $f$ is not identically zero. Then for an $m-1$-tuple $u$ over a generic residue class, $f\circ\sigma_u$ is $z_m$-distinguished in $\mathcal{A}^m(r).$
I emphasize that because we can choose $u$ over a generic residue class, given any finite collection of functions $f_k$ and given any finite number of radii $r_\ell\in|\mathbf{F}^\times|$ with $r_\ell\le R,$ we can find an automorphism $\sigma_u$ so that the $f_k\circ\sigma_u$ are all simultaneously $z_m$-distinguished in each of the rings $\mathcal{A}^m(r_\ell).$ In fact, we can do this simultaneously for all $r\le R,$ but we will not need that.
Write $$f\circ\sigma_u(z_1,\dots,z_m)=\sum_{j=0}^\infty
B_j z_m^j.$$ Each $B_j$ is a power-series with integer coefficients in the $a_\gamma,$ in the $u_j,$ and in the variables $z_1,\dots,z_{m-1}.$ Those coefficients $a_\gamma$ which appear in $B_j$ are precisely those with where $|\gamma|\ge j$ and $\gamma_m\le j.$ Let $\mu$ be the largest multi-index in the graded lexicographical order such that $|a_\mu|r^{|\mu|}=|f|_r.$
Consider $j>|\mu|.$ In this case, all the coefficients $a_\gamma$ appearing in $B_j$ are such that Thus, for $j>|\mu|,$ $$|B_j|_rr^j \le
\sup_{\gamma}|a_\gamma|r^{|{\gamma}|} < |f|_r,$$ where the $\sup$ is taken over those $\gamma$ with $a_\gamma$ appearing in $B_j,$ all of which have graded lexicographical order greater than $\mu.$
For $j=|\mu|,$ note that any term appearing in $B_j$ that involves any of the variables $z_1,\dots,z_{m-1}$ will include a coefficient $a_\gamma$ with $\gamma$ greater that $\mu$ in the graded lexicographical ordering, and thus will have $$|a_\gamma| r^{|\gamma|} < |a_\mu| r^{|\mu|}.$$ On the other hand, one of the constant terms appearing in $B_j$ is $$a_\mu u_1^{\mu_1}\cdots u_{m-1}^{\mu_{m-1}}.$$ Thus, keeping in mind we are considering $j=|\mu|,$ $$|B_j|_r r^j = |a_\mu| r^{|\mu|} = |f|_r,$$ provided none of the other constant terms in $B_j$ reduce the norm of $B_j,$ and this is true for $u$ over a generic residue class. Also, because the norm of the constant term in $B_j$ dominates all the norms of the variable terms, $B_j$ is a unit in $\mathcal{A}^{m-1}(r)$ by Proposition \[unit\]. Note that here we only use the implication in Propostion \[unit\] that we have already proven.
For $j<\mu,$ we have $$|B_j|_rr^j \le
\sup_{\gamma}|a_\gamma|r^{|{\gamma}|}
\le |a_\mu|r^{\mu}=|f|_r,$$ where again the $\sup$ is taken over those $\gamma$ appearing in $B_j.$ Hence, we conclude that, for $u$ over a generic residue class, $|f\circ\sigma_u|_r = |f|_r$ and that $f\circ\sigma_u$ is $z_m$-distinguished in $\mathcal{A}^m(r).$
Recall that we are in the situation where $$u(z)=1+\sum_{|\gamma|\ge1}a_\gamma z^\gamma.$$ We need to show that if $$\label{supgeone}
\sup_{|\gamma|\ge1} |a_\gamma|r^{|\gamma|}\ge1,$$ then $u$ is not a unit. By Proposition \[changevars\], we may assume that $u$ is $z_m$-distinguished, and of positive degree by (\[supgeone\]). Theorem \[wprep\] then says that we can write $u=vW,$ where $v$ is a unit and $W$ is a Weierstrass polynomial of positive degree. Propostion \[Wnotunit\] then implies that $u$ is not a unit.
\[ufd\] The ring $\mathcal{A}^m(r)$ is factorial.
This was proven by Salmon in [@Salmon].
\[relprime\] Let $r<R$ with $r$ and $R$ in $|\mathbf{F}^\times|.$ Let $f_1$ and $f_2$ be analytic functions in If $f_1$ and $f_2$ are relatively prime in the ring $\mathcal{A}^m(R),$ they remain relatively prime when considered as elements of the bigger ring $\mathcal{A}^m(r),$ in other words when they are restricted to $\mathbf{B}^m(r).$
This is a standard argument. See, for instance [@Krantz Prop. 6.4.11], where the analogous result for germs of analytic functions on a domain in $\mathbf{C}^m$ is proven.
We can multiply by units and make changes of variables without changing the question as to whether two functions are relatively prime. Hence, using Proposition \[changevars\] (and the remark following it) and Theorem \[wprep\], we may assume without loss of generality that $f_1$ and $f_2$ are Weierstrass polynomials relatively prime in $\mathcal{A}^m(R),$ and that they are $z_m$-distinguished in $\mathcal{A}^m(r).$
We now claim that $f_1$ and $f_2$ are relatively prime in $\mathcal{A}^{m-1}(R)[z_m].$ The novice reader should think about why this is not an entirely trivial statement because although $\mathcal{A}^{m-1}(R)[z_m]$ is a smaller ring than $\mathcal{A}^m(R),$ there are also fewer units. Indeed, suppose there is a non-trivial common factor $h$ and functions $g_1$ and $g_2$ in $\mathcal{A}^{m-1}(R)[z_m]$ such that $f_j=hg_j.$ Then by Proposition \[wpfactor\], $h,$ $g_1$ and $g_2$ are also Weierstrass polynomials, up to units. By assumption $h$ is not a unit in $\mathcal{A}^{m-1}(R)[z_m],$ and hence is not of degree 0. Because, up to a unit, $h$ is a Weierstrass polynomial, this means $h$ is also not a unit in $\mathcal{A}^m(R)$ contradicting our original assumption that $f_1$ and $f_2$ are relatively prime in $\mathcal{A}^m(R).$
Now, by Gauss’s Lemma, $f_1$ and $f_2$ are relatively prime in $\mathcal{M}^{m-1}(R)[z_m].$ This is important, because $\mathcal{M}^{m-1}(R)[z_m],$ being a one-variable polynomial ring over a field, is a principal ideal domain. Hence, there exist $G_1$ and $G_2$ in $\mathcal{M}^{m-1}(R)[z_m]$ such that $$1=G_1 f_1 + G_2 f_2.$$ Clearing denominators, we find functions $h,$ $g_1$ and $g_2$ in $\mathcal{A}^{m-1}(R)$ such that $$h=g_1f_1+g_2f_2.$$
Finally, suppose that $f_1$ and $f_2$ are not relatively prime in $\mathcal{A}^m(r).$ Then, there is a non-trivial common factor $\bar f$ of $f_1$ and $f_2$ in $\mathcal{A}^m(r).$ Since $f_1$ is $z_m$-distinguished in $\mathcal{A}^m(r),$ we know by Theorem \[wprep\] that it is a unit times a Weierstrass polynomial in $\mathcal{A}^{m-1}(r)[z_m].$ Because $\bar f$ is a factor of $f_1,$ we can then use Proposition \[wpfactor\] to conclude that $\bar f$ is a unit times a Weierstrass polynomial. Thus, we might as well assume $\bar f$ is a Weierstrass polynomial. But $\bar f,$ being a common factor of $f_1$ and $f_2,$ divides $h,$ which does not depend on $z_m.$ Hence $\bar f$ has degree zero as a Weierstrass polynomial, and is therefore a unit in $\mathcal{A}^{m-1}(r).$
\[gcd\] Let $r<R$ with $r$ and $R$ in $|\mathbf{F}^\times|.$ Let $f_1,\dots,f_k$ be analytic functions in If $G$ is a greatest common divisor of the $f_j$ in $\mathcal{A}^m(R)$ and if $g$ is a greatest common divisor of the $f_j$ in $\mathcal{A}^m(r).$ Then considering $g$ and $G$ as elements of $\mathcal{A}^m(r),$ they differ by a unit in $\mathcal{A}^m(r).$
By induction, we need only consider the case $k=2.$ Clearly $G$ divides $g.$ By assumption $f_1/G$ and $f_2/G$ are relatively prime in $\mathcal{A}^{m}(R).$ By the proposition they remain relatively prime in $\mathcal{A}^m(r).$ Hence $g$ divides $G.$
Let $r<R$ with both $r$ and $R$ in $|\mathbf{F}^\times|.$ Let $P$ be an irreducible element in $\mathcal{A}^m(R).$ If we restrict $P$ to an element of $\mathcal{A}^m(r),$ one of three things can happen: $P$ may remain irreducible, $P$ may become a unit, or $P$ may become reducible. As an example of the second case, consider $P(z)=1-z$ in one variable. If $r<1<R,$ then $P$ is irreducible in $\mathcal{A}^1(R)$ but a unit in $\mathcal{A}^1(r).$ The third possibility is strictly a several variable phenomenon. For example, consider $P(z_1,z_2)=z_2^2-z_1^2(1-z_1).$ Then, $P$ is irreducible for $R$ large. However, for $r<1,$ we can find an analytic branch of $\sqrt{1-z_1},$ and hence $P$ factors as $$P(z_1,z_2)=(z_2-z_1\sqrt{1-z_2})(z_2+z_1\sqrt{1-z_1}).$$ However, we do have the following useful corollary.
\[irreducible\] Let $r<R$ be in $|\mathbf{F}^\times|.$ Let $f$ be an element of $\mathcal{A}^m(R).$ Let $q$ be an irreducible factor of $f$ in $\mathcal{A}^m(r).$ Then, up to multiplication by unit in $\mathcal{A}^m(R),$ there exists a unique irreducible factor $Q$ of $f$ in $\mathcal{A}^m(R)$ such that $q$ divides $Q$ in $\mathcal{A}^m(r).$ Moreover, $q$ divides $Q$ with exact multiplicity $1,$ and $Q$ divides $f$ in $\mathcal{A}^m(R)$ with the same exact multiplicity with which $q$ divides $f$ in $\mathcal{A}^m(r).$
Using Theorem \[ufd\], write $f=p_1^{d_1}\cdots p_s^{d_s}$ in $\mathcal{A}^m(r)$ and $f=P_1^{e_1}\cdots P_t^{e_t}$ in $\mathcal{A}^m(R),$ with the $p_i$ and the $P_j$ distinct irreducible elements. Without loss of generality, assume $q=p_1.$
I will first show that $q$ divides at most one $P_j$ in $\mathcal{A}^m(r).$ Since $P_j$ and $P_k$ are irreducible in $\mathcal{A}^m(R),$ they are relatively prime in $\mathcal{A}^m(R).$ If $q$ were to divide $P_j$ and $P_k$ with $k\ne j,$ then $P_j$ and $P_k$ would not be relatively prime in $\mathcal{A}^m(r),$ which would contradict Proposition \[relprime\].
Since $q$ is irreducible in $\mathcal{A}^m(r)$ and divides $f=P_1^{e_1}\cdots P_t^{e_t},$ it must clearly divide one of the $P_j,$ which, without loss of generality, we will assume is $P_1.$
It remains to check that $q$ divides $P_1$ with multiplicity $1,$ as this will then imply that $d_1=e_1.$ Because $P_1$ is not a unit and irreducible, there exists a $j$ with $1\le j \le m$ such that $\partial P_1/\partial z_j\not\equiv0.$ In characteristic zero, this follows from the fact that any non-constant analytic function has at least one partial derivative which does not vanish identically. In positive characteristic $p,$ if all the partial derivatives vanish identically, then the analytic function is a pure $p$-th power, and hence not irreducible. Since $P_1$ is irreducible, it must be relatively prime to $\partial P_1/\partial z_j.$ Again, by Propositon \[relprime\], $P_1$ and $\partial P_1/\partial z_j$ remain relatively prime in $\mathcal{A}^m(r).$ Thus, no irreducible element in $\mathcal{A}^m(r)$ can divide $P_1$ with multiplicity greater than one.
I now present an argument of Lütkebohmert [@Lu].
[\[lu\]]{} For $i=1,2,3,\dots,$ let $r_i$ be an increasing sequence of elements in $|\mathbf{F}^\times|$ such that $r_i\to\infty.$ Suppose that for each $i,$ we are given analytic functions $g_i$ in $\mathcal{A}^m(r_i)$ and for each $i<j,$ we are given units $u_{i,j}$ in $\mathcal{A}^m(r_i)$ such that in $\mathcal{A}^m(r_i)$ we have $$g_i = u_{i,j}g_j.$$ Then, there exists an entire function $G$ on $\mathbf{F}^m$ and units $v_i$ in $\mathcal{A}^m(r_i)$ such that $g_i=Gv_i$ in $\mathcal{A}^m(r_i).$
Since $g_i=Gv_i,$ we see that for $j\ge i,$ $$\label{eqeqn}
g_iv_i^{-1}=G=g_jv_j^{-1} \qquad\textnormal{in~}
\mathcal{A}^m(r_i),$$
If one of the $g_i$ is identically zero, then they all are, and we can clearly take $g\equiv0$ and $v_i\equiv1.$ Thus, we may assume that there exists a point $z_0$ in $\mathbf{B}^m(r_1)$ such that $g_1(z_0)\ne0,$ and hence $g_j(z_0)\ne0$ for all $j$ since $g_1$ and $g_j$ differ by a unit. Without loss of generality, we may adjust the $g_i$ by multiplicative constants so that $g_i(z_0)=1$ for all $i.$ This of course implies that $u_{i,j}(z_0)=1$ for all $i<j$ too. Now, expand $u_{i,i+1}$ as a power series about $z_0$ to get $$u_{i,i+1}(z)=1+\sum_{|\gamma|\ge1}a_\gamma(z-z_0)^\gamma,
\qquad\textnormal{with~}|a_\gamma|r_i^{|\gamma|} < 1
\textnormal{~for all~}\gamma,$$ by Proposition \[unit\]. Hence, for $j>i,$ $$|u_{j,j+1}-1|_{r_i} < \frac{r_i}{r_j}.$$ Fixing $i$ and letting $j\to\infty,$ we have $r_i/r_j\to0,$ and so we can use an infinite product to define a unit $v_i$ in $\mathcal{A}^m(r_i)$ by $$v_i=\prod_{k=i}^\infty u_{k,k+1}.$$ For $j>i,$ note that $$\begin{aligned}
g_jv_i&=g_j\prod_{k=i}^\infty u_{k,k+1}\\
&=g_j\left(\prod_{k=1}^{j-1}u_{k,k+1}\right)
\left(\prod_{k=j}^\infty u_{k,k+1}\right) = g_iv_j.\end{aligned}$$ Therefore, for all $i\le j,$ we have $$g_iv_i^{-1}=g_jv_j^{-1} \qquad\textnormal{in~}
\mathcal{A}^m(r_i),$$ which is precisely (\[eqeqn\]) and which also means that the $g_iv_i^{-1}$ converge to an entire function $G$ such that $G=g_iv_i^{-1}$ in $\mathcal{A}^m(r_i).$
Now using Lütkebohmert’s argument as presented in the appendix to [@CY], we get the key result of this note and what was needed in [@CY].
\[entire\] Greatest common divisors exist in the ring of entire functions on $\mathbf{F}^m.$ Moreover, if $G$ is the greatest common divisor of the entire functions $f_1,\dots,f_k$ in the ring of entire functions, then $G$ is also the greatest common divisor of $f_1,\dots,f_k$ in the ring $\mathcal{A}^m(r)$ for all $r\in|\mathbf{F}^\times|.$
It suffices to prove the theorem when $k=2.$
Let $f_1$ and $f_2$ be two entire functions on $\mathbf{F}^m.$ If $f_1$ is identically zero, then clearly $f_2$ is a greatest common divisor of $f_1$ and $f_2.$ Thus, we now assume $f_1$ is not identically zero.
Let $r_i\in|\mathbf{F}|$ for $i=1,2,3,\dots$ be an increasing sequence with $r_i\to\infty.$ Of course $f_1$ and $f_2$ are also elements of each of the factorial rings $\mathcal{A}^m(r_i).$ Hence, for each $i,$ there exist analytic functions $g_i$ in $\mathcal{A}^m(r_i)$ such that $g_i$ is a greatest common divisor of $f_1$ and $f_2$ in $\mathcal{A}^m(r_i).$ For any $i<j,$ by Corollary \[gcd\], there exists a unit $u_{i,j}$ in $\mathcal{A}^m(r_i)$ such that $$g_i=u_{i,j}g_j$$ in $\mathcal{A}^m(r_i).$ Now, let $v_i$ and $G$ be as in Lemma \[lu\]. Since $g_i=Gv_i,$ we see that $G$ and $g_i,$ differing by a unit, are both greatest common divisors of $f_1$ and $f_2$ in $\mathcal{A}^m(r_i).$ By Corollary \[gcd\], this also implies that $G$ is the greatest common divisor of $f_1$ and $f_2$ in $\mathcal{A}^m(r)$ for all $r$ in $|\mathbf{F}^\times|$ with $r\le r_i.$
It remains to show that $G$ is a greatest common divisor for $f_1$ and $f_2$ in the ring of entire functions on $\mathbf{F}^m.$ We first check that $G$ divides $f_1.$ Since $g_i$ is a factor of $f_1$ in $\mathcal{A}^m(r_i),$ there exist analytic functions $h_i$ in $\mathcal{A}^m(r_i)$ such that $f_1=g_ih_i.$ By (\[eqeqn\]), $h_iv_i$ converge to an entire function $H$ such that $f_1=GH,$ and hence $G$ is a factor of $f_1.$ Similarly, $G$ is a factor of $f_2,$ and so $G$ is a common factor.
Now let $g$ be any other entire common factor of $f_1$ and $f_2.$ Because $g_i$ is a greatest common factor in $\mathcal{A}^m(r_i),$ there exist analytic functions $\omega_i$ such that in $\mathcal{A}^m(r_i).$ Thus, $g_iv_i^{-1}=g\omega_iv_i^{-1}$ in $\mathcal{A}^m(r_i).$ Because $\mathcal{A}^m(r_i)$ is an integral domain, equation (\[eqeqn\]) implies that if $i\le j$, then $\omega_iv_i^{-1}=\omega_jv_j^{-1}$ in $\mathcal{A}^m(r_i),$ and so $\omega_iv_i^{-1}$ converges to an entire function $\Omega$ on $\mathbf{F}^m$ such that $G=g\Omega.$
A ring is factorial if each element in the ring can be uniquely written, up to a permuation and multiplication by units, as a finite product of irreducible elements. Although the ring of entire functions on $\mathbf{F}^m$ is plainly not factorial, I will conclude this note by showing that it is almost as good as factorial. Namely, any entire function can be written as a (possibly infinite) product of irreducible entire functions, and the irreducible factors and multiplicities in the product are unique, up to permutation and multiplication by units.
\[almostufd\] Let $f$ be a non-zero entire function on $\mathbf{F}^m.$ Then, there exists a countable index set $I,$ for each $i$ in $I,$ there exist irreducible elements $P_i$ in the ring of entire functions on $\mathbf{F}^m,$ and for each $i$ in $I,$ there exist natural numbers $e_i$ such that such that if $i\ne j$, then $P_i$ and $P_j$ are relatively prime, and such that $$f = \prod_{i\in I} P_i^{e_i}.$$ Moreover, if $J$ is a countable index set, if for each $j$ in $J,$ there are irreducible entire functions $Q_j,$ and if for each $j$ in $J,$ there are natural numbers $d_j$ such that $$f=\prod_{j\in J}Q_j^{d_j}$$ and such that for $i\ne j$ in $J,$ we have $Q_i$ and $Q_j$ relatively prime, then there is a bijection $\sigma:I\rightarrow J$ such that $P_i = Q_{\sigma(i)}$ and $e_i=d_{\sigma(i)}.$
I recall here that the meaning of the infinite products in Theorem \[almostufd\] is that the finite partial products converge to $f$ in $\mathcal{A}^m(r)$ for all $r\in|\mathbf{F}^\times|.$
I will begin with a proposition describing how to find the irreducible factors.
\[existence\] Let $f$ be a non-zero entire function on $\mathbf{F}^m.$ Let $r_i$ be an increasing sequence of elements of $|\mathbf{F}^\times|$ such that $r_i\to\infty.$ Let $p_{i_0}$ be an irreducible factor of $f$ in $\mathcal{A}^m(r_{i_0}).$ Then, up to multiplication by a unit, there exists a unique irreducible entire function $P$ such that $p_{i_0}$ divides $P$ and such that $P$ divides $f.$
I will begin by proving existence. By Corollary \[irreducible\], for each $i\ge i_0,$ there exists a unique irreducible factor $p_i$ of $f$ in $\mathcal{A}^m(r_i)$ such that $p_{i_0}$ divides $p_i$ in $\mathcal{A}^m(r_{i_0}).$ I now claim that if $j>i,$ then $p_i$ divides $p_j$ in $\mathcal{A}^m(r_i).$ Indeed, using Corollary \[irreducible\] again, there is a unique irreducible factor $q_j$ of $f$ in $\mathcal{A}^{m}(r_j)$ such that $p_i$ divides $q_j$ in $\mathcal{A}^m(r_i).$ But then $p_{i_0}$ divides $q_j$ in $\mathcal{A}^m(r_{i_0}),$ and so by uniqueness, $p_j=q_j.$ Now, for each $i$ and for each $j\ge i,$ the function $p_j$ is a factor of $f$ in $\mathcal{A}^m(r_i),$ and so a finite product of finitely many irreducible factors with bounded multiplicity. Hence, for each $i,$ there exists $J_i$ such that for all $j,k\ge J_i$ we have that $p_j$ and $p_k$ differ by a unit in $\mathcal{A}^m(r_i).$ For each $i,$ let $f_i$ be $p_j$ restricted to $\mathcal{A}^m(r_i)$ for some $j\ge J_i.$ Now, for each $j\ge i \ge i_0,$ we have units $u_{i,j}$ in $\mathcal{A}^m(r_i)$ such that $f_i = u_{i,j}f_j.$ By Lemma \[lu\], there exists an entire function $P$ and units $v_i$ in $\mathcal{A}^m(r_i)$ such that $f_jv_j^{-1}=P.$ I claim that $P$ is an irreducible entire function which divides $f$ and such that $p_{i_0}$ divides $P$ in $\mathcal{A}^m(r_{i_0}).$
To see that $P$ divides $f$ note that each $f_i$ divides $f$ in $\mathcal{A}^m(r_i).$ That means there exist functions $h_i$ in $\mathcal{A}^m(r_i)$ such that $f_ih_i=f.$ But then for $j\ge i,$ we have $Pv_ih_i=f_ih_i=f=f_jh_j=Pv_jh_j,$ which implies $h_iv_i=h_jv_j.$ Hence, $h_iv_i$ converges to an entire function $H$ such that $PH=f$ since $PH=Ph_iv_i=f$ in $\mathcal{A}^m(r_i)$ for all $i.$
Since $P$ restricted to $\mathcal{A}^m(r_{i_0})$ is $f_{i_0}v_{i_0}^{-1},$ clearly $p_{i_0}$ divides $P$ in $\mathcal{A}^m(r_{i_0}).$
To see that $P$ is irreducible, suppose that there exist entire functions $g$ and $h$ such that $P=gh.$ Since $p_{i_0}$ divides $P$ in $\mathcal{A}^m(r_{i_0}),$ we must have that $p_{i_0}$ divides $g$ or $h,$ so assume without loss of generality that it divides $g.$ But this implies that $p_i$ divides $g$ in $\mathcal{A}^m(r_i)$ for all $i\ge i_0,$ and hence $f_i$ divides $g$ for all $i\ge i_0.$ Thus, $P$ divides $g$ in $\mathcal{A}^m(r_i)$ for all $i\ge i_0.$ In other words, there exist $g_i$ in $\mathcal{A}^m(r_i)$ such that $g=Pg_i$ in $\mathcal{A}^m(r_i).$ But, $P=gh=Pg_ih,$ and so $g_ih=1$ for all $i\ge i_0.$ It then follows that $h$ is a unit in the ring of entire functions, and so $P$ must be irreducible.
Finally, it remains to check uniqueness. Let $P$ be as constructed above and suppose there is another irreducible entire function $Q$ such that $p_{i_0}$ divides $Q$ in $\mathcal{A}^m(r_{i_0})$ and such that $Q$ divides $f$ in the ring of entire functions. As above, since $p_{i_0}$ divides $Q,$ we have that $p_i$ divides $Q$ for all $i\ge i_0.$ Hence $f_i$ divides $Q$ for all $i\ge i_0.$ Hence $P$ divides $Q,$ in which case $P$ and $Q,$ both being irreducible, differ by a unit, as was to be shown.
For $k=1,2,\dots,$ let $r_k$ be an increasing sequence of elements in $|\mathbf{F}^\times|$ such that $r_k\to\infty.$ Since $f$ is not identically zero, let $z_0$ be an element of $\mathbf{B}^m(r_1)$ such that $f(z_0)\ne0.$ Without loss of generalizty, assume that $f(z_0)=1.$ We proceed to inductively construct a countable set $\mathcal{P}$ of ordered pairs $(P,e),$ where $P$ is an irreducible entire factor of $f$ and $e$ is a natural number. Start by setting $\mathcal{P}=\emptyset.$ Now we add to $\mathcal{P}$ as follows. Let $k$ be the smallest natural number such that there is an irreducible factor $p$ of $f$ in $\mathcal{A}^m(r_k)$ which does not divide any of the $P\in\mathcal{P}.$ By Proposition \[existence\], there exists a unique irreducible entire function $P$ such that $p$ divides $P$ in $\mathcal{A}^m(r_k)$ and such that $P$ divides $f.$ Since $f(z_0)\ne0,$ we also have $P(z_0)\ne0,$ and so we may assume, without loss of generalizty, that $P(z_0)=1.$ Now if $e$ is the multiplicity with which $p$ divides $f$ in $\mathcal{A}^m(r_k),$ for a reason similar to the analagous statement in Corollary \[irreducible\], $P$ divides $f$ with exact multiplicity $e$ in the ring of entire functions on $\mathbf{F}^m.$ Thus, add the ordered pair $(P,e)$ to the set $\mathcal{P},$ and repeat the process. As we have only countably many $r_k$ and only finitely many irreducible factors of $f$ in each $\mathcal{A}^m(r_k),$ this process will terminate with a countable set $\mathcal{P}.$
I claim that, up to a unit, $$f = \prod_{(P,e)\in\mathcal{P}} P^e.$$ Index the elements $(P_i,e_i)$ of $\mathcal{P}$ by a countable index set $I.$ Since any finite product $$\prod_{i=1}^s P_i^{e_i}$$ divides $f,$ we have entire functions $g_s$ such that $$f = g_s \prod_{i=1}^s P_i^{e_i}.$$ Also, for each $k,$ there exists $S_k$ such that for all $s\ge S_k,$ we have that $g_s$ is a unit in $\mathcal{A}^m(r_k).$ Since $g_s(z_0)=1,$ we conclude, by Propostion \[unit\], that for $k\ge j$ and for $s\ge S_k$ that $$|1-g_s|_{r_j} < \frac{r_j}{r_k},$$ which will tend to zero as $k$ tends to infinity. Since only finitely many of the $P_i$ are not units in $\mathcal{A}^m(r_j),$ there exists $s_0$ such that for $s> s_0,$ we have, again by Proposition \[unit\], that $|P_s|_{r_j}=1.$ Hence, we find that for $k\ge j$ and $s\ge S_k,$ that $$\left|f-\prod_{i=1}^sP_i^{e_i}\right|_{r_j}
= \left|\prod_{i=1}^{s_0}P_i^{e_i}\right|_{r_j}\cdot
\left|\prod_{i=s_0+1}^sP_i^{e_i}\right|_{r_j}\cdot
|g_s-1|_{r_j}
< \left|\prod_{i=1}^{s_0}P_i^{e_i}\right|_{r_j}\frac{r_j}{r_k}
\to 0,$$ and hence the product converges to $f,$ as was to be shown.
To show uniqueness, it suffices to show that if $Q$ is an irreducible entire function dividing $f,$ then $Q$ is, up to multiplication by a unit, equal to one of the $P_i$ constructed above. So, suppose $Q$ is an irreducible entire function dividing $f.$ Let $r_k$ be large enough so that $Q$ is not a unit in $\mathcal{A}^m(r_k).$ Then, there is an irreducible factor $p$ of $f$ in $\mathcal{A}^m(r_k)$ which divides $Q.$ By construction, there is a unique $P_i$ such that $p$ divides $P_i.$ Also, by construction (and by Proposition \[existence\]), $P_i$ divides $Q$ in the ring of entire functions since $p$ divides $Q.$ But since $P_i$ and $Q$ are both irreducible entire functions, we then have that $P_i$ and $Q$ differ by a unit, as was to be shown.
[\[M-F\]]{} S. Bosch, U. Güntzer and R. Remmert, [*Non-Archimedean Analysis,*]{} Springer-Verlag, 1984; MR0746961. W. Cherry and Z. Ye, Non-Archimedean Nevanlinna theory in several variables and the non-Archimedean Nevanlinna inverse problem, *Trans. Amer. Math. Soc.* **349** (1997), 5043–5071; MR1407485. A. Escassut, *Ultrametric Banach Algebras,* World Scientific, 2003; MR1978961. S. Krantz, *Function Theory of Several Complex Variables,* Second Edition, Wadsworth & Brooks/Cole, 1992; MR1162310 M. Lazard, Les zéros des fonction analytiques d’une variable sur un corps valué complet, *Inst. Hautes Études Sci. Publ. Math.* **14** (1962), 47–75; MR0152519. W. Lütkebohmert, Letter to R. Remmert, 1995. P. Salmon, Sur les séries formelles restreintes, *C. R. Acad. Sci. Paris* **255** (1962), 227–228; MR0144924.
[^1]: Partial financial support provided by the National Security Agency under Grant Number H98230-07-1-0037. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.
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{
"pile_set_name": "ArXiv"
}
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---
author:
- 'Bart van Ginkel[^1]'
- 'Frank Redig[^2]'
bibliography:
- 'refs.bib'
date: ' TU Delft\'
title: Equilibrium fluctuations for the Symmetric Exclusion Process on a compact Riemannian manifold
---
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors thank Richard Kraaij for helpful discussions and for pointing out the reference [@jakubowski1986skorokhod]. The support of the grant 613.009.112 of the Netherlands Organisation for Scientific Research (NWO) is gratefully acknowledged.
[^1]: [email protected]
[^2]: [email protected]
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{
"pile_set_name": "ArXiv"
}
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---
author:
- 'M. Büttiker and T. Christen'
title: Basic Elements of Electrical Conduction
---
Introduction
============
Many phenomena of electrical conduction in small mesoscopic conductors have been successfully explained within the framework of the scattering approach [@Review]. The main emphasis of this work is an extension of this approach to time-dependent phenomena. Of particular interest are the basic requirements which a dynamic conductance of a mesoscopic conductor has to satisfy. There is a gap in the way dynamic conduction is treated in a large body of the physics literature and in more applied discussions based on simple circuit theory. For an electric network we know that its time-dependent behavior is crucially determined by its capacitive and inductive elements. In contrast to this, time-dependent phenomena are often treated by a response theory of non-interacting carriers. Unfortunately, this approach to ac-conductance which has been employed by some of the leading condensed matter physicists [@MOTT] is prevalent. We emphasize that in order to obtain a reasonable answer it is not sufficient to consider non-interacting electrons, a Fermi liquid or even a Luttinger liquid with short range interactions. An analysis based on an electric circuit model gives an answer which conserves the total charge and which has the property that all frequency dependent currents at the input and output nodes of such a circuit add up to zero. To obtain results which are [*charge and current conserving*]{}, it is necessary to consider the implication of the long range Coulomb interaction. This leads to a theory which can be considered to be an extension of the work of Pines and Nozières [@PINES] and Martin [@MARTIN] on bulk systems to mesoscopic conductors. Below we present a discussion [@MBCAP] which for simplicity focuses on the low-frequency transport. The benefit of this restriction is its considerable generality, and that it can be used to analyze the self-consistency of the dc scattering-approach (Sect. \[potential\]), the leading order nonlinearities of the current voltage characteristic (Sect. \[nonlinear\]), and the low-frequency dynamical conductance (Sect. \[frequency\]).\
For a two-terminal conductor the low-frequency admittance is of the form $$\begin{aligned}
G(\omega) = G(0)-i\omega E_{\mu} + O(\omega^{2})\;\; .
\label{eqI1}\end{aligned}$$ where $G(0)$ is the dc-conductance and $E_{\mu}$ is called the [*emittance*]{} [@MBCAP]. The emittance describes the current response in the leading order with respect to frequency and can be associated with the displacement charge passing a contact. For conductors with poor transmission as, e.g., a condenser or a metallic diffusive wire, the emittance is positive: the current follows the voltage as is characteristic of a capacitive response. On the other hand, it turns out that the emittance is negative in samples with good transmission. The current leads the voltage as is characteristic of circuits with an inductance.
To illustrate the physics contained in this response coefficient $E_{\mu}$, we present now a few results. Their detailed derivation is presented in Sect. \[examples\].\
Consider first the case of a mesoscopic condenser [@WEISS; @BTPC] as shown in Fig. 1 with a geometrical capacitance $C_{0}$. The measured electrochemical capacitance $C_{\mu}$ is determined by the geometrical capacitance in series with quantum capacitances $e^{2} dN_{1}/dE$, $e^{2} dN_{2}/dE.$ Here $dN_{1}/dE$ and $dN_{2}/dE$ are the total densities of states of the mesoscopic plates in the regions into which the electrical field penetrates. Thus the electrochemical capacitance of a mesoscopic condenser is [@BTPC] $$\begin{aligned}
C_{\mu}^{-1} =
C^{-1}_{0} + (e^{2} dN_{1}/dE)^{-1}+ (e^{2}
dN_{2}/dE)^{-1}.
\label{cmu}\end{aligned}$$ As it must be, in the macroscopic limit, where the densities of states diverge, one finds $E_{\mu}=C_{0}$. On the other hand, if the density of states at the surfaces of the plates are sufficiently small, the geometrical capacitance can be neglected. For the symmetric case this leads to the emittance $E_{\mu}= (1/4)e^{2}(dN/dE)$, where $dN/dE = dN_{1}/dE+ dN_{2}/dE$ is the total density of states of the two surfaces.\
Next consider a sample which permits transmission of carriers between the contacts. To be definite, first consider a metallic diffusive wire connecting two reservoirs (see Fig. 2). Experiments have been carried out by Pieper and Price [@PIEP1] and related theoretical work can be found in Refs. [@PIEP2; @LIU; @YSHAI; @KRAMER]. We determine the local potential with the help of Thomas-Fermi screening (a local charge-neutrality condition). In the presence of an applied dc-voltage, the ensemble averaged potential drops linearly. The total density of states in the volume over which the electric field is non-vanishing is denoted by $dN/dE$. We find that the emittance of such a metallic diffusive wire is positive like a capacitance and is given by [@MBUNP] $$\begin{aligned}
E_{\mu} = (1/6) e^{2} (dN/dE)\;\; .
\label{eqI2}\end{aligned}$$ It is not the total density of states that counts! Without screening and in the presence of a dc potential-drop, one-half of the states would be filled. Screening, however, reduces the naively expected result $(1/2) (dN/dE)$ by a factor $1/3$. Such a $1/3$-reduction is familiar from the theory of shot noise of metallic diffusive conductors were it is found that the actual shot noise is $1/3$ of the full Poisson noise [@BEE13; @NAGA; @DEJON]. The physical origin of the factor 1/3 in these two problems is quite different but from a purely mathematical point of view the origin [@NAGA] of the factor 1/3 is the same.\
Next consider a ballistic wire of length L. Now, for dc transport the ensemble-averaged potential drops as we enter the wire from the reservoir, it is constant in the wire, and it drops again as we leave the wire. We find then a negative (inductive) emittance $$\begin{aligned}
E_{\mu} = - (1/4) e^{2} (dN/dE)\;\;,
\label{eqI3}\end{aligned}$$ where $dN/dE$ is the total density of states of the wire in the volume where the potential is uniform.\
As a further example, consider a resonant double barrier [@PRICE; @FU; @MBHTP]. Suppose for simplicity that it is symmetric and suppose that it is reasonable to determine the potential in the well from a charge neutrality condition. If we denote by $T$, $R=1-T$, and $dN/dE$ the transmission probability, the reflection probability, and the density of states in the well, respectively, the emittance can be written in the form [@MBHTP; @CHRIS] $$\begin{aligned}
E_{\mu} = (1/4) (R-T) e^{2}(dN/dE)\;\;.
\label{eqI4}\end{aligned}$$ Note that Eq. (\[eqI4\]) interpolates between the emittance of a symmetric condenser with large geometrical capacitance and a ballistic wire. The emittance is negative (inductive) at resonance $T=1$ and crosses zero at $T = R = 1/2$ where the Fermi level is a half-width of the resonant level above (or below) the energy of the resonant state. If the Fermi level is further than a half-width away from the resonant level the emittance is positive (capacitive). The case of a double barrier which is asymmetric has been considered in Ref. [@MBHTP]. A result for single tunneling barriers which goes beyond Thomas-Fermi screening will be discussed elsewhere [@CHRIS].\
Finally, consider a two-dimensional electron gas patterned into the shape of a Corbino disk (Fig. 3a) or a Hall bar (Fig. 3b) with two metallic contacts. In the range of magnetic fields over which the Hall conductance is quantized the dc-resistances can be evaluated solely by considering edge states [@MB88]. In a non-interacting theory the edge states intercept the Fermi energy with a finite slope and have a local density of states per unit length given by $dn(s)/dE =1/hv(s)$, where $s$ is a coordinate along the edge state and where $v(s)$ is the velocity of carriers at point $s$. The integrated density of states along the edge channel $k$ is thus $dN_{k}/dE = \int ds (dn_{k}/dE)$. Suppose for simplicity that the magnetic field is such that we have only one pair of edge states. The Corbino disk acts then as an insulator with vanishing dc conductance $G(0) = 0$. The long-range Coulomb interaction between the inner and the outer edge state can be described by a geometrical capacitance $C_{0}$ (which depends logarithmically on the width of the sample). As one expects, we find that the Corbino-disk exhibits an emittance which equals the electrochemical capacitance [@TCMB95] $E_{\mu} = C_{\mu}$, where $C_{\mu}$ is given by Eq. (\[cmu\]). A topologically different conductor of the same width and length but with its contacts arranged at its ends is shown in Fig. 3b. Now the two-terminal dc-conductance is quantized and given by $G(0) = e^{2}/h$ for spin-split Landau levels. Obviously, the low frequency response of this conductor is not capacitive but dominated by transmission of carriers between different reservoirs. In particular, we find the inductive-like emittance [@TCMB95] $E_{\mu}=-C_{\mu}$. Thus the ac-response of a quantized Hall conductor is determined by the way the edge states are connected to reservoirs. In the Corbino disk each edge state returns to the reservoir from which it emanates. In contrast, in the two-terminal Hall-bar, each edge state connects to a reservoir that is different from the one from which it emanates. We do not discuss this example anymore in the sequel. A forthcoming publication [@TCMB95] presents a general formula for the emittance which is applicable to a wide variety of samples with different edge-state topologies.
The Scattering Approach {#transmission}
=======================
Consider a conductor connected to a number of contacts [@MB86]. The contacts are labeled with the greek indices $\alpha$. We assume that the distance between these contacts is so small that transmission from one contact to another one can be considered to be phase coherent. Thus we assume that scattering inside the conductor is purely elastic. For small deviations of the electrochemical potentials away from their equilibrium value the dc-current $I_{\alpha}$ at probe $\alpha$ is given by [@MB86] $$\begin{aligned}
I_{\alpha} = (e/h) \int dE (-\frac{df}{dE})
\left( (N_{\alpha} - R_{\alpha \alpha}) \mu_{\alpha}
- \sum_{\beta} T_{\alpha \beta} \mu_{\beta} \right) .
\label{eq01}\end{aligned}$$ Here, $N_{\alpha}$ is the number of quantum channels with thresholds below the equilibrium electrochemical potential in contact $\alpha$, and $f$ is the Fermi function. Carriers in the $N_{\alpha}$ incident channels have a total combined probability $R_{\alpha \alpha}$ for reflection back into contact $\alpha$. Carriers incident in contact $\beta$ have a total probability $T_{\alpha \beta}$ to traverse the sample into contact $\alpha .$ Equation (\[eq01\]) is a quantum-mechanical Kirchhoff law. It states the conservation of the current at an arbitrary intersection of a mesoscopic wire and that the currents are only a function of voltage differences. In the present context these features are a consequence of the unitarity of the scattering matrix ${\bf S}$ and its behavior under time reversal. The fact that the scattering matrix is unitary and that the microscopic equations are reversible implies that under a reversal of magnetic field the scattering matrix has the symmetry ${\bf S}^{T}(B) = {\bf S} (-B)$. The scattering matrix ${\bf S}$ for the conductor can be arranged such that it is composed of sub-matrices ${\bf s}_{\alpha \beta }$ which relate the incident current amplitudes in probe $\beta$ to the outgoing current amplitudes in probe $\alpha$. In terms of these scattering matrices the probabilities introduced above are simply $T_{\alpha \beta} = Tr({\bf s}^{\dagger}_{\alpha \beta }
{\bf s}_{\alpha \beta })$ if $\alpha \ne \beta$ and $R_{\alpha \alpha} = Tr({\bf s}^{\dagger}_{\alpha \alpha}
{\bf s}_{\alpha \alpha})$. The trace is over the quantum channels in each probe. As a consequence of the unitary properties of the scattering matrix, we have $N_{\alpha} =
R_{\alpha \alpha} +\sum_{\beta} T_{\alpha \beta} $ and $N_{\alpha} = R_{\alpha \alpha} +\sum_{\beta} T_{\beta \alpha}$ which imply current conservation $\sum_{\alpha} I_{\alpha} = 0$. Furthermore, the currents depend only on differences of the electrochemical potentials.
Let us restate these properties in terms of conductance coefficients $G_{\alpha\beta} = I_{\alpha} /V_{\beta}.$ Taking into account that in a reservoir the voltages and electrochemical potentials move in synchronism, $\delta \mu_{\alpha} =eV_{\alpha}$, we have [@MB86] $$\begin{aligned}
G_{\alpha\alpha}(0) =\frac{e^{2}}{h}\int dE (- \frac{df}{dE})
(N_{\alpha} - R_{\alpha \alpha})
= \frac{e^{2}}{h} \int dE (- \frac{df}{dE})
\sum_{\beta} T_{\alpha \beta} \;\;.
\label{eq02}\end{aligned}$$ for $ \alpha = \beta $ and $$\begin{aligned}
G_{\alpha\beta} (0) = - \frac{e^{2}}{h} \int dE (- \frac{df}{dE})
\: T_{\alpha \beta} \;\;.
\label{eq03}\end{aligned}$$ Current conservation means for the conductance matrix $$\begin{aligned}
\sum_{\alpha} G_{\alpha\beta}(0) = 0
\label{eq04}\end{aligned}$$ for any $\beta$. On the other hand, the statement that the currents can only depend on voltage differences leads to $$\begin{aligned}
\sum_{\beta} G_{\alpha\beta}(0) = 0\;\;.
\label{eq05}\end{aligned}$$ We emphasize the simple properties (\[eq04\]) and (\[eq05\]) since later on we point out that they are also valid for $G_{\alpha\beta}(\omega).$
Before proceeding we briefly comment on closely related work. In his work Landauer determined the voltage drop across an obstacle with the help of a local charge neutrality argument applied to the perfect sections of the conductor on either side of an obstacle [@LA57]. For an obstacle with transmission probability T, this leads to a resistance proportional to $(1-T)/T$. This resistance has the obviously correct property that it vanishes in the limit of perfect transmission. Already Engquist and Anderson pointed out that voltages measured at contacts are determined not by a charge neutrality condition but by adjusting the electrochemical potential of a voltage probe in such a manner that the voltmeter draws zero net current [@EA81]. Due to the special geometry considered, the restriction to a one channel conductor, and the restriction to a phase-incoherent voltage measurement [@MB89], the answer found by Engquist and Anderson is the same as that found by Landauer. The key notion that the resistance across an obstacle should vanish if the obstacle permits perfect transmission persisted long after the work of Engquist and Anderson. The task seemed, therefore, to be to find resistance formulae [@AZ81] of the type $(1-T)/T$. In a work that largely centers on this notion, Imry [@IM86] observed that if one considers in the discussion of Engquist and Anderson not the voltage drop across the obstacle but the voltage drop between the current source and sink one obtains a resistance that is proportional to $T^{-1}$, i. e. a conductance that is proportional to $T$. Equation (\[eq01\]), in contrast, was motivated by experiments by Benoit et al. [@BE86] in which the role of the current and voltage source was exchanged. Such experiments imply that we need a formulation of electrical transport in which apriori all contacts are treated equivalently and on an equal footing [@MB86]. In a specific arrangement of current and voltage sources the voltages are a posteriori determined by a zero current condition. This leads to conductances which are electrochemical quantities and which obey the basic requirements of a transport theory. These basic requirements include a reciprocity symmetry [@MB86; @BE86; @VANH] (which is a consequence of microreversibility and the irreversibility of thermodynamic electron reservoirs) and include a fluctuation-dissipation theorem [@MB92]. Below we discuss in more detail the electrochemical requirements which lead to Eq. (\[eq01\]).
It is sometimes implied in the theoretical literature, that the derivation of conductance formulae is just a question of applying formal linear response theory correctly to this problem. This is not borne out by the history of this field: Depending on one’s preconceived notions quite different results can be derived [@THOU]. But it is certainly true, that the derivation of these results from linear response has made them more acceptable. This is, however, neither a question of rigour nor depth but simply a consequence of the fact that even simple results are often only accepted if they are embellished by a sufficiently complicated derivation. For very concise linear response discussions of Eq. (\[eq01\]) and some still open related questions we refer the reader to Ref. [@SHEP; @MBAPT; @NOECK].
Potential Distribution in a Mesoscopic Conductor {#potential}
================================================
The scattering approach as discussed above seems like a simple black-box approach. If we know through which contact carriers leave the sample and know the current that is injected by a reservoir we can find the total currents. The issue which is not trivial is the fact that the voltages at the contacts must be well-defined in order for Eq. (\[eq02\]) and Eq. (\[eq03\]) to be the correct answer. In many articles on the subject one finds a conductance attributed to a wire which is strictly one-dimensional, or one finds pictures of mesoscopic conductors with leads that are narrower then the mesoscopic sample itself and are called reservoirs. Such geometries, as we now show, do in fact not lead to the conductances given above. This criticism is not novel: it has been aired in a number of papers by Landauer [@LA87] and one of the authors [@MB92] - unfortunately without much success. The following discussion is most closely related to work by Levinson [@LE89] but in detail follows Ref. [@MBCAP].\
For Eqs. (\[eq02\]) and (\[eq03\]) to be valid, the electric potential $U_{\alpha}$ in the reservoirs $\alpha $ must follow the electrochemical potential $\mu_{\alpha}$ in this contact. To generate transport we must consider a non-equilibrium situation. The electrochemical potentials in the contacts $\mu_{\alpha}$ must be allowed to differ from the equilibrium chemical potential $\mu _{0}$. Suppose that the increment of the electrochemical potential in contact $\alpha$ is $\delta \mu_{\alpha} = \mu_{\alpha} - \mu _{0}$. The electric potential changes from its equilibrium configuration $U_{eq}({\bf r})$ to a new non-equilibrium configuration $U([\mu_{\alpha}],{\bf r})$. Here the argument $[\mu_{\alpha}]$ indicates that the non-equilibrium potential depends on the electrochemical potentials in the contacts. Equations (\[eq02\]) and (\[eq03\]) presuppose that deep in contact $\alpha$ the electrochemical and electric potential change in synchronism, i. e. $\delta \mu_{\alpha} = e\delta
U([\mu_{\alpha}],{\bf r})$. This is a consequence of the charge neutrality deep in the reservoirs. The difference between the electrochemical potential and the electrostatic potential is the Fermi energy, $E_{F\alpha} ({\bf r}) = \mu_{\alpha} -eU({\bf r})$. This chemical potential $E_{F\alpha}$ determines the charge in the neighborhood of ${\bf r}$. Deep in the reservoir the charge cannot change, even if we bias the conductor. Consequently we have $0\equiv \delta E_{F\alpha} ({\bf r}) = \delta \mu_{\alpha} -e
\delta U([\mu_{\alpha}],{\bf r})$. Note that we have taken here the chemical potential to be space depended. However, if the quantum coherence of the wave functions is taken serious then also deep in the reservoir, i. e. in the wide wire, a single energy $E_{F\alpha}$ is all that is needed to specify the chemical potential. The energy dispersions in a wide wire are of the type $E_{\alpha n}(k) = \hbar^{2} k^{2}/2m + E_{\alpha n}^{0} + eU_{\alpha} $. Here $\hbar^{2} k^{2}/2m$ is the longitudinal energy for motion along the wire, $E_{\alpha n}^{0}$ is the energy for transverse motion (the channel threshold) and $eU_{\alpha}$ is the equilibrium potential (the bottom of the conduction band) in contact $\alpha$. The energy dispersion depends in an explicit manner only on a single spatially independent constant $eU_{\alpha}$. Instead of the spatially dependent relationship between the electrochemical, chemical, and electrostatic potentials we have for coherent conductors the relation $E_{F\alpha} = \mu_{\alpha} -eU_{\alpha} $. In either case the differential relationship is spatially independent in the lead. In order to discuss the validity of Eqs. (\[eq01\])-(\[eq03\]), we have thus to find the electrostatic potential and the conditions under which it changes synchronously with the electrochemical potential in the reservoirs.
Characteristic Potentials {#characteristic}
-------------------------
The electrostatic potential $U([\mu_{\alpha}], {\bf r})$ for mesoscopic conductors is a function of the electrochemical potentials of the contacts, and a complicated function of position. Small increases in the electrochemical potentials $\delta \mu_{\alpha}$ bring the conductor to a new state (see Fig. 4) with an electrostatic potential $U([\mu_{\alpha}+\delta \mu_{\alpha}],{\bf r})$. The difference $\delta U$ between these two potentials can be expanded in powers of the increment in the electrochemical potential. To linear order we have $$\begin{aligned}
e\: \delta U([\mu_{\alpha}], {\bf r}) =
\sum_{\alpha} u_{\alpha}({\bf r}) \: \delta \mu_{\alpha}\;\;.
\label{eq06}\end{aligned}$$ Here, $u_{\alpha}({\bf r}) =$ $e \: \partial U([\mu_{\alpha}], {\bf r})/
\partial \mu_{\alpha}|_{\mu_{\alpha} = \mu_{0}}$, with $\alpha = 1, 2$ are the [*characteristic potentials*]{} [@MBCAP] which determine the electrostatic potential inside the sample in response to a variation of an electrochemical potential at a contact.\
Suppose for a moment that we increase all electrochemical potentials simultaneously and by the same amount, $\delta \mu _{\alpha }\equiv
\delta \mu$. Both before and after the change the conductor is at equilibrium, hence the physical properties of the conductor remain unchanged.
Consequently, the shift of the electrochemical potentials must be accompanied by a shift $e\delta U \equiv \delta \mu$ of the electrical potential. Imposing this condition on Eq. (\[eq06\]) implies that the sum of all characteristic potentials is equal to one at every space point [@MBCAP], $$\begin{aligned}
\sum_{\alpha} u_{\alpha}({\bf r}) \equiv 1\;\;.
\label{eq07}\end{aligned}$$ Equation (\[eq07\]) is a consequence of the long-range Coulomb interaction. It is the most important result of this work: the conservation of charge under the application of a dc or ac bias and the conservation of current are, as we will show, a consequence of Eq. (\[eq07\]).\
Let us now return to our original problem and consider what happens if we increase just one electrochemical potential, say in reservoir $\alpha $, by $\delta \mu _{\alpha }$. Obviously, the condition that the electrochemical potential and the electrostatic potential move in synchronism deep inside reservoir $\alpha$ implies that the characteristic function $u_{\alpha}({\bf r})=1$ for ${\bf r}$ deep inside reservoir $\alpha$. Together with Eq. (\[eq06\]), this implies that Eqs. (\[eq01\])-(\[eq03\]) are valid if and only if the characteristic potentials have the property that $u_{\alpha}({\bf r}) = 1$ for ${\bf r}$ deep in contact $\alpha$ and $u_{\alpha}({\bf r}) = 0$ for ${\bf r}$ deep in any other contact.\
The electrostatic potentials are determined by the charge distribution in the sample. As we increase the chemical potential of contact $\alpha$ [*keeping all electrostatic potentials fixed*]{}, the additional charge $$\begin{aligned}
\delta n({\bf r}) =
(dn({\bf r}, \alpha)/dE) \: \delta \mu_{\alpha}
\label{eq08}\end{aligned}$$ enters the conductor. Here, $dn({\bf r}, \alpha)/dE$ is the [*injectivity*]{} of contact $\alpha$ into point ${\bf r}$ of the sample. With the help of the scattering states $\Psi_{\alpha n}({\bf r})$ which have unit amplitude in the incident channel $n$ in lead $\alpha$, the injectivity can be expressed as $$\begin{aligned}
dn({\bf r}, \alpha)/dE = \sum_{n} (hv_{\alpha n})^{-1}
|\Psi_{\alpha n}({\bf r})|^{2}
\label{eq09}\end{aligned}$$ where $v_{\alpha n}$ is the velocity of carriers at the Fermi energy in channel $n$ in contact $\alpha$. Equation (\[eq08\]) gives of course not the true density variation. The injected charges induce a change in the electrostatic potential which in turn implies an induced contribution to the density, $\delta
n_{ind}$, which has to be determined self-consistently. The total charge density is $$\begin{aligned}
\delta n({\bf r}) =
(dn({\bf r}, \alpha)/dE)\: \delta \mu_{\alpha} + \delta n_{ind}({\bf r})\;\;.
\label{eq10}\end{aligned}$$ The induced charge density is connected to the electrostatic potential via the response function $\Pi({\bf r},
{\bf r}^{\prime})$ (Lindhard function): $$\begin{aligned}
\delta n_{ind} ({\bf r}) = - \int d^{3}r^{\prime} \:
\Pi({\bf r}, {\bf r}^{\prime})\: e \delta U({\bf r}^{\prime})\;\;.
\label{eq11}\end{aligned}$$ The response function can be expressed in terms of the scattering states (Green’s function of the Schrödinger equation). Note that the Lindhard function describes the variation of the charge density not only of the mobile electrons which can be reached from the contacts but also of the localized states. For the purpose of our discussion we simply assume that this response function has been calculated and is known. There is one property of the Lindhard function which is needed later on and which is a simple consequence of the invariance of the electrical system under a global potential shift. A simultaneous change in all electrochemical potentials injects a charge $$\begin{aligned}
\delta n({\bf r}) = \sum_{\alpha}
(dn({\bf r}, \alpha)/dE) \delta \mu + \delta n_{ind}({\bf
r})\equiv 0 \;\;.
\label{eq12}\end{aligned}$$ Taking into account Eq. (\[eq11\]) and $e\delta U = \delta \mu$, one concludes [@MBCAP] $$\begin{aligned}
dn({\bf r})/dE = \int d^{3}r^{\prime} \;\;
\Pi({\bf r}, {\bf r}^{\prime}) \;\;.
\label{eq13}\end{aligned}$$ The quantity on the left hand side side of Eq. (\[eq13\]), $$dn({\bf r})/dE= \sum_{\alpha}
(dn({\bf r}, \alpha)/dE)\;\;,
\label{ldos}$$ is the local density of states which equals the sum of all injectivities. Equation (\[eq13\]) connects a chemical response quantity, the local density of states, to the Lindhard function $\Pi$. It is in this regard similar to the Einstein relation between a diffusion constant and a conductivity.\
Now we come back to the case where the voltage is changed only in contact $\alpha $. By inserting Eq. (\[eq10\]) into Poisson’s equation and taking $e \delta U({\bf r})
= u_{\alpha}({\bf r}) \delta \mu_{\alpha}$ into account, we find that the characteristic potential $u_{\alpha}({\bf r})$ is the solution of a field equation with a non-local screening kernel and a source term given by the [*injectivity*]{} of contact $\alpha$, $$\begin{aligned}
-\Delta u_{\alpha} ({\bf r}) + 4 \pi e^{2}
\int d^{3}r^{\prime}
\Pi ({\bf r}, {\bf r}^{\prime})
u_{\alpha} ({\bf r}^{\prime}) =
4 \pi e^{2} (dn({\bf r},\alpha )/dE) \;\;.
\label{eq14}\end{aligned}$$ We define the Green’s function $g({\bf r}, {\bf r}_{0})$ as the solution of Eq. (\[eq14\]) with the source term $e\: dn({\bf r},\alpha)/dE$ replaced by a localized test charge $ e\delta({\bf r}-{\bf r}_{0})$ at point ${\bf r}_{0}$. The characteristic potential $u_{\alpha}({\bf
r}) $ can then be written in the form $$\begin{aligned}
u_{\alpha} ({\bf r}) =
\int d^{3}r^{\prime} \: g ({\bf r}, {\bf r}^{\prime})\:
(dn({\bf r}^{\prime},\alpha )/dE) \;\;.
\label{eq15}\end{aligned}$$ Using Eqs. (\[eq07\]) and (\[ldos\]), a summation over $\alpha $ implies for Green’s function the property [@MBCAP] $$\begin{aligned}
\int d^{3}r^{\prime} g ({\bf r}, {\bf r}^{\prime})
\sum_{\alpha} (dn({\bf r}^{\prime},\alpha )/dE) =
\int d^{3}r^{\prime} g ({\bf r}, {\bf r}^{\prime})\:
(dn({\bf r}^{\prime})/dE ) \equiv 1 \;\;.
\label{eq16}\end{aligned}$$ The same relationship follows from the condition that the sum of all induced charge densities plus the test charge is zero.\
Now we find the condition for the electrical self-consistency of Eqs (\[eq01\]). According to Eq. (\[eq16\]) the characteristic potential is equal to unity if the Green’s function is convoluted with the local density of states. Therefore, we must have that the injectivity $dn({\bf r},\alpha )/dE$ deep in contact $\alpha$ is equal to the local density of states $dn({\bf r})/dE$. This requires that nearly all (in a thermodynamic sense) electrons approaching the contact $\alpha$ be reflected into the reservoir. If the conductor and the reservoir consist of the same material then the reservoir must be wide compared to the mesoscopic conductor. In semiconductor samples with metallic contacts, on the other hand, the contact might be actually narrow compared to the dimensions of the semiconductor since the density of states at the Fermi energy of the metal is much larger then that of the semiconductor. This is the case, for instance, in Ga/As-samples used to measure the quantized Hall effect [@MB88]. Our emphasis that self-consistency requires geometries which are of a wide-narrow-wide geometry deserves further discussion: the notion that a portion of length $L$ of a purely one-dimensional conductor has a conductance $G = (e^{2}/h) T$ (per spin) seems to be widely accepted. In contrast, from the point of view taken here a strictly one-dimensional conductor cannot be characterized by a conductance.
Charge Conservation {#conservation}
-------------------
Let us now demonstrate that if the conditions of self-consistency are met, then the application of an external bias to the conductor preserves the total charge in the system. To be more precise, imagine a volume $\Omega$ which encloses the entire conductor including a portion of the reservoirs which is so large that at the place were the surface of $\Omega$ intersects the reservoir all the characteristic potentials are either zero or unity. We demonstrate charge conservation for the case that we increase $\mu_{1}$ by a small value $\delta \mu_{1}$ above the equilibrium chemical potential. The change in density is given by Eq. (\[eq10\]). Expressing the induced charge with the help of the Lindhard function $\Pi({\bf r},{\bf r^{\prime}})$ and using the Green’s function to relate the characteristic potentials to the injected density gives $$\begin{aligned}
\delta n({\bf r}) = \left(
\frac{dn({\bf r}, 1)}{dE} -
\int d^{3}r^{\prime} d^{3}r^{\prime \prime}
\Pi({\bf r}, {\bf r}^{\prime}) g ({\bf r}^{\prime}, {\bf r}^{\prime \prime})
\frac{dn({\bf r}^{\prime \prime},1 )}{dE} \right) \delta \mu_{1}
\label{eq17}\end{aligned}$$ The total variation in charge is $\int_{\Omega} d^{3}r \: \delta n({\bf r}) =$ $$\begin{aligned}
\int_{\Omega} d^{3}r^{\prime}
\left( \frac{dn({\bf r^{\prime}}, 1)}{dE} -
\int_{\Omega} d^{3}r^{\prime \prime}
\frac{dn({\bf r}^{\prime})}{dE}
g ({\bf r}^{\prime}, {\bf r}^{\prime \prime})
\frac{dn({\bf r}^{\prime \prime},1)}{dE} \right) \delta \mu_{1}
\label{eq18}\end{aligned}$$ where we used Eq. (\[eq13\]) and the symmetry property $\Pi({\bf r},{\bf r^{\prime}}) = \Pi({\bf r^{\prime}},{\bf r})$ of the Lindhard function. Equation (\[eq16\]) then implies that the total variation in charge inside the volume $\Omega $ vanishes. This can be understood in the following way. According to the law of Gauss the charge included in a volume $\Omega$ is $\int_{ \partial \Omega} {\bf E} d{\bf S} = 4 \pi Q$. The charge in $\Omega$ is conserved if the electric flux through the surface of $\Omega$ vanishes. This means that for this conductor all electric field-lines which are generated when we bias the sample have their sources and sinks within the volume $\Omega$. Application of a bias voltage to an electrical conductor results in a redistribution of the charge within our sample but not in an overall change of the charge. If the conductor is poor, i. e. nearly an insulator, the reservoirs act like plates of capacitors. In this case long-range fields exist which run from one reservoir to the other and from a reservoir to a portion of the conductor. But if we chose the volume $\Omega$ to be large enough then all field lines stay within this volume.\
In all these considerations we have implicitly assumed that our conductor and the reservoirs is all that counts. Such a situation might be realized for the metallic mesoscopic structures fabricated on insulating substrates. But this picture is certainly not complete if we deal with modern mesoscopic semiconductor structures which are often defined with the help of a number of nearby gates. In such a case we must take a broader view and include inside our volume $\Omega$ not only the conductor of interest but also all of the nearby gates [@MBCAP]. >From an electrostatic point of view, we deal then not only with the mesoscopic object of interest but we have to take into account the nearby electrical bodies used to define this object. In such a case the overall charge is still conserved, even though the total charge on the mesoscopic conductor of interest varies with the applied bias. The theory presented here can easily be extended to this case [@MBCAP].
Nonlinear I-V Characteristic {#nonlinear}
============================
As an application of the discussion given above, let us consider the nonlinear I-V characteristic. The discussion presented here can also be carried out in terms of an external and internal response and provides a nice illustration of these concepts [@MBCAP]. Nonlinearities in metallic mesoscopic samples have been analyzed by Al’tshuler and Khmelnitskii [@KHME] using diagrammatic techniques without a self-consistent potential. For transmission through a tunnel contact the effect of a potential which changes with increasing applied bias has been investigated by Frenkel [@FREN]. Landauer has pointed to the necessity of a self-consistent treatment of the internal potential [@LAND]. In this section we derive the current-voltage characteristic taking into account nonlinearities which are a consequence of the increase of the external electrochemical potential differences as well as the changing internal (self-consistent) potential distribution. We focus on the leading nonlinear correction of the low-voltage ohmic behavior of the sample. The I-V characteristic of a mesoscopic sample is in general rectifying, i. e. $I(V) \ne - I(-V)$. Furthermore, since rectification also depends on the internal potential and since the internal potential in conductor $k$ depends on the charge distribution of other nearby conductors, the rectification properties of a small sample are dependent on its entire electric environment. Nevertheless, we consider for simplicity a conductor which is in electric isolation. Reference [@MBCAP] presents a more general result being valid if there are additional nearby conductors like gates or capacitors.\
To proceed we view the scattering matrices as a functional of the potential distribution ${\bf s}_{\alpha\beta}(E,U([\mu_{\alpha}], {\bf r}))$ and expand $U$ away from the equilibrium potential-distribution. The scattering matrix in the neighborhood of the equilibrium reference-state (index $0$) is ${\bf s}_{\alpha\beta}(E, eU({\bf r}))
={\bf s}^{0}_{\alpha\beta}(E) + \int d^{3}r^{\prime}
(\delta {\bf s}^{0}_{\alpha \beta}/
e \delta U({\bf r^{\prime}})) e\delta U({\bf r^{\prime}}).$ Here, $e \delta U({\bf r^{\prime}})$ can be expressed in terms of the characteristic potentials and the electrochemical potentials of the reservoirs. The total current at probe $\alpha $ can be found by the same considerations that lead to Eq. (\[eq01\]). However, we stop short of linearizing the resulting expression in the electrochemical potentials. The neglect of any inelastic scattering in the presence of large applied voltages is of course a limitation, but since we only focus on the quadratic term in the voltages this limitation might be not be so serious. The current is [@MB92] $$\begin{aligned}
I_{\alpha} =
(e/h)
\sum_{\beta}\int dE f_{\beta}
Tr \left[{\bf 1}_{\alpha}
\delta_{\alpha\beta}
- {\bf s}^{\dagger}_{\alpha\beta}(E, U({\bf r}))
{\bf s}_{\alpha\beta}(E, U({\bf r}))
\right] \;\;,
\label{eq30} \end{aligned}$$ where $f_{\beta }$ is the Fermi function belonging to reservoir $\beta $. The sum over all currents at all terminals is still zero due to the unitarity of the scattering matrix, hence the current is conserved. In order that the current depends on voltage differences only, it is necessary to treat the potentials self-consistently. We expand Eq. (\[eq30\]) in powers of the electrochemical potential deviations $\delta \mu_{\alpha}=
eV_{\alpha}$, $$\begin{aligned}
I_{\alpha} = \sum_\beta g_{\alpha\beta} V_{\beta}
+ (1/2) \sum_{\beta\gamma} g_{\alpha\beta\gamma} V _{\beta}
V_{\gamma} \;\;.
\label{eq31} \end{aligned}$$ The terms linear in the electrochemical potentials are determined by the dc-conductances $g_{\alpha\beta} = (e^{2}/h) \int (-df/dE) Tr \left[{\bf 1}_{\alpha\beta}
- {\bf s}^{\dagger}_{\alpha\beta}(E)
{\bf s}_{\alpha\beta}(E)
\right]$ which are a functional of the equilibrium reference-potential only. The leading order nonlinear terms are given by transport coefficients which are composed of an external and an internal response: $$g_{\alpha\beta\gamma} =
g^{e}_{\alpha\beta\gamma} +
g^{i}_{\alpha\beta\gamma} \;\; .
\label{gige}$$ The external response arises from the expansion of the Fermi functions in powers of the electrochemical potentials and is given by [@MBCAP] $$\begin{aligned}
g^{e}_{\alpha\beta\gamma} =
- \frac{e^{3}}{h}
\delta_{\beta\gamma}
\int dE (-\frac{df_{0}}{dE})
Tr \left[
{\bf s}^{\dagger}_{\alpha\beta}
\frac{d{\bf s}_{\alpha\beta}}{dE}
+\frac{d{\bf s}^{\dagger}_{\alpha\beta}}{dE}
{\bf s}_{\alpha\beta}
\right]
\label{eq32} \end{aligned}$$ where $f_{0}$ is the equilibrium Fermi function. Let us examine the external response for the case that we have a two-terminal conductor. >From the unitarity of the scattering matrix we find that all the non-vanishing second-order conductance coefficients are equal in magnitude, $$\begin{aligned}
g^{e}_{111} = - g^{e}_{122} = - g^{e}_{211} = g^{e}_{222} \;\; .
\label{eq33} \end{aligned}$$ Furthermore, these coefficients can also be expressed just as the energy derivative of the transmission probability, $$\begin{aligned}
g^{e}_{111} = \frac{e^{3}}{h} \int dE (-\frac{df_{0}}{dE})
\frac{dT}{dE}
\label{eq34} \end{aligned}$$ Thus up to the second order the contribution of the external response to the current is $$\begin{aligned}
I_{1} = - I_{2} = g (V_{1} - V_{2}) + (1/2) g^{e}_{111}
(V^{2}_{1} - V^{2}_{2})\;\;.
\label{eq35} \end{aligned}$$ Despite the fact that currents are conserved this is an unphysical result. The quadratic term depends not only on the voltage difference but on the individual voltages. Equation (\[eq35\]) would predict that we should observe a different current depending on whether we rise the voltage of the left contact by $\delta \mu_{1} = eV_{1}$ or whether we decrease the voltage on the right contact by $ \delta \mu _{2}= - eV_{1}$ as compared to the equilibrium value of the electrochemical potential. This is a simple example which demonstrates why the calculation of a nonlinear current voltage characteristic without the self-consistent adjustment of the electrostatic potential makes no sense.\
Let us now consider the internal response. The internal response is a consequence of the change in the potential distribution and is given by [@MBCAP] $$\begin{aligned}
g^{i}_{\alpha\beta\gamma} =
-\frac{e^{3}}{h}
\int dE \: (-\frac{df_{0}}{dE})
\int d^{3}r
Tr \left[
{\bf s}^{\dagger}_{\alpha\beta}
\frac{\delta
{\bf s}_{\alpha\beta}}
{\delta eU({\bf r})}
+ \frac{\delta
{\bf s}^{\dagger}_{\alpha\beta}}
{\delta eU({\bf r})}
{\bf s}_{\alpha\beta}
\right]
u_{\gamma} ({\bf r})
\nonumber \\
- \frac{e^{3}}{h}
\int dE \: (-\frac{df_{0}}{dE})
\int d^{3}r
Tr \left[
{\bf s}^{\dagger}_{\alpha\gamma}
\frac{\delta
{\bf s}_{\alpha\gamma}}
{\delta eU({\bf r})}
+ \frac{\delta
{\bf s}^{\dagger}_{\alpha\gamma}}
{\delta eU({\bf r})}
{\bf s}_{\alpha\gamma}
\right]
u_{\beta}({\bf r})\;.
\label{eq36} \end{aligned}$$ Note that the internal response contributes only to quadratic order in the voltage. The linear conductance is a purely [*external*]{} response. If we now add external and internal response, take into account Eq. (\[eq07\]) and that the integral over $\Omega$ of an internal response term with $u = 1$ is equal to minus the external response with the functional derivative replaced by an energy derivative, we find [@MBCAP] $$\begin{aligned}
\sum_{\alpha} g_{\alpha\beta\gamma} =
\sum_{\beta} g_{\alpha\beta\gamma} =
\sum_{\gamma} g_{\alpha\beta\gamma} = 0\;\; .
\label{eq37} \end{aligned}$$ For a two-terminal conductor the second-order conductance coefficients obey $ g_{111} = $ $- g_{112} =$ $ - g_{121} = $ $g_{122} = $ $- g_{211} = $ $g_{212} =$ $ - g_{221} = $ $- g_{222}$ Consequently, the currents are $$\begin{aligned}
I_{1} = - I_{2} = g (V_{1} - V_{2}) + (1/2) g_{111} (V_{1} - V_{2})^{2}
+... \;\;.
\label{eq39} \end{aligned}$$ Now, the current depends only on the voltage difference as it must be. In contrast to the external response which could simply be expressed in terms of energy derivatives of the transmission probability, the total response depends on the charge distribution inside the conductor.\
As a simple example, we consider an asymmetric resonant double barrier. The long lived state has a decay width $\Gamma_{1}$ to the left and $\Gamma_{2}$ to the right. For simplicity, assume that the potential in the well is determined by a local charge-neutrality argument (see Sec. 6). The characteristic potentials in the well are $u_{1} = \Gamma_{1}/\Gamma$ and $u_{2} = \Gamma_{2}/\Gamma,$ where $\Gamma = \Gamma_{1} + \Gamma_{2}$ is the decay width. >From Eqs. (\[eq32\]) and (\[eq36\]) we find for the second order conductance coefficient $g_{111} = ({e^{3}}/{h}) (dT/dE) (1-2u_{1})$ and hence [@MBUNP; @CHRIS] $$g_{111}= ({e^{3}}/{h}) (dT/dE) (\Gamma_{2} - \Gamma_{1})/\Gamma \;\;.
\label{g111}$$ In summary, we emphasize that the nonlinearity cannot be discussed without a concern for the way the potential drops in the interior of the conductor.
Frequency Dependent Conductance {#frequency}
===============================
External Response {#external}
-----------------
We are interested in the dynamical response of the conductor. A time-dependent voltage $ \propto \exp(-i\omega t)$ can be applied across two terminals, between a terminal and a nearby gate, or between two nearby gates. We want to know the currents which appear as a consequence of these oscillating voltages at the contacts of the conductor or at the contacts to the nearby gates. We are seeking the admittance matrix $$\begin{aligned}
G_{\alpha\beta}(\omega) = I_{\alpha}(\omega)/ V_{\beta}(\omega)\;\;.
\label{eq40} \end{aligned}$$ Again we consider for simplicity the case of a two-terminal conductor in electrical isolation. The case where there are a number of nearby gates (capacitors) or other conductors has been the subject of a number of discussions [@MBCAP; @MBROM]. Thus the indices $\alpha$, $\beta$ take the values $1$ and $2$ for the left and right contact, respectively. If there are no other nearby electrical conductors then all electric field lines which emanate from the conductor also return to the conductor or to the reservoirs. Even in the presence of time-dependent voltages applied to this conductor the reservoirs remain locally charge neutral, i. e. the electric field lines emanate from a reservoir only in the region where the transition to the conductor occurs. If the conductor is very short then the reservoirs act like the plates of a capacitor and, due to long range Coulomb intercation, field lines connect the surfaces of the two reservoirs facing each other. As in the dc-case there exists, therefore, a volume $\Omega$ which is so large that there is no electric flux through its surface. Consequently, if we include all components of the system within $\Omega$ then the total charge within this volume is zero, i. e. all currents at the terminals must add up to zero. Furthermore, since a potential which is uniform over the entire volume $\Omega$ is of no physical consequence the resulting currents must depend on the potential differences only. Therefore, Eqs. (\[eq04\]) and (\[eq05\]) hold also for the dynamic conductance: the rows and columns of the dynamic conductance matrix $G_{\alpha
\beta} (\omega)$ must add up to zero. We call such a discussion of the ac-transport a [*charge*]{} and [*current conserving*]{} theory. Below we illustrate the features of such a theory for the case of low frequencies only. But the extension to a larger range of frequencies and to nonlinearities must follow the very same line of thought.\
Decomposition of the density of states {#decomposition}
--------------------------------------
The total density of states in the conductor inside the volume $\Omega$ is a sum of [*four*]{} contributions [@MBAPT], $$\begin{aligned}
dN/{dE} = \sum_{\alpha\beta} (dN_{\alpha\beta}/{dE})\;\;,
\label{eq42}\end{aligned}$$ where $$\begin{aligned}
\frac{dN_{\alpha\beta}} {dE} =
\frac{1}{4\pi i}\,
{\rm Tr} \left[ {\bf s}^{\dagger}_{\alpha\beta}
\frac{d{\bf s}_{\alpha\beta} }{dE}
-\frac{d{\bf s}^{\dagger}_{\alpha\beta} }{dE}
{\bf s}_{\alpha\beta}\right]
\label{eq43}\end{aligned}$$ are the [*partial densities of states*]{}. Fig. 5 gives a schematic representation of the partial density of states. The partial density of states $dN_{11}/dE$ consists of carriers that originate in contact $1$ and return to contact $1$. The partial density of states $dN_{21}/dE$ consists of carriers that originate in contact $1$ and are transmitted to contact $2$. It turns out that the external response is determined exactly by these four partial densities of states. The partial densities of states represent a decomposition of the total density of states both with respect to the origin of the carriers (injecting contact, right index) and the final destination of the carriers (emitting contact, left index).\
At [*constant electrostatic potential*]{} the total charge injected into the conductor under a simultaneous and equal increase of the chemical potentials at its contacts $\delta E_{F1} = \delta E_{F2} = e \delta \mu$ is $\delta Q ^{e} = e \sum_{\alpha\beta}
(dN_{\alpha\beta}/dE) \delta \mu$. To find the current at contact $\alpha$ we need to know which portion of this charge enters or leaves the conductor through contact $\alpha $, i. e. how the total charge is partitioned on the two contacts. The answer to this question was found by one of the authors, Pretre and Thomas [@MBAPT] using a linear response calculation. The following simple argument leads to the same result. The scattering matrix ${\bf s}_{\alpha\beta}$ determines the current amplitudes of the outgoing waves in contact $\alpha$ as a function of the current amplitudes of the incident waves in contact $\beta$. The charge which is injected by an increase of the Fermi energy at contact $1$ is $(e dN_{11}/{dE}+
e dN_{21}/{dE})
e V_{1}(\omega)$. Only the additional charge $\delta Q_{1}^{e}(\omega) = e (dN_{11}/{dE}) e V_{1}(\omega)$ leads to a current at contact 1, whereas $\delta Q_{2}^{e}(\omega) = e (dN_{21}/{dE})
e V_{1}(\omega)$ is determined by carriers which leave the conductor through contact $2$. Therefore, a variation of the Fermi levels of the contacts $\beta $ causes the current $$\begin{aligned}
\delta I_{\alpha}^{e}(\omega) = -i \omega e^{2} \sum _{\beta }
(dN_{\alpha\beta}/{dE})
V_{\beta}(\omega)
\label{eq44}\end{aligned}$$ at contact $\alpha $. Since direct transmission between contact $1$ and $2$ is possible, an oscillating voltage causes in addition at these contacts a current determined by the dc-conductance. Thus the leading [*external*]{} low-frequency current response to an oscillating chemical potential $eV_{\alpha}(\omega)$ is given by [@MBAPT] $$\begin{aligned}
G^{e}_{\alpha\beta}(\omega) = G_{\alpha\beta}(0)
-i \omega e^{2} (dN_{\alpha\beta}/{dE})\;\;.
\label{eq45}\end{aligned}$$ This external response is not current conserving. Since the dc-conductances satisfy $\sum_{\alpha} G_{\alpha\beta}(0)=
\sum_{\beta} G_{\alpha\beta}(0)=0$, one finds that to leading order in frequency $\sum_{\alpha} G_{\alpha\beta}^{e}(\omega)$ is proportional to the total charge injected from contact $\beta$ into the conductor. The injected charges create an internal, time-dependent electric potential $\delta U({\bf r}, t)$ which in turn causes additional currents. In the following subsection, we investigate the response to such an internal electrostatic potential. It turns out that this requires a detailed knowledge of the charge distribution.\
The density of states (\[eq42\]) has been obtained assuming a perturbation which is (in a mathematical sense) asymptotically far away from the sample. The derivative with respect to the energy $E$ is a consequence of the asymptotic nature of this perturbation. Physically what counts is the density of states in a finite volume $\Omega$. To obtain these densities it is better to first calculate the local densities corresponding to Eq. (\[eq43\]) and to integrate these local densities over the volume $\Omega$. As shown below the local densities are not given by energy derivatives of the scattering matrix but by functional derivatives with respect to the local potential $eU({\bf r}).$ The densities determined by integration of such local densities of states differ in general from a simple energy derivative by a quantum correction [@Aronov; @Gasp]. The difference vanishes in the semi-classical (WKB) limit. The same is of course valid for the derivatives in Eq. (\[eq32\]).\
We conclude this section with a remark on localized states. Equation (\[eq42\]) is not complete. A conductor might also contain a contribution to the density of states from localized states, in addition to the extended scattering states considered so far. For the external response the localized states play no role, but later when we consider the screening the localized states are important. To be brief, however, we do not discuss here their role in detail.
Response to an oscillating electrostatic potential {#response}
--------------------------------------------------
To investigate the self-consistency of dc-transport we already discussed the local charge distribution inside the conductor. In Eq. (\[eq08\]) we introduced a [*local partial density of states*]{} which we called the injectivity (see Fig. 6). Now we introduce additional local partial densities of states which permit eventually to write the dynamical conductivity in a simple and transparent manner. We are interested in the currents generated at the contacts of a sample in the presence of an oscillating potential $\delta U({\bf r}, t).$ We can Fourier transform this potential with respect to time and consider a perturbation of the form $$\begin{aligned}
\delta U({\bf r}, t) = u({\bf r}) (U_{+\omega} \exp(-i\omega t) +
U_{-\omega} \exp(+i\omega t))\;\;.
\label{eq46}\end{aligned}$$ Since the potential is real we have $U_{-\omega}= U^{\ast}_{+\omega}$. The response to such a potential can be treated using a scattering approach [@MBHTP]: due to the oscillating internal potential a carrier incident with energy $E$ can gain or loose modulation energy $\hbar \omega$ during reflection at the sample or during transmission through the sample. The amplitude of an outgoing wave is a superposition of carriers incident at energy $E$ and at the side-band energies, $E \pm \hbar \omega.$ In the low-frequency limit the amplitudes of the out going waves can be obtained by considering the scattering matrix ${\bf s}_{\alpha\beta} (U({\bf r}, t), E)$ to be a slowly varying function of the potential $U({\bf r}, t).$ Since the deviations of the actual potential away from the (time-independent) equilibrium potential $U_{eq}({\bf r})$ are small, we can expand the scattering matrix in powers of $\delta U({\bf r}, t) =
U({\bf r}, t) -
U_{eq}({\bf r})$ to linear order $$\begin{aligned}
{\bf s}_{\alpha\beta}
(U({\bf r}, t), E) =
{\bf s}_{\alpha\beta}
(U_{eq}({\bf r}), E) +
(\delta {\bf s}_{\alpha\beta}/
\delta U({\bf r}))
\delta U({\bf r}, t)\;\;.
\label{eq47}\end{aligned}$$
Evaluation of the current at contact $\alpha$ gives [@MBHTP] $$\begin{aligned}
\delta I_{\alpha}^{i}(\omega) = i e^{2} \omega
\int d^{3}{r} (dn(\alpha, {\bf r})/dE)
u({\bf r})
U_{+\omega}\;\;.
\label{eq48}\end{aligned}$$ Here we have introduced the local partial density of states [@MBCAP; @MBHTP] $$\begin{aligned}
\frac{dn(\alpha ,{\bf r})}{dE} =
- \frac{1}{4 \pi i} \sum_{\beta}
Tr\left[{\bf s}^{\dagger}_{\alpha\beta}
\frac{\delta {\bf s}_{\alpha\beta}}{e\delta U({\bf r})}
- \frac{\delta {\bf s}^{\dagger}_{\alpha\beta}}{e\delta U({\bf r})}
{\bf s}_{\alpha\beta} \right]
\label{eq49}\end{aligned}$$ which we call the [*emissivity*]{} (see Fig. 7). It describes the local density of states of carriers at point ${\bf r}$ which are emitted by the conductor at probe $\alpha$. A more detailed derivation of Eq. (\[eq48\]) can be found in Ref. [@MBHTP]. It is useful to express the response to the internal potential in the from of a conductance defined as $\delta I_{\alpha}^{i}(\omega) = G^{i}_{\alpha}(\omega)
U_{+\omega}$. Comparison with Eq. (\[eq45\]) gives for the internal conductances $$\begin{aligned}
G^{i}_{\alpha}(\omega) = i e^{2}
\omega
\int d^{3}{r} \: (dn(\alpha, {\bf r})/dE)\: u({\bf r}) \;\;.
\label{eq50}\end{aligned}$$ Below we use this internal response to complete the calculation of the total current. Before doing this it seems useful to pause for a moment and to discuss in more detail the local density of states which determine the internal response.
Decomposition of the local density of states {#locpart}
--------------------------------------------
In Eq. (\[eq09\]) we have expressed the injectivity (\[eq08\]) with the help of scattering states. Now we give an expression of the injectivity in terms of derivatives of the scattering matrix. Expressions which relate wave functions to functional derivatives are known from the discussion of the characteristic times occurring in tunneling processes [@LEVE; @EAVES]. Consider for a moment a one-dimensional scattering problem with a potential $V(x)$ in an interval $(-a, a)$. The scattering matrices are $s_{\alpha\beta}$ where $\alpha$ and $\beta$ take the values $1$ and $2$ to designate left and right, respectively. Of interest is the time a particle [*dwells*]{} in this region irrespective of whether it is ultimately transmitted or whether it is ultimately reflected. There are two dwell times $\tau_{D \alpha}$, for the particles arriving from the left or from the right. In terms of the scattering states $\Psi_{\alpha}(x)$ and the incident current $I$, the dwell time is given by [@EAVES] $$\begin{aligned}
\tau_{D \alpha} = \int_{x}^{x+a} dx |\Psi_{\alpha}(x)|^{2}/I \;\;.
\label{eq51}\end{aligned}$$ To find the time a particle dwells in an interval $(x, x+a)$ an infinitesimal uniform perturbation $dV$ is added in this region to the potential $V(x)$. It is found that the dwell time is then related to the scattering matrix via the following relationship [@LEVE; @EAVES] $$\begin{aligned}
\tau_{D \alpha}= \hbar \: {\rm Im} \left( \: |s_{1\alpha}|^{2}
\frac{d\ln s_{1\alpha}}{dV} + |s_ {2\alpha}|^{2}
\frac{d\ln s_{2\alpha}}{dV}\: \right) \;\; .
\label{eq52}\end{aligned}$$ where Im denotes the imaginary part. For a plane-wave scattering state with wave vector $k$ the current is $v=\hbar k/m$. Thus a comparison with Eq. (\[eq09\]) shows that for a single quantum channel the dwell time is related to the injectivity by $$\begin{aligned}
\tau_{D \alpha}/h= \int_{x}^{x+a} dx \: (dn(x,\alpha)/dE)\;\; .
\label{eq53}\end{aligned}$$ This means that their exists a simple relationship between local density of states and derivatives of the scattering matrix with respect to potentials. It is easy to extend this relation to the case of an arbitrary space dependent potential and to an arbitrary number of channels. The final result is that the injectivity is given by [@MBCAP; @MBHTP] $$\begin{aligned}
dn({\bf r},\alpha)/dE =
- \frac{1}{4 \pi i} \sum_{\beta}
Tr\left[{\bf s}^{\dagger}_{\beta\alpha}
\frac{\delta {\bf s}_{\beta\alpha}}{e\delta U({\bf r})}
- \frac{\delta {\bf s}^{\dagger}_{\beta\alpha}}{e\delta U({\bf r})}
{\bf s}_{\beta\alpha}\right]\;\;.
\label{eq54}\end{aligned}$$
The injectivity contains information about the origin of the particles: it is important through which contact the carriers enter the conductor. Consequently, the summation is over the first index of the scattering matrices. In contrast, the [*emissivity*]{} defined in Eq. (\[eq49\]) contains information about the future of the carriers: it is important through which contact the carriers leave the sample. The summation is thus over the second index of the scattering matrix. The sum of all the injectivities (see Fig. 6) or the sum of all the emissivities (see Fig. 7) is equal to the local density of states, $$\begin{aligned}
dn({\bf r})/dE = \sum_{\alpha} dn(\alpha,{\bf r})/dE = \sum_{\alpha} dn({\bf r},\alpha)/dE
\;\;.
\label{eq55}\end{aligned}$$ Equation (\[eq55\]) represents a decomposition of the local density of states into emissivities and injectivities. We mention that the emissivities and injectivities are not independent of one another. In fact, in the absence of a magnetic field they are equal, $dn(\alpha,{\bf r})/dE = dn({\bf r},\alpha)/dE$. In the presence of a magnetic field the microreversibility of the scattering matrix implies that the emissivity into contact $\alpha$ in magnetic field $B$ is equal to the injectivity of contact $\alpha$ if the magnetic field is reversed, $$\begin{aligned}
dn(\alpha ,{\bf r};B)/dE = dn({\bf r},\alpha;-B)/dE\;\; .
\label{eq56}\end{aligned}$$ While the local density of states at equilibrium is an even function of the magnetic field, i. e. $dn( {\bf r};B)/dE = dn({\bf r} ;-B)/dE$, the injectivities and emissivities are in general [*not*]{} even functions of $B$. This has some peculiar physical consequences, as has been shown recently by a low-frequency measurement of capacitances in a quantum Hall system [@CHEN].
Combined external and internal response {#combined}
---------------------------------------
We need to find an expression of the current response generated by the electric potential oscillations caused by the external potentials $\delta \mu_{\alpha} (\omega)$ $ \exp(-i\omega t)$. We are interested in the response to first order in $\omega$, and since the currents in Eqs. (\[eq44\]) and (\[eq48\]) are proportional to $\omega$ it is sufficient to know the quasi-static nonequilibrium state discussed in Sect. \[potential\]. We express the deviation of the potential away from the equilibrium potential with the help of the characteristic potentials defined in Eq. (\[eq06\]). >From Eq. (\[eq48\]) we find $$\begin{aligned}
\delta I_{\alpha}^{i}(\omega) = i e\omega \sum_{\beta}
\int d^{3}{r} \: (dn(\alpha , {\bf r})/dE) \:
u_{\beta}({\bf r})\delta \mu_{\beta }(\omega)\;\; .
\label{eq57}\end{aligned}$$ Thus the induced potentials give rise to a conductance $$\begin{aligned}
G^{i}_{\alpha \beta} (\omega) = i e^{2} \omega
\int d^{3}{r}\: (dn(\alpha , {\bf r})/dE) \: u_{\beta}({\bf r})
\:\:,
\label{eq58}\end{aligned}$$ which can be written in terms the Green’s function and the injectivity, $$\begin{aligned}
G^{i}_{\alpha \beta} (\omega) =
i e^{2} \omega
\int d^{3}{ r}
\int d^{3}r^{\prime}
(dn(\alpha, {\bf r})/dE)
g ({\bf r}, {\bf r}^{\prime})
(dn({\bf r}^{\prime},\beta)/dE)\;\;.
\label{eq59}\end{aligned}$$ Eq. (\[eq59\]) tells us that the internal response $G^{i}_{\alpha \beta} (\omega)$ is a consequence of the charge injected from contact $\beta$ which generates a potential determined by the Green’s function and that this potential in turn generates a current at $\alpha$ determined by the emissivity into that contact. The total response is the sum of the external response (\[eq45\]) and the internal response (\[eq59\]), $$\begin{aligned}
G_{\alpha \beta} (\omega) = G^{e}_{\alpha \beta} (\omega)
+ G^{i}_{\alpha \beta} (\omega)\;\;.
\label{eq60}\end{aligned}$$ We express it in the form $$\begin{aligned}
G_{\alpha \beta} (\omega) = G_{\alpha \beta} (0)
-i \omega E_{\alpha \beta} + O(\omega^{2})
\label{eq61}\end{aligned}$$ and call $E_{\alpha \beta}$ the (screened) [*emittance*]{} of the conductor. It is given by [@MBCAP] $$\begin{aligned}
E_{\alpha \beta} = e^{2}\frac{dN_{\alpha\beta}}{dE} -
e^{2} \int d^{3}{ r} \int d^{3}r^{\prime}\:
\frac{dn(\alpha, {\bf r})}{dE}\:
g ({\bf r}, {\bf r}^{\prime}) \:
\frac{dn({\bf r}^{\prime},\beta)}{dE}.
\label{eq62}\end{aligned}$$ Before we apply this result in the next section to a few simple problems, we want to demonstrate that $G_{\alpha \beta}$ is indeed current conserving, i. e. that the rows and columns of the emittance matrix add up to zero. Consider the first column. If we add $E_{11}$ and $E_{21}$ the first terms in the emittance give the total charge $dN_{11}/{dE} + dN_{21}/{dE}$ injected from contact $1.$ In the second term the two emissivities add to give the local density of states. Now Eq. (\[eq16\]) is used. What remains is the integral over the entire volume of the injectivity which is just the total injected charge. Thus for a two terminal conductor in electrical isolation the emittance matrix satisfies $E_{\mu} \equiv E_{11} = -E_{12} = -E_{21} = E_{22}$.
Examples
========
Emittance of a metallic diffusive conductor {#metallic}
-------------------------------------------
Let us consider a mesoscopic metallic conductor connecting two reservoirs. In a metallic conductor charge accumulations are screened over a Thomas-Fermi screening length (apart from miniscule and more subtle Friedel-like long-range effects [@LE89]). If we assume in addition that the density varies not to rapidly then the local potential is directly determined by the local density. The local potential $\delta U({\bf r})$ generated by an injected charge $\delta n_{in}({\bf r})$ is determined by $(dn({\bf r})/dE) \: e\delta U({\bf r}) = \delta n_{in}({\bf r}).$ This corresponds to a Green’s function which is a delta function in space and with a weight inversely proportional to the local density of states $ g({\bf r}, {\bf r}^{\prime}) = (dn({\bf r})/dE)^{-1}
\delta ({\bf r}-{\bf r}^{\prime})$. Using this in Eq. (\[eq62\]) gives an emittance in terms of densities only [@MBHTP], $$E_{\alpha \beta} (\omega) = e^{2}\frac{dN_{\alpha\beta}}{dE} -
e^{2} \int d^{3}{r} \:
\frac{dn(\alpha, {\bf r})}{dE}\:
(\frac{dn({\bf r})}{dE})^{-1} \:
\frac{dn({\bf r},\beta)}{dE} \;\;.
\label{eq63}$$ There are no electric field lines outside the conductor.\
The wire with cross-section $A$ ranges from $x=-L/2$ to $x=L/2$. The mean distance between the impurities is $l$. The reflection and the transmission probability per channel are $R=1-l/L$ and $T=l/L$, respectively. The partial densities of states of reflected carriers are $dN_{11}/dE = dN_{22}/dE = (1/2) (1-l/L) (dN/dE)$ where $dN/dE$ is the total density of states in the volume $\Omega = AL$. The partial densities of states of transmitted carriers are $dN_{12}/dE = dN_{21}/dE = (1/2)
(l/L) (dN/dE)$. The diffusion equation for the diffusive metallic conductor implies the ensemble averaged and over the cross section averaged injectivities $dn(x,1)/dE = (1/2L)(dN/dE) (1- 2x/L)$ and $dn(x,2)/dE = $ $(1/2L)$ $(dN/dE)$ $(1+ 2x/L)$. In the absence of a magnetic field, the emissivities are given by the same expressions. The linear dependence of the injectivities gives an ensemble averaged potential which drops also linearly. With the help of these expressions, we find for the emittance [@MBUNP] $$E_{\mu} = (1/6 -l/2L) (dN/dE) \;\;.
\label{emmet}$$ Already for metallic wires longer then $3l$ the emittance is a positive quantity. Since typically $l/L$ is much smaller than $1/3$ the metallic diffusive wire at low frequencies responds like a capacitor with an ensemble averaged capacitance given by Eq. (\[eqI2\]).
Emittance of a perfect ballistic wire {#ballistic}
-------------------------------------
Consider a ballistic wire of length L with N quantum channels. We apply again Thomas-Fermi screening. However, this is not very well justified and permits to obtain an estimate only. We also ignore the variation of the potentials near the contacts, which in a more realistic treatment might well give us a capacitive contribution. Each quantum channel contributes with a density of states $2L/hv_{n}$, were $v_{n}$ is the channel velocity evaluated at the Fermi energy. The total density of states per spin is $dN/dE = \sum_{n} 2L/hv_{n}$. In the absence of backscattering it holds $dN_{11}/dE = dN_{22}/dE = 0$ and $dN_{21}/dE =dN_{12}/dE = (1/2)\:
dN/dE$. The injectivities $dn(x,1)/dE = (1/2L)(dN/dE)$ and $dn(x,2)/dE = (1/2L)(dN/dE)$ are independent of the space coordinate. This corresponds to a potential in the ballistic wire which is constant and midway between the electrochemical potentials at the contacts. Hence, using Eqs. (\[eq62\]) we find that an ideal perfect wire has a negative emittance given by Eq. (\[eqI3\]). A ballistic wire responds like a conductor which classically is represented by a resistance and an inductance in series, and where the emittance can be viewed as a kinetic inductance.
Emittance of a resonant double barrier {#barrier}
--------------------------------------
As a third example we consider a resonant double barrier [@MBHTP]. The scattering matrix is $s_{\alpha\beta} = (\delta_{\alpha\beta}-i \Gamma_{\alpha} \Gamma_{\beta}
/\Delta)\exp(i\delta_{\alpha}+ i\delta_{\beta})$, where $\delta_{\alpha\beta}$ is the Kronecker symbol and the $\delta_{\alpha}$ are phases whose energy dependence can be neglected compared to the rapid variation of the resonant denominator $\Delta = E - E_{r} - e\delta U - i \Gamma /2$ with $\Gamma = \Gamma_{1}+\Gamma_{2}$. Here, $E_{r}$ is the resonant energy at equilibrium and $e\delta U$ is the deviation of the electrostatic potential away from its equilibrium value at the site of the long lived state in the presence of transport. With the help of the injectances $dN_{\alpha}/dE =
(\Gamma _{\alpha}/2 \pi |\Delta|^{2})$ and the total density of states $dN/dE =
(\Gamma/2 \pi |\Delta|^{2})$ the partial densities of state can be expressed in the following manner: For $\alpha = \beta $ we find $$\frac{dN_{\alpha\alpha}}{dE} = \frac{R}{2} \frac{dN}{dE}
\pm \frac{1}{2}(\frac{dN_{1}}{dE} -\frac{dN_{2}}{dE})
\label{eq66}$$ where the plus and the minus sign correspond to $\alpha = 1$ and $\alpha = 2$, respectively. Here, $R=1-T$ is the reflection probability. For $\alpha \neq \beta$, one has $$\frac{dN_{\alpha \beta}}{dE} = \frac{T}{2}\frac{dN}{dE}\;\;.
\label{eq67}$$ The unscreened injectances (emittances) are found by integrating the injectivities (emissivities) over the volume of the localized state, i. e. over the well. They are given by $$\begin{aligned}
dN_{1\alpha}/dE + dN_{2\alpha}/dE & = & dN_{\alpha}/dE\;\; .
\label{eq70}\end{aligned}$$ In a Thomas-Fermi approach the characteristic potentials $u_{\alpha}$ in the well are determined by $(dN/dE) u_{\alpha} = dN_{\alpha}/dE .$ This gives $u_{\alpha} = \Gamma_{\alpha}/\Gamma .$ Using Eq. (\[eq63\]) gives an emittance [@MBHTP] $$E_{\mu} = - e^{2} \: \frac{(dN_{1}/dE)\:(dN_{2}/dE)}{dN/dE}
\left(
\frac{\Gamma^{2} /2 - |\Delta|^{2}}{|\Delta|^{2}}\right)
\label{eq71}$$ For a symmetric resonant tunneling barrier Eq. (\[eq71\]) simplifies and is given by Eq. (\[eqI4\]). At resonance the emittance is negative reflecting kinetic (inductive) behavior, it is zero at half-width of the resonance, and it is positive (capacitive) if the Fermi level is more than a half width above (below) the resonant energy. Clearly, Thomas-Fermi screening is not very realistic for such a conductor. Moreover, the quantization of charge in the well might play a decisive role. Nevertheless, these considerations indicate the character of the results that a more realistic treatment might yield and hopefully stimulate work in that direction.
Summary
=======
We have developed a self-consistent discussion of mesoscopic electrical conduction. The determination of the electrical potential in the presence of a dc-current, although unimportant for the discussion of the dc-conductances itself, permits to discuss the conditions under which the dc-conductance formulae are valid, it permits to calculate the first nonlinear corrections of the purely linear response, and permits to find the ac-conductances to first order in frequency. We applied the results to some simple examples.\
We have emphasized the case of two terminal conductors in electric isolation. The theory permits, however, also to discuss the effect of nearby capacitors and gates and in fact provides a mesoscopic description of capacitances [@MBCAP; @MBROM]. Some implications of a mesoscopic theory of capacitance, like Aharonov-Bohm oscillations in capacitance coefficients, the gate voltage dependence of persistent currents have already been the subject of recent works [@MBREK; @MBCS].\
The self-consistent nature of electrical transport is a consequence of the long-range Coulomb interaction of carriers. A self-consistent description must, therefore, tackle an interacting many-particle problem. Consequently, the results of such a theory depend somewhat on the sophistication that is used to treat the many-particle problem. Here we have used a simple Hartree approach. Since density-functional theory is nothing but an improved Hartree theory it gives results which look formally very similar to the results presented here [@MBTRI]. A stronger modification of the results discussed here can be expected in situations where one must take the quantization of charge into account [@MBCS; @BS].
The theory presented here, demonstrates that interesting results can be obtained by investigating nonlinearities and ac-conductances. The theory demonstrates that it is necessary to treat nonlinearities and the ac-response self-consistently to conserve both charge and current. We are confident that experiments will eventually demonstrate the close connection between electrostatic questions and nonlinearities and ac-response.
[99]{}
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Al’tshuler, B. L. and Khmelnitskii, D. E., JETP Lett. [**42**]{}, 359 (1985).
Frenkel, J., Phys. Rev. [**36**]{}, 1604 (1930).
Landauer, R., in [*Nonlinearity in Condesned Matter*]{}, edited by A. R. Bishop, et al., (Berlin, Springer, 1987). p. 2.
Aronov, A. G., Gasparian, V. M., and Gummich, U., J. Phys. Condens. Matter [**3**]{}, 3023 (1991)
Gasparian, V. M., Christen, T., and Büttiker, M., (unpublished).
Leavens, C. R. and Aers, G. C., Solid State Commun. [**67**]{}, 1135 (1988); Phys. Rev. B [**39**]{}, 1202 (1989).
Büttiker, M., Phys. Rev. B[**27**]{}, 6187 (1983); in “Electronic Properties of Multilayers and low Dimensional Semiconductors”,\
edited by J. M. Chamberlain, L. Eaves, and J. C. Portal, (Plenum, New York, 1990). p. 297-315. For recent works on the Larmor clock see, Steinberg, A. M., Phys. Rev. Lett. [**74**]{}, 2405 (1995); Gasparian, V., Ortuno, M., Ruiz, J., and Cuevas, E., Phys. Rev. Lett. [**75**]{}, 2312 (1995).
Chen, W., Smith, T. P., Büttiker, M., and Shayegan, M., Phys. Rev. Lett. [**73**]{}, 146 (1994).
Büttiker, M., Il Nuovo Cimento, B [**110**]{}, 509-522 (1995); Jpn. J. Appl. Phys. [**34**]{}, 4279 (1995).
Büttiker, M., Physica Scripta, T [**54**]{}, 104 - 110, (1994).
Büttiker, M. and Stafford, C. A., Phys. Rev. Lett. (in press), (1995).
Büttiker, M., in “Quantum Dynamics and Submicron Structures”, edited by H. Cerdeira, G. Schön, and B. Kramer, (Kluwer Academic Publishers, Dordrecht, 1995). p. 657 -672.
Bruder, C. and Schöller, H., Phys. Rev. Lett., 1076 (1994).
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WUE-ITP-97-050\
SPhT-T97/139
[**$|V_{ub}|$ and Perturbative QCD Effects** ]{}\
[**in the $B\to\pi$ Transition Form Factor**]{} [^1]\
\
[ $^a$ Centre d’Etudes de Saclay, $^b$ Universität Würzburg, $^c$ Budker INP Novosibirsk. ]{}\
[ We report on recent improvements for the $B\to\pi$ form factor. The updated value of $|V_{ub}|$ is presented.]{}\
[**1. Motivation.**]{} The semileptonic decay $B\to\pi l\nu_l $ is one of the most important reactions for the determination of the CKM parameter $V_{ub}$. However, in order to extract $V_{ub}$ from data one needs an accurate theoretical calculation of the hadronic matrix element $$\begin{aligned}
\label{formdef}
\langle\pi (q)|\bar u\gamma_{\mu}b |B(p+q)\rangle
=2f^+(p^2)q_{\mu}+(f^+(p^2)+
f^-(p^2))p_{\mu},\end{aligned}$$ where $p+q$, $q$ and $p$ denote the $B$ and $\pi$ four-momenta and the momentum transfer, respectively, and $f^{\pm}$ are two independent form factors. A very reliable approach to calculate $f^{\pm}$ in the framework of QCD is provided by the operator product expansion (OPE) on the light-cone \[1,2,3\] in combination with QCD sum rule techniques. The sum rule for the form factor $f^{+}(p^2)$ has been obtained in the leading order (LO) in $\alpha_s$ in \[4,5\] taking into account twist 2, 3 and 4 operators. The most important missing elements of these calculations are the perturbative QCD corrections to the correlation function. Here we report on a results of the calculation of the $O(\alpha_s)$ correction to $f^+$ \[13,14\] which eliminates one of the main uncertainty in the sum rule results.
[**2. Sum rule.**]{} The general idea of the sum rule method is to consider the correlation function of two heavy-light currents, $$\begin{aligned}
F_{\mu}(p,q)&=&i\int dxe^{ip\cdot x}
\langle\pi (q)|T\{\bar u (x)\gamma_\mu b(x) , m_b\bar b(0)i\gamma_5
d(0)\}|0\rangle
\label{corr} \end{aligned}$$ which can be calculated in the region $(p+q)^2<0 $ and $p^2 < m_b^2 - O($1GeV$^2)$ using OPE near the light-cone, i.e. at $x^2 \simeq 0$. In this note we focus on the leading twist 2 contribution. The sum rule for $f^+$ in LO in $\alpha_s$ is given by $$\begin{aligned}
f_Bf^+(p^2)=\frac{f_\pi m_b^2}{2m^2_B}\int^{s_0}_{m_b^2} \varphi_\pi (u_0)
e^{\frac{m^2_B-s}{M^2}}ds +\mbox{higher twists}~,
\label{sr}\end{aligned}$$ Here $\varphi_\pi (u)$ is the pion wave function, $m_b,M_B$ are the masses of the heavy quark and meson, $M^2$ is the Borel parameter, $s_0$ is the threshold of the continuum, $f_\pi=132$MeV, $u_0=\frac{m_b^2-p^2}{s-p^2}$. The calculation has several aspects which are worth pointing out. Firstly, the sum rule is actually derived for the product $f_Bf^+$, $f_B$ being the $B$ meson decay constant defined by $
\langle B|\bar bi\gamma_5 d|0 \rangle =m^2_Bf_B/m_b~.
$ The form factor $f^+$ itself is then obtained by dividing out $f_B$ taking the value determined from the corresponding two-point QCD sum rule. In previous estimates, the $O(\alpha_s)$ correction to $f_B$ was thereby ignored for consistency because of the lack of the $O(\alpha_s)$ correction to $f_Bf^+$. Our calculation now allows to take into account the correction to $f_B$ which is known to be sizeable. Secondly, knowing the $O(\alpha_s)$ corrections, also the heavy quark mass entering the sum rule can be properly defined. The calculation for a finite quark mass is new and will have numerous applications. [**3. QCD correction.**]{} The correlator can be written as a convolution of a hard amplitude $T(p^2,(p+q)^2,u)$ calculable within perturbation theory, with the pion wave function $\varphi_{\pi}(u)$ containing the long-distance effects: $$\begin{aligned}
\label{represent}
F(p^2,(p+q)^2)=-f_\pi\int^1_0 du \varphi_\pi (u) T(p^2,(p+q)^2,u).\end{aligned}$$ The evolution of the light-cone wave function $\varphi_\pi(u)$ is controlled by the Brodsky-Lepage equation \[2\] $$d\varphi_\pi (u,\mu)/d\ln \mu = \int^1_0 d\omega V(u,\omega)
\varphi_\pi(\omega,\mu)
\label{BLL}$$ The first step is to calculate the $O(\alpha_s)$ correction to the hard amplitude $T$. The calculation is performed in general covariant gauge in order to have a possibility to check the gauge invariance of the result. Both the ultraviolet (UV) and infrared divergences are regularized by dimensional regularization and renormalized in the $\overline{MS}$ scheme with totally anticommuting $\gamma_5$. This choice is motivated by the fact that the same scheme is used in the calculation of the NLO evolution kernel of the wave function $\varphi_\pi(u)$ \[6\]. After UV renormalization, IR factorization and reexpressing of the $\overline{MS}$ mass by the pole mass, we have obtained $$T(r_1,r_2,u,\mu) =
\frac1{\rho-1}
+\frac{\alpha_s(\mu)C_F}{4\pi}
\Bigg\{\frac1{\rho-1}( -4+3
\ln \frac{m_b^{*2}}{\mu^2})
+\frac{2}{\rho-1} \left[
2 G\left(\rho\right) - G\left(r_1\right) - G\left(r_2\right)
\right]$$ $$+\frac{2}{(r_1-r_2)^2} \left(
\frac{1-r_2}{u} \left[ G\left(\rho\right) -
G\left(r_1\right)\right]
+ \frac{1-r_1}{1-u} \left[ G\left(\rho\right) -
G\left(r_2\right)\right] \right)$$ $$+\frac{\rho+(1-\rho)\ln\left(1-\rho\right)}{\rho^2}
+\frac{2}{\rho-1} \frac{(1-r_2)\ln\left(1-r_2\right)}{r_2}
-\frac{2}{\rho-1}$$ $$-\frac{2}{(1-u)(r_1-r_2)}
\left( \frac{ (1-\rho) \ln\left(1-\rho\right)}{\rho}
- \frac{ (1-r_2) \ln\left(1-r_2\right)}{r_2} \right)
\Bigg\} ~.
\label{result}$$ We used convenient dimensionless variables $r_1 = p^2/m_b^2$ and $r_2=(p+q)^2/m_b^2$ and $$\begin{aligned}
&&\Delta = \frac{2}{4-d}-\gamma_E+\ln (4\pi ),\quad \quad
\rho = r_1 + u (r_2-r_1), \\
G \left(\rho \right)
& = & \mbox{Li}_2(\rho) + \ln^2(1-\rho) +\ln(1-\rho)
\left(\ln\frac{m_b^2}{\mu^2}
+1 \right),\nonumber \end{aligned}$$ $\mbox{Li}_2(x)=-\int\limits^x_0\frac{dt}t \ln(1-t)$ being the Spence function. The UV renormalization scale and the factorization scale of the collinear (COL) divergences are taken to be equal and denoted by $\mu$. As an additional check on the origin of the various divergent terms we have performed additional explicit calculations. In particular, we have used mass regularization by giving the light quarks a small but finite mass, and momentum regularization keeping the light quarks off mass shell. [**4. Numerical results.**]{} The next step is to determine the decay constant $f_B$ and the pion wave function $\varphi_\pi(u,\mu)$ in NLO. For that purpose we have analyzed the two-point sum rule for $f_B$ obtained from the renormalization-group-invariant correlation function\
$m_b^2\langle 0\mid T\{J_5^+(x)J_5(0)\} \mid 0 \rangle$ in $O(\alpha_s)$ \[7\]. For the running coupling constant we use the two-loop expression with $N_f=4$ and $\Lambda^{(4)}=234$ MeV \[8\] corresponding to $\alpha_s(M_Z)= 0.112$. For $\mu^2$ we take the value $\mu^2_B= m_B^2-m_b^{*2}$. corresponding to the average virtuality of the correlation function. With this choice the following correlated results are extracted from the two-point sum rule: $$\begin{aligned}
\label{fbb}
f_{B}=180\pm 30\quad\mbox{MeV} \qquad
m_b^*=4.7\mp0.1\quad\mbox{GeV},\qquad
s_0=35\pm 2\quad\mbox{GeV}^2.\end{aligned}$$ In the following, we adopt the central values in the above intervals. Note that without $O(\alpha_s)$ correction one obtains $f_B = 140 \pm 30 $ MeV. The remaining parameters entering the sum rules are directly measured: $m_B=5.279 $ GeV and $f_{\pi}=132$ MeV.
For the wave function $\varphi_\pi$ we adopt the ansatz suggested in \[9\]: $$\begin{aligned}
\varphi_\pi(u,\mu_0)=\Psi_0(u)+a_2(\mu_0) \Psi_2(u)
+a_4(\mu_0) \Psi_4(u),\end{aligned}$$ where $ \quad \Psi_{n}(u) = 6u (1-u) C_{n}^{3/2}(2 u -1)$. The coefficients $a_2(\mu_0)=2/3$ and $a_4(\mu_0)=0.43$ at the scale $\mu_0=500$ MeV have been extracted \[9\] from a two-point QCD sum rule for the moments of $\varphi_\pi(u)$ \[1\].
Now we are ready to perform a numerical analysis of the sum rule. In Fig. 1, the product $f_Bf^+(0)$ is plotted as a function of the Borel parameter $M^2$. The $O(\alpha_s)$ correction turns out to be large, between 30% and 35% , and stable under variation of $M^2$. Fig. 2 shows the momentum dependence of the form factor $f^+(p^2)$ in the region $0<p^2<15\div17$ GeV$^2$ for $M^2=10$ GeV$^2$, where the sum rule is expected to be valid. Note the almost complete cancellation of the NLO correction in $f^+$. Finally, it is interesting to compare the $\mu$ dependence in LO and NLO (Fig.3). The very mild $\mu$ -dependence in LO results from the evolution of the wave function. In NLO, the $\mu$-dependence is stronger than in LO but similar to the $\mu$-dependence of $f_B$. As a result, the residual scale dependence of $f^+$ is again mild.
[**5. Application.**]{} The above results refer to the leading twist 2 approximation. The thorough numerical analysis of the NLO sum rules have been performed in \[10\] taking into account LO twist 3 and 4 contributions. Here we give preliminary numbers. The final result for the form factor $f^+(r), r=\frac{p^2}{m_B^2}$ can be approximate by the function \[10\] (see also \[14,15\]) $$f^+(r)=\frac{f^+(0)}{1-ar+br^2}.$$ with $a=1.5, b=0.52$ and $$f^+(0) = 0.27\pm 0.02 \pm 0.02 ~.$$ The first uncertainty is connected with the unknown perturbative corrections to the twist-2 ($O(\alpha_s^2)$)and twist-3 ($O(\alpha_s)$) contributions and the second one is connected with the wave functions. This value is $10\% $ lower than the $LO$ estimate $f^+(0) = 0.30 $ obtained in \[4,11\]. Integrating over momentum one obtains the decay width \[10\] $$\Gamma (B^0\to\pi^-e^+\nu_e) =(7.5 \pm 2)|V_{ub}|^2 ps^{-1}.$$ And finally, using the current CLEO number for the $Br(B^0\to\pi l\nu)=(1.8\pm 0.4)
\cdot10^{-4}$ \[12\] and the world average of the $B^0$ lifetime $\tau_{B^0}=(1.56\pm 0.06)$ ps \[8\] one obtains $$|V_{ub}|^{B\to\pi}=0.0039\pm 0.0005_{exp}\pm 0.0005_{th}.$$ where we indicate the theoretical and experimental uncertainty.
[**6. Outlook.**]{} Here we stress that the light cone sum rule gives a reliable estimation for the $f^+(p^2)$ form factor. The present accuracy of the result is estimated to be $15-20\%$ and can be improved up to $10\% $ by including the unknown perturbative $O(\alpha_s^2)$ correction to twist-2 and the $O(\alpha_s)$ correction to the twist-3 contributions and by more accurate extraction of the pion wave functions from the data.\
[**7. Acknowledgements.**]{} We are grateful to R. Rückl, A. Khodjamirian, Ch. Winhart for the collaboration. We would like to thank P. Ball, V. Braun, A. Grozin, M. Neubert, K. Melnikov and A. Vainshtein for useful discussions.
[^1]: This work is supported by BMBF under contract number 05 7WZ91P(0);\
Talk given by O. Yakovlev at IV International Workshop on Progress in Heavy Quark Physics, Rostock, September 20-22.
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---
abstract: |
We continue our study of the Fourier-Stieltjes algebra associated to a twisted (unital, discrete) C\*-dynamical system and discuss how the various notions of equivalence of such systems are reflected at the algebra-level. As an application, we show that the amenability of a system, as defined in our previous work, is preserved under Morita equivalence.
0.9cm 2020 *Mathematics Subject Classification*: Primary 46L55; Secondary 37A55, 43A35, 46H25.
*Keywords*: Fourier-Stieltjes algebra, $C^*$-dynamical system, equivariant representation, cocycle conjugacy, Hilbert bimodule, Morita equivalence, amenability.
author:
- 'Erik Bédos, Roberto Conti\'
title: 'The Fourier-Stieltjes algebra of a $C^*$-dynamical system II'
---
Introduction {#Intro}
============
The classical notion of Fourier-Stieltjes algebra of a locally compact group $G$ [@Eym] was extended in [@BeCo6] to a (unital, discrete) twisted C\*-dynamical system $\Sigma = (A, G, \alpha, \sigma)$. In short, the outcome is a Banach algebra $B(\Sigma)$ attached to $\Sigma$ with a rich analytical structure that can be better described in terms of coefficients of the so-called equivariant representations of $\Sigma$. In the case where $A$ is trivial, any such a representation is nothing but a unitary representation of $G$ on a Hilbert space, and one therefore recovers the Fourier-Stieltjes algebra $B(G)$. Some aspects of the classical theory survive to the new setting, notably the inclusion of $B(\Sigma)$ in the completely bounded full/reduced multipliers of $\Sigma$, as well as the fact that $B(\Sigma)$ is spanned by the $\Sigma$-positive definite functions, which themselves give rise to completely positive maps of the full and reduced twisted crossed product $C^*$-algebras associated to $\Sigma$. We note here that in the case of an untwisted system our concept of $\Sigma$-positive definiteness can be reformulated using the notion of completely positive Herz-Schur $\Sigma$-multiplier (cf. [@MSTT]). We also recall that $B(G)$ continuously embeds into $B(\Sigma)$, although these two algebras differ significantly from each other for it can be shown that, under mild assumptions, $B(\Sigma)$ is always noncommutative (actually, $B(G)$ is contained in the center of $B(\Sigma)$). Finally, we mention that one can use the aforementioned coefficients of equivariant representations of $\Sigma$ to introduce suitable approximation properties for $\Sigma$, such as amenability (cf. [@BeCo6]) and the Haagerup property (cf. [@MSTT]), that parallel the analogous notions for $G$ and provide intrinsic features of the dynamical system $\Sigma$.
The main motivation for this paper was to explore to which extent the Fourier-Stieltjes algebra $B(\Sigma)$ depends on $\Sigma$. We recall that if $G_1$ and $G_2$ are locally compact groups, then Walter showed in [@Wal] that $B(G_1)$ and $B(G_2)$ are isometrically isomorphic as Banach algebras if and only if $G_1$ and $G_2$ are topologically isomorphic. Hence one may hope that $B(\Sigma)$ is better suited to characterize $\Sigma$ than other algebras associated to it. Now there are several natural notions of equivalence between two dynamical systems $\Sigma = (A, G, \alpha, \sigma)$ and $\Theta = (B, H, \beta, \theta )$, most notably exterior equivalence, conjugacy, and cocycle conjugacy, but also Morita equivalence (in the case where $G=H$). It is immediate that the first two notions are stronger than the third one, which is itself stronger than the last one. We show in Theorem \[coc-conj-isom\] that $B(\Sigma)$ and $B(\Theta)$ are isometrically isomorphic whenever $\Sigma$ and $\Theta$ are cocycle conjugate (up to a group isomorphism), in a way that preserves the classical Fourier-Stieltjes algebras of the corresponding groups, and also the canonical copies of the corresponding algebras. In connection with this result, we also note that the Fourier-Stieltjes algebra of a system does not detect a perturbation of the system by a ${{\mathbb{T}}}$-valued group $2$-cocycle, cf. Remark \[coc-pert2\]. In the case of Morita equivalent systems, the connection between the Fourier-Stieltjes algebras remains somewhat more elusive, but we are at least able to show that these algebras can be determined from each other, see Corollary \[M-eq-FS\]. However, as a byproduct of this study, we obtain an interesting consequence for Morita equivalent systems, namely we show in Theorem \[Amen\] that the amenability of a system (as defined in [@BeCo6]) is preserved under such an equivalence.
The paper is organized as follows. After some preliminaries in Section 2, we review in Section 3 some of the natural notions of equivalence for twisted $C^*$-dynamical systems (exterior equivalence, (group) conjugacy, and cocycle (group) conjugacy) and prove that the Fourier-Stieltjes algebra is invariant, up to isometric isomorphism, under cocycle group conjugacy (which is the most general among these notions). In Section 4 we consider two Morita equivalent systems and point out that there is, up to isomorphism, a one-to-one correspondence between the equivariant representations of the respective systems. We use this to show that the corresponding Fourier-Stieltjes algebras can then be recovered from each other. Finally, in Section 5, we recall our definition of amenability for a system and show that this property is Morita invariant.
Preliminaries {#Preliminaries}
=============
We only consider *unital* $C^*$-algebras in this paper, and a homomorphism between two such algebras will always mean a unit preserving $*$-homomorphism. Isomorphisms and automorphisms between $C^*$-algebras are therefore also assumed to be $*$-preserving. The group of unitary elements in a $C^*$-algebra $A$ will be denoted by ${{\mathcal U}}(A)$, the center of $A$ by $Z(A)$, and the group of automorphisms of $A$ by ${\rm Aut}(A)$. The identity map on $A$ will be denoted by ${\rm id}$ (or ${\rm id}_A$). If $B$ is another $C^*$-algebra, $A\otimes B$ will denote their minimal tensor product.
By a Hilbert $C^*$-module, we will always mean a *right* Hilbert $C^*$-module, unless otherwise specified, and follow the notation introduced in [@La1]. In particular, all inner products will be assumed to be linear in the second variable, ${{\mathcal L}}_B(X, Y)$ will denote the space of all adjointable operators between two Hilbert $C^*$-modules $X$ and $Y$ over a $C^*$-algebra $B$, and ${{\mathcal L}}_B(X) = {{\mathcal L}}_B(X,X)$. A representation of a $C^*$-algebra $A$ on a Hilbert $B$-module $Y$ is then a homomorphism from $A$ into the $C^*$-algebra ${{\mathcal L}}_B(Y)$. If $Z$ is another Hilbert $C^*$-module (over $C$), we will let $\pi \otimes \iota : A \to {{\mathcal L}}_{B\otimes C}(Y \otimes Z)$ denote the amplified representation of $A$ on $Y \otimes Z$ given by $(\pi \otimes \iota)(a) = \pi(a) \otimes I_Z$, where the Hilbert $B\otimes C$-module $Y \otimes Z$ is the external tensor product of $Y$ and $Z$ (cf. [@La1]), and $I_Z$ denotes the identity operator on $Z$. If $Z$ is a Hilbert space, then we consider $Y\otimes Z$ as a Hilbert $B$-module.
The quadruple $\Sigma = (A, G, \alpha,\sigma)$ will always denote a *twisted unital discrete $C^*$-dynamical system*. This means that $A$ is a $C^*$-algebra with unit $1_A$, $G$ is a discrete group with identity $e$ and $(\alpha,\sigma)$ is a *twisted action* of $G$ on $A$ (sometimes called a cocycle $G$-action on $A$), that is, $\alpha$ is a map from $G$ into ${\rm Aut}(A)$ and $\sigma: G \times G \to {{\mathcal U}}(A)$ is a normalized $2$-cocycle for $\alpha$, such that $$\begin{aligned}
\alpha_g \alpha_h & = {\rm Ad}(\sigma(g, h)) \alpha_{gh}, \\
\sigma(g,h) \sigma(gh,k) & = \alpha_g(\sigma(h,k)) \sigma(g,hk), \\
\sigma(g,e) & = \sigma(e,g) = 1_A \end{aligned}$$ for all $g,h,k \in G$. Of course, ${\rm Ad}(u)$ denotes here the (inner) automorphism of $A$ implemented by the unitary $u$ in ${{\mathcal U}}(A)$. If $\sigma=1$ is the trivial $2$-cocycle, that is, $\sigma(g,h)=1_A$ for all $g,h \in G$, then $\alpha$ is a genuine action and $\Sigma$ is an ordinary $C^*$-dynamical system (see e.g. [@Wi; @BrOz]), usually denoted by $\Sigma=(A, G, \alpha)$. If $\sigma$ is *central*, that is, it takes values in ${{\mathcal U}}(Z(A))$, then $\alpha$ is also a genuine action of $G$ on $A$, and this is the case studied in [@ZM]. In the sequel we will often just use the word system to mean a discrete unital twisted $C^*$-dynamical system.
An *equivariant representation* of $\Sigma$ on a Hilbert $A$-module $X$ (see e.g. [@BeCo3; @BeCo4]) is a pair $(\rho, v)$ where $\rho : A \to {{\mathcal L}}_A(X)$ is a representation of $A$ on $X$ and $v$ is a map from $G$ into the group $\mathcal{I}(X)$ of all $\mathbb{C}$-linear, invertible, bounded maps from $X$ into itself, which satisfy:
- $\rho(\alpha_g(a)) = v(g) \rho(a) v(g)^{-1}, \quad \quad g\in G, a \in A,$
- $v(g) v(h) = {\rm ad}_\rho(\sigma(g,h)) v(gh), \quad \quad g, h \in G,$
- $\alpha_g\big(\langle x, x' \rangle\big) = \langle v(g) x, v(g) x' \rangle , \quad \quad g\in G, x, x' \in X,$
- $v(g)(x \cdot a) = (v(g) x)\cdot \alpha_g(a),
\quad \quad g \in G, x\in X, a \in A$.
In (ii) above, $ {\rm ad}_\rho(\sigma(g,h)) \in \mathcal{I}(X) $ is defined by $${\rm ad}_\rho(\sigma(g,h)) x = \big(\rho(\sigma(g,h)) x \big)\cdot \sigma(g,h)^*, \quad g, h \in G, x \in X.$$ Note that the equivariant representations of $\Sigma$ may instead be presented in terms of $(\Sigma$,$\Sigma$)-compatible actions, as in [@EKQR-0; @EKQR], cf. Remark \[equirep-action\]. Note also that condition (iii) implies that each $v(g)$ is isometric. For completeness, we mention some examples of equivariant representations. First, the *trivial equivariant representation* of $\Sigma$, which is the pair $(\ell, \alpha)$ acting on $A$, considered as a right $A$-module over itself in the canonical way, where $\ell:A\to {{\mathcal L}}_A(A)$ is given by left-multiplication. Next, let $A^G := \ell^2(G,A)$ denote the right $A$-module given by $$A^G = \Big\{\xi:G\to A \mid \sum_{g\in G} \xi(g)^{*} \xi(g) \text{ is norm-convergent in $A$}\Big\},$$ with the obvious right $A$-module structure, and inner product given by $$\langle \xi, \eta\rangle = \sum_{g\in G} \xi(g)^* \eta(g).$$ Then the *regular equivariant representation* of $\Sigma$ on $A^G$ is the pair $(\check{\ell}, \check{\alpha})$ acting on $A^G$ defined by $$(\check{\ell}(a) \xi)(h) = a \xi(h), \quad
(\check{\alpha}(g)\xi)(h) = \alpha_g(\xi(g^{-1}h))$$ for $a \in A, \xi \in A^G$ and $ g, h \in G$.
More generally, if $(\rho, v)$ is an equivariant representation of $A$ on a right Hilbert $A$-module $X$ and $w$ is a unitary representation of $G$ on some Hilbert space $\mathcal H$, then $(\rho \otimes \iota, v \otimes w)$ is an equivariant representation of $\Sigma$ on $X \otimes {\mathcal H}$.
One can also form the tensor product of equivariant representations. Assume that $(\rho_1,v_1)$ and $(\rho_2,v_2)$ are equivariant representations of $\Sigma$ on some Hilbert $A$-modules $X_1$ and $X_2$, respectively. We can then form the internal tensor product $X_1\otimes_{\rho_2} Y$, which is a right Hilbert $A$-module (cf. [@La1]); we will suppress $\rho_2$ in our notation and denote $X_1\otimes_{\rho_2} X_2$ by $X_1\otimes_A X_2$, as it is quite common in the literature. Then the tensor product $(\rho_1,v_1) \otimes (\rho_2,v_2)$ acts on $X_1 \otimes_{A} X_2$ as follows. For $a \in A$, let $(\rho_1 \otimes \rho_2)(a)\in {{\mathcal L}}_A(X_1 \otimes_{A} X_2) $ be the map determined on simple tensors by $$(\rho_1 \otimes \rho_2)(a) (x_1 \dot\otimes x_2) = \rho_1(a)x_1 \dot\otimes x_2 \quad \text{for } x_1 \in X_1 \text{and } x_2 \in X_2.$$ Moreover, for every $g \in G$, let $(v_1 \otimes v_2)(g)$ in ${{\mathcal I}}(X_1 \otimes_{A} X_2)$ be the map determined on simple tensors by $$(v_1 \otimes v_2)(g) (x_1 \dot\otimes x_2) = v_1(g)x_1 \dot\otimes v_2(g)x_2\quad \text{for } x_1 \in X_1\text{ and } x_2 \in X_2 .$$ Then $(\rho_1,v_1)\otimes(\rho_2,v_2):=(\rho_1 \otimes \rho_2, v_1 \otimes v_2)$ is an equivariant representation of $\Sigma$ on the right Hilbert $A$-module $X_1 \otimes_{A} X_2$ (cf. [@EKQR; @BeCo3]).
Let $(\rho,v)$ be an equivariant representation of $\Sigma$ on a Hilbert $A$-module $X$ and let $x, y \in X$. Then we define $T_{\rho,v,x,y}:G\times A \to A$ by $$T_{\rho,v,x,y}(g,a) = \big\langle x, \rho(a) v(g) y \big \rangle \quad \text{for } a \in A, g \in G,$$ and think of $T_{\rho, v, x, y}$ as an $A$-valued coefficient function associated with $(\rho, v)$.
The *Fourier-Stieltjes algebra* $B(\Sigma)$ is defined in [@BeCo6] as the collection of all the maps from $G\times A$ into $A$ of the form $T_{\rho, v, x, y}$ for some equivariant representation $(\rho,v)$ of $\Sigma$ on a Hilbert $A$-module $X$ and $x, y \in X$. Then $B(\Sigma)$ becomes a unital subalgebra of $L(\Sigma)$, where $$L(\Sigma) =\{ T:G\times A \to A\mid T \text{ is linear in the second variable}\}$$ is equipped with its natural algebra structure: for $T, T' \in L(\Sigma)$ and $\lambda \in {{\mathbb{C}}}$, we let $T+T', \lambda T$, $ T\cdot T'$ and $I_\Sigma$ be the maps in $L(\Sigma)$ defined by $$\begin{aligned}
(T+T')(g,a) & := T(g,a)+T'(g,a) \\
(\lambda T) (g,a) & := \lambda T(g,a) \\
(T \cdot T')(g,a) & := T(g,T'(g,a)) \\
I_\Sigma(g,a) & := a\end{aligned}$$ for $g\in G$ and $a \in A$. Given $T\in L(\Sigma)$ and $g \in G$, we will sometimes write $T_g$ for the linear map from $A$ into itself given by $T_g(a)=T(g, a)$ for all $a\in A$.
If $T\in B(\Sigma)$, letting $\|T\|$ denote the infimum of the set of values $\|x\| \|y\|$ associated with the possible decompositions of $T$ of the form $T = T_{\rho, v, x,y}$, one gets a norm on $B(\Sigma)$ such that $B(\Sigma)$ is a unital Banach algebra w.r.t. $\|\cdot\|$.
We also recall that there is a canonical way of embedding $B(G)$ into $B(\Sigma)$ (cf. [@BeCo6 Proposition 3.2]): For $f \in B(G)$, define $T^f \in L(\Sigma)$ by $T^f(g, a) = f(g) a$ for $g\in G$ and $a \in A$. Then $T^f \in B(\Sigma)$, and the map $f \to T^f $ gives an injective, contractive, algebra-homomorphism of $B(G)$ into $B(\Sigma)$.
The Fourier-Stieltjes algebra $B(\Sigma)$ also contains a copy of $A$. Indeed, for $b \in A$, let $T^b \in L(\Sigma)$ be given by $T^b(g,a) = ba$ for all $g\in G$ and $a\in A$. Then we have that $T^b = T_{\ell, \alpha, b^*\!, 1_A} \in B(\Sigma)$ and $\|T^b\| \leq \|b\|$. From this, one readily deduces that the map $b\to T^b$ gives an isometric algebra-homomorphism from $A$ into $B(\Sigma)$.
Finally, we recall that, as in the classical case, $B(\Sigma)$ is spanned by its positive definite elements (cf. [@BeCo6 Corollary 4.5]). For the ease of the reader, we review how positive definiteness is defined in our setting. Let $T\in L(\Sigma)$. Then $T$ is [called *positive definite*]{} (w.r.t. $\Sigma$), or *$\Sigma$-positive definite*, when for any $n\in {{\mathbb{N}}}$, $g_1, \ldots, g_n \in G$ and $a_1, \ldots, a_n \in A$, the matrix $$\Big[ \alpha_{g_i}\Big(T_{g_i^{-1}g_j}\big(\alpha_{g_i}^{-1}\big(a_i^*a_j\sigma(g_i, g_i^{-1}g_j)^*\big)\big)\Big) \sigma(g_i, g_i^{-1}g_j)\Big]$$ is positive in $M_n(A)$ (the $n\times n$ matrices over $A$). As shown in [@BeCo6 Corollary 4.4], which is an analogue of the Gelfand-Raikov theorem, this is equivalent to requiring that $T$ may be written as $T = T_{\rho, v, x, x}$ for some equivariant representation $(\rho, v)$ of $\Sigma$ on some Hilbert $A$-module $X$ and some $x\in X$. It then follows that $$\|T\|_\infty:=\sup \{\|T_g\|\mid g\in G\} = \|T_e(1_A) \|= \|\langle x, x\rangle_A\|$$ (cf. [@BeCo6 Corollary 4.3]). We set $$P(\Sigma) =\big\{T\in L(\Sigma)\mid T \text{is positive definite $($w.r.t. } \Sigma )\big\}.$$
Cocycle group conjugate systems
===============================
There are various notions of equivalence for $C^*$-dynamical systems in the literature. In this section we will study how the notions of exterior equivalence, (group) conjugacy and cocycle (group) conjugacy are reflected at the level of the Fourier-Stieltjes algebras.
Consider a system $\Sigma = (A, G, \alpha, \sigma)$, and let $w:G\to \mathcal{U}(A)$ be a normalized map, that is, such that $w(e)=1_A$. Then it is well known (cf. [@PaRa Section 3]) that we get another twisted action $(\alpha^w, \sigma^w)$ of $G$ on $A$ by setting $$\alpha_g^w = {\rm Ad}(w(g)) \circ \alpha_g \quad \text{and} \quad \sigma^w(g,g') = w(g)\alpha_g(w(g'))\sigma(g,g') w(gg')^*$$ for all $g, g' \in G$. We then set $\Sigma^w:= (A, G, \alpha^w, \sigma^w)$ and call $\Sigma^w$ a *perturbation of $\Sigma$ by $w$*.
\[coc-pert\] Another way to perturb a system $\Sigma = (A, G, \alpha, \sigma)$ is as follows. Let $\alpha'$ denote the restriction of $\alpha$ to a (genuine) action of $G$ on $Z(A)$, and let $\eta : G\times G \to \mathcal{U}(Z(A))$ be a normalized $2$-cocycle for $\alpha'$. (For example, we can let $\eta : G\times G \to {{\mathbb{T}}}$ be any normalized $2$-cocycle for the group $G$ and consider $\eta$ as a $2$-cocycle for $\alpha'$.) Then we get a twisted action $(\alpha , \sigma_\eta )$ of $G$ on $A$ by setting $$\sigma_\eta (g, g') := \sigma(g,g')\eta (g, g')$$ for all $g, g' \in G$. The system $\Sigma(\eta) := (A, G, \alpha, \sigma_\eta )$ is called a *perturbation of $\Sigma$ by $\eta $*.
Two systems $\Sigma = (A, G, \alpha, \sigma)$ and $\Theta =(A, G, \beta, \theta )$ are called *exterior equivalent*, and we write $\Sigma \sim_e \Theta$, when $\Theta = \Sigma^w$ for some map $w:G\to \mathcal{U}(A)$ (which is then necessarily normalized).
\[ext-eq\] Let $\alpha$ and $\beta$ be two genuine actions of $G$ on $A$ and set $\Sigma = (A, G, \alpha, 1)$ and $\Theta =(A, G, \beta, 1)$. We recall that a map $w: G\to \mathcal{U}(A)$ is called a *$1$-cocycle for $\alpha$* when it satisfies that $w(gg') = w(g)\alpha_g(w(g'))$ for all $g, g'\in G$. Then we have that $ \Sigma \sim_e \Theta$ if and only if there exists some $1$-cocycle $w: G\to \mathcal{U}(A)$ for $\alpha$ such that $\beta_g = {\rm Ad}(w(g)) \circ \alpha_g $ for all $g\in G$. One usually says that $\beta$ is a perturbation of $\alpha$ by $w$ in this case.
Assume now that $\alpha$ and $\beta$ agree up to inner automorphisms, that is, they satisfy that $\beta_g = {\rm Ad}(u(g)) \circ \alpha_g$ for some map $u:G\to \mathcal{U}(A)$, which may be assumed to be normalized. Set $$\partial u(g, h):= u(g)\alpha_g(u(h))u(gh)^*$$ for all $g, h \in G$. Then it can easily be checked that $\partial u$ is a $2$-cocycle for $\beta$ taking its values in $\mathcal{U}(Z(A))$. If $\partial u \neq 1$, i.e., $u$ is not a $1$-cocycle for $\alpha$, then we get that $(\beta, \partial u)$ is a twisted action of $G$ on $A$ satisfying that $\Sigma = (A, G, \alpha, 1) \sim_e (A, G, \beta, \partial u)$. Similarly, $\Theta \sim_e (A, G, \alpha, \partial u^*)$, where $u^*(g):=u(g)^*$ for all $g \in G$.
We note that if the map $u$ above takes its values in $ \mathcal{U}(Z(A))$ (so we have $\beta=\alpha$), and $\alpha'$ denotes the restriction of $\alpha$ to an action of $G$ on $Z(A)$, then $\partial u$ is a normalized $2$-cocycle for $\alpha'$ (called a *coboundary* for $\alpha'$). A perturbation of $\Sigma$ by $u$ is then clearly the same as a perturbation of $\Sigma$ by $\partial u$ (in the sense of Remark \[coc-pert\]), i.e., we have $ \Sigma^u = \Sigma(\partial u) $, and we get that $\Sigma \sim_e \Sigma(\partial u)$ in this case.
Next, consider $\Sigma = (A, G, \alpha, \sigma)$ and note that if $\phi:A\to B$ is an isomorphism of $C^*$-algebras and $\varphi:G\to H$ is an isomorphism of groups, then we get a new system $\Theta = (B, H, \beta, \theta )$ by setting $$\beta_{h}= \phi \circ \alpha_{\varphi^{-1}(h)} \circ \phi^{-1} \quad \text{and} \quad
\theta (h, h') = \phi\big(\sigma(\varphi^{-1}(h), \varphi^{-1}(h'))\big)$$ for all $h, h' \in H$. This motivates the following notion.
Two systems $\Sigma = (A, G, \alpha, \sigma)$ and $\Theta =(B, H, \beta, \theta )$ are said to be *group conjugate* if there exist an isomorphism $\phi:A\to B$ and an isomorphism $\varphi:G\to H$ such that
- $\beta_{\varphi(g)}= \phi \circ \alpha_g \circ \phi^{-1}$,
- $\theta \big(\varphi(g), \varphi(g')\big) = \phi\big(\sigma(g, g')\big)$
for all $g, g' \in G$, in which case we write $ \Sigma \sim_{gc} \Theta$. In the case where $H=G$, we will say that $\Sigma$ and $\Theta $ are *conjugate*, and write $ \Sigma \sim_{c} \Theta$, if $\varphi$ can be chosen to be the identity map.
Two systems $\Sigma = (A, G, \alpha, \sigma)$ and $\Theta =(B, H, \beta, \theta )$ are said to be *cocycle group conjugate* if $\Sigma^w \sim_{gc} \Theta$ for some normalized $w:G\to \mathcal{U}(A)$, in which case we write $ \Sigma \sim_{cgc} \Theta $. Equivalently, as one readily checks, $ \Sigma \sim_{cgc} \Theta$ if and only if $\Theta $ is exterior equivalent to some group conjugate of $\Sigma$. In the case where $H=G$, we will say that $\Sigma$ and $\Theta $ are *cocycle conjugate*, and write $ \Sigma \sim_{cc} \Theta $, if $\Sigma^w$ is conjugate to $\Theta $ for some normalized $w:G\to \mathcal{U}(A)$.
Discarding set-theoretical problems, one may show without much trouble that $\sim_{cgc}$ (resp. $\sim_{cc}$) satisfies the properties of an equivalence relation. Moreover, it is evident from the definitions that (group) conjugacy and exterior equivalence are stronger notions than cocycle (group) conjugacy.
\[excocycony\] Assume again $\alpha$ and $\beta$ are genuine actions of $G$ on $A$. Then we have $(A, G, \alpha, 1) \sim_{cc} (A, G, \beta, 1)$ if and only if $(A, G, \alpha^w, 1^w) \sim_{c} (A, G, \beta, 1)$ for some normalized $w:G\to\mathcal{U}(A)$, in which case we get $1 = 1^w(g,g')
= w(g) \alpha_g(w(g')) w(gg')^*$ for all $g\in G$, so that $w$ is a $1$-cocycle for $\alpha$. Hence $(A, G, \alpha, 1) \sim_{cc} (A, G, \beta, 1) $ if and only if there is a perturbation of $\alpha$ by a $1$-cocycle for $\alpha$ which is conjugate to $\beta$, i.e., $\alpha$ is cocycle conjugate to $\beta$ (as defined for example in [@Bl II.10.3.18]).
It is part of the folklore that the $C^*$-crossed products associated to cocycle conjugate systems are isomorphic, both in the full and in the reduced case, via an isomorphism that preserves the “diagonal” algebra (for partial results in this direction, see e.g. [@PaRa Lemma 3.2] and [@Wi Lemma 2.68]). In our setting, we have:
\[coc-conj-isom\] Assume $\Sigma = (A, G, \alpha, \sigma)$ and $\Theta =(B, H, \beta, \theta )$ are cocycle group conjugate. Then $B(\Sigma)$ and $B(\Theta )$ are isometrically isomorphic.
More precisely, there exists an algebra-isomorphism $\Psi: B(\Theta) \to B(\Sigma)$ such that
- $\Psi$ is isometric$;$
- $\Psi$ maps the copy of $B(H)$ inside $B(\Theta)$ isometrically onto the copy of $B(G)$ inside $B(\Sigma)$ $($w.r.t. the norms of $B(G)$ and $B(H)$$)$$;$
- $\Psi$ restricts to an isomorphism from the copy of $B$ inside $B(\Theta)$ onto the copy of $A$ inside $B(\Sigma)$, and the associated map from $B$ to $A$ is $*$-preserving $($hence isometric$)$.
It clearly suffices to prove the result in the two separate cases where $\Sigma$ and $\Theta$ are group conjugate or exterior equivalent.
Assume first that $\Sigma \sim_{gc} \Theta $ via isomorphisms $\phi:A\to B$ and $\varphi:G\to H$. Then the reader should have no trouble in verifying that the map $\Psi: B(\Theta )\to B(\Sigma)$ given by $$[\Psi(S)](g,a) = \phi^{-1}\big(S(\varphi(g), \phi(a))\big)$$ for $S \in B(\Theta )$, $g \in G$ and $ a\in A$, is a well-defined algebra-isomorphism satisfying 1), 2) and 3).
Next, assume that $\Sigma$ and $\Theta$ are exterior equivalent, so we have $\Theta = \Sigma^w$ for some normalized map $w:G\to \mathcal{U}(A)$, where $\Sigma^w= (A, G, \alpha^w, \sigma^w)$. Noting that $L(\Sigma^w)=L(\Sigma)$, it is straightforward to check that the map $\Pi: L(\Sigma)\to L(\Sigma^w)$ given by $$[\Pi(T)](g,a) = T\big(g, aw(g)\big)w(g)^*$$ for $T \in L(\Sigma)$, $g \in G$ and $ a\in A$, is an algebra-isomorphism.
Now, let $T \in B(\Sigma)$, so $T=T_{\rho, v, x, y}$ for some equivariant representation $(\rho,v)$ of $\Sigma$ on a Hilbert $A$-module $X$ and $x, y \in X$. Then set $\widetilde\rho = \rho$ and define $\widetilde{v}:G \to \mathcal{I}(X)$ by $\widetilde{v}(g) = {\rm ad}_\rho(w(g)) v(g)$, i.e., for each $g\in G$, $$\widetilde{v}(g)x = \big(\rho(w(g))v(g)x\big)\cdot w(g)^*$$ for all $x \in X$. We claim that $(\widetilde\rho,\widetilde{v})$ is an equivariant representation of $\Sigma^w$ on $X$.
Indeed, let $ g, h\in G, a \in A$ and $x, y \in X$. Then, using the properties of $(\rho, v)$ repeatedly, we get:
- $$\begin{aligned}
\widetilde\rho\big(\alpha^w_g(a)\big) \widetilde{v}(g) x
&=\rho\big(w(g)\alpha_g(a)w(g)^*\big) \Big( \big(\rho(w(g)) v(g)x\big)\cdot w(g)^*\Big) \\
&=\big(\rho(w(g)) \rho(\alpha_g(a)) v(g) x \big)\cdot w(g)^* \\
&=\big(\rho(w(g)) v(g) \rho(a) x \big)\cdot w(g)^* \\
&= \widetilde{v}(g) \widetilde\rho(a) x,\end{aligned}$$
- $$\begin{aligned}
& \widetilde{v}(g) \widetilde{v}(h)x = \big(\rho(w(g)) v(g) \widetilde{v}(h) x \big) \cdot w(g)^* \\
& = \Big(\rho(w(g)) v(g) \big( (\rho(w(h)) v(h) x) \cdot w(h)^* \big) \Big) \cdot w(g)^* \\
& = \Big(\rho(w(g)) \Big( \big( (v(g) \rho(w(h)) v(h) x) \cdot \alpha_g(w(h))^* \big) \Big) \Big) \cdot w(g)^* \\
& = \Big(\rho(w(g)) \big( (\rho(\alpha_g((w(h))) v(g) v(h) x) \cdot \alpha_g(w(h))^* \big) \Big) \cdot w(g)^* \\
& = \Big( \big( \rho(w(g)) \rho(\alpha_g(w(h))) v(g) v(h) x \big) \cdot \alpha_g(w(h))^* \Big) \cdot w(g)^* \\
& = \Big( \rho(w(g)) \rho(\alpha_g(w(h))) \big( (\rho(\sigma(g,h)) v(gh) x) \cdot \sigma(g,h)^* ) \big) \Big) \cdot \alpha_g(w(h))^* w(g)^* \\
& = \Big( \rho(\sigma^w(g,h)) \rho(w(gh)) v(gh) x \Big) \cdot \sigma(g,h)^* \alpha_g(w(h))^* w(g)^* \\
& = \Big( \rho(\sigma^w(g,h)) \rho(w(gh)) v(gh) x \Big) \cdot w(gh)^* w(gh) \sigma(g,h)^* \alpha_g(w(h))^* w(g)^* \\
& = \Big( \rho(\sigma^w(g,h)) \big( ( \rho(w(gh)) v(gh) x ) \cdot w(gh)^* \big) \Big) \cdot \sigma^w(g,h)^* \\
& = \Big( \rho(\sigma^w(g,h)) \big( \widetilde{v}(gh)x \big) \Big) \cdot \sigma^w(g,h)^* \\
& = {\rm ad}_{\widetilde\rho}(\sigma^w(g,h)) \widetilde{v}(gh)x, \\\end{aligned}$$
- $$\begin{aligned}
\alpha^w_g\big(\langle x , y \rangle\big) &= w(g) \alpha_g\big(\langle x , y \rangle\big) w(g)^*\\
&= w(g) \big\langle v(g) x , v(g) y \big\rangle w(g)^* \\
&= w(g) \big\langle \rho(w(g)) v(g) x, \rho(w(g)) v(g) y \big\rangle w(g)^* \\
&= \big\langle \big(\rho(w(g)) v(g) x\big)\cdot w(g)^*, \big(\rho(w(g)) v(g) y\big) \cdot w(g)^* \big\rangle \\
&= \big\langle \widetilde{v}(g) x, \widetilde{v}(g) y \big\rangle,\end{aligned}$$
- $$\begin{aligned}
\widetilde{v}(g)(x \cdot a) & = \big( \rho(w(g)) v(g) (x \cdot a) \big) \cdot w(g)^* \\
& = \rho(w(g)) \big( (v(g) x) \cdot (\alpha_g(a) w(g)^*) \big) \\
& = \rho(w(g)) \big( (v(g)x) \cdot (w(g)^* \alpha_g^w(a)) \big) \\
& = \Big( \rho(w(g))\big((v(g)x) \cdot w(g)^*\big)\Big) \cdot \alpha_g^w(a) \\
& = (\widetilde{v}(g)x) \cdot \alpha_g^w(a),\end{aligned}$$
as claimed. Now for all $g \in G$ and $a\in A$ we have $$\begin{aligned}
[\Pi(T)](g,a) &= T_{\rho, v, x, y}\big(g, aw(g)\big)w(g)^* = \big\langle x, \rho(aw(g)) v(g) y\big\rangle w(g)^*\\
&= \big\langle x, \big(\rho(a)\rho(w(g)) v(g) y\big)\cdot w(g)^*\big\rangle = \big\langle x, \widetilde\rho(a)\widetilde{v}(g) y\big\rangle\\
&= T_{\widetilde\rho, \widetilde{v}, x, y} (g,a),\end{aligned}$$ so we get that $\Pi$ maps $B(\Sigma)$ into $B(\Sigma^w)$ and that $\|\Pi(T)\| \leq \|x\| \|y\|$. Since this inequality holds for any $\rho,v, x,y$ such that $T= T_{\rho, v, x, y}$, it follows that $\|\Pi(T)\| \leq \|T\|$. By symmetry, we then see that $\Pi$ restricts to an isometric algebra-isomorphism between $B(\Sigma)$ and $B(\Sigma^w)$. It follows that $\Psi := \Pi^{-1}$ is an algebra-isomorphism from $B(\Theta) = B(\Sigma^w)$ onto $B(\Sigma)$ such that 1) holds. In passing, we note that one can also easily deduce that $\Pi(T)$ is $\Sigma^w$-positive definite whenever $T$ is $\Sigma$-positive definite, either by a direct computation, or using what we just have done in combination with the Gelfand-Raikov characterization of positive definiteness (cf. [@BeCo6 Corollary 4.4]).
Let now $f\in B(G)$ and consider $T^f \in B(\Sigma)$. Then we have that $$\Pi(T^f)(g, a) = T^f\!\big(g, aw(g)\big)w(g)^* = \big(f(g) aw(g)\big)w(g)^* = f(g) a = T^f\!(g,a)$$ for all $g\in G$ and $a\in A$, which shows that $\Pi(T^f) = T^f \in B(\Sigma^w)$. Thus it is clear that $\Pi$ restricts to the identity map from $B(G)$ (inside $B(\Sigma)$) into $B(G)$ (inside $B(\Sigma^w)$), hence that $\Psi = \Pi^{-1}$ satisfies 2).
Finally, let $b\in A$ and consider $T^b \in B(\Sigma)$. Then we have that $$\Pi(T^b)(g, a) = T^b\big(g, aw(g)\big)w(g)^* = baw(g)w(g)^* = b a = T^b(g,a)$$ for all $g\in G$ and $a\in A$. Thus it is clear that $\Pi$ restricts to the identity map from $A$ (inside $B(\Sigma)$) into $A$ (inside $B(\Sigma^w)$), hence that $\Psi = \Pi^{-1}$ satisfies 3).
The converse of Theorem \[coc-conj-isom\] is not true in general. Indeed, set $$Z^2(G, {{\mathbb{T}}}) = \{ \omega:G\times G\to {{\mathbb{T}}}\mid \omega \text{ is a normalized $2$-cocycle on $G$}\}.$$ Then let $\omega \in Z^2(G, {{\mathbb{T}}})$ and consider the systems $\Sigma =({{\mathbb{C}}}, G, {\rm triv}, 1)$ and $\Theta=({{\mathbb{C}}}, G, {\rm triv}, \omega)$, where ${\rm triv}$ denotes the obvious action of $G$ on ${{\mathbb{C}}}$. Then we have that $B(\Sigma) = B(G) = B(\Theta)$, but $\Sigma$ is not cocycle group conjugate to $\Theta$ if $\omega$ is not a coboundary.
\[coc-pert2\] In order to look for a converse of Theorem \[coc-conj-isom\] one option is to weaken cocycle group conjugacy as follows. If $\Sigma = (A, G, \alpha, \sigma)$ is a system and $\omega \in Z^2(G,{{\mathbb{T}}})$, then we may regard $\omega$ as a normalized $2$-cocycle for the restriction of $\alpha$ to $Z(A)$ and perturb $\Sigma$ by $\omega$ (cf. Remark \[coc-pert\]). Obviously, $\Sigma$ and $\Sigma(\omega)= (A, G, \alpha, \sigma_\omega)$ have then the same equivariant representations, so we have that $B(\Sigma)=B(\Sigma(\omega))$.
If $\Theta =(B, H, \beta, \theta )$ is another system, let us say that $\Sigma$ and $\Theta$ are *weakly cocycle group conjugate* if $\Sigma(\omega)$ is cocycle group conjugate to $\Theta$ for some $\omega \in Z^2(G, {{\mathbb{T}}})$. Using Theorem \[coc-conj-isom\] we get that $B(\Theta)$ is then isomorphic to $B(\Sigma(\omega))= B(\Sigma)$ via an algebra-isomorphism satisfying 1), 2) and 3).
Let us now assume that the conclusion of Theorem \[coc-conj-isom\] holds. One may then wonder under which additional requirements it would be possible to conclude that $\Sigma$ and $\Theta$ are weakly cocycle group conjugate. A result in this direction goes as follows.
By invoking Walter’s theorem recalled in the introduction we get from 2) that $\Psi$ determines an isomorphism $\varphi:G\to H$, while 3) gives that there is a $*$-isomorphism $\phi:A \to B$. For each $g \in G$, set $\gamma_g := \phi^{-1}\beta_{\varphi(g)} \phi \in {\rm Aut}(A)$. Then one may check whether $\gamma_g$ and $\alpha_g$ agree up to inner automorphisms for every $g \in G$. Assume that this happens to be the case, i.e., there exists some normalized map $w:G\to{{\mathcal U}}(A)$ such that $\gamma_g = {\rm Ad}(w(g)) \alpha_g$ for all $g\in G$. Then, letting $u:G\times G \to {{\mathcal U}}(A)$ be defined by $$u(g, g')= \phi^{-1}\big(\theta(\varphi(g), \varphi(g'))\big) \quad \text{ for all } g, g' \in G,$$ we get a twisted action $(\gamma, u)$ of $G$ on $A$. Define then a map $\omega:G\times G \to {{\mathcal U}}(A)$ by $$\omega(g, g') := u(g, g') \sigma^w(g, g')^*$$ for all $g, g' \in G$. Then, using the two expressions for $\gamma$ and making use of some cocycle identities, one verifies that $\omega$ takes its values in $Z(A)$, and that it is a $2$-cocycle for $\alpha'$ (the restriction of $\alpha$ to $Z(A)$). Since $u = (\sigma^w)_\omega$, it follows that $$\Theta = (B, H, \beta, \theta) \sim_{gc} (A, G, \gamma, u) = (A, G, \alpha^w,
(\sigma^w)_\omega) = \Sigma^w(\omega)$$ (using notation as in Remark \[coc-pert\]). Hence, if $A$ (and therefore $B$) has trivial center, we get that $\omega \in Z^2(G, {{\mathbb{T}}})$ and $\Theta$ is group conjugate to $\Sigma^w(\omega)$, which is exterior equivalent to $\Sigma(\omega)$. Thus, $\Sigma$ and $\Theta$ are weakly group cocycle conjugate in this case.
As a consequence, we obtain the following.
Consider two systems $\Sigma = (A, G, \alpha, \sigma)$ and $\Theta = (B, H, \beta, \theta )$. Assume that there exists an algebraic isomorphism $\Pi: B(\Sigma) \to B(\Theta)$ satisfying that $$\label{eqPi}
\Pi(T)\big(\varphi(g),\phi(a)\big) = \phi\big(T(g,aw(g))w(g)^*\big) \quad \text{ for all} \ g \in G, a\in A,$$ for some isomorphism $\varphi: G \to H$, some $*$-isomorphism $\phi: A \to B$ and some map $w: G \to {{\mathcal U}}(A)$, which also satisfies $$\label{eqPi2}
\Pi(T_{\ell_A,\alpha,x,y}) = T_{\ell_B,\beta,\phi(x),\phi(y)}$$ for all $x,y \in A$. If the center of $A$ is trivial, then $\Sigma$ and $\Theta$ are weakly cocycle group conjugate.
Using (\[eqPi\]) and (\[eqPi2\]), one deduces that $ \phi^{-1}\beta_{\varphi(g)}\phi={\rm Ad}(w(g)) \alpha_g$ for all $g \in G$. We are then in the position to proceed as we did above, and the desired assertion follows at once.
On Morita equivalent systems
============================
Let us consider two twisted unital discrete $C^*$-dynamical systems $\Sigma=(A,G,\alpha,\sigma)$ and $\Theta=(B,G,\beta,\theta)$ over the same group $G$. (We will briefly discuss the more general situation in Remark \[weak ME\].) Our main aim in this section is to show that if $\Sigma$ and $\Theta$ are *Morita equivalent* in the sense of [@Bui; @Kal], then the Fourier-Stieltjes algebras $B(\Sigma)$ and $B(\Theta)$ can be determined from each other. Morita equivalence for (untwisted) $C^*$-dynamical systems goes at least back to [@Com]. For the ease of the reader, we review the definitions of the concepts that we will use.
Following [@EKQR], we say that a right Hilbert $B$-module $Z$ is a *right Hilbert $A$-$B$ bimodule* if there is a homomorphism $\kappa: A \to {{\mathcal L}}_B(Z)$.[^1] We set $a\cdot z = \kappa(a)z$ for $a\in A$ and $z \in Z$, and frequently write $_AZ_B$ for $Z$. A *right Hilbert $A$-$B$ bimodule isomorphism* $\Phi : _A\!\!Z_B \to _A\!\!W_B$ between two right $A$-$B$ Hilbert bimodules $Z$ and $W$ (or simply an isomorphism, for short) is a bimodule isomorphism such that $\langle\Phi(z), \Phi(z')\rangle_B = \langle z, z'\rangle_B$ for $z, z' \in _A\!\!Z_B$. Left Hilbert $A$-$B$ bimodules and their isomorphisms are defined in a similar way.
Let $_AZ_B$ be a right Hilbert $A$-$B$-bimodule. A map $\delta$ from $G$ into ${{\mathcal I}}(Z)$ (the group of invertible ${{\mathbb{C}}}$-linear bounded maps from $Z$ into itself) is called a $(\Sigma,\Theta)$-*compatible action of $G$ on $_AZ_B$* when the following conditions are satisfied for $g \in G, a \in A$, $z,\zeta \in Z$ and $b \in B$:
- $\delta(g)(a\cdot z) = \alpha_g(a) \cdot (\delta(g)z)$,
- $\delta(g)(z\cdot b) = (\delta(g)z)\cdot \beta_g(b)$,
- $\delta(g) \delta(h) z = \sigma(g,h) \cdot (\delta(gh)z) \cdot \theta(g,h)^*$,
- $\big\langle \delta(g)z,\delta(g)\zeta \big\rangle_B = \beta_g(\langle z,\zeta\rangle_B)$.
We will let $S_{\delta, z, \zeta}: G\times A\to B$ be the map defined by $$ S_{\delta, z, \zeta}(g,a) = \big\langle z, a\cdot (\delta(g)\zeta)\big\rangle_B$$ for all $g\in G$ and $a\in A$. Clearly, if $g\in G$ is fixed, the map $a\to S_{\delta, z, \zeta}(g,a)$ from $A$ into $B$ is linear; moreover, it is bounded, since one easily shows that $$\|S_{\delta, z, \zeta}(g,a)\| \leq \|z\| \|\zeta\| \|a\|$$ for all $a\in A$.
Two $(\Sigma,\Theta)$-compatible actions $\delta$ and $\delta'$ of $G$, acting respectively on $_AZ_B$ and $_AZ'_B$, are called *equivariantly isomorphic* if there exists an isomorphism of right Hilbert $A$-$B$-bimodules between $_AZ_B$ and $_AZ'_B$ which intertwines $\delta$ and $\delta'$.
\[equirep-action\] If $(\rho,v)$ is an equivariant representation of $\Sigma$ on a right Hilbert $A$-module $X$, then $X$ is a right Hilbert $A$-$A$-bimodule (using $\rho$ as the left action of $A$ on $X$) and $v$ is a $(\Sigma,\Sigma)$-compatible action of $G$ on $_A X_A$. Conversely, if $v$ is a $(\Sigma,\Sigma)$-compatible action of $G$ on a right Hilbert $A$-$A$-bimodule $X$, where the left action of $A$ on $X$ is given by some homomorphism $\rho:A\to {{\mathcal L}}_A(X)$, then $(\rho, v)$ is an equivariant representation of $\Sigma$ on $X$. For example, if we consider $A$ as a right Hilbert $A$-$A$-bimodule in the obvious way, then the map $\alpha: G \to {{\mathcal I}}(A)$ is a $(\Sigma,\Sigma)$-compatible action of $G$ on $_AA_A$, corresponding to the trivial equivariant representation $(\alpha, \ell)$ of $\Sigma$ on $A$.
We recall that a right Hilbert $B$-module $X$ is called *full* when $\overline{\langle X, X\rangle} = B$. Fullness of a left Hilbert $C^*$-module is defined in a similar way. An *$A$-$B$ imprimitivity bimodule* $Z={} _AZ_B$ (sometimes called an equivalence $A$-$B$-bimodule) is a full right Hilbert $A$-$B$-bimodule w.r.t. a $B$-valued inner product $\langle \cdot, \cdot \rangle_B$, which is also a full left Hilbert $A$-$B$-bimodule w.r.t. to an $A$-valued inner product $_A\langle \cdot, \cdot \rangle$, in such a way that $$_A\langle z, z'\rangle\cdot z'' = z\cdot \langle z', z''\rangle_B$$ for all $z, z', z'' \in Z$. It then follows that $\|_A\langle z, z\rangle\| =\|\langle z, z\rangle_B\|$ for all $z \in Z$, hence that the two norms on $Z$ associated to the left and the right inner products coincide.
Following [@Bui; @Kal], we say that the two systems $\Sigma$ and $\Theta$ are *Morita equivalent* when there exist an $A$-$B$ imprimitivity bimodule $Z$ together with a $(\Sigma,\Theta)$-compatible action $\delta$ of $G$ on $Z$; we then write $\Sigma \sim _{(Z,\delta)} \Theta$. We note that $\delta$ automatically satisfies
- $_A\big\langle \delta(g)z,\delta(g)\zeta \big\rangle = \alpha_g(_A\langle z,\zeta\rangle),$
see e.g. the argument given in [@EKQR Remark 2.6 (2)].
It is easy to check that $\Sigma$ and $\Theta$ are Morita equivalent whenever they are cocycle conjugate (see e.g. [@Com Section 9] for the untwisted case). Moreover, Morita equivalent twisted $C^*$-dynamical systems have Morita equivalent $C^*$-crossed products (see [@Bui Theorem 2.3] for the full case, and [@CELY Sections 2.5.4 and 2.8.6] for the reduced case). We also mention the following result, which is probably a part of the folklore on this topic.
\[MR-comm\] Assume that $\Sigma=(A,G,\alpha,\sigma)$ and $\Theta=(B,G,\beta,\theta)$ are Morita equivalent, and that $A$ and $B$ are commutative. Then the action $\alpha$ of $G$ on $A$ is conjugate to the action $\beta$ of $G$ on $B$, i.e., there exists an isomorphism $\phi$ from $A$ onto $B$ which intertwines these actions. Moreover, $\Sigma$ is conjugate to the system $(B, G, \beta, \sigma_\phi)$, while $\Theta $ is conjugate to the system $(A, G, \alpha, \theta_{\phi^{-1}})$, where $\sigma_\phi(g, h):= \phi(\sigma(g, h))$ and $\theta_{\phi^{-1}}(g,h):= \phi^{-1}(\theta(g,h))$ for all $g, h \in G$.
The assumption says that $\Sigma \sim _{(X,\delta)} \Theta$ for some $A$-$B$ imprimitivity bimodule $Z$ and some $(\Sigma,\Theta)$-compatible action $\delta$ of $G$ on $Z$. In particular, $A$ and $B$ are Morita equivalent. As $A, B$ are both commutative, we can then apply [@BCL Theorem 2.24] to conclude that there is a unique isomorphism $\phi:A \to B$ satisfying that $$\label{phi-prop}
\phi(_A\langle z, z'\rangle ) = \langle z', z\rangle_B \quad \text{for all } z, z'\in Z,$$ and we also have that $a\cdot z = z\cdot \phi(a) $ for all $a\in A$ and $z\in Z$. Using properties of $\delta$ in combination with (\[phi-prop\]) we get $$\begin{aligned}
\phi\big(\alpha_g(_A\langle z, z'\rangle) \big)&= \phi\big(_A\langle\delta(g) z, \delta(g)z'\rangle \big)= \langle\delta(g) z', \delta(g)z\rangle_B \\
& = \beta_g( \langle z', z\rangle_B) = \beta_g\big( \phi(_A\langle z, z'\rangle)\big)\end{aligned}$$ for all $g \in G$ and $z, z'\in Z$. Since $Z$ is full as a left Hilbert $A$-module, it follows that $\phi \alpha_g = \beta_g \phi$ for every $g\in G$, hence that $\alpha$ and $\beta$ are conjugate. This shows the first part of the proposition. The second part follows immediately.
In the setting of Proposition \[MR-comm\], it is not clear that $\Sigma$ and $\Theta$ are conjugate. However, this is certainly the case when $\sigma$ and $\theta$ are both trivial:
\[com-M-eq\] Suppose that $(A, G, \alpha)$ and $(B, G, \beta)$ are $($untwisted discrete unital$)$ $C^*$-dynamical systems with both $A$ and $B$ commutative. Then these systems are Morita equivalent if and only if they are conjugate, in which case the associated Fourier-Stieltjes algebras are isometrically isomorphic.
This follows from Proposition \[MR-comm\] and Theorem \[coc-conj-isom\].
Assume now that $\Omega=(C, G, \gamma, \omega)$ is another twisted discrete unital $C^*$-dynamical system, $\delta$ is a $(\Sigma, \Theta)$-compatible action of $G$ on $_AX_B$ and $\eta$ is a $(\Theta, \Omega)$-compatible action of $G$ on $_BY_C$. If $\pi:B\to {{\mathcal L}}_C(Y)$ denotes the left action of $B$ on $Y$, we can form the internal tensor product $X\otimes_\pi Y$, which is a right Hilbert $C$-module (cf. [@La1]); we will suppress $\pi$ in our notation and denote $X\otimes_\pi Y$ by $X\otimes_B Y$ in the sequel, as is common in the literature. Moreover, $X\otimes_B Y$ can be turned into a right Hilbert $A$-$C$ bimodule, the left action of $A$ on $X\otimes_B Y$ being given on simple tensors by $a\cdot(x\dot\otimes y) = (a\cdot x) \dot\otimes y$, and we can define a $(\Sigma, \Omega)$-compatible *product action* $\delta\otimes_B\eta$ of $G$ on $_A(X\otimes_B Y)_C$, which is given on simple tensors by $(\delta\otimes_B\eta)(g)(x\dot\otimes y) = \delta(g)x \dot\otimes \eta(g)y$. Indeed, as a sample, consider $g, h \in G$, $x \in X$ and $y \in Y$. Then we have $$\begin{aligned}
\big((\delta\otimes_B\eta)&(g) (\delta\otimes_B\eta)(h)\big) (x\dot\otimes y)
= (\delta\otimes_B\eta)(g)\big( \delta(h)x \dot\otimes \eta(h)y\big) \\
& =\delta(g)\delta(h)x \dot\otimes \eta(g)\eta(h)y\\
&=\big(\sigma(g,h) \cdot (\delta(gh)x) \cdot \theta(g,h)^*\big) \dot\otimes \big(\theta(g,h) \cdot (\eta(gh)x) \cdot \omega(g,h)^*\big)\\
&= \big(\big(\sigma(g,h) \cdot (\delta(gh)x) \cdot \theta(g,h)^*\big)\cdot \theta(g,h)\big)\dot\otimes \big(\big(\eta(gh)y\big) \cdot \omega(g,h)^*\big) \\
&= \sigma(g,h) \cdot \big(\delta(gh)x \dot\otimes \eta(gh)y\big) \cdot \omega(g,h)^* \\
&= \sigma(g,h) \cdot \big((\delta\otimes_B\eta)(gh)(x\dot\otimes y)\big) \cdot \omega(g,h)^* \end{aligned}$$ Thus, by continuity, it follows that $\delta\otimes_B\eta$ satisfies the third property required for being a $(\Sigma, \Omega)$-compatible action. The reader will find more details about this construction and its properties in [@EKQR-0; @EKQR]. These articles deal with the untwisted case, but it is easy to adapt the proofs to our setting. In particular, arguing as in the proof of [@EKQR Theorem 2.8 and Remark 2.9], we obtain that the following facts hold:
- Up to equivariant isomorphism, the product of compatible actions is associative.
- Recalling that $\alpha$ is a $(\Sigma, \Sigma)$-compatible action of $G$ on $_AA_A$, the $(\Sigma, \Theta)$-compatible product action $\alpha\otimes_A\delta$ of $G$ on $_A(A\otimes_AX)_B$ is equivariantly isomorphic to $\delta$. In a similar way, the product action $\delta\otimes_B\beta$ of $G$ on $_A(X\otimes_B B)_B$ is equivariantly isomorphic to $\delta$.
- Assume that $\Sigma$ and $\Theta$ are Morita equivalent with $\Sigma \sim _{(Z,\delta)} \Theta$. Then we have:
- $\Theta \sim _{(\widetilde{Z},\widetilde\delta)} \Sigma$, where $\widetilde{Z}$ is the right Hilbert $B$-$A$ bimodule conjugate (or reverse) to $Z$ and $\widetilde{\delta}$ is the $(\Theta, \Sigma)$-compatible action of $G$ on $\widetilde{Z}$ given by $\widetilde\delta(g) \widetilde{z} = \widetilde{\delta(g) z}$.
- The product action $\delta\otimes_B \widetilde{\delta}$ of $G$ on $_A(Z\otimes_B\widetilde{Z})_A$ is equivariantly isomorphic, as a $(\Sigma, \Sigma)$-compatible action, to $\alpha$.
- The product action $\widetilde\delta\otimes_A \delta$ of $G$ on $_B(\widetilde{Z}\otimes_A Z)_B$ is equivariantly isomorphic, as a $(\Theta, \Theta)$-compatible action, to $\beta$.
Next, consider a $(\Sigma, \Sigma)$-compatible action $v$ of $G$ on a right Hilbert $A$-$A$ bimodule $X$. We will use the same notation as in [@EKQR] and let $[X,v]$ denote the class of all pairs $(X', v')$ where $v'$ is a $(\Sigma, \Sigma)$-compatible action of $G$ on a right Hilbert $A$-$A$-module $X'$ such that $v'$ is equivariantly isomorphic to $v$. Further, we will let $\mathcal{A}(\Sigma)$ denote the collection of these equivalence classes. Using the above properties, one sees that $\mathcal{A}(\Sigma)$ can be equipped with an associative product given by $$[X_1, v_1] [X_2,v_2] := [ X_1\otimes_A X_2, v_1\otimes_A v_2],$$ and that $[A, \alpha]$ acts as a unit in $\mathcal{A}(\Sigma)$. Moreover, one readily gets the following result.
\[Mor-eq-action\] Assume that the systems $\Sigma$ and $\Theta$ are Morita equivalent with $\Sigma \sim _{(Z,\delta)} \Theta$, and let $v$ be a $(\Sigma, \Sigma)$-compatible action on a right Hilbert $A$-$A$ bimodule $X$.
Then $w:=(\widetilde{\delta}\otimes_A v)\otimes_A \delta$ is a $(\Theta, \Theta)$-compatible action on the right Hilbert $B$-$B$-bimodule $Y:= (\widetilde{Z}\otimes_AX)\otimes_A Z$.
Moreover, the action $\delta\otimes_B (w\otimes_B \widetilde{\delta})$ on the right Hilbert $A$-$A$-bimodule $ \widetilde{Z} \otimes_B (Y\otimes_B Z)$ is equivariantly isomorphic to $v$.
Hence, the map $[X,v] \mapsto [Y,w]$ gives a one-to-one correspondence between $\mathcal{A}(\Sigma)$ and $ \mathcal{A}(\Theta) $ which preserves products.
Taking into account Remark \[equirep-action\] this result says that, up to isomorphism, the equivariant representations of two Morita equivalent systems are in a one-to-one correspondence. As we will soon see, this has some relevance for the associated Fourier-Stieltjes algebras. By isomorphism of equivariant representations of a system, we mean the following.
Let $(\rho,v), (\rho',v')$ be equivariant representations of $\Sigma$ on right Hilbert $A$-modules $X$ and $X'$, respectively. Then $(\rho,v)$ and $ (\rho',v')$ are said to be *isomorphic* if $v$ and $v'$ are equivariantly isomorphic as $(\Sigma,\Sigma)$-compatible actions of $G$, i.e., there exists an isomorphism of right Hilbert $A$-modules $\phi: X \to X'$ which intertwines $v$ and $v'$, as well as $\rho $ and $\rho'$. We note that in this case we have $$T_{\rho, v, x, y} = T_{\rho', v', \phi(x), \phi(y)}$$ for all $x, y \in X$. Indeed, for each $a\in A$ and $g\in G$, we have $$\begin{aligned}
T_{\rho, v, x, y} (g, a) &= \big\langle x, \rho(a)v(g) y\big\rangle = \big\langle \phi(x), \phi\big(\rho(a)v(g)y\big) \big\rangle' \\
&= \big\langle \phi(x), \rho'(a)v'(g)\phi(y) \big\rangle' = T_{\rho', v', \phi(x), \phi(y)} (g, a).
\end{aligned}$$
The following notation will be useful. If $S:G\times A \to B, T:G\times A \to A$ and $R: G\times B \to A$ are maps, then we let $S\cdot T: G\times A \to B$ and $T\cdot R: G\times B\to A$ be the maps given by $$(S\cdot T)(g, a) = S(g, T(g,a)), \quad (T\cdot R)(g, b)= T(g, R(g,b))$$ for all $g\in G, a\in A$ and $b\in B$. Moreover, we let $S\cdot T \cdot R: G\times B\to B$ be given by $$S\cdot T \cdot R:= (S\cdot T) \cdot R= S \cdot (T\cdot R).$$
\[Morita1\] Assume that the systems $\Sigma$ and $\Theta$ are Morita equivalent with $\Sigma \sim _{(Z,\delta)} \Theta$, and let $(\rho,v)$ be an equivariant representation of $\Sigma$ on a right Hilbert $A$-module $X$. Let $x, x' \in X$ and $z, z', \zeta, \zeta' \in Z$. Then the map $$S_{\delta, z', \zeta'} \cdot T_{\rho, v, x, x'} \cdot S_{\tilde{\delta}, \tilde{z}, \tilde{\zeta}}:G\times B \to B$$ belongs to $B(\Theta)$. Thus we get a linear map $F_{z, z', \zeta, \zeta'} : B(\Sigma) \to B(\Theta)$ given by $$F_{z, z', \zeta, \zeta'}(T) = S_{\delta, z', \zeta'}\cdot T \cdot S_{\tilde{\delta}, \tilde{z}, \tilde{\zeta}}$$ for every $T\in B(\Sigma)$. Similarly, the assignment $T' \mapsto S_{\tilde{\delta}, \tilde{z}, \tilde{\zeta}}\cdot T'\cdot S_{\delta, z', \zeta'}$ gives a linear map from $B(\Theta)$ into $B(\Sigma)$.
Let $Y= (\widetilde{Z}\otimes_AX)\otimes_A Z$ and $w=(\widetilde{\delta}\otimes v)\otimes \delta: G \to \mathcal{I}(Y)$ be as in Proposition \[Mor-eq-action\], and let $\tau:B \to {{\mathcal L}}_B(Y)$ denote the homomorphism coming from the left action of $B$ on $Y$, so $(\tau, w)$ is an equivariant representation of $\Theta$ on the right Hilbert $B$-module $Y$.
Let $g\in G$ and $b \in B$. Then we have $$\label{eq-imp-map} T_{\tau, w, (\tilde z\dot\otimes x) \dot \otimes z'\:, (\tilde\zeta\dot\otimes x') \dot \otimes \zeta'} (g, b) =
\Big\langle z' , T_{\rho, v, x, x'}\big( g, _A\big\langle z\cdot b, \delta(g)\zeta\big\rangle\big) \cdot \delta(g)\zeta'\Big\rangle_B.$$ Indeed, $$\begin{aligned}
T_{\tau, w, (\tilde z\dot\otimes x) \dot \otimes z', (\tilde\zeta \dot\otimes x') \dot \otimes \zeta'} &(g, b)
= \Big\langle (\tilde z\dot\otimes x) \dot \otimes z',\tau(b)w(g) (\tilde\zeta\dot\otimes x') \dot \otimes \zeta'\Big\rangle_B\\
&= \Big\langle (\tilde z\dot\otimes x) \dot \otimes z', \big( ((\delta(g)\zeta)\cdot b^*)^{\widetilde{}} \dot\otimes v(g)x'\big) \dot \otimes \delta(g)\zeta'\Big\rangle_B\\
&= \Big\langle z', \big\langle \tilde z\dot\otimes x,((\delta(g)\zeta)\cdot b^*)^{\widetilde{}} \dot\otimes v(g)x'\big\rangle_A \cdot \delta(g)\zeta'\Big\rangle_B\\
&= \Big\langle z', \big\langle x,\langle \tilde z, ((\delta(g)\zeta)\cdot b^*)^{\widetilde{}}\rangle_A \cdot v(g)x'\big\rangle_A \cdot \delta(g)\zeta'\Big\rangle_B\\
&= \Big\langle z', \big\langle x, _A\langle z, (\delta(g)\zeta)\cdot b^*\rangle \cdot v(g)x'\big\rangle_A \cdot \delta(g)\zeta'\Big\rangle_B\\
&= \Big\langle z', \big\langle x, \rho\big(_A\langle z\cdot b, \delta(g)\zeta\rangle\big) v(g)x'\big\rangle_A \cdot \delta(g)\zeta'\Big\rangle_B\\
&=\Big\langle z' , T_{\rho, v, x, x'}\big( g, _A\big\langle z\cdot b, \delta(g)\zeta\big\rangle\big) \cdot \delta(g)\zeta'\Big\rangle_B,\end{aligned}$$ as asserted. Since $$\begin{aligned}
S_{\tilde{\delta}, \tilde{z}, \tilde{\zeta}} (g,b)&= \big\langle \tilde z, b \cdot (\tilde\delta(g) \tilde\zeta)\big\rangle_A =
\big\langle \tilde z, \big((\delta(g)\zeta)\cdot b^*\big)\widetilde{}\big\rangle_A \\
&= {}_A\big\langle z, (\delta(g)\zeta)\cdot b^*\big\rangle = {}_A\big\langle z\cdot b, \delta(g)\zeta\big\rangle,\end{aligned}$$ we get that $$\begin{aligned}
\big(S_{\delta, z', \zeta'} \cdot T_{\rho, v, x, x'} \cdot S_{\tilde{\delta}, \tilde{z}, \tilde{\zeta}}\big)(g,b)&=
S_{\delta, z', \zeta'}\big(g, T_{\rho, v, x, x'}\big(g, {}_A\big\langle z\cdot b, \delta(g)\zeta\big\rangle\big)\big)\\
&=\big\langle z' , T_{\rho, v, x, x'}\big( g, _A\big\langle z\cdot b, \delta(g)\zeta\big\rangle\big) \cdot \delta(g)\zeta'\big\rangle_B.\end{aligned}$$ This shows that $$S_{\delta, z', \zeta'} \cdot T_{\rho, v, x, x'} \cdot S_{\tilde{\delta}, \tilde{z}, \tilde{\zeta}} =T_{\tau, w, (\tilde z\dot\otimes x) \dot \otimes z', (\tilde\zeta \dot\otimes x') \dot \otimes \zeta'} \in B(\Theta)$$ and the first claim follows. The remaining claims are then easily obtained.
\[M-eq-FS\] Assume $\Sigma$ and $\Theta$ are Morita equivalent with $\Sigma \sim _{(Z,\delta)} \Theta$. Then $B(\Theta)$ can be determined from $B(\Sigma)$ and $Z$ $($and similarly for the other way around$)$. Indeed, we have $$\label{BT}
B(\Theta) = {\rm Span} \Big\{ F_{z, z', \zeta, \zeta'}(T) \mid T\in B(\Sigma), z, z', \zeta, \zeta' \in Z\Big\}.$$
Using Proposition \[Morita1\] we get that the right-hand side of (\[BT\]) is contained in $B(\Theta)$. To show the reverse inclusion, we first observe that for $z, z', \zeta, \zeta' \in Z$, $g \in G$ and $b\in B$ we have $$\begin{aligned}
\big(S_{\delta, z', \zeta'} \cdot S_{\tilde{\delta}, \tilde{z}, \tilde{\zeta}}\big)(g,b)&=
S_{\delta, z', \zeta'} \big(g, S_{\tilde{\delta}, \tilde{z}, \tilde{\zeta}}(g,b)\big) \\
&=S_{\delta, z', \zeta'} \big(g, {}_A\big\langle z\cdot b, \delta(g)\zeta\big\rangle\big)\\
&= \big\langle z', {}_A\big\langle z\cdot b, \delta(g)\zeta\big\rangle\cdot (\delta(g)\zeta') \big\rangle_B\\
&= \big\langle z', z\cdot b \cdot \big\langle \delta(g)\zeta, \delta(g)\zeta' \big\rangle_B\big\rangle_B\\
&= \big\langle z', z\big\rangle_B b \big\langle \delta(g)\zeta, \delta(g)\zeta' \big\rangle_B\\
&= \big\langle z, z'\big\rangle_B^{ *} b \beta_g\big(\langle \zeta, \zeta' \rangle_B\big)\end{aligned}$$ Now, since $Z$ is full as a right Hilbert $B$-module, we can use Lemma 2.5 in [@BCL] to find $z_1, z'_1, \ldots, z_n, z'_n\in Z$ such that $$\label{unit}
\sum_{i=1}^n \langle z_i, z'_i\rangle_B = 1_B\quad \text{(the unit of $B$)}.$$ (In fact, proceeding as in [@GBVF p. 90], one may even choose $z'_j= z_j$ for all $j=1, \ldots, n$, but we won’t need this). We note that $$\label{Identity} \sum_{i, j=1}^n F_{z_i, z_i', z_j, z_j'}(I_\Sigma) = \sum_{i, j=1}^n S_{\delta, z_i', z_j'} \cdot S_{\tilde{\delta}, \tilde{z_i}, \tilde{z_j}} = I_\Theta.$$ Indeed, for $g\in G$ and $b\in B$, using (\[unit\]), we get $$\begin{aligned}
\Big(\sum_{i, j=1}^n S_{\delta, z_i', z_j'} \cdot S_{\tilde{\delta}, \tilde{z_i}, \tilde{z_j}}\Big)(g,b) &=
\sum_{i, j=1}^n \big\langle z_i, z_i'\big\rangle_B^{ *} b \beta_g\big(\langle z_j, z_j' \rangle_B\big)\\
&= \big(\sum_{i=1}^n \big\langle z_i, z_i'\big\rangle_B\big)^{*} b \beta_g\big(\sum_{j=1}^n \langle z_j, z_j' \rangle_B\big)= b.\end{aligned}$$ Let $T'\in B(\Theta)$. For each $i,j,k,l \in \{1, \ldots, n\}$, set $$T'_{i,j,k, l} := S_{\tilde{\delta}, \tilde{z_i}, \tilde{z_j}}\cdot T'\cdot S_{\delta, z_k', z_l'},$$ which belongs to $B(\Sigma)$ (by Proposition \[Morita1\]). Then, using (\[Identity\]), we get that $$\begin{aligned}
\sum_{i,j,k,l=1}^n F_{z_k, z_i', z_l, z_j'} & \big(T'_{i,j,k,l}\big) = \sum_{i,j,k,l}^n S_{\delta, z_i', z_j'} \cdot S_{\tilde{\delta}, \tilde{z_i}, \tilde{z_j}}\cdot T'\cdot S_{\delta, z_k', z_l'} \cdot S_{\tilde{\delta}, \tilde{z_k}, \tilde{z_l}}\\
&=\Big(\sum_{i,j=1}^n S_{\delta, z_i', z_j'} \cdot S_{\tilde{\delta}, \tilde{z_i}, \tilde{z_j}}\Big) \cdot T'\cdot \Big(\sum_{k,l=1}^n S_{\delta, z_k', z_l'} \cdot S_{\tilde{\delta}, \tilde{z_k}, \tilde{z_l}}\Big)\\
&= I_\Theta \cdot T' \cdot I_\Theta = T',\end{aligned}$$ which shows that $T' \in {\rm Span} \Big\{ F_{z, z', \zeta, \zeta'}(T) \mid T\in B(\Sigma), z, z', \zeta, \zeta' \in Z\Big\}$, as desired.
In view of the last statement of Corollary \[com-M-eq\], one might wonder under which assumptions the Fourier-Stieltjes algebras associated to Morita equivalent systems are actually (isometrically) isomorphic, cf. Theorem \[coc-conj-isom\] (see also Remark \[triv\]). Also, it would be interesting to investigate whether in general those Fourier-Stieltjes algebras could be Morita equivalent as Banach algebras in some suitable sense (see e.g. [@Gro] or [@Par]). However, elaborating on this topic would require the development of additional machinery, and we won’t discuss this here.
\[triv\] It may be worth to point out that in general Morita equivalence of systems is not sufficient to ensure that the associated Fourier-Stieltjes algebras are isomorphic. Indeed, consider $\Sigma = ({{\mathbb{C}}}, G, {\rm triv}, 1)$ and $\Theta = (M_2({{\mathbb{C}}}), G, {\rm triv}, 1)$ for some discrete group $G$ (where ${\rm triv}$ denotes the trivial action in both cases). It is then easy to see that $\Sigma$ and $\Theta$ are Morita equivalent. On the other hand, $B(\Sigma) = B(G)$ is commutative, while $B(\Theta)$ is not as it contains a copy of $M_2({{\mathbb{C}}})$.
\[weak ME\] Consider two systems $\Sigma=(A,G,\alpha,\sigma)$ and $\Theta=(B,H,\beta,\theta)$ where $H$ might be different from $G$, as in the previous section. If $\varphi: G\to H$ is an isomorphism, we obtain a new system $\Theta^\varphi= (B, G, \beta^\varphi, \theta^\varphi)$ by setting $\beta^\varphi_g = \beta_{\varphi(g)}$ and $\theta^\varphi(g, g') = \theta(\varphi(g), \varphi(g'))$. One easily checks that $B(\Theta)$ is isometrically isomorphic to $B(\Theta^\varphi)$. Now, let us say that $\Sigma$ and $\Theta$ are *weakly Morita equivalent* if there exist some $\omega \in Z^2(G, {{\mathbb{T}}})$ and some isomorphism $\varphi: G\to H$ such that $\Sigma(\omega)$ is Morita equivalent to $\Theta^\varphi$. Corollary \[M-eq-FS\] gives then that $B(\Sigma) = B(\Sigma(\omega))$ can be determined from $B(\Theta^\varphi)$, hence from $B(\Theta)$, and vice-versa. Finally, we mention that $\Sigma$ and $\Theta$ are weakly Morita equivalent whenever they are cocycle group conjugate, as the reader will easily verify.
An application to amenable systems
==================================
Amenability is an important topic within operator algebras, and it has received a good deal of attention, also in connection with $C^*$-dynamical systems (see e.g. [@AD1; @AD2; @Ex; @ExNg; @BrOz; @BeCo3; @Ex4; @BeCo6; @MSTT; @BEW; @ABF2; @BEW2] and references therein). Using the technique used in the proof of Corollary \[M-eq-FS\], we will show that amenability of a system, as defined in [@BeCo6], is preserved under Morita equivalence. As before, we let $\Sigma=(A,G,\alpha,\sigma)$ and $\Theta=(B,G,\beta,\theta)$ denote two twisted unital discrete $C^*$-dynamical systems. We recall that $\Sigma$ is said to be *amenable* whenever there exists a net $\{T^{\nu}\} $ in $P(\Sigma)$ such that
- each $T^{\nu}$ is finitely supported, i.e., the set $\{ g\in G \mid T^{\nu}_g \neq 0\}$ is finite for each $\nu$,
- $\{T^{\nu}\} $ is uniformly bounded, i.e., $\sup_{\nu} \|T^{{\nu}}\|_\infty
< \infty,$
- $\lim_{\nu} \|T^{\nu}_g(a) - a\| = 0$ for every $g \in G$ and $a\in A$.
Assume for example that $\Sigma$ has *Exel’s $($positive$)$ approximation property* [@Ex; @Ex4; @ExNg], that is, there exists a net $\{\xi_\nu\}$ of finitely supported functions from $G$ into $A$ such that
- $\sup_\nu \big\| \sum_{g\in G} \xi_\nu(g)^*\xi_\nu(g)\big\| < \infty$;
- $\lim_\nu \big\| \sum_{h\in G} \xi_\nu(h)^*a\alpha_g\big(\xi_\nu(g^{-1}h)\big) - a\big\| = 0
$ for all $g \in G$ and $a \in A$.
Then $\Sigma$ is amenable because setting $T^\nu_g(a) = \sum_{h\in G} \xi_\nu(h)^*a\alpha_g\big(\xi_\nu(g^{-1}h)\big)$ for all $g\in G$ and $a\in A$ gives a net $\{T^\nu\}$ satisfying the required properties. Note that if all $\xi_\nu$’s take their values in $Z(A)$, then (b) is equivalent to $$\lim_\nu \big\| \sum_{h\in G} \xi_\nu(h)^*\alpha_g\big(\xi_\nu(g^{-1}h)\big) - 1_A\big\| = 0$$ for all $g \in G$. Thus it readily follows that if $\sigma =1$, then $\Sigma$ is amenable whenever the action $\alpha$ is amenable in the sense of [@BrOz], a notion that is stronger than Anantharaman-Delaroche’s original definition of amenability of $\alpha$ in [@AD1]. Notice also that as long as $\sigma$ is scalar-valued then the amenability of $\Sigma$ does not depend on $\sigma$. As shown in [@BeCo6 Theorem 4.6], amenability of $\Sigma$ implies that $\Sigma$ is *regular*, i.e., the full and the reduced $C^*$-crossed products associated to $\Sigma$ are canonically isomorphic. Several other notions of amenability (for untwisted systems) are discussed in [@BEW; @BEW2]. We note that if $A$ is commutative, $G$ is exact and $\sigma =1$, then it follows readily from [@BEW Theorem 5.2] that all existing notions of amenability for $\Sigma$ (including ours, and regularity) are equivalent.
Strong and weak equivalence of Fell bundles over groups are studied in [@AF; @ABF1; @ABF2]. Having in mind that $\Sigma$ gives rise to a Fell bundle over $G$ in a canonical way (cf. [@Ex3]), one may for instance deduce from [@ABF1 Corollary 4.5] and [@ABF2 Theorem 6.23] that regularity and Exel’s approximation property are preserved under Morita equivalence of systems. We prove below that this is also true for amenability in our sense.
\[Amen\] Assume that the systems $\Sigma$ and $\Theta$ are Morita equivalent, with $\Sigma \sim _{(Z,\delta)} \Theta$. Then $\Theta$ is amenable whenever $\Sigma$ is amenable.
Assume that $\Sigma$ is amenable. As in the proof of Proposition \[Morita1\], we can find $z_1, z'_1, \ldots, z_n, z'_n\in Z$ such that $$\sum_{i=1}^n \langle z_i, z'_i\rangle_B = 1_B.$$ For later use, we set $K= \big(\sum_{i=1}^n \|z_i\| \|z_i'\|)^2$. Let then $F:B(\Sigma) \to B(\Theta)$ be the linear map given by $$F=\sum_{i,j=1}^n F_{z_i, z'_i, z_j, z'_j}.$$ We first note that $F$ maps $P(\Sigma) $ into $P(\Theta)$. To show this, we use the notation introduced in the proof of Proposition \[Morita1\]. Let $T= T_{\rho, v, x, x} \in P(\Sigma)$ and set $y:= \sum_{i=1}^n(\tilde z_i\dot\otimes x) \dot \otimes z'_i\in Y:= (\tilde Z \otimes_A X) \otimes_A Z$. Then we have $$\begin{aligned}
F(T) &= \sum_{i,j=1}^n F_{z_i, z'_i, z_j, z'_j}(T_{\rho, v, x, x}) =
\sum_{i,j=1}^n T_{\tau, w, (\tilde z_i\dot\otimes x) \dot \otimes z'_i, (\tilde z_j\dot\otimes x) \dot \otimes z'_j}\\& = \sum_{j=1}^n T_{\tau, w, y, (\tilde z_j\dot\otimes x) \dot \otimes z'_j} = T_{\tau, w, y, y} \in P(\Theta).\end{aligned}$$ This computation also gives that $$\begin{aligned}
\|F(T)\|_\infty &= \| \langle y, y\rangle_B\| = \|y\|^2 \leq \Big( \sum_{i=1}^n \| (\tilde z_i\dot\otimes x) \dot \otimes z'_i\|\Big)^2 \\
&\leq \Big( \sum_{i=1}^n \|\tilde z_i\|\| x\|\| z'_i\|\Big)^2 = K \|x\|^2 = K \|T\|_\infty\end{aligned}$$ (since $\|\tilde{z}\| = \|z\|$ for all $z\in Z$). Further, we note that $F(T)$ is easily seen to be finitely supported whenever $T\in B(\Sigma)$ is finitely supported.
Let now $\{T^{\nu}\}$ be a net in $P(\Sigma)$ witnessing the amenability of $\Sigma$. Then $\{F(T^{\nu})\}$ is clearly a net of finitely supported elements in $P(\Theta)$. Moreover, we have that $$\sup_\nu \|F(T^{\nu})\|_\infty \leq K \sup_\nu \|T^{\nu}\|_\infty < \infty,$$ so $\{F(T^{\nu})\}$ is uniformly bounded. Finally, let $g \in G$ and $b\in B$. Then $$\begin{aligned}
F(T^{\nu})(g, b) &= \sum_{i,j=1}^n \big(F_{z_i, z'_i, z_j, z'_j}(T^{\nu})\big)(g,b)
= \sum_{i,j=1}^n \big(S_{\delta, z_i', z_j'} \cdot T^{\nu} \cdot S_{\tilde{\delta}, \tilde{z_i}, \tilde{z_j}}\big)(g,b)\\
&= \sum_{i,j=1}^n S_{\delta, z_i', z_j'}\big(g, T^{\nu}_g\big(S_{\tilde{\delta}, \tilde{z_i}, \tilde{z_j}}(g,b)\big)\big).\end{aligned}$$ Consider now $i,j\in \{1, \ldots, n\}$. Using that the map $a \to S_{\delta, z_i', z_j'}\big(g, a\big)$ from $A$ into $B$ is continuous, we get that $$\begin{aligned}
\lim_\nu S_{\delta, z_i', z_j'}\big(g, T^{\nu}_g\big(S_{\tilde{\delta}, \tilde{z_i}, \tilde{z_j}}(g,b)\big)\big)
&= S_{\delta, z_i', z_j'}\big(g, \lim_\nu T^{\nu}_g\big(S_{\tilde{\delta}, \tilde{z_i}, \tilde{z_j}}(g,b)\big)\big)\\
&=S_{\delta, z_i', z_j'}\big(g, \big(S_{\tilde{\delta}, \tilde{z_i}, \tilde{z_j}}(g,b)\big)\big)\\
&= \big(S_{\delta, z_i', z_j'} \cdot S_{\tilde{\delta}, \tilde{z_i}, \tilde{z_j}}\big)(g,b).\end{aligned}$$ Hence, using Equation , we get that $$\begin{aligned}
\lim_\nu F(T^{\nu})(g, b) &= \sum_{i,j=1}^n \lim_\nu S_{\delta, z_i', z_j'}\big(g, T^{\nu}_g\big(S_{\tilde{\delta}, \tilde{z_i}, \tilde{z_j}}(g,b)\big)\big)\\
&= \sum_{i,j=1}^n \big(S_{\delta, z_i', z_j'} \cdot S_{\tilde{\delta}, \tilde{z_i}, \tilde{z_j}}\big)(g,b) = I_{\Theta}(b) = b.\end{aligned}$$ This shows that $\Theta$ is amenable, as desired.
An immediate consequence of this result is that amenability of a system is also preserved under weak Morita equivalence (as defined in Remark \[weak ME\]).
A result of a nature similar to Theorem \[Amen\] is Theorem 2.2.17 in [@ADR], which says that topological amenability of locally compact groupoids is invariant under topological equivalence (whose definition is hinted by Morita equivalence of $C^*$-algebras).
Acknowledgements {#acknowledgements .unnumbered}
----------------
Most of the present work has been done during the visits made by E.B. at the Sapienza University of Rome and by R.C. at the University of Oslo in the period 2018-2019. Both the authors are grateful to the hosting institutions for the kind hospitality, and to the Trond Mohn Foundation (TMS) for financial support. We are also indebted to the referees for their many helpful comments and suggestions, leading to an improved version of this article.
[EKQR-0]{}
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[^1]: We recall that by our standing assumptions, $\kappa$ is then unit preserving, hence nondegenerate, as required in [@EKQR].
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'M.Spite'
- 'F.Spite'
- 'P.Bonifacio'
subtitle: 'an observer’s perspective'
title: The cosmic lithium problem
---
Introduction
============
Over the past decade WMAP measurements of the cosmic microwave background (CMB) radiation allowed a precise determination of the cosmological constants: @LDH11, @KSD11. As a consequence the cosmic baryon density of the Universe $\rm \Omega_{b}$ that, in the standard model governs the Big Bang nucleosynthesis (BBN) of the elements during the first few minutes in the evolution of the Universe, is now known with a good precision and therefore, the quantity of $\rm^{3}He,~ ^{4}He, D,~ ^{6}Li~ and~ ^{7}Li$ produced by the standard BBN.
These predictions may be compared with the primordial abundances of these elements inferred from observational data, but, at least for lithium ($\rm ^{7}Li~ and~ ^{6}Li$), they do not agree.
The aim of this meeting is to explore the ways to reconcile predictions of the BBN to observations. In this introductory review we will try to provide an update of the “lithium problem”.
A Brief summary of the predictions of the standard BBN {#bbn}
======================================================
Broad reviews of primordial nucleosynthesis and its relation to cosmology and lithium problems can be found e.g. in @CFO08, @IMM09, @CocVan10, @Fields11, @CocGX12, @gary.
In the standard BBN, the production of the elements depends only on the baryon to photon ratio $\eta$. At low energy scales, the value of $\eta$ is simply related to $\rm \Omega_{b}$. The recent seven-year WMAP data release leads to $\rm \Omega_{b} h^{2} = 2.249 \times 10^{-2}$ and $\eta = 6.19 \pm 0.15 \times 10^{-10}$ [@LDH11].
With this value of $\eta $, BBN is expected to produce a quantity of lithium corresponding to $\rm A(^{7}Li) = ~2.72$ [^1] and\
$\rm A(^{6}Li) = -2.00$\
with thus an extremely low value of the isotopic ratio $\rm ^{6}Li/^{7}Li \approx 2 \times 10^{-5}$.
How to observe the primitive Galactic matter ?
==============================================
In the first approximation, from the theory of stellar atmospheres, the chemical composition of the atmosphere of dwarf stars is a good witness of the chemical composition of the cloud that formed these stars. Therefore, if such low mass stars (with $M<0.9M_{\odot}$) were formed at the very beginning of the Galaxy, since their lifetime is larger than the Hubble time, they are still shining today and the chemical composition of their atmosphere is similar to the the chemical composition of the primitive cloud from which they have been formed.
However, lithium is an exception because it is very fragile, destroyed as soon as the temperature is higher than $2.5 \times 10^{6}$ K for and even $2.0 \times 10^{6}$ K for through the reactions\
$\rm^{7}Li +p \to ~^{4}He +~^{4}He$\
$\rm^{6}Li +D \to \,^{4}He +~^{4}He$\
$\rm^{6}Li +p \to ~^{4}He +~^{3}He$\
These temperatures are never reached in the stellar atmospheres but if the convection zone is deep, lithium in the atmosphere is swept along to hot deep layers and destroyed little by little; its atmospheric abundance decreases. This happens for example in the Sun and in cool low mass stars.
But in warm metal-poor stars, like turn-off stars, with $\rm T_{eff } \ge 5900\,K$ the convection is not so deep, and lithium should be preserved. Therefore, the lithium abundance in the atmosphere of these stars is expected to be a good witness of the abundance of lithium in the primitive Galactic matter.
Abundance of in field Galactic stars
======================================
Since lithium abundance in the atmosphere of metal-poor stars has been considered as one of the diagnostic to constrain the description of the primordial Universe, a wealth of data is available in the literature, and in particular since 2005: @ChaPr05, @ALN06, @AsMe08, @GonBL08, @GarCR08, @AokBB09, @HosRG09, @SboBC10, @MelCR10, @SchK12. The temperature is a crucial parameter to determine the abundance of lithium in the atmosphere of the stars. Sometimes the authors have used the colors of the stars, sometimes the dependance of the iron abundance on the excitation potential of the lines, and often the wings of the line.\
If we consider all the stars hotter than 5900K and with a metallicity in the interval $\rm -2.8<[Fe/H]<-2.0$, the lithium abundance in this sample is constant: $\rm A(Li)\approx 2.2 \pm 0.06$ (Fig. \[plat\]). In spite of the inhomogeneity of the temperature determinations by various authors, the scatter is very small.\
In these intervals of temperature and metallicity the lithium abundance does not depend on the metallicity nor on the temperature of the star. But this “plateau” is at a lithium abundance three times lower than the Big Bang prediction (see Fig. \[plat\]).
In this range of temperature and metallicity, two stars G122-69 and G139-08 are significantly below the lithium plateau. Both stars are well known lithium-poor stars [@Tho92]. @Boes07 could measure the beryllium abundance in one of them (G139-08). The star is also Be-poor and thus is considered as a blue straggler.
Recently, with the advent of very efficient new stellar surveys (Hamburg-ESO, SDSS...) many extremely metal-poor (EMP) stars were discovered and thus the abundance of lithium could be measured in a number of turn-off stars with $\rm [Fe/H]<-2.8$. If we add these EMP stars to our previous sample (Fig. \[plat2\]), many stars have a lithium abundance significantly lower than the “plateau”, and the scatter compared to Fig. \[plat\] increases dramatically, suggesting the influence of metallicity.
In figure \[LiFe\] we have plotted the abundance of lithium in the same sample of stars as a function of $\rm ~N(Fe)/N(Fe)_{\odot}$ , with a linear scale. For more convenience we have added just above the X axis the corresponding values of \[Fe/H\]. The low scatter of the lithium abundance for $\rm N(Fe)/N(Fe)_{\odot} > 1.6 ~10^{-3}$ (or $\rm [Fe/H] > -2.8$, dotted line on the figure), is clearly seen with a mean value A(Li)=2.2. Then for $\rm [Fe/H] < -2.8$ the plateau suddenly breaks down. But none of the stars has a lithium abundance significantly higher than the plateau.
If we want to solve the problem we have to explain that:\
\
$\bullet$ for $\rm -2.8 <[Fe/H]< -2.0$\
the lithium abundance in turn-off stars with $\rm T_{eff } > 5900K$, is constant at a level three times lower than the SBBN+WMAP prediction.\
\
$\bullet$ for $\rm [Fe/H]< -2.8$\
the lithium abundance is strongly scattered below the lithium plateau. This scatter does not seem to depend on the temperature as soon as $ > 5900$K (Fig. \[plat2\]).\
Is it possible to explain the behavior of by stellar processes ?
==================================================================
Two different ways were proposed to explain, in the frame of standard BBN, the unexpected behaviour of in metal-poor stars. Primordial Li would be depleted, either within the matter that formed the stars observed today, or during the lifetime of those stars.
Destruction of Galactic in massive stars ?
--------------------------------------------
It was proposed by @PiauBB06 that after a primordial production of lithium compatible with the WMAP predictions, the Galactic matter would have been rapidly processed in massive Pop III stars in the mass range $10 M_{\odot}<M<40 M_{\odot}$ destroying two thirds of the primordial lithium.
Therefore, the EMP stars observed today would have been formed from a matter already depleted in lithium.\
But [@Pran07] remarked that these massive Pop III stars would have produced heavy elements like carbon, oxygen... and thus the metallicity of the second generation of stars, (the old stars we observe today), would be much higher than the one observed in the EMP stars. Let us note that an ultra metal-poor star, without carbon enrichment has been found recently with \[Fe/H\]=-5.0 and thus with $Z/Z_{\odot} \approx 4.\times 10^{-5}$ [@Caff11; @Caff12].\
This interpretation that explains the avoidance zone (no lithium abundance in warm turn-off stars significantly higher than A(Li)=2.2, the value of the lithium plateau), is not able to explain the break of this plateau at a metallicity of $\rm [Fe/H]=-2.8$ in Fig. \[LiFe\].
On the other hand, the lithium abundance has been also measured in turn-off stars of $\omega$ Centauri by @MonBS10. This globular cluster is generally considered as the remnant of a dwarf spheroidal galaxy captured by the Milky Way, and thus, there is no reason that the same quantity of matter be processed by massive Pop III stars in $\omega$ Centauri and in the Milky Way. However the lithium abundance in the turn-off stars of $\omega$ Centauri is the same as in the field turn-off stars.\
As a consequence, the interpretation of @PiauBB06, compatible with the Standard BBN, does not seem to be well grounded.\
Lithium plateau induced by turbulent diffusion ?
------------------------------------------------
Atomic diffusion is a slow gravitational settling of the elements below the convective zone. Following @Mich84 *“it is always present in stars. It cannot be turned off. It can only be rendered inefficient by sufficient mass motion either due to meridional circulation or turbulence”*. For a long time, it was supposed that diffusion was indeed rendered inefficient in metal-poor turn-off stars [@DelDK90].
But @RichMR05 computed the influence of diffusion on the lithium abundance with different turbulent parameters, after 13.5Gyr starting with a Li abundance compatible with the SBBN.\
Without turbulence (Fig. \[richard\], dashed green line) the abundance of lithium would decrease when the temperature of the star increases in contradiction with the existence of a plateau.\
But with an ad hoc model of turbulence (model “T6.25” where the turbulent diffusion coefficient, $D_{T}$ is 400 times larger than the He atomic diffusion coefficient at $\log T_{0} = 6.25$ and varies as $\rho^{-3}$) it is possible to represent the plateau (solid green line in Fig. \[richard\]).
Up to now, the cause of this turbulence is unknown. The diffusion coefficient is an ad hoc parameter, but at least with this parameter it becomes possible to represent the observed plateau with an original lithium abundance of 2.7 (as predicted by the standard BBN). Such a turbulence should exist to compensate the effect of diffusion. However, to date, turbulent diffusion does not explain that the plateau suddenly breaks down at $\rm [Fe/H] < -2.8$.
There were recently several attempts to determine this “turbulent parameter” in globular clusters: and in particular in NGC 6397: @KorGR07, @GonBC09, @LinPC09. Since in globular clusters all the stars (at least the “first generation” stars) have the same age, the same metallicity and a well known evolutionary stage it is possible to check the small variations of the abundances as a function of the evolution of the stars. The result of these studies depends strongly on the adopted temperature scales, and up to now, they led to different turbulent parameters. In all these studies the estimation of the initial lithium abundance $\rm A(Li)_{0}$ is still not in complete agreement with the latest predictions of the SBBN.
Moreover, if turbulent diffusion is responsible for the formation of a plateau of the lithium abundance at a level three times lower than the prediction of WMAP, we would expect that at least some stars (because, for example, they would have a little higher turbulence) would have kept a value of the lithium abundance close to the initial value and thus would lay in Fig. \[plat\] and \[plat2\], between the WMAP value and the observed plateau. But no field star has been observed in this “forbidden zone”.
One turn-off star in NGC6397 [@KochLR11; @Koch12], and one in M4 [@MonVB12; @Mon12] have been found Li-rich (respectively A(Li)=4.03 and A(Li)=2.87). But both stars are also Na-rich and are thus suspected to be polluted by the ejecta of a first generation of stars. The lithium abundance in these stars would not represent the abundance in the cloud which formed these globular clusters.
[l@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c]{}\
& &\
&\
Star & & $\log g$ & \[Fe/H\] & A() & /& & $\log g$ & \[Fe/H\] & A() & /\
\
BD +26 3578 & 6335& 4.0& $-2.26$ &2.22& $0.010\pm0.013$ & 6239& 3.9 & $-2.33$ & 2.21& $0.004\pm 0.028$\
BD –04 3208 & 6298& 4.0& $-2.30$ &2.19& $0.048\pm0.019$ & 6338& 4.0 & $-2.28$ & 2.27& $0.047\pm 0.039$\
G64-37 & 6368& 4.4& $-3.08$ &1.97& $0.111\pm0.032$ & 6318& 4.2 & $-3.12$ & 2.04& $0.006\pm 0.039$\
\
Does the lithium abundance on the plateau represent the BBN production ?
========================================================================
The observations could finally reflect a production of lithium by a non standard BBN at the level of the plateau. Atomic diffusion would be inhibited within the temperature and metallicity range of the plateau (turbulence, meridional circulation. . .) as it was supposed, before the WMAP measurements [@DelDK90].
However even in this hypothesis, the breakdown of the plateau for $\rm [Fe/H] < -2.8$ has to be explained.\
We know that this depletion of lithium in stars with $\rm [Fe/H] < -2.8$, is the result of an internal stellar depletion. @GonBL08 indeed, could measure the abundance of lithium in the two components of CS 22876-32. This turn-off binary star is extremely metal-poor, \[Fe/H\]=–3.6, and the abundance of observable elements (iron, calcium, oxygen etc.) have been found similar in both components. There is only one exception: the abundance of lithium is significantly different: in CS 22876-32A (=6500K) A(Li)=2.22, but in CS 22876-32B (=5900K) A(Li)=1.75 (see Fig. \[plat2\] and \[LiFe\], where the two components of CS 22876-32 are represented by an asterisk). A similar situation, albeit less extreme, has been found for the double-lined spectroscopic binary BD +26 2606 at \[Fe/H\]$=-2.5$ [@AokIT12; @Aokmem12]. The two components have temperatures of 6350K and 5830K and the lithium abundances are A(Li)=2.23 and 2.11, respectively. This may suggest that the differential depletion between two stars of different mass may be present even at higher metallicities than that of CS 22876-32, but the depletion is much less than that observed in CS 22876-32. Alternatively this could simply mean that at metallicity –2.5 the Li depletion by “normal” convection starts already around 5800K and for this reason the lower mass component has a slightly depleted lithium content.
Since the components of binaries formed from the same cloud, the only possibility to explain this difference is that lithium has been depleted during the lifetime of the star (at least in CS22876-32B and BD +26 2606B). A possible mechanism could perhaps be turbulent diffusion in the secondary, erased in the primary because of its evolution towards the subgiant phase.
A number of possible non standard BBN have been proposed, producing a lower primordial lithium abundance, at the level of the plateau (see e.g. @Oliv12 and @Kaj12 in this meeting). Such BBN would explain the plateau and also the strange “forbidden zone” described above. But any depletion along the plateau (by diffusion or other process) has then to be suppressed (by a strong turbulence? or adequate circulations such as convection, meridional circulation? by adequate mixing?) At very low metallicities ($\rm [Fe/H] < -2.8$) aleatory depletions would be more or less at work (diffusion + turbulence? + others?)
Abundance of in field metal-poor stars
========================================
Several teams tried recently to measure the abundance of in metal-poor halo stars, in particular: @ALN06, @StefCB09, @GarAI09, @Stef12. This is a very difficult exercise because the lines of and are superimposed and that, moreover, the hyperfine structure of the lines of (widening the profile) has to be taken into account (Fig. \[li6profile\]). As a consequence very high quality spectra are needed: a resolution $R \approx 100\,000$ with a $S/N \approx 500$ are necessary.
@ALN06 and @AsMe08 measured the lithium abundance in a sample of 27 metal-poor stars and they claimed that they have detected in twelve of them (Fig. \[aspli6\]). In all the stars the abundance of was compatible with A()= 0.8, a value corresponding to a ratio / = 5%, not compatible with the predictions of the standard BBN ($\rm ^{6}Li/^{7}Li \le 2 \times 10^{-5}$, see section \[bbn\]). This could have been produced by cosmic rays (spallation) after the BBN. But in this case we would expect that the abundance of increases with the metallicity as it is for Be or B formed also by spallation. The constancy of the abundance of is not compatible with a spallation origin of the observed in metal-poor stars.
@CaySC07, @StefCB09 and @Stef12 have shown that, when lines asymetries generated by convection are taken into account, the resulting abundance of is strongly reduced.
On the other hand, @GarAI09 have measured with the HDS spectrograph mounted on the Subaru telescope ($R\approx95000$, $S/N=500$) the and abundances in five turn-off stars. In all these stars the abundance of is compatible with an absence of . Three stars are in common with @ALN06 and @AsMe08: BD+26 3578 (HD338529), BD–04 3208 (G13-09) and G64-37. These observations are compared in Table \[comp6li\]. In particular G64-37, was found -rich by @AsMe08 with / = 0.11 $\pm 0.03$, when @GarAI09 found for this star / = 0.01 $\pm 0.04$%: no .
@GarAI09 conclude that the abundance of is very sensitive to the assumptions made for the continuum location, the residual fringing treatment and even the wavelength range covered in the analysis, and that the observational error is often underestimated.
At this conference @Lind12 have shown some new measurements, based on high quality spectra. Their analysis method relies on lines other than to estimate the rotational broadening, when such lines are treated in NLTE, the resulting broadening implies that no is present in the stars analysed by them.
To date, the presence of in the most metal-poor turn-off stars of our Galaxy does not seem to be firmly established.
Conclusion
==========
The considerable observational progress (large telescopes, high resolution spectra, multiplex spectrographs, large surveys of metal-poor stars detecting many extremely metal-poor starsÉ) have specified the contour of the lithium problem, but no completely satisfactory solution has been found yet. Additional observations will be useful. The detailed observations of binary stars will certainly bring essential data. Precise distances (GAIA) and therefore luminosities, evolutionary status and ages of field stars will bring essential complements to the available information about the lithium problem.
It is a pleasure to acknowledge many helpful discussions with R. Cayrel about Li abundances.
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[^1]: with the usual notation A(Li)= $\rm \log (N(Li)/N(H)) +12$
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1em
HUB-EP-97/32\
\
.7in
[**Chiral Multiplets in $N=1$ Dual String Pairs**]{}
.3in
Gottfried Curio[^1]\
1.2cm
[*Humboldt-Universität zu Berlin, Institut für Physik, D-10115 Berlin, Germany*]{}
.1in
.2in
> We compare the spectrum of chiral multiplets in the $N=1$ vacua of the heterotic string on a Calabi-Yau together with an $E_8\times E_8$ vector bundle and $F$-theory on a smooth Calabi-Yau fourfold. Under suitable restrictions we show agreement using an index-computation.
Two of the most interesting classes of models coming from string theory compactifications with $N=1$ supersymmetry in four dimensions are obtained from the heterotic string on a threedimensional Calabi-Yau $Z$ with a vector bundle $V$ respectively from $F$-theory on an elliptically fibered Calabi-Yau fourfold $X^4 \rightarrow B^3$. These cases are of even greater interest in view of the duality philosophy that one should get dual models by adiabatically extending the eightdimensional duality between the heterotic string on $T^2$ and $F$-theory on $K3$ \[\[V\]\]. So we will furthermore assume that the heterotic Calabi-Yau is elliptically fibered $Z \rightarrow B^2$ and that the $F$-theory fourfold Calabi-Yau is $K3$ fibered $X^4 \rightarrow B^2$. This involves also the existence of a $P^1$ fibration $B^3 \rightarrow B^2$.\
In this note we will show a duality matching among the concerned moduli assuming that we are in the case without an unbroken gauge group. So we will assume that $V$ is an $E_8 \times E_8$ vector bundle and that $X^4$ is a smooth Weierstrass model; for worked out examples cf. \[\[ACL\]\]. The comparison works technically similar to the matching \[\[FMW\]\] of the number $\chi /24$ of spacetime-filling threebranes \[\[SVW\]\] occuring in the course of tadpole-cancellation to the number $h$ of fivebranes (wrapping the fibre $F$ of the elliptic fibration $Z \rightarrow B^2$) occuring in the context of the heterotic string anomaly cancellation
(V\_1)+(V\_2)+h \[F\]=c\_2(Z)in connection with a $E_8 \times E_8$ vector bundle $V=V_1 \times V_2$. Note also that at the end of this paper we remark on circumstances that the actual computation carried out here will find its proper place when embedded in an enlarged context.
So we consider $F$-theory on a smooth elliptically fibered Calabi-Yau fourfold $X^4$ with base $B^{3}$ which can be represented by a smooth Weierstrass model. We assume even more that the rank $v$ of the gauge group in four dimensions is zero (,i.e. also no $U(1)$ factors). Now one can expect that the spectrum would get contributions from Kaehler and complex structure parameters related to $h^{1,1}-2$ (not counting the unphysical $F$-theory elliptic fibre as well as not counting the class corresponding to the heterotic dilaton) and $h^{3,1}$ respectively as well as from $h^{2,1}$ giving in total $h^{1,1}-2+h^{2,1}+h^{3,1}$ parameters. One can see from a $M$-theory versus $F$-theory consideration that these contributions divide themselves in 4$D$ between chiral and vector multiplets according to whether or not they come from the threefold base $B^3$ of the $F$-theory elliptic fibration (just as in the analogous 6$D$ $N=2$ case \[\[MV2\]\]; we will assume here $v=0$ anyway).
So following this line of reasoning one would expect for the rank $v$ of the $N=1$ vector multiplets (unspecified hodge numbers relate to $X^4$) (cf. \[\[Mo\]\],\[\[CL\]\],\[\[ACL\]\])
v=h\^[1,1]{}-h\^[1,1]{}(B\^3)-1+h\^[2,1]{}(B\^3) and for the number $c$ of $N=1$ neutral chiral (resp. anti-chiral) multiplets
c&=&h\^[1,1]{}(B\^3)-1+h\^[2,1]{}-h\^[2,1]{}(B\^3)+h\^[3,1]{}\
&=&h\^[1,1]{}-2+h\^[2,1]{}+h\^[3,1]{}-v
Note that we are speaking here of the generic gauge group and not of the situation where one tunes the deformations of the Calabi-Yau to unhiggs an unbroken gauge group and correspondingly counts only the number of deformations [*preserving*]{} a particular singular locus (cf. \[\[BJPS\]\]).\
Now one has \[\[SVW\]\]
-8=h\^[1,1]{}-h\^[2,1]{}+h\^[3,1]{}so that one finds
c=-10+2h\^[2,1]{}-vNow one can compute the Euler number of X in terms of topological data of the base $B^3$ of the elliptic fibration according to \[\[SVW\]\]
$$\begin{aligned}
\frac{\chi}{24}=12+15\int_{B^3}c_{1}^3(B^3)\nonumber\end{aligned}$$
One can go even further: assuming, in the light of the application to duality with the heterotic string, that $X^4$ is a $K3$ fibration over a twofold base $B^2$ (so that $B^3$ is a $P^1$ fibration over $B^2$) one gets \[\[FMW\]\]
$$\begin{aligned}
\frac{\chi}{24}=12+90\int_{B^2}c_{1}^2(B^2) +30\int_{B^2} t^2\nonumber\end{aligned}$$
where $t$ encodes the $P^1$ fibration structure \[\[FMW\]\]: assume the $P^1$ bundle over $B^2$ given by the projectivization of a vector bundle $Y={\cal O}\oplus {\cal T}$, with ${\cal T}$ a line bundle over $B^2$; then $t=c_1({\cal T})$. So one gets in the case with no unbroken gauge group
$$\begin{aligned}
c_F=\frac{\chi}{6}-10+2 h^{2,1}=
38+360\int_{B^2}c_{1}^2(B^2)+120\int_{B^2} t^2+2h^{2,1}\nonumber \end{aligned}$$
On the other hand we have to count the moduli on the heterotic side. Here one has contributions $m_{geo}=h^{1,1}(Z)+h^{2,1}(Z)$ from geometrical moduli plus the bundle moduli. Concerning the former let us assume that the dual heterotic Calabi-Yau threefold $Z$ is smooth so that \[\[Klemm\]\]
$$\begin{aligned}
\chi (Z)=-60\int_{B^2}c_1^2(B^2)\nonumber\end{aligned}$$
and assume furthermore that its smooth Weierstrass model is general \[\[Gr\]\], i.e. has only one section (a typical counterexample being the $CY^{19,19}=B_9\times _{P^1}B_9$ with the del Pezzo $B_9$, the blow up of $P^2$ in the nine points of intersection of two cubics, cf. \[\[DGW\]\], \[\[CL\]\]). So $h^{1,1}(Z)=h^{1,1}(B^2)+1$; this gives using Noethers formula that
$$\begin{aligned}
h^{1,1}(Z)=\int_{B^2}c_2(B^2)-1=11-\int_{B^2}c_1(B^2)^2\nonumber \end{aligned}$$
So one gets for the number of geometrical moduli
$$\begin{aligned}
m_{geo}&=&h^{1,1}(Z)+h^{2,1}(Z)=2 h^{1,1}(Z)-\frac{\chi}{2}\nonumber\\
&=&22+28\int_{B^2}c_1^2(B^2)\nonumber\end{aligned}$$
Concerning the contribution of the bundle moduli of the $E_8 \times E_8$ bundle $V$ on $Z$ let us recall the setup of the index-computation in \[\[FMW\]\]. As the usual quantity suitable for index-computation $\sum_{i=o}^3 (-1)^i h^i(Z,V)$ vanishes by Serre duality one has to introduce a further twist and to compute a character-valued index. Now because of the elliptic fibration structure one has on $Z$ the involution $\tau$ coming from the “sign-flip” in the fibers which we furthermore assume has been lifted to an action on the bundle. The character-valued index
$$\begin{aligned}
I=-\frac{1}{2}\sum_{i=o}^3 (-1)^i Tr_{H^i(Z,V)} \tau \nonumber\end{aligned}$$
simplifies by the vanishing of the ordinary index to
$$\begin{aligned}
I=-\sum_{i=o}^3 (-1)^i h^i(Z,V)_e \nonumber\end{aligned}$$
where the subscript “e” (resp. “o”) indicates the even (resp. odd) part. As we have the gauge group completely broken one finds
$$\begin{aligned}
I=n_e-n_o \nonumber\end{aligned}$$
denoting by $n_{e/o}$ the number $h^1(Z,V)_{e/o}$ of massless even/odd chiral superfields. Now one gets with Wittens index formula \[\[W-un\]\],\[\[ACL\]\]
$$\begin{aligned}
I=16+332\int_{B^2}c_1^2(B^2) +120\int_{B^2} t^2 \nonumber\end{aligned}$$
that one has for the number of the bundle moduli $m_{bun}=n_e+n_o=I+2n_o$
$$\begin{aligned}
m_{bun}=16+332\int_{B^2}c_1^2(B^2) +120\int_{B^2} t^2 +2n_o \nonumber\end{aligned}$$
So adding up one gets in total
$$\begin{aligned}
c_{het}&=&h^{1,1}(Z)+h^{2,1}(Z)+I+2n_o\nonumber\\
&=&38+360\int_{B^2}c_1^2(B^2)+120\int_{B^2} t^2+2n_o\nonumber\end{aligned}$$
Now on the $F$-theory side the modes odd under the involution $\tau ^{\prime}$ corresponding to the heterotic involution $\tau$ correspond to the $h^{2,1}(X^4)$ classes \[\[FMW\]\]. Let us assume that no 4-flux was turned on (which is not a free decision \[\[W4fl\]\] in general); otherwise there are further twistings possible (cf. \[\[FMW\]\], sect. 4.4, and also \[\[AM\]\]) which account for a possible multi-component structure of the bundle moduli space (cf. also \[\[BJPS\]\] for the case of $SU(n)$ bundles). This clearly deserves further study to embed the simple picture employed here in the more general context.\
So one gets complete matching with $n_o=h^{2,1}(X^4)$.\
I would like to thank B. Andreas and E. Witten for discussion.
References {#references .unnumbered}
==========
1. \[V\] C. Vafa, [*Evidence for F-theory*]{}, Nucl. Phys. [**B 469**]{} (1996) 493, hep-th/9602022.
2. \[FMW\] R. Friedman, J. Morgan and E. Witten, [*Vector Bundles and F-Theory*]{}, hep-th/9701162.
3. \[SVW\] S. Sethi, C. Vafa and E. Witten, [*Constraints on Low-Dimensional String Compactifications*]{}, Nucl. Phys. [**B 480**]{} (1996) 213, hep-th/9606122.
4. \[MV2\] D. R. Morrison and C. Vafa, [*Compactification of F-theory on Calabi-Yau Threefolds II*]{}, Nucl. Phys. [**B 476**]{} (1996) 437, hep-th/9603161.
5. \[Gr\] A. Grassi, [*Divisors on elliptic Calabi-Yau 4-folds and the superpotential in F-theory*]{}, alg-geom/9704008.
6. \[Mo\] K. Mohri, [*F- Theory Vacua in Four Dimensions And Toric Threefolds*]{}, hep-th/9701147.
7. \[Klemm\] A. Klemm, B. Lian, S.-S. Roan and S.-T. Yau, [*Calabi-Yau fourfolds for $M$- and $F$-Theory Compactifications*]{}, hep-th/9701023.
8. \[W-un\] E. Witten, unpublished notes.
9. \[W4fl\] E. Witten, [*On Flux Quantization in M Theory and the Effective Action*]{}, hep-th/9609122.
10. \[DGW\] R. Donagi, A. Grassi and E. Witten, [*A Nonperturbative Superpotential with $E_8$ Superpotential*]{}, Mod. Phys. Lett. [**A 11**]{} (1996) 2199, hep-th/9607091.
11. \[CL\] G. Curio and D. Lüst, [*A Class of $N=1$ Dual String Pairs and its Modular Superpotential*]{}, to appear in Int. Journ. of Mod. Phys. [**A**]{}, hep-th/9703007.
12. \[ACL\] B. Andreas, G. Curio and D. Lüst, [*$N=1$ Dual String Pairs and their Massless Spectra*]{}, hep-th/9705174.
13. \[AM\] P.S. Aspinwall and D.R. Morrison, [*Pointlike Instantons on K3 Orbifolds*]{}, hep-th/9705104.
14. \[BJPS\] M. Bershadsky, A. Johansen, T. Pantev and V. Sadov, [*On Four-Dimensional Compactifications of F-Theory*]{}, hep-th/9701165.
[^1]: email: [email protected]
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---
abstract: |
We developed a method to make <span style="font-variant:small-caps;">gem</span> foils with a spherical geometry. Tests of this procedure and with the resulting spherical <span style="font-variant:small-caps;">gem</span>s are presented. Together with a spherical drift electrode, a spherical conversion gap can be formed. This would eliminate the *parallax error* for detection of x-rays, neutrons or <span style="font-variant:small-caps;">uv</span> photons when a gaseous converter is used. This parallax error limits the spatial resolution at wide scattering angles. The method is inexpensive and flexible towards possible changes in the design.
We show advanced plans to make a prototype of an entirely spherical triple-<span style="font-variant:small-caps;">gem</span> detector, including a spherical readout structure. This detector will have a superior position resolution, also at wide angles, and a high rate capability. A completely spherical gaseous detector has never been made before.
author:
- '[^1] [^2] [^3]'
title: 'Spherical <span style="font-variant:small-caps;">gem</span>s for parallax-free detectors'
---
Motivation {#intro}
==========
Position sensitive radiation detectors with a spherical geometry can be attractive for various applications, for several reasons. The most common reason in the case of gas detectors is to eliminate the *parallax error* arising from the uncertainty of how deep radiation penetrates the sensitive volume before causing ionization. Figure shows that the depth of interaction of x-rays and thermal neutrons in a gas volume ranges from few to many millimeters, even when using the most efficient gases known for these purposes. The situation is similar for gaseous <span style="font-variant:small-caps;">vuv</span> photoconverters such as tetrakis dimethylamine ethylene (<span style="font-variant:small-caps;">tmae</span>) or triethylamine (<span style="font-variant:small-caps;">tea</span>) [@CharpakParallax]. If the electric field in the conversion region of a gas detector is not parallel to the direction of irradiation, an uncertain conversion depth will give rise to an error in position reconstruction, see Fig. .
![Absorption length of various x-ray and thermal neutron conversion gases at atmospheric pressure, as a function of the energy of x-ray photons (lower horizontal scale) or neutrons (upper scale). Calculated from cross-section data [@X-rayCross-sections; @NeutronDataBooklet].[]{data-label="Absorption_Gases"}](AbsorptionLengths){width="\columnwidth"}
Methods that have been used to suppress parallax error include:
- Arranging small area flat detectors in such positions as to approximate a spherical shape [@LobsterPaper].
- Creating an almost spherical conversion region with foils and meshes, then transferring the charge to a planar wire chamber [@CharpakPaper; @Comparison].
- Having a spherical cathode with an otherwise flat detector, while reducing the conversion depth by using an efficient conversion gas at a high pressure ($\sim 3$ bar) [@MoscowPaper; @BrukerPaper].
- Imitating a spherical cathode by dividing a flat electrode into concentric circular segments and controlling the voltage applied to each sector with a resistive divider [@SegmentedCathodePaper].
- Using a pulsed radiation source or additional hardware to calculate the depth of interaction per event, then correct the position reconstruction [@NeutronDiffraction].
![The cause of a parallax error in a gas detector with a homogeneous drift field.[]{data-label="ParallaxError"}](ParallaxError){width="\columnwidth"}
Each of these methods has its limitations. In all cases the challenge of making a fully spherical detector, however desirable, is avoided. We started an effort that should lead to the first fully spherical gas detector, based on spherical GEMs, cathode and readout board. We developed a method that allows us to make a spherical <span style="font-variant:small-caps;">gem</span> from a flat standard <span style="font-variant:small-caps;">gem</span> [@firstGEM], apparently without affecting its properties significantly.
{width="\textwidth"}
First performance tests of single spherical <span style="font-variant:small-caps;">gem</span>s will be done in a setup with a spherical entrance window (which serves as drift electrode as well) and a flat readout structure, see the left side of Fig. . The electric field in the conversion region is truly radial. The amplitude of signals from conversions in the non-radial induction region will be suppressed by the gain of the <span style="font-variant:small-caps;">gem</span>.
Designs are being prepared to make a prototype of a wide-angle spherical triple <span style="font-variant:small-caps;">gem</span>, see the right side of Fig. .
{width=".9\textwidth"}
This will be the first entirely parallax-free gaseous detector. Additional challenges of this design compared to the one on the left are the spherical readout structure and the narrow spacing between <span style="font-variant:small-caps;">gem</span>s, for which curved spacers are developed.
Procedure *&* tooling
=====================
For the manufacturing of a spherical <span style="font-variant:small-caps;">gem</span> we start with a flat <span style="font-variant:small-caps;">gem</span> foil. The shape of the electrodes is designed for the purpose; otherwise the foils used are no different from <span style="font-variant:small-caps;">gem</span>s used for other applications. Thus, our base material is of proven reliability and we know its properties well. Starting with this flat <span style="font-variant:small-caps;">gem</span> foil we use a method similar to thermoplastic heat forming; the foil is forced into a new shape by stretching it over a spherical mold, see Fig. . After a heat cycle it keeps this spherical shape. Heat forming is routinely done with thermoplastic polymers, where above the so-called *glass transition temperature* monomers can migrate freely, and polymerize again upon cooling down. However the polyimide substrate of <span style="font-variant:small-caps;">gem</span>s is a thermoset polymer which has no well-defined glass transition temperature. Strongly heating a polyimide leaves the polymer chains intact, but allows cross-links between chains to break or dislocate, thus relieving mechanical stresses. Starting from a certain temperature the polyimide will start to degrade irreversibly; it becomes weak, brittle and dark-colored, see Fig. . We found that $^\circ$C is the highest temperature we can apply to foils without damaging the quality of the polymer significantly.
Figure shows the simple setup designed for forming <span style="font-variant:small-caps;">gem</span>s spherically. It consists of a spherical mold, a ring-and-plate structure to hold the foil to be formed, and four rods along which the plate can slide down toward the mold. Weights are applied to the plate to control the force that stretches the foil over the mold. Note that, contrary to thermoplastic heat forming, the foil cannot be allowed to slip between the ring and plate where it is mounted. The degree to which monomers migrate during the forming is much less than for thermoplastics; foils that accidentally slipped during forming were consistently wrinkled. Also, the forming process for polyimide is apparently very slow: we found that the shortest heat cycle that gave satisfying results took hours.
![The result of overheating a foil when trying to form it. The electrodes of this <span style="font-variant:small-caps;">gem</span> were removed beforehand to avoid their oxidation. During the forming procedure it was heated up to $^\circ$C.[]{data-label="TooHot"}](TooHot){width=".75\columnwidth"}
Partly due to this long heating time, the copper electrodes get fully oxidized in the process. From such deep oxidation, electrodes cannot be recovered by etching. The oxidation also causes some delamination of copper from the polyimide substrate. Therefore the procedure should be carried out in an oxygen-free atmosphere.
![Deposits on the electrode.[]{data-label="Deposits"}](Deposit2){width="\columnwidth"}
Tests performed in a gas-tight enclosure with a constant flow of argon show that oxidation can indeed be largely avoided. However, after forming the electrodes are covered with a thin film of an unknown substance, see Fig. . Samples taken from these deposits will be analyzed to identify the material and to understand the processes leading to its formation. The very loose attachment to the electrodes suggests that outgassing of polyimide and/or copper plays a role. These deposits can be removed afterwards, but as it involves mechanical brushing or rather strong spraying this is likely to affect the spherical shape and should better be avoided.
![Spherical <span style="font-variant:small-caps;">gem</span> made in a vacuum of $\sim$ mbar.[]{data-label="UnderVacuum"}](UnderVacuum){width="\columnwidth"}
To avoid formation of these deposits as much as possible the procedure should be carried out in a vacuum. A test done in a moderate vacuum of $\sim$ mbar gave encouraging results, see Fig. . However, the foil still tends to attach to the mold, making it difficult to dismount the <span style="font-variant:small-caps;">gem</span> from the setup without damaging it. A carbon spray coating of the mold is considered as an anti-adhesive layer to prevent this in the future. In order to exclude outgassing of the polyimide to contribute to this attachment, foils are heated in a vacuum before they are mounted in the setup, until the outgassing process diminishes. Outgassing studies on polyimides [@outgassing] suggest that this point is reached after $\sim$ hours, at a temperature of $^\circ$C. This procedure need not be a great complication; contrary to the spherical forming this can be done with many foils at the same time. In addition, <span style="font-variant:small-caps;">gem</span>s treated in this way have such benign outgassing behavior that they can be used in *sealed* detectors, where the gas is not flowing [@x-rayAstronomy]. This opens the way to use more efficient gases such as xenon (see Fig. ), which would otherwise be prohibitively expensive.
Properties of spherical <span style="font-variant:small-caps;">gem</span>s
==========================================================================
<span style="font-variant:small-caps;">Gem</span>s bent with the methods described above hold high voltages up to $\sim$ V in air with a few nA leakage current, just like before the forming procedure. When discharges occur, they are randomly distributed over the area of the foil. This suggests that deformations due to the change in shape are spread homogeneously over the foil. Observing holes at various locations on a spherical <span style="font-variant:small-caps;">gem</span> through a microscope confirms that, although they must have become wider, their shape has not become elliptical.
Assuming homogeneous stretching, one can estimate the change in relevant dimensions. The active area of the foil before (flat) and after stretching (curved) is: $$\begin{aligned}
A_\textrm{\scriptsize flat} &=& \frac{\pi d^2}{4}=\pi r^2\sin^2\theta_{1/2}\\
A_\textrm{\scriptsize curved}&=& 2\pi \int_0^{\theta_{1/2}} r^2\sin\theta\textrm{d}\theta=2\pi r^2\left( 1-\cos\theta_{1/2} \right)\end{aligned}$$ Where $r$ is the radius of curvature of the sphere, and $\theta_{1/2}$ is half the opening angle as indicated in Fig. . Then the surface stretching factor is: $$\frac{A_\textrm{\scriptsize curved}}{A_\textrm{\scriptsize flat}}=2\frac{1-\cos\theta_{1/2}}{1-\cos^2\theta_{1/2}}$$ It depends only on the opening angle. If we substitute the numbers from Fig. , we obtain % and % of increase in area for the left and right detectors respectively.
These stretching factors will have an effect on the aspect ratio of <span style="font-variant:small-caps;">gem</span> holes (defined as *depth/width* of a hole), which in turn influences the amplifying behavior of the <span style="font-variant:small-caps;">gem</span>. The change in width of a hole is proportional to the square root of the stretching factor, and the change in depth is inversely proportional to the stretching factor. This results in aspect ratios reduced to % and % respectively, compared to their state before forming. This will be compensated by changing the diameter and pitch of holes from the standard / microns to / and / microns respectively.
Another effect to be taken into account is the increase in capacitance of a foil due to stretching, as this will also increase the power generated in a discharge. As the area of the foil increases with the stretching factor, the thickness of the dielectric decreases with the same factor. Hence the capacitance increases with the square of the stretching factor: % and % respectively. Capacitance measurements of the spherical <span style="font-variant:small-caps;">gem</span>s produced so far confirm these calculations.
Work in progress
================
![Extraction field versus angle for the spherical <span style="font-variant:small-caps;">gem</span> with planar readout of Fig. .[]{data-label="ExtractionField"}](ExtractionField){width="\columnwidth"}
At the time of writing the detectors shown in Fig. are in the design phase. Simulation studies are being made to understand the charge transfer between the spherical <span style="font-variant:small-caps;">gem</span> and the flat readout board (Fig. , left). For instance, Fig. shows how the electrostatic field just below the spherical <span style="font-variant:small-caps;">gem</span> decreases at wider angle. This is the extraction field for secondary electrons from the holes, which influences the effective gain of the <span style="font-variant:small-caps;">gem</span>. The resulting $\theta$-dependence of the gain can be corrected for numerically.
![The principle of a conical field cage made from a multilayer <span style="font-variant:small-caps;">pcb</span>.[]{data-label="FieldCage"}](ConicalFieldCage){width="\columnwidth"}
The field quality in the drift region is more critical, as the elimination of the parallax error depends on it. Therefore a spherical field cage is designed and manufactured to maintain a good radial field until the edge of the active area. This field cage is a conical enclosure of the conversion region, and is made from a standard multilayer <span style="font-variant:small-caps;">pcb</span>, see Fig. . A resistive divider defines the voltages supplied to each layer. It will also serve as a rigid mechanical fixture to which the <span style="font-variant:small-caps;">gem</span> can be glued, and as a high voltage distributor which supplies the <span style="font-variant:small-caps;">gem</span>.
![The first conical field cage produced. Below a detail where the circuitry (without components) and the exposed annular electrodes are visible.[]{data-label="FirstFieldCage"}](FieldCage){width="\columnwidth"}
![Principle of spiral-pattern readout structure for a spherical gas detector.[]{data-label="spirals"}](Spiral2DReadout){width="\columnwidth"}
{width="\textwidth"}
The use of spacers in the drift region is considered, in case the spherical <span style="font-variant:small-caps;">gem</span> is not sufficiently self-supporting. In case of a spherical multiple <span style="font-variant:small-caps;">gem</span> structure, spacers between the <span style="font-variant:small-caps;">gem</span>s cannot be avoided. For planar <span style="font-variant:small-caps;">gem</span> detectors, spacers are easily made from plate materials. The design is made with any 2<span style="font-variant:small-caps;">d</span> graphics package, and the plates are machined with a <span style="font-variant:small-caps;">cnc</span> bench. For a spherical detector curved spacers are needed, which must be designed with 3<span style="font-variant:small-caps;">d</span> software. Also the manufacturing is less straightforward. The spacers for these prototypes are being made by *stereolithography*, a fast 3<span style="font-variant:small-caps;">d</span> prototyping technique that uses a <span style="font-variant:small-caps;">uv</span> laser to polymerize a liquid epoxy in a selective and accurate way.
In order to arrive at a fully spherical detector, a spherical readout board must be developed. This is a considerable technical challenge, as many possibilities are excluded: vias will crack, adhesives for multi-laminates do not support the high forming temperatures, standard rigid substrates cannot be formed like <span style="font-variant:small-caps;">gem</span>s. We foresee a strip readout in a spiral pattern, where strips printed on top and bottom of a <span style="font-variant:small-caps;">gem</span>-like structure are mutually orthogonal (i.e. clockwise and counter-clockwise spirals), see Fig. . The sharing of charge between top and bottom strips can be tuned by varying a low voltage between the sides.
Conclusions and outlook
=======================
We have shown that is it feasible to make spherical <span style="font-variant:small-caps;">gem</span>s, using a reasonably inexpensive method. Many forming tests have been done to gain a good understanding of the parameters that lead to satisfying results, in a reproducible way. The <span style="font-variant:small-caps;">gem</span>s made this way hold high voltage with the same low leakage as ordinary flat foils. They will soon be tested with a spherical cathode and a planar readout structure.
The next challenge will be to develop a spherical readout board and make a fully spherical detector. Detailed designs have already been prepared for this detector (see Fig. on the next page), which will be the first entirely parallax free gas detector.
[00]{} E. Storm and H.I. Israel, *Photon cross sections from 1 keV to MeV for elements Z=1 to Z=.* Nuclear Data Tables, A7, , –.
A. Dianoux, G. Lander, *Neutron Data Booklet.* Institut Laue-Langevin, .
G. Charpak, *Parallax-free, high-accuracy gaseous detectors for x-ray and <span style="font-variant:small-caps;">vuv</span> localization.* Nucl. Inst. & Meth. , , pp. –.
J.K. Black et al., *The imaging x-ray detector for Lobster-<span style="font-variant:small-caps;">iss</span>.* Nucl. Inst. & Meth. A, vol. , , pp. –.
Z. Hajduk, G. Charpak, A. Jeavons, R. Stubbs, *The spherical drift chamber for x-ray imaging applications.* Nucl. Inst. & Meth. I, , pp. –.
P.F. Christie, E. Mathieson and K.D. Evans, *An x-ray imaging proportional chamber incorporating a radial field drift chamber.* Journal of Physics E: Scientific Instruments, vol. 9, , pp. –.
S.P. Chernenko et al., *Test results of the parallax-free x-ray area detector <span style="font-variant:small-caps;">cd</span>- in the diffractometer <span style="font-variant:small-caps;">card</span>-7.* Nucl. Inst. & Meth. A, vol. , , pp. –.
Y. Diawara et al., *A parallel-plate resistive-anode gaseous detector for x-ray imaging.* <span style="font-variant:small-caps;">Ieee</span> transactions on nuclear science, vol. , June .
P. Rehak, G.C. Smith, B.Yu, *a method for reduction of parallax broadening in gas-based position sensitive detectors.* <span style="font-variant:small-caps;">Ieee</span> transactions on nuclear science, Vol. , June .
B. Guérard et al., *Advances in detectors for single crystal neutron diffraction.* Nucl. Inst. & Meth. A, vol. , , pp. –.
F. Sauli, *<span style="font-variant:small-caps;">Gem</span>: A new concept for electron amplification in gas detectors.* Nucl. Inst. & Meth. A, vol. , issues 2–3, /2/, pp. –.
K. Kurvinen et al., *Outgassing analysis of various detector materials.* <span style="font-variant:small-caps;">Ieee-nss</span> conference record, vol. 5, pp. –. K. Kurvinen et al., *<span style="font-variant:small-caps;">Gem</span> detectors for x-ray astronomy.* Nucl. Instr. & Meth. A, vol. , issues 1–2, 1 November , pp. –.
[^1]:
[^2]:
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{
"pile_set_name": "ArXiv"
}
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---
author:
- 'K. T. Kono,'
- 'T. T. Takeuchi,'
- 'S. Cooray,'
- 'A. J. Nishizawa,'
- 'K. Murakami'
bibliography:
- 'Kono2020b.bib'
title: A Study on the Baryon Acoustic Oscillation with Topological Data Analysis
---
Introduction {#sec:intro}
============
Characterizing the Matter Distribution in the Universe
------------------------------------------------------
The distribution of matter and galaxies at each epoch of the Universe contains fundamental information on the formation and evolution of the cosmic structures in general [e.g., @1980lssu.book.....P; @1983FCPh....9....1E; @2002PhR...367....1B]. A vast number of sophisticated methods have been proposed to characterize the statistical properties of the fluctuation in the Universe [e.g., @1980lssu.book.....P; @martinez2001statistics; @2002PhR...367....1B]. Among them, the $n$-point correlation functions are the most popular and well-studied methods in the analysis of observational galaxy distribution [e.g., @1980lssu.book.....P]. Mathematically, the infinite sequence of $n$-point correlation functions are sufficient to specify stochastic random fields [e.g., @adler1981geometry; @adler2009random].
However, in practice, estimating higher-order $n$-point correlations becomes unrealistically difficult, and other approaches that can treat the information of higher-order correlations have been considered. One of the popular methods in cosmology and astrophysics is the so-called “Minkowski Funcionals” [e.g., @2003PASJ...55..911H; @2003ApJ...584....1M; @2019MNRAS.485.1708S]. In the applications of the Minkowski Functionals to the Large-Scale Structure, the topology of smoothed density field of galaxy distribution is analyzed by examining the “genus” or number of holes in the density field. Since basically the smoothed random field contains the whole information on the infinitely high-order moments of the field, in principle the Minkowski Functionals can specify the galaxy density field completely. Indeed this method has been applied very extensively to the analysis of the cosmic density field , as well as redshift distortion [e.g., @2018ApJ...863..200A], weak lensing cosmology , cosmic reionization [@2017MNRAS.465..394Y; @2018MNRAS.477.1984B; @2019MNRAS.485.2235B; @2019ApJ...885...23C], test of cosmological/gravity theories [@2015PhRvD..92f4024L; @2015PhRvD..92d3505J; @2016PhRvD..94d3506S; @2017PhRvL.118r1301F], and the cosmic microwave background (CMB: e.g., ).
From Minkowski Functionals to topological data analysis
-------------------------------------------------------
In fact, however, the method of Minkowski Functionals, or sometimes called the genus statistics, is a part of more general framework to characterize the properties of a point data set, known in the field of topological data analysis (TDA: e.g., [@edelsbrunner2002; @10.1145/997817.997870; @wasserman2018]). We briefly overview the method of TDA in the following.
Intuitively, the topological information is characterized by the holes of the object. For the current interest, the object to be analyzed is constructed from $n$-dimensional sphere with radius $r$ from discrete data points, i.e., $N$-body particles and/or galaxies (and some other astronomical objects, in general). With the discovery of the 3-dim structure in galaxy distribution [e.g., @1986ApJ...302L...1D], an idea of using the geometry (topology) of smoothed density field was proposed to quantify the large scale structure and specify the physics behind it . The basic idea of these early works was an application of random percolation: they set a sphere around each galaxy with an increasing radius, and examine the geometry of connected components in a considered volume. This approach has been developed in relation to the theoretical development of perturbation theory of the cosmic structure [e.g., @1994ApJ...434L..43M; @1996ApJ...457...13M; @1996ApJ...460...51M; @1996ApJ...463..409M; @2003ApJ...584....1M; @2009PhRvD..80h1301P; @2012PhRvD..85b3011G]. The expectation of the number of genus of the smoothed field as a function of the threshold density is referred to as the genus curve, and it can be expressed analytically with these theories. Since this curve can distinguish between Gaussian and other random fields, it became one of the most popular tools for the analysis of the large-scale structure.
In spite of the great development of the theories of the Minkowski Functionals in cosmology, it was not well recognized that it consists a part of much larger, richer framework studied in the topological data analysis (TDA). The birth of the primitive concept of the TDA dates back to 90’s, but its actual birth was in the new millennium [see e.g., @edelsbrunner2002; @DBLP:books/daglib/0025666]. The concept directly related to the Minkowski Functionals is the persistent homology (PH) in TDA. Similarly to the random percolation, the PH considers a sphere (solid ball) of radius $r$ around each point in a point data set. More sophisticatedly, the PH handles the birth and death of loop structures in various dimensions. This is schematically described in Fig. \[fig:persistent\_homology\_schematic\]. Consider a set of three points. When the radius $r$ is small, we simply have a set of three balls (left of Fig. \[fig:persistent\_homology\_schematic\]. Then, at some $r$, we have a connected structure, a loop (middle), regarded as the birth of a hole. When $r$ becomes larger, the hole in the loop disappears, the death of the hole (right). For general point data set, we trace such a life history of loops with dimension $0, 1, 2, 3, \dots, q$, corresponding to a number of connected components (0-dim), number of loops (1-dim), number of hollow voids (2-dim), $\dots$, number of $q$-dim holes, respectively. Thus, we can easily imagine that, if we plot the birth and death of these quantities as a function of $r$, this will contain very rich information of the point set. Such a plot is referred to as the persistent diagram (PD) and is the central tool for the PH. Complete definitions and methodology will be thoroughly presented in Section \[sec:TDA\].
In spite of the vast number of applications of Minkowski Functionals as mentioned above, extensive applications of the TDA has not been very frequently found in astrophysics and cosmology yet, though it has been gradually known for recent several years. Pranav et al. [@2017MNRAS.465.4281P; @2019MNRAS.485.4167P] and Feldgrugge et al. [@2019JCAP...09..052F] presented a detailed formulation on the stochastic homology of Gaussian and non-Gaussian random fields, gearing at the application to the stochastic field. By the method, they claimed a detection of an anomalous behavior the the CMB fluctuation . Cole and Shiu [@2018JCAP...03..025C] also developed a method to apply the TDA to the evaluation of the non-Gaussianity of the CMB. Xu et al. applied the PH for a large galaxy data set generated from $N$-body simulation [@2018MNRAS.473.1195L] and found 23 voids in the simulation data.
Here, in this work, we introduce the method with the emphasis on the [*inverse analysis*]{} of the detected topological structures, and apply the PH to the signal of the baryon acoustic oscillation.
![A schematic description on the concept of the persistent homology. Topological information is characterized by “holes” constructed from $n$-dimensional spheres with a radius $r$ from discrete data points. []{data-label="fig:persistent_homology_schematic"}](schematic.pdf){width="90.00000%"}
The baryon acoustic oscillation
-------------------------------
Baryons evolve in a very complicated way via electromagnetic interactions (with radiative heating/cooling, gas pressure, fluid dynamical processes, etc.) A typical example of such a nontrivial phenomenon is the baryon acoustic oscillation (BAO) generated in the matter-radiation fluid in the early Universe . Here we briefly review the BAO and related observables.
Consider a point-like initial perturbation in the primordial matter-photon plasma. In the plasma, the matter and photons are locked into a single fluid. Since the photons are so hot and numerous, the combined fluid has an tremendous pressure with respect to its density. The pressure tries to equalize itself with the surroundings, which results in an expanding spherical sound wave. The sound speed at this early epoch is evaluated by $$\begin{aligned}
\label{eq:sound_speed_decoupling}
c_{\rm s} = \frac{\displaystyle c}{\displaystyle \sqrt{3\left( 1 + \frac{3\rho_{\rm B}}{4\rho_{\rm R}} \right)}}\end{aligned}$$ where $c$ is the speed of light, and the subscript B and R stand for baryon and radiation, respectively. Equation \[eq:sound\_speed\_decoupling\] shows that the sound speed before the matter-radiation decoupling can be $\sim 57$ % of the speed of light at early epoch. Sound horizon (final radius) $r_{\rm s}$ is obtained by $$\begin{aligned}
r_{\rm s} = \int_0^{t_{\rm dec}} (1+z) c_{\rm s} {\rm d} t =
\int_{z_{\rm dec}}^{\infty} \frac{c_{\rm s} }{H_0\sqrt{\Omega_{\rm R,0}(1+z)^4+\Omega_{\rm M,0}(1+z)^3 + \Omega_{\Lambda, 0}}}{\rm d} z\end{aligned}$$ where subscript dec stands for the time of the decoupling between photons and matter. What we actually observe is the superposition of many acoustic waves imprinted on the large-scale structure emerged from the primordial fluctuations. Though the BAO is a baryonic phenomenon by definition, the density enhancement structure of baryons and dark matter becomes the same at the final state [e.g., @MaBertschinger:1995; @2007ApJ...664..675E].
The BAO length scale is constant in comoving coordinates. Then, in principle, we can detect the signal on the galaxy 2-point correlation function. However, since the BAO scale is very large compared to the typical scale of the large-scale structure, we need a very large galaxy sample with dense sampling, since we must measure the signal at such a large scale, where usually the 2-point correlation is very weak. The SDSS is the largest optical photometric and spectroscopic surveys ever existed, covering one-third of the whole sky. Then, the advent of the SDSS finally made it realistic to detect the BAO signal on the 2-point correlation function. Eisenstein et al. [@2005ApJ...633..560E] first detected the signal around 150 Mpc on the 2-point correlation function. Since then, still this analysis is only possible with SDSS data.
The BAO can be used for various directions of cosmological studies. Then, if a more flexible and easier method for the detection and quantifying the BAO signal exists, it would be desirable. We introduce the PH as such an ideal method for the analysis of BAO. This paper is organized as follows: We introduce the definitions and concepts of the TDA relatively rigorously in Section \[sec:TDA\]. Especially, we introduce the method of inverse analysis of the PH. Then we examine the performance of the PH as a tool for the studies of BAO by using simulation datasets with and without baryon physics in Section \[sec:simulation\]. After knowing its power, we apply the PH to quasar data extracted from the extended Baryon Oscillation Spectroscopic Survey (eBOSS) [@2016AJ....151...44D] LSS catalog [@2018MNRAS.473.4773A] in the SDSS DR14 in Section \[sec:SDSS\]. Section \[sec:summary\] is devoted to our summary and discussion on the result and future prospects of the presented methodology.
In this study, we adopt cosmological parameters $h = H_0/100\ [{\rm km\ s^{-1}\ Mpc^{-1}}]= 0.6766$, $\Omega_{\Lambda,0} = 0.6889$, $\Omega_{\rm M,0}=0.3111$, and $\ln (10^{10} A_s) = 3.047$, as constrained by the latest CMB observation by Planck [@Planck+CosPar:2018] throughout this paper.
Topological Data Analysis (TDA) {#sec:TDA}
===============================
Geometric modeling of point cloud data
--------------------------------------
In the topological data analysis (TDA), we often face a problem to examine topological properties of a finite sequence of points $P$ in $N$-dim Euclidean space $\mathbb{R}^N$[^1]. Since $P$ is a finite set, a simple application of traditional topological discussion cannot extract any nontrivial information on the features of $P$. Instead, in TDA, a geometric model with a “filtration”, $\mathcal{X} = \{ X^a ; a \in \mathbb{R} \}$, is constructed from the finite point cloud, and topological methods are applied to the model. Here the parameter of the filtration $a$ is considered to control the “resolution” or “scale” of the input data. The TDA extract meaningful topological characteristics of the point data by examining the persistent properties along the change of the resolution [e.g., @edelsbrunner2002; @boissonnat_chazal_yvinec_2018]. We introduce such geometric models that play a fundamental role in the TDA.
### Simplex and simplicial complex
![An example of 0, 1, 2, and 3-dim simplex. []{data-label="fig:simplex"}](simplex.pdf){width="80.00000%"}
A simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. More precisely, it is defined as follows. Points of a finite set $P = \{ p_0, p_1, \dots, p_k \}$ in $\mathbb{R}^N$ is affinely independent if they are not contained in any affine subspace of dimension less than $k$.
\
Suppose a set of $k+1$ points, $P = \{ p_i\}\; (i = 0, \dots, k)$ in $N$-dimensional Euclidean space $\mathbb{R}^N$, arranged to be affinely independent. If we consider $k$ vectors $\overrightarrow{p_0 p_1}, \dots, \overrightarrow{p_0 p_k}$, the convex hull that contains these vectors $$\begin{aligned}
\left| p_0 p_1\dots p_k \right| \equiv \left\{ x \in \mathbb{R}^N | x = \lambda_0 p_0 + \dots + \lambda_k p_k, \lambda_i \geq 0\,, \; \sum_{i=0}^{k} \lambda_i = 1 \right\}\end{aligned}$$ is called $k$-simplex, often denoted as $\sigma$. The dimension of $k$-simplex is defined as $k$.
For example, a $0$-simplex is a point, a $1$-simplex is a line segment, a $2$-simplex is a triangle, and a $3$-simplex is a tetrahedron.
\
Consider a $k$-simplex. The convex hull of any nonempty subset of the $\ell+1$ points ($\ell \leq k$) that define an $\ell$-simplex is called a face of the simplex.
A face is itself a simplex. The convex hull of a subset of size $\ell+1$ of the $k+1$ points is defined as $\ell$-simplex, called an $\ell$-face of the $k$-simplex. A $0$-face is called the vertex, and a 1-face is the edge. Figure \[fig:simplex\] shows the examples of simplices with dimension of 0, 1, 2, and 3. For example, a $2$-simplex in Fig. \[fig:simplex\] has seven surfaces $|p_0|, |p_1|, |p_2|, |p_0 p_1|, |p_0 p_2|, |p_1 p_2|$, and $|p_0 p_1 p_2|$.
Based on these concepts, we can define the simplicial complex as
\[def:simplicial\_complex\] \
A simplicial complex $K$ in $\mathbb{R}^N$ is a set of simplices that satisfies the following conditions:
1. Every face $\sigma$ of a simplex from $K$ is also in $K$.
2. The non-empty intersection of any two simplices $\sigma_{i}, \sigma_{j} \in K$ is a face of both $\sigma _{i}$ and $\sigma_{j}$ $(0 \leq i, j \leq k)$.
The maximal dimension of simplices in $K$ is called the dimension of the simplicial complex $K$, denoted as $\mbox{dim}\;K$. An example of a simplicial complex is presented in Fig. \[fig:simplicial\_complex\].
\
Consider a union of all the simplices in $K$, $$\begin{aligned}
|K| \equiv \bigcup_{\sigma \in K} \sigma \, , \end{aligned}$$ a geometrical shape in $\mathbb{R}^N$ is obtained. This shape is referred to as a polyhedron defined by a simplicial complex $K$.
![An example of simplicial complex. []{data-label="fig:simplicial_complex"}](simplicial_complex.pdf){width="40.00000%"}
\
If a family $\tilde{K}$ of non-empty finite subsets of a finite set $V$ satisfies the following condition, $(V, \tilde{K})$ is an abstract simplicial complex:
1. For every set $v \in V$, $\{ v\} \in \tilde{K}$,
2. A subset of $\tau \subset \tilde{K}$, $\sigma \subset \tau$, also belongs to $\tilde{K}$.
Here a subset $\tau$ of $\tilde{K}$, $\tau = \{ v_0, \dots, v_k\}$, is called $k$-simplex, whose dimension is defined as $k$. The maximal dimension of simplices in $\tilde{K}$ is called the dimension of the abstract simplicial complex, denoted as $\mbox{dim}\; (V, \tilde{K})$. As trivially follows from the definition, a simplex consisting an abstract simplicial complex does not need to be in $\mathbb{R}^N$.
A simplicial complex can be treated as an abstract simplicial complex. Inversely, a simplicial complex in $\mathbb{R}^N$ can be assigned to an abstract simplicial complex. Consider a point $$\begin{aligned}
p_i = (0, \dots, 0, \underset{i}{1}, 0, \dots, 0) \in \mathbb{R}^N\end{aligned}$$ (i.e., only the $i$-th component is $1$ and all the others are $0$) for each element $v_i$ ($i = 1, \dots, N$) of an abstract simplicial complex $(V, \tilde{K})$. By these points $\{ p_i\}$, we define a simplex $|p_0, \dots, p_k|$ for $\{v_0, \dots, v_k \} \in \tilde{K}$. If we define $K$ as a set of all the simplices in $\mathbb{R}^N$ obtained as above, $K$ satisfies Definition \[def:simplicial\_complex\]. Namely, $K$ is a simplicial complex. The polyhedron $|K|$ is called a geometric realization of $(V, \tilde{K})$ (e.g., [@boissonnat_chazal_yvinec_2018]).
### Nerve and nerve theorem {#subsec:nerve}
As mentioned above, simplicial complices can be seen as purely combinatorial objects, as well as a topological space at the same time. We introduce an important concept “nerve” here.
\[def:open\_cover\] \
An open cover of a set $X$ is a collection $\mathcal{U} = \{ U_i \}\; (i = 1, \dots, m)$ of open subsets $U_i \subseteq X$, such that $$\begin{aligned}
X = \bigcup_{i=1}^{m} U_i \; .\end{aligned}$$
Then, we define a nerve as follows.
\[def:nerve\] \
Given a cover of a set $X$, $\mathcal{U} = \{ U_i \} \in \mathbb{R}^N \; (i=1, \dots, m)$, its nerve is the abstract simplicial complex $\tilde{K} (\mathcal{U})$ whose vertex set is $\mathcal{U}$ and $$\begin{aligned}
\sigma = [ U_{i_0}, U_{i_1}, \dots, U_{i_k}] \in \tilde{K}(\mathcal{U}) \end{aligned}$$ if and only if $$\begin{aligned}
\bigcap_{j=1}^{k} U_{i_j} \neq \emptyset \; .\end{aligned}$$ We denote the nerve of $\mathcal{U}$ as $\mathcal{N}(\mathcal{U})$.
In general, the nerve $\mathcal{N}(\mathcal{U})$ does not always accurately reflect the topology of $X$, i.e., $\mathcal{N}(\mathcal{U})$ is not homotopic equivalent to $X$. However, if an intersection of the $U_i$s is either empty or contractible (see Definition \[def:contractible\]), namely a “good cover”, the following important theorem holds.
\[theo:nerve\] \
Let $\mathcal{U} = \{ U_i \} \in \mathbb{R}^N \; (i=1, \dots, m)$ be a finite open cover of a subset $X \in \mathbb{R}^N$ such that any intersection of the $U_i$s is either empty or contractible. Then $X$ and the nerve $\mathcal{N}(\mathcal{U})$ are homotopy equivalent.
### Čech complex {#subsubsec:cech_complex}
![An example of Čech complex. Left: a union of balls with radius $r$ around $x_i$, Right: the corresponding Čech complex. []{data-label="fig:cech_complex"}](ball.pdf "fig:"){width="30.00000%"} ![An example of Čech complex. Left: a union of balls with radius $r$ around $x_i$, Right: the corresponding Čech complex. []{data-label="fig:cech_complex"}](cech_complex.pdf "fig:"){width="30.00000%"}
Now we relate the filtration and simplicial complex.
\
A filtration of a finite simplicial complex $K$ is a nested sequence of sub-complices $\emptyset \subset K^1 \subset \dots \subset K^m = K$ such that $$\begin{aligned}
K^{i+1} = K^i \cup \sigma^{i+1} \end{aligned}$$ where $\sigma^{i+1}$ is a simplex of $K$.
We note that a filtration of $K$ is just an ordering of the simplices, it may be natural to index the simplices by an increasing sequence of real numbers $\{ \alpha_i \} \in \mathbb{R} \, , \; (i = 1, \dots, m)$, $$\begin{aligned}
\emptyset \subset K^{\alpha_0} \subset K^{\alpha_1} \subset \dots \subset K^{\alpha_m} = K. \end{aligned}$$
We define the Čech complex as follows.
\[def:cech\_complex\] \
Consider a point set $P \in \mathbb{R}^N$, $P = \{ x_i \in \mathbb{R}^N, i = 1, \dots, m\}$. Set a sphere with radius $r$ around each $x_i$, $$\begin{aligned}
B_r (x_i) = \left\{ x \in \mathbb{R}^N; \|x - x_i\| \leq r \right\} \; ,\end{aligned}$$ where $\|x\|$ stands for the Euclid norm. The nerve of the collection of these balls $\mathcal{B} = \{ B_r(x_i), x_i \in X \}$, $$\begin{aligned}
\mathcal{C}(P, r) \equiv \mathcal{N}(\mathcal{B}) = \left\{ |x_{i_0}, \dots, x_{i_k}|; \bigcap_{j=0}^{m} B_r(x_{i_j}) \neq \emptyset \right\} \end{aligned}$$ is the Čech complex of $P$ with radius $r$.
Since a ball satisfies the condition of Theorem \[theo:nerve\], we have $$\begin{aligned}
X_r \equiv \bigcup_{i=1}^{m} B_r(x_i) \simeq \mathcal{C}(P,r) \; .\end{aligned}$$ This means that if we want to examine homotopy invariant characteristics of a set $X$ that satisfies the condition of Theorem \[theo:nerve\], we can do it equivalently on its nerve $\mathcal{N}(\mathcal{U})$. Since the nerve is an abstract simplicial complex, it is much more computer-friendly to calculate such properties. A schematic example of a Čech complex is shown in Fig. \[fig:cech\_complex\]
The necessary and sufficient condition that $k$-simplex of a Čech complex is $$\begin{aligned}
\bigcap_{j=0}^{k} B_r(x_{i_j}) \neq \emptyset \; .\end{aligned}$$ Note that this hols for $r' > r$, namely, $$\begin{aligned}
\bigcap_{j=0}^{k} B_r(x_{i_j}) \neq \emptyset \; \Rightarrow
\bigcap_{j=0}^{k} B_{r'}(x_{i_j}) \neq \emptyset , \quad (r < r') \; .\end{aligned}$$ Thus $$\begin{aligned}
\mathcal{C} (P, r) \subset \mathcal{C} (P, r') , \quad (r < r') \; ,\end{aligned}$$ and therefore, for an increasing sequence $r_0 < r_1 < \dots < r_T$, we have an increasing sequence of Čech complices $$\begin{aligned}
\mathcal{C} (P, r_0) \subset \dots \subset \mathcal{C} (P, r_i) \subset \dots \subset \mathcal{C} (P, r_T) \; .\end{aligned}$$ This defines the Čech complex filtration. This enables us to deal not only with the topological information of $\mathcal{C}(P, r_i)$ at a radius $r_i$ but also their transition and persistence with changing $r$.
### Voronoi diagrams, Delauney complex and alpha complex
![Schematic description of the relation between a point set (top left), its Čech complex (top right), Delauney complex (bottom left), and alpha complex (bottom right). []{data-label="fig:summary_complex"}](summary_complex.pdf){width="90.00000%"}
Though the definition of Čech complex is straightforward, computationally it is rather demanding when the data size is large. A more convenient type of complex is desirable for this aspect. We introduce the alpha complex in the following [@10.1145/174462.156635; @10.1145/160985.161139]. We first define the Voronoi diagram of a point set.
\[def:voronoi\] \
Consider a point set $P \in \mathbb{R}^N$, $P = \{ x_i \in \mathbb{R}^N, i = 1, \dots, m\}$. A Voronoi diagram can be expressed with Voronoi cells $V_i\; (i=1, \dots, m)$ for each point $x_i$ as $$\begin{aligned}
&&V_i = \left\{ x \in \mathbb{R}^N; \|x - x_i\| \leq \|x - x_j\| , 1 \leq j \leq m, j \neq i \right\} \; , \\
&&\mathbb{R}^N = \bigcup_{i=1}^{m} V_i \; . \label{eq:voronoi_tessellation}\end{aligned}$$ A region defined for each $x_i$ is the Voronoi cell, and the tessellation (eq. (\[eq:voronoi\_tessellation\])) is referred to as the Voronoi diagram.
Then, we define Delauney complex.
Delauney complex $\mathcal{D}(P)$ is the nerve $\mathcal{N}(\mathcal{V})$ of the Voronoi diagram, $$\begin{aligned}
\mathcal{D} (P) = \mathcal{N}(\mathcal{V})\end{aligned}$$ where $\mathcal{V}$ is a convex closed set $$\begin{aligned}
\mathcal{V} = \{ V_i ; i = 1, \dots, m\} \; .\end{aligned}$$
Further, we consider the most important concept for this analysis.
\
We define an intersection of $B_r(x_i)$ and $V_i$ denoted as $W_i$, $$\begin{aligned}
&&W_i \equiv B_r(x_i) \cap V_i \; , \\
&&X_r = \bigcup_{i=1}^{m} W_i \; .\end{aligned}$$ The alpha complex $\mathcal{A}(P, r)$ for a set of the center of spheres $P$ can be defined as $$\begin{aligned}
\mathcal{A} (P, r) \equiv \mathcal{N} (\mathcal{W}) \; , \end{aligned}$$ where $\mathcal{W} = \{ W_i ; i=1, \dots, m\}$.
The motivation to introduce the Alpha complex was to introduce a simplicial complex model to connect a point sequence and its convex hull with spatial resolution [@10.1145/174462.156635; @10.1145/160985.161139]. We note that the dimension of $\mathcal{A}(P, r)$ is at most the dimension $N$ of the embedding space.
Simplicial homology
-------------------
We introduce the basic notions of simplical homology that are necessary to define topological persistence. According to [@boissonnat_chazal_yvinec_2018], we restrict the homology with coefficients in the finite field $\mathbb{Z}_2 \equiv \mathbb{Z}/2\mathbb{Z}=\{0, 1 \}$. In this work, we did out best to avoid to introduce complicated algebraic concepts. Readers who prefer a rigorous introduction are guided to read some textbooks on abstract algebra [e.g., @hatcher2002; @dummit2004].
### Space of $k$-chains
For any non-negative integer $k \in \mathcal{N}_0$, the space of $k$-chains is a vector space of all the formal sums with coefficients in $\mathbb{Z}_2$ of $k$-dim simplices of $K$. Precisely, if $\{ \sigma_1, \dots, \sigma_p\}$ is a set of $k$-simplices of $K$, any $k$-chain is uniquely written as $$\begin{aligned}
c = \sum_{i=1}^{p} \varepsilon_i \sigma_i \end{aligned}$$ with $\varepsilon_i \in \mathbb{Z}_2$.
\
The space of $k$-chain is a set of $C_k(K)$ of the simplicial $k$-chains of $K$ with operations of a sum and a scalar product as $$\begin{aligned}
c+c' &=& \sum_{i=1}^{p} (\varepsilon_i +\varepsilon_i') \sigma_i\; , \\
\lambda c &=& \sum_{i=1}^{p} (\lambda \varepsilon_i) \sigma_i \, .\end{aligned}$$ This forms a $\mathbb{Z}_2$-vector space whose zero element is $$\begin{aligned}
0 = \sum_{i=1}^{p} 0 \sigma_i \; .\end{aligned}$$
Note that the sums and products are modulo 2. The set of $k$-simplices of $K$ consists the basis of $C_i(K)$.
Chains with coefficient in $\mathbb{Z}_2$ have a straightforward geometric interpretation. Since any $k$-chain is uniquely written as $c = \sigma_{i_1} + \dots + \sigma_{i_m}$ where $\sigma_{i_j}$ are $k$-simplices, $c$ can be regarded as the union of the simplices $\{ \sigma_{i_j}\}$.
### Boundary operator and homology group
\
A boundary $\partial(\sigma)$ of a $k$-simplex $\sigma$ is the sum of its $(k-1)$-faces. This is a $(k-1)$-chain. let $\sigma_i = \left[ v_0, \dots, v_k \right]$ a $k$-simplex, then $$\begin{aligned}
\partial \sigma_i \equiv \sum_{i=0}^{k} \left[ v_0, \dots, \check{v_i}, \dots, v_k \right]\end{aligned}$$ ($\check{v_i}$ means $i$-th component is removed).
The boundary operator defined on the simplices of $K$ can be extended linearly to $C_k(K)$.
\[def:boundary\] \
The boundary operator $\partial_k: C_k(K) \rightarrow C_{k-1}(K)$ is defined as $$\begin{aligned}
\partial[v_1,v_2,\cdots,v_n] = \sum^n_{i=0} [v_1,v_2,\cdots,\check{v}_i,\cdots,v_n] \;.\end{aligned}$$
The boundary operator satisfies the following important property.
\[theo:boundary\] \
The boundary of a boundary of a chain is always zero, i.e., For all $k \in \mathbb{N}$, $$\begin{aligned}
\partial_{k-1} \circ \partial_k = 0 \; .\end{aligned}$$
The boundary operator defines a sequence of linear maps between the spaces of chains.
\[def:chain\_complex\] \
A chain complex associated with a complex $K$ of dimension $N$ is a sequence of linear operators as $$\begin{aligned}
\{ 0 \} \xrightarrow{} C_N(K) \xrightarrow{\partial_N} C_{N-1}(K) \xrightarrow{\partial_{N-1}} \dots \xrightarrow{\partial_{k+1}}C_k\xrightarrow{\partial_{k}}C_{k-1}\xrightarrow{\partial_{k-1}} \cdots \xrightarrow{\partial_{2}}C_1\xrightarrow{\partial_{1}}C_0\xrightarrow{\partial_{0}} \{0\} \,. \nonumber \\\end{aligned}$$
For $k \in [0, \dots, N]$, We further introduce the following.
$$\begin{aligned}
Z_k(K) &\equiv& {\rm Ker}\{\partial: C_{k}\to C_{k-1}\} = \{c \in C_k(K); \partial_k(c) = 0 \} \; ,\\
B_k(K) &\equiv& {\rm Im}\{\partial: C_{k+1}\to C_{k}\} = \{c \in C_k(K); \partial_{k+1}(c'), c' \in C_{k+1}(K)\} \; \end{aligned}$$
are the $k$-cycles and $k$-boundaries of a $k$-chain, respectively.
By definition, $Z_k$ means it does not have a boundary, and the $k$-boundary is the boundary of a $k+1$-chain. It follows from Theorem \[theo:boundary\] that $$\begin{aligned}
B_k(X) \subset Z_k(X) \subset C_k(K) \; . \end{aligned}$$
Thus, the quotient group $Z_k/B_k$ can measure the difference of the two, and enables us to extract the $k$-complices without boundaries but not boundaries of a set of $k+1$-complices.
\[def:homology\_group\] \
A quotient vector space $$\begin{aligned}
H_k(K) = Z_k(K)/B_k(K)\end{aligned}$$ is the $k$-th homology group of a simplicial complex $K$. Elements of the vector space $H_k(K)$ is the homology class of $K$. The dimension $\beta_k(K)$ of $H_k(K)$ is the $k$-th Betti number of $K$.
Informally, the homology class represents a hole[^2].
Persistent homology {#sec:PH}
-------------------
The concept of persistent homology was introduce by [@edelsbrunner2002]. Intuitively, the persistent homology aims at pursuing the track of all subcomplices of a filtration, and to pair the creation (birth) and destruction (death) of homology classes appearing during the process.
We define the persistence for a filtration of a simplical complex. Let $K$ be a $N$-dim simplicial complex and let $$\begin{aligned}
\emptyset = K^0 \subset K^1 \subset \dots \subset K^m = K\end{aligned}$$ be a filtration of $K$ such that, for any $i = 0, \dots, m-1$, $K^{i+1} \equiv K^i \cup \sigma^{i+1}$ where $\sigma^{i+1}$ is a simplex. For any $0 \leq n \leq m$, we denote the set of $k$-chains of $K$ (with coefficients in $\mathbb{Z}_2$) by $C_k^n$. The restriction of the boundary operator to $C_k^n$ has its image contained in $C_{k-1}^{n-1}$. We denote the sets of $k$-cycles and $k$-boundaries of $K^n$ by $Z_k^n$ and $B_k^n$, respectively.
\
The $k$-th persistent homology group of $K^n$ is $$\begin{aligned}
H_k^n \equiv Z_k^n/B_k^n \, . \end{aligned}$$
With these definitions, we have $$\begin{aligned}
&&Z_k^0 \subset Z_k^1 \subset \dots \subset Z_k^n \subset \dots \subset Z_k^m = Z_k(K) \; , \\
&&B_k^0 \subset B_k^1 \subset \dots \subset B_k^n \subset \dots \subset B_k^m = B_k(K) \; . \end{aligned}$$
For $p \in \{ 0, \dots, m \}$and $\ell \in \{ 0, \dots, m^p \}$, the $k$-th persistent Betti number of $K^\ell$ is the dimension of the vector space $$\begin{aligned}
\label{eq:betti}
H_k^{\ell, p}(K) = Z_k^\ell/\left( B_k^{\ell + p} \cap Z_k^{\ell}\right) \; .\end{aligned}$$ Here $Z_k^\ell$ and $B_k^{\ell + p}$ stands for a $k$-cycle group on the chain group of $X^\ell$ and a $k$-boundary group of $X^{\ell+p}$, respectively.
The $k$-th persistent Betti number of $K^\ell$ represents the number of independent homology classes of $k$-cycles in $K^\ell$ that are not boundaries in $K^{\ell+p}$. A $k$-cycle in $K^\ell$ generating a nonzero element in $H_k^{\ell+p}$ is a cycle, that has appeared in the filtration before the step $\ell + 1$ and that is still not a boundary at step $\ell+p$. Namely, for a sequence of homology determined from the filtration $$\begin{aligned}
H_k(K^0) \to H_k(K^1) \to \dots \to H_k(K^\ell) \to \dots \to H_k(K^{\ell+p}) \to \dots \; , \end{aligned}$$ it can be examined whether an element of $H_k(K^\ell)$ still exists in $H_k(K^{\ell+p})$ that is $p$-steps ahead along the filtration.
Thus, an element of $H_k(X^\ell)$ that survives long is regarded as an important topological feature, while an element that lives short (i.e., during a small interval $\ell$) is treated as topological noise. Intuitively, persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of spatial scales and are more likely to represent “true” features of the underlying point cloud rather than artifacts of sampling, noise, or some particular choice of parameters.
![Schematic persistence diagram. Data points laying close to the diagonal line should be regarded as a topological noise, while those far from the diagonal represent significant and robust structures in the data point cloud. []{data-label="fig:PD_schematic"}](PD_schematic.pdf){width="40.00000%"}
Persistence diagram {#subsec:PD}
-------------------
### Persistent pair
To make use of the persistent homology, we define a few more notions.
\
Let $K$ be a $d$-sim simplical complex and $$\begin{aligned}
\emptyset = K^0 \subset K^1 \subset \dots \subset K^m = K \, \end{aligned}$$ be a filtration of $K$. A simplex $\sigma^i$ is called positive if it is contained in a $(k+1)$-cycle in $K^i$, and negative otherwise.
We have the following useful theorem.
\[theo:positive\] \
Let $\sigma = \sigma^i$ be a positive $k$-simplex in the filtration of $K$. There exists a unique $k$-cycle $c$ that is not a boundary in $K^i$, that contains $\sigma$ and does not contain any other positive $k$-simplex.
The $k$-cycles associated to the positive $k$-simplices in Theorem \[theo:positive\] allow to maintain a basis of the $k$-dim homology groups of the subcomplices of the filtration. At the beginning, the basis of $H_k(K^0)$ is empty, and the basis of the $H_k(K^i)$ are then built inductively. Then, assume that the basis of $H_k^{j-1}$ is built and the $j$-th simplex $\sigma^j$ is negative with dim of $k+1$. Let $c^{i_1}, \dots, c^{i_p}$ be the cycles associated to the positive simplices $\sigma^{i_1}, \dots, \sigma^{i_p}$ whose homology classes form a basis of $H_k^{j-1}$. Since the boundary $\partial \sigma^j$ is a $k$-cycle in $K^{j-1}$, which is not a boundary in $K^{j-1}$ but is a boundary in $K^j$, we can write $$\begin{aligned}
\partial \sigma^j = \sum_{k=1}^{p} \varepsilon_k c^{i_k} + b\end{aligned}$$ where $\varepsilon_k \in \mathbb{Z}_2 = \{0,1\}$ and $b$ is a boundary of simplices with dimension $< j$. We then denote $\ell(j) = \max \{ i_k; \varepsilon_k = 1 \}$ and we remove the homology class of $c^\ell(j)$ from the basis of $H_k^{j-1}$.
\[def:persistence\_pair\] \
The pairs of simplices $(\sigma^{\ell(j)}, \sigma^j)$ are called the persistence pairs of the filtration of $K$.
The homology class created by $\sigma^{\ell(j)} \in K^{\ell(j)}$ is destroyed by $\sigma^j \in K^j$. The persistence (duration) of this pair is, then, $j - \ell(j) -1$.
### Persistence diagram {#persistence-diagram}
For a fixed $k$, the persistence pairs of simplices of respective dimensions $k$ and $k+1$ are conveniently represented by a diagram on $\mathbb{R}^2$. Each pair $(\sigma^{\ell(j)}, \sigma^j)$ is represented by the point of coordinates $(\ell(j), j)$. For each positive simplex $\sigma^i$ which is not paired to any negative simplex in the filtration, we associate the pair $(\sigma^i, +\infty)$. Generally, if the filtration is indexed by a non-decreasing real numbers as $$\begin{aligned}
\emptyset = K^{\alpha_0} \subset K^{\alpha_1} \subset \dots \subset K^{\alpha_m} = K \, , \alpha_0 \leq \alpha_1 \leq \dots \leq \alpha_m \; , \end{aligned}$$ a persistent pair of simplices $(\sigma_i, \sigma_j)$ is represented as the point of coordinates $(\alpha_i, \alpha_j)$. In this case, since the sequence $\{\alpha_i\}$ is non-decreasing, some pairs can be associated to the same point on the plane. This can be dealt with defining a multiset.
\[def:multiset\] \
A pair of a subset $D$ of a set $S$ and a function on $D$, $m: D \to \mathbb{N}\cup \{ +\infty \}$ is called a multiset. For $x \in D$, $m(x)$ is the multiplicity of $x$.
The support of $D$, i.e. the subset considered without the multiplicities, is denoted by $|D|$. Equivalently, $D$ can be represented as a disjoint union $$\begin{aligned}
D \equiv \bigcup_{x \in |D|} \coprod_{i=1}^{m(x)} x \; ,\end{aligned}$$ where $\coprod$ stands for a coproduct. Then, we can formally define a (general) persistence diagram.
\[def:general\_PD\] \
Set $$\begin{aligned}
&&\Delta_+ \equiv \{ (x, y) \in \mathbb{R}^2 ; y \geq x \} \\
&&\Delta \equiv \{ (x, x) \in \Delta_+ ; x \in \mathbb{R}\} \; . \end{aligned}$$ A general persistence diagram is defined as a multiset on $\Delta_+$ such that $\Delta$ is involved, the points in $\Delta_+ - \Delta$ are finite and have a finite multiplicity, and the points on $\Delta$ has infinite multiplicity.
In our case, we plot a set of persistence pairs on the persistence diagram (PD).
Since $\alpha_i < \alpha_j)$, all the points in the persistence diagram lie above the diagonal line. By definition, points lying near the diagonal line have a short interval, indicating that they quickly die after their birth. In contrast, points far from the diagonal line represent the homology classes that can survive long. This is schematically indicated in Fig. \[fig:PD\_schematic\].
### Bottleneck distance
We often need a method to compare the difference of two PDs for various applications. Since the PD is not treated a vector, we should be careful for this comparison. The bottleneck distance is one of the appropriate measures [@DBLP:books/daglib/0025666; @10.1145/3064175; @cohen_steiner2007; @chazal:inria-00292566].
\[def:bottleneck distance\] \
The bottleneck distance is defined as, $$\begin{aligned}
d_{\rm b}(D_1,D_2) \equiv \inf_{m} \sup_{x_1 \in |D_1|} \|x_1 - m(x_1)\|_{\infty}\; ,\end{aligned}$$ where $\|x\|_{\infty}$ is a Chebyshev distance $L^{\infty} \equiv \max\{|x_1|,|x_2|\}$, $|D| \equiv D \cup \Delta$, $m:|D_1| \to |D_2|$ is a multi-bijection between two PDs and $\Delta$ is a multiset of diagonal line.
\[def:multi\_bijection\] \
A multi-bijection between two multisets $D_1$ and $D_2$ is a bijection $$\begin{aligned}
\phi : \bigcup_{x_1 \in |D_1|} \coprod_{i=1}^{m(x_1)} x_1 \to \bigcup_{x_2 \in |D_2|} \coprod_{j=1}^{m(x_2)} x_2 \; .\end{aligned}$$
When two persistent diagrams are completely the same, bottleneck distance takes its minimum value.
A schematic description of the bottleneck distance is presented in Fig. \[fig:bottleneck\_distance\]. The stability of the bottleneck distance is guaranteed and we can safely use it for the examination of a PD ([@cohen_steiner2007]).
![A schematic description how to calculate the bottleneck distance between two persistence diagrams (PDs). A point without a counterpart in the other PD is regarded as corresponding to the diagonal line. []{data-label="fig:bottleneck_distance"}](bottleneck_distance.pdf){width="70.00000%"}
Inverse analysis
----------------
In the conventional BAO analysis with two point correlation function, we cannot locate the position nor specify the shape of cosmological features in a data set. Since this type of analysis represent an statistically features of the correlation, we can only access an average feature of the data set. However, these features contain essential information on the application to cosmology, or astrophysics in general. Therefore, solving the inverse problem from a given persistence diagram is of a vital importance for fundamental discussions on the related physics. For a persistent set $(\ell_i, \ell_i+p_i)$ at certain time $i$ in the filtration, corresponding simplicial complexes $(\sigma_{\ell_i},\sigma_{\ell_i+p_i})$ can be obtained. As we have discussed in Section \[subsec:PD\], the simplicial complex at the death $\ell_i+p_i$ shows the position and its shape. In this study, we calculated $p$-values of detected holes with making use of SCHU[^3] along with `R TDA` package. This enables us to specify the location and shape of the detected persistent homology classes.
We adopted this approach in this work mainly because of its conciseness, but since this field is still developing rapidly, various sophisticated methods have been proposed (see, e.g., [@Obayashi2018]). It will be meaningful to explore such method for future analysis.
Performance verification by cosmological simulations {#sec:simulation}
====================================================
Data
----
We performed a set of $N$-body simulations consists of only dark matter in order to accommodate a verification of our method. We use the publicly available $N$-body simulation code `Gadget-2` [@Springel:2005], and the initial condition is produced by the second order Lagrangian perturbation theory, `2LPT` [@Crocce+:2006]. The initial condition is generated at redshift $z=20$ with the box size being 2 Gpc, which is sufficiently larger than the typical BAO scale of $\sim 150 \;{\rm Mpc}$. In order to detect the BAO from the simulation, the mass resolution is not important and thus we include only $256^3$ particles in the simulation box, which roughly corresponds to the particle mass $6.4\times 10^{13} M_{\odot}$. Although our simulation only include dark matter particles, the signature of BAO can be imprinted in the initial conditions since the epoch we are focusing on is well after the decoupling of photon and baryon and only we need to see is the gravitational force. We compute the initial power spectrum using `CLASS` [@Julien:2011]. We take a snapshot of simulation at $z=0$ and find that the BAO wiggle can be significantly detected with this data set using power spectrum analysis. Figure \[fig:matter\_power\] show the power spectra for our simulation set with and without baryons, normalized by smoothed power spectrum without baryon oscillation features [@EisensteinHu:1998].
![Measured power spectra at $z=0$ normalized by a smooth power spectrum without baryon wiggle of [@EisensteinHu:1998]. The sold symbol is a simulation with baryon oscillation and solid line is a halo-fit model for the same set of the cosmological parameters [@Takahashi+:2012]. Open square is the one for without baryon oscillation.[]{data-label="fig:matter_power"}](matter_power.pdf){width="90.00000%"}
As a control sample, we also generated an exactly same set of simulation but replacing the initial power spectrum without BAO wiggle. This can be done by reducing the amount of baryons keeping the total amount of matter (CDM + baryon) unchanged when we compute the initial power spectrum by `CLASS`. As expected, the measured power spectrum does not exhibit the BAO wiggle for this control sample. The assumed baryon density parameter for each case is $\Omega_{B} = 0.049$ (w/ baryon) and $\Omega_{\rm B} = 0.002204$ (w/o baryon), respectively. Figure \[fig:data\_simulation\] shows the snapshot of our simulation with and without baryon oscillation feature. Since we use the same random seed for both simulation set, it is fairly difficult to see the difference between them by a visual inspection.
In this paper, for the TDA analysis, we randomly extracted 2000 particles from the parent sample of $256^3$ particles simply because of the computational limitation in our system. In our future works we will upgrade our algorithm to handle much larger data set by parallelize the code or use of more powerful resources.
![Simulation datasets at $z=0$ with (right) and without (left) baryon oscillation. The insert is a close up of the region of 200–500 Mpc box. The density fluctuations are projected along $z$-axis with 100 Mpc width, and take `arcsinh` for illustrative purpose. []{data-label="fig:data_simulation"}](BAO_simulation2.pdf){width="90.00000%"}
Result of the simulation analysis
---------------------------------
We discuss the PDs from the simulations. In order to evaluate the statistical significance of the detected signals, we need a method to obtain the confidence intervals. One of the ways to determine the confidence intervals on a PD is to use the bottleneck distance. In this study, we measure the distance between a pair of persistent diagrams obtained from bootstrap resampling samples by the bottleneck distance.
The obtained PDs for the simulations are shown in Fig. \[fig:PD\_simulation\]. The upper panel is for (w/o) baryon and the lower panel is for (w/) baryon. The black points, red triangles and blue diamonds are $H_0$, $H_1$ and $H_2$ points. The red and blue dotted diagonal lines are 90 % confidence bands for $H_1$ and $H_2$, respectively. These confidence bands for this PD were calculated by bootstrap resampling with $N_{\rm boot}=30$. In this study, we regard points lying above the corresponding confidence bands as being statistically significant.
As can be seen in the bottom panel of Fig. \[fig:PD\_simulation\], we detected four significant $H_2$, $3\mbox{-}$dimensional holes (blue diamonds) in the (w/) baryon sample. However, there is no significant hole in the (w/o) baryon sample. The $r_{\rm birth}$ and $r_{\rm death}$ for significant holes detected for (w/) baryon sample are displayed in Tab. \[tab:PD\_simulaiton\]. The mean $r_{\rm death}$ is $150.16\pm 8.46\ [{\rm Mpc}]$. Although the number of detected hole is only four, obtained $\bar{r}_{\rm death}$ is consistent with the radius that is expected for BAO signal. The position an the shape of detected $H_2$ is displayed in Fig. \[fig:inverse\_analysis\_simulation\].
For $H_1$ homology (red triangles), we obtained significant features from both of the sample. We detected 17 significant $H_1$ homology whose $p\mbox{-}$value is less than $0.2$ from the simulation data with baryon. The mean $r_{\rm death}$ for $H_1$ homology is $99.00\pm 2.26\ {\rm [Mpc]}$. For the sample without baryon, we detected 34 $H_1$ homology and $\bar{r}_{\rm death}=100.49\pm 3.24\ {\rm [Mpc]}$. The agreement in the characteristic scale in $H_1$ suggests that there are loop-like structures whose scale is not affected by the existence of baryon in $2\mbox{-}$dimensional space. Cosmic filament can be a candidate for these structure. Meanwhile, there is a difference in $\bar{r}_{\rm birth}$ between the simulation setup. The mean birth radius is $46.13\pm 2.24\ [{\rm Mpc}]$ for (w/) baryon sample while $\bar{r}_{\rm birth}= 62.34\pm 2.84\ [{\rm Mpc}]$ for (w/o) baryon sample. From this analysis, the separation between galaxies that construct loops become smaller with baryon.
For $H_0$ homology (black points), we found clear difference in distribution of PH between (w/) and (w/o) baryon sample. The $H_0$ corresponds to sequential structure, such as galaxy clusters and filaments. For the sample (w/o) baryon, $r_{\rm birth}$ distributes almost uniformly along the diagonal line. On the other hand, the distribution of $r_{\rm birth}$ concentrates to low value for (w/) baryon sample. This means the separation between two points in this sample is roughly the same. As for $H_0$ and $H_1$, further analysis is clearly needed to interpret what we discovered from the PD. This is left for our subsequent future works.
\[tab:PD\_simulaiton\]
No. $r_{\rm birth}\ {\rm [Mpc]}$ $r_{\rm death}\ {\rm [Mpc]}$
----- ------------------------------ ------------------------------
1 116.34 130.65
2 127.42 141.80
3 140.17 151.70
4 144.60 176.47
: The $r_{\rm birth}$ and $r_{\rm death}$ for holes detected in our simulated data whose $p\mbox{-}$value is less than 0.2.
![Persistent diagrams (PDs) of the simulation datasets. Black dots, red triangles and blue diamonds are $H_0$, $H_1$ and $H_2$ respectively. Diagonal black solid line is $r_{\rm birth}=r_{\rm death}$. Upper: PD of the simulation without baryon effect, Lower: PD of the simulation with baryon effect. []{data-label="fig:PD_simulation"}](PD_wobaryon.pdf "fig:"){width="70.00000%"} ![Persistent diagrams (PDs) of the simulation datasets. Black dots, red triangles and blue diamonds are $H_0$, $H_1$ and $H_2$ respectively. Diagonal black solid line is $r_{\rm birth}=r_{\rm death}$. Upper: PD of the simulation without baryon effect, Lower: PD of the simulation with baryon effect. []{data-label="fig:PD_simulation"}](PD_wbaryon.pdf "fig:"){width="70.00000%"}
![ An example of inverse analysis. We show the real space structure for 4 highest significant $H_2$ in the simulated data. The red shaded region is a convex hull of all the constituent particles. []{data-label="fig:inverse_analysis_simulation"}](Inverse_analysis2.pdf){width="90.00000%"}
Analysis with Sloan Digital Sky Survey Data Release 14 {#sec:SDSS}
======================================================
Data
----
We this study we used quasar data from the extended Baryon Oscillation Spectroscopic Survey (eBOSS) [@2016AJ....151...44D] LSS catalog [@2018MNRAS.473.4773A] in the SDSS DR14 . This is a part of the SDSS-IV [@2017AJ....154...28B]. The observed bands are $u$, $g$, $r$, $i$, and $z$. The sky coverage area is $2044\ \deg^2$ and covers a redshift range $0.8<z<2.2$. The effective area is $1288\ {\deg}^2$ the Northern Galactic Cap (NGC) and $995\ {\deg}^2$ Southern Galactic Cap (SGC). The quasar selection was done in [@2015ApJS..221...27M]. We used the data The BAO signal in this sample is already examined and confirmed in the literature [@2018MNRAS.473.4773A].
Although the whole sample contains 147,000 quasars, we randomly sampled 2000 galaxies. Namely, we do not use the full dataset but a sparsely drawn subsample of the SDSS red galaxies.
Result
------
![Persistent diagrams (PDs) of the SDSS data. []{data-label="fig:SDSS"}](PD_SDSS.pdf){width="70.00000%"}
The PD for SDSS DR14 data is displayed in Fig. \[fig:SDSS\]. We obtained four significant $H_2$ homology, “shells”, from SDSS DR14 data. The mean $r_{\rm death}$ is, $146.6\pm 2.0\ {\rm [Mpc]}$. Although the number of detected $H_2$ homology is low, the $\bar{r}_{\rm birth}$ is consistent with the expected value for BAO signal. The result of inverse analysis for $H_2$ is displayed in Fig.\[fig:inverse\_analysis\_SDSS\]. In this analysis, 19 $H_1$ homology are detected as significant loop. The mean $r_{\rm death}$ for $H_1$ homology is $101.82\pm 3.54\ {\rm [Mpc]}$. This radius agreed with the $\bar{r}_{\rm birth}$ which we have obtained from the simulation data. The birth radius for the data is $57.92\pm3.05\ {\rm [Mpc]}$. Comparing with the results from simulation data, this value agrees with $\bar{r}_{\rm birth}$ without baryon. The distribution of $H_0$ homology is similar to that of (w/o) baryon sample from simulation. This means the separations between two galaxies takes some range in their values. We need to note that we did not detected any significant $H_0$ from the data. Therefore we can only discuss about the trend of distribution in PD.
![ An example of inverse analysis. We show the real space structure for 4 highest significant $H_2$ in the simulated data. The red shaded region is a convex hull of all the constituent particles. []{data-label="fig:inverse_analysis_SDSS"}](SDSS_Inverse_analysis.pdf){width="90.00000%"}
\[tab:PD\_data\]
No. $r_{\rm birth}\ {\rm [Mpc]}$ $r_{\rm death}\ {\rm [Mpc]}$
----- ------------------------------ ------------------------------
1 107.00 141.98
2 107.83 143.30
3 114.86 150.20
4 117.56 150.98
: The $r_{\rm birth}$ and $r_{\rm death}$ for holes detected in SDSS DR14 data whose $p\mbox{-}$value is less than 0.2.
Summary and discussion {#sec:summary}
======================
In this study, we verified the usefulness of TDA on detecting BAO signal from galaxy distribution. PH constructs filtration defined by simplicial complex for given scale and dimension. This property is particularly useful when we analyse the scale and shape of holes.
We examined the effect of baryon on PHs with $N$-body simulation with $N=256^3$. We limited the number of particles to 2,000 due to the limitation of our system. Simulations with and without baryon physics showed drastically different PHs. Not only the 2-dim shells but also the behavior of 0- and 1-dim loops was found to be quantitatively different. Although we detected four significant $H_2$ homology from (w/) baryon sample, no $H_2$ was detected from (w/o) baryon sample. We also found clear difference in the distribution of $H_1$ and $H_0$ homology. While the $\bar{r}_{\rm death}$ of $H_1$ has equivalent value, $\bar{r}_{\rm birth}$ significantly differs. Similarly, $\bar{r}_{\rm birth}$ for $H_0$ has different trend between (w/) and (w/o) baryon sample. Obviously, we need to increase the number of particles that is used in the PD analysis. This lack of particle will cause to miss accurate structure in the data set.
We succeeded in detecting the BAO signal only with a 2000 subsample of quasars from SDSS. The obtained $\bar{r}_{\rm death}$ is $146.6\pm 2.0\ {\rm [Mpc]}$ for $H_2$ which is consistent with the scale of BAO signal. This means that the PH is computationally far much less expensive than the correlation function method. Clearly the PH approach will shed light to the cosmological analysis of BAOs.
Our successful detection of the BAO makes a clear contrast to the traditional 2-point correlation function method. The 2-point correlation requires a very large and dense-sampled galaxy data, while the PH can detect the BAO signal only with sparse-sampled data of 2000 galaxies. This means that, with the PH analysis, a sparse sampled galaxy survey can be used for cosmological studies. It will bring a new insight for the design and strategy of next generation cosmological galaxy surveys.
Further, we can visualize the BAO structure by the inverse analysis of the PH. We stress that this particular function of the inverse analysis in the TDA is worthy of close attention, indeed a distinctly different feature of the method. Since we can straightforwardly specify the structure contributing to the BAO signal, it makes various cosmological discussions much easier than previous analysis. More detailed results and cosmological tests are left to be presented in our subsequent works.
We thank Shiro Ikeda, Kenji Fukumizu, Satoshi Kuriki, and Yoh-ichi Mototake for fruitful discussions and suggestions. This work has been supported by JSPS Grants-in-Aid for Scientific Research (17H01110 and 19H05076), and Grant-in-Aid for Scientific Research on Innovative Areas “Cosmic Acceleration” (15H05890 and 16H01096). This work has also been supported in part by the Sumitomo Foundation Fiscal 2018 Grant for Basic Science Research Projects (180923), and the Collaboration Funding of the Institute of Statistical Mathematics “New Development of the Studies on Galaxy Evolution with a Method of Data Science”.
Algebraic preliminaries
=======================
We introduce basic terminologies to understand homotopy in algebra.
\[def:equivalence\_relation\] \
A given binary relation $\sim$ on a set $X$ is an equivalence relation if and only if it is reflexive, symmetric and transitive. That is, for all $a$, $b$ and $c \in X$,
1. Reflexivity: $a \sim a$,
2. Symmetry: $a \sim b$ if and only if $b \sim a$,
3. Transitivity: if $a \sim b$ and $b \sim c$ then $a \sim c$.
\[def:contractible\] \
When a set $X$ is homotopy equivalent to a one-point space, $X$ is contractible.
\
For continuous maps $f, g : X \rightarrow Y$, a continuous map $F$ from a product space of a closed interval $I = [0,1]$ and $X$ to $Y$ satisfying $$\begin{aligned}
F_{X\times\{0\}} = f, \quad F_{X\times \{1\}} = g
\end{aligned}$$ exists, $f$ and $g$ are said to be homotopic, denoted as $f \simeq g$. Here $$\begin{aligned}
F_{X\times\{t\}}(x) \equiv F(x, t)\end{aligned}$$ $F$ is referred to as the homotopy from $f$ to $g$.
It follows that two homotopic maps $f$ and $g$ can be transformed continuously to each other by changing $t$ via $F$. Then the binary relation $\simeq$ is an equivalence relation (see Definition \[def:equivalence\_relation\]).
A continuous map $f : X \rightarrow Y$ is a homotopy equivalence if there exist a continuous map $g : Y \rightarrow X$ such that $$\begin{aligned}
g \circ f \simeq 1_X \mbox{ and } f \circ g \simeq 1_Y \; , \end{aligned}$$ where $1_X$ and $1_Y$ are identity maps on $X$ and $Y$, respectively. If a homotopy equivalence exits, $X$ and $Y$ are homotopy equivalent, denoted as $X \simeq Y$.
Intuitively, it means that $X$ and $Y$ are homotopy equivalent if they can be transformed into each other by bending, shrinking and expanding. The homotopy equivalence is fundamentally important because many concepts in algebraic topology are homotopy invariant, that is, they respect the relation of homotopy equivalence. Particularly in the TDA, the homology group is homotopy invariant. The homology group of a simplicial complex is compatible with computers and easy to calculate. Then, even if a homology group of a certain shape is difficult to calculate directly, we can calculate it by constructing the homotopy equivalent simplicial complex. This guarantees all the analysis presented in this article.
[^1]: Throughout this paper, $\mathbb{R}, \mathbb{N}$, and $\mathbb{N}_0$ stand for a set of real number, set of natural number, and non-negative natural number, respectively.
[^2]: The homology classes with $k=0, 1, 2$ are mainly discussed in the analysis with persistent homology. We expect one would not be confused between the $k=0$ homology class and the Hubble parameter $H_0$.
[^3]: URL: [https://github.com/xinxuyale/SCHU]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Molecular motors play pivotal roles in organizing the interior of cells. A motor efficient in cargo transport would move along cytoskeletal filaments with a high speed and a minimal error in transport distance (or time) while consuming a minimal amount of energy. The travel distance of the motor and its variance are, however, physically constrained by the free energy being consumed. A recently formulated thermodynamic principle, called the *thermodynamic uncertainty relation*, offers a theoretical framework for the energy-accuracy trade-off relation ubiquitous in biological processes. According to the relation, a measure $\mathcal{Q}$, the product between the heat dissipated from a motor and the squared relative error in the displacement, has a minimal theoretical bound ($\mathcal{Q} \geq 2 k_B T$), which is approached when the time trajectory of the motor is maximally regular for a given amount of free energy input. Here, we use the uncertainty measure ($\mathcal{Q}$) to quantify the transport efficiency of biological motors. Analyses on the motility data from several types of molecular motors reveal that $\mathcal{Q}$ is a complex function of ATP concentration and load ($f$). For kinesin-1, $\mathcal{Q}$ approaches the theoretical bound at $f\approx 4$ pN and over a broad range of ATP concentration (1 $\mu$M – 10 mM), and is locally minimized at \[ATP\] $\approx$ 200 $\mu$M. In stark contrast to the wild type, this local minimum vanishes for a mutant that has a longer neck-linker, and the value of $\mathcal{Q}$ is significantly greater, which underscores the importance of molecular structure. Transport efficiencies of the biological motors studied here are semi-optimized under the cellular condition (\[ATP\] $\approx 1$ mM, $f=0-1$ pN). Our study indicates that among many possible directions of optimization, cytoskeletal motors are designed to operate at a high speed with a minimal error while leveraging their energy resources.'
author:
- Wonseok Hwang
- Changbong Hyeon
title: 'Energetic costs, precision, and efficiency of a biological motor in cargo transport'
---
Introduction
============
Biological systems are in nonequilibrium steady states (NESS) in which the energy and material currents flow constantly in and out of the system. Subjected to incessant thermal and nonequilibrium fluctuations, cellular processes are inherently stochastic and error-prone. Biological systems adopt a plethora of error-correcting mechanisms that utilize energy to fix any error deleterious to their functions [@sartori2015PRX]. Trade-off relations between the energetic cost and information processing are ubiquitous in cellular processes, and have been a recurring theme in physics and biology for many decades [@Hopfield74PNAS; @ehrenberg1980BJ; @bennett1982IJTP; @AlbertsBook; @mehta2012PNAS; @Lan2012NaturePhysics; @banerjee2017PNAS].
A recent study by Barato and Seifert [@barato2015PRL] has formulated a concise inequality known as the *thermodynamic uncertainty relation*, which quantifies the trade-off between free energy consumption and precision of an observable from dissipative processes in NESS. In their study, the uncertainty measure $\mathcal{Q}$ is defined as the product between the energy consumption (heat dissipation, $Q(t)$) of a driven process in the steady state and the squared relative error of an output observable from the process $X(t)$, $\epsilon^2_X(t)=\langle\delta X^2\rangle/\langle X\rangle^2$. It has been further conjectured that for an arbitrary chemical network formulated by Markov jump processes $\mathcal{Q}$ cannot be smaller than $2k_BT$, $$\begin{aligned}
\mathcal{Q}=Q(t)\times \epsilon_X^2(t)\geq 2k_BT.
\label{eqn:inequality1} \end{aligned}$$ The measure $\mathcal{Q}$ quantifies the uncertainty of a dynamic process. The smaller the value of $\mathcal{Q}$, the more regular and predictable is the trajectory generated from the process, improving the precision of the output observable. In the presence of large fluctuations inherent to cellular processes, harnessing energy into precise motion is critical for accuracy in cellular computation. The uncertainty measure $\mathcal{Q}$ can be used to assess the efficiency of suppressing the uncertainty in dynamical process via energy consumption. The proof and physical significance of this inequality have been discussed [@barato2015PRL; @Gingrich2016RPL; @pietzonka2016PRE; @Pigolotti2017PRL; @Hyeon2017PRE; @Proesmans:2017]. Among others, we have shown that the minimal bound of $\mathcal{Q}$, $2k_BT$, is attained when heat dissipated from the process is normally distributed, such that $P(Q)\sim e^{-Q^2}$ [@Hyeon2017PRE].
Historically, the efficiency of heat engines has been discussed in terms of the thermodynamic efficiency, the aim of which is to maximize the amount of work extracted from two heat reservoirs with different temperatures [@lee2017SciReport]. For nonequilibrium machines in general driven by chemical forces that are constantly regulated in the live cell, the power production could be a more pertinent quantity to maximize. Meanwhile, for transport motors in the cell, the uncertainty measure $\mathcal{Q}$ can be used to assess the transport efficiency of a motor (or motors) [@dechant2017Arxiv]. Because the displacement $l(t)$ (or travel distance) is a natural output observable of interest in the cargo transport, we set $X=l(t)$, which recasts Eq.\[eqn:inequality1\] into $$\begin{aligned}
\mathcal{Q}=\dot{Q}\frac{2D}{V^2}\geq 2k_BT.
\label{eqn:inequality2}\end{aligned}$$ $\mathcal{Q}$ is minimized by a motor that transports cargos (i) at a high speed ($V\sim \langle l(t)\rangle/t$), (ii) with a small error ($D\sim \langle\delta l(t)^2\rangle/t$) in the displacement (or punctual delivery to a target site), and (iii) with a small energy consumption ($\dot{Q}$). Thus, a molecular motor efficient in the cargo transport is characterized by a small $\mathcal{Q}$ with its minimal bound $2k_BT$.
In the present study, we assess the “transport efficiency” of several biological motors (kinesin-1, KIF17 and KIF3AB in kinesin-2 family, myosin-V, dynein, and F$_1$-ATPase) in terms of $\mathcal{Q}$, and study how it changes with varying conditions of load ($f$) and \[ATP\]. Of particular interest is to identify the optimal condition for motor efficiency, if any, that $\mathcal{Q}$ is minimized. To evaluate $\mathcal{Q}$, one should know $\dot{Q}$, $D$, and $V$ of the system (see Eq.\[eqn:inequality2\]), which can be obtained using a suitable kinetic network model that can delineate the dynamical characteristics of the system [@Hwang2017JPCL]. Our analyses on motors show that $\mathcal{Q}$, demonstrating a complex functional dependence on $f$ and \[ATP\], is sensitive to a subtle variation in motor structure. Transport and rotory motors studied here are semi-optimized in terms of $\mathcal{Q}$ under the cellular condition, which alludes to the role of evolutionary pressure that has shaped the current forms of molecular motors in the cell.
Results
=======
Chemical driving force, steady state current, and heat dissipation of double-cycle kinetic network for kinesin-1
----------------------------------------------------------------------------------------------------------------
To study the transport properties of molecular motors, we used experimental data, available in the literature, of $V$ and $D$ under varying conditions of $f$ and \[ATP\]. Once a set of kinetic rate constants $\{k_{ij}\}$ that defines the network model is determined by fitting the data of $V(\{k_{ij}(f,[\text{ATP}])\})$ and $D(\{k_{ij}(f,[\text{ATP}])\})$, it is straightforward to calculate $\dot{Q}(f,[\text{ATP}])$ [@Hwang2017JPCL], and hence $\mathcal{Q}(f,[\text{ATP}])$.
For the case of kinesin-1, we empoloy the 6-state kinetic network model [@Liepelt07PRL], consisting of two cycles, $\mathcal{F}$ and $\mathcal{B}$ (Fig. \[fig\_kinesin\_6state\]A). Although the conventional (N=4)-state unicyclic kinetic model [@Fisher1999_PNAS; @Fisher01PNAS] confers a similar result with the 6-state double-cycle network model at small $f$ (compare Figs.\[fig\_kinesin\_6state\] and \[fig\_kinesin\_uni\]), the unicyclic model is led to a physically problematic interpretation especially when the molecular motor is stalled and starts taking backsteps at large hindering load [@Astumian1996BJ; @Liepelt07PRL; @Hyeon09PCCP]. As explicated previously in Ref. [@Hyeon09PCCP], the backstep in the unicyclic network, by construction, is produced by a reversal of the forward cycle, which implies that the backstep is always realized via the synthesis of ATP from ADP and P$_i$. More importantly, in calculating $\dot{Q}$ from kinetic network, the unicyclic network results in $\dot{Q}=0$ under the stall condition, which however contradicts the physical reality; an idling car still burns fuel and dissipates heat, thus $\dot{Q}\neq 0$. To build a more physically sensible model that considers the possibility of ATP-induced (fuel-burning) backstep, we extend the unicyclic network into a multi-cyclic one which takes into account an ATP-consuming stall, i.e., a futile cycle [@Liepelt07PRL; @Yildiz08Cell; @Hyeon09PCCP; @clancy2011NSMB].
The proposed double-cycle network scheme is physically more sensible and general than unicyclic schemes in that it can accommodate 4 different possibilities for the kinetic paths: (i) ATP-hydrolysis induced forward step; (ii) ATP-hydrolysis induced backward step; (iii) ATP-synthesis induced forward step; (iv) ATP-synthesis induced backward step. With the kinetic rate constants determined for kinesin-1 using the double-cycle model, the kinesin-1 predominantly moves forward through the $\mathcal{F}$-cycle under small hindering ($f>0$) or assisting load ($f<0$), whereas it takes a backstep through the $\mathcal{B}$-cycle under a large hindering load. In principle, the steady-state reaction current within the $\mathcal{F}$-cycle, $J_{\mathcal{F}}$, itself is decomposed into the forward ($J^+_{\mathcal{F}}$) and backward current ($J^-_{\mathcal{F}}$), such that $J_{\mathcal{F}}=J^+_{\mathcal{F}}-J^-_{\mathcal{F}}>0$. Although a backstep could be realized through an ATP synthesis [@Hackney05PNAS], corresponding to $J_{\mathcal{F}}^-$, a theoretical analysis [@Hyeon09PCCP] on experimental data [@Nishiyama02NCB; @Cross05Nature] suggest that such backstep current (ATP synthesis induced backstep, $J^-_{\mathcal{F}}$) is negligible in comparison with $J^+_{\mathcal{B}}$ (ATP hydrolysis induced backstep).
We illuminate the dynamics realized in the double-cycle network by calculating $J_{\mathcal{F}}$ and $J_{\mathcal{B}}$ with increasing $f$ (see Fig.\[fig\_kinesin\_6state\]B). Without load ($f=0$), kinesin-1 predominantly moves forward ($J_{\mathcal{F}}\gg J_{\mathcal{B}}$). This imbalance diminishes as $f$ is increased. At stall conditions, the two reaction currents are balanced ($J_\mathcal{F} = J_\mathcal{B}$), so that the net current $J$ associated with the mechanical stepping defined between the states (2) and (5) vanishes ($J=J_\mathcal{F}-J_\mathcal{B}=0$), but nonvanishing current due to chemistry still remains along the cycle of $\rightarrow (2)\rightarrow (3)\rightarrow (4)\rightarrow (5)\rightarrow (6)\rightarrow (1)\rightarrow (2)\rightarrow$ (see Fig.\[fig\_kinesin\_6state\]A). A further increase of $f$ beyond the stall force renders $J_{\mathcal{F}}< J_{\mathcal{B}}$, augmenting the likelihood of backstep.
For a given set of rate constants, it is straightforward to calculate the rates of heat dissipation ($\dot{Q}$), work production ($\dot{W}$), and total energy supply ($\dot{E}$). The total heat generated from the kinetic cycle depicted in Fig. \[fig\_kinesin\_6state\]A is decomposed into the heat generated from two subcycles, $\dot{\mathcal{Q}}_{\mathcal{F}}$ and $\dot{\mathcal{Q}}_{\mathcal{B}}$, each of which is the product of reaction current and affinity [@Liepelt07PRL; @barato2015PRL; @Seifert2012RPP; @Qian2005_BiophyChem; @Qian_PRE_2004; @Qian2010PRE; @Wachtel:2015:PRE] $$\label{eq:dotQ}
\dot{Q} = \dot{Q}_\mathcal{F} + \dot{Q}_\mathcal{B}= J_\mathcal{F} \mathcal{A}_\mathcal{F} + J_\mathcal{B} \mathcal{A}_\mathcal{B}.$$ Here, the affinities (driving forces) for the $\mathcal{F}$ and $\mathcal{B}$ cycles are $$\begin{aligned}
\mathcal{A}_\mathcal{F} = k_B T \log \left( \frac{ k_{12} k_{25} k_{56} k_{61}}{k_{21} k_{16} k_{65} k_{52} } \right) = (-\Delta \mu_{\text{hyd}}) - f d_0
\label{eq:AF}\end{aligned}$$ and $$\begin{aligned}
\mathcal{A}_\mathcal{B} = k_B T \log \left( \frac{ k_{23} k_{34} k_{45} k_{51}}{k_{32} k_{25} k_{54} k_{43} } \right) = (-\Delta \mu_{\text{hyd}}) + f d_0.
\label{eq:AB}\end{aligned}$$ The explicit forms of $J_{\mathcal{F}}$ and $J_{\mathcal{B}}$ as a function of $\{k_{ij}\}$ are available (see Eq. \[eq:JF\]) but the expression is generally more complicated than the affinity. It is of note that at $f=0$, the chemical driving forces for $\mathcal{F}$ and $\mathcal{B}$ cycles are identical to be $-\Delta \mu_{\text{hyd}}$. The above decomposition of affinity associated with each cycle into the chemical driving force and the work done by the motor results from the Bell-like expression of transition rate between the states (2) and (5): $k_{25} = k_{25}^o e^{-\theta f d_0 / k_B T}, k_{52} = k_{52}^o e^{(1-\theta) f d_0 / k_B T}$ [@Liepelt07PRL; @Fisher1999_PNAS; @Fisher01PNAS] (see [**Materials and Methods**]{}). From Eqs. \[eq:dotQ\], \[eq:AF\], and \[eq:AB\], $\dot{Q}$ can be decomposed into the total free energy input ($\dot{E} = (J_\mathcal{F} + J_\mathcal{B} ) (-\Delta \mu_{\text{hyd}})$) and work production ($\dot{W} = (J_\mathcal{F} - J_\mathcal{B} ) f d_0$). Hence, $$\begin{aligned}
\dot{Q} &= (J_\mathcal{F} + J_\mathcal{B} ) (-\Delta \mu_{\text{hyd}}) - (J_\mathcal{F} - J_\mathcal{B} ) f d_0 \nonumber\\
&= \dot{E} - \dot{W}. \end{aligned}$$
A few points are noteworthy from the dependences of $V$, $D$, $\dot{Q}$, and $\dot{W}$ for kinesin-1 on $f$ and \[ATP\] (see Fig. \[fig\_kinesin\_6state\]): (i) The stall condition, indicated with a white dashed line in each map, divided all the 2D maps of $V$, $D$, $\dot{Q}$, and $\dot{W}$ into two regions; (ii) In contrast to $V$ (Fig. \[fig\_kinesin\_6state\]D) and $D$ (Fig. \[fig\_kinesin\_6state\]E), which decrease monotonically with $f$, $\dot{Q}$ and $\dot{W}$ display non-monotonic dependence on $f$ (Figs. \[fig\_kinesin\_6state\]F, G). At high \[ATP\], $\dot{Q}$ is locally maximized at $f=10$ pN, whereas $\dot{W}$ is maximized at 5 pN and locally minimized at 10 pN (see Fig. \[fig\_kinesin\_6state\]I calculated at \[ATP\]=2 mM). At the stall force $f_\text{stall}$ ($\approx 7$ pN) (Figs. \[fig\_kinesin\_6state\]B, \[fig\_kinesin\_6state\]I, black dashed line), the reaction current of the $\mathcal{F}$-cycle is exactly balanced with that of $\mathcal{B}$-cycle ($J= J_\mathcal{F}- J_\mathcal{B} = 0$), giving rise to zero work production ($\dot{W} = f V = f d_0J= 0$). The numbers of forward and backward steps taken by kinesin motors are identical, and hence there is no net directional movement ($V=0$) [@Cross05Nature]. Importantly, even at the stall condition, kinesin-1 consumes the chemical free energy of ATP hydrolysis, dissipating heat in both forward and backward steps, and hence rendering $\dot{Q}=(J_{\mathcal{F}}+J_{\mathcal{B }})(-\Delta \mu_{\text{hyd}})$ always positive.
Next, branching of reaction current of the $\mathcal{F}$-cycle into $\mathcal{B}$-cycle with increasing $f$ gives rise to non-monotonic changes of $\dot{Q}$ and $\dot{W}$ with $f$. At small $f$, $\dot{Q}$ decreases with $f$ because exertion of load gradually deactivates the $\mathcal{F}$-cycle via the decrease of $\mathcal{A}_\mathcal{F}$ (Eq.\[eq:dotQ\], Eq.\[eq:AF\], Fig. \[fig\_QWVD6\_supple\]C) and $J_\mathcal{F}$ (Fig. \[fig\_QWVD6\_supple\]A). By contrast, $\mathcal{A}_\mathcal{B}$ and $J_\mathcal{B}$ increase with $f$ (Eq.\[eq:AB\], Figs. \[fig\_QWVD6\_supple\]A and \[fig\_QWVD6\_supple\]D), which leads to an increase of $\dot{Q}_{\mathcal{B}}$. The non-monotonic dependence of $\dot{W}$ on $f$ can be analyzed in a similar way. The forward and backward currents along the cycles $\mathcal{F}$ and $\mathcal{B}$ (i.e., $J_\mathcal{F}$ and $J_\mathcal{B}$) are negatively correlated (Figs. \[fig\_QWVD6\_supple\]A and \[fig\_QWVD6\_supple\]B).
Motor head distortion at high external stress hinders the binding and hydrolysis of ATP in the catalytic site [@Uemura03NSB; @Hyeon07PNAS; @Hyeon11BJ]; $k_{ij}=0$ when ATP cannot be processed. This effect is modeled into the rate constants such that $k_{ij} = 2k_{ij}^o (1 + e^{\chi_{ij} f d_0 /k_B T})^{-1}$ with $\chi_{ij} > 0$ ($k_{ij} \neq k_{25}, k_{52}$) [@Liepelt07PRL]. Thus, it naturally follows that $\dot{Q}=0$ when $f$ is much greater than $f_{\text{stall}}$ (Figs.\[fig\_kinesin\_6state\]B, F, I).
![$\mathcal{Q}$ calculated based on kinesin-1 data [@Visscher99Nature] using the 6-state double-cycle model [@Liepelt07PRL] at varying $f$ and \[ATP\], where $f>0$ and $f<0$ signify the hindering and assisting load, respectively. [**A.**]{} 2-D contour plot of $\mathcal{Q} = \mathcal{Q}(f, [\text{ATP}])$. A suboptimal point $\mathcal{Q}_{\text{opt}}\approx 4$ $k_BT$ is found at $f=4.1$ pN and $[\text{ATP}]\approx 210$ $\mu$M. The solid lines in magenta are the loci of locally optimal $\mathcal{Q}$ at varying \[ATP\] for a given $f$ ([**B**]{}). The dashed lines in magenta are the loci of locally optimal $\mathcal{Q}$ at varying $f$ for a given \[ATP\] ([**C**]{}). The star symbol indicates the cellular condition of $[\text{ATP}]\approx 1$ mM and $f\approx 1$ pN. []{data-label="fig_Qa_contour"}](r_fig_Qa_contour.pdf)
Quantification of $\mathcal{Q}$ for kinesin-1
---------------------------------------------
Unlike $V$, $D$, and $\dot{Q}$, which are maximized at large \[ATP\] and small $f$ (Figs. \[fig\_kinesin\_6state\]D, E, F), the uncertainty measure $\mathcal{Q}(f, \text{[ATP]})$ displays a complex functional dependence (Fig. \[fig\_Qa\_contour\]A). (i) Small $\mathcal{Q}$ at low \[ATP\] and $f$, which approaches the lower bound of 2$k_B T$, is a trivial outcome of the detailed balance condition where \[ATP\] is balanced with \[ADP\] and \[P$_i$\]. The motor, without chemical driving force and only subjected to thermal fluctuations ($\dot{Q}\rightarrow 0$), is on average motionless ($V\rightarrow 0$); $\mathcal{Q}$ is minimized in this case ($\mathcal{Q}\rightarrow 2k_BT$). (ii) $\mathcal{Q}$ is generally smaller below the stall condition, $f<f_{\text{stall}}([\text{ATP}])$, demarcated by the white dashed lines in Fig.\[fig\_kinesin\_6state\]. In this case, the reaction current along the $\mathcal{F}$-cycle is more dominant than that above the stall. At the stall, $\mathcal{Q}$ diverges because of $V\rightarrow 0$ and $\dot{Q}\neq 0$. (iii) Notably, a suboptimal value of $\mathcal{Q}\approx 4$ $k_BT$ is identified at \[ATP\] $= 210$ $\mu$M and $f = 4.1$ pN (Figs. \[fig\_Qa\_contour\]). (iv) At $f \approx 4$ pN, $4 k_B T \lesssim \mathcal{Q} \lesssim 6 k_B T$ over the broad range of \[ATP\] ($=1$ $\mu$M$-$10 mM) remaining close to the local minimum value $4 k_B T$ (Fig. \[fig\_Qa\_contour\]C), which indicates that kinesin-1 works robustly against the variation of \[ATP\] in the cell.
Comparison of $\mathcal{Q}$ between different types of kinesins
---------------------------------------------------------------
The dynamic property of molecular motor differs from one motor type to another. Effect of modifying motor structure on the transport properties as well as on the directionality and processivity of molecular motor has been of great interest because it provides glimpses into the design principle of a motor at molecular level [@liao2009JMB; @Bryant2007PNAS; @Hyeon11BJ; @Jana2012PLoS; @hinczewski2013PNAS2]. To address how modifications to motor structure alter the transport efficiency of motor, we analyze single-molecule motility data of a mutant of kinesin-1 (Kin6AA), and homodimeric and heterotrimeric kinesin-2 (KIF17 and KIF3AB).
Data of Kin6AA, a mutant of kinesin-1 that has a longer neck-linker domain, were taken from Ref. [@clancy2011NSMB]. Six amino-acid residues inserted to the neck-linker reduce the internal tension along the neck-linker which plays a critical role for regulating the chemistry of two motor heads and coordinating the hand-over-hand motion [@Hyeon07PNAS]. Disturbance to this motif is expected to affect $V$ and $D$ of the wild type. We analyzed the data of Kin6AA again using the 6-state network model (Figs. \[fig\_kinesin\_6state\]A, \[fig\_exp\_Kin6AA\], \[fig\_Kin6AA\_various\], Table \[table\]. See [**SI**]{} for detail), indeed finding reduction of $V$ and $D$ (Fig. \[fig\_Kin6AA\_various\]A) as well as its stall force (Fig. \[fig\_Kin6AA\_various\]A, white dashed line). Of particular note is that the rate constant $k_{25}$ associated with the mechanical stepping process is reduced by two order of magnitude (Table \[table\]). In $\mathcal{Q}(f,[\text{ATP}])$ (Fig. \[fig\_Qa\_other\_kinesins\]A), the suboptimal point observed in kinesin-1 (Fig. \[fig\_Qa\_contour\]A) vanishes (Fig. \[fig\_Qa\_other\_kinesins\]A), and $\mathcal{Q}$ diverges around $\sim$ 4 pN due to the decreased stall force. Finally, overall, the value of $\mathcal{Q}$ has increased dramatically. This means that compared with that of kinesin-1 ($\mathcal{Q}\approx 7$ $k_BT$), the trajectory of Kin6AA is less regular and unpredictable ($\mathcal{Q}\approx 20$ $k_BT$) at $f=1$ pN and \[ATP\] = 1 mM, that roughly represents the cellular condition [@Welte:1998; @Shubeita:2008]. Thus, Kin6AA is three fold less efficient than the wild-type in cargo transport. Next, the values of $\mathcal{Q}$ were calculated for two active forms of vertebrate kinesin-2 class motors responsible for intraflagellar transport (IFT). KIF17 is a homodimeric form of kinesin-2, and KIF3AB is a heterotrimeric form made of KIF3A, KIF3B, and a nonmotor accessory protein, KAP. To quantify their motility properties, we digitized single-molecule motility data from Ref. [@Milic:2017:PNAS] and fitted them to the 6-state double-cycle model (Figs. \[fig\_exp\_KIF17\], \[fig\_exp\_KIF3AB\]) (See [**SI**]{} for detail). $\mathcal{Q}(f,[\text{ATP}])$’s of KIF17 and KIF3AB are qualitatively similar to that of kinesin-1 with some variations. $\mathcal{Q}$ for KIF17 forms a shallow local minimum of $\mathcal{Q}\approx 9.2$ $k_BT$ at \[ATP\] = 200 $\mu$M and $f = 1.5$ pN (Fig. \[fig\_Qa\_other\_kinesins\]B), whereas such suboptimal condition vanishes in KIF3AB (Fig. \[fig\_Qa\_other\_kinesins\]C). KIF3AB, however, display a local valley of $\mathcal{Q}$ around $f \sim $ 4 pN and $1$ $\mu$M $\lesssim \text{[ATP]} \lesssim 10$ mM in which $\mathcal{Q} \approx 4 k_B T$.
The plots of $\mathcal{Q} (\text{[ATP]})$ at fixed $f$ and $\mathcal{Q} (f)$ with fixed \[ATP\] in Fig. \[fig\_Qa\_other\_kinesins\]D recapitulate the difference between different classes of kinesins more clearly. The following features are noteworthy. (i) An extension of neck-linker domain (Kin6AA, orange lines) dramatically increases $\mathcal{Q}$ compared with the wild type (Kinesin WT, magenta lines). (ii) Non-monotonic behaviors of $\mathcal{Q} (\text{[ATP]})$ are qualitatively similar for all kinesins although $\mathcal{Q}$ is, in general, the smallest for kinesin-1. (iii) The movement of KIF3AB (black lines) becomes the most regular at low \[ATP\] ($ \lesssim 10$ $\mu$M). (iv) $\mathcal{Q}$ for kinesin-1 analyzed using the ($N$=4)-state unicyclic model (dashed magenta lines) displays only small deviations as long as $0\lesssim f\lesssim 4$ pN $\ll f_{\text{stall}} \approx 7$ pN.
Comparison of $\mathcal{Q}$ among different types of motors
-----------------------------------------------------------
We further investigate $\mathcal{Q}(f, \text{[ATP]})$ for other motor types, myosin-V, dynein, and F$_1$-ATPase using the kinetic network models proposed in the literature [@Bierbaum:2011:BPJ; @Sarlah:2014:BPJ; @Gerritsma:2010]. [*Myosin-V*]{}: The model studied in Ref. [@Bierbaum:2011:BPJ] consists of chemomechanical forward cycle $\mathcal{F}$, dissipative cycle $\mathcal{E}$, and pure mechanical cycle $\mathcal{M}$ (Fig. \[fig\_Qa\_other\_motor\_proteins\]A). In $\mathcal{F}$-cycle, myosin-V either moves forward by hydrolyzing ATP or takes backstep via ATP synthesis. In $\mathcal{M}$-cycle, myosin-V moves backward under the load without involving chemical reactions. The $\mathcal{E}$-cycle, consisting of ATP binding \[$(2) \rightarrow (5)$\], ATP hydrolysis \[$(5) \rightarrow (6)$\], and ADP release \[$(6) \rightarrow (2)$\] (Fig. \[fig\_Qa\_other\_motor\_proteins\]B), was originally introduced to connect the two cycles $\mathcal{F}$ and $\mathcal{M}$. The calculation of $J_\mathcal{E}$ and $J_\mathcal{F}$ reveals that a gradual deactivation of $\mathcal{F}$-cycle with decreasing \[ATP\] activates the $\mathcal{E}$-cycle (Fig. \[fig\_myosin\_various\_70\_1000\]A, $100$ $ \mu$M $\lesssim \text{[ATP]} \lesssim 1$ mM, $f \lesssim 1$ pN). Thus, $\mathcal{E}$-cycle can be regarded a futile $\mathcal{F}$-cycle, which is activated when chemical driving force is balanced with a load $f$ at low \[ATP\]. $\mathcal{Q}( f,\text{[ATP]})$ calculated at \[ADP\] = 70 $\mu$M and \[P$_i$\] = 1 mM using the rate constants from Ref. [@Bierbaum:2011:BPJ] (see **SI** for details and Fig. \[fig\_myosin\_various\_70\_1000\]) reveals no local minimum in this condition. However, at \[ADP\] = 0.1 $\mu$M and \[P$_i$\] = 0.1 $\mu$M, which is the condition used in Ref. [@Bierbaum:2011:BPJ], a local minimum with $\mathcal{Q}$ = 6.5 $k_B T$ is identified at $f =$ 1.1 pN and \[ATP\] = 20 $\mu$M (Fig. \[fig\_myosin\_various\]D, Table \[table\_opt\]). Both values of $f$ and \[ATP\] at the suboptimal condition of myosin-V are smaller than those of kinesin-1 (Table \[table\_opt\]). In (N=2)-unicyclic model for myosin-V (Fig. \[fig\_myosin\_uni\]) [@Kolomeisky:2003:BPJ], $\mathcal{Q}$ has local valley around $f\sim 2$ pN and \[ATP\] $\sim 10$ $\mu$M. Similar to the result from the multi-cyclic model with \[ADP\] = 0.1 $\mu$M and \[P$_i$\] = 0.1 $\mu$M, the values of $f$ and \[ATP\] along the valley of $\mathcal{Q}$ are smaller than the values optimizing $\mathcal{Q}$ for the kinesin-1 (Table \[table\_opt\]).\
[*Dynein*]{}: Dynein is a family of $(-)$-end directed cytoskeletal motor. There are two groups of dyneins: cytoplasmic and axonemal dyneins. Cytoplasmic dyneins involve the transport of cellular cargoes whereas axonemal dyneins are responsible for generating the beating motion of cilia or flagella by sliding microtubles in the axonemes. Here we study cytoplasmic dyneins whose locomotion along microtubules is pertinent to the issue discussed here.
$\mathcal{Q}(f,[\text{ATP}])$ for cytoplasmic dyneins was evaluated by considering (N=7)-unicyclic kinetic model (Fig. \[fig\_Qa\_other\_motor\_proteins\]B) based on a previous study [@Sarlah:2014:BPJ]. The original model (Fig. 5A of Ref. [@Sarlah:2014:BPJ]) describes the major pathway of tightly coupled dimeric dynein whose linker connecting the head domains of two dynein monomers is short and stiff. This major pathway is found in Ref. [@Sarlah:2014:BPJ] from the kinetic simulation of their elastomechanical model whose the transitions between chemical states and the mechanical movements of motors are described by elastic-energy- and load-dependent rate constants. Although the futile cycle, which branches out of the major pathway, is expected at large hindering loads [@Sarlah:2014:BPJ], we consider a simpler unicycle model; as shown in kinesin-1 (Fig. \[fig\_Qa\_other\_kinesins\]D), as long as $f$ is small ($f<f_c$), the system under the unicycle model behaves similarly to a more complicated model. The model consists of 7 states: dissociation of Pi \[$(1) {\rightarrow}(2)$\]; dissociation of ADP \[$(2){\rightarrow}(3)$\]; ATP binding \[$(3) {\rightarrow}(4)$\]; dissociation of microtuble binding domain (MTBD) from the filament \[$(4) {\rightarrow}(5)$\]; power stroke \[$(5) {\rightarrow}(6)$\]; linker swinging to the pre-power stroke state \[$(6) {\rightarrow}(7)$\]; MTBD binding to the filament \[$(7) {\rightarrow}(1)$\]. We also assume only the rate constants describing the mechanical transition of dynein depend on $f$. The more detailed description of the model is given in [**SI**]{}. $\mathcal{Q}(f,[\text{ATP}])$ calculated from the model is locally minimized to $\mathcal{Q}\approx 5.2 ~ k_B T$ at $f$ = 3.9 pN, \[ATP\] = 200 $\mu$M (Fig. \[fig\_Qa\_other\_motor\_proteins\]D, \[fig\_dynein\_various\]D). This condition of local minimum is compatible with that of kinesin-1 (Table \[table\_opt\]).\
[*F$_1$-ATPase*]{}: F$_1$-ATPase is a rotary molecular motor. In vivo, it combines with F$_0$ subunit and synthesizes ATP by using proton gradient across membrane. $\mathcal{Q}(\tau,[\text{ATP}])$ calculated using the (N=2)-state unicyclic model in Ref. [@Gerritsma:2010] (also see [**SI**]{} for the detailed description of the model) reveals that there is a valley around torque $\tau \approx -10$ pN$\cdot$nm and \[ATP\] $\approx 10$ $\mu$M reaching $\mathcal{Q} \approx 4$ $k_B T$ (Fig. \[fig\_Qa\_other\_motor\_proteins\]C). Notably, $\mathcal{Q}$ for F$_1$-ATPase is optimized at hindering load ($\tau <0$) in which ATP is synthesized, which comports well with the biologically known role of F$_1$-ATPase as an ATP synthase *in vivo*.\
To highlight the difference between the motors, we plot $\mathcal{Q} (\text{[ATP]})$ at fixed $f$ and $\mathcal{Q} (f)$ at fixed \[ATP\] in Fig. \[fig\_Qa\_other\_motor\_proteins\]D, which find $\mathcal{Q}_{\text{kinesin-1}} < \mathcal{Q}_{\text{dynein}} < \mathcal{Q}_\text{myosin-V}$ over the broad range of $f$ and \[ATP\]. We note that at a special condition (\[ATP\] ($=1-10$ $\mu$M) and $f$ ($=1 - 2$ pN)) $\mathcal{Q}_\text{myosin-V}$ is smaller than the values of other motors.
![Plots of $Q$ versus $\epsilon_l(=\sqrt{\langle (\delta l)^2\rangle}/\langle l\rangle)$ for various motors under the cellular condition, \[ATP\] = 1 mM [@Milo:2015] and $f = 1$ pN [@Welte:1998; @Shubeita:2008]. The plot for the ideal motor ($\mathcal{Q}=2$ $k_BT$) is plotted with a dashed line. $\mathcal{Q}/k_BT=7.2$ (kinesin-1), 7.7 (F$_1$-ATPase), 9.1 (dynein), 9.9 (KIF17), 9.9 (KIF3AB), 13 (myosin-V), 19 (Kin6AA). For F$_1$-ATPase, zero torque and \[ATP\] $=1$ mM are assumed as the cellular condition. []{data-label="fig_Qa_scheme"}](r_fig_Qa_scheme.pdf)
Discussion
==========
Biological motors are far superior to macroscopic machines in harnessing free energy into linear movement. The thermal noise is utilized to rectify the ATP binding/hydrolysis-coupled conformational dynamics into unidirectional movement, which conceptualizes the Brownian ratchet [@Astumian97Science], but it also comes with a cost of overcoming the thermal noise that makes the movement of biological motors inherently stochastic and error-prone. Mechanism of harnessing energy into faster and more precise motion is critical for the accuracy of cellular computation. The uncertainty measure $\mathcal{Q}$ assesses the efficiency of improving the speed and regularity of dynamics for a given energetic cost.
Here, we have quantified the uncertainty measure $\mathcal{Q}$ for various biological motors. We found that the values of $\mathcal{Q}$ for various motors are all semi-optimized near the cellular condition (star symbols marking $f\approx 1$ pN and \[ATP\]$=1$ mM in Figs. \[fig\_Qa\_contour\], \[fig\_Qa\_other\_kinesins\], \[fig\_Qa\_other\_motor\_proteins\]). $Q$ versus $\epsilon_l$ plots (Fig.\[fig\_Qa\_scheme\], $\epsilon_{\theta}$ for F$_1$-ATPase motor) for the various motors, sorting the motors in the increasing order of $\mathcal{Q}$, and their lower bound dictated by $\mathcal{Q}=Q\epsilon_l^2=2k_BT$ are reminiscent of the recent study on the free-energy cost of accurate biochemical oscillations [@Cao2015NatPhys]. The plots indicate that kinesin-1 is the best motor whose $\mathcal{Q}$($\approx 7.2$ $k_BT$) approaches the bound of ideal case ($\mathcal{Q}=2$ $k_BT$). Note that $\mathcal{Q}(\approx 19$ $k_BT$) for the mutant kinesin-1 (Kin6AA) is significantly greater than that for the wild-type. The structure of $\mathcal{Q}(f,[\text{ATP}])$ and the suboptimal condition of $\mathcal{Q}$ differ from one motor type to another.
![Various quantities calculated for kinesin-1 at varying conditions of $f$ and \[ATP\]. [**A**]{}. Transport efficiency $\eta_T(=2k_BT/\mathcal{Q})$. A suboptimal point $\eta_T^*\approx 0.48$ (indicated by $\times$) is formed at $f=4.1$ pN and $[\text{ATP}]=210$ $\mu$M. [**B**]{}. Transport speed $V(f,[\text{ATP}])$, [**C**]{}. Work production $\dot{W}(f,[\text{ATP}])$, [**D**]{}. Power efficiency calculated using $\eta \equiv \dot{W}/\dot{E}$. For $f > f_\text{stall}$, we set $\eta=0$ for convenience because the motor moves backward and $\dot{W}<0$. At the cellular condition ($f\approx 1$ pN and \[ATP\] $\approx 1$ mM), indicated by the star symbol in each panel, $\eta_T = 0.28$, $V = 0.74$ $\mu$m/s, $\dot{W} = 182$ $k_B T/s$, and $\eta = 0.12$. []{data-label="efficiency"}](r_efficiency.pdf)
Minimizing $\mathcal{Q}$ towards its lower bound $2k_BT$ for optimal transport is equivalent to maximizing the transport efficiency, which can be defined as [@dechant2017Arxiv] $$\begin{aligned}
\eta_T(f,[\text{ATP}])=\frac{2k_BT}{\mathcal{Q}(f,[\text{ATP}])}, \end{aligned}$$ where $\eta_T$ is bounded in the interval $0\leq\eta_T\leq 1$. It is of particular note that the structure of $\eta_T(f,[\text{ATP}])$ (equivalently $\mathcal{Q}(f,[\text{ATP}])$) differs significantly from that of other quantities such as the flux $J(f,[\text{ATP}])$ [@Brown:2017:PNAS] (equivalent to $V(f,[\text{ATP}])$), work production (power) $\dot{W}(f,[\text{ATP}])$, and the power efficiency $\eta(f,[\text{ATP}])\equiv \dot{W}(f,[\text{ATP}])/\dot{E}(f,[\text{ATP}])$ (Fig.\[efficiency\] for kinesin-1. See **SI** Figures for other motors). Remarkably, only $\eta_T(f,[\text{ATP}])$, not $V$, $\dot{W}$, nor $\eta$, displays a suboptimal peak near the cellular condition.
To what extent can our findings on the *in vitro* single motor properties be generalized into those in live cells? First, the force hindering the motor movement varies with cargo size and subcellular location; the load or viscoelastic drag exerted against motors inside the cell varies dynamically [@Narayanareddy:2014; @Wortman:2014]. Yet, actual forces opposing the cargo movement in cytosolic environment are $\lesssim$ 1 pN [@Welte:1998; @Shubeita:2008]. Since $\mathcal{Q}$’s for microtubule-binding motors, kinesin-1, kinesin-2, and dynein, are narrowly tuned, varying only a few $k_B T$ over the range of $0 \leq f \leq 4$ pN at \[ATP\] = 1 mM (Figs. \[fig\_Qa\_other\_kinesins\]D, \[fig\_Qa\_other\_motor\_proteins\]D), our discussion on the *in vitro* single motor property can be extended to the cargo transport in cytosolic environment. Next, a team of motors is often responsible for cargo transport in the cell [@li2016BJ]. It has, however, been shown that the extent of coordination between two kinesin motors attached to a cargo is not significant under low load and saturating ATP [@carter2008BJ]. Although trajectories generated by multiple motors have not been analyzed here, extension of the present analysis to such cases is straightforward.
In the axonal transport, of particular importance is the fast and timely delivery of cellular material, the failure of which is linked to neuropathology [@mandelkow2002TCB; @Hafezparast2003Science]. Since there are already numerous regulatory control mechanisms as well as other motors, it could be argued that the role played by the optimized single motor transport efficiency is redundant in light of the overall function of axonal transport. Yet, given that cellular regulations are realized through multiple layers of checkpoints [@AlbertsBook], the optimized transport efficiency of motors at single molecule level can also be viewed as one of the checkpoints that assure the optimal cargo transport.
Taken together, the thermodynamic uncertainty relation, a general principle for dissipative processes in nonequilibrium steady states, offers quantitative insight into the energy-speed-precision trade-off relation for biological systems. Here, we have adapted this principle to assess the transport efficiency of biological motors in terms of $\mathcal{Q}$. With a multitude of time traces of a biological motor generated at varying conditions at hand, it is straightforward to calculate the uncertainty measure $\mathcal{Q}$ as well as other dynamic quantities of interest by mapping the dynamics of the motor to an adequate kinetic network. Given that there are many possible directions involving the design principle of biological motors, it is significant to find that biological motors indeed possess a semi-optimal transport efficiency under the cellular condition. Finally, it is of great interest to extend the proposed concept and analysis using $\mathcal{Q}$ to other energy-consuming biological processes.
Materials and Methods
=====================
To calculate the uncertainty measure $\mathcal{Q}$ of a motor, we first define a chemical network model that can describe the dynamics of the motor in terms of a set of rate constants $\{k_{ij}\}$, where $k_{ij}(f, [\text{ATP}])$ is the transition rate from the $i$-th to $j$-th state and depends on both load $f$ and \[ATP\]. Next, we fit the experimental data of $V$ and $D$ obtained under varying conditions of $f$ and \[ATP\] to the formal expressions of $V(\{k_{ij}(f,[\text{ATP}])\})$ and $D(\{k_{ij}(f,[\text{ATP}])\})$ [@Koza1999; @Lebowitz:1999; @barato2015PRL] (the details of the procedure is described in the next paragraph and in [**SI**]{}). The fit determines the set of rate constants $k_{ij}(f, [\text{ATP}])$ [@Hwang2017JPCL], and allows us to calculate the reaction current ($J$), current fluctuation ($\delta J^2$), affinity ($\mathcal{A}$, net driving force), heat dissipation ($\dot{Q}$), and hence $\mathcal{Q}$ (Eq.\[eqn:inequality2\]) associated with the network.
To define the motor’s mechanical step using the transitions in chemical state space, we assign a set of distance metric $\{d_{ij}\}$ to each transition. For example, if the transition from the $i$-th to $j$-th state is made at time $t$, the increment of displacement is $l(t+dt) = l(t) + d_{ij}$, where we assign nonzero value $d_{ij}\neq 0$ ($d_{ij} = - d_{ji}$) for the transition associated with physical movement along the filament, and $d_{ij}=0$ for pure chemical transitions. Next, to calculate the probability density of motor along the spatial and chemical state space we extend the method of generating function used in ref. [@Koza1999] (see [**SI**]{} for details). The time evolution of the generating function described in terms of the generalized variable $z$ is described by the transition matrix $$\label{eq:Gamma_manu}
\Gamma_{i j}(z) =
\begin{cases}
k_{i j} e^{z d_{i j}} \text{, for $i \neq j$}\\
- \sum_{m=1(\neq i)}^N k_{i m}\text{, for $i = j$}.
\end{cases}$$ $\Gamma_{i j}(0)$ is a usual transition matrix for a master equation. $V$ and $D$, defined at the asymptotic limit ($t \gg 1$), can be obtained from the derivatives of the largest eigenvalue $\lambda_0(z)$ of $\Gamma(z)$ [@Koza1999] (see [**SI**]{}): $$\begin{aligned}
V &= \lambda_0'(0)\end{aligned}$$ and $$\begin{aligned}
D &= \frac{\lambda_0''(0)}{2},\end{aligned}$$ where the prime denotes a partial derivative with respect to $z$. This method can be employed to calculate the transport property associated with a subcycle of an arbitrarily complex chemical network (see [**SI**]{}).
For kinesin-1, we considered a 6-state double-cycle kinetic network (Fig.\[fig\_kinesin\_6state\]A). The load dependence of kinetic rate was modeled using $k_{25}(f) = k_{25}^o e^{- \theta f d_0 / k_B T}$ and $k_{52}(f) = k_{52}^o e^{(1-\theta) f d_0 / k_B T}$ for the mechanical step between the states (2) and (5), and $k_{ij} (f) = 2 k_{ij}^o (1 + e^{\chi_{ij} f d_0 / k_B T})^{-1}$ for other steps associated with ATP chemistry ($ij\neq 25, 52$). The condition of $\chi_{ij} = \chi_{ji}$ makes only the mechanical transition contribute to the work [@Liepelt07PRL]. The rate constants determined for the $\mathcal{F}$-cycle are copied to the corresponding chemical steps in the $\mathcal{B}$-cycle [@Liepelt07PRL]. For example, ADP dissociation rate constant $k_{23}$ of $\mathcal{B}$-cycle is equal to $k_{56}$ which describes ADP dissociation in $\mathcal{F}$-cycle. Similarly, $k_{32} = k_{65}, k_{34} = k_{61}, k_{43} = k_{16}, k_{45}(=k_{45}^{bi}[\text{ATP}]) =k_{12}(= k_{12}^{bi}[\text{ATP}]), \chi_{23} = \chi_{56}, \chi_{34} = \chi_{61}, \chi_{45} = \chi_{12}$. Since the ATP hydrolysis free energy that drives the $\mathcal{F}$- and $\mathcal{B}$-cycle is identical, $ (k_{12} k_{25} k_{56} k_{61} / k_{21} k_{52} k_{65} k_{16} )= (k_{23} k_{34} k_{45} k_{52} / k_{32} k_{43} k_{54} k_{25})$; thus $k_{54} (= k_{21} ( k_{52}/k_{25} )^2)$ [@Liepelt07PRL]. Because of the paucity of data at high load condition that activates the $\mathcal{B}$-cycle, it is not easy to determine all the parameters for $\mathcal{F}$ and $\mathcal{B}$ cycles simultaneously using the existing data [@Visscher99Nature]. To circumvent this difficulty, we fit the data using the following procedure. First, the affinity $\mathcal{A}$ at $f=0$ was determined from our previous study that employed the (N=4)-state unicyclic model [@Hwang2017JPCL]. Even though $\mathcal{B}$-cycle is not considered in Ref. [@Hwang2017JPCL], $J_\mathcal{B} \approx 0$ at $f \sim 0$, which justifies the use of unicyclic model at $f \ll f_{\text{stall}}$. Next, the range of parameters were constrained during the fitting procedure (Table \[table\_fit\]) based on the values obtained in [@Hyeon09PCCP; @Liepelt07PRL]). To fit the data globally, we employed the `minimize` function with ‘L–BFGS–B’ method from the scipy library. For Kin6AA, KIF17, and KIF3AB, motility data digitized from Ref. [@clancy2011NSMB; @Milic:2017:PNAS] were fit to the same 6-state double-cycle network model used for kinesin-1. For myosin-V, dynein, and F$_1$-ATPase, we employed kinetic network models and corresponding rate constants used in Ref. [@Bierbaum:2011:BPJ; @Sarlah:2014:BPJ; @Gerritsma:2010]. Further details are provided in [**SI**]{}.\
[**Acknowledgements.**]{} We thank Steven P. Gross for insightful comments on cargo transport in live cells. This work was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. We acknowledge the Center for Advanced Computation in KIAS for providing computing resources.
Calculation of $V$ and $D$ of kinesin-1 in 6-state multi-cyclic model
=====================================================================
To obtain the expression of $V$ and $D$ for multicyclic kinetic network model in terms of a set of rate constants $\{k_{ij}\}$, we have generalized the technique by Koza [@Koza1999] (Alternatively, technique based on the large deviation theory can be used. See Ref. [@Wachtel:2015:PRE; @Lebowitz:1999]). We define the generating functions for the given network model.
In the 6-state double-cycle kinetic network (Fig. \[fig\_kinesin\_6state\]A), we define the three distinct generating functions for ${\mathcal{F}}$, $\mathcal{B}$, and $\mathcal{X}$ cycles. The two generating functions for the subcycles, $\mathcal{F}$ and $\mathcal{B}$-cycles, are convenient to calculate the chemical current $J_{\mathcal{F}}$ and $J_{\mathcal{B}}$ in each subcycle. To calculate $V$ and $D$ in a convenient way, we have defined another generating function for $\mathcal{X}$-cycle, which is not explicit in the kinetic scheme in Fig. \[fig\_kinesin\_6state\]A. The $\mathcal{X}$-cycle differs from ${\mathcal{F}}$, $\mathcal{B}$-cycle in that the former explicitly considers the physical location of the motor along the 1D track. Although $V$ is obtained either evaluating $V=d_0(J_{\mathcal{F}}-J_{\mathcal{B}})$ or $V=d_0J_{\mathcal{X}}$, it is not straightforward to decompose the diffusivity of motor $D$ into the contributions from ${\mathcal{F}}$ and $\mathcal{B}$-cycle.
The expressions of $J_{\mathcal{F}}(\{k_{ij}\})$, $J_{\mathcal{B}}(\{k_{ij}\})$, $V (\{k_{ij}\})$ and $D(\{k_{ij}\})$ can be obtained by considering an asymptotic limit ($t {\rightarrow}\infty$) of the corresponding generating function.
In what follows, we provide the derivation of generating function in details. In order to derive the generating function, we introduce a generalized index for reaction cycle ${\mathcal{I}}$, with ${\mathcal{I}}={\mathcal{F}}$, $\mathcal{B}$, or $\mathcal{X}$.\
Master equation
-----------------
For a system with $N$ chemical states ($\{1, 2, \cdots, N\}$), a generalized state $\mu^{\mathcal{I}}(t)$ is defined by using the chemical state of the motor at time $t$ and the number of completed ${\mathcal{I}}$-cycles ($n_c^{\mathcal{I}}(t)$). For kinesins whose dynamics can be mapped onto the 6-state double-cycle kinetic network model, if the motor is in the $i$-th chemical state ($i\in \{1, 2, \cdots, N\}$ with $N=6$) at time $t$, the generalized state of the motor in the ${\mathcal{I}}$-cycle is $\mu^{\mathcal{I}}(t) = i + N \times n_c^{\mathcal{I}}(t)$, where ${\mathcal{I}}$ could denote either ${\mathcal{F}}$, $\mathcal{B}$, or $\mathcal{X}$ depending on reader’s interest. $P(\mu^{\mathcal{I}}, t)$ that represents the probability of the system being in $\mu^{\mathcal{I}}$ at time $t$, satisfies
$$\begin{aligned}
\frac{ \partial P(\mu^{\mathcal{I}}, t) } {\partial t} = \sum_{\xi} K_{\mu^{\mathcal{I}}-\xi, \mu^{\mathcal{I}}} P( \mu^{\mathcal{I}}-\xi, t ) - K_{\mu^{\mathcal{I}}, \mu^{\mathcal{I}}-\xi} P(\mu^{\mathcal{I}}, t),
\label{eq:master}
\end{aligned}$$
where $K_{\mu, \nu} = \sum_{\alpha} k^\alpha_{\mu, \nu}$ and $k^\alpha_{ij}$ denotes the rate of transition from state $i$ to state $j$ that follows the $\alpha$-th pathway. Here, the periodicity of network model imposes $k^\alpha_{\mu+N, \nu+N} = k^\alpha_{\mu, \nu}$, $K_{\nu + N, \mu+N} = K_{\nu, \mu}$, and $k^\alpha_{\mu, \nu} = k^\alpha_{i,j}$ for $\mu = i ~ \text{(mod N)}$ and $\nu = j ~ \text{(mod N)}$. The range of (integer) summation index $\xi$ depends on the existing pathways for ${\mathcal{I}}$-cycle. Hereafter, the superscript ${\mathcal{I}}$ on $\mu$ shall be omitted for simplicity.
Following Ref. [@Koza1999], we define $$P_j(\mu, t)\equiv P(\mu,t) \delta^N_{\mu,j}$$ where, $$\delta^N_{\mu,j} =
\begin{cases}
1 \text{, if } j = \mu \text{ (mod $N$) }\\
0 \text{, otherwise}
\end{cases}$$ Here, $j \in \{1, 2, \cdots, N\}$. Multiplying $\delta^N_{\mu, j}$ on both sides of Eq.\[eq:master\] and using the equality $\delta_{\mu,j}^N = \delta_{\mu-\xi, j-\xi}^N$, we get
$$\begin{aligned}
\label{eq:ME}
\frac{\partial P_j(\mu, t)}{\partial t}= \sum_{\xi} K_{j -\xi, j} P_{j-\xi}( \mu -\xi, t) - K_{j, j-\xi} P_j(\mu, t).
\end{aligned}$$
Generating function
---------------------
We define a generating function to derive $V$ and $D$. The generating function for ${\mathcal{I}}$-cycle is defined by $$\label{eq:genQ}
\mathcal{G}^{\mathcal{I}}_j (z,t) \equiv \sum_{\mu=-\infty}^{\infty} e^{z X^{\mathcal{I}}_\mu} P_j (\mu, t).$$ where $X^{\mathcal{I}}_\mu$ denotes the generalized coordinate for ${\mathcal{I}}$-cycle at generalized state $\mu$. Then Eq.(\[eq:ME\]) and the equality $(X^{\mathcal{I}}_{\mu} - X^{\mathcal{I}}_{\mu-\xi}) P_{j-\xi}(\mu-\xi, t) = (X^{\mathcal{I}}_j- X^{\mathcal{I}}_{j-\xi})P_{j-\xi}(\mu-\xi, t) $ with $\delta_{\mu,j}^N = \delta_{\mu-\xi, j-\xi}^N$ lead to
$$\begin{aligned} \label{eq:MEq}
\frac{\partial \mathcal{G}^{\mathcal{I}}_j(z, t)}{\partial t}
&=\sum_{\xi} \left( \sum_{\mu=-\infty}^{\infty} e^{z X^{\mathcal{I}}_\mu} K_{j -\xi, j} P_{j-\xi}( \mu -\xi, t) \right)
- \sum_{\xi} K_{j, j-\xi} \mathcal{G}^{\mathcal{I}}_j(z, t) \\
&= \sum_{\xi} e^{ z d^\mathcal{I}_{j- \xi, j} } K_{j -\xi, j} \mathcal{G}^{\mathcal{I}}_{j-\xi}( z, t)
- \sum_{\xi} K_{j, j-\xi} \mathcal{G}^{\mathcal{I}}_j(z, t) \\
\end{aligned}$$
where $d^\mathcal{I}_{\mu \nu} \equiv X^{\mathcal{I}}_\nu - X^{\mathcal{I}}_\mu$. In general, different cycle has different $\{d^\mathcal{I}_{\mu, \nu}\}$. For example, for the $\mathcal{F}$-cycle in Fig. \[fig\_kinesin\_6state\]A, $$\begin{aligned}
d^\mathcal{F}_{i, j} =
\begin{cases}
1 \text{, for $i=6, j=1$} \\
-1 \text{, for $i=1, j=6$} \\
0 \text{, otherwise},
\end{cases}
\end{aligned}$$ for the $\mathcal{B}$-cycle, $$\begin{aligned}
d^\mathcal{B}_{i, j} =
\begin{cases}
1 \text{, for $i=3, j=4$} \\
-1 \text{, for $i=4, j=3$} \\
0 \text{, otherwise},
\end{cases}
\end{aligned}$$ and for the $\mathcal{X}$-cycle, $$\begin{aligned}
d^\mathcal{X}_{i, j} =
\begin{cases}
1 \text{, for $i=2, j=5$} \\
-1 \text{, for $i=5, j=2$} \\
0 \text{, otherwise}.
\end{cases}
\end{aligned}$$ In fact, Eq. (\[eq:MEq\]) can be expressed more succinctly as $$\label{eq:MEqM}
\begin{aligned}
\partial_t \mathcal{G}^{\mathcal{I}}_j(z, t) = \sum_{i=1}^{N}\Gamma^\mathcal{I}_{i j} \mathcal{G}^{\mathcal{I}}_i(z,t)
\end{aligned}$$ where $$\label{eq:Gamma}
\Gamma^\mathcal{I}_{i j}(z) =
\begin{cases}
\sum_{\alpha} k^\alpha_{i j} e^{z d^{\mathcal{I},\alpha}_{i j}} \text{, if $i \neq j$}\\
- \sum_{m=1(\neq i)}^N \sum_{\alpha} k^\alpha_{i m}\text{, if $i = j$}
\end{cases}$$ With $\alpha$, an index to discern the pathways, $\Gamma$ can be written in the form of $N \times N$ matrix.
Generating function at the asymptotic limit
-------------------------------------------
Here we consider the asymptotic limit ($t {\rightarrow}\infty$) in which $V$ and $D$ are well defined for an arbitrary chemical network model. The general solution of Eq.(\[eq:MEqM\]) can be written as [@Koza1999] $$\begin{aligned}
\mathcal{G}^{\mathcal{I}}_j(z,t) = \sum_m T^\mathcal{I}_{m j}(z, t) e^{\lambda^\mathcal{I}_m(z) t}
\end{aligned}$$ where $\lambda^\mathcal{I}_m(z)$’s ($m=0,1,2,\ldots$, $N$) are the eigenvalues of $\Gamma^\mathcal{I}(z)$. For a system in (unique) steady state, the eigenvalues satisfy $\lambda^\mathcal{I}_0(0) = 0$ and $\lambda^\mathcal{I}_m(0) < 0$ for $m\neq 0$. Thus, at $t \rightarrow \infty$ and when $z \sim 0$, $$\label{eq:QApprox}
\begin{aligned}
\lim_{t \rightarrow \infty} \mathcal{G}^{\mathcal{I}}_j(z,t) \sim T^\mathcal{I}_{0 j}(z,t) e^{\lambda^\mathcal{I}_0(z) t}
\end{aligned}$$ Now, summed over the index $j$, Eq.(\[eq:genQ\]) is led to $$\begin{aligned}
\sum_{j=1}^N \mathcal{G}^{\mathcal{I}}_j(z,t) &= \sum_{j=1}^N \sum_{\mu=-\infty}^{\infty} e^{z X^\mathcal{I}_\mu} P_j(\mu, t) \\
&= \sum_{\mu=-\infty}^{\infty} e^{z X^\mathcal{I}_\mu} P(\mu, t) \\
&\equiv \mathcal{G}^\mathcal{I}(z,t).
\end{aligned}$$ From Eq.(\[eq:QApprox\]), at $t \rightarrow \infty$, we have $$\begin{aligned} \label{eq:Q}
\mathcal{G}^{\mathcal{I}}(z, t) &\sim \sum_j {T^\mathcal{I}_{0 j}(z,t)} e^{\lambda^{\mathcal{I}}_0(z) t} = h^{\mathcal{I}}(z,t) e^{\lambda^{\mathcal{I}}_0(z) t}
\end{aligned}$$ where $h^{\mathcal{I}}(z,t) \equiv \sum_j T_{0 j}^\mathcal{I}(z,t)$. Since $\mathcal{G}^{\mathcal{I}}(0,t) = 1$ and $\lambda^\mathcal{I}_0(z=0)=0$, $h^{\mathcal{I}}(0,t) \sim 1$ at $t \rightarrow \infty$.
Velocity and Diffusion coefficient
-----------------------------------
In this section, we first define the flux $J^{\mathcal{I}}$ and the diffusion coefficient $D^{\mathcal{I}}$ of ${\mathcal{I}}$-cycle using $X^{\mathcal{I}}(t)$ at $t {\rightarrow}\infty$. Then by using the asymptotic form of the generating function, we will get the relation between $J^{\mathcal{I}}$ and $D^{\mathcal{I}}$, and the lowest eigenvalue $\lambda_0^{\mathcal{I}}(z)$.
The mean value of the generalized coordinate $X^{\mathcal{I}}(t)$ can be obtained using $$\begin{aligned}
{\left\langle X^{\mathcal{I}}(t) \right\rangle}
&= \partial_z \mathcal{G}^{\mathcal{I}}(z,t) \vert_{z=0}\sim (h^{\mathcal{I}})'
+ t ~ (\lambda^{\mathcal{I}}_0)'
\end{aligned}$$ where Eq.(\[eq:Q\]) was used and the prime denotes a partial derivative with respect to $z$ at $z=0$. The flux of ${\mathcal{I}}$-cycle is defined by $$\begin{aligned}
J_{\mathcal{I}} &\equiv \lim_{t \rightarrow \infty} \frac{ {\left\langle X^{\mathcal{I}}(t) \right\rangle}}{t}=(\lambda^{\mathcal{I}}_0)'.
\end{aligned}$$ $J_\mathcal{X}$ multiplied by the step size $d_0$ corresponds to the velocity $V$ of motor
Similarly, the diffusion coefficient $D_{\mathcal{I}}$ is obtained by considering the second moment of $X^{\mathcal{I}}$. $$\begin{aligned}
&{\left\langle (X^{\mathcal{I}})^2 \right\rangle} = \partial_z^2 \mathcal{G}^{\mathcal{I}}(z,t)\nonumber\\
&= (h^{\mathcal{I}})'' + 2 t (h^{\mathcal{I}})' (\lambda^{\mathcal{I}}_0)' + t (\lambda^{\mathcal{I}}_0)'' + ((\lambda^{\mathcal{I}}_0)')^2 t^2, \end{aligned}$$ which gives $$\begin{aligned}
D_\mathcal{I} =\lim_{t \rightarrow \infty} \frac{{\left\langle (X^\mathcal{I}(t))^2 \right\rangle} - {\left\langle X^\mathcal{I} (t) \right\rangle}^2}{2 t} = \frac{(\lambda^{\mathcal{I}}_0)''}{2}.
\end{aligned}$$ Thus, the diffusion coefficient of motor is obtained: $D = d_0^2 D_\mathcal{X}$.\
Characteristic polynomial
-------------------------
To express the derivatives of $\lambda^{\mathcal{I}}_0$ in terms of rates $\{k_{ij}\}$, we use the characteristic polynomial of $\Gamma^\mathcal{I}(z)$ [@Koza1999], $$\begin{aligned}
\det( \lambda^\mathcal{I}_0 \mathbb{I} - \Gamma^\mathcal{I}(z) ) &= \sum_{n=0}^{N} (\lambda^{\mathcal{I}}_0)^n C_n(z) = 0.
\label{eqn:det}\end{aligned}$$ By differentiating both side of Eq.\[eqn:det\] with respect to $z$ and setting $z=0$, we get $$\label{eq:cha_d1}
\begin{aligned}
C_0' + C_1 (\lambda^{\mathcal{I}}_0)' &= 0, \\
\end{aligned}$$ and $$\label{eq:cha_d2}
\begin{aligned}
C_0'' + 2 C_1' ( \lambda^{\mathcal{I}}_0)' + C_1 (\lambda^{\mathcal{I}}_0)'' + 2 C_2 (( \lambda^{\mathcal{I}}_0)' )^2= 0.
\end{aligned}$$ From Eqs.(\[eq:cha\_d1\]) and (\[eq:cha\_d2\]), we get $$\label{eq: V in C}
J_{\mathcal{I}} = (\lambda^{\mathcal{I}}_0)' = - \frac{ C_0' }{ C_1 }$$
$$\label{eq: D in C}
D_{\mathcal{I}} = \frac{(\lambda^{\mathcal{I}}_0)''}{2} = - \frac{ C_0'' + 2 C_1' (\lambda^{\mathcal{I}}_0)' + 2 C_2 (\lambda^{\mathcal{I}}_0)'}{2 C_1} = - \frac{C_0'' + 2 J_\mathcal{I} + 2 C_2 J_\mathcal{I} }{ 2 C_1}$$
$C_n$’s and their derivatives, which depend on the choice of $X^{\mathcal{I}}$, can readily be found by differentiating the characteristic polynomial with respect to $\lambda^\mathcal{I}_0(z)$ with $\lambda^{\mathcal{I}}_0(0)=0$ [@Koza1999].
Explicit expression of $J_\mathcal{F}$
----------------------------------------
The expression of reaction current in each subcycle $\mathcal{F}$ and $\mathcal{B}$ in terms of $\{k_{ij}\}$ can be obtained by considering the corresponding generating function $\mathcal{G}^{{\mathcal{I}}\in\{{\mathcal{F}},\mathcal{B}}\}$ Here, we provide the expression of $J_\mathcal{F}$ in terms of rate constants $\{k_{ij}\}$ for the 6-state double-cycle kinetic network.
$$\label{eq:JF}
\begin{aligned}
J_\mathcal{F}&=J_\mathcal{F}^+-J_\mathcal{F}^-\nonumber\\
&=\bigg(k_{12} (k_{25} k_{32} k_{43} + k_{23} k_{34} k_{45} + k_{25} (k_{32} + k_{34}) k_{45}) k_{56} k_{61}
-k_{16} k_{21} (k_{34} k_{45} k_{52}
+ k_{32} (k_{43} + k_{45}) k_{52} + k_{32} k_{43} k_{54}) k_{65}\bigg) \bigg/
\\
&\bigg(((k_{25} k_{32} k_{43} + k_{23} k_{34} k_{45} + k_{25} (k_{32} + k_{34}) k_{45}) k_{56}
+ k_{21} (k_{34} k_{45} (k_{52} + k_{56})
\\& + k_{32} (k_{45} (k_{52} + k_{56}) + k_{43} (k_{52} + k_{54} + k_{56})))) k_{61}
+ k_{21} (k_{34} k_{45} k_{52} + k_{32} (k_{43} + k_{45}) k_{52} + k_{32} k_{43} k_{54}) k_{65}
\\&+k_{16} ((k_{25} k_{32} k_{43} + k_{23} k_{34} k_{45}
+ k_{25} (k_{32} + k_{34}) k_{45}) k_{56} + (k_{32} k_{43} k_{52} + k_{32} k_{45} k_{52}
+ k_{34} k_{45} k_{52} + k_{32} k_{43} k_{54}
\\&+
k_{25} (k_{34} k_{45} + (k_{34} + k_{43}) k_{54} + k_{32} (k_{43} + k_{45} + k_{54})) +
k_{23} ((k_{43} + k_{45}) k_{52} + k_{43} k_{54} + k_{34} (k_{45} + k_{52} + k_{54}))) k_{65}
\\&+
k_{21} (k_{34} k_{45} (k_{52} + k_{56}) + k_{43} k_{54} k_{65} + k_{34} (k_{45} + k_{54}) k_{65} +
k_{32} (k_{54} k_{65} + k_{45} (k_{52} + k_{56} + k_{65})
\\& +
k_{43} (k_{52} + k_{54} + k_{56} + k_{65})))) +
k_{12} ((k_{34} k_{45} (k_{52} + k_{56}) +
k_{32} (k_{45} (k_{52} + k_{56}) +
k_{43} (k_{52} + k_{54} + k_{56}))) k_{61}
\\&+ (k_{34} k_{45} k_{52} +
k_{32} (k_{43} + k_{45}) k_{52} + k_{32} k_{43} k_{54}) k_{65} +
k_{23} (k_{45} (k_{52} + k_{56}) k_{61}
\\&+ k_{43} (k_{52} + k_{54} + k_{56}) k_{61} +
k_{45} k_{52} k_{65} + k_{43} (k_{52} + k_{54}) k_{65}
+
k_{34} ((k_{52} + k_{54} + k_{56}) k_{61} + (k_{52} + k_{54}) k_{65}
\\& +
k_{45} (k_{56} + k_{61} + k_{65}))) +
k_{25} (k_{43} k_{54} (k_{61} + k_{65}) +
k_{34} (k_{54} (k_{61} + k_{65}) + k_{45} (k_{56} + k_{61} + k_{65}))
\\&+
k_{32} (k_{54} (k_{61} + k_{65}) + k_{43} (k_{56} + k_{61} + k_{65}) +
k_{45} (k_{56} + k_{61} + k_{65}))))\bigg)
\end{aligned}$$
Similarly, $J_\mathcal{B}$ and $D_\mathcal{\mathcal{X}}$ can also be expressed in terms of $\{k_{ij}\}$.\
Analysis of other types of kinesins
=====================================
Kinesin-1 mutant (Kin6AA)
---------------------------
Single molecule motility data digitized from Ref. [@Clancy:2011:NSMB] was fitted to 6-state network model (Figs. \[fig\_kinesin\_6state\]A, \[fig\_exp\_Kin6AA\]) by using the same method employed for the analysis of kinesin-1 data ([**Materials and Methods** ]{}). However, 4 additional initial conditions for $k_{25}$ ($\{300, 3000, 30000, 3000000\}$), thus total 245 initial conditions, were explored. The rate constants estimated from this procedure are provided in Table \[table\].
Kinesin-2 (KIF17, KIF3AB)
---------------------------
Single molecule motility data digitized from Ref. [@Milic:2017:PNAS] was again fitted to the 6-state double-cycle kinetic model (Fig. \[fig\_kinesin\_6state\]A, \[fig\_exp\_KIF17\], \[fig\_exp\_KIF3AB\]) following the identical procedure employed in the analysis of kinesin-1 data ([**Materials and Methods**]{}). However, two additional initial conditions for $k_{25}$ ($\{30000, 3000000\}$) were explored, which results in total 147 initial conditions. The rate constants are shown in Table \[table\].
Myosin-V
========
Here we summarize the multi-cyclic model for myosin-V [@Bierbaum:2011:BPJ] which consists of ATP-dependent chemomechanical forward cycle $\mathcal{F}$, dissipative cycle $\mathcal{E}$, and ratcheting cycle (ATP independent stepping cycle) $\mathcal{M}$ (Fig. \[fig\_Qa\_other\_motor\_proteins\]A). We first provide the explanation of how $V$ and $D$ of myosin-V are calculated. Next, the affinity and heat production ($\dot{Q}$) are expressed in terms of a set of rates $\{k_{ij}\}$. Finally, $\mathcal{Q}$ shall be calculated using $V$, $D$, and $\dot{Q}$.
Calculation of $V$ and $D$
--------------------------
The $\mathcal{M}$-cycle consisting of a single state (Fig. \[fig\_Qa\_other\_motor\_proteins\]A) prevents the application of Eq.(\[eq:Gamma\]). To circumvent this difficulty, the model with additional state ($5'$) is considered (Fig. \[fig\_myosin\_model\_aug\]). The $(5')$-state is chemically equivalent to the state (5), but describes motor in different position on actins, such that $X(5) = X_0$ and $X(5') = X_0 \pm d_0$ where $d_0 = 36$ nm for myosin-V. In this new network, the rate constants $\kappa_{ij}$’s are [ $$\begin{aligned}
\kappa_{2, 5} &= \kappa_{2, 5'} = \frac{k_{25}}{2} \\
\kappa_{6, 5} &= \kappa_{6, 5'} = \frac{k_{65}}{2} \\
\kappa_{5, 2} &= \kappa_{5', 2} = k_{52} \\
\kappa^f_{5,5'} &= \kappa^f_{5', 5} = k_{55, f} \\
\kappa^b_{5,5'} &= \kappa^b_{5', 5} = k_{55, b} \\
\end{aligned}$$ ]{}\
where the subscripts $f$ and $b$ denote the forward and backward motion, respectively. Other rate constants satisfy $\kappa_{ij} = k_{ij}$. This modification can be justified by considering stochastic movement of myosin-V on the chemical network [@Gillespie:1977]: $\kappa_{i,5}, \kappa_{i,5'}$ are set to $k_{i5}/2$, such that the outgoing fluxes from the states $i=$ (2), (6) to the state (5) remain identical in the both networks depicted in Fig. \[fig\_Qa\_other\_motor\_proteins\]A and Fig.\[fig\_myosin\_model\_aug\]. Next, we set $\kappa_{5,i} = \kappa_{5', i}$ to keep the inward fluxes toward (6), (2) identical for the two networks. Finally, $\kappa^{f}_{5, 5'} = \kappa^{f}_{5', 5} = k_{55, f}$ and $\kappa^{b}_{5, 5'} = \kappa^{b}_{5', 5} = k_{55, b}$. These modification of rate constants enable us to describe transitions within the ${\mathcal{M}}$-cycle.
Now, the elements of distance matrix scaled by $d_0$ are [ $$\begin{aligned}
d^\mathcal{X}_{3,4} & = 1,\\
d^\mathcal{X}_{4,3} & = -1, \\
d^{\mathcal{X},f}_{5, 5'} &= 1,\\
d^{\mathcal{X},f}_{5', 5} &= 1,\\
d^{\mathcal{X},b}_{5, 5'} &= -1, \\
d^{\mathcal{X},b}_{5', 5} &= -1. \\
\end{aligned}$$ ]{} Other elements ($d^\mathcal{X}_{i,j}$) are all zero. Thus, $\Gamma_{i,j}^\mathcal{X}$ is written as (with $(7) \equiv (5')$)\
$$\begin{aligned}
\left(
\begin{smallmatrix} \label{eq:Gamma_mV}
-{\kappa_{12}}-{\kappa_{14}} & {\kappa_{12}} & 0 & {\kappa_{14}} & 0 & 0 & 0 \\
{\kappa_{21}} & -{\kappa_{21}}-{\kappa_{23}}-{\kappa_{25}}-{\kappa_{26}}-{\kappa_{27}} & {\kappa_{23}} & 0 & e^z {\kappa_{25}} & {\kappa_{26}} & {\kappa_{27}} \\
0 & {\kappa_{32}} & -{\kappa_{32}}-{\kappa_{34}} & {\kappa_{34}} & 0 & 0 & 0 \\
{\kappa_{41}} & 0 & {\kappa_{43}} & -{\kappa_{41}}-{\kappa_{43}} & 0 & 0 & 0 \\
0 & e^{-z} {\kappa_{52}} & 0 & 0 & -{\kappa_{52}}-{\kappa_{56}}-{\kappa_{57,b}}-{\kappa_{57,f}} & {\kappa_{56}} & {\kappa_{57,b}}+{\kappa_{57,f}} \\
0 & {\kappa_{62}} & 0 & 0 & {\kappa_{65}} & -{\kappa_{62}}-{\kappa_{65}}-{\kappa_{67}} & {\kappa_{67}} \\
0 & {\kappa_{72}} & 0 & 0 & {\kappa_{75,b}}+{\kappa_{75,f}} & {\kappa_{76}} & -{\kappa_{72}}-{\kappa_{75,b}}-{\kappa_{75,f}}-{\kappa_{76}}
\end{smallmatrix}
\right)
\end{aligned}$$
Now, the travel velocity $V$ and the diffusion coefficient $D$ of myosin-V are readily acquired by using Eqs.(\[eq: V in C\], \[eq: D in C\], and \[eq:Gamma\_mV\]). The rate constants used in the calculation are summarized in Table. \[table\_mV\].
Affinities and heat production
------------------------------
The affinities of individual cycles are
[ $$\begin{aligned}
{\mathcal{A}}_{\mathcal{F}}&= k_B T \log {\left( \frac{k_{12} k_{23} k_{34} k_{41} }{ k_{21} k_{32} k_{43} k_{41} } \right)},
\\
{\mathcal{A}}_{\mathcal{E}}&= k_B T \log {\left( \frac{k_{25} k_{56} k_{62} } {k_{52} k_{65} k_{26} } \right)}, \\
{\mathcal{A}}_{\mathcal{M}}&= k_B T \log {\left( \frac{k_{55,f} } {k_{55,b} } \right)}. \\
\end{aligned}$$ ]{}\
Only the following rate constants depend on the load ($f$): [ $$\begin{aligned}
k_{34} &= k_{34}^o e^{- \theta d_m f / k_B T } \\
k_{43} &= k_{43}^o e^{(1-\theta) d_m f / k_B T }\\
k_{56} &= k_{56}^o \frac{ 1 + e^{-\chi d_m f_c / k_B T } }{ 1 + e^{ \chi d_m (f-f_c) / k_B T } }\\
k_{52} &= k_{52}^o \frac{ 1 + e^{-\chi d_m f_c / k_B T } }{ 1 + e^{ \chi d_m (f-f_c) / k_B T } }\\
k_{55,b} &= \frac{ D'}{k_B T } \frac{ f d_m - U}{d_m^2} \frac{1}{1 - e^{ {\left( U - f d_m \right)} /k_B T }} \\
k_{55,f} &= k_{55,b} e^{- f d_m / k_B T}
\end{aligned}$$ ]{} where [ $$\begin{aligned}
\theta &= 0.65\\
\chi &= 4\\
f_c & = 1.6 \text{ pN } \\
U &= \text{20 }k_B T \\
D' & = 4.7 \times 10^{-4} \mu m/s^2
\end{aligned}$$ ]{} as described in Ref. [@Bierbaum:2011:BPJ]. Thus, the affinities can be written as [ $$\begin{aligned}
{\mathcal{A}}_{\mathcal{F}}&= k_B T \log {\left( \frac{k_{12}^o k_{23}^o k_{34}^o k_{41}^o }{ k_{21}^o k_{32}^o k_{43}^o k_{41}^o} \right)} - f d_0 \\
{\mathcal{A}}_{\mathcal{E}}&= {\mathcal{A}}_{{\mathcal{E}}, f=0}\\
{\mathcal{A}}_{\mathcal{M}}&= - f d_0.
\end{aligned}$$ ]{} The relation ${\mathcal{A}}_{\mathcal{M}}= - f d_0$ results from the fact that ${\mathcal{M}}$-cycle is ATP-independent and activated by the load. Thus, the heat production rate of the system is [ $$\begin{aligned}
\dot{Q} &= J_{\mathcal{F}}{\mathcal{A}}_F + J_{\mathcal{E}}{\mathcal{A}}_E + J_{\mathcal{M}}{\mathcal{A}}_M \\
\end{aligned}$$ ]{} $J_{\mathcal{F}}, J_{\mathcal{E}}$, and $J_{\mathcal{M}}$ can be calculated by using Eqs. (\[eq: V in C\]) and (\[eq:Gamma\_mV\]). Finally, $\mathcal{Q}$ for myosin-V is given by [ $$\begin{aligned}
\mathcal{Q}_{\text{Myosin-V} } &= \dot{Q}\frac{ 2 D } {V^2} \\
\end{aligned}$$ ]{} where $D = D_\mathcal{X} d_0^2$ and $V = J_\mathcal{X} d_0$. \[table\_dynein\]
Dynein
=======
$(N=7)$-unicyclic model is considered based on the model of cytoplasmic dimeric dynein studied in Ref. [@Sarlah:2014:BPJ]. Only the major forward pathway, where the transitions between the states are denoted by solid black lines in Fig. 5A of Ref. [@Sarlah:2014:BPJ], is considered. The values of rate constants obtained from Ref. [@Sarlah:2014:BPJ] are summarized in Table \[table\_dynein\]. To describe the force-dependence of power-stroke, we model the rate constant for forward and reverse strokes ($k_{+PS}$ and $k_{-PS}$) as follows. $$\label{eq:f_dynein}
\begin{aligned}
k_{+PS} &= k_{+PS, f=0} e^{-\theta \frac{f d_0}{k_B T} }\\
k_{-PS} &= k_{-PS, f=0} e^{(1-\theta) \frac{f d_0}{k_B T} }\\
\end{aligned}$$ where $\theta = 0.3$ is selected based on the previous studies [@Singh:2005:PNAS; @Wagoner:2016:JPCB]. In the original literature [@Sarlah:2014:BPJ], all the rate constants depend on both elastic energy originated from the interaction between two monomer units of dynein, and $f$. Although this approach will better describe the details of dynein dynamics, it is not possible to calculate elastic energy without explicit simulation of the motion of dyneins which are modeled as elastic materials [@Sarlah:2014:BPJ]. Thus, for simplicity, we assumes only $k_{\pm PS}$ changes significantly by $f$. Again, $V$ and $D$ were calculated using Eqs. \[eq: V in C\], \[eq: D in C\].
Affinity and heat production
------------------------------
The affinity for unicyclic model is written as [@Qian2005_BiophyChem; @Seifert:2005:PRL; @Hwang2017JPCL] [ $$\begin{aligned}
\mathcal{A} &= k_B T \log \prod_{i=1}^N \frac{ k_{i,i+1}} {k_{i+1, i}}
\\& = -\Delta \mu_{\text{hyd}} - f d_0
\end{aligned}$$ ]{} where $d_0 = 8.2$ nm. The second term, describing force-dependence, is originated from the use of Eq. \[eq:f\_dynein\]. Finally, $\mathcal{Q}$ is [ $$\begin{aligned}
\mathcal{Q} &= \frac{2 D}{V d_0} \mathcal{A}.
\end{aligned}$$ ]{}\
F$_1$-ATPase
============
Here, we summarize the unicyclic model developed for F$_1$-ATPase in Ref. [@Gerritsma:2010]. The model is $(N=2)$ unicyclic model (Fig. \[fig\_Qa\_other\_motor\_proteins\]C) where 3 cycles in chemical state space correspond to a single rotation in real space (angle changes by 90$\degree$ upon transition from the state (1) to state (2) whereas transitions from the state (2) to $(1)'$ induce 30$\degree$ rotation (Fig. \[fig\_Qa\_other\_motor\_proteins\]C). The model is valid when the torque applied to F$_1$-ATPase is small enough ($\tau \lesssim 30$ pN$\cdot$nm) that the mechanical cycle is tightly coupled to the chemical reaction [@Gerritsma:2010]. The dependences of rate constants on the torque are [ $$\begin{aligned}
k_{12}(\tau, \zeta) & = k^{\text{bi}}_{12} (\tau, \zeta) \times [\text{ATP}] \\
&= \frac{1}{ e^{a_{k_{12}}(\tau)} + \zeta e^{b_{k_{12}}(\tau)} }\times [\text{ATP}] \\
k_{21'}(\tau, \zeta) & = \frac{1}{ e^{a_{k_{21'}}(\tau)} + \zeta e^{b_{k_{21'}}(\tau)} } \\
k_{1'2}(\tau, \zeta) & = k_{1'2}^{\text{bi}}(\tau, \zeta) \times [\text{ADP}][\text{Pi}] \\
&= \frac{1}{ e^{a_{k_{1'2}}(\tau)} + \zeta e^{b_{k_{1'2}}(\tau)} } \times [\text{ADP}][\text{Pi}]\\
k_{21}(\tau, \zeta) &= \frac{ k_{12}(\zeta, \tau) u_2(\zeta, \tau) }{k_{21}(\zeta, \tau)} e^{(\Delta \mu_{\text{hyd}}^0 + \frac{2 \pi}{3} \tau)/k_B T}
\end{aligned}$$ ]{}\
where $\Delta \mu_{\text{hyd}}^0 =-12.5$ $k_B T \approx -50$ pN$\cdot$nm, $\zeta$ is the friction coefficient (for example, if the $\gamma$-shaft of F$_1$-ATPase is attached to a bead of radius $r$, $\zeta = 2 \pi \eta r^3 ( 4 + 3 \sin^2 \pi/6 )$ [@Gerritsma:2010] with the water viscosity $\eta = 1$ cP $= 10^{-9}$ pN$\times$s$\times$ nm$^{-2}$. In our calculation, $r=40$ nm as in Ref. [@Gerritsma:2010]), and $a_i(\tau), b_i(\tau)$ are polynomial function of $\tau$ defined in Ref. [@Gerritsma:2010]. The expressions of $a_i(\tau)$, $b_i(\tau)$ and the coefficients of the polynomials are given in Table. \[table\_f1\].
$V$, $D$, affinities, and heat production
-------------------------------------------
For (N=2)-unicyclic model, the speed of rotation $V$, diffusion coefficient $D$, and affinity $\mathcal{A}$ are [@Derrida1983_JSP; @Koza1999; @Fisher1999_PNAS; @Qian07ARPC; @Qian2005_BiophyChem; @Seifert2012RPP; @Hwang2017JPCL] $$\begin{aligned}
\label{eq:V,D,A_f1}
V &= d_R \frac{k_{1,2} k_{2,1'} - k_{2,1} k_{1', 2}}{k_{1,2} + k_{2,1'} + k_{2,1} + k_{1', 2}} \equiv d_R J,\nonumber\\
D&=\frac{d_R^2}{2}\left[\frac{k_{1,2}k_{2,1'}}{k_{2,1}k_{1', 2}}+1-2\left(\frac{k_{1,2}k_{2,1'}}{k_{2,1}k_{1', 2}}-1\right)^2\frac{k_{2,1}k_{1', 2}}{\sigma^2}\right]\nonumber\\
&\times\frac{k_{2,1}k_{1', 2}}{\sigma}, \nonumber\\
{\mathcal{A}}&= k_B T \log {\left( \frac{ k_{1,2} k_{2,1'} }{ k_{2,1} k_{1', 2} } \right)}\nonumber\\
&={\left( -\Delta \mu_{\text{hyd}}^0 + k_B T \log {\left( \frac{ [ATP] }{[ADP][Pi]} \right)} \right)} - \frac{2 \pi }{3} \tau \nonumber\\
&= -\Delta \mu_{\text{hyd}} - W,\end{aligned}$$ where $d_R = \frac{2 \pi}{3} $ is the radian distance that motor travels upon ATP hydrolysis, $\sigma=k_{12} + k_{2 1'} + k_{21} + k_{1'2}$, and $W \equiv \frac{2 \pi}{3} \tau$ denotes the work done by the motor. Here, $\tau>0$ implies the motor performs work against the hindering load. Thus, $\mathcal{Q}$ is given by [ $$\begin{aligned}
\mathcal{Q} = \frac{ 2 D }{ V d_R } \mathcal{A}
\end{aligned}$$ ]{} .
Unicyclic kinetic model for kinesin-1
======================================
To analyze the kinesin-1 data, we also considered $(N=4)$-unicyclic model (Fig. \[fig\_kinesin\_uni\]A) which was used in our previous study [@Hwang2017JPCL]. Briefly, the model consists of four forward rates $\{u_1, u_2, u_3, u_4\}$ and four backward rates $\{w_1, w_2, w_3, w_4\}$. Only $u_1 (= k^{bi} [\text{ATP}])$ depends on \[ATP\]. Barometric dependence of the rates on forces are assumed: $u_n = u_n^o e^{-f d_0 \theta_n^+ / k_B T}$ and $w_n = w_n^o e^{f d_0 \theta_n^- / k_B T}$ with $\sum_{n=1}^N\left( \theta_n^+ + \theta_n^- \right) = 1$ [@Fisher99PNAS; @Fisher01PNAS]. $V$, $D$, $A$, and a set of rate constants used in the calculation of $\mathcal{Q}$ are provided in Table. \[table\_kinesin\_uni\] [@Hwang2017JPCL].
Unicyclic kinetic model for myosin-V
=====================================
For myosin-V, we also considered the ($N=2$) unicyclic model from Ref. [@Kolomeisky:2003:BPJ]. Briefly, the model consists of two forward rates $\{u_1, u_2\}$ and two backward rates $\{w_1, w_2\}$. Only $u_1 (= k [\text{ATP}])$ and $w_2 (= k'[\text{ATP}]^\alpha)$ depend on \[ATP\]. Here, $\alpha = 1/2$. Different choice of $\alpha$ introduces only minor difference in the results as argued in [@Kolomeisky:2003:BPJ]. Barometric dependences of the rates on forces are assumed again: $u_n = u_n^o e^ 3{-f d_0 \theta_n^+ / k_B T}$ and $w_n = w_n^o e^{f d_0 \theta_n^- / k_B T}$ with $\sum_{n=1}^N\left( \theta_n^+ + \theta_n^- \right) = 1$ [@Fisher99PNAS; @Fisher01PNAS]. The parameters used in the calculation are available in Eqs. (12), (13) in Ref. [@Kolomeisky:2003:BPJ] and summarized in Table. \[table\_mV\_uni\]. Identical expressions for $V$, $D$, and $\mathcal{A}$ from Eq. \[eq:V,D,A\_f1\] were used for the calculation except for $W = f d_0$.
The lower bound of $\mathcal{Q}$ for unicyclic model
=====================================================
The analytic expression for the lower bound of the uncertainty measure $\mathcal{Q}$ is available for unicyclic models [@barato2015PRL]. For $(N)$-state unicyclic model, the lower bound of $\mathcal{Q}$ is $$\label{eq:Qb}
\mathcal{Q}_b = \frac{ \mathcal{A} }{N} \coth {\left( \frac{\mathcal{A}} { 2 N k_B T} \right)} \geq 2 k_B T.$$ The $\mathcal{Q}_b$ and the $\Delta \mathcal{Q} \equiv \mathcal{Q} - \mathcal{Q}_b$ of the motors as a function of $f$ and \[ATP\] are calculated in Figs. \[fig\_kinesin\_uni\]D (kinesin-1), \[fig\_f1\_various\]D (F$_1$-ATPase), and \[fig\_myosin\_uni\]D (myosin-V).
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--------------------------------------------- --------------- ------------ --------------- ------------------------------ ------------ ------------ --------------
Kinesin-1 Kinesin-1 KIF17 Myosin-V Myosin-V Dynein F$_1$-ATPase
(multi-cycle) (unicycle) (multi-cycle) (multi-cycle) (unicycle) (unicycle) (unicyclic)
\[ADP\]=\[P$_i$\]=0.1 $\mu$M
$f$ (pN) 4.1 3.2 [^1] 1.5 1.1 0.03 3.9 8.6 [^2]
($\mu$M) 210 460 200 20 17 200 16
$\mathcal{Q}_{\text{min}} ~ (k_B T)$ 4.0 4.5 9.2 6.5 14 5.2 4.2
$\Delta \mathcal{Q}_{\text{min}} ~ (k_B T)$ n/a 1.6 n/a n/a 0 2.6 0
--------------------------------------------- --------------- ------------ --------------- ------------------------------ ------------ ------------ --------------
\[table\_opt\]
Kinesin-1 Kin6AA KIF17 KIF3AB
--------------- ---------------------- ---------------------- ---------------------- -----------------------
$k_{12}^{bi}$ $2.8$ 10 10 10
$k_{21}$ $4200$ 92 3600 500
$k_{25}$ $1.6\times 10^6$ $1.6\times 10^4$ $4.0 \times 10^4$ $6.2\times 10^6$
$k_{52}$ $1.1$ 3.4 0.079 7.2
$k_{56}$ $190$ 680 590 92
$k_{65}$ $10$ 4.1 13 37
$k_{61}$ $250$ 58 310 320
$k_{16}$ $230$ 260 1100 750
$k_{54}$ $2.1 \times 10^{-9}$ $4.3 \times 10^{-6}$ $1.4 \times 10^{-8}$ $6.8 \times 10^{-10}$
$\theta$ $0.61$ 0.59 0.34 0.82
$\chi_{12}$ $0.15$ 0.12 0.15 0.09
$\chi_{56}$ $0.0015$ 0.0 0.012 0.021
$\chi_{61}$ $0.11$ 0.18 0.17 0.16
\[table\]
--------------- ------------------------- ------------- ------------------------------------------------------ ------------- ----------------------------------------------------- ------------- ------------------------------------
$k_{12}^{bi}$ $0.5 \leq 1.8 \leq 10$ $k_{56}$ $10 \leq 200 \leq 10^4$ $k_{61}$ $10 \leq 200 \leq 10^4$ $k_{25}$ $10^4 \leq 3\times 10^5 \leq 10^7$
$k_{21}$ $10 \leq 100 \leq 10^4$ $k_{65}$ $10^{-4} \leq 10^{[-3,- 2, -1, 0, 1, 2,3]}\leq 10^4$ $k_{16}$ $0^{-4} \leq 10^{[-3, -2, -1, 0, 1, 2,3]}\leq 10^4$
$\theta$ $0 \leq 0.3 \leq 1$ $\chi_{12}$ $0 \leq 0.25 \leq 1$ $\chi_{56}$ $0 \leq 0.05 \leq 1$ $\chi_{61}$ $0 \leq 0.05 \leq 1$
--------------- ------------------------- ------------- ------------------------------------------------------ ------------- ----------------------------------------------------- ------------- ------------------------------------
\[table\_fit\]
Description value
--------------- --------------------------- ----------------------
$k_{12}$ ADP release 1.2
$k_{21}^{bi}$ ADP binding 4.5
$k_{23}^{bi}$ ATP binding 0.9
$k_{32}$ ATP release $2 \times 10^{-5}$
$k_{34}$ step 7000
$k_{43}$ reverse step 0.65
$k_{56}^{bi}$ ATP binding 0.9
$k_{65}$ ATP release $2 \times 10^{-5}$
$k_{55,f}$ step (mechanical) $1.5 \times 10^{-8}$
$k_{55,b}$ reverse step (mechanical) $1.5 \times 10^{-8}$
\[table\_mV\]
Description value
--------------- -------------------------------- -------
$k_{12}$ Pi release 5000
$k_{21}^{bi}$ Pi binding 0.01
$k_{23}$ ADP release 160
$k_{32}^{bi}$ ADP binding 2.7
$k_{34}^{bi}$ ATP binding 2
$k_{43}$ ATP release 50
$k_{45}$ MT release in poststroke state 500
$k_{54}$ MT binding in poststroke state 100
$k_{56}$ Power stroke 5000
$k_{65}$ Reverse stroke 10
$k_{67}$ linker swing to prestroke 1000
$k_{76}$ linker swing to poststroke 100
$k_{71}$ MT binding in prestroke state 10000
$k_{17}$ MT release in prestroke state 500
\[table\_dynein\]
$i=k_{12}$ $i=k_{21'}$ $i=k_{1'2}$ Unit
------------- ----------------------- ----------------------- ----------------------- ----------------
$a_i^{(0)}$ -16.952 -5.973 -19.382 -
$a_i^{(1)}$ $9.8 \times 10^{-4}$ $1.7 \times 10^{-4}$ 0.129 (pN nm)$^{-1}$
$a_i^{(2)}$ $5.8 \times 10^{-4}$ $1.0 \times 10^{-3}$ $2.8 \times 10^{-4}$ (pN nm)$^{-2}$
$b_i^{(0)}$ -16.352 -2.960 -18.338 -
$b_i^{(1)}$ $-6.6 \times 10^{-2}$ $-2.7 \times 10^{-2}$ $5.9 \times 10^{-3}$ (pN nm)$^{-1}$
$b_i^{(2)}$ $1.0 \times 10^{-3}$ $3.6 \times 10^{-4}$ $-2.1 \times 10^{-4}$ (pN nm)$^{-2}$
\[table\_f1\]
------------------ -------- ------------------ -------- ------------------ -------- ------------------ --------
$u_1^0$ $2.3$ $u_2$ $600$ $u_3$ $400$ $u_4$ $190$
$\theta_{1}^{+}$ $0.00$ $\theta_{2}^{+}$ $0.04$ $\theta_{3}^{+}$ $0.01$ $\theta_{4}^{+}$ $0.02$
$w_1$ $20$ $w_2$ $1.4$ $w_3$ $1.7$ $w_4$ $120$
$\theta_{1}^{-}$ $0.14$ $\theta_{2}^{-}$ $0.15$ $\theta_{3}^{-}$ $0.5$ $\theta_{4}^{-}$ $0.14$
------------------ -------- ------------------ -------- ------------------ -------- ------------------ --------
\[table\_kinesin\_uni\]
-------------- ----------------------
$k$ 0.70
$w_1$ $6 \times 10^-6$
$u_2$ 12
$k'$ $5.0 \times 10^{-6}$
$\theta_1^+$ -0.01
$\theta_1^-$ 0.045
$\theta_2^+$ 0.385
$\theta_2^-$ 0.58
-------------- ----------------------
\[table\_mV\_uni\]
[^1]: \[dQ\]Condition for local minimization of $\Delta \mathcal{Q} = \mathcal{Q}- \mathcal{Q}_b$.
[^2]: For F$_1$-ATPase, we consider a resisting torque ($\tau$ with the unit of pN$\cdot$nm) against the rotation of the motor.
|
{
"pile_set_name": "ArXiv"
}
|
---
address: |
$^{\ast}$University of Warsaw\
Krakowskie Przedmieście 26/28, 00-927 Warsaw, Poland\
[email protected]
bibliography:
- 'spellcor.bib'
title: Evaluation of basic modules for isolated spelling error correction in Polish texts
---
Introduction
============
Spelling error correction is a fundamental NLP task. Most language processing applications benefit greatly from being provided clean texts for their best performance. Human users of computers also often expect competent help in making spelling of their texts correct.
Because of the lack of tests of many common spelling correction methods for Polish, it is useful to establish how they perform in a simple scenario. We constrain ourselves to the pure task of isolated correction of non-word errors. They are traditionally separated in error correction literature [@kukich1992techniques]. Non-word errors are here incorrect word forms that not only differ from what was intended, but also do not constitute another, existing word themselves. Much of the initial research on error correction focused on this simple task, tackled without means of taking the context of the nearest words into account.
It is true that, especially in the case of neural networks, it is often possible and desirable to combine problems of error detection, correction and context awareness into one task trained with a supervised training procedure. In language correction research for English language also grammatical and regular spelling errors have been treated uniformly with much success [@Ge2018ReachingHP].
However, when more traditional methods are used, because of their predictability and interpretability for example, one can mix and match various approaches to dealing with the subproblems of detection, correction and context handling (often equivalent to employing some kind of a language model). We call it a modular approach to building spelling error correction systems. There is recent research where this paradigm was applied, interestingly, to convolutional networks trained separately for various subtasks [@dronen]. In similar setups it is more useful to assess abilities of various solutions in isolation. The exact architecture of a spelling correction system should depend on characteristics of texts it will work on.
Similar considerations eliminated from our focus handcrafted solutions for the whole spelling correction pipeline, primarily the LanguageTool [@milkowski]. Its performance in fixing spelling of Polish tweets was already tested [@ogr:kop:17]. For our purposes it would be given an unfair advantage, since it is a rule-based system making heavy use of words in context of the error.
Problems of spelling correction for Polish
==========================================
Published work on language correction for Polish dates back at least to 1970s, when simplest Levenshtein distance solutions were used for cleaning mainframe inputs [@subieta1976; @subieta1985simple]. Spelling correction tests described in literature have tended to focus on one approach applied to a specific corpus. Limited examples include works on spellchecking mammography reports and tweets [@mammografia; @ogr:kop:17]. These works emphasized the importance of tailoring correction systems to specific problems of corpora they are applied to. For example, mammography reports suffer from poor typing, which in this case is a repetitive work done in relative hurry. Tweets, on the other hand, tend to contain emoticons and neologisms that can trick solutions based on rules and dictionaries, such as LanguageTool. The latter is, by itself, fairly well suited for Polish texts, since a number of extensions to the structure of this application was inspired by problems with morphology of Polish language [@milkowski].
These existing works pointed out more general, potentially useful qualities specific to spelling errors in Polish language texts. It is, primarily, the problem of leaving out diacritical signs, or, more rarely, adding them in wrong places. This phenomenon stems from using a variant of the US keyboard layout, where combinations of `AltGr` with some alphabetic keys produces characters unique to Polish. When the user forgets or neglects to press the AltGr key, typos such as writing *\*olowek* instead of *ołówek* appear. In fact, [@ogr:kop:17] managed to get substantial performance on Twitter corpus by using this ”diacritical swapping” alone.
Methods
=======
Baseline methods
----------------
The methods that we evaluated are baselines are the ones we consider to be basic and with moderate potential of yielding particularly good results. Probably the most straightforward approach to error correction is selecting known words from a dictionary that are within the smallest edit distance from the error. We used the Levenshtein distance metric [@levenshtein1966bcc] implemented in Apache Lucene library [@pylucene]. It is a version of edit distance that treats deletions, insertions and replacements as adding one unit distance, without giving a special treatment to character swaps. The SGJP – Grammatical Dictionary of Polish [@sgjp] was used as the reference vocabulary.
Another simple approach is the aforementioned diacritical swapping, which is a term that we introduce here for referring to a solution inspired by the work of [@ogr:kop:17]. Namely, from the incorrect form we try to produce all strings obtainable by either adding or removing diacritical marks from characters. We then exclude options that are not present in SGJP, and select as the correction the one within the smallest edit distance from the error. It is possible for the number of such diacritically-swapped options to become very big. For example, the token *Modlin-Zegrze-Pultusk-Różan-Ostrołęka-Łomża-Osowiec* (taken from PlEWi corpus of spelling errors, see below) can yield over $2^{29}=536,870,912$ states with this method, such as *Módłiń-Żęgrzę-Pułtuśk-Roźąń-Óśtróleką-Lómzą-Óśówięć*. The actual correction here is just fixing the *ł* in *Pułtusk*. Hence we only try to correct in this way tokens that are shorter than 17 characters.
Vector distance
---------------
A promising method, adapted from work on correcting texts by English language learners [@NAGATA2017474], expands on the concept of selecting a correction nearest to the spelling error according to some notion of *distance*. Here, the Levenshtein distance is used in a weighted sum to cosine distance between word vectors. This is based on the observation that trained vectors models of distributional semantics contain also representations of spelling errors, if they were not pruned. Their representations tend to be similar to those of their correct counterparts. For example, the token *enginir* will appear in similar contexts as *engineer*, and therefore will be assigned a similar vector embedding.
The distance between two tokens $a$ and $b$ is thus defined as
$$\operatorname{D}(a, b) = \frac{\operatorname{LD}(a,b) + \operatorname{CD}(\operatorname{V}(a), \operatorname{V}(b))}{2}.$$
Here $\operatorname{LD}$ is just Levenshtein distance between strings, and $\operatorname{CD}$ – cosine distance between vectors. $\operatorname{V}(a)$ denotes the word vector for $a$. Both distance metrics are in our case roughly in the range \[0,1\] thanks to the scaling of edit distance performed automatically by Apache Lucene. We used a pretrained set of word embeddings of Polish [@mykowiecka_wektory], obtained with the flavor word2vec procedure using skipgrams and negative sampling [@mikolov].
Recurrent neural networks
-------------------------
Another powerful approach, if conceptually simple in linguistic terms, is using a character-based recurrent neural network. Here, we test uni- and bidirectional Long Short-Term Memory networks [@Hochreiter:1997:LSM:1246443.1246450] that are fed characters of the error as their input and are expected to output its correct form, character after character. This is similar to traditional solutions conceptualizing the spelling error as a chain of characters, which are used as evidence to predict the most likely chain of replacements (original characters). This was done with n-gram methods, Markov chains and other probabilistic models [@araki1994evaluation]. Since nowadays neural networks enjoy a large awareness as an element of software infrastructure, with actively maintained packages readily available, their evaluation seems to be the most practically useful. We used the PyTorch [@Paszke2017AutomaticDI] implementation of LSTM in particular.
The bidirectional version [@Schuster:1997:BRN:2198065.2205129] of LSTM reads the character chains forward and backwards at the same time. Predictions from networks running in both directions are averaged.
In order to provide the network an additional, broad picture peek at the whole error form we also evaluated a setup where the internal state of LSTM cells, instead of being initialized randomly, is computed from an ELMo embedding [@DBLP:journals/corr/abs-1802-05365] of the token. The ELMo embedder is capable of integrating linguistic information carried by the whole form (probably often not much in case of errors), as well as the string as a character chain. The latter is processed with a convolutional neural network. How this representation is constructed is informed by the whole corpus on which the embedder was trained. The pretrained ELMo model that we used [@Che2018TowardsBU] was trained on Wikipedia and Common Crawl corpora of Polish.
The ELMo embedding network outputs three layers as matrices, which are supposed to reflect subsequent compositional layers of language, from phonetic phenomena at the bottom to lexical ones at the top. A weighted sum of these layers is computed, with weights trained along with the LSTM error-correcting network. Then we apply a trained linear transformation, followed by $\operatorname{ReLU}$ non-linearity: $$\operatorname{ReLU}(x) = \max (0, x)$$ (applied cellwise) in order to obtain the initial setting of parameters for the main LSTM. Our ELMo-augmented LSTM is bidirectional.
**Method** **Accuracy** **Perplexity** **Loss (train)** **Loss (test)**
-------------------- -------------- ---------------- ------------------ -----------------
Edit distance 0.3453 - - -
Diacritic swapping 0.2279 - - -
Vector distance 0.3945 - - -
LSTM-1 net 0.4183 **907** 0.3 0.41
LSTM-2 net 0.6634 11182 0.1 **0.37**
LSTM-ELMo net **0.6818** 706166 **0.07** 0.38
**Layer I** **Layer II** **Layer III**
------------- -------------- ---------------
0.036849 0.08134 0.039395
: Discovered optimal weights for summing layers of ELMo embedding for initializing an error-correcting LSTM. The layers are numbered from the one that directly processes character and word input to the most abstract one.[]{data-label="table:elmo_weights"}
Experimental setup
==================
PlEWi [@grundkiewicz:automatic] is an early version of WikEd [@Grundkiewicz2014TheWE] error corpus, containing error type annotations allowing us to select only non-word errors for evaluation. Specifically, PlEWi supplied 550,755 \[error, correction\] pairs, from which 298,715 were unique. The corpus contains data extracted from histories of page versions of Polish Wikipedia. An algorithm designed by the corpus author determined where the changes were correcting spelling errors, as opposed to expanding content and disagreements among Wikipedia editors.
The corpus features texts that are descriptive rather than conversational, contain relatively many proper names and are more likely to have been at least skimmed by the authors before submitting for online publication. Error cases provided by PlEWi are, therefore, not a balanced representation of spelling errors in written Polish language. PlEWi does have the advantage of scale in comparison to existing literature, such as [@ogr:kop:17] operating on a set of only 740 annotated errors in tweets.
All methods were tested on a test subset of 25% of cases, with 75% left for training (where needed) and 5% for development.
The methods that required training – namely recurrent neural networks – had their loss measured as cross-entropy loss measure between correct character labels and predictions. This value was minimized with Adam algorithm [@DBLP:journals/corr/KingmaB14]. The networks were trained for 35 epochs.
Results
=======
The experimental results are presented in Table \[table:results\]. Diacritic swapping showed a remarkably poor performance, despite promising mentions in existing literature. This might be explained by the already mentioned feature of Wikipedia edits, which can be expected to be to some degree self-reviewed before submission. This can very well limit the number of most trivial mistakes.
On the other hand, the vector distance method was able to bring a discernible improvement over pure Levenshtein distance, comparable even with the most basic LSTM. It is possible that assigning more fine-tuned weights to edit distance and semantic distance would make the quality of predictions even higher. The idea of using vector space measurements explicitly can be also expanded if we were to consider the problem of contextualizing corrections. For example, the semantic distance of proposed corrections to the nearest words is likely to carry much information about their appropriateness. Looking from another angle, searching for words that seem semantically off in context may be a good heuristic for detecting errors that are not nonword (that is, they lead to wrong forms appearing in text which are nevertheless in-vocabulary).
The good performance of recurrent network methods is hardly a surprise, given observed effectiveness of neural networks in many NLP tasks in the recent decade. It seems that bidirectional LSTM augmented with ELMo may already hit the limit for correcting Polish spelling errors without contextual information. While it improves accuracy in comparison to LSTM initialized withrandom noise, it makes the test cross-entropy slightly worse, which hints at overfitting. The perplexity measures actually increase sharply for more sophisticated architectures. Perplexity should show how little probability is assigned by the model to true answers. We measure it as
$$\operatorname{perplexity}(P, x) = 2^{-\frac{1}{N}\sum_{i \leqslant N} \log P(x_i)},$$
where $x$ is a sequence of $N$ characters, forming the correct version of the word, and $P(x_i)$ is the estimated probability of the $i$th character, given previous predicted characters and the incorrect form. The observed increase of perplexity for increasingly accurate models is most likely due to more refined predicted probability distributions, which go beyond just assigning the bulk of probability to the best answer.
Interesting insights can be gained from weights assigned by optimization to layers of ELMo network, which are taken as the word form embedding (Table \[table:elmo\_weights\]). The first layer, and the one that is nearest to input of the network, is given relatively the least importance, while the middle one dominates both others taken together. This suggests that in error correction, at least for Polish, the middle level of morphemes and other characteristic character chunks is more important than phenomena that are low-level or tied to some specific words. This observation should be taken into account in further research on practical solutions for spelling correction.
Conclusion
==========
Among the methods tested the bidirectional LSTM, especially initialized by ELMo embeddings, offers the best accuracy and raw performance. Adding ELMo to a straightforward PyTorch implementation of LSTM may be easier now than at the time of performing our tests, as since then the authors of ELMoForManyLangs package [@Che2018TowardsBU] improved their programmatic interface. However, if a more interpretable and explainable output is required, some version of vector distance combined with edit distance may be the best direction. It should be noted that this method produces multiple candidate corrections with their similarity scores, as opposed to only one “best guess“ correction that can be obtained from a character-based LSTM. This is important in applications where it is up to humans to the make the final decision, and they are only to be aided by a machine.
It is desirable for further reasearch to expand the corpus material into a wider and more representative set of texts. Nevertheless, the solution for any practical case has to be tailored to its characteristic error patterns. Works on language correction for English show that available corpora can be ”boosted” [@Ge2018ReachingHP], i.e. expanded by generating new errors consistent with a generative model inferred from the data. This may greatly aid in developing models that are dependent on learning from error corpora.
A deliberate omission in this paper are the elements accompanying most real-word error correction solutions. Some fairly obvious approaches to integrating evidence from context include n-grams and Markov chains, although the possibility of using measurements in spaces of semantic vectors was already mentioned in this article. Similarly, non-word errors can be easily detected with comparing tokens against reference vocabulary, but in practice one should have ways of detecting mistakes masquerading as real words and fixing bad segmentation (tokens that are glued together or improperly separated). Testing how performant are various methods for dealing with these problems in Polish language is left for future research.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Consider a one dimensional diffusion process on the diffusion interval $I$ originated in $x_0\in I$. Let $a(t)$ and $b(t)$ be two continuous functions of $t$, $t>t_0$ with bounded derivatives and with $a(t)<b(t)$ and $a(t),b(t)\in I$, $\forall t>t_0$. We study the joint distribution of the two random variables $T_a$ and $T_b$, first hitting times of the diffusion process through the two boundaries $a(t)$ and $b(t)$, respectively. We express the joint distribution of $T_a, T_b$ in terms of $P(T_a<t,T_a<T_b)$ and $P(T_b<t,T_a>T_b)$ and we determine a system of integral equations verified by these last probabilities. We propose a numerical algorithm to solve this system and we prove its convergence properties. Examples and modeling motivation for this study are also discussed.'
title: |
Joint densities of first hitting times of a\
diffusion process through two time dependent boundaries
---
Introduction {#Sect:1}
============
Exit times of diffusion processes from a strip play an important role in a variety of application ranging from computer science to engineering, from biology to metrology or finance (cf. [@Giorno; @DL; @NFK; @PTZ; @Sm]). According to the features of the model, constant or time dependent thresholds may bound the considered process. Typical examples are quality models with two tolerance bands. Some parameter may control exit times from the strip, with different effects on the exit time from the upper or the lower bound. The knowledge of the joint exit times pdf clarifies the role of these parameters. Another example is given by the survival probability of a population in a finite capacity environment or by tumor growth models (cf. [@Giorno]). Similar problems arise in metrology when we need to maintain the atomic clock error bounded by two tolerance bands. Moreover, avoiding an excessive increase of the error is of primary importance to improve GPS instruments (cf. [@GTSZ]). In this setting the knowledge of the relationship between exit times from the upper and the lower boundary may suggest improvements to the clock reliability by acting on some parameters of the model involved in the joint distribution. Other possible applications can be found in finance where the interest focuses on the dependency between the times to sell or buy options when the level of gain or loss is preassigned. A large literature exists for the study of the first passage time of one dimensional diffusion processes through a boundary and analytical, numerical and simulation methods have been studied both for the direct (cf. [@BNR; @BSZ; @GS; @GS1; @GSZ; @Pe-1]) and the inverse problem (cf. [@ZucSac]). However, most of these papers focuses on the one boundary problem, while for the two boundary case the few analytical results published rely either on the Brownian motion (cf. [@Ors]) or particular time dependent boundaries, corresponding to special symmetries, for specific diffusions (cf. [@BGNR; @DGNR]). The existing results generally focus on the first exit time from the strip, while our interest lies in the joint distribution of the times when the process first attains the upper and the lower boundary, respectively. This paper aims to cover this subject considering the joint distribution between these times. Some results, presented in a recent paper [@GNR], are related with those on the Laplace transforms presented in this paper. However their focus is not the joint distribution of exit times from a strip.
The notation and the existing results that will be used in this paper are introduced in Section 2, while Sections 3 and 4 are devoted to the presentation of our results. In Section 3 we determine the expression of the joint distribution of the exit times from the upper and the lower boundary. The results are expressed in terms of first hitting time through a single boundary and of the probability of crossing the upper (lower) boundary for the first time at some instant preceding $t$ before crossing the lower (upper) boundary. Note that these probabilities are generally unknown. We prove then that they are the unique solution of a system of Volterra integral equations of the first kind. We also show that there exists an equivalent system of Volterra equations of the second type. When the boundaries are constant the Laplace transform method can be applied to solve the system, since the integrals of such system are of convolution type. Here we introduce three equivalent representations of the Laplace transform. In the case of the Brownian motion and constant boundaries a closed form expression for the joint distribution of the exit times from a strip is known (cf. [@Borod]).
In Section 4 we propose a numerical scheme for the solution of the system of integral equations and we determine the order of convergence. This method works for both constant and time dependent boundaries. In the case of two constant boundaries the Laplace transforms (cf. [@Abate]) of the probability of crossing the upper (lower) boundary for the first time at some instant preceding $t$ before crossing the lower (upper) boundary can be numerically inverted. Finally in Section 5, we present a set of examples.
Mathematical Background and Notations {#Sect:2}
=====================================
Let $X=\{X(t),t\geq t_0\}$ be a one-dimensional regular time homogeneous diffusion process defined on a suitable probability space $(\Omega, \mathcal{A}, \mathbb{P})$ such that $P(X(t_0)=x_0)=1$ and with diffusion interval $I$, where $I$ is an interval of the form $(r_1,r_2)$, $(r_1,r_2]$, $[r_1,r_2)$ or $[r_1,r_2]$ where $r_1=-\infty$ and/or $r_2=+\infty$ are admissible when the diffusion interval is open. If not specified, the diffusion interval is open and the endpoints $r_1$ and $r_2$ are natural boundaries. Let $F_{X(t)}(x|y,\tau)=P(X(t)\leq x|X(\tau)=y)$ be the transition probability distribution function (pDf) of the process $X$ and let $f_{X(t)}(x|y,\tau)$ be the corresponding transition probability density function (pdf).
Let $a(t)$ be a continuous functions with bounded derivatives. We denote as $T_a$ the first hitting time of the stochastic process $X$ across a boundary $a(t)\in I$ $$\label{eq:2.5}
T_a=\inf\{t\geq t_0,X(t)=a(t)\}$$ Its pDf is $$F_{T_a}(t|x_0,t_0)=P(T_a\leq t|X(t_0)=x_0)$$ and $f_{T_a}(t|x_0,t_0)$ is the corresponding pdf. In the case of two boundaries $a(t)<b(t)$, $\forall t$, we indicate with $T_a$ and $T_b$ the first hitting times of the stochastic process $X$ across the boundaries $a(t)$ and $b(t)$ respectively. Aim of this paper is to study the dependency properties of $(T_a,T_b)$, i.e. to determine $$F_{T_a,T_b}(t,s|x_0,t_0)=P(T_a\leq t, T_b\leq s|X(t_0)=x_0),$$ the joint pDf of $(T_a,T_b)$ and the corresponding joint pdf $f_{T_a,T_b}(t,s|x_0,t_0)$.
We define the following densities that distinguish the first boundary reached between the two ones delimiting the strip $$\label{ga}
g_{a}(t|x_0,t_0)dt=P(T_a\in dt, T_a<T_{b}|X(t_0)=x_0)$$ and $$\label{gb}
g_{b}(t|x_0,t_0)dt=P(T_b\in dt, T_a>T_{b}|X(t_0)=x_0).$$
For a standard Brownian motion $W=\{W(t),t\geq t_0\}$ the two densities $g_{a}(t|x_0,t_0)$ and $g_{b}(t|x_0,t_0)$ are known in closed form (cf. [@Borod]) when the boundaries are constant $a(t)=a$ and $b(t)=b$ $$\begin{aligned}
\label{Wiener}
g_{a}(t|x_0,t_0)&=&\sum_{k=-\infty}^{\infty} \frac{x_0-a+2k(b-a)}{\sqrt{2\pi (t-t_0)^3}}e^{-\frac{(x_0-a+2k(b-a))^2}{2(t-t_0)}}\\
g_{b}(t|x_0,t_0)&=&\sum_{k=-\infty}^{\infty}\frac{b-x_0+2k(b-a)}{\sqrt{2\pi (t-t_0)^3}}e^{-\frac{(b-x_0+2k(b-a))^2}{2(t-t_0)}}\nonumber\end{aligned}$$ and the pdf and pDf of $T_a$ are $$\begin{aligned}
\label{T_a pdf}
f_{T_a}(t|x_0,t_0)&=&\frac{|a-x_0|}{\sqrt{2\pi (t-t_0)^3}}e^{-\frac{(a-x_0)^2}{2(t-t_0)}}\\
F_{T_a}(t|x_0,t_0)&=&1-\text{erf}\left(\frac{|a-x_0|}{\sqrt{2t}}\right).\end{aligned}$$
The quantities (\[ga\]) and (\[gb\]) are useful for the computation of the joint density function of $T_a$ and $T_b$. Two different instances arise according to the location of the starting point $X(t_0)=x_0$ with respect to the boundaries (cf. Figure \[Fig:2cases\]).
![Sample path of a stochastic process through two constant boundaries $a$ and $b$: i) $x_0\in (a,b)$; ii) $x_0\notin (a,b)$.[]{data-label="Fig:2cases"}](Fig1.pdf){height="6cm"}
It holds:
\[lem\_general\] Let $X=\{X(t),t\geq t_0\}$ be a diffusion process such that $X(t_0)=x_0$ and let $a(t)$ and $b(t)$ be two continuous time dependent boundaries.
1. If $x_0<a(t_0)<b(t_0)$ and $a(t)<b(t)$ for each $t>t_0$ or $b(t_0)<a(t_0)<x_0$ and $b(t)<a(t)$ for each $t>t_0$, then $$\label{densit1}
f_{T_{a},T_{b}}\left(t,s|x_0,t_0\right)=\left\{
\begin{array}{ll}
0 &t\geq s \\
f_{T_{a}}(t|x_0,t_0)f_{T_{b}}(s|a(t),t) &t<s \\
\end{array} .\right.$$
2. If $a(t_0)<x_0<b(t_0)$ and $a(t)<b(t)$ for each $t>t_0$, then $$\label{densit2}
f_{T_{a} T_{b}}\left(t,s|x_0,t_0\right)=\left\{
\begin{array}{ll}
f_{T_{b}}(s|a(t),t) g_{a}(t|x_0,t_0) &t<s \\
0 &t=s \\
f_{T_{a}}(t|b(s),s) g_{b}(s|x_0,t_0) &t>s \\
\end{array}
.\right.$$
We omit the proof that is straightforward using the strong Markov property.
Note that the FPT pdf verifies the initial condition $$\begin{aligned}
\label{limit}
\lim_{s\rightarrow t} f_{T_{b}}(s|a(t),t)=\lim_{t\rightarrow s}f_{T_{a}}(t|b(s),s)=0.\end{aligned}$$ Furthermore, due to the differentiability of the boundaries, it holds (cf. [@RSS]) $$\begin{aligned}
\lim_{t\rightarrow s}F_{X(t)}(b(t)|b(s),s)&=&\lim_{t\rightarrow s}F_{X(t)}(a(t)|a(s),s)=\frac{1}{2}\label{limit Fa}\\
\lim_{t\rightarrow s}[1-F_{X(t)}(b(t)|a(s),s)]&=&\lim_{t\rightarrow s}F_{X(t)}(a(t)|b(s),s)=0\label{limit Fb}.\end{aligned}$$
For some applications it might be interesting to determine the copula function (cf.[@Nelsen]) between $T_a$ and $T_b$. When the two densities $g_{a}(t|x_0,t_0)$ and $g_{b}(t|x_0,t_0)$ are known, use of (\[densit1\]) and (\[densit2\]) allows to determine this function.
System of integral equations {#Sect:3}
============================
The computation of $f_{T_{a} T_{b}}\left(t,s|x_0,t_0\right)$ involves the transition pdfs $f_{T_{b}}(s|a(t),t)$ and $f_{T_{a}}(t|b(s),s)$ and the terms $g_{a}(t|x_0,t_0)$ and $g_{b}(t|x_0,t_0)$. When the process is a linear regular diffusion, the transition pdf is available in closed form and, if the process is strictly linear, it is Gaussian. In the literature, transition pdf is also available for other regular diffusion processes, such as the Cox-Ingersoll-Ross model (also known as Feller process), the Bessel process or some instances of the Raleigh process (cf. [@GS; @GNRS]). Further examples arise from space-time transformation of the Brownian motion (cf. [@R1]) or of the Cox-Ingersol-Ross process (cf. [@CR]). When closed form solutions are not available, the transition pdf is evaluated resorting to numerical methods, such as the numerical solution of the Kolmogorov equation [@LP; @Sm1] or the numerical inversion of Fourier transforms [@VL]. Unfortunately closed form expressions for the densities $g_{a}(t|x_0,t_0)$ and $g_{b}(t|x_0,t_0)$ are known only for the Brownian motion with constant boundaries (cf. [@Borod] formula 3.0.6) or for processes related to it through suitable transformations. Use of the following theorem helps to overcome this problem.
Let $X=\{X(t),t\geq t_0\}$ be a diffusion process such that $X(t_0)=x_0$. Let $a(t)$ and $b(t)$ be two time dependent boundaries with bounded derivatives such that $a(t_0)<x_0<b(t_0)$ and $a(t)<b(t)$ for each $t>t_0$. The pdf’s $g_{a}(t|x_0,t_0)$ and $g_{b}(t|x_0,t_0)$ are solution of the following system of Volterra first kind integral equations
\[Fortet2\] $$\begin{aligned}
1-F_{X(t)}(b(t)|x_0,t_0)&=\int_{t_0}^t \left[1-F_{X(t)}(b(t)|a(\tau),\tau)\right]g_{a}(\tau|x_0,t_0) d\tau\label{Fortet2a}\\
&+\int_{t_0}^t \left[1-F_{X(t)}(b(t)|b(\tau),\tau)\right]g_{b}(\tau|x_0,t_0)d\tau \nonumber\\
F_{X(t)}(a(t)|x_0,t_0)&=\int_{t_0}^t F_{X(t)}(a(t)|a(\tau),\tau)g_{a}(\tau|x_0,t_0) d\tau\label{Fortet2b}\\
&+ \int_{t_0}^t F_{X(t)}(a(t)|b(\tau),\tau)g_{b}(\tau|x_0,t_0)d\tau.\nonumber\end{aligned}$$
For $t\in [0,\infty]$, conditioning on the boundary first attained by the process, for $x\notin[a(t),b(t)]$ we get $$\begin{aligned}
\nonumber
P\left(X(t)\leq x|X(t_0)=x_0\right)
&=& \int^{t}_{t_0}P\left(X(t)\leq x|T_{a}<T_{b},T_{a}=\tau,X(t_0)=x_0\right)g_{a}(\tau|x_0,t_0) d\tau\nonumber\\
&+&\int^{t}_{t_0}P\left(X(t)\leq x|T_{a}>T_{b},T_{b}=\tau,X(t_0)=x_0\right)g_{b}(\tau|x_0,t_0)d\tau\nonumber\\
&=& \int^{t}_{t_0}P\left(X(t)\leq x|X(\tau)=a(\tau),X(t_0)=x_0\right)g_{a}(\tau|x_0,t_0) d\tau\nonumber\\
&+&\int^{t}_{t_0}P\left(X(t)\leq x|X(\tau)=b(\tau),X(t_0)=x_0\right)g_{b}(\tau|x_0,t_0)d\tau\nonumber.\end{aligned}$$ Differentiating with respect to $x$ we obtain $$\begin{aligned}
\label{Fortetdensity}
f_{X(t)}(x|x_0,t_0)&=&\int_{t_0}^t f_{X(t)}(x|a(\tau),\tau)g_{a}(\tau|x_0,t_0) d\tau\\
&+& \int_{t_0}^t f_{X(t)}(x|b(\tau),\tau)g_{b}(\tau|x_0,t_0)d\tau.\nonumber\end{aligned}$$ Integrating (\[Fortetdensity\]) with respect to $x$ on the two subdomains $[b(t),\infty]$ and $[-\infty,a(t)]$ respectively, we get (\[Fortet2a\]) and (\[Fortet2b\]).
Differentiating (\[Fortet2a\]) and (\[Fortet2b\]) with respect to $t$ and recalling and one gets
\[Fortet2\_2\] $$\begin{aligned}
g_{b}(t|x_0,t_0)&=-2\frac{\partial F_{X(t)}(b(t)|x_0,t_0)}{\partial t}\label{Fortet2a_2}\\
&+\int_{t_0}^t 2\left(\frac{\partial F_{X(t)}(b(t)|a(\tau),\tau)}{\partial t}g_{a}(\tau|x_0,t_0)+ \frac{\partial F_{X(t)}(b(t)|b(\tau),\tau)}{\partial t}g_{b}(\tau|x_0,t_0)\right)d\tau.\nonumber\\
g_{a}(t|x_0,t_0)&=2\frac{\partial F_{X(t)}(a(t)|x_0,t_0)}{\partial t}\label{Fortet2b_2}\\
&-\int_{t_0}^t 2\left(\frac{\partial F_{X(t)}(a(t)|a(\tau),\tau)}{\partial t}g_{a}(\tau|x_0,t_0)+ \frac{\partial F_{X(t)}(a(t)|b(\tau),\tau)}{\partial t}g_{b}(\tau|x_0,t_0)\right)d\tau.\nonumber
\end{aligned}$$
This system of Volterra integral equations coincides with the one proposed in [@BGNR] if the kernel of the two equations is regularized.
It holds
The system (\[Fortet2\]) has a unique continuous solution for $t>t_0$.
The system of Volterra integral equations of the first kind (\[Fortet2\]) is equivalent to the system of Volterra integral equations of the second kind (\[Fortet2\_2\]) that can be written in matricial form $$\text{\bfseries{g}}(t)=\text{\bfseries{h}}(t)+\int_{t_0}^t \text{\bfseries{k}}(t,\tau)\text{\bfseries{g}}(\tau) d\tau$$ where $$\text{\bfseries{g}}(t)=\left[
\begin{array}{c}
g_{a}(t|x_0,t_0) \\
g_{b}(t|x_0,t_0)\end{array}
\right],$$ $$\text{\bfseries{h}}(t)=\left[
\begin{array}{c}
-2\frac{\partial F_{X(t)}(b(t)|x_0,t_0)}{\partial t} \\
2\frac{\partial F_{X(t)}(a(t)|x_0,t_0)}{\partial t}\end{array}
\right],$$ $$\text{\bfseries{k}}(t,\tau)=\left[
\begin{array}{cc}
-2\frac{\partial F_{X(t)}(b(t)|a(\tau),\tau)}{\partial t} & -2\frac{\partial F_{X(t)}(b(t)|b(\tau),\tau)}{\partial t}\\
2\frac{\partial F_{X(t)}(a(t)|a(\tau),\tau)}{\partial t}& 2\frac{\partial F_{X(t)}(a(t)|b(\tau),\tau)}{\partial t}
\end{array}
\right],$$ Since the kernel $\text{\bfseries{k}}(t,\tau)$ is singular in $\tau=t$, we introduce an equivalent system with continuous kernel. Mimicking the method presented in [@BNR], we introduce two couple of functions $\gamma_i(t)$ and $\eta_i(t)$, $i=1,2$, continuous in $[t_0,+\infty]$. Combining (\[Fortet2\]), (\[Fortetdensity\]) and (\[Fortet2\_2\]), together with $\gamma_i(t)$ and $\eta_i(t)$ we obtain $$\begin{aligned}
\label{eqconKR}
g_{b}(t|x_0,t_0)=-\Psi^1(b(t)|x_0,t_0)+\int_{t_0}^t \left( \Psi^1(b(t)|a(\tau),\tau)g_{a}(\tau|x_0,t_0)+ \Psi^1(b(t)|b(\tau),\tau)g_{b}(\tau|x_0,t_0)\right)d\tau.\nonumber\\
g_{a}(t|x_0,t_0)=\Psi^2(a(t)|x_0,t_0)-\int_{t_0}^t \left( \Psi^2(a(t)|a(\tau),\tau)g_{a}(\tau|x_0,t_0)+ \Psi^2(a(t)|b(\tau),\tau)g_{b}(\tau|x_0,t_0)\right)d\tau.\end{aligned}$$ where $$\begin{aligned}
\label{Psi}
\Psi^1(b(t)|x,s)&=&-2\frac{\partial F_{X(t)}(b(t)|x,s)}{\partial t}+\gamma_1(t)f_{X(t)}(b(t),t|x,s)+\eta_1(t) [1-F_{X(t)}(b(t)|x,s)]\nonumber\\
\Psi^2(a(t)|x,s)&=&2\frac{\partial F_{X(t)}(a(t)|x,s)}{\partial t}-\gamma_2(t)f_{X(t)}(a(t),t|x,s)-\eta_2(t) F_{X(t)}(a(t)|x,s)\end{aligned}$$
A suitable choice of $\gamma_i(t)$ and $\eta_i(t)$, $i=1,2$, makes $\Psi^1(b(t)|b(\tau),\tau)$ and $\Psi^2(a(t)|a(\tau),\tau)$ not singular. On the other hand, since (cf. [@BGNR]) $$\begin{aligned}
\lim_{\tau\rightarrow t}f_{X(t)}(b(t),t|a(\tau),\tau)&=\left.\lim_{\tau\rightarrow t}\frac{\partial}{\partial x}f_{X(t)}(x,t|a(\tau),\tau)\right|_{x=b(t)}=0\\
\lim_{\tau\rightarrow t}f_{X(t)}(a(t),t|b(\tau),\tau)&=\left.\lim_{\tau\rightarrow t}\frac{\partial}{\partial x}f_{X(t)}(x,t|b(\tau),\tau)\right|_{x=a(t)}=0\nonumber,\end{aligned}$$ the kernels $\Psi^1(b(t)|a(\tau),\tau)$ and $\Psi^2(a(t)|b(\tau),\tau)$ are not singular. This makes possible to apply Theorem 3.11 of [@Li] to get the thesis.
The functions $\gamma_i(t)$ and $\eta_i(t)$, $i=1,2$ can be determined. For example, for an Ornstein Uhlenbeck process characterized by the drift $\mu(t,x)=\alpha x+\beta$ and infinitesimal variance $\sigma(t,x)=\sigma$, where $\alpha$, $\beta$ and $\sigma>0$ are arbitrary real constants. The functions that regularize the kernels are $\gamma_1(t)=1/2[\alpha b(t)+\beta -b'(t)]$, $\gamma_2(t)=1/2[\alpha a(t)+\beta -a'(t)]$ and $\eta_i(t)\equiv0$, $i=1,2$ (cf. [@BNR]).
When the boundaries are constant it holds:
\[Cor Laplace\] Let $X=\{X(t),t\geq t_0\}$ be a diffusion process such that $X(t_0)=x_0$ and let $a$ and $b$ be two constant boundaries such that $a<x_0<b$, then the following three expressions are equivalent for $g_{a}^{\lambda}(x_0)=\int_{t_0}^{+\infty}e^{-\lambda t}g_{a}(t|x_0,t_0)dt$ and $g_{b}^{\lambda}(x_0)=\int_{t_0}^{+\infty}e^{-\lambda t}g_{b}(t|x_0,t_0)dt$:
\[LaplaceItoMcKean\] $$\begin{aligned}
g_{a}^{\lambda}(x_0)&=\frac{f_{T_b}^{\lambda}(x_0)f_{T_a}^{\lambda}(b)-f_{T_a}^{\lambda}(x_0)}{f_{T_a}^{\lambda}(b)f_{T_b}^{\lambda}(a)-1} \label{LaplaceOtoMcKeana}\\
g_{b}^{\lambda}(x_0)&=\frac{f_{T_a}^{\lambda}(x_0)f_{T_b}^{\lambda}(a)-f_{T_b}^{\lambda}(x_0)}{f_{T_a}^{\lambda}(b)f_{T_b}^{\lambda}(a)-1} \label{LaplaceOtoMcKeanb}\end{aligned}$$
\[LaplaceFortet2\] $$\begin{aligned}
g_{a}^{\lambda}(x_0)&=\frac{\left[1-\lambda F_X^{\lambda}(b|x_0)\right]F_X^{\lambda}(a|b)-\left[1-\lambda F_X^{\lambda}(b|b)\right]F_X^{\lambda}(a|x_0)}{\left[1-\lambda F_X^{\lambda}(b|a)\right]F_X^{\lambda}(a|b)-\left[1-\lambda F_X^{\lambda}(b|b)\right]F_X^{\lambda}(a|a)} \label{LaplaceFortet2a}\\
g_{b}^{\lambda}(x_0)&=\frac{\left[1-\lambda F_X^{\lambda}(b|x_0)\right]F_X^{\lambda}(a|a)-\left[1-\lambda F_X^{\lambda}(b|a)\right]F_X^{\lambda}(a|x_0)}{\left[1-\lambda F_X^{\lambda}(b|b)\right]F_X^{\lambda}(a|a)-\left[1-\lambda F_X^{\lambda}(b|a)\right]F_X^{\lambda}(a|b)} \label{LaplaceFortet2b}\end{aligned}$$
\[Laplacedens\] $$\begin{aligned}
g_{a}^{\lambda}(x_0)&=\frac{f_X^{\lambda}(x_1|x_0)f_X^{\lambda}(x_2|b)-f_X^{\lambda}(x_1|b)f_X^{\lambda}(x_2|x_0)}{f_X^{\lambda}(x_1|a)f_X^{\lambda}(x_2|b)-f_X^{\lambda}(x_1|b)f_X^{\lambda}(x_2|a)}\nonumber\\
&=\frac{v_1(\alpha,x_0)v_2(\alpha,b)-v_2(\alpha,x_0)v_1(\alpha,b)}{v_1(\alpha,a)v_2(\alpha,b)-v_2(\alpha,a)v_1(\alpha,b)} \label{Laplacedensa}\\
g_{b}^{\lambda}(x_0)&=\frac{f_X^{\lambda}(x_1|a)f_X^{\lambda}(x_2|x_0)-f_X^{\lambda}(x_1|x_0)f_X^{\lambda}(x_2|a)}{f_X^{\lambda}(x_1|a)f_X^{\lambda}(x_2|b)-f_X^{\lambda}(x_1|b)f_X^{\lambda}(x_2|a)}\nonumber\\
&=\frac{v_1(\alpha,a)v_2(\alpha,x_0)-v_2(\alpha,a)v_1(\alpha,x_0)}{v_1(\alpha,a)v_2(\alpha,b)-v_2(\alpha,a)v_1(\alpha,b)} \label{Laplacedensb}\end{aligned}$$
where $$\begin{aligned}
F_X^{\lambda}(x|x_0)&=&\int_0^{+\infty}e^{-\lambda t}F_{X(t)}(x|x_0,t_0)dt\\
f_X^{\lambda}(x|x_0)&=&\int_0^{+\infty}e^{-\lambda t}f_{X(t)}(x|x_0,t_0)dt\\
f_{T_a}^{\lambda}(x_0)&=&\int_0^{+\infty}e^{-\lambda t}F_{T_a}(t|x_0,t_0)dt
\end{aligned}$$ and the functions $v_i(\alpha,x)$, $i=1,2$ are fundamental solutions of (8.13b) in [@RicSat90].
Generalizing the standard calculation of p. 30 in [@IM] for an arbitrary regular diffusion we obtain (\[LaplaceItoMcKean\]).
Applying Laplace transform to (\[Fortet2\]) and using the convolution theorem we get (\[LaplaceFortet2\]), a result recently published in [@GNR].
Applying Laplace transform to (\[Fortetdensity\]) together with the convolution theorem for two generic points $x_1>b$ and $x_2<a$ we get the first equality in (\[Laplacedens\]). The use of (8.22) in [@RicSat90] allows to get the second equality.
Furthermore, recalling that (cf. [@R]) $$f_{T_a}^{\lambda}(x_1|x_0)=\frac{f_X^{\lambda}(x|x_0)}{f_X^{\lambda}(x|a)},$$ the first equality in (\[Laplacedens\]) becomes (\[LaplaceItoMcKean\]).
Finally, remembering that (cf. [@R]) $$\begin{aligned}
f_{T_a}^{\lambda}(x_1|x_0)=\left\{
\begin{array}{lll}
\frac{1-\lambda F_X^{\lambda}(a|x_0)}{1-\lambda F_X^{\lambda}(a|a)} &\text{if} &a>x_0\\
\frac{F_X^{\lambda}(a|x_0)}{F_X^{\lambda}(a|a)} &\text{if} &a<x_0
\end{array}\right.\end{aligned}$$ the equations (\[LaplaceItoMcKean\]) become (\[LaplaceFortet2\]). This implies that the three formulations are equivalent.
The above results also hold for diffusion processes bounded by one or two reflecting boundaries when the diffusion interval is characterized by non natural boundaries, i.e. for Cox-Ingersoll-Ross whose diffusion interval is $I=[0,\infty)$ or for the reflected Brownian Motion.
Algorithms for $P(T_a\in dt,T_a<T_b)$ and $P(T_b\in dt,T_a>T_b)$ {#Sect:4}
================================================================
In this section we describe two approaches to determine the density functions $g_{a}(t|x_0,t_0)$ and $g_{b}(t|x_0,t_0)$.
When the boundaries are constant the densities $g_{a}(t|x_0,t_0)$ and $g_{b}(t|x_0,t_0)$ are obtained from the Laplace transforms (\[LaplaceFortet2\]) by inverting them numerically using, for example, Euler method [@Abate].
Alternative methods become necessary when the boundaries $a(t)$ and $b(t)$ are time dependent or when the Laplace inversion presents numerical difficulties. For example in the case of the Ornstein Uhlenbeck process the expression of $g_{a}^{\lambda}(x_0)$ and $g_{b}^{\lambda}(x_0)$ involve parabolic cylinder function (cf. [@GS1]). Their numerical inversion requests efforts specific for this instance. Furthermore there are processes for which $F_X^{\lambda}(a|x_0)$ and $F_X^{\lambda}(b|x_0)$ are not known in the literature. Their computation is possible however it requests the solution of specific second order differential equations (cf. [@R]).
Here we propose the following numerical method that can be applied both for constant and time depending boundaries. Let us introduce a time discretization $t_i=t_0+ih$, $i=1,2,\ldots$ where $h$ is a positive constant. To determine the two pdf’s $g_{a}(t|x_0,t_0)$ and $g_{b}(t|x_0,t_0)$ at the finite set of knots $t_i$ for $i=1,\ldots,n$, we use Euler method [@Li] to approximate the integrals on the r.h.s. of (\[Fortet2a\]) and (\[Fortet2b\]). Hence we get
\[Fortet2discret\] $$\begin{aligned}
1-F_{X(t_i)}(b(t_i)|x_0,t_0)&=\sum_{j=1}^i \left[1-F_{X(t_i)}(b(t_i)|a(t_j),t_j)\right]\hat{g}_{a}(t_j|x_0,t_0) h\label{Fortet2discreta}\\
&+\sum_{j=1}^i \left[1-F_{X(t_i)}(b(t_i)|b(t_j),t_j)\right]\hat{g}_{b}(t_j|x_0,t_0)h \nonumber\\
F_{X(t_i)}(a(t_i)|x_0,t_0)&=\sum_{j=1}^i F_{X(t_i)}(a(t_i)|a(t_j),t_j)\hat{g}_{a}(t_j|x_0,t_0) h\label{Fortet2discretb}\\
&+ \sum_{j=1}^i F_{X(t_i)}(a(t_i)|b(t_j),t_j)\hat{g}_{b}(t_j|x_0,t_0)h.\nonumber
\end{aligned}$$
The densities $g_{a}(t|x_0,t_0)$ and $g_{b}(t|x_0,t_0)$ can be evaluated in the knots $t_i$ for $i=1,\ldots,n$ by means of the following algorithm.
[**Step 1**]{}
\[step1\] $$\begin{aligned}
\hat{g}_{b}(t_1|x_0,t_0)&=\frac{2(1-F_{X(t_1)}(b(t_1)|x_0,t_0))}{h}\\
\hat{g}_{a}(t_1|x_0,t_0)&=\frac{2F_{X(t_1)}(a(t_1)|x_0,t_0)}{h}.\end{aligned}$$
[**Step $i$, $i=2,3,\ldots$**]{}
\[step i 1\] $$\begin{aligned}
\hat{g}_{b}(t_i|x_0,t_0)&=\frac{2(1-F_{X(t_i)}(b(t_i)|x_0,t_0))}{h}\\
&-2\sum _{j=1}^{i-1}\left[(1-F_{X(t_i)}(b(t_i)|a(t_j),t_j))\hat{g}_{a}(t_j|x_0,t_0)\right.\nonumber\\
&+\left. (1-F_{X(t_i)}(b(t_i)|b(t_j),t_j))\hat{g}_{b}(t_j|x_0,t_0) \right]\nonumber\\
\hat{g}_{a}(t_i|x_0,t_0)&=\frac{2F_{X(t_i)}(a(t_i)|x_0,t_0)}{h}\\
&-2\sum _{j=1}^{i-1}\left[F_{X(t_i)}(a(t_i)|a(t_j),t_j)\hat{g}_{a}(t_j|x_0,t_0)\right.\nonumber\\
&+ \left.F_{X(t_i)}(a(t_i)|b(t_j),t_j)\hat{g}_{b}(t_j|x_0,t_0) \right]\nonumber\end{aligned}$$
where we used and .
The choice of equally spaced knots is motivated by the simplification of the notation but the method can be easily extended to non constant $h$.
\[error\] If constants $c_1$ and $c_2$ exist, such that for all $h>0$ $$\begin{aligned}
\max_{1<i<n, 0<j\leq i-1}, |F_{X(t_i)}(b(t_i)|a(t_j),t_j)-F_{X(t_{i-1})}(b(t_{i-1})|a(t_j),t_j)|\leq c_1 h\label{ip1}\\
\max_{1<i<n, 0<j\leq i-1}, |F_{X(t_i)}(a(t_i)|b(t_j),t_j)-F_{X(t_{i-1})}(a(t_{i-1})|b(t_j),t_j)|\leq c_2 h\label{ip2}\end{aligned}$$ then the absolute value of the errors $\epsilon_{a,i}$ and $\epsilon_{b,i}$ of the proposed algorithm at the discretization knots $t_i$, $i = 1, 2,\ldots$ $$\begin{aligned}
\epsilon_{a,i}&:=&\hat{g}_{a}(t_i|x_0,t_0)-g_{a}(t_i|x_0,t_0)\\
\epsilon_{b,i}&:=&\hat{g}_{b}(t_i|x_0,t_0)-g_{b}(t_i|x_0,t_0)\end{aligned}$$ are $O(h)$.
The Euler method applied to (\[Fortet2\]) gives
\[error1\] $$\begin{aligned}
1-F_{X(t_i)}(b(t_i)|x_0,t_0)&=\sum_{j=1}^i \left[1-F_{X(t_i)}(b(t_i)|a(t_j),t_j)\right]g_{a}(t_j|x_0,t_0) h\label{error1a}\\
&+\sum_{j=1}^i \left[1-F_{X(t_i)}(b(t_i)|b(t_j),t_j)\right]g_{b}(t_j|x_0,t_0)h \nonumber\\
&+\delta_1(h,t_n)\nonumber\\
F_{X(t_i)}(a(t_i)|x_0,t_0)&=\sum_{j=1}^i F_{X(t_i)}(a(t_i)|a(t_j),t_j)g_{a}(t_j|x_0,t_0) h\label{error1b}\\
&+ \sum_{j=1}^i F_{X(t_i)}(a(t_i)|b(t_j),t_j)g_{b}(t_j|x_0,t_0)h\nonumber\\
&+\delta_2(h,t_n).\nonumber\end{aligned}$$
where $\delta_1(h,t_i)$ and $\delta_2(h,t_i)$ are the differences between the integrals on the r.h.s. of (\[Fortet2\]) and the finite sums on the r.h.s. of (\[error1\]).
On subtracting from and from we get
\[error2\] $$\begin{aligned}
\delta_1(h,t_i)&=h\sum_{j=1}^i \left[1-F_{X(t_i)}(b(t_i)|a(t_j),t_j)\right]\epsilon_{a,j} +h\sum_{j=1}^i \left[1-F_{X(t_i)}(b(t_i)|b(t_j),t_j)\right]\epsilon_{b,j}\\
\delta_2(h,t_i)&=h\sum_{j=1}^i F_{X(t_i)}(a(t_i)|a(t_j),t_j)\epsilon_{a,j}+ h\sum_{j=1}^i F_{X(t_i)}(a(t_i)|b(t_j),t_j)\epsilon_{b,j}.\end{aligned}$$
Differencing and recalling and we obtain
\[error3\] $$\begin{aligned}
\delta_1(h,t_i)-\delta_1(h,t_{i-1})&=\frac{h}{2}\epsilon_{b,i}+ h \sum_{j=1}^{i-1} \left(\left[1-F_{X(t_i)}(b(t_i)|a(t_j),t_j)\right]-\left[1-F_{X(t_{i-1})}(b(t_{i-1})|a(t_j),t_j)\right]\right)\epsilon_{a,j}\nonumber\\ &+h\sum_{j=1}^{i-1} \left(\left[1-F_{X(t_i)}(b(t_i)|b(t_j),t_j)\right]-\left[1-F_{X(t_{i-1})}(b(t_{i-1})|b(t_j),t_j)\right]\right)\epsilon_{b,j}\\
\delta_2(h,t_i)-\delta_2(h,t_{i-1})&=\frac{h}{2}\epsilon_{a,i}+ h\sum_{j=1}^{i-1} \left(F_{X(t_i)}(a(t_i)|a(t_j),t_j)-F_{X(t_{i-1})}(a(t_{i-1})|a(t_j),t_j)\right)\epsilon_{a,j}\nonumber\\
&+ \sum_{j=1}^{i-1} \left(F_{X(t_i)}(a(t_i)|b(t_j),t_j)-F_{X(t_{i-1})}(a(t_{i-1})|b(t_j),t_j)\right)\epsilon_{b,j}\end{aligned}$$
that can be rewritten as
\[error4\] $$\begin{aligned}
\epsilon_{b,i}&=-2\sum_{j=1}^{i-1} \left(\left[1-F_{X(t_i)}(b(t_i)|a(t_j),t_j)\right]-\left[1-F_{X(t_{i-1})}(b(t_{i-1})|a(t_j),t_j)\right]\right)\epsilon_{a,j}\\
&-2\sum_{j=1}^{i-1} \left(\left[1-F_{X(t_i)}(b(t_i)|b(t_j),t_j)\right]-\left[1-F_{X(t_{i-1})}(b(t_{i-1})|b(t_j),t_j)\right]\right)\epsilon_{b,j}\nonumber\\
&+\frac{2}{h}\left( \delta_1(h,t_i)-\delta_1(h,t_{i-1})\right)\nonumber\\
\epsilon_{a,i}&=-2\sum_{j=1}^{i-1} \left(F_{X(t_i)}(a(t_i)|a(t_j),t_j)-F_{X(t_{i-1})}(a(t_{i-1})|a(t_j),t_j)\right)\epsilon_{a,j}\\
&-2\sum_{j=1}^{i-1} \left(F_{X(t_i)}(a(t_i)|b(t_j),t_j)-F_{X(t_{i-1})}(a(t_{i-1})|b(t_j),t_j)\right)\epsilon_{b,j}\nonumber\\
&+\frac{2}{h}\left(\delta_2(h,t_i)-\delta_2(h,t_{i-1})\right)\nonumber.\end{aligned}$$
Let us now consider the global error $$\xi_i=|\epsilon_{a,i}|+|\epsilon_{b,i}|.$$ When the hypotheses and are fulfilled $$\begin{aligned}
|\xi_i|&\leq&\left|(c_1+c_2)h\right|\sum_{j=1}^{i-1}\left|\xi_j\right|\\
&+&\frac{2}{h}\left| |\delta_1(h,t_i)-\delta_1(h,t_{i-1})|+|\delta_2(h,t_i)-\delta_2(h,t_{i-1})|\right|\nonumber.\end{aligned}$$ Observing that Euler method errors are $|\delta_1(h,t)|=|\delta_2(h,t)|=O(h^2)$ and applying Theorem 7.1 of [@Li] we get $|\xi_i|=O(h^2)$ and hence the thesis.
A better result on the errors can be obtained by improving the integral discretization rule, i.e. using the midpoint formula instead of Euler method. Other integration rules can improve the order of the error but strongly increase the computational complexity of the algorithm.
The two methods are equivalent in terms of computational time when the Laplace transform expression is a well behaved function. Nevertheless, the generalization of the method for a time dependent boundary $S(t)$ is possible only for the numerical method.
Examples {#Sect:5}
========
In this section we discuss a set of examples of interest for the applications, i.e. standard Brownian motion, Geometric Brownian motion, Ornstein Uhlenbeck process. We apply the algorithms of Section \[Sect:4\] for numerical evaluations. When the joint densities are known in closed form, we use them to illustrate the reliability of the algorithms.
Standard Brownian motion {#SubSect:1}
------------------------
Let us consider a standard Brownian motion with constant boundaries. It is a time and space homogeneous diffusion process, hence we can rewrite its joint density functions (\[densit1\]) and (\[densit2\]) in closed form as
1. If $x_0<a<b$ or $b<a<x_0$ then $$\label{densit1BM}
f_{T_{a} T_{b}}\left(t,s\right)=\left\{
\begin{array}{ll}
0 &t\geq s \\
f_{T_{a}}(t|0,0)f_{T_{b-a}}(s-t|0,0) &t<s \\
\end{array} .\right.$$
2. If $a<x_0<b$ then $$\label{densit2BM}
f_{T_{a} T_{b}}\left(t,s\right)=\left\{
\begin{array}{ll}
f_{T_{b-a}}(s-t|0,0) g_{a}(t|0,0) &t<s \\
0 &t=s \\
f_{T_{a-b}}(t-s|0,0) g_{b}(s|0,0) &t>s \\
\end{array}
.\right.$$
where $g_{a}(t|x_0,t_0)$ and $g_{b}(t|x_0,t_0)$ are given by (\[Wiener\]) and $f_{T_{a}}(t|0,0)$ is given by (\[T\_a pdf\]).
Figure \[Fig:Wiener\] shows the joint pdf of the first hitting times of a standard Brownian motion through two constant boundaries $b=-a=1$ and the corresponding copula together with the contour lines. Figures \[Fig:Wiener2\] illustrates the case of constant boundaries $a=-1$ and $b=1.5$ asymmetric with respect to $x_0$. The asymmetry of the boundaries location determines peaks of different height. Note that the maximum of the joint density have inverted height in the corresponding copula.
Furthermore the Laplace transforms of $g_{a}^{\lambda}(x_0)$ and $g_{b}^{\lambda}(x_0)$ can be found in [@Borod] or applying Corollary \[Cor Laplace\].
The stability of the algorithms introduced in Section \[Sect:4\], already proved by Theorem \[error\], is confirmed by the standard Brownian motion case where the pdf’s $g_{a}(t|x_0,t_0)$ and $g_{b}(t|x_0,t_0)$ are available in closed form. We apply the algorithms to the standard Brownian motion with constant boundaries $a=-1$ and $b=2$ with discretization step $h=0.01$ and we compare the results with the closed form densities (\[Wiener\]) with the series truncated to $N=10^3$ steps. The inversion of the Laplace transform with the Euler method gives a mean square deviation $MSE_a= 6.02\cdot10^{-19}$ and $MSE_b=9.83\cdot10^{-20}$. The numerical algorithm gives a mean square deviation $MSE_a=3.23\cdot10^{-6}$ and $MSE_b=5.11\cdot10^{-8}$. It confirms the reliability of the new algorithm. The higher precision of the Laplace inversion with respect to the numerical method is determined by the simple expression of the involved Laplace transforms. However we cannot infer an analogous property for the other diffusions.
\[Wmu\] The extension of the above results to a Brownian motion with diffusion coefficient $\sigma \neq 1$ is straightforward. Indeed, a Brownian motion with diffusion coefficient $\sigma$ can be transformed in a standard Brownian motion via the space transformation $x=x'/ \sigma$ and the boundaries $a$ and $b$ becomes $a/ \sigma$ and $b/ \sigma$ respectively. When $\mu>0$ and $a(0)<x_0<b(0)$ one can determine $g_b(t|x_0,t_0)$. In this case the crossing of the boundary $a(t)$ is not a sure event and the study of $g_a(t|x_0,t_0)$ requests a suitable normalization. Similarly the case of $\mu<0$ and $a(0)<x_0<b(0)$ is analogous interchanging the role of the two boundaries.
Indicating with $C^{\sigma}_{T_a,T_b}$ the copula of $(T_a,T_b)$ for a Brownian motion with diffusion coefficient $\sigma$ and with $C_{T_a,T_b}$ the copula in the case $\sigma=1$, recalling the transformation $x=x'/ \sigma$, the relationship $C^{\sigma}_{T_a,T_b}=C_{T_{a/\sigma},T_{b/\sigma}}$ holds. Geometric Brownian motion can be obtained by a standard Brownian motion via the space transformation $x'=\exp(\sigma x)$. The corresponding copula, $C^{GBM}_{T_a,T_b}$, is related with the copula of the standard Brownian motion through $C^{GBM}_{T_a,T_b}=C_{T_{\ln a /\sigma},T_{\ln a /\sigma }}$.
The more general transformation $x'=\exp(\mu t+\sigma x)$ is not interesting from the point of view of the exit times from a strip because it corresponds to transform the process into a Brownian motion with drift that has not a sure crossing, as stated in Remark \[Wmu\].
![First hitting times of a Brownian motion through two constant boundaries $b=-a=1$: i) Joint pdf ii) Contour lines of the joint pdf iii) Density of the copula iv) Contour lines of the density of the copula.[]{data-label="Fig:Wiener"}](Fig2.pdf){height="10cm"}
![First hitting times of a Brownian motion through two constant boundaries $a=-1$ and $b=1.5$: i) Joint pdf ii) Contour lines of the joint pdf iii) Density of the copula iv) Contour lines of the density of the copula.[]{data-label="Fig:Wiener2"}](Fig3.pdf){height="10cm"}
As a further example we consider a standard Brownian motion with the following boundaries $b(t)=1+0.1 \cos (\pi t)$ and $a(t)=-1 +0.1 \cos (\pi t+\pi)$. Since the boundaries are time dependent, Laplace transform inversions cannot be applied. Figure \[Fig:Wiener\_osc\] shows the joint pdf of the first hitting times and the corresponding contour lines obtained with the proposed numerical algorithm.
![First hitting times of a Brownian motion through two time dependent boundaries $b(t)=1+0.1 \cos (\pi t)$ and $a(t)=-1 +0.1 \cos (\pi t+\pi)$: i) Joint pdf ii) Contour lines of the joint pdf .[]{data-label="Fig:Wiener_osc"}](Fig4.pdf){height="8cm"}
Ornstein Uhlenbeck Process {#SubSect:3}
--------------------------
Consider as a further example the Ornstein Uhlenbeck process, described by the stochastic differential equation
$$\begin{aligned}
\label{OU}
dX(t)&=&\left(-\frac{X(t)}{\theta}+\mu\right)dt+\sigma dW_t\\
X(t_0)&=&x_0.\end{aligned}$$
For this process representations and numerical methods are available and can be used to evaluate the first hitting time pdf $f_{T_a}(t|x_0,t_0)$ [@APP; @BNR; @L]. On the other side, the density $g_a(t|x_0,t_0)$ is not known in closed form. Here we have applied classical numerical algorithms (cf. [@BNR]) to evaluate the first hitting time pdf and the algorithms of Section 4 to compute the second. Figure \[Fig:OU\] shows the joint pdf and the corresponding copula of the first hitting times of an Ornstein Uhlenbeck process with parameter $\theta=10$, $\mu=0$, $\sigma=1$ and $x_0=0$ through two constant boundaries $b=-a=1$. Figures \[Fig:OU2\] illustrates the case of asymmetric w.r.t. $x_0$ constant boundaries $a=-1$ and $b=1.5$. Note that $x_0$ represents the symmetry axis of the Ornstein Uhlenbeck process. The height of the peaks of the joint density and of the copula behaves as the Brownian motion case.
![First hitting times of an Ornstein Uhlenbeck process through two constant boundaries $b=-a=1$: i) Joint pdf ii) Contour lines of the joint pdf iii) Density of the copula iv) Contour lines of the density of the copula.[]{data-label="Fig:OU"}](Fig5.pdf){height="10cm"}
![First hitting times of an Ornstein Uhlenbeck process through two constant boundaries $a=-1$ and $b=1.5$: i) Joint pdf ii) Contour lines of the joint pdf iii) Density of the copula iv) Contour lines of the density of the copula.[]{data-label="Fig:OU2"}](Fig6.pdf){height="10cm"}
The Laplace transforms $g_{a}^{\lambda}(x_0)$ and $g_{b}^{\lambda}(x_0)$ for the OU process can be found in [@Borod]. However the presence of the parabolic cylinder functions in their expression discourage their numerical inversion.
We are grateful to an anonymous referee for his interesting and constructive comments to improve the paper. Work supported in part by University of Torino Grant 2012 Stochastic Processes and their Applicationsand by project A.M.A.L.F.I. - Advanced Methodologies for the AnaLysis and management of the Future Internet (Università di Torino/Compagnia di San Paolo).
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The properties of their hosts provide important clues to the progenitors of different classes of gamma-ray bursts (GRBs). The hosts themselves also constitute a sample of high-redshift star-forming galaxies which, unlike most other methods, is not selected on the luminosities of the galaxies themselves. We discuss what we have learnt from and about GRB host galaxies to date.'
address:
- 'Department of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH. United Kingdom'
- 'Department of Physics, University of Warwick, Coventry, CV4 7AL. United Kingdom'
author:
- 'N. R. Tanvir'
- 'A. J. Levan'
title: The population of GRB hosts
---
Introduction {#intro}
============
Pinpointing the first long-duration GRB afterglows quickly resolved the debate over their distance scale [@jvp97; @Metzger97], and the realisation that their hosts were high-redshift star-forming galaxies was one of the first pieces of evidence suggesting their progenitors were massive stars [@Bohdan98].
Subsequently, the characteristics of their (likely) host galaxies have formed an equally important line of argument regarding the nature of the short-duration bursts [@Gehrels05; @Hjorth05; @Bloom06].
In addition to helping understand GRBs themselves, hosts are becoming increasingly important as a high-redshift population selected only by its star forming properties, and not dependent on the luminosity of individual galaxies, which is the case for most other samples.
In this contribution we discuss the latest developments in our understanding of the population of GRB hosts, and consider likely future directions.
Long-duration bursts {#longs}
====================
As mentioned above, the fact that no long-duration bursts (LGRBs) have been found in early-type galaxies was a strong argument in favour of their association with massive star death. The question naturally arises exactly what properties a star must have in order to produce a GRB at the end of its life. This is related to the important question of whether the properties, rate and/or luminosity, of GRBs depends on the characteristics of the stellar populations which produce them. In particular, if GRB properties depend on the chemical makeup of their progenitor stars, or other aspects of its galactic environment, that will influence the statistical properties of the sample of host galaxies they select.
GRBs and metallicity
--------------------
In the popular “collapsar" model for the production of GRBs, it has been argued that high (around solar and above) metallicity single Wolf-Rayet stars will lose too much mass and angular momentum to produce the rapidly rotating massive cores that ultimately collapse to produce GRBs [@Heger03].
A number of observational studies are consistent with the idea that GRBs are preferentially produced by stellar populations that are at least moderately poor (sub-solar) in heavy metals. @Fynbo03 first noted that the high proportion of GRB host galaxies above redshift $z\approx2$ that show Lyman-$\alpha$ in emission suggests they are low dust, low metal systems. Subsequently, studies of samples of GRB host galaxies in the mid- and far-IR and submm [@Berger03; @Tanvir04; @Lefloch06; @CastroCeron06] have found fewer hosts are bright in these bands than expected, given the large amount of obscured star formation expected to be taking place in such galaxies. Again, a preference for lower metallicities would help explain this finding if the very high star-formation rate galaxies correspond to higher metallicity systems as is thought. Note there is potentially a selection effect here, since some GRB hosts will not be identified in the first place if the optical light of the GRB is extincted by dust. However, several such “dark bursts" have sufficiently good positions to identify their hosts, and were included in the samples studied [eg. @Barnard03]. A similar argument has been made for the five lowest redshift GRBs, which are all in rather small, metal-poor galaxies when compared to the population of low-$z$ star-forming galaxies [@Stanek06].
Most recently, @Fruchter06 compared the host galaxies of GRBs and core-collapse supernovae in a similar redshift range, roughly $z=0.5$–1 (the average and spread of redshift was also similar between the two). The characteristics of the galaxies and also the positions of the exploding stars, differed significantly between the samples. In figures 1 and 2 we show somewhat updated mosaics of the GRB and SNe hosts respectively. The supernova hosts are clearly more likely to be brighter, frequently grand-design spirals, while the GRB hosts are typically smaller and have irregular/merger morphologies [see also @Conselice05; @Wainwright07]. This could also be explained if the GRBs are preferentially formed from lower-metallicity core-collapse supernovae. A bias against finding GRBs enshrouded in dusty systems should be more than matched by the same bias against finding supernovae hidden by dust (recalling that GRBs can burn through significant columns of intervening dust and so may sometimes be found optically even when enshrouded [eg. @Waxman00]). However, @Wolf07 argue that the same data, whilst compatible with a mild metallicity dependence of GRB rate/luminosity, would not be consistent with a strong effect.
The chemical abundances of gas in the hosts of GRBs can also be estimated directly via absorption line spectroscopy of the GRB afterglow itself. Again, there could be a bias against higher metallicity, dusty galaxies, since the afterglow must be optically bright to perform this analysis. However, when such abundances have been determined they show a wide range from about 1% solar to nearly solar [eg. @Vreeswijk04; @Prochaska07].
GRB host samples
----------------
The immense luminosity of GRBs means that they can be detected in principle to very high redshifts. Thus they can be used to select and characterise galaxies from very early times up to the present.
If the rate of GRB production were the same for all young stellar populations, then GRB host samples should allow us to discriminate the proportions of global star formation arising in different galaxy types, and more generally map the history of star formation in the universe [eg. @Wijers98; @Trentham02]. As we have discussed above, it seems unlikely that GRBs do trace star formation in a completely unbiased way. However, GRB selection will favour hosts with high star formation rates (and probably, lower metallicities), but otherwise is not biased against small faint galaxies, which are typically missed in other flux-limited catalogues. Redshifts, metallicities and gas dynamics can be determined in many cases from the afterglow spectroscopy. A good example of this power was GRB 020124, whose host was undetected to $R\sim29.5$ in HST imaging [@Berger02], but was found to be a high column density DLA at $z=3.2$ from the afterglow [@Hjorth03].
A number of attempts have been made to compare GRB hosts as a whole to other high redshift populations. For example, @Jakobsson05 demonstrated that around $z\sim3$ the bright end of the host luminosity function is consistent with that expected by weighting by star formation the Lyman-break galaxy luminosity function.
Many authors have noted that whilst occasional bursts have been found in very red (ERO) star-forming galaxies [eg. @Levan06a; @Berger07a], the bulk of GRB hosts are sub-L\*, blue, low-dust, apparently young galaxies with relatively strong line-emission and a high specific rate of star formation [eg. @Fruchter99; @Lefloch03; @Christensen04]. Qualitatively these are similar characteristics to the population of galaxies found in emission-line surveys for Lyman-$\alpha$. An interesting comparison is with the wide area survey of @Gawiser06, for Ly-$\alpha$ emitters around $z\approx3.1$. In broad terms this population is very like the GRB host sample in the same redshift range, albeit that the Ly-$\alpha$ equivalent width is, unsurprisingly, somewhat higher on the average. Figure 3 shows the cumulative histograms of R-band continuum luminosity (rest frame UV) for this sample together with the published GRB hosts with $2.6<z<3.6$, illustrating their similarity.
Short-duration bursts {#shorts}
=====================
The first few short-duration GRB afterglows seemed to paint a picture of being in galaxies at redshifts of a few tenths and some of which contained little or no young stellar population. This was widely interpretted as being consistent with the neutron-star neutron-star (or neutron-star black-hole) binary coalescence model for GRB production.
Since then the picture has become murkier. Several apparently short-duration GRBs have been found where the host is hard to identify, and most likely is at much higher redshift $z>1$. In particular, GRB 060121 had a red afterglow and host galaxy indicating a likely redshift $z>4$ and almost certainly $z>1.5$ [@Levan06b; @AdUP06]. The host and energetics of this burst are much more typical of LGRBs, and the possibility remains that it was actually a member of that class, despite the short duration.
Although in many individual cases there can be ambiguity over whether a given burst should be in the short or long class [@Levan07b; @Bloom07], the weight of several likely high-z short bursts has led to speculation that they form a separate sub-class [@Berger07b]. It is worth commenting, though, that so-far all redshifts for short bursts have come from their putative host rather than the afterglow, and one consequence of a NS-NS progenitor would be the possibility that the burst occurs well beyond the optical extent of it’s host, making definite association unclear in some cases.
Short-duration bursts from nearby galaxies
------------------------------------------
In a parallel development @Tanvir05 have shown that there is a weak cross-correlation signal between the distribution of BATSE short-duration bursts and galaxies in the nearby universe. In particular, they used the PSCz galaxy redshift survey, which provides uniform selection over 85% of the sky, and found a positive signal with the sample cut at various recession velocities out to 8000 km s$^{-1}$ (approximately 110 Mpc). Simulations suggested that this level of signal could be produced if between 10 and 25% of BATSE short bursts were coming from nearby galaxies.
These on average must be considerably weaker bursts than those found at cosmological distances. The most likely progenitors are giant flares from soft gamma-ray repeaters. At least one such flare from an SGR in the Milky Way (SGR 1806-20) was bright enough that it could have been detected by BATSE to several tens of Mpc [@Palmer05; @Hurley05]. In fact, only a very low rate of roughly one per millenium per Milky-Way sized galaxy is sufficient to explain the BATSE observed rate [@Levan07a; @Ofek07].
If of order 10% of BATSE bursts were really from low redshift galaxies it remains surprising that amongst those well-localised by Swift and HETE-II there aren’t any clear-cut examples. The best candidate is the weak burst GRB 050906 whose BAT error circle contained the outer parts of an actively star-forming galaxy IC328 at a distance of about 130 Mpc [@Levan07a], although the spectrum of the burst was significantly softer than previous giant flares. Possibly the softer sensitivity of Swift/BAT and HETE-II compared to BATSE makes it less likely that they will detect SGR giant flares, which, on the basis of only three events, seem to be typically hard (and thermal).
Interestingly, though, there are two candidates for low-redshift short-duration bursts localised by the Inter-Planetary Network (IPN). Specifically, GRB 051103 was determined to have occurred in a thin error region which lay close to the outskirts of the galaxy M81 and at that distance the burst would have been quite consistent energetically with being an SGR giant flare comparable to that from SGR 1806-20 [@Ofek06; @Frederiks07]. An even more compelling case may be GRB 070201, which was found to overlap the outer part of the disk of M31 [@Perley07; @Hurley07]. This was an extremely bright burst, and in that regard, again, quite consistent with a very energetic SGR flare at the distance of M31. The only concern we might have is that two such rare events should occur in neighbouring large spiral galaxies (M31 and the Milky Way) within only two years of each other!
Conclusions
===========
The characteristics of their hosts has provided important clues to the nature of GRB progenitors. Several lines of evidence suggest that LGRBs show some preference for lower-metallicity hosts. Particularly at high redshifts GRBs may be the root to identifying and studying low-metallicity star formation, and especially the faint end of the galaxy luminosity function that is generally missed in other surveys. To fully realise the power of GRBs to select high-z populations, it is important that statistical samples of bursts and hosts with redshifts be as complete as possible. As it is, optical/nIR afterglows have been found for nearly 80% of well-positioned Swift LGRBs, but redshifts for only about 50% [@Tanvir07].
Our understanding of the class of short-duration bursts is at an earlier stage, but has seen huge progress in the past two years. Hosts have proved crucial to these breakthroughs, not least because redshifts have yet to be found directly for any short-burst afterglow. By way of illustration of the current, rather confusing, state of affairs, we show in figure 4 a panel of hosts (or candidate hosts) of various short duration GRBs, which range from the very nearby candidates for SGR giant flare bursts, via the intermediate redshift likely NS-NS progenitors, to the new “population" of apparently high redshift short bursts whose nature remains controversial.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Current-induced magnetic domain wall motion, induced by transfer of spin transfer effect due to exchange interaction, is expected to be useful for next generation high-density storages. We here show that efficient domain wall manipulation can be achieved by introduction of Rashba spin-orbit interaction, which induces spin precession of conduction electron and acts as an effective magnetic field. Its effect on domain wall motion depends on the wall configuration. We found that the effect is significant for Bloch wall with the hard axis along the current, since the effective field works as $\beta$ or field-like term and removes the threshold current if in extrinsic pinning is absent. For Néel wall and Bloch wall with easy axis perpendicular to Rashba plane, the effective field induces a step motion of wall corresponding to a rotation of wall plane by the angle of approximately $\pi$ at current lower than intrinsic threshold. Rashba interaction would therefore be useful to assist efficient motion of domain walls at low current.'
author:
- 'Katsunori Obata$^{1}$'
- 'Gen Tatara$^{1,2}$'
title: 'Current-induced domain wall motion in Rashba spin-orbit system'
---
Introduction
============
In recent years, magnetic-memory devices like hard disk drives have been utilized for various products such as portable music players and home appliances. With the diversification of the use of the devices, they deal with a much larger amount of information and need to be more miniaturized and have higher capacity. However, it is believed that the conventional magnetic-memory devices will reach the limit of downsizing and high capacity in the near future. The reason is because the magnetization so far is controlled by a magnetic field and the more miniaturized and the higher capacity they get, the higher necessary field they will need. Therefore, a new method which replaces the magnetic field is required. One of the methods expected is the control by electric current, i.e. current-induced magnetization reversal. The current-induced magnetization reversal was pointed out by Berger [@Berger78; @Berger86; @Berger96] and Slonczeski. [@Slonczewski96] The torque is caused by the $s$-$d$ exchange interaction via spin transfer effect arising from conservation of spin angular momentum between the current and the magnetization.
To realize devices using the current-induced magnetization reversal, it is absolutely essential to reduce the current necessary. In this paper, we demonstrate that the problem can be solved by introduction of Rashba spin-orbit interaction. [@Rashba] A spin precession induced by Rashba interaction is expected to affect the spin transfer torque mechanism, and to lead to efficient control of magnetization. The interaction has been proposed on two-dimensional electron systems realized at the interface of semiconductors, but is now known to arise quite generally when inversion symmetry is broken. [@Lashel96] For instance, significant Rashba effect can arise on surface of heavy metals. [@Nakagawa07; @Ast07] Our results therefore would apply to systems of magnetic semiconductors [@yamano] with gate for Rashba interaction attached, or on magnetic metallic thin films with heavy ions doped. [@Ast07]
Recent theoretical studies revealed that current-driven domain wall motion is significantly affected by spin relaxation process. Actually, spin relaxation triggers a torque perpendicular to the adiabatic spin transfer torque [@Zhang04; @Thiaville05] and this torque, $\beta$ torque [@Thiaville05], acts as an effective force on domain wall, which deletes intrinsic pinning effects[@Thiaville05; @TTKSNF06]. Microscopic analysis of spin relaxation was done in the case of spin flip scattering by random impurity spins and value of $\beta$ was found to be of similar magnitude with (but not necessarily equal to) the spin relaxation contribution to Gilbert damping parameter $\alpha$ [@Tserkovnyak06; @KTS06; @KS07; @Duine07]. On the other hand, role of spin-orbit interaction, which also causes spin relaxation, on the wall dynamics has not been much studied so far. Spin-orbit interaction in magnetic semiconductors was recently studied based on Kohn-Luttinger Hamiltonian, and large enhancement of wall velocity due to spin-orbit interaction was found [@Nguyen07]. The Rashba spin-orbit interaction we are going to study here turns out also to assist wall motion to a large extent. In this paper, we study current-driven domain wall motion in the presence of Rashba interaction. Rashba interaction acts on $x$-$y$ plane and electric current is applied along $x$ axis. The spin system we consider is with easy and hard axes. The wall is treated as planar and rigid, which is justified if easy axis energy gain, $K$, is larger than hard axis anisotropy energy, $K_\perp$[@begt2; @TKSLK07; @TKS08]. We consider three cases where easy axis direction (we call $\eta$) is $x$, $y$ and $z$. The wall structure in these three cases are Néel for $\eta=x$ and Bloch wall for $\eta=y, z$. We will see that dynamics of Néel wall and Bloch wall with $\eta=z$ (called Bloch(z)) are essentially the same, while Bloch with $\eta=y$ (Bloch(y)) is different. The effect of current is calculated for each anisotropy configuration using gauge transformation in spin space, assuming adiabatic limit. The Rashba interaction is treated perturbatively to the second order. Expansion with respect to Rashba interaction is justified here since electron has large spin polarization $\Delta$. (This is in contrast to the spin Hall case without polarization, where perturbative treatment is not allowed [@Inoue].) Below, we derive the effective Hamiltonian for local spin, derive the equation of motion for domain wall, and then discuss wall dynamics solving the equation.
Model and method
================
Local spin
----------
Domain wall is described by a Lagrangian of local spins given by where local spin direction is expressed by polar coordinates ($\theta , \phi$). The first term describes time evolution of local spin (spin Berry phase term). The Hamiltonian of local spin we consider is given as follows; The first term is ferromagnetic exchange interaction between local spins, the second and third terms are magnetic anisotropy energies. $S_{\|}$ is the easy axis component of spin and $K(>0)$ is the corresponding anisotropic energy, and $S_{\bot}$ and $K_{\bot}$ are hard axis ones.
In the absence of Rashba interaction, we can choose space coordinate and spin coordinate independently. When Rashba spin-orbit interaction is switched, spin and space coordinates correlate each other, and the spin torque and gauge field depend on the choice of magnetic easy axis, which we call $\eta$ axis. We consider three different cases with easy axis $\eta = x,y$ and $z$. Domain wall structure then becomes Néel and Bloch walls. We will call domain wall configuration with the easy axis label, such as Bloch(z) for a Bloch wall with $\eta =z$ (see Table. \[dwhard\]).
Easy ($\eta$) axis ($K$) Hard axis ($K_{\bot}$)
---------- -------------------------- ------------------------
Bloch(z) z y
Neel(x) x z
Bloch(y) y x
: Configuration of magnetic anisotropy and corresponding domain wall structure.[]{data-label="dwhard"}
In this paper Rashba interaction acts within $xy$-plane and current is applied always along $x$. Instead, our definition of polar angle in spin space depends on the wall configuration. We define $\theta$ as the angle measured from the easy axis (i.e., $S_{\parallel} = \cos{\theta}$), and $\phi$ as the angle in the plane perpendicular to easy axis. The hard axis is given by $\phi = \frac{\pi}{2}$ and so the hard axis component is written as $S_{\bot} = \sin{\theta} \sin{\phi}$. Note therefore that ($\theta , \phi$) for Néel and Bloch(y) walls below are different from standard definition measured from $z$-axis. Lagrangian $L_s$ is thus written in terms of polar angle as for any type of walls considered here.
Conduction electron
-------------------
The Hamiltonian of the conduction electron is given by the following four terms. The first term is the free electron part, represented as where $c$ and $c^\dagger$ are electron annihilation and creation operators, $\Ef$ is the Fermi energy and $m$ is the effective mass. The second is the exchange interaction between electron and local spin, where $\Delta$ is the magnitude of the interaction, $\vet{S}$ is the local spin vector and $\vet{\sigma}$ are Pauli matrices. The third one, Rashba spin-orbit interaction, is given as where $\overleftrightarrow{\nabla}$ acts on both sides and $\la$ is the magnitude of Rashba interaction (the dimensions is \[J $\cdot$ m\]). When electric field is applied in the $x$-direction, the momentum, $-i \hbar \ave{\cxd \nax{x} \cx}$, grows, and Rashba interaction then changes the electron spin direction toward $y$-direction, inducing electron spin precession.
Since we are interested in a response of local spins to applied current, we introduce the interaction with electric field, given by (neglecting $O(E^2)$) where $\bm{A}_{\rm{EM}} = \frac{\bm{E}}{i \Omega} e^{i \Omega t} $ is a $U(1)$ gauge field, $\bm{E}$ is applied electric field. Velocity operator is given as The field $\bm{E}$ is spatially uniform but has finite frequency $\Omega$. This frequency is introduced for calculation purpose and is chosen as $\Omega=0$ at the end of calculation, as is usually the case of linear response calculation.
Gauge transformation
--------------------
The exchange interaction, $H_{\rm{ex}}$, has in general off-diagonal components. Besides the local ground state of conduction electron varies at each lattice point if magnetization is non-uniform. In this case, local gauge transformation in spin space [@korenman; @gt2] which diagonalizes the exchange interaction is useful. The spatial change of local spin is then represented by a gauge field, which is proportional to spatial spin variation, $\partial_{\mu} \vet{S}$. We consider the case when local spin profile is slowly varying (called the adiabatic limit), and then gauge field is small. We thus examine only the first-order contribution of gauge field. A new electron operator ,$a$, after gauge transformation is defined by use of $2 \times 2$ unitary matrix $U$ as where electron operators here have two spin components like $c\equiv(c_+,c_-)$. The matrix $U$ is expressed using Pauli matrix as where $\vet{m}(x)$ is a vector which characterize the gauge transformation. We denote the spin easy axis as $\eta$ (e.g., $\eta=x$ for Néel wall). The gauge transformation is defined so that conduction electron spins are polarized along magnetic easy axis $\eta$, i.e. to satisfy $\frac{\vet{S}(x)}{S} \cdot \lc{\cxd \vet{\sigma} \cx} = \axd \pax{\eta} \ax$. (The transformation thus differs for different wall configuration.) The Hamiltonian given by Eqs. (\[freepart\])(\[eq:hex1\])(\[rashbapart\]) is written in $a$-electron representation as where gauge field is given as The electromagnetic interaction after gauge transformation is given as where $\overline{\pax{\mu}} \equiv U^{\dagger} \pax{\mu} U = 2 m^{\mu} \lc{\bm{m} \cdot \vet{\sigma}} - \pax{\mu}$ and the electric field is applied in $x$-direction.
The total electron Hamiltonian is therefore obtained as
Effective Hamiltonian
---------------------
The current-induced part of the effective Hamiltonian for local spin is directly obtained from Eq. (\[eq:ron24\]) as where electron properties are represented by the following expectation values, Here $\left\langle \ \right\rangle$ denotes expectation value evaluated using non-perturbed Hamiltonian $H_a$ defined as and including to linear order the effect of current,
Current-induced part of the effective Lagrangian is given by
In calculating the expectation values, Rashba spin-orbit interaction is treated perturbatively to the second-order, $\la^2$. This approximation correspond to assuming $\frac{\la k_f}{\Ef} \ll 1$ with Fermi wave vector $k_F$. This expansion with respect to $\la$ is justified by the presence of $\Delta$, in contrast to the non-perturbative nature of unpolarized Rashba system[@Inoue]. We also assume that the effect of impurities is weak and the electron lifetime is long, i.e. $\frac{1}{\Ef \tau} \ll 1$. In this case, the term in Eq. (\[eq:heff35\]) including electric field turns out to be small by a factor of $\frac{1}{\Ef \tau}$ compared with dominant contributions. We will thus evaluate the effective Hamiltonian given by
Bloch(z) case
=============
In this section, we derive the effective Hamiltonian for the anisotropy configuration of Bloch(z) type, namely, magnetic easy axis is in the $z$-direction and hard axis is in the $y$-direction.
![Polar coordinates of local spin in the Bloch(z) case.[]{data-label="fig:spz"}](blochz.eps){width="8cm"}
![Polar coordinates of local spin in the Bloch(z) case.[]{data-label="fig:spz"}](spinz.eps){width="4cm"}
In this case, domain wall configuration is Bloch(z) wall shown in Fig. \[fig:bdwz\]. Polar coordinates are defined by (as in Fig. \[fig:spz\]) $$\begin{aligned}
\vet{S}(x,t)=S(\sin{\theta}\cos{\phi}, \sin{\theta}\sin{\phi}, \cos{\theta}) .\end{aligned}$$
In the case of Bloch(z)-type anisotropy, the gauge transformation is defined by $\vet{S} (\uxdn \vet{\sigma} U) =S \pax{z}$. This is satisfied by choosing Gauge field is then given as $$\begin{aligned}
A_{\mu}=\begin{pmatrix} \sgax{\mu}{x} \\ \sgax{\mu}{y} \\ \sgax{\mu}{z}
\end{pmatrix} =
\frac{1}{2} \begin{pmatrix} -\partial_{\mu}\theta \sin{\phi} - \sin{\theta} \cos{\phi} \partial_{\mu}\phi \\
\partial_{\mu}\theta\cos{\phi}-\sin{\theta}\sin{\phi}\partial_{\mu}\phi \\
\big(1-\cos{\theta}\big) \partial_{\mu} \phi \end{pmatrix}. \end{aligned}$$ The unperturbed Hamiltonian, given by choosing $\eta=z$ in Eq. (\[Hadef\]), reads This Hamiltonian $H_a$ has the off-diagonal elements in the spin space due to Rashba interaction, and so we will diagonalize it using a unitary transformation in momentum space, $\akd = d_{k}^{\dagger} T_k^{\dagger}$, where $d_{k}^{\dagger}$ is a new creation operator and $T_k$ is $2 \times 2$ unitary matrix. The Hamiltonian $H_a$ in momentum space reads $$\begin{aligned}
H_a = \sum_{k} d_{k}^{\dagger} T_k^{\dagger} \left( \begin{matrix} \Ek -\Delta & -i \la k_{-} \\ i \la k_{+} & \Ek - \Delta \end{matrix} \right) T_k d_k ,\label{eq:haz2}\end{aligned}$$ where $k_{\pm} \equiv k_x \pm i k_y$. Diagonalization of $H_a$ is done by chosen the unitary matrix $T_k$ as, $$\begin{aligned}
T_k \equiv \frac{1}{\sqrt{A_k^2+\la^2 k^2}} \left( \begin{matrix} A_k & i \la k_{-} \\ -i \la k_{+} & -A_k \end{matrix} \right) ,\end{aligned}$$ where $A_k = \Delta + Z_k, Z_k = \sqrt{\Delta^2+\la^2 k^2}$. The result is $$\begin{aligned}
H_a = \sum_{k} d_k^{\dagger} \left( \begin{matrix} \Ek - Z_k & 0 \\ 0 & \Ek + Z_k \end{matrix} \right) d_k \label{eq:mha65}.\end{aligned}$$ After diagonalization, the energy of conduction electrons is spin split as where $\sigma = (+,-)$ is spin polarization.
Now, let us estimate expectation values. The electron density $n$ (Fig. \[figdiag\]) is given as where tr is a trace in spin space and $\tpax{\al}\equiv T_k^{\dagger} \pax{\al} T_k$.
![Diagramatic representation of electron density ($n$), spin density ($s^\alpha$) and spin current ($j_{{\rm s},i}^\alpha$) at the linear order in the applied electric field, $\bm{E}$. Solid line represents electron Green function with Rashba interaction included. Vertex denoted by $\times$ is $1$, $\sigma^\alpha$ and $k_i\sigma^\alpha$, for $n$, $s^\alpha$ and $j_{{\rm s},i}^\alpha$, respectively. []{data-label="figdiag"}](diag.eps){width="0.7\linewidth"}
Keldysh Green function [@keld; @smith] with Rashba interaction included, $\ti{G}_{k,\w}^{<}$, is defined as $$\begin{aligned}
\notag
\ti{G}_{k,\w}^{<} \equiv & i \ave{d_{k,\ome}^{\dagger} d_{k,\ome}} \\
= & \begin{pmatrix} g_{k,\w,\up}^{<} & 0 \\
0 & \ti{g}_{k,\w,\dw}^{<}
\end{pmatrix},\end{aligned}$$ where component $\tgwl{k,\ome}{\sigma}$ is given as $f( \ome )\equiv \frac{1}{e^{\beta\omega}+1}$ being Fermi distribution function ($\beta$ is inverse temperature). Summation over $k$ is carried out in two-dimensions by replacing by energy integration, $\sum_{k} = \frac{V m}{2 \pi \hbar^2} \int d\E$. Products of Keldysh Green functions is calculated using relations where $\ti{G}_{k,\w}^{r}$ ($\ti{G}_{k,\w}^{a}$) is retarded (advanced) Green function, and Expanding with respect to $\la$ to the second order, we obtain the density as We see that there is no effect from the applied electric field here.
The electron spin density $s^{\g}$ is similarly calculated as The result is We see that the electric field induces perpendicular components $s^x$ and $s^y$, but $\abs{s^y} \gg \abs{s^x}$ since $\frac{1}{\Delta \tau} \ll 1$. We will thus approximate $\abs{s^x}\simeq0$. The $z$-component of spin in Eq. (\[eq:spz3\]) is the adiabatic contribution, which is not affected by applied field.
Spin current, $\js{\mu}{\al}$, is estimated using as We see that spin current are generated in $x$ and $y$ direction by the Rashba interaction without electric field and that $\js{y}{x}$ is equal to $- \js{x}{y}$. This is due to the symmetry of Rashba spin orbit interaction. In contrast, $\js{x}{z}$ is induced by applied electric field. We define the current density (divided by $e$) and electron density without spin-orbit interaction as ($\sigma$ is Boltzmann conductivity) and In term of these parameters, the above result reads $$\begin{aligned}
n =& n_0 \bi{1 + \frac{m \la^2}{2 \hbar^2 \Ef}} \nonumber \\
\lc{s^x, s^y, s^z} =& \lc{ 0, - \frac{m \la}{\hbar \Ef} J, n_0 \frac{\Delta}{\Ef}}\nonumber \\
\lc{\js{x}{y}, \js{x}{z}, \js{y}{x}} =& \lc{- \frac{\la}{\hbar} n_0 , \bi{\frac{\Delta}{\Ef} - \frac{m \la^2}{2 \Delta \hbar^2}} J, \frac{\la}{\hbar} n_0}. \label{eq:js44}\end{aligned}$$ Other components of spin current vanish.
The effective Hamiltonian for Bloch(z) case is therefore obtained from Eq. (\[eq:hs41\]) as We see that applied current induces $\js{x}{z}$ (Eq. (\[eq:js44\])) and this induces when coupled with Rashba interaction an effective magnetic field in $y$-direction as indicated by the second term. The first term of Eq. (\[eq:ey128\]) represents standard spin transfer torque (with current modified by Rashba interaction). The third term is independent of applied current and is a modification of magnetic anisotropy by Rashba interaction.
Néel(x) case
============
![Definition of polar coordinates for Néel(x) case.[]{data-label="fig:spx"}](neelx.eps){width="7cm"}
![Definition of polar coordinates for Néel(x) case.[]{data-label="fig:spx"}](spinx.eps){width="4cm"}
In this section, we consider a case of Néel wall realized when magnetic easy axis and hard axis are in the $x$ and $z$ direction, respectively (Fig. \[fig:dwx1\]). Polar coordinates are defined differently from standard definition as (Fig. \[fig:spx\])
Derivation of effective Hamiltonian is done similarly to Bloch(z) case. Difference is in definition of gauge transformation, $\vet{S} (U^{\dagger} \vet{\sigma} U) =S \pax{x}$. Vector $\vet{m}$ is accordingly chosen as and $\sgax{\mu}{\al}$ is given as $$\begin{aligned}
A_{\mu}=
\frac{1}{2}\left( \begin{matrix}
(1-\cos{\theta}) \de{\mu}\phi \\
-\sin{\phi} \de{\mu}\theta - \sin{\theta} \cos{\phi} \de{\mu} \phi \\
\de{\mu}\theta \cos{\phi} - \sin{\theta} \sin{\phi} \de{\mu}\phi
\end{matrix} \right) .\end{aligned}$$ Hamiltonian $H_a$ is also different from Bloch(z) case since uniform spin polarization is now along $x$-direction. It is given as $$\begin{aligned}
H_a = \sum_k
\akd \left( \begin{matrix} \Ek & -b_- \\ -b_+ & \Ek \end{matrix} \right) \ak ,
\label{eq:hax132}\end{aligned}$$ where $b_{\pm} \equiv (\Delta \mp i \la k_{\pm})$. The diagonalization of $H_a$ is carried out as $\akd = \dkd T_k^{\dagger}$ where $$\begin{aligned}
T \equiv \frac{1}{\sqrt{2}Z_k} \left(\begin{matrix} Z_k & b_- \\ b_+ & -Z_k \end{matrix}\right) ,\end{aligned}$$ where The Hamiltonian after diagonalization reads $$\begin{aligned}
H_a = \sum_{k} \dkd \left( \begin{matrix} \Ek - Z_k & 0 \\ 0 & \Ek + Z_k \end{matrix} \right) \dk \label{eq:hax1}.\end{aligned}$$ The expectation values are calculated similarly to Bloch(z) case and we obtain where other components of spin and spin current vanish.
The current-induced effective Hamiltonian is then obtained as The applied current coupled with spin-orbit interaction induces an effective magnetic field along hard, i.e., $y$ axis (second term of Eq. (\[eq:he195\])), as in the Bloch(z) case.
Bloch(y) case
=============
![Polar coordinates for Bloch(y) configuration. []{data-label="fig:spy"}](blochy.eps){width="7cm"}
![Polar coordinates for Bloch(y) configuration. []{data-label="fig:spy"}](spiny.eps "fig:"){width="4cm"}\
Finally, we consider a case of magnetic easy axis and hard axis along $y$ and $x$ direction, respectively. Domain wall in this case is Bloch(y) wall (Fig. \[fig:dwy1\]). Polar coordinates are define as (Fig. \[fig:spy\]) Gauge transformation is given by $\vet{S} (U^\dagger \vet{\sigma} U) =S \pax{y}$, and vector $\vet{m}$ consequently becomes and gauge field is $$\begin{aligned}
A_{\mu}=
\frac{1}{2}\left( \begin{matrix}
\de{\mu}\theta \cos{\phi} - \sin{\theta} \sin{\phi} \de{\mu}\phi \\
(1-\cos{\theta}) \de{\mu}\phi \\
-\sin{\phi} \de{\mu}\theta - \sin{\theta} \cos{\phi} \de{\mu} \phi \\
\end{matrix} \right) .\end{aligned}$$ Hamiltonian $H_a$ is given as $$\begin{aligned}
H_a = \sum_{k} \akd
\left(\begin{matrix} \Ek & -i b_- \\ -i b_+ & \Ek \end{matrix} \right) \ak ,\end{aligned}$$ where $b_{\pm} = (\Delta \mp \la k_{\pm})$ (is different from the Néel case). Hamiltonian $H_a$ is diagonalized as $$\begin{aligned}
H_a = \sum_{k} \dkd \left( \begin{matrix} \Ek - Z_k & 0 \\ 0 & \Ek + Z_k \end{matrix} \right) \dk \label{eq:hay01},\end{aligned}$$ by defining $\akd = d^{\dagger} T_k^{\dagger}$ with $$\begin{aligned}
T \equiv \frac{1}{\sqrt{2}Z_k} \left( \begin{matrix} -Z_k & i b_{-} \\ -i b_{+} & Z_k \end{matrix} \right) ,\end{aligned}$$ where Expectation values which appears in the effective Hamiltonian are estimated as The effective Hamiltonian is therefore obtained as
Analysis of effective Hamiltonian
=================================
The effective Hamiltonian representing the effect of current obtained above is summarized as follows.\
Bloch(z) case: Neel(x) case: Bloch(y) case: Here $\es{\al}$ is local spin component in $\al$-direction. The first term of each effective Hamiltonian represents spin transfer torque, which is enhanced by Rashba interaction in Bloch(z) and Bloch(y) cases at the second order in spin-orbit interaction. The enhancement is thus small but independent of sign of $\la$. The second term of the effective Hamiltonian, $\es{y} \la \frac{\Delta}{\Ef} J$, indicates that the effective magnetic field arises in the $y$-direction in all three cases. This is a result of spin current induced by Rashba interaction and applied current. The last term of each Hamiltonian exists without electric field and thus represents magnetic anisotropy modified by Rashba interaction. This change of magnetic anisotropy is at the second-order in $\la$ and is small. Furthermore, in reality, this static contribution should be contained already in the anisotropy parameters $K , K_{\bot}$. We therefore do not take it into account in the following analysis.
The effects of Rashba interaction are summarized in Table. \[dwsoi1\].
Bloch(z) Néel(x) Bloch(y)
------------------------------------- ------------ ------------ ------------
Enhancement of spin transfer torque $O(\la^2$) $\times$ $O(\la^2$)
Effective magnetic field $O(\la$) $O(\la$) $O(\la$)
Change of magnetic anisotropy $O(\la^2$) $O(\la^2$) $O(\la^2$)
: Summary of effects of Rashba interaction for three configurations represented by the order of the effects in $\la$. Symbol $\times$ denotes the absence of the effect.[]{data-label="dwsoi1"}
Equation of motion of Domain Wall
=================================
Let us discuss how Rashba spin-orbit interaction affects current-induced domain wall motion based on the effective Hamiltonian we derived. For this Lagrangian formalization is convenient. We consider rigid planar domain wall. The Lagrangian for domain wall is then obtained as $L = \dot{X} \phi - (H_S+H_{\rm{eff}})|_{X,\phi}$ where $X$ is domain wall position, $\phi$ is local spin angle in magnetic easy plane and $(H_S+H_{\rm{eff}})|_{X,\phi}$ is effective spin Hamiltonian evaluated for domain wall configuration[@begt2]. Explicitly, domain wall Lagrangian is given as follows.\
Bloch(z) wall: Neel(x) wall: Bloch(y) wall: Here we introduced following dimensionless parameters , where $N$ is a number of local spin in the domain wall and $\ell$ is domain wall thickness given as $\ell \equiv \sqrt{\frac{J}{K}}$. (Dimensionless time is $\ti{t} \equiv \frac{v_c}{\ell}t$). Here a velocity $v_c \equiv \frac{K_{\bot} S \ell}{2 \hbar}$ correspond to drift velocity of electron at intrinsic threshold current without Rashba interaction[@begt2].
Equations of motion for domain wall is obtained taking account of dissipation as,($Q = \ti{X}, \phi $) where $\tilde{W_s}$ is dimensionless dissipation function written as $\tilde{W_s} = \frac{\al}{2} \bi{\tilde{\dot{X}}^2 + \tilde{\dot{\phi}}^2 }$[@TKS08].
In deriving the equation of motion, we neglect contribution of conduction electron density ($s^{\g} \ll 1$) since they turns out to be small in actual situations. The equation of motion is obtained as follows.\
Bloch(z) wall: where time evolution of $\phi$ reduces to a single equation of Neel(x) wall: which result in Bloch(y) wall: and the time evolution of $\phi$ is obtained as
We here see a large difference between Bloch(y) and other cases. In fact, the angle $\phi$ in Bloch(y) wall is directly driven by current and Rashba interaction as indicated by the right-side of Eq. (\[eq:eyq302\]). Such effect of current has been known as $\beta$ terms in the case of electron spin relaxation due to random spin [@Li04; @Zhang04]. In the present Rashba case, the parameter $\be$ is then given by for Bloch(y) ($\be = 0$ for Bloch(z) and Neel(x) walls). This result can be explained by noting that $\be$ terms is effectively equivalent to an external magnetic magnetic field and that Rashba interaction induces an effective magnetic field, which coinsides with the easy axis for Bloch(y) wall. (Eq. (\[heffbly\]). We will see below that this coefficient $\be$ is quite large even assuming standard semiconducting systems and reduces much the threshold current and enhances the wall velocity. The effect would be even stronger if the systems has giant Rashba effect as realized in metallic surfaces[@Ast07].
The effect of effective magnetic field due to Rashba interaction in Bloch(z) and Neel(x) cases is to induce anisotropy within $\phi$-plane as seen as $\cos\phi$ and $\sin \phi$ terms in Eqs. (\[eq:eqm291\]) and (\[eq:eqmxx\]), respectively. This anisotropy energy turns out to drive stepwise wall motion at low current.
We also see from Eqs. (\[eq:eqm291\]) and (\[eq:eyy106\]) that spin transfer torque effect (represented by the second term of right-hand side) is enhanced by Rashba interaction, at the second oder of $\la$ in Bloch(z) and Bloch(y) cases, but not in Neel(x) case. Numerically, these second order effects are negligibly small as we will show below.
Domain wall dynamics
====================
We first note that domain wall velocity $\tdt{X}$ is closely related to $\tdt{\phi}$ [@begt2]. In fact, from Eqs. (\[eq:eqm292\]),(\[eq:eqx104\]),(\[eq:eyq302\]), we see that and
Here, we show numerical results based on the equation of motion. Parameter are chosen as to simulate actual semiconductor systems [@yamano]. (We thus have $\frac{\hbar^2}{m\ell^2 \Ef}\simeq 0.007$, and so the second order contribution from Rashba interaction is very small like $\tilde\lambda^2 \frac{\hbar^2}{m\ell^2 \Ef}\simeq 0.7\times 10^{-4}$ for $\tilde\lambda=0.1$.) We calculated the domain wall position $\ti{X}(t)$ after current is applied at $t = 0$.
![Wall position at $t=1\mu$s as function of dimensionless current, $\tilde{J}=J/v_c$. Solid line represents the case $\tilde\lambda=0$, and dashed and dotted lines represent the case $\tilde\lambda=0.1$ for Bloch(z), Néel(x) and Bloch(y) walls, respectively. []{data-label="fig:grawall01"}](kyori.eps "fig:"){width="16cm"}\
Fig. \[fig:grawall01\] shows the wall position at $t = 1 \mu m$ after current is applied. (Current is normalized by $v_c$, the electron drift velocity at intrinsic threshold.) The case without Rashba interaction is shown as solid line. We see here the intrinsic pinning due to hard axis anisotropy[@begt2], since we do not consider $\beta$ term of non-Rashba origin. Wall motion in the presence of Rashba interaction with $\tilde\lambda=0.1$ is plotted by lines marked by $\times$ for Bloch(z) and Neel(x) and $\ast$ for Bloch(y) walls. Bloch(z) and Neel(x) walls behaves essentially the same.
Bloch(y) wall
-------------
We immediately see that Rashba interaction affects Bloch(y) wall drastically, resulting in vanishing of threshold current and very high velocity. This is due to a large effective $\be$ term induced by Rashba interaction, Eq. (\[eq:beta67\]). (Note that we do not consider extrinsic pinning.) For the present parameters, its ratio to $\alpha$ is given as $\frac{\beta}{\alpha}=\frac{\tilde{\lambda}}{\alpha}$, and so very large value of $\beta/\alpha\sim 10$ can be realized in actual experiment with $\tilde\lambda\sim0.1$ and $\alpha=0.01$. Terminal wall velocity of Bloch(y) wall is plotted for different values of $\tla$ in Fig. \[fig:gravelo\].
![Plot of wall velocity as function is current for $\tla=0,0.001,0.005,0.01$ and $0.1$. Damping constant is $\al = 0.01$. []{data-label="fig:gravelo"}](j-velo.eps "fig:"){width="14cm"}\
The behavior is consistent with previous studies including $\be$ term [@Thiaville05; @TTKSNF06] indicating that the present motion of Bloch(y) is governed by $\be$ term induced by Rashba interaction. It is natural since $\be$ term is linear in $\tla$ Eq. (\[eq:beta67\]) while other effects of Rashba interaction enter at the second order (see Eq. (\[eq:eyy106\])). Within our analysis, which neglects the extrinsic pinning, threshold current of Bloch(y) is zero due to $\be$ term. In reality, however, finite threshold will appear from pinning potential [@Thiaville05; @TTKSNF06]. Due to the $\beta$ term, we have obtained very large wall velocity (like 100m/s) at very small current (like 10% of intrinsic threshold current). In reality, Gilbert damping ($\al$) is also modified by Rashba interaction, which might slow the velocity $v$ according to $v\propto\beta/\alpha$[@TTKSNF06]. The modification of $\alpha$ is, however, second order in $\la$ [@hanki07; @HK08] and thus small, hence the high wall velocity would remain unchanged.
Bloch(z) and Neel(x) walls
--------------------------
Behaviors of Bloch(z) and Néel(x) walls are essentially the same for present values of parameters, and are distinct from Bloch(y) wall. These walls have a plateau in $v$-$J$ curve for $0.8 \leq \ti{J} \leq 1.3$ as seen in Fig. \[fig:grawall01\]. This plateau is due to a step motion of wall induced by the anisotropy field within $\phi$-plane arising from Rashba interaction. This stepwise motion is induced above threshold current of $\sim 0.8$, and the distance the wall moves is about 22$\mu m$ regardless of current density (for $0.8 \leq \ti{J} \leq 1.3$).
Let us see the mechanism of the plateau in detail. Dynamics near intrinsic pinning threshold is described by a potential for $\phi$[@begt2], obtained from the Lagrangian (Eq. (\[eq:lez274\]) and Eq. (\[eq:lex275\])) as follows: These potentials are plotted in Fig. \[fig:grapote\].
In the absence of current, the potential is solely by magnetic anisotropy, and energy minimum is at $\phi =0$ (Fig. \[fig:grapote\](1)). When current is applied, the potential tilts (Fig. \[fig:grapote\] (2)). When local minimum near $\phi = 0$ vanishes, the wall starts to move, and this gives the intrinsic threshold current of $\tilde{J_c}=1$ in the absence of Rashba interaction. [@begt2] When Rashba interaction is switched, $\pi$-periodicity of potential is broken due to the last terms in Eqs. (\[eq:bpz305\])(\[eq:npx306\]). The way of deformation depends on Bloch(z) and Néel(x) cases. Let us consider a Néel(x) wall case where the local energy barrier around $\pi\sim\frac{\pi}{2}$ is lowered by Rashba interaction. The variable $\phi$ (and wall) starts to move at threshold current (where the local minimum disappears), which is lowered by Rashba interaction. But this motion stops if current is not large enough since $\phi$ is trapped by the next local minimum near $\phi \sim \pi$, which has a large energy barrier around $\pi\sim \frac{3\pi}{2}$. Thus in this regime, $\phi$ hops roughly by the amount $\Delta \phi = O(\pi)$ and this corresponds by Eq. (\[eq:beta35\]) to a wall shift of $\Delta X = \frac{\ell}{\al} \Delta \phi$. This distance corresponds to the distance $\sim 20 \mu$ m seen in Fig. \[fig:grawall01\]. If current is sufficiently large to remove the second local minimum, the wall motion continues and terminal velocity grows (for $\tilde{J}>1.3$ in Fig. \[fig:grawall01\]). This is the mechanism of plateau.
An interesting consequence of the step motion is an asymmetric ratchet motion. We consider a Néel wall initially at $\phi=0$ (we choose $\tilde\lambda>0$). When we apply a negative current, $\tilde J<0$, the local minimum at $\phi=0$ is lifted and the energy barrier for right direction is lowered as we saw in Fig. \[fig:grapote\]. In contrast, when current is positive, $\tilde J>0$, the local minimum is lowered ($\propto -\tilde\lambda \tilde J$ by Eq. (\[eq:npx306\])) and then the effective energy barrier in the left direction becomes higher. Therefore, the threshold current for step motion differs by amount $\Delta\tilde J\sim \tilde\lambda$ for the two current directions (starting from fix $\phi$), and thus the wall behaves as a ratchet moving only in one way if current is small enough. These features are common for Bloch(z) wall case if $\phi=0$ is replaced by $\phi=\pi$.
The plateau region has finite initial velocity (but zero terminal velocity). Figure \[fig:bvaz\] shows the initial and terminal velocities for Bloch(z) and Néel(x) walls (the velocity is the same for two walls).
![The initial and terminal velocity as function of dimensionless current in the case of Bloch(z) and Neel(x) wall. The velocity is the same for the two walls. []{data-label="fig:bvaz"}](blochzvj.eps){width="15cm"}
We see that even plateau region shows a high velocity comparable to terminal velocity above $\ti{J} = 1$. For device application, motion over a distance of $20 \mu $m is large enough and so the plateau region would be quite useful for low current switching.
Conclusion
==========
We have theoretically calculated the effect of the Rashba spin-orbit interaction on the spin transfer torque. The effective Hamiltonian of local spin under current was calculated using gauge transformation, and the equation of motion for domain wall was derived. We considered three cases with different magnetic easy and hard axes, where domain wall structures realized are called Bloch(z), Neel(x) and Bloch(y) walls. We found there are three influences of Rashba spin-orbit interaction, namely, inducing effective magnetic field, increasing spin transfer torque and modification of magnetic anisotropy. The major effect is that of effective magnetic field, which arises at the first-order in Rashba interaction. Applying voltage in $x$-direction, the effective magnetic field is induced in $y$-direction via Rashba interaction. In case of Bloch(y) wall, where $y$-direction is the easy axis direction, we showed this field acts as a force which pushes the wall, or in other words, effective $\be$ term arises. Threshold current thus vanishes. The value of $\beta$ is large even if evaluated for common semiconductor systems, and wall velocity is enhanced greatly. In contrast, in the cases of Bloch(z) and Neel(x), the effective field is perpendicular the easy axis ($z$ and $x$ directions, respectively), and step motion of wall over a distance of $\Delta X \sim O(\frac{\pi \ell}{\al})$ arises at low current regime, corresponding to a change of the angle out of easy plane, $\Delta \phi \sim O(\pi)$. The current necessary for this step motion is lower than the case without Rashba interaction (by $20 \%$ at $\tla = 0.1$). The initial velocity of step motion is high enough (the same order as steady motion slightly above intrinsic threshold). Wall motion in the step motion regime is asymmetric with respect to current direction, i.e., wall behaves as a ratchet.
Other effects by Rashba interaction, modification of spin transfer torque and anisotropy, appears at second order, $\tla^2$, and are negligibly small. Change of spin transfer is due to the change of effective electron spin polarization by Rashba interaction.
Rashba interaction arises quite generally when inversion symmetry is broken, e.g., on surface of metals doped with heavy ions[@Nakagawa07; @Ast07]. Such systems would be suitable to realize quite high wall velocity at very small current we have predicted for Bloch(y).
The authors are grateful to H. Kohno, J. Shibata, J. Ohe and E. Saitoh for valuable discussion.
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|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Dendritic flux avalanches is a frequently encountered instability in the vortex matter of type II superconducting films at low temperatures. Previously, linear stability analysis has shown that such avalanches should be nucleated where the flux penetration is deepest. To check this prediction we do numerical simulations on a superconducting rectangle. We find that at low substrate temperature the first avalanches appear exactly in the middle of the long edges, in agreement with the predictions. At higher substrate temperature, where there are no clear predictions from the theory, we find that the location of the first avalanche is decided by fluctuations due to the randomly distributed disorder.'
author:
- 'J. I. Vestg[å]{}rden'
- 'Y. M. Galperin'
- 'T. H. Johansen'
title: 'Dendritic flux avalanches in rectangular superconducting films – numerical simulations'
---
Introduction
============
Dendritic flux avalanches have been observed in many kinds of type II superconducting films at low temperatures [@altshuler04]. The origin of the avalanches is a thermomagnetic instability mechanism between the Joule heating created by vortex motion and the reduction of the critical current density as temperature increases [@mints81; @rakhmanov04; @denisov05].
When transverse applied field is gradually increased, a critical state is formed from the edges, with almost constant current density, and non-zero magnetic flux density. The flux penetration is gradual and smooth until the conditions for onset of instability is fulfilled. Then, an avalanche is nucleated and large amounts of magnetic flux rushes in from the edges and forms a complex tree-like structure. The onset conditions for such events are determined by a competition between the Joule heating, heat diffusion and heat removal to the substrate [@denisov05]. From this theory, it is expected that instabilities are most likely to happen where the electric field is highest and the flux penetration is deepest. Since rectangles in low applied fields experience a flux penetration which is slightly deeper in the middle of the long edges [@brandt95], we hence expect the first avalanches to be nucleated there. At higher fields, the flux front straightens out, and it is less clear where the most favorable nucleation location will be.
In this work, we will run numerical simulations of a thermomagnetically unstable superconducting rectangle in applied transverse field. The location of the avalanches will be discussed in context of prediction from linear stability analysis. The simulation method is described in Ref. [@vestgarden11].
Model
=====
Consider a type II superconducting film subjected to an applied magnetic field transverse to the film plane, ramped at a constant rate $\dot H_a$. The shape of the sample is a rectangle with dimensions $2a\times 2b$.
A thermomagnetic instability is a consequence of the nonlinear material characteristics of type II superconductors, which conventionally is approximated by a power law $$\label{EJ}
\mathbf E = \frac{\rho_0}{d}\left(\frac{J}{J_c}\right)^{n-1}\mathbf J
,$$ where $\mathbf E$ is electric field, $\mathbf J$ is sheet current, $J=|\mathbf J|$, $\rho_0$ is a resistivity constant, $d$ is sample thickness, $J_c$ is critical sheet current, and $n$ is the creep exponent. The temperature dependencies are taken as $$J_c=J_{c0}(1-T/T_c),~~~ n=n_1/T,$$ where $T_c$ is the critical temperature. The electrodynamics must be supplemented by the heat diffusion equation $$\label{dotT}
c\dot T = \kappa \nabla^2T-\frac{h}{d}(T-T_0)+\frac{1}{d}JE
,$$ where $c$ is specific heat, $\kappa$ is thermal conductivity, $h$ is the coefficient for heat removal to the substrate, and $T_0$ is the substrate temperature.
Eq. must be solved together with Maxwell’s equations and the material law, Eq. . The description of the simulation method is in Ref. [@vestgarden11] and here only the key ingredients are outlined. The main challenge is to invert the Biot-Savart law in an efficient way. This is done by including also the vacuum outside the sample in the simulation formalism. At the cost of including the extra space, one can use a real space/Fourier space hybrid method with the very attractive performance scaling of $O(N\log
N)$, where $N$ is the number of grid points.
Quenched disorder is important in order to give realistic nucleation conditions, and it is introduced as a $5\%$ reduction of $J_{c0}$ in randomly selected $5\%$ of the grid points.
![ Distributions of magnetic flux density, $B_z$, at $H_a/J_{c0}=0.052$(a), 0.064(b), and 0.078(c). Substrate temperature is $T_0=0.17T_c$. The magnetic field is highest at the edges, seen as a white rim, while the black central region is still flux-free, i.e., $B_z=0$. The avalanches are all small, most of them are fingers, while in (c) some avalanches also have two or three branches. \[fig:B17\] ](fig1.pdf){width="\columnwidth"}
{width="18cm"}\
The parameters correspond to typical values for MgB$_2$ films [@schneider01]: $T_c = 39$ K, $j_{c0} = 1.2\times 10^{11}$ A/m$^2$, $\rho_n = 7~\mu\Omega$cm, $\kappa = 170$ W/Km$\times (T/T_c)^3$, $c = 35$ kJ/Km$^3\times (T/T_c)^3$, $h = 200$ kW/Km$^2\times (T/T_c)^3$, and $n_1=15$. Here $\rho_n$ is normal resistivity, and we have $J_{c0}=dj_{c0}$, and $\rho_0=\rho_n$. The sample dimensions are $2a = 8~$mm, $2b = 4~$mm, and $d = 0.4~\mu$m, while the total simulated area is $12\times 6$ mm$^2$, where the extra space outside the sample is used to implement the boundary conditions. The total area is discretized on a $768 \times 384$ equidistant grid.
Analytical prediction
=====================
The linear stability analysis in Ref. [@denisov06] determines the condition for instability onset as $$l^*=\frac{\pi}{2}\sqrt{\frac{d\kappa}{|J_c'|E}}\left(1-\sqrt{\frac{h}{n|J_c'|E}}\right)^{-1}$$ where $l^*$ is the threshold flux penetration depth, $|J_c'|=J_{c0}T_0/T_c$, and $E$ is the background electric field. For a given $E$, the sample is unstable when the flux front exceeds $l^*$, which means that avalanches should appear first where the flux penetration is deepest. In a rectangle, this is in the middle of the two long sides and consequently it is expected that the first avalanches appear there.
Simulation results
==================
The simulations start from initially zero-field cooled state, and the applied field is increased at constant rate, $\mu_0\dot H_a=10~$T/s. The high ramp rate was chosen by performance reasons, since a high ramp rate closes the gap between the velocity of the avalanches and the normal flux penetration. Yet, there is clear separation of time scales, since full penetration is reached in approximately $J_{c0}/\dot H_a=6$ ms, while the duration of the avalanches is less than $0.1~\mu$s.
Fig. \[fig:B17\] shows the distributions of magnetic flux density transverse to the film plane, $B_z$, at $H_a/J_{c0}=0.053$(a), 0.064(b), and 0.078(c). The substrate temperature is $T_0=0.17T_c$, and at this low temperature the threshold field for avalanche activity is low, $H_\text{th}=0.052J_{c0}$. Image (a) is just above the threshold, where two small fingers have appeared symmetrically in the middle of the two long edges, just as expected from the theory. In image (b) many more avalanches have come at the long sides, and for the first time there are avalanches appearing at the two short sides, also these in the middle. In image (c) there are even more avalanches and they now cover most of the boundary, except close to the corners. All avalanches are small, and will consequently be seen as small, jumps towards zero in magnetization curves. With increasing field, it seems like a trend of increasing avalanche size.
Fig. \[fig:B22\] shows $B_z$ for a simulation run at a higher substrate temperature, $T_0=0.22T_c$. The figure contains just one avalanche, which appeared at $H_\text{th}=0.12J_{c0}$. Both the increased $H_\text{th}$ and the much larger size of the avalanche is as expected for higher $T_0$ [@denisov06]. Also expected is the complex branching pattern [@vestgarden11]. The location is not entirely symmetric. The reason is that at deeper flux penetrations the flux front straightens out, as seen in regular Bean-state of the upper part of the image. Hence, the location of the avalanche is instead determined by the fluctuations in electric field due to the quenched disorder. The avalanche location is typical for what is seen experimentally, e.g., by magneto-optical images in NbN rectangles [@yurchenko07].
Conclusions
===========
We have simulated dendritic flux avalanches in superconducting films in the shape of a rectangle. The results confirm the prediction from linear stability analysis that avalanches will first appear in the middle of the long sides, at least for low $T_0$. At higher $T_0$, the avalanche location was not symmetric. In general, the results regarding avalanche size, threshold field, and time between avalanches follow the typical dependency on $T_0$, i.e., that increasing $T_0$ gives larger $H_\text{th}$, larger avalanches, more branches, and longer time between the avalanches. Hence, the simulations of this work also confirms the correctness of the simulation method of Ref. [@vestgarden11].
[1]{} url \#1[`#1`]{}urlprefix
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A. L. Rakhmanov, D. V. Shantsev, Y. M. Galperin, T. H. Johansen, Finger pattern produced by thermomagnetic instability in superconductors, Phys. Rev. B 70 (2004) 224502.
D. V. Denisov, A. L. Rakhmanov, D. V. Shantsev, Y. M. Galperin, T. H. Johansen, Dendritic and uniform flux jumps in superconducting films, Phys. Rev. B 73 (1) (2006) 014512.
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J. I. Vestg[å]{}rden, D. V. Shantsev, Y. M. Galperin, T. H. Johansen, Dynamics and morphology of dendritic flux avalanches in superconducting films, Phys. Rev. B 84 (2011) 054537.
M. Schneider, D. Lipp, A. Gladun, P. Zahn, A. Handstein, G. Fuchs, [S.-L. Drechsler]{}, M. Richter, [K.-H. M[" u]{}ller and H. Rosner]{}, Heat and charge transport properties of [MgB$_2$]{}, Physica C 363 (2001) 6.
D. V. Denisov, D. V. Shantsev, Y. M. Galperin, E.-M. Choi, H.-S. Lee, S.-I. Lee, A. V. Bobyl, P. E. Goa, A. A. F. Olsen, T. H. Johansen, Onset of dendritic flux avalanches in superconducting films, Phys. Rev. Lett. 97 (2006) 077002.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper we propose a dispersive method to describe two-body scattering with unitarity imposed. This approach is applied to elastic $\pi\pi$ scattering. The amplitudes keep single-channel unitarity and describe the experimental data well, and the low-energy amplitudes are consistent with that of chiral perturbation theory. The pole locations of the $\sigma$, $f_0(980)$, $\rho(770)$ and $f_2(1270)$ and their couplings to $\pi\pi$ are obtained. A virtual state appearing in the isospin-two S-wave is confirmed. The correlations between the left (and right) hand cut and the poles are discussed. Our results show that the poles are more sensitive to the right hand cut rather than the left hand cut. The proposed method could be used to study other two-body scattering processes.'
author:
- 'Ling-Yun Dai'
- 'Xian-Wei Kang'
- Tao Luo
- 'Ulf-G. Mei[ß]{}ner'
title: 'A study on the correlation between poles and cuts in $\pi\pi$ scattering'
---
=3.5mm
Introduction {#sec:introduction}
============
In a two-body scattering system, for example two hadrons, the general principals that we know are unitarity, analyticity, crossing, the discrete symmetries, etc. The resonances that appear as the intermediate states in such system are important. Among them the lightest scalar mesons, related to $\pi\pi$ scattering, have the same quantum numbers as the QCD vacuum and are rather interesting, for some early references, see [@MRP10; @jaffe4q; @Meissner:1990kz]. The $\pi\pi$ scattering amplitude is also crucial to clarify the hadronic contribution to the anomalous magnetic moment of the muon, see e.g. [@Colangelo:2018mtw; @Danilkin:2018qfn]. To study the resonances in a given scattering process, one needs dispersion relations to continue the amplitude from the real $s$-axis (the physical region) to the complex-$s$ plane [@Kang:2013jaa; @DLY-MRP14; @Chen2015; @Hanhart:2016pcd], where the pole locations and their couplings are extracted. Following this method, some work on the light scalars can be found in [@zheng00; @colangelo01; @zhou04; @Moussallam06; @caprini06; @PelaezPRL], where the accurate pole locations and residues of the $\sigma$ and $\kappa$ mesons are given. For the dispersive methods, a key problem is how to determine the left hand cut (l.h.c.) and the right hand cut (r.h.c.), with the unitarity kept at the same time. In Refs. [@DLY-MRP14; @Chen2015] the l.h.c is estimated by crossed-channel exchange of resonances, where chiral effective field theory ($\chi$EFT) is used to calculate the amplitude. And the contribution of r.h.c. is represented by an Omnès function, with unitarity kept. In the well-known Roy equations, crossing symmetry and analyticity are perfectly combined together as the l.h.c is represented by the unitary cuts of the partial waves. The single channel unitarity is also well imposed by keeping the real part of the partial wave amplitudes the same as what is calculated by the phase shift directly, which could be obtained by fitting to the experimental data in some analyses. Until now, Roy and Roy-Steiner equations certainly give the most accurate description of the two-body scattering amplitude and the information of resonances appearing as the intermediate states, such as $\pi\pi$, $\pi K$ scattering and the pole locations and residues of the $\rho$, $\sigma$, $f_0(980)$ and $\kappa$, etc., see e.g. [@Moussallam06; @caprini06; @PelaezPRL]. In addition, Ref. [@caprini06] shows that the l.h.c. can not be ignored for the determination of the pole location of the $\sigma$. By removing the parabola term of the l.h.c., the $\sigma$ pole location is changed by about 15% accordingly, while the unitarity is violated due to the removal of the l.h.c.. And thus the method to get the poles on the second Riemann sheet, calculated from the zeros of the S-matrix, is not reliable any more, as the method is based on the continuation implemented by unitarity. Here, we focus on obtaining a quantitative relation between cuts and poles, with unitarity imposed and the l.h.c. and r.h.c. are correlated with each other.
This paper is organized as follows: In Sect. II we establish a dispersive method based on the phase. In the physical region we also represent the amplitudes by an Omnès function of the phase above threshold. In Sect. III we fit the $\pi\pi$ scattering amplitudes up to 1 GeV in a model-independent way, including the $IJ=00,02,11,20$ waves, where $I$ denotes the total isospin and $J$ the angular momentum. The fit results are the same as those given by the Omnès function representation and comparable with those of chiral perturbation theory ($\chi$PT) in the low-energy region. The poles and couplings are also extracted. In Sect. IV we give the estimation of the relation between poles and cuts, including both the l.h.c and the r.h.c. . We end with a brief summary.
Scattering amplitude formalism {#sec:formalism}
==============================
A dispersive representation {#sec:sub;single}
---------------------------
The two-body scattering amplitude can be written as: T(s)= f(s) e\^[i (s)]{}, \[eq:T\] with $\varphi(s)$ the phase and $f(s)$ a real function. By writing a dispersion relation for $\ln T(s)$, one has: T(s)&=&f(s\_0) +\_L\
&+&\_R .\[eq:T;DR\] Here, $s_0$ is chosen at a specific point where the amplitude is real, and L’ denotes the l.h.c. and R’ stands for the r.h.c.. The amplitude turns into T(s)=T(s\_0)\_L(s)\_R(s).\[eq:T;Omnes\] On the other hand, unitarity is a general principal required for the scattering amplitude. In the single channel case one has T(s)=(s) |T(s)|\^2 ,\[eq:unitarity\] where $s$ is in the elastic region and $\rho(s)$ is the phase space factor. Substituting Eq. (\[eq:T;Omnes\]) into Eq. (\[eq:unitarity\]), we obtain a representation (in the elastic region) for a single channel scattering amplitude T(s)=-.\[eq:unitarity;ph\] Also, the Omnès function of the phase for the l.h.c. is correlated with that of the r.h.c. \_L(s)=-,\[eq:unitarity;lhc\] which is again valid in the elastic region. A simple way to get the two-body scattering amplitude proceeds in two steps: First, we follow Eq. (\[eq:unitarity;ph\]) to fit the Omnès function of the r.h.c. to experimental data, and then use Eq. (\[eq:unitarity;lhc\]) and other constraints below the threshold to fit the Omnès function of the l.h.c. . Note that Eq. (\[eq:unitarity;ph\]) does not only work for the single channel case, but also for the coupled channel case in the physical region.
On $\pi\pi$ scattering {#sec:sub;th}
----------------------
In the equations above, the threshold factor is not included. Considering such factors, we need to change the amplitudes into: T\^I\_J(s)=(s-z\^I\_J)\^[n\_J]{}f\^I\_J(s)e\^[i\^[IJ]{}(s)]{}.\[eq:T;th\] Here and in what follows, we take $\pi\pi$ scatering as an example. Thus one has $z^I_J=4 M_\pi^2$ for the P-, D-, and higher partial waves, and $z^I_J$ is the Adler zero for the S-waves. $n_J$ is one for S- and P-waves and two for D waves. We define a reduced amplitude \^I\_J(s)=f\^I\_J(s)e\^[i\^[IJ]{}(s)]{},\[eq:T;red\] and again we can write a dispersion relation for $\ln\tilde{T}^I_J(s)$, so that we have \^I\_J(s)&=&f\^I\_J(s\_0) +\_[-]{}\^[0]{}\
&+&\_[4M\_\^2]{}\^ .\[eq:T;DR;red\] Here, $s_0$ could be chosen from the range $[0,4 M_\pi^2]$. For the r.h.c., we cut off the integration somewhere in the high energy region, see discussions in the next sections. We have f\^I\_J(s\_0)=, and T\^I\_J(s)=T\^I\_J(s\_0)()\^[n\_J]{}\^[IJ]{}\_L(s)\^[IJ]{}\_R(s).\[eq:T;Omnes;red\] The $T^I_J(s_0)$ could be fixed by $\chi$PT or scattering lengths, or other low-energy constraints. For simplicity, we choose $s_0=0$. Combining unitarity, embodied by Eq. (\[eq:unitarity;ph\]), we have a correlation between Omnès functions of l.h.c. and r.h.c. in the elastic region \^[IJ]{}\_L(s)=-.\[eq:unit;lhc\] This is similar to Eq. (\[eq:unitarity;lhc\]). Substituting Eq. (\[eq:unit;lhc\]) into Eq. (\[eq:T;Omnes;red\]), we still have Eq. (\[eq:unitarity;ph\]). Since we know the $\pi\pi$ scattering amplitudes well in the region \[$4M_\pi^2$, 2 GeV$^2$\] and $\chi$PT describes the amplitudes well in the low-energy region, we have to fit the l.h.c. to both Eq. (\[eq:unitarity;lhc\]) and $\chi$PT.
Phenomenology
==============
Fits
----
For the $\pi\pi$ scattering amplitude, we can parametrize the phase caused by the l.h.c. by a conformal mapping \^[IJ]{}\_L(s)=\_[n=1]{}\^k c\^[IJ]{}\_n [Im]{}\[(s)\]\^n ,\[eq:phiL\] with (s)=. \[eq:comformal\] Notice that ${\rm Im}~\omega(s)$ behaves as $\sqrt{-s^3}$ around $s=0$, which is consistent with that of$\chi$PT, see Ref. [@zhou04] and references therein.
As concerns the r.h.c., it is less known in the high energy region. However, these distant r.h.c. have less important effects in the low-energy region, especially in the region $s\leq 1$ GeV$^2$. We choose three kinds of $\Omega^{IJ}_R(s)$ to test the stability and uncertainty caused by the distant r.h.c.. In Case A, the phases [@DLY-MRP14] are cut off at $s=2.25$ GeV$^2$. In Case B, the phases are given by [@DLY-MRP14], up to $s=22$ GeV$^2$. In Case C, the phases/Omnès functions of the r.h.c. are given by [@Dai:2017tew; @Dai:2017uao] and references therein, up to $s=22$ GeV$^2$. Here, the phases are fitted to the experimental data [@CERN-Munich; @OPE1973] up to $\sqrt{s}=2$ GeV and constrained by unitarity up to $\sqrt{s}=4$ GeV. Notice that in Case A and B the phase of the isospin-one P-wave is given by CFDIV [@KPY], and we continue it to the higher energy region by means of the function \^[11]{}\_[R,h]{}(s)=\^[11]{}\_+B\[k,n\]()\^k+C\[k,n\] ()\^n, with B\[k,n\]&=&\^[11]{}\_[R]{}(s\_R)-\^[11]{}\_-C\[k,n\],\
C\[k,n\]&=&. The function (and also its first derivative) is smooth at the point $s_R$. We set $k=1$, $n=2$, $s_R=1.4^2$ GeV$^2$ and $\varphi^{11}_{\infty}=160^\circ$, which is close to $\varphi^{11}_R((1.4\, \text{GeV})^2)=170^\circ$ and ensures that the phase in the high energy region behaves smoothly. The upper limits of the integration of the r.h.c. of isospin-one P-wave are the same as the other partial waves.
The parameters of our fits for all the Cases are given in Tab. \[tab:para\].
The $c^{IJ}_n$ are determined by the following procedure. In the elastic region, we choose one or two mesh points’, depending on how many coefficients $c^{IJ}_n$ we require. Combining Eqs. (\[eq:unit;lhc\],\[eq:phiL\]), we can build a matrix and solve for $c^{IJ}_n$. This strategy gives a good description of the amplitudes, with unitarity kept. See the fit results shown in Fig. \[Fig:T\].
![\[Fig:T\] Fit of the $\pi\pi$ scattering amplitudes for Case B. Notice that $\tilde{E}=\rm{sgn}(s)\sqrt{s}$. The solid lines denote the real part of the amplitudes and the dotted, dashed, dash-dotted and dash-dot-dotted lines denote the imaginary part. The black lines are from our fit. The red lines are from a K-matrix fit [@DLY-MRP14] for the isospin-zero S-wave, and the violet lines are from CFDIV [@KPY] for other waves. The borders of the cyan and green bands in the low-energy region are from $SU(2)$ and $SU(3)$ $\chi$PT, respectively. The CERN-Munich data are from Ref. [@CERN-Munich], and the OPE and OPE-DP data are from [@OPE1973].](T.eps){width="48.00000%" height="0.35\textheight"}
Here all the partial waves refer to Case B, in which the phase is cut off at $s=2.25$ GeV$^2$. The amplitudes from the other Cases are quite close to this one, except for the inelastic region and the distant l.h.c. ($\tilde{E}\leq-0.4$ GeV$^2$). Our fit, both the real part (black solid line) and imaginary (black dotted line) part of the amplitudes shown in Fig. \[Fig:T\], is indistinguishable from that given by the K-Matrix [@DLY-MRP14] or CFDIV [@KPY]. Note that the amplitudes given by Eq. (\[eq:unitarity;ph\]) are exactly the same as those of the K-Matrix or CFDIV from $\pi\pi$ threshold to the inelastic threshold. This implies that the unitarity is respected. To test it quantitatively, we define \_[ T\^I\_J]{}&=&\_[n=1]{}\^N. \[eq:Delta;T\] $\Delta T^I_J(s_n)$ is the difference between our amplitude and that of Eq. (\[eq:unitarity;ph\]). Here we choose $s_n=0.1-0.9$ GeV$^2$ for the S-waves, $s_n=0.1-0.8$ GeV$^2$ for the P-wave, and $s_n=0.1-1.0$ GeV$^2$ for the D-wave, with step of 0.1 GeV$^2$. These points are located between the $\pi\pi$ and the inelastic thresholds. From here on all the steps are chosen to be 0.1 GeV$^2$ (or 0.1 GeV for $\tilde{E}$). We find that $\mathcal{R}_{ T^0_S}=0.1\%$, $\mathcal{R}_{ T^1_P}=0.1\%$, $\mathcal{R}_{ T^2_S}=1.4\%$, and $\mathcal{R}_{ T^0_D}=1.4\%$. The violation of unitarity is rather small.
For $T^I_J(0)$, $\chi$PT could be used to fix it. The analytical $SU(3)$ 1-loop $\chi$PT amplitudes of each partial waves, are recalculated and given in Appendix. \[app:ChPT\]. The low-energy constants are given by [@Bijnens:2014lea]. Those of $SU(2)$ 2-loop $\chi$PT amplitudes are given by [@Bijnens:1997vq; @zhou04] and references therein. All the values of $T^I_J(0)$ in Tab. \[tab:para\] are close to the prediction of $\chi$PT or our earlier analyses [@Dai:2017tew; @Dai:2017uao]. In the isospin-zero S-wave, the magnitude of our $T^0_0(0)$ is a bit larger than that of $\chi$PT. This is consistent with what is known about this scattering length, where the one-loop $\chi$PT calculation gives a smaller result than what is obtained by dispersive methods, Roy equations or in experiment, see e.g. the review [@Bijnens:2014lea]. A better comparison would be given with the 2-loop $\chi$PT amplitudes.
In the isospin-zero D-wave, the $T^0_2(0)$ varies more in the different Cases. The reason is that some fine-tuning is needed as the inelastic r.h.c. is difficult to be implemented well. The amplitudes given by Eq. (\[eq:unitarity;ph\]) are much different from that of CFDIV in the inelastic region where the $f_2(1270)$ appears. Notice further that the value of $T^0_D(0)$ is very small, one order smaller than that of the other waves.
Pole locations and couplings
----------------------------
With these amplitudes given by a dispersion relation, the information of the poles can be extracted. The pole $s_R$ and its coupling/residue $g_{f\pi\pi}$ on the second Riemann sheet are defined as T\^[II]{}(s)=.\[eq;g\] Note that the continuation of the $T(s)$ amplitude to the second Riemann sheet is based on unitarity, T\^[II]{}(s+i)=T\^[I]{}(s-i)=.\[eq;T;II\] The poles and couplings/residues for Cases A,B,C are given in Table \[tab:poles;case\].
[|@c @|c | c |c |@c @|@c@ |]{}
------------------------------------------------------------------------
&
------------------------------------------------------------------------
& pole locations &\
------------------------------------------------------------------------
& & (MeV) & $~|g_{f\pi\pi}|~(GeV)~$ & $~\phi~(^\circ)~$\
& A & $432.5-i269.8$ & $0.46$ & $-77$
------------------------------------------------------------------------
\
& B & $442.7-i270.5$ & $0.48$ & $-74$
------------------------------------------------------------------------
\
& C & $438.2-i270.6$ & $0.47$ & $-75$
------------------------------------------------------------------------
\
& A & $997.5-i19.0$ & $0.25$ & $-81$
------------------------------------------------------------------------
\
& B & $997.6-i21.6$ & $0.27$ & $-83$
------------------------------------------------------------------------
\
& C & $997.6-i20.5$ & $0.26$ & $-82$
------------------------------------------------------------------------
\
& A & $1260.9 -i111.2$ & $0.55$ & $-10$
------------------------------------------------------------------------
\
& B & $1294.1 -i57.9$ & $0.52$ & $11$
------------------------------------------------------------------------
\
& C & $1266.0 -i99.5$ & $0.54$ & $-8$
------------------------------------------------------------------------
\
& A & $761.1 -i70.6 $ & $0.34$ & $-12$
------------------------------------------------------------------------
\
& B & $763.0 -i73.3 $ & $0.35$ & $-11$
------------------------------------------------------------------------
\
& C & $761.3 -i71.7 $ & $0.34$ & $-12$
------------------------------------------------------------------------
\
& A & $29.8 $ & $9.8\times10^{-3}$ & $90$
------------------------------------------------------------------------
\
& B & $29.8 $ & $9.8\times10^{-3}$ & $90$
------------------------------------------------------------------------
\
& C & $32.3 $ & $11.0\times10^{-3}$ & $90$
------------------------------------------------------------------------
\
All the poles and residues of the different Cases are close to each other, except for the pole location of the $f_2(1270)$. The reason is that the $f_2(1270)$ is located outside the elastic unitary cut of $T^0_D(s)$, while Eq. (\[eq:unit;lhc\]) only works in the elastic region. For this partial wave one needs a more dedicated method to study, including coupled-channel unitarity. For the poles of the $\sigma$, the differences between the different Cases is a also bit larger than those of other resonances such as the $\rho(770)$ and the $f_0(980)$. This is because the $\sigma$ is far away from the real axis. For the virtual state in the isospin-two S-wave, the poles and residues are a bit different from Cases A and B to Case C. This situation is comparable with that of $T^2_S(0)$, where in Cases A and B $T^2_S(0)$ is 0.055 and in Case C it is 0.060, respectively.
In addition, we also find that there exists a virtual state in the isospin-two S-wave very close to $s=0$ [^1]. According to Eq. (\[eq;T;II\]), the virtual state a the zero of the S-matrix below the threshold. This zero equals to the intersection point between two lines: $T^2_S(s)$ and $i/2\rho(s)$. As shown in Fig. \[Fig:T;2S\], the line of the $T^2_S(s)$ and the line of $i/2\rho(s)$ will always intersect with each other and the crossing point always lies in the energy region of \[0,$s_a$\], where $s_a$ is the Adler zero. This is the virtual state. Since the scattering length is negative and the Adler zero (only one) is below threshold, one would expect that the amplitude of $T^2_S(s)$, from $s=4 M_\pi^2$ to $s=0$, will always cross the real axis of $s$ and arrive at the positive vertical axis. In all events, it will intersect with that of the $i/2\rho(s)$. Thus the existence of the virtual state is confirmed. This inference is model-independent, only the sign of the scattering length, [^2] the Adler zero, and analyticity are relevant.
![\[Fig:T;2S\] The lines of $T^2_S(s)$ for the different Cases and $i/2\rho(s)$. Note that all of them are real. The intersection point corresponds to the virtual state. ](T2S.eps){width="48.00000%" height="0.25\textheight"}
For a general discussion of the virtual state arising from a bare discrete state in the quantum mechanical scattering, we recommend readers to read [@Xiao2016; @Xiao:2016mon] and references therein. We suggest that the isospin-two S-wave amplitude could be checked in the future measurement of $\Lambda_c^+\to\Sigma^-\pi^+\pi^+$. Its branching ratio [@Ablikim:2017iqd] is large enough.
The average values of the poles and residues of all the Cases define our central values. The deviations of the different Cases to the central values are used to estimate the uncertainties. The results are shown in Tab. \[tab:poles\].
[|@c @ | c |c |@c @|@c@ |]{}
------------------------------------------------------------------------
& pole locations &\
------------------------------------------------------------------------
& (MeV) & $~|g_{f\pi\pi}|~(GeV)~$ & $~\phi~(^\circ)~$\
$\sigma/f_0(500)$ & $437.8(52)-i270.3(5)$ & $0.47(1)$ & $-75(2)$
------------------------------------------------------------------------
\
$f_0(980)$ & $997.6(1)-i20.3(13)$ & $0.26(1)$ & $-82(1)$
------------------------------------------------------------------------
\
$f_2(1270)$ & $1273.7(179)-i89.5(280)$ & $0.53(2)$ & $-3(12)$
------------------------------------------------------------------------
\
$\rho(770)$ & $761.8(11) -i71.9(14) $ & $0.34(1)$ & $-12(1)$
------------------------------------------------------------------------
\
$2S~~v.s.$ & $30.6(15) $ & $10.2(7)\times10^{-3}$ & $90(0)$
------------------------------------------------------------------------
\
These are very similar from those of previous analyses [@caprini06; @DLY-MRP14; @PelaezPRL; @Rusetsky2011; @PDG16]. The $f_2(1270)$ has a much larger uncertainty compared to the other resonances, just as discussed before. The residues of all resonances have roughly similar magnitude at the region \[0.25,0.55\] GeV, except for that of the virtual state in the isospin-two S-wave, which is much weaker. But their phases are quite different. The phases of $\rho(770)$ and $f_2(1270)$ are close to zero, while those of the $\sigma$ and $f_0(980)$ are close to $-90^\circ$, and the virtual state one is close to $90^\circ$. This may imply that $\rho(770)$ and $f_2(1270)$ are normal $\bar{q}q$ states but that the $\sigma$ and $f_0(980)$ have large molecular components.
The correlation between poles and cuts
--------------------------------------
It is interesting to find the correlation between the poles and cuts. We focus here on the isospin-zero S-wave and isospin-one P-wave, as the $f_2(1270)$ is far away from the l.h.c and the virtual state is too close to the l.h.c.. Also, the light scalars are more difficult to understand. All the fits of different Cases about these two partial waves are shown in Fig. \[Fig:T;lhc\].
![\[Fig:T;lhc\] Comparison of different solutions of the $\pi\pi$ scattering amplitudes. The solid lines are the real part of the amplitudes and other lines are the imaginary part. The green lines are from Case A, the cyan lines are from Case B, and the black lines are from Case C. The red lines are from K-Matrix [@DLY-MRP14] for isospin 0 S-wave, and the violet lines are from CFDIV [@KPY] for isospin 1 P-wave. Note that the lines of K-matrix and/or CDFIV are overlapped with our fits in the elastic region, or even a bit further in the inelastic region. ](T0S.eps "fig:"){width="23.00000%" height="0.15\textheight"} ![\[Fig:T;lhc\] Comparison of different solutions of the $\pi\pi$ scattering amplitudes. The solid lines are the real part of the amplitudes and other lines are the imaginary part. The green lines are from Case A, the cyan lines are from Case B, and the black lines are from Case C. The red lines are from K-Matrix [@DLY-MRP14] for isospin 0 S-wave, and the violet lines are from CFDIV [@KPY] for isospin 1 P-wave. Note that the lines of K-matrix and/or CDFIV are overlapped with our fits in the elastic region, or even a bit further in the inelastic region. ](T1P.eps "fig:"){width="23.00000%" height="0.15\textheight"}
In our approach only unitarity is used to constrain the amplitudes, but the low-energy amplitudes are consistent with those of $\chi$PT. Only in Cases B and for the isospin-zero S-wave, the $T^0_S(s)$ amplitude is inconsistent with that of $\chi$PT at $\tilde{E}<-0.4$ GeV. This implies that unitarity has a strong constraint on the low-energy amplitudes below $\tilde{E}=0$. This could also be simply checked by using Eq. (\[eq:Delta;T\]), with $\tilde{E}=-0.1$ to $-0.4$ GeV and $s_n=-\tilde{E}_n^2$. Here, $\Delta T^I_J(s_n)$ is the difference between our amplitude and that of SU(2) $\chi$PT. Typically, in Case B, $\mathcal{R}^L_{ T^0_S}=21\%$, $\mathcal{R}^L_{ T^1_P}=16\%$, and they are quite close to those of other Cases. Note that the real parts of the amplitudes are more consistent with those of $\chi$PT, while the imaginary parts have a bit larger deviation (as is expected as imaginary parts start later in the chiral expansion).
To see the variation of the l.h.c. in the different solutions, we apply Eq. (\[eq:Delta\]) on ${\rm Im}T^I_J(s_n)$, with $\tilde{E}=-0.1$ to $-0.6$ GeV. Notice that at $\tilde{E}=0$ all of the l.h.c are zero and behave as $\sqrt{-s^3}$, this partly ensures the l.h.c to be consistent with that of $\chi$PT in the low-energy region. We fix the average value of all Cases as the central value, and calculate the relative deviation for each point. At last we avarage these relative deviations to estimate the variation of the cuts. The variation of cuts and poles are defined as \_[[Im]{} T\^I\_J]{}&=&\_[n=1]{}\^N,\
\_[pole]{}&=&,\[eq:Delta\] with $s_p$ the pole on the second Riemann Sheet. Finally, we collect the uncertainties in Tab. \[tab:para;C\]. And we define the correlation between poles and cuts as C\_[pole]{}=.\[eq:C\] The simple meaning of the correlation $C_{pole}$ is to answer the following question: When the cut is changed by 100%, how much would the pole location be changed?
To test the correlation between poles and the r.h.c., we simply set $\varphi_{test}(s)
= 1.04\varphi(s)$, and check the variation of poles and cuts, respectively. The relative uncertainty of the r.h.c. is also estimated by Eq. (\[eq:Delta\]), with $s_n=0.1-0.9$ GeV$^2$ for the isospin-zero S-wave and $s_n=0.1-0.8$ GeV$^2$ for the isospin-one P-wave. The relative uncertainty of the poles and the correlation are calculated in the same way as that of the l.h.c, see Eqs. (\[eq:Delta\],\[eq:C\]).
From Tab. \[tab:para;C\], we find that $C_\sigma^R$ is roughly two orders larger than that of $C_\sigma^L$, though $\sigma$ is rather close to the l.h.c.. Comparing to Ref. [@caprini06], which has roughly 15% contribution from l.h.c, we have a rather smaller contribution from the l.h.c, caused by the constraint of unitarity on the l.h.c. . Also, $C_{f_0(980)}^R$ is roughly three orders larger than that of $C_{f_0(980)}^L$, and $C_{\rho(770)}^R$ is roughly one order larger than that of $C_{\rho(770)}^L$. These indicate that the correlation between the unitarity cut and the poles is much larger than that of the l.h.c. and poles. Note that in our case the l.h.c. is not arbitrary but correlated with the r.h.c., constrained by unitarity and analyticity, see Eq. (\[eq:unit;lhc\]). For each Case, $C_\sigma^{L}$ is larger than $C_{f_0(980)}^{L}$. This is not surprising as the $\sigma$ is much closer to the l.h.c. . Also, $C_\sigma^{R}$ is larger than $C_{f_0(980)}^{R}$. The reason is that the $\sigma$ is farther away from the real axis, the uncertainty of the pole is larger as the amplitude is continued from the physical region deeper into the complex-$s$ plane. It is interesting to see that in average $C_\sigma^{L}$ is roughly two times larger than $C_{\rho(770)}^L$. And for the distance between these poles and l.h.c (simply set $s=0$), $|s_\sigma|$ is one half of that of $|s_\rho|$, this tells us that the correlation between poles and l.h.c is inversely proportional to their distance. In contrast, $C_\sigma^{R}$ is roughly one order larger than $C_{\rho(770)}^R$. For the distance between these poles and r.h.c. (simply set $s={\rm Re}~s_{pole}$), $|{\rm Im}~s_\sigma|$ is two times larger than $|{\rm Im}~s_\rho|$, this tells us that the correlation between poles and r.h.c. is proportional to their distance. These conclusions are still kept when comparing the $\sigma$ and the $f_0(980)$.
Summary {#sec:summary}
=======
We proposed a dispersive method to calculate the two-body scattering amplitude. It is based on the Omnès function of the phase, including that of the left hand cut and the right hand cut. The input of the r.h.c. is given by three kinds of parametrizations, and the l.h.c is solved by Eq. (\[eq:unit;lhc\]), with unitarity and analyticity respected. The pion-pion $IJ=00,02,11,20$ waves are fitted within our method and the poles and locations are extracted. They are stable except for that of the $f_2(1270)$, which lies in the inelastic region. The r.h.c. has much larger contribution to the poles comparing to that of the l.h.c.. This method could be useful for the studies of strong interactions in two-body scattering, and the $\pi\pi$ scattering amplitudes obtained here could be used for the future studies when one has $\pi\pi$ final state interactions, see e.g. [@AMP-FSI; @Dai:2012pb; @Gonzalez-Solis:2018xnw; @Ropertz:2018stk; @Cheng:2019hpq], and/or to multi-pions, see e.g. [@Guo:2015zqa; @Dumm:2009va].
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to Zhi-Yong Zhou for helpful discussions and for supplying us the files of the $SU(2)$ 2-loop $\chi$PT amplitudes. This work is supported by National Natural Science Foundation of China (NSFC) with Grant Nos.11805059, 11805012, 11805037, and Fundamental Research Funds for the Central Universities. TL also thanks support from the Joint Large Scale Scientific Facility Funds of the NSFC and Chinese Academy of Sciences (CAS) under Contract No. U1832121, and from Shanghai Pujiang Program under Grant No.18PJ1401000, Open Research Program of Large Research Infrastructures (2017), CAS. UGM acknowledges support from the DFG (SFB/TR 110, “Symmetries and the Emergence of Structure in QCD”), from the Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative (PIFI) (Grant No. 2018DM0034) and from VolkswagenStiftung (Grant No. 93562).
Analytical amplitudes of partial waves within $\chi$PT {#app:ChPT}
======================================================
The analytical 1-loop amplitudes of these partial waves within of $SU(3)$ $\chi$PT are recalculated. For reader’s convenience, they are given below. We have the IJ=00 waves up to $\mathcal{O}(p^4)$:
T\^[(2)]{}\_[0S]{}\[s\]&=&,\
t\_[0S,1]{}\[s\]&=&(3 s-4 M\_\^2) A\[M\_K\]+ (3 s-4 M\_\^2) A\[M\_\] + M\_\^4 B\[s,M\_\]+ s\^2 B\[s,M\_K\]\
&&+ (2 s-M\_\^2)\^2B\[s,M\_\]+(-40M\_\^2 s+44 M\_\^4+11 s\^2)L\_1\
&&+ (-20 M\_\^2 s+28 M\_\^4+7 s\^2)L\_2+(-40M\_\^2 s+44 M\_\^4+11 s\^2) L\_3\
&&+256 \^2 M\_\^2 (s-3M\_\^2)(2L\_4+L\_5)+1280\^2 M\_\^4 (2L\_6+L\_8)\
&&-(9 s M\_K\^2-12 M\_\^2 M\_K\^2+8 M\_\^2 s-16 M\_\^4+2 s\^2),\
t\_[0S,2]{}\[s\]&=&-{ H\_2\[4M\_\^2-s,M\_K\]+2( s-M\_K\^2-M\_\^2) H\_1\[4M\_\^2-s,M\_K\]+ H\_2\[4M\_\^2-s,M\_\].\
&&+2(- s M\_K\^2+ M\_K\^2 M\_\^2)H\_0\[4M\_\^2-s,M\_K\]+2(s - M\_\^2) H\_1\[4M\_\^2-s,M\_\]\
&&.+2(- s M\_\^2+ M\_\^4)H\_0\[4M\_\^2-s,M\_\]+M\_\^4H\_0\[4M\_\^2-s,M\_\]},\
T\^[(4)]{}\_[0S]{}\[s\]&=&(t\_[0S,1]{}\[s\]+).\[eq:ChPT;0S\] Here the superscript of $T$ in the bracket means the chiral order, and subscripts represent for Isospin and spin, respectively. Note that for reader’s convenience we also give the analytical forms of the imaginary part (r.h.c.) of the amplitudes. The I=2 S wave is T\^[(2)]{}\_[2S]{}\[s\]&=&-,\
t\_[2S,1]{}\[s\]&=&( M\_\^2-s) A\[M\_K\]+( M\_\^2-s) A\[M\_\]+ B\[s,M\_\] (s-2 M\_\^2)\^2\
&&+(9 s M\_K\^2-12 M\_\^2 M\_K\^2-16 M\_\^4+2 s\^2+8 M\_\^2 s)+ (-2 M\_\^2 s+4 M\_\^4+s\^2)L\_1\
&&+ (-7 M\_\^2 s+8 M\_\^4+2 s\^2) L\_2-128 \^2 M\_\^2 s (2L\_4+L\_5)\
&&+ (-2 M\_\^2 s+4 M\_\^4+ s\^2)L\_3+512 \^2 M\_\^4(2L\_6+L\_8),\
t\_[2S,2]{}\[s\]&=&-2{H\_2\[4M\_\^2-s,M\_K\]+(+-)H\_1\[4M\_\^2-s,M\_K\].\
&&+(- M\_\^2 M\_K\^2)H\_0\[4M\_\^2-s,M\_K\]+H\_2\[4M\_\^2-s,M\_\] +(--)H\_1\[4M\_\^2-s,M\_\]\
&&.+(-)H\_0\[4M\_\^2-s,M\_\]+H\_0\[4M\_\^2-s,M\_\]},\
T\^[(4)]{}\_[2S]{}\[s\]&=&(t\_[2S,1]{}\[s\]+). \[eq:ChPT;2S\] The I=1 P wave is T\^[(2)]{}\_[1P]{}\[s\]&=&,\
t\_[1P,1]{}\[s\]&=& (s-4 M\_\^2) { 3+B\[s,M\_K\](s-4M\_K\^2)+2B\[s,M\_\] (s-4 M\_\^2)},\
t\_[1P,2]{}\[s\]&=&-{H\_3\[4M\_\^2-s,M\_K\]+((2s-4M\_K\^2-4 M\_\^2)+(s-4M\_\^2)) H\_2\[4M\_\^2-s,M\_K\].\
&&+( (-8 s M\_K\^2+16 M\_K\^2 M\_\^2)+(s-4M\_\^2)(2s-4M\_K\^2-4 M\_\^2))H\_1\[4M\_\^2-s,M\_K\]\
&&+(s-4M\_\^2)(-8 s M\_K\^2+16 M\_K\^2 M\_\^2)H\_0\[4M\_\^2-s,M\_K\]+H\_3\[4M\_\^2-s,M\_\]\
&&+((s+2 M\_\^2)+(s-4M\_\^2)) H\_2\[4M\_\^2-s,M\_\]+H\_1\[4M\_\^2-s,M\_\]\
&&+(-(4 s M\_\^2+M\_\^4)+(s+2 M\_\^2)(s-4M\_\^2)) H\_1\[4M\_\^2-s,M\_\]\
&&.-(4 s M\_\^2+M\_\^4)(s-4M\_\^2)H\_0\[4M\_\^2-s,M\_\]+(s-4m\_\^2)H\_0\[4M\_\^2-s,M\_\]},\
T\^[(4)]{}\_[1P]{}\[s\]&=&(t\_[1P,1]{}\[s\]+).\[eq:ChPT;1P\] And the I=0 D wave is T\^[(2)]{}\_[0D]{}\[s\]&=&0,\
t\_[0D,1]{}\[s\]&=&(s-4 M\_\^2)\^2 (384 \^2 (2 L\_1+4 L\_2+L\_3)+1),\
t\_[0D,2]{}\[s\]&=&-2{H\_4\[4M\_\^2-s,M\_K\]+((-2M\_K\^2-2M\_\^2+s)+(s-4M\_\^2))H\_3\[4M\_\^2-s,M\_K\].\
&&+((8 M\_\^2 M\_K\^2-4 s M\_K\^2)+(s-4M\_\^2)(-2M\_K\^2-2M\_\^2+s)+(s-4M\_\^2)\^2)H\_2\[4M\_\^2-s,M\_K\]\
&&+((s-4M\_\^2)(8 M\_\^2 M\_K\^2-4 s M\_K\^2)+(s-4M\_\^2)\^2(--+))H\_1\[4M\_\^2-s,M\_K\]\
&&+(s-4M\_\^2)\^2( M\_\^2 M\_K\^2-)H\_0\[4M\_\^2-s,M\_K\]+10 H\_4\[4M\_\^2-s,M\_\]\
&&+((2s-32M\_\^2)+10(s-4M\_\^2))H\_3\[4M\_\^2-s,M\_\]\
&&+((37 M\_\^4-8M\_\^2 s)+(s-4M\_\^2)(2s-32M\_\^2)+(s-4M\_\^2)\^2)H\_2\[4M\_\^2-s,M\_\]\
&&+((s-4M\_\^2)(37 M\_\^4-8M\_\^2 s)+(s-4M\_\^2)\^2(-))H\_1\[4M\_\^2-s,M\_\]\
&&+(s-4M\_\^2)\^2(-)H\_0\[4M\_\^2-s,M\_\] +H\_2\[4M\_\^2-s,M\_\]\
&&.+(s-4M\_\^2)H\_1\[4M\_\^2-s,M\_\]+(s-4M\_\^2)\^2 H\_0\[4M\_\^2-s,M\_\]},\
T\^[(4)]{}\_[0D]{}\[s\]&=&(t\_[0D,1]{}\[s\]+).\[eq:ChPT;0D\] It should be noted that in all these partial waves, $2L_4+L_5$ and $2L_6+L_8$ appear together [@DLY11]. The $A$, $B$, $H$ functions are given as below A\[m\]&=&m\^2(1-),\
B\[s,m\]&=&2--(s,m)(),\
H\_[0]{}\[t,m\]&=&-t +m\^2 \^2()- t (t,m)()+3 t ,\
H\_[1]{}\[t,m\]&=& t\^2 (5-2 )-m\^2 t+m\^4 \^2()- t (t-2 m\^2) (t,m)(),\
H\_[2]{}\[t,m\]&=& t\^3 (7-3 )- m\^2 t\^2- t (t,m) (-t m\^2 -6 m\^4+t\^2) ()\
&&-2 m\^4 t+2 m\^6 \^2(),\
H\_[3]{}\[t,m\]&=&- t (t,m) (-2 t\^2 m\^2-10 t m\^4-60 m\^6+3 t\^3) ()\
&&+ t\^4 (9-4 )- t\^3 m\^2- t\^2 m\^4-5 t m\^6+5 m\^8 \^2(),\
H\_[4]{}\[t,m\]&=& t\^5 (11-5 )- t\^4 m\^2- t\^3 m\^4- t\^2 m\^6-14 t m\^8+14 m\^[10]{} \^2()\
&&-t (t,m) (-3 t\^3 m\^2-14 t\^2 m\^4-70 t m\^6-420 m\^8+6 t\^4) (), with $\rho(t,m)=\sqrt{1-4 m^2/t}$. Notice that our amplitudes are calculated in the formalism of $\overline{MS}$, while that of [@Gasser1984; @Pelaez02; @Guo2011] is done in $\overline{MS}-1$. The relation between our LECs ($L_i$) and that of the latter one ($\tilde{L}_i$) is $\tilde{L}_i=L_i+\frac{\Gamma_i}{32\pi^2}$.
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[^1]: It has already been discussed in [@Ang:2001bd], within a unitarized $\chi$PT method. Here we use dispersion approach and re-confirm it, but we do not have the extra poles caused by unitarization.
[^2]: Recently, Lattice QCD gives negative scattering length as $-0.0412(08)(16)M_\pi^{-1}$ [@Beane:2011sc], $-0.04430(25)(40)M_\pi^{-1}$ [@Fu:2013ffa]. $-0.04430(2)(^{+4}_{-0})M_\pi^{-1}$[@Helmes:2015gla] These values are consistent with that of $\chi$PT [@Bijnens:2014lea], the Roy equations matched to $\chi$PT [@Colangelo:2000jc] and a dispersive analysis [@KPY].
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
**A.K.Aringazin$^{\dagger\ddagger}$, K.M.Aringazin$^{\dagger}$, **S.Baskoutas$^{*}$,\
**G.Brodimas$^{*}$, A.Jannussis$^{*\ddagger}$ and E.Vlachos$^{*}$\
$^{\dagger}$Department of Theoretical Physics, Karaganda State University,\
Karaganda 470074, Kazakhstan\
$^*$ Department of Physics, University of Patras,\
Patras 26110, Greece\
$^{\ddagger}$The Institute for Basic Research, P.O. Box 1577,\
Palm Harbor, FL 34682, U.S.A.******
date: '[*Proc. Intern. Conf. “Advances in Fundamental Physics”, Olympia, Greece, 27-30 Sept. 1993, Eds. M.Barone and F.Selleri, Hadronic Press, 1995, pp. 329-348*]{}'
title: ' q-DEFORMED HARMONIC OSCILLATOR IN PHASE SPACE'
---
Introduction
============
Phase space formulation of quantum mechanics since pioneering papers by Weyl[@Weyl-27], Wigner[@Wigner-32] and Moyal[@Moyal-49] who were motivated by the obvious reason to realize quantum mechanics as some extended version of the Hamiltonian mechanics rather than somewhat sharp step to the theory of operators acting on Hilbert space, now becomes of special interest at least for two major reasons.
The first reason is that the quantum mechanics in phase space represents an example of theory with non-commutative geometry. Moyal[@Moyal-49] and Bayen [*et al.*]{}[@Bayen-78] developed non-commutative algebra of functions on phase space which is aimed to represent non-commutative property of the operators. In turn, the operators are sent to functions on phase space - symbols - due to the symbol map[@Weyl-27; @Wigner-32], which is well defined one-to-one map. So, the symbol calculus[@Hormander-79; @Berezin-80] provides a reformulation of whole machinery of quantum mechanics in terms of non-commutative functions on phase space.
Also, Bopp[@Bopp-61] and Kubo[@Kubo-64] extended the phase space and introduced non-commutative variables in terms of which they expressed the Wigner operator[@Wigner-32; @JP-75] and the Wigner density operator[@Kubo-64; @Feynman-72; @BJP-76]. The Bopp-Kubo formulation deals with functions on the extended phase space which are also non-commuting due to the non-commutative character of the variables.
Recent studies of the non-commutative phase space[@Wess-90]-[@Dimakis-92] are much in the spirit of modern non-commutative geometry[@Connes-85]-[@Coquereaux-92]. Exterior differential calculus in the quantum mechanics in phase space has been proposed recently by Gozzi and Reuter[@GR-93; @GR-93a]. They studied in detail algebraic properties of the symbol calculus, and have found[@GR-93b], particularly, quantum analogue of the classical canonical transformations. Gozzi and Reuter have argued that the quantum mechanics in phase space can be thought of as a smooth deformation of the classical one.
Jannussis, Patargias and Brodimas[@JPB-78] have constructed creation and annihilation operators in phase space, and studied harmonic oscillator in phase space[@JP-77]. Various problems related to the Wigner operator, Wigner distribution function, and the density matrix in phase space have been investigated in a series of papers by Jannussis [*et al.*]{}[@GJP-77]-[@JLPFFV-82]. The second reason of the importance of the quantum mechanics in phase space is that the resulting formalism is very similar to the Hamiltonian formulation of classical mechanics (not surprise certainly).
An obvious advantage of the phase space formulation of quantum mechanics is that it arises to a tempting possibility to exploit this formal similarity, provided by the smooth deformation, to extend some of the useful notions and tools, such as action-angle variables, ergodicity, mixing, Kolmogorov-Sinai entropy, and chaos, which had been elaborated in Hamiltonian mechanics to quantum mechanics. The only thing that one should keep in mind here is that the phase space quantum mechanics deals with the non-commutative symplectic geometry rather than the usual symplectic geometry. So, one should take care of this, primarily because the usual notion of phase space points is lost in non-commutative case so that one is forced to work mostly in algebraic terms rather than to invoke to geometrical intuition. For example, it is not obvious what is an analogue of the Lyapunov exponents when there are no classical trajectories.
However, as a probe in this direction, we attempt to formulate, in this paper, the extention of the classical ergodicity condition.
We should emphasize here that, clearly, it is highly suitable to have at disposal the phase space formulation before going into details of quantum mechanical analogues of the classical chaos and related phenomena.
As to chaos in dynamical sytems, it should be noted that the evolution equations, both in the classical and quantum mechanics in phase space, are Hamiltonian flows, which are deterministic in the sense that there are no source terms of stochasticity. In view of this, chaos can be still thought of as an extreme sensitivity of the long-time behavior of the probability density and, therefore, of the other observables of interest, to initial state. Another fundamental aspect of this consideration is the process of measurements. However, we shall not discuss this problem here.
As a specific example of quantum mechanical system in phase space, we consider, in this paper, one-dimensional harmonic oscillator.
We study also the [*$q$-deformed*]{} oscillator in phase space which is now of special interest in view of the developments of quantum algebras[@Drinfeld-86]-[@Bernard-90]. We should note here that the quantum algebras are particular cases of the Lie-admissible algebras[@J-91]-[@JBB-92]. The algebra underlying the properties of the $q$-oscillator in phase space appears to be the algebra $su_{q}(2)$[@Biedenharn-89]-[@Fiore-93]. The paper is organized as follows.
In Sec 2.1, we briefly recall the Bopp-Kubo formulation of quantum mechanics in phase space.
Sec 2.2 is devoted to Weyl-Wigner-Moyal symbol-calculus approach to quantum mechanics the main results of which are sketched.
In Sec 2.3, we discuss, following Gozzi and Reuter[@GR-93b], modular conjugation and unitary transformations. We show that the Bopp-Kubo formulation and the symbol calculus are explicitly related to each other. We give an interpretation of the quantum mechanics in phase space in terms of non-commutative geometry. Quantum mechanical extention of the classical ergodicity condition is proposed.
In Sec 2.4, we study translation operators in phase space. Commutation relations of the Bopp-Kubo translation operators, both in Hamiltonian and Birkhoffian cases, are presented.
The results presented in Sec 2 are used in Sec 3 to study the harmonic ($q-$)oscillator in phase space.
In Sec 3.1, we present the main properties of the one-dimensional oscillator in terms of annihilation and creation operators in phase space. We identify fundamental $2D$-lattice structure of the phase space resulting from the commutation relations of the Bopp-Kubo translation operators. The Fock space for the oscillator is found to be related to the double Hilbert space of the Gelfand-Naimark-Segal construction.
In Sec 3.2, we study $q$-deformed harmonic oscillator in phase space. The Wigner operator is found to be proportional to the 3-axis spherical angular momentum operator of the algebra $su_{q}(2)$. Also, the Wigner density operator appeared to be related to the 3-axis hyperbolical angular momentum operator of the algebra $su_{q}(1,1) \approx sp_{q}(2,R)$.
Phase space formulation of quantum mechanics
============================================
Bopp-Kubo formulation. Non-commutative coordinates
--------------------------------------------------
In studying Wigner representation[@Wigner-32] of quantum mechanics, Bopp[@Bopp-61] and Kubo[@Kubo-64] started from classical Hamiltonian $H(p,q)$ and used the variables (see also [@JP-78; @JPLFFSV-82])
$$\begin{aligned}
\label{PQ}
P= p - \frac{i\hbar}{2}\frac{\partial}{\partial q},\quad
Q= q + \frac{i\hbar}{2}\frac{\partial}{\partial p} \\ \label{PQ*}
P^*= p + \frac{i\hbar}{2}\frac{\partial}{\partial q},\quad
Q^*= q - \frac{i\hbar}{2}\frac{\partial}{\partial p}\end{aligned}$$
instead of the usual $(p,q)$ and obtained the Wigner operator, $W_-$, and the Wigner density operator, $W_+$, in the following form: $$W_{\pm} = H(P,Q) \pm H(P^*,Q^*) \label{W}$$ These operators enter respectively the Wigner equation $$i\hbar\partial_t \rho = W_- \rho \label{Wigner}$$ and the Bloch-Wigner equation[@Kubo-64] $$\partial_\beta F + \frac{1}{2} W_+ F = 0 \qquad\qquad
\beta = \frac{1}{kT} \label{BlochWigner}$$ Here, $\rho=\rho (p,q)$ is the Wigner distribution function[@Wigner-32; @JLPFFV-82] and $F=F(p,q,p',q';\beta )$ is the Wigner density matrix[@BJP-76].
As it is wellknown, the Wigner equation (\[Wigner\]) is a phase space counterpart of the usual von Neumann equation of quantum mechanics while the Bloch-Wigner equation (\[BlochWigner\]) describes quantum statistics in phase space[@Feynman-72; @GJP-77].
In view of the definitions (\[PQ\])-(\[PQ\*\]) of the variables, it is quite natural to treat the above formulation in terms of non-commutative geometry[@Connes-90].
With the usual notation, $\phi^i = (p_{1} , \dots , p_n ,q^1 , \dots , q^n )$, $\phi^i \in M_{2n}$, the first step is to extend the phase space $M_{2n}$ to the (co-)tangent phase space $TM_{2n}$ and define the complex coordinates, $$\Phi^{i}_{\pm} = \phi^i \pm
\frac{i\hbar}{2}\omega^{ij}\frac{\partial}{\partial \phi^{j}}
\label{Phi}$$ where $\omega^{ij}$ is a fundamental symplectic tensor, $\omega^{ij}=-\omega^{ji}; \ \omega_{ij}\omega^{jk}=\delta^{k}_{i}$.
We observe immediately that these coordinates are non-commutative, $$\bigl[\Phi^{i}_{\pm},\Phi^{j}_{\pm}\bigr] = \pm i\hbar\omega^{ij}
\qquad \bigl[\Phi^{i}_{\pm},\Phi^{j}_{\mp}\bigr] = 0 \label{comff}$$ and do not mix under time evolution. The natural projection $TM_{2n} \rightarrow M_{2n}$ comes with the classical limit $\hbar \rightarrow 0$.
Commutation relations (\[comff\]) imply that the “holomorphic” functions, $f(\Phi_-)$, and “anti-holomorphic” functions, $f(\Phi_+)$, form two mutually commuting closed algebras on space of functions $C(TM_{2n})$.
Thus, the holomorphic, $H(\Phi_{-})$, and anti-holomorphic, $H(\Phi_{+})$, Hamiltonians define two separate dynamics, which are not mixed. Wigner operators (\[W\]) are simply sum and difference between these two Hamiltonians, respectively, $$W_{\pm} = H(\Phi_- ) \pm H(\Phi_+ ) \label{WPhi}$$ So, physical dynamics comes with the combinations of these two Hamiltonians. In the classical limit, the Wigner operators cover the Liouvillian $L$ and the Hamiltonian, $$W_{-} = -i\hbar L + O(\hbar^2 ) \qquad \label{Wclass}
W_{+} = 2H(p,q) + O(\hbar^2 )$$ where $L \equiv \ell_{h}=-h^{i}\partial_{i}$ is a Lie derivative along the Hamiltonian vector field $h^{i} = \omega^{ij}\partial_{j}H$[@Arnold-78; @Abraham-78].
According to complex character of the variables (\[Phi\]), one can define the involution ${\cal J}$ acting simply as complex conjugation $${\cal J}:\ \Phi^{i}_{\pm} \rightarrow \Phi^{i}_{\mp} \label{JPhi}$$ This involution may be thought of as a conjugation interchanging the two pieces of the physical dynamics.
Weyl-Wigner-Moyal formulation. Symbol calculus
----------------------------------------------
In order to achieve phase space formulation of quantum mechanics, Weyl[@Weyl-27] and Wigner[@Wigner-32] introduced symbol map associating with each operator $\hat A$, acting on Hilbert space, a symbol $A(\phi )$, function on phase space, $A(\phi ) = symb(\hat{A})$, due to $$A(\phi ) =
\int \frac{d^{2n}\phi_{0}}{(2\pi\hbar)^n}
exp\Bigl[\frac{i}{\hbar}\phi^{i}_{0}\omega_{ij}\phi^{j}\Bigr]
Tr\bigl( \hat T (\phi_{0})\hat A \bigr) \label{symbol}$$ with $$\hat T (\phi_{0}) =
exp\Bigl[\frac{i}{\hbar}\phi^{i}_{0}\omega_{ij}\hat\phi^{j}\Bigr]
\label{TWeyl}$$ The symbol map is well defined invertible one-to-one map from space of operators, ${\cal O}$, to space of functions depending on phase space coordinates[@Hormander-79; @GR-93a], ${\cal O} \rightarrow C(M_{2n})$. Particularly, Hermitean operators are mapped to real functions, and vice versa.
The key property of the symbol calculus is that the ordinary pointwise product of the functions is appropriately generalized to reproduce the non-commutative product of the operators. The product on $C(M_{2n})$, making the symbol map an algebraic homomorphism, is the Moyal product[@Moyal-49; @Berezin-80], $$\begin{aligned}
\label{mp}
(A*B)(\phi ) &=&
symb(\hat A \hat B) \nonumber \\
&=&
A(\phi )
exp\bigl[\frac{i}{2\hbar}\bar\partial_{i}\omega_{ij}\vec\partial_{j}\bigr]
B(\phi ) \\
&=&
A(\phi )B(\phi ) + O(\hbar ) \nonumber\end{aligned}$$ The Moyal product is associative but apparently non-commutative, and represents, in $C(M_{2n})$, non-commutative property of the algebra of operators, and non-local character of quantum mechanics.
The Moyal bracket[@Moyal-49] $$\begin{aligned}
\{ A, B \}_{mb} &=& \label{mb}
symb(\frac{1}{i\hbar}\bigl[ A, B ] ), \nonumber \\
&=&
\frac{1}{i\hbar}( A*B - B*A) \\
&=&
\{ A, B \}_{pb} + O(\hbar^2) \nonumber\end{aligned}$$ is a symbol of commutator between two operators, and reduces to the usual Poisson bracket $\{ .,. \}_{pb}$ in the classical limit. Thus, the algebra $(C(M_{2n}), \{ ,\}_{mb})$ is an algebra of quantum observables, and it can be continuously reduced to the algebra $(C(M_{2n}), \{ ,\}_{pb})$ of classical observables.
Symbol map of the von Neumann’s equation is written as[@GR-93a] $$\begin{aligned}
\label{me}
\partial_t \rho (\phi , t) &=&
-\{ \rho, H \}_{mb} \nonumber \\
&=&
-\ell_h \rho + O(\hbar^2 )\end{aligned}$$
In the classical limit, this equation covers the Liouville equation of classical mechanics[@Koopman-31; @Neumann-32].
To summarize, the symbol calculus can be treated as a smooth deformation of classical mechanics linking non-associative Poisson-bracket algebra of classical observables, $A(\phi), \dots ,$ and an associative commutator algebra of quantum observables, $\hat A , \dots $. Full details of the symbol calculus may be found in [@GR-93a; @GR-93b] and references therein.
Chiral symmetry and unitary transformations
-------------------------------------------
Gozzi and Reuter[@GR-93b] have investigated recently the algebraic properties of the quantum counterpart of the classical canonical transformations using the symbol-calculus approach to quantum mechanics.
They found, particularly, that the operators $L_f$ and $R_f$ acting as the left and right multiplication with symbol $f$, respectively, $$L_{f}g = f*g \qquad R_{f}g = g*f \label{LR}$$ form two mutually commuting closed algebras, ${\cal A}_{L}$ and ${\cal A}_{R}$, (cf. [@JSPSV-77]) $$\label{LL}
\bigl[ L_{f_1} ,L_{f_2} \bigr] = i\hbar L_{\{f_1 f_2\}_{mb} } \qquad
\bigl[ R_{f_1} ,R_{f_2} \bigr] =-i\hbar R_{\{f_1 f_2\}_{mb} } \qquad
\bigl[ L_{f_1} ,R_{f_2} \bigr] = 0$$ which are explicitly isomorphic to the original Moyal-bracket algebra on $C(M_{2n})$.
Also, $L_f$ and $R_f$ can be presented by virtue of the Moyal product (\[mp\]) as[@GR-93b] $$L_{f} = : f(\Phi^{i}_{+}) : \qquad
R_{f} = : f(\Phi^{i}_{-}) : \label{LRPhi}$$ where $\Phi^{i}_{\pm}$ are defined by (\[Phi\]), and $:\dots :$ means normal ordering symbol (all derivatives $\partial_{i}$ should be placed to the right of all $\Phi$’s).
It has been shown[@GR-93b] that the linear combinations of the above operators, $$V^{\pm}_{f} = \frac{1}{i\hbar}(L_f \pm R_f ) \label{V}$$ for real $f$, generate non-unitary, $\hat g \rightarrow \hat{U}\hat g \hat{U}$, and unitary, $\hat g \rightarrow \hat{U}\hat g \hat{U^{-1}}$, transformations, respectively ($\hat{U}$ is an unitary operator).
We see that the Wigner operators, $W_{\pm}$, given in the Bopp-Kubo formulation by (\[WPhi\]), are just $$W_{\pm} = L_H \pm R_H = i\hbar V^{\pm}_{H} \label{WLR}$$ so that the Wigner equation (\[Wigner\]) can be written as $$\partial_t \rho = V^{-}_{H} \rho \label{Wigner2}$$ where $H$ is the Hamiltonian. So, we arrive at the conclusion that the representations (\[LRPhi\]) provide the relation between the Bopp-Kubo and Weyl-Wigner-Moyal formulations.
Various algebraic properties of the generators $V^{-}_{f}$ have been found by Gozzi and Reuter[@GR-93b]. Particularly, they found that in two dimensional phase space the generators $V^{-}_{f}$, in the basis $V_{\vec m} = -exp(i\vec m \vec \phi),\
\vec m = (m_1 , m_2) \in Z^2 $, on torus $M_2 = S^1 \times S^1$, satisfy a kind of the $W_{\infty}$-algebra commutation relations, $$\bigl[ V^{-}_{\vec m}, V^{-}_{\vec n} \bigr] =
\frac{2}{\hbar} \label{Winfty}
sin(\frac{\hbar}{2}m_{i}\omega^{ij}m_{j})V^{-}_{\vec m + \vec n}$$ which are deformed version of the $w_{\infty}$-algebra of the classical $sdiff(T^2 )$, area preserving diffeimorphisms on the torus.
Also, an important result shown in [@GR-93b] is that $V^{-}_{f}$ is invariant under the modular conjugation operator defined on symbols by $${\cal J} f = f^* \qquad
{\cal J}(f*g) = {\cal J}(g)*{\cal J}(f) \label{J}$$ Namely, $${\cal J} L_f {\cal J} = R_f \qquad \label{JLJ}
{\cal J} R_f {\cal J} = L_f \qquad
{\cal J} V^{-}_f{\cal J} = V^{-}_f \qquad
{\cal J} V^{+}_f{\cal J} = -V^{+}_f$$ This symmetry resembles the [*chiral*]{} symmetry and seems to be broken in the classical mechanics. This argument is supported by the fact that the Moyal product (\[mp\]) becomes commutative in the classical limit. Indeed, in the classical case, the difference between the left and right multiplications on $C(M_{2n})$ disappears so that there is no room for the modular conjugation operator ${\cal J}$, and the original algebra ${\cal A}_{L}\otimes{\cal A}_{R}$ is contracted to its diagonal subalgebra[@GR-93b].
The operator $V^{+}_f$ seems to have no analogue in the geometry of phase space of classical mechanics since $V^{+}_f$ blows up at $\hbar \rightarrow 0$ due to the factor $1/i\hbar$ in the definition (\[V\]). However, $i\hbar V^{+}_{f}$ is ${\cal J}$-invariant and has the classical limit $i\hbar V^{+}_{f} = 2f + O(\hbar^{2})$ so that $i\hbar V^{+}_{H} = W_{+}= 2H + O(\hbar^{2})$ is simply two times Hamiltonian. In the Bloch-Wigner equation (\[BlochWigner\]), $i\hbar V^{+}_f$ plays the role of Hamiltonian defining the density matrix in quantum statistics[@Feynman-72].
The operator $V^{-}_f$ has an explicit interpretation[@GR-93b] as a [*quantum deformed Lie derivative along the hamiltonian vector field*]{} in accordance with (\[Wigner2\]). Furthermore, in quantum mechanics the ${\cal J}$-invariance of $V^{-}_f$ provides [*unitary*]{} time evolution due to the Wigner equation (\[Wigner2\]).
The structure of the Weyl-Wigner-Moyal calculus, which deals with non-commutative algebra, may be seen in a more refined way from the non-commutative geometry[@Connes-90; @Coquereaux-92] point of view as follows.
First, recall that usual definition of topological space $M$ is equivalent to definition of commutative algebra ${\cal A}$ due to the identification ${\cal A} = C(M)$, with the algebra $C(M)$ of continuous complex valued functions on $M$ (Gelfand correspondence). Conversely, $M$ can be understood as the spectrum of algebra ${\cal A}$, [*i.e.*]{} points $x \in M$ are irreducible representations owing to the relation $x[f] = f(x)$ when $f \in {\cal A}$. Next step is that one is free to assume that the algebra ${\cal A}$ is non-commutative in general, and then think about a non-commutative version of the space $M$. Particularly, classical notion of point $x \in M$ is modified, in non-commutative geometry, due to the basic relation mentioned above.
Specific example we will consider for our aims is a non-commutative vector bundle. In classical geometry, sections of a vector bundle $E$ above a manifold $M$ play, in physical context, the role of matter fields. Here, an important point to be noted is that the space ${\cal E}$ of the sections is a bimodule over the algebra ${\cal A} = C(M)$ of the functions on $M$. In the non-commutative case, there are [*left*]{} and [*right*]{} modules over non-commutative algebra ${\cal A}$ instead of the bimodule. That is, for $\sigma \in {\cal E}$ and $f \in {\cal A}$, $f\sigma$ and $\sigma f$ are not both made sense as elements of ${\cal E}$. One may choose, for convenience, the right module, and then characterize the non-commutative vector bundle as a quotient of free module $A^{m}$, [*i.e.*]{} as the (right) projective module over the algebra ${\cal A}$, $\, {\cal E} = P{\cal A}^{m}$, for some projector $P, \, P^{2} = P$, and some $m$.
In the symbol calculus, we have, obviously, ${\cal E} = {\cal A}$ itself, where ${\cal A} = (C(M_{2n}), *)$ is the non-commutative algebra endowed with the Moyal product. The sections are functions on $M_{2n}$ acting by the left and right multiplications and forming, respectively, left and right ${\cal A}$-modules. The modular conjugation acts due to $${\cal J}: {\cal A}\otimes {\cal A} \rightarrow {\cal A}\otimes {\cal A}$$ $$(L,R) \mapsto (R,L)$$ and ${\cal E}$ is the quotient, ${\cal E}= {\cal A}\otimes {\cal A} /{\cal J}$.
The ${\cal A}$-modules to be [*unital*]{} one has to put $I_{L}*f = f$ and $f*I_{R} = f$, $\forall f \in C(M_{2n})$, with $I_{L,R}$ being the left and right “identity” elements of ${\cal E}$. Because ${\cal E}={\cal A}$, we have actually $I_{L}=I_{R} = I \in C(M_{2n})$ so that the above conditions imply $$f*I - I*f = 0 \qquad
f*I + I*f = 2f \qquad$$ According to definitions (\[LR\]) and (\[WLR\]), these equations can be rewritten as $$V^{-}_{f}I = 0 \qquad \label{unital}
i\hbar V^{+}_{f}I = 2f \quad\qquad \forall f \in C(M_{2n})$$ The question arises as to existence of such unique function $I$ that the both equations (\[unital\]) are satisfied for any function $f$. We observe that in the classical case the last two equations have correct limits at $\hbar \rightarrow 0$, and are satisfied for any $f$ identically only if $I(\phi ) = 1$, as it was expected ($1f = f1 = f$).
The Bopp-Kubo representation provides a realization of representation space of the algebra ${\cal A}$, with the variables $\Phi^{i}_{\pm}$, which extends the usual $M_{2n}$ for the non-commutative case.
In the remainder of this section, we will consider the extention of the classical [*ergodicity*]{} condition[@Arnold-68]. Quantum mechanical analogue of the classical condition of ergodicity can be written as $$V^{-}_{H}\rho = 0 \label{ergoGozzi}$$ due to comparison of (\[me\]) and (\[Wigner2\]), with the solution $\rho$ being non-degenerate, at least at the classical level.
In the classical limit, this equation covers the usual equation, $L\rho = 0$, where $L$ denotes the Liouvillian, whose non-degenerate eigenfunctions with zero eigenvalues describe ergodic Hamiltonian systems[@Arnold-68], which are characterized by the only constant of motion, energy $H$. As to solutions, recent studies[@GRT-91a]-[@Aringazin-93b] of the classical ergodicity condition within the path integral approach to classical mechanics show that the solution is given specifically by the Gibbs state form.
The condition (\[ergoGozzi\]) can be rewritten in the Bopp-Kubo representation as $$H(\Phi_{+})\rho(\phi ) = H(\Phi_{-})\rho(\phi ) \label{ergoBopp}$$ where we have used the relations (\[WPhi\]) and (\[WLR\]), that means that the holomorphic and anti-holomorphic Hamiltonians have the same spectrum. Also, it is remarkable to note that the equation (\[ergoGozzi\]) is similar to the first equation of (\[unital\]), with $f(\phi ) = H(\phi )$ and $I(\phi ) = \rho(\phi )$.
We pause here with the further discussion stating that more analysis is needed to verify the proposed extention of the ergodicity condition (\[ergoGozzi\]) which may be made elsewhere.
Translation operators
---------------------
The operator $T(\phi_0)$ defined by (\[TWeyl\]) and used to represent the Weyl symbol map (\[symbol\]) has a meaning of the operator of translations in phase space. Bopp[@Bopp-61] has introduced such an operator in $\Phi^{i}_{\pm}$-variables representation and Jannussis [*et al.*]{} [@JPB-78] have studied their properties.
Let us define the translation operators, in the Bopp-Kubo formulation, $$T_{\pm}(\phi_{0}) =
exp\bigl[ \pm\frac{i}{\hbar}\phi^{i}_{0}\omega_{ij}\Phi^{j}_{\pm}\bigr]
\label{T}$$ where $\Phi^{i}_{\pm}$ are defined by (\[Phi\]). It is easy to verify that due to the fundamental commutation relations (\[comff\]) they build up two mutually commuting algebras, $$\begin{aligned}
\label{TT}
\bigl[ T_{\pm}(\phi_{1}),T_{\pm}(\phi_{2})\bigr] =
\pm 2i\ sin(\frac{1}{\hbar}\phi^{i}_{1}\omega_{ij}\phi^{j}_{2})
T_{\pm}(\phi_{1} + \phi_{2}) \\
\bigl[ T_{\pm}(\phi_{1}),T_{\mp}(\phi_{2})\bigr] = 0\end{aligned}$$
In the case of Birkhoffian generalization of Hamiltonian mechanics[@Aringazin-93a]-[@GRT-91b] one supposes that the symplectic 2-form $\omega$ depends on phase space coordinates, $\omega = \omega (\phi)$, but it is still non-degenerate and closed, $d\omega=0$. Consistency of the Birkhoffian mechanics is provided by the Lie-isotopic construction[@Santilli-88]-[@Aringazin-mono-91]. In this case, the fundamental commutation relations (\[comff\]) are essentially modified, $$\begin{aligned}
\label{comffBirk}
\bigl[ \Phi^{i}_{\pm} , \Phi^{j}_{\pm} \bigr] &=&
\pm i\hbar\omega^{ij}
+ (\frac{i\hbar}{2})^2\omega^{mn}\omega^{ij}_{\ \ ,m}\partial_{n} \\
\bigl[ \Phi^{i}_{\pm} , \Phi^{j}_{\mp} \bigr] &=&
\mp (\frac{i\hbar}{2})^2\omega^{mn}\omega^{ij}_{\ \ ,m}\partial_{n} \nonumber\end{aligned}$$ Consequently, the commutation relations (\[TT\]) for the translation operators are also changed. Tedious calculations show that $$\begin{aligned}
\label{TTBirk}
\bigl[ T_{\pm}(\phi_{1}), T_{\pm}(\phi_{2})\bigr] =
\pm 2i\ sin \Bigl( \frac{1}{\hbar}\phi^{i}_{1}\phi^{j}_{2}
(\omega_{ij} + \frac{1}{2}\omega_{ij,m}\phi^{m}) \Bigr)
T_{\pm}(\phi_{1}+\phi_{2}) \\
\bigl[ T_{\pm}(\phi_{1}),T_{\mp}(\phi_{2})\bigr] = \nonumber \\
\pm 2i\ sin \Bigl( \frac{1}{\hbar}\phi^{i}_{1}\phi^{j}_{2}
(\omega_{im,j} - \frac{1}{2}\omega_{ij,m})\phi^{m} \Bigr)
exp\bigl[\pm \frac{i}{\hbar}
(\phi^{i}_{1}\omega_{ij}\Phi^{j}_{\pm}
-\phi^{i}_{2}\omega_{ij}\Phi^{j}_{\mp})\bigr]\end{aligned}$$ Here, we have used the identity $\omega^{im}\omega^{jk}_{\ \ ,m} +
\omega^{jm}\omega^{ki}_{\ \ ,m} +
\omega^{km}\omega^{ij}_{\ \ ,m} = 0$, and denote $\omega_{ij,m} = \partial_{m}\omega_{ij}$. We see that in the Birkhoffian case the holomorphic and anti-holomorphic functions do not form two mutually commuting algebras, in contrast to the Hamiltonian case characterized by $\omega_{ij,m} = 0$. Evidently, the Birkhoffian generalization is important for the case when the symplectic manifold can not be covered by [*global*]{} chart with constant symplectic tensor $\omega_{ij}$. This is, for example, the case of $M_{2n}$ with a non-trivial topology. However, it should be noted that the symplectic manifold can be always covered by local charts with constant $\omega_{ij}$ due to Darboux theorem.
$q$-deformed harmonic oscillator in phase space
===============================================
Harmonic oscillator in the Bopp-Kubo phase space representation
---------------------------------------------------------------
Instead of studying the harmonic oscillator in phase space via the Wigner equation (\[Wigner\]) it is more convenient to exploit corresponding creation and annihilation operators in the phase space.
Jannussis, Patargias and Brodimas[@JPB-78] have defined the following two pairs of the creation and annihilation operators following the Bopp-Kubo formulation: $$\begin{aligned}
\label{a-}
a_{-} = \frac{1}{\sqrt 2}
\bigl(\sqrt{\frac{m\omega}{\hbar}}Q+i\sqrt{\frac{1}{m\omega\hbar}}P\bigr) \\
a^{+}_{-} = \frac{1}{\sqrt 2} \label{a+-}
\bigl(\sqrt{\frac{m\omega}{\hbar}}Q-i\sqrt{\frac{1}{m\omega\hbar}}P\bigr) \\
a_{+} = \frac{1}{\sqrt 2} \label{a+}
\bigl(\sqrt{\frac{m\omega}{\hbar}}Q^*
+i\sqrt{\frac{1}{m\omega\hbar}}P^*\bigr)\\
a^{+}_{+} = \frac{1}{\sqrt 2} \label{a++}
\bigl(\sqrt{\frac{m\omega}{\hbar}}Q^* -
i\sqrt{\frac{1}{m\omega\hbar}}P^*\bigr)\end{aligned}$$ These operators obey the following usual commutation relations: $$\bigl[a_{\pm}, a^{+}_{\pm}\bigr] = 1 \quad \label{aa}
\bigl[a_{\pm}, a^{+}_{\mp}\bigr] =
\bigl[a_{\pm}, a_{\mp}\bigr] =
\bigl[a^{+}_{\pm}, a^{+}_{\mp}\bigr] = 0$$ The Bopp-Kubo holomorphic and anti-holomorphic Hamiltonians for the harmonic oscillator then read $$\begin{aligned}
\label{H}
H(P,Q) =
\frac{P^2}{2m} + \frac{m}{2}\omega^2 Q^2
=
\hbar\omega (a^{+}_{-}a_{-} + \frac{1}{2}) \\
H(P^* ,Q^* ) =
\frac{P^{*2}}{2m} + \frac{m}{2}\omega^2 Q^{*2}
=
\hbar\omega (a^{+}_{+}a_{+} + \frac{1}{2}) \\\end{aligned}$$ and the Wigner operator due to (\[W\]) takes the form $$\begin{aligned}
\label{Waa}
W_{-} &=& \hbar\omega (a^{+}_{+}a_{+} - a^{+}_{-}a_{-}) \\
\label{Wnn}
&\equiv& \hbar\omega (\hat{n_1} - \hat{n_2})\end{aligned}$$ In the two-particle Fock space ${\cal F}_{1} \otimes{\cal F}_{2}$ with the basis $|n_1 \, n_2\rangle$, the pairs of operators (\[a-\])-(\[a++\]) act due to $$\begin{aligned}
\label{aa+-}
a^{+}_{-}|n_1 \, n_2\rangle = \sqrt{n_1 +1}|n_1 +1 \, n_2\rangle \\
\label{aa-}
a^{ }_{-}|n_1 \, n_2\rangle = \sqrt{n_1} |n_1 -1 \, n_2\rangle \\
\label{aa++}
a^{+}_{+}|n_1 \, n_2\rangle = \sqrt{n_2 +1}|n_1 \, n_2 +1\rangle \\
\label{aa+}
a^{ }_{+}|n_1 \, n_2\rangle = \sqrt{n_2} |n_1 \, n_2 -1\rangle\end{aligned}$$ Then, the Wigner operator (\[Wnn\]) has the following eigenvalues $$W_{-}|n_1 \, n_2\rangle = (n_1 - n_2)|n_1 \, n_2 \rangle \label{W-eigen}$$ The eigenfunctions of the Wigner operator (\[Wnn\]) have the following form[@JPB-78; @JP-77]: $$\varphi_{n_1 n_2}(p,q) = \label{varphi}
\int dp_0 dq_0\ T_{+}(p_0 , q_0 )\varphi_{0n_2}(p,q)\varphi_{n_1 0}(p,q)$$ where $$\varphi_{n_1 0} =
\frac{1}{\pi\sqrt{\hbar}}\frac{1}{\sqrt{n_1 !}}
\Bigl(\frac{2m\omega}{\hbar}\Bigr)^{n_1 /2}
\Bigl( q - i\frac{p}{m\omega}\Bigr)^{n_1}
exp\Bigl(- \frac{2H(p,q)}{\hbar\omega}\Bigr) \label{varphi1}$$ and the same for $\varphi_{0 n_2}$ with the replacement $n_1 \rightarrow n_2$ in the r.h.s. of (\[varphi1\]).
The vacuum is characterized by the Gibbs state form $$\varphi_{00} = \frac{1}{\pi\sqrt{\hbar}}
exp\Bigl(- \frac{2H(p,q)}{\hbar\omega}\Bigr) \label{vacuum}$$ The action of the Bopp translation operators on the functions (\[varphi1\]) can be easily determined, and the result is $$T_{\pm}(p_0 , q_0 )\varphi_{n_1 0}(p,q)=
exp\Bigl(\pm \frac{i}{\hbar}(p_0 q - q_0 p)\Bigr)
\varphi_{n_1 0}(p+p_0 , q+q_0 ) \label{Tvarphi}$$ The commutators (\[TT\]) take the form $$\label{TT2dim}
\bigl[ T_{\pm}(p_{1},q_{1}), T_{\pm}(p_{2},q_{2})\bigr] =
\pm 2i\ sin\frac{1}{\hbar}(p_{1}q_{2}-q_{1} p_{2})
T_{\pm}(p_{1}+p_{2},q_{1}+q_{2})$$ The translation operators in (\[TT2dim\]) commute when $$\frac{1}{\hbar}(p_{1}q_{2} - q_{1}p_{2}) = \pi l
\qquad l \in Z \label{flux}$$ This condition is similar to the one of quantization of magnetic flux for $2D$-electron gas in uniform magnetic field[@JPB-78].
This means that the phase space asquires $2D$-lattice structure with the basic unit-cell vectors $\vec \phi_{1} = (p_{1},q_{1})$ and $\vec \phi_{2} = (p_{2},q_{2})$ obeying (\[flux\]), i.e. $$\vec n \cdot\vec \phi_{1}\times\vec \phi_{2} = l\Psi_{0}
\qquad \Psi_{0} = \pi\hbar \label{aflux}$$ The degeneracy of the energy levels of the harmonic oscillator in the phase space is then related to the lattice structure. Namely, the representation (\[varphi\]) of $\varphi_{n_{1}n_{2}}$ means that one “smears” the product $\varphi_{n_{1}0}\varphi_{0n_{2}}$ (a “composite state” of two identical systems) over all the phase space. So, $\varphi_{n_{1}n_{2}}$ remain to be eigenfunctions with the same eigenvalues under the translations of the form $\vec R = N_{1}\vec \phi_{2} + N_{2}\vec \phi_{2},\ N_{1,2}\in Z,$ leaving the lattice invariant. This is a kind of the magnetic group periodicity[@Zak-64]-[@Cristofano-91]. To implement the lattice structure of the phase space explicitly one may start with the vacuum state (\[vacuum\]), which is characterized by zero angular momentum, to define four sets of functions $$\varphi^{(\alpha )}_{\vec k}(\vec \phi ) =
\sum_{\vec R^{\alpha}} exp(i\vec k \vec R^{\alpha}) \label{varphiR}
T_{+}(\vec R^{\alpha})\varphi_{00}(\vec \phi )$$ where $$\begin{aligned}
\label{R}
\vec R^{\alpha} = \vec R_{0} + I^{\alpha}_{i} \qquad
\vec R_{0} = N_{1}\vec \phi_{2} + N_{2}\vec \phi_{2} \qquad
\alpha = 0,1,2,3 \\
I^{0}_{i}= (0,0)\qquad
I^{1}_{i}= (1,0)\qquad
I^{2}_{i}= (0,1)\qquad
I^{3}_{i}= (1,1) \nonumber\end{aligned}$$ and the sum is over all four-sets of the $2D$-lattice points. The unit cell in the definition of each $\vec R^{\alpha}$ has $4l$ flux quanta $\Psi_{0}$ passing through it.
Gozzi and Reuter[@GR-93b] have argued that there is a close relation between the symbol-calculus formalism and the Gelfand-Naimark-Segal construction[@Thirring-79]. In general, the GNS construction is specifically aimed to define non-commutative measure and topology[@Coquereaux-92].
The GNS construction provides bra-ket-type averaging, instead of the usual trace averaging, in the thermo field theory[@Umezava-82] when one deals with [*mixed*]{} states. This construction assumes a double Hilbert space representation of states, $||\hat A \rangle\rangle =
\sum A_{\alpha\beta}|\alpha\rangle \otimes|\beta\rangle
\in {\cal H}\otimes{\cal H}$. So, particularly, the average of $\hat A$ is given by $\langle \hat A \rangle =
\langle\langle \hat \rho^{1/2}||
\hat{A} \otimes \hat{I}
||\hat \rho^{1/2}\rangle\rangle $, with $\hat{I}$ being identity operator. The modular conjugation operator ${\cal J}$ acts on the double Hilbert space by interchanging the two Hilbert spaces[@GR-93b].
Time evolution of the GNS density is given by $i\hbar\partial_{t}||\hat \rho^{1/2}\rangle\rangle
= H^{-}||\hat \rho^{1/2}\rangle\rangle$, Here, the GNS Hamiltonian $H^{-} = \hat{H}\otimes I - {\cal J}(\hat{H}\otimes I){\cal J}$ can be evidently associated with the Wigner operator $W_{-} = i\hbar V^{-}_{H}$.
In view of the analysis of the oscillator in phase space, the eigenfunctions $\varphi_{n_{1}0}$ and $\varphi_{0n_{2}}$ can be ascribed to the two pieces of the GNS double Hilbert space. Also, the GNS double Hilbert space is associated to the double Fock space ${\cal F}_{1}\otimes{\cal F}_{2}$, with the modular conjugation operator ${\cal J}$ acting on ${\cal F}_{1}\otimes{\cal F}_{2}$ by interchanging the two Fock spaces.
$q$-deformed harmonic oscillator in phase space
-----------------------------------------------
The $q$-deformation of the commutation relations (\[aa\]) for the Bopp-Kubo creation and annihilation operators (\[a-\])-(\[a++\]) reads $$b_{-}b^{+}_{-} - \frac{1}{q}b^{+}_{-}b_{-} = q^{\hat n_{1}} \qquad\label{bqb}
b_{+}b^{+}_{+} - \frac{1}{q}b^{+}_{+}b_{+} = q^{\hat n_{2}}$$ The bozonization procedure of the above operators according to Jannussis [*et al.*]{}[@JPB-78] yields the following expressions for the $q$-deformed operators ($q$-bosons): $$\begin{aligned}
\label{boson}
b_{-}=\sqrt{\frac{\bigl[\hat n_{1} +1\bigr]}{\hat n_{1} +1}}a_{-} \qquad
b^{+}_{-}=a^{+}_{-}\sqrt{\frac{\bigl[\hat n_{1} +1\bigr]}{\hat n_{1} +1}}\\
b_{+}=\sqrt{\frac{\bigl[\hat n_{2} +1\bigr]}{\hat n_{2} +1}}a_{+} \qquad
b^{+}_{+}=a^{+}_{+}\sqrt{\frac{\bigl[\hat n_{2} +1\bigr]}{\hat n_{2} +1}}\end{aligned}$$ where $\bigl[ x\bigr] = (q^{x}-q^{-x})/(q-q^{-1})$ and $a_{\pm}$ and $a^{+}_{\pm}$ are given by (\[a-\])-(\[a++\]). Due to these definitions, we can directly find that $$\begin{aligned}
\label{bn}
b_{-}b^{+}_{-} = \bigl[ \hat n_{1} +1\bigr] \qquad
b^{+}_{-}b_{-} = \bigl[ \hat n_{1} \bigr] \\
b_{+}b^{+}_{+} = \bigl[ \hat n_{2} +1\bigr] \qquad
b^{+}_{+}b_{+} = \bigl[ \hat n_{2} \bigr]\end{aligned}$$ Clearly, $b^{+}_{\pm}=(b_{\pm})^{\dagger}$ if $q \in R$ or $q \in S^{1}$. The actions of the $q$-boson operators on the Fock space ${\cal F}_{1}\otimes{\cal F}_{2}$ with the basis $$| n_{1}\, n_{2}\rangle =
\frac{(b^{+}_{-})^{n_{1}}(b^{+}_{+})^{n_{2}}}
{\sqrt{n_{1}!}\sqrt{n_{2}!}}| 0\, 0\rangle \label{basis}$$ have the form $$\begin{aligned}
\label{b-action}
b_{-} | n_{1}\, n_{2}\rangle =
\sqrt{\bigl[n_{1}\bigr]} | n_{1}-1\, n_{2}\rangle \qquad
b^{+}_{-}| n_{1}\, n_{2}\rangle =
\sqrt{\bigl[n_{1}+1\bigr]}| n_{1}+1\, n_{2}\rangle \\
b_{+} | n_{1}\, n_{2}\rangle =
\sqrt{\bigl[n_{2}\bigr]} | n_{1}\, n_{2}-1\rangle \qquad
b^{+}_{+}| n_{1}\, n_{2}\rangle =
\sqrt{\bigl[n_{1}+1\bigr]}| n_{1}\, n_{2}+1\rangle\end{aligned}$$ In the following we consider the algebra implied by the generators $$J_+ = b_{-}b^{+}_{+} \qquad
J_- = b_{+}b^{+}_{-} \label{J+-}$$ It is a matter of straightforward calculations to find that $$\bigl[J_{+},J_{-}\bigr] = \bigl[2J_{3}\bigr] \qquad
\bigl[J_{3},J_{\pm}\bigr] = \pm J_{\pm} \label{J+J-}$$ where $$2J_{3} = \hat n_{1} - \hat n_{2} \label{J3}$$ One can recognize that the above relations are standard quantum algebra $su_{q}(2)$ commutation relations, in the Kulish-Reshetikhin-Drinfeld-Jimbo realization[@Biedenharn-89]-[@SK-92] according to which $su_{q}(2)$ can be realized by two commuting sets of $q$-bosons ($q-$deformed version of the Jordan-Schwinger approach to angular momentum). Hereafter, we write $su_{q}(2)$ to denote the quantum algebra which is in fact $U_{q}(su(2))$.
Comparing (\[J3\]) with (\[Wnn\]) we see that the Wigner operator for harmonic oscillator is just proportional to the 3-axis projection of the ($q$-deformed) spherical angular momentum operator, $$W_{-} = 2\hbar\omega J_{3} \label{WJ}$$
Indeed, in the $su(2)$ notations[@Macfarlane-89] for basis vector $|n_{1}n_{2}\rangle$, $$|j\, m\rangle = |n_{1}\, n_{2}\rangle \qquad \label{jm-state}
j = \frac{1}{2}(n_{1}+n_{2}) \qquad
m = \frac{1}{2}(n_{1}-n_{2})$$ the operators $J_{\pm}$ and $J_{3}$ act on ${\cal F}_{1}\otimes{\cal F}_{2}$ according to $$\begin{aligned}
\label{J-action}
J_{-}|j\, m\rangle =
\sqrt{\bigl[j+m\bigr]\bigl[j-m+1\bigr]}|j\, m-1\rangle \nonumber\\
J_{+}|j\, m\rangle =
\sqrt{\bigl[j-m\bigr]\bigl[j+m-1\bigr]}|j\, m+1\rangle \\
J_{3}|j\, m\rangle =
m|j\, m\rangle \nonumber\end{aligned}$$ For a fixed value $2j \in Z$, vector $|j\ m\rangle$ span the irrep $(j)$ of the quantum algebra $su_{q}(2)$. We assume that $q$ is not root of unity. Acordingly, the charge operator $J = \frac{1}{2}(\hat{n_{1}}+\hat{n_{2}})$ commutes with $J_{\pm ,3}$, and $J|j\, m\rangle = j|j\, m\rangle$.
The indication of the Wigner density operator $W_{+}$ may be seen from the following. The basic fact[@SK-92] is that the vector $|n_{1}n_{2}\rangle \equiv |j\ m\rangle$ can be represented also as a basis vector $|k\ l\rangle$ for the irrep beloning to the positive discrete series of $su_{q}(1,1) \approx sp_{q}(2,R)$ with the hyperbolic angular momentum, $k = \frac{1}{2}(n_{1}-n_{2}-1)= m-\frac{1}{2}$, and 3-axis projection, $l = \frac{1}{2}(n_{1}+n_{2}+1)= j+\frac{1}{2}$. The generators of $su_{q}(1,1)$ are $$K_{+} = b^{+}_{+}b^{+}_{-} \qquad
K_{-} = b_{+}b_{-} \qquad
K_{3} = J + \frac{1}{2}$$ Particularly, the 3-axis hyperbolic angular momentum operator $K_{3}$ acts due to $$K_{3}|k\ l\rangle = l|k\ l\rangle$$ Thus, the Wigner density operator $W_+ = H(\Phi_{-}) + H(\Phi_{+}) = \hbar\omega (\hat n_{1}+\hat n_{2}+1)$ can be immediately identified with $K_{3}$, $$W_{+} = 2\hbar\omega K_{3} \label{WK}$$
To summarize, we note that the harmonic ($q$-)oscillator in phase space naturally arises to the Jordan-Schwinger approach to ($q$-deformed) angular momentum, with the Wigner operator $W_{-}$ ($W_{+}$) being identified with the 3-axis spherical (hyperbolical) angular momentum operator.
As a final remark, we notice that there are ways to give geometrical interpretation of the quantum algebras and its representations. Namely, one may follow the line of reasoning by Fiore[@Fiore-93] and construct a realization of the quantum algebra within $Diff(M_{q})$, where $M_{q}$ is a $q$-deformed version of the ordinary manifold. For example, in the context of the $q$-oscillator in phase space it is highly interesting to find such a realization for the algebra $su_{q}(1,1) \approx sp_{q}(2,R)$, which is concerned the $q$-deformed phase space.
Also, there is a possibility[@Franco-93] to give a geometric interpretation of the representations of $su_{q}(2)$ following the lines of the standard Borel-Weyl-Bott theory[@Wallach-73; @Fulton-91].
Conclusions
===========
We studied the relation between the Bopp-Kubo formulation and the Weyl-Wigner-Moyal calculus of quantum mechanics in phase space which is found to arise from the fact that the Moyal product of functions on phase space, $f(\phi )*g(\phi )$, can be rewritten equivalently as the product of functions defined on the extended phase space, $f(\Phi )g(\phi )$.
From the non-commutative geometry point of view, the phase-space formulation of quantum mechanics is an example of the theory with non-commutative geometry. The non-commutative algebra ${\cal A}$ is the algebra of functions on phase space endowed with the Moyal product. The right and left ${\cal A}$-modules are interchanged by the modular conjugation ${\cal J}$ so that the space of sections ${\cal E} ={\cal A} \otimes {\cal A}/{\cal J}$, and there is a kind of chiral symmetry due to the non-commutativity.
Due to a similarity between the phase-space formulation of quantum mechanics and Hamiltonian formulation of classical mechanics, there is an attractive possibility to extend useful classical notions and tools to quantum mechanics. An attempt is made to formulate the quantum extension of the classical ergodicity condition.
We studied one-dimensional harmonic ($q$-)oscillator in phase space. The phase space has a $2D$-lattice structure, similar to the one of the magnetic group periodicity for the $2D$-electron gas in magnetic field.
The Fock space for the oscillator is related to the double Hilbert space of the Gelfand-Naimark-Segal construction in accordance with the relation between the symbol calculus and the GNS construction.
For the $q$-oscillator, the Wigner operator $W_{-}$($W_{+}$) is found to be proportional to the 3-axis spherical (hyperbolical) angular momentum operator of the $q$-deformed algebra $su_{q}(2)$ ($su_{q}(1,1)\approx sp_{q}(2,R)$).
The analysis of this paper is valid at fixed value of time. Its reformulation in a form invariant under time evolution will be studied in a future paper.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present the first overview of the *Herschel* observations of the nearby high-mass star-forming region NGC 7538, taken as part of the *Herschel* imaging study of OB Young Stellar objects (HOBYS) Key Programme. These PACS and SPIRE maps cover an approximate area of one square degree at five submillimeter and far-infrared wavebands. We have identified 780 dense sources and classified 224 of those. With the intention of investigating the existence of cold massive starless or class 0-like clumps that would have the potential to form intermediate- to high-mass stars, we further isolate 13 clumps as the most likely candidates for followup studies. These 13 clumps have masses in excess of 40 [M$_{\odot}$]{} and temperatures below 15 K. They range in size from 0.4 pc to 2.5 pc and have densities between $3\times10^{3}$ cm$^{-3}$ to $4\times10^{4}$ cm$^{-3}$. Spectral energy distributions are then used to characterize their energetics and evolutionary state through a luminosity-mass diagram. NGC 7538 has a highly filamentary structure, previously unseen in the dust continuum of existing submillimeter surveys. We report the most complete imaging to date of a large, evacuated ring of material in NGC 7538 which is bordered by many cool sources.'
author:
- 'C. Fallscheer, M. A. Reid, J. Di Francesco, P. G. Martin, M. Hennemann, T. Hill, Q. Nguyen-Luong, F. Motte, A. Men’shchikov, Ph. André, D. Ward-Thompson, M. Griffin, J. Kirk, V. Konyves, K. L. J. Rygl, M. Sauvage, N. Schneider, L. D. Anderson, M. Benedettini, J.-P. Bernard, S. Bontemps, A. Ginsburg, S. Molinari, D. Polychroni, A. Rivera-Ingraham, H. Roussel, L. Testi, G. White, J. P. Williams, C. D. Wilson, M. Wong, A. Zavagno'
title: '*Herschel*[^1] Reveals Massive Cold Clumps in NGC 7538'
---
Introduction
============
The European Space Agency’s *Herschel* Space Observatory [@pilb] probes the cold, dusty precursors of stars in unprecedented numbers and detail. *Herschel*’s submillimeter and far-infrared wavebands span the peak of the spectral energy distribution (SED) where cold cores emit the bulk of their radiation. The combination of *Herschel*’s spectral coverage and high angular resolution allows, for the first time, the identification of large numbers of potential star-forming cores ($\sim$0.1 pc) and clumps ($\sim$0.5 pc) in high-mass star-forming regions and to measure their dust properties. In this paper, we report the first *Herschel* results on the nearby high-mass star-forming region NGC 7538 observed as part of the *Herschel* imaging study of OB Young Stellar Objects (HOBYS, @hobys) Key Programme. At a trigonometric parallax distance of only 2.7 kpc from the solar system [@mos09], is a relatively nearby region of high-mass star formation and an excellent place to search for the precursors of future high-mass stars.
Although is best known for the bright region surrounding the source known as IRS 5 [@cgg78], star-formation is distributed widely throughout the larger molecular cloud. Each of the three brightest infrared sources IRS 1, 2, and 3 has its own associated compact region [@ww74]. Collectively, these three sources are embedded in a region rich with outflows and photodissociation fronts [@davis1998]. Observations of a 12$\times$8 region at 850 [$\mu$m]{} [@rw05] showed that all of these sources are embedded in an extensive network of filaments and compact sources comprising several thousand solar masses of gas and dust. Elsewhere in the region, @fries07 found a filamentary complex in an 18$\times$18 map of C$^{18}$O(2–1) around the cold cloud G111.80+0.58 (at $\alpha,\delta$ (J2000) = 23:16:22, 61:22:47; G111 hereafter). The G111 complex contains about a dozen candidate cold, high-mass clumps, several with masses exceeding 100 [M$_{\odot}$]{} [@fries07]. All of these regions are labeled in Figure \[fig:threecol\].
The *Herschel* data presented here expand our spatial coverage of this region by nearly an order of magnitude, revealing the full extent of the filamentary structure in the area as well as a large population of compact, potentially star-forming sources. *Herschel*’s ability to both detect and characterize cold, high-mass sources make it an especially exciting tool for the study of high-mass star formation in nearby regions such as NGC 7538.
Observations
============
*Herschel* Data {#sec:data}
---------------
The observations of a $\sim 1\degr \times 1\degr$ portion of were made by *Herschel* on 14 December 2009 as part of the HOBYS Key Programme. The data were acquired using the PACS [@pacs] and SPIRE [@spire] cameras working in parallel mode with a scanning speed of 20/second (OBSIDs: 134218808, 1342188089). Images were obtained simultaneously with PACS at 70 [$\mu$m]{} and 160 [$\mu$m]{} and SPIRE at 250 [$\mu$m]{}, 350 [$\mu$m]{}, and 500 [$\mu$m]{}. These five wavebands range in angular resolution from 5.6 at 70 [$\mu$m]{} to 36 at 500 [$\mu$m]{}. The calibration and deglitching of the Level 0 PACS and SPIRE data were done using HIPE[^2] version 9.0. The level 1 data were then used to produce maps with version 18 of the Scanamorphos software package [@rous2012]. Figure \[fig:threecol\] shows a three-color image made from SPIRE and PACS data. The stretch and contrast of each waveband have been manipulated slightly to accentuate color variations. The bright region which dominates the western half of the image coincides with the void in the region. This bright complex contains the aforementioned IRS sources as well as the young stellar object NGC 7538S ($\alpha,\delta$ (J2000)= 23:13:45, +61:26:51). The two brightest emission peaks in the image saturated the detectors in the SPIRE 250 [$\mu$m]{} band. These peaks are coincident with IRS 1–3 (at $\alpha,\delta$ (J2000)= 23:13:45, +61:28:10) and IRS 11 (at $\alpha,\delta$ (J2000)= 23:13:44, +61:26:49). Both peaks were re-observed with *Herschel* in bright source mode to fill in the holes in the map (OBSID: 1342239268). These observations were taken on 13 Feb 2012 and the saturated pixels were replaced in the images using the method described in Nguyen Luong et al. (subm.). Figure \[fig:threecol\] also highlights the highly filamentary nature of the emission. Using the 160 [$\mu$m]{}, 250 [$\mu$m]{}, 350 [$\mu$m]{}, and 500 [$\mu$m]{} data, we constructed H$_{2}$ column density and dust temperature maps of the region using the IDL $\chi^{2}$ minimization fit routine *mpfitfun*. These maps are shown in Figure \[fig:dentemp\] and include labels of all of the sources mentioned above.
JCMT data
---------
To obtain CO(3–2) emission observations, we observed NGC 7538 with the HARP instrument at the James Clerk Maxwell Telescope[^3] (JCMT). HARP is a 4 $\times$ 4 array of heterodyne receivers that can observe 325-375 GHz. HARP was tuned to observe CO(3–2) at 345.7959899 GHz [@pick98], and the JCMT’s ACSIS correlator was configured to observe the line over a 1 GHz wide band with 2048 channels having a velocity resolution of 0.42 km s$^{-1}$ channels. The NGC 7538 region was divided into 36 fields each 12$^{\prime}$ $\times$ 12$^{\prime}$ in size, and spaced in R.A. or decl. by 10$^{\prime}$, that covered essentially the same region observed by *Herschel*. Each field was observed in on-the-fly mode using an offset position found to be free of CO(3–2) emission located at 23$^{h}$20$^{m}$55.72$^{s}$, +60$^{\circ}$50$^{\prime}$00.4$^{\prime\prime}$ (J2000). After each scan along the length of a field, the array was moved in either R.A. or decl. by 1/4 of its extent (i.e., $\sim$30$^{\prime\prime}$) between scans to obtain samples of each position by several receivers. The observations were obtained throughout semesters 07B, 09B, and 10B.
The JCMT data were reduced using standard procedures within the Starlink package. Integrations from each receiver were checked visually for baseline ripples or extremely large spikes, and any such affected data were removed from the ensemble. Each integration had its baseline subtracted, its frequency axis converted to velocities in the local standard of rest frame, and its outer velocities trimmed using scripts kindly supplied by T. Van Kempen. The data for each field were co-added into final spectral cubes, and then arranged into a final mosaic of all fields of about 1$^{\circ}$ $\times$ 1$^{\circ}$ in extent. The typical 1 $\sigma$ rms sensitivity reached was $\sim$0.6 K per channel on the $T_{A}^{*}$ scale. Spectra were converted to main beam brightness temperature using an efficiency measured from planetary observations of 0.75 (P. Friberg, private communication).
![Three-color image of an approximately 50$\times$ 50 portion of NGC 7538. The wavebands included are SPIRE 250 [$\mu$m]{} (*red*), PACS 160 [$\mu$m]{} (*green*), and PACS 70 [$\mu$m]{} (*blue*). The eastern half of the image is dominated by a prominent ring-like feature of uncertain origin. \[fig:threecol\]](fig-3.png)
Results
=======
SED Fitting
-----------
Figure \[fig:dentemp\] shows the column density (N$_{H_2}$) and dust temperature maps of NGC 7538. To produce these maps, we corrected the arbitrary zero-point flux offset for each PACS and SPIRE map with data from the IRAS and Planck telescopes [see @bern2010]. After applying the offsets, each PACS and SPIRE map was convolved to the 500 micron beam size (36$\arcsec$) and regridded to the same pixel resolution as the 500 [$\mu$m]{} map. We used the IDL routine mpfitfun to fit a modified blackbody function to each pixel, assuming the dust spectral index, $\beta$=2.0 and the dust opacity per unit mass column density, $\kappa_{\nu}$=0.1 cm$^2$/g for a reference wavelength of 300 [$\mu$m]{}. For these fits, we exclude the 70 [$\mu$m]{} data since the emission from this short wavelength more likely results from warmer material rather than the cold dust component traced by the longer wavelengths which we are most interested in [e.g. @hill2011]. For the NGC 7538 region, line-of-sight dust temperatures predominantly range between 12 K and 25 K with a mean temperature of 17 K and column densities vary from 3$\times$10$^{21}$ cm$^{-2}$ to 4$\times$10$^{23}$ cm$^{-2}$. From the column density map, we calculate a total mass of the region to be nearly 4$\times$10$^{5}$ [M$_{\odot}$]{} which is in agreement with @unge2000. Over half of this mass is contained in high-column density structures ($>$10$^{22}$ cm$^{-2}$).
The Ring Structure {#sec:ring}
------------------
A striking feature of the NGC 7538 maps is the existence of ring-like features, especially the nearly complete ring which dominates the eastern section of the *Herschel* map (see Figure \[fig:threecol\]). A zoomed-in view of this region is shown in Figs. \[fig:ring\] and \[fig:jcmt\]. Portions of this ring were previously detected by @fries07 in C$^{18}$O(2–1) emission and by @fries08 in the *Spitzer* continuum bands. Here, we also present JCMT CO(3–2) data which include the entire ring structure. Such a large, well-defined ring has not been seen in the other regions observed in the HOBYS survey, e.g., Rosette [@schn2010], W48 [@luong], or Cygnus X [@henn2012], emphasizing the peculiarity of this feature[^4]. As shown in Figure \[fig:ring\], the ring is prominent in thermal dust emission at wavelengths of 160 [$\mu$m]{} and longer. It is visible as a string of point sources in the PACS 70 [$\mu$m]{} image and the *Spitzer* MIPS 24 [$\mu$m]{} image (see Figure \[fig:ring\]). The ring is a nearly complete ellipse with major and minor axes of approximately 10.6 pc and 7.4 pc, respectively. The ring may be the edges of a bubble produced by an internal energetic source, but its origin is not yet clear. According to an exhaustive catalog of both known and candidate O and B stars [@reed][^5], no such stars lie within the ring. Although there is an A0 type star within the ring, the nearest known massive star is a B star about 4 pc east of the ring. A search of archival data has so far revealed no MSX, IRAS, or radio continuum sources within the ring which might account for its existence.
The JCMT CO(3–2) data also show no indication of objects present that may be responsible for forming the ring. In the channel map presented in Figure \[fig:jcmt\], the emission from the channels spanning the ring’s velocity range is shown. At this longer wavelength (870 [$\mu$m]{}), a driving source is still not evident.
Aside from a small feature most prominent in the \[-56,-54\] km s$^{-1}$ channel, the region within the ring appears completely devoid of material in its entire 17 km s$^{-1}$ velocity range. A spherically symmetric object such as a bubble would likely exhibit emission within the ring at the higher velocities in the channel map, which we do not see. It is more likely that we are observing a bubble that has broken out of the molecular cloud, producing a ring of CO emission (similar to smaller molecular rings seen by @beau2010 around the Churchwell Spitzer bubbles.) Such a void (with a typical column density of 6$\times$10$^{22}$ cm$^{-2}$) is in stark contrast to the rest of the NGC 7538 region as well as the other HOBYS fields which are dominated by low levels of diffuse emission (typically higher than 10$^{22}$ cm$^{-2}$) throughout the maps. This lack of diffuse continuum [*a*nd]{} line emission indicates that the ring was created through different mechanisms than the methods by which other filaments in these star formation regions formed. Of course, it is possible that the ring could be a coincidental alignment of curved filaments, but the lack of diffuse gas within the ring would suggest otherwise.
While the ring appears elliptical in the *Herschel* data and at the redder velocities in the CO data, it becomes even more elongated in the northeast-southwest direction at the bluer CO velocities. The integrated intensity contours overplotted in the bottom right panel of Figure \[fig:jcmt\] retain this dominant oblong shape. Although the ring is brightest at the southern end closest to G111, these integrated intensity contours establish the nearly-closed morphology of this ring structure.
![Images of the ring structure from *Spitzer*’s MIPS (24 [$\mu$m]{}) camera, as well as *Herschel*’s PACS (70 [$\mu$m]{} and 160 [$\mu$m]{}) and SPIRE (250 [$\mu$m]{}, 350 [$\mu$m]{}, and 500 [$\mu$m]{}) instruments. The sources circled in the 160 [$\mu$m]{} image are discussed further in Section \[sec:getsources\]. \[fig:ring\]](fig-4.png){width="6.0in"}
![Integrated intensity maps of the ring region in JCMT CO(3–2) data. The central pixel is at RA=23h16m26.5s decl.=61$^{\circ}$29’56” (J2000). For each panel, contours start at 1$\sigma$ and continue in steps of 1$\sigma$ [**(**with the exception of the \[-50,-48\] channel which starts at 2 $\sigma$ and increases in steps of 2 $\sigma$)]{} where $\sigma$ is 2.0, 4.5, 5.9, 4.8, 4.9, and 2.9 K km s$^{-1}$ for the panels in order of increasing velocity. The red contours in the upper left panel are the column density from 5$\times$10$^{21}$ cm$^{-2}$ to 5$\times$10$^{22}$ cm$^{-2}$ in steps of 5$\times$10$^{21}$ cm$^{-2}$. The blue contour lines in the bottom right panel are for the ring region integrated over the entire velocity range (-58 km s$^{-1}$ to -41 km s$^{-1}$) of emission. These contours start at 1 $\sigma$ and continue in steps of 1.5 $\sigma$ where $\sigma$=16.0 K km s$^{-1}$. These intensities are on the $T_{A}^{*}$ scale. \[fig:jcmt\]](fig-1.png){width="4.5in"}
Source Extraction {#sec:getsources}
-----------------
To identify compact sources in the field, we used version 1.120916 of the [[getsources]{}]{} algorithm [@gs1; @gs2]. Since the PACS and SPIRE instruments are so sensitive, we detect significant extended, diffuse emission which makes source extraction more difficult. Rather than subtract a global background from the entire image, as is sometimes done, [[getsources]{}]{} separates emission across a wide range of angular scales and then uses this information to identify sources as peaks relative to their local backgrounds. Thus, compact sources are extracted without imposing any parametrization on either their structure or that of the diffuse background. The algorithm assigns, to each source identified in each waveband, a shape, size, peak flux, total flux and significance value. Note that it is possible for sources detected at longer wavelengths to be resolved into multiple objects at shorter wavelengths due to increasing resolution. However, [[getsources]{}]{}passes information on from the higher resolution images to the extractions of lower resolution images. For each object detected in the higher resolution maps, the final catalog assigns a single non-overlapping flux value at all the wavelengths for which a significant detection is made. The significance parameter is analogous to a signal-to-noise value in a single waveband. Using this method, we have preliminarily identified 780 compact sources in NGC 7538. Of these, many were detected in fewer than the five possible wavebands. In such cases, the subset of wavebands in which the sources were detected varied. A common pattern was that sources were detected with SPIRE, but not with PACS. This result is partly due to the low luminosity of cold sources in the PACS wavebands and partly due to the wide range of angular resolutions among the five wavebands.
For our analysis, we consider only those sources which could be identified with [[getsources]{}]{} in at least two different wavebands with a significance of at least 7. Before fitting a modified black body function to the SED to a source, we additionally stipulated that a source must be detected in at least one more waveband with a significance of at least 5. These criteria were satisfied by 224 of the $\sim$800 sources. Figure \[fig:threecolsources\] shows the molecular hydrogen column density image overlaid with the sources extracted by [[getsources]{}]{}. The image shows that the compact sources divide into two groups: tight clusters of point-like sources which are bright at 70 [$\mu$m]{} (blue and green in Figure \[fig:threecolsources\]) and long, filamentary chains of sources which are typically brighter at the longer wavelengths (red in Figure \[fig:threecolsources\]). Observations by @schn2012 support the idea that the most massive YSOs preferentially form in clusters at the junction of filaments. However, the observed distribution of sources in this region may be due more to their proximity to high-mass versus low-mass sites of star formation.
Figure \[fig:sedfig\] shows the SEDs of three sources identified in NGC 7538 and labeled with triangles in Figure \[fig:threecolsources\]. These sources are a representative sample of the sources which define the ring structure in eastern NGC 7538 discussed in Section \[sec:ring\] and are well-fit by single-temperature grey body SEDs. Source A ($\alpha,\delta$ (J2000) = 23:17:10.3, +61:31:14) is luminous ($L = 600$ [L$_{\odot}$]{}) and extended (diameter $\simeq$ 0.6 pc). We estimate its mass to be 20$\pm$5 [M$_{\odot}$]{}, which, combined with its high temperature of 26$\pm$2 K, suggests it might be forming a high-mass star or a small cluster. Source B (at $\alpha,\delta$ (J2000) = 23:16:07.5, +61:25:14) is a more typical low-mass core, with a luminosity and diameter of 38 [L$_{\odot}$]{} and 0.4 pc, respectively. Having a relatively high temperature of 20$\pm$3 K, it is likely already undergoing star formation. Source C (at $\alpha,\delta$ (J2000) = 23:16:10.8, +61:22:52) is one of the candidate cold, high-mass clumps identified in this study and is listed in Table \[tab:hmdc\] as HMDC 8. With a luminosity and diameter of 18 [L$_{\odot}$]{} and 0.4 pc, respectively, it has a mass of 76$\pm$27 [M$_{\odot}$]{}, and a temperature of 12$\pm$1 K. Thus, it is within the $L/M <
1$ [L$_{\odot}$]{}/[M$_{\odot}$]{} regime where unevolved high-mass sources are thought to lie (see Section \[sec:props\] below). Note that because it is cold, Source C is barely visible in the 70 [$\mu$m]{} image (see Figure \[fig:ring\]) but becomes much brighter at longer wavelengths.
![Spectral energy distributions for the three sources around the edge of the ring in eastern NGC 7538 that are discussed in Section \[sec:getsources\]. The points show the integrated fluxes of each source as measured by [[getsources]{}]{} after the application of the flux scaling discussed in Section \[sec:props\]. The lines show the best-fitting single-temperature grey body of the form shown in Eqn. \[eq:gb\]. The source labels–A, B, and C–correspond to the sources marked in Figs. \[fig:ring\] and \[fig:threecolsources\]. The temperature and mass of each source derived from the grey body fit to the four longest wavelengths as discussed in Section \[sec:props\] are indicated. \[fig:sedfig\]](fig-5.png){width="5" height="6"}
Discussion
==========
The Ring
--------
We determine an order-of-magnitude estimate of 500 M$_{\odot}$ for the mass of the ring by assuming a temperature of 15 K and converting the observed 250 [$\mu$m]{} flux in the ring based on a dust opacity at 250 [$\mu$m]{} of 2 as given by @h83. Using a simple order-of-magnitude energy calculation, we determine that the energy required to move that mass from a centrally concentrated sphere out to a spherical shell of radius 8 pc based on an assumed expansion rate of 1 km s$^{-1}$ would be on the order of 10$^{45}$ erg. While the observations suggest that a two dimensional geometry may be more appropriate, we choose spherical symmetry for this simple illustrative first-order approximation. The assumed expansion rate of 1 km s$^{-1}$ is typical for a supernova remnant near the end of the radiative expansion phase when the expansion rate becomes that of the sound speed of the ambient medium. This energy is several orders of magnitude lower than the $\sim10^{51}$ erg released in a typical supernova collapse of a massive star [@wils1985]. It may instead be similar to the amount of energy contributed by the stellar wind of a massive star, but we see no indication for the presence of such a source within the ring. We also look into the possibilities that a runaway O star that may have originated from within the ring, or alternatively, that a nearby windy O-star blew out the cavity within a bubble. The strongest candidate for either of these scenarios is HIP 115424, an O8 star approximately 50 ($\sim$40 pc) northeast from the center of the ring. This star is classified as a runaway star with a peculiar tangential velocity of 30 km s$^{-1}$ [@moff1998]. Despite its relative proximity, the position and kinematics of this star make it unlikely that it originated within the ring. At its present location, the winds of an O8 star are likely too weak to have had much influence on the ring. Investigation into the accuracy of the spectral type classification of this star is necessary to explore this possibility further. Based on its geometry, we suggest that this ring might be an example of triggered star formation. Although there is no region associated with this ring as in the case of the triggered star formation in the HOBYS study of N49 [@zava2010], for example, Figures \[fig:ring\] and \[fig:threecolsources\] show that most of the ring is delineated by compact sources and cool dust and show that it is a coherent velocity structure. Our SED fits show that very few of these sources have temperatures exceeding 30 K and that many are somewhat colder. Several of our massive cold clumps—sites with the potential for intermediate- to high-mass star formation—lie along the ring and especially in the cluster of sources to the south. @fries07 also located several candidate sites of early high-mass star formation in this cluster. This cluster of sources make up the Infrared Dark Cloud (IRDC) G111.80+0.58 [@fries08]. Our data reveal that this IRDC is not merely part of a filament, but actually at the intersection of a ring-like structure with surrounding filaments.
Source Properties and Energetics {#sec:props}
--------------------------------
Given the wide range of angular resolutions, and the likely presence of thermal substructure within each source, each waveband is sensitive to slightly different physical components within each source. For example, the warmer and more compact central regions of protostellar objects would be more prominent at shorter wavelengths within the smallest beams, while longer wavelengths with their larger beams are more sensitive to their cooler and more extended envelopes. To account for this size-wavelength dependence, we adopt the scaling prescription of @hobys, which was elaborated further by @luong. In general, this technique is applied to the regions in the HOBYS program because of the large distances involved, but it is unnecessary for resolved cores in the nearby low-mass star formation clouds. According to this prescription, we set the deconvolved source size using a fit to the 160 [$\mu$m]{} image and then scale the 250 [$\mu$m]{}, 350 [$\mu$m]{}, and 500 [$\mu$m]{} fluxes down according to
$$S_{{\rm int},\lambda}^{\rm scaled} (< r_{\lambda}) = S_{{\rm
int},\lambda}^{\rm original} \left( \frac{r_{160~\mu m}}{r_{\lambda}} \right), \label{eq:scale}$$
where $r_{\lambda}$ is the deconvolved radius of the source over which the flux, $S_{{\rm
int},\lambda}$, is integrated. This flux scaling method assumes that the source size obtained in the 160 micron image is accurate, that the emission at the longer wavelengths (250 [$\mu$m]{}, 350 [$\mu$m]{}, and 500 [$\mu$m]{}) is optically thin, and that the flux varies linearly with angular radius. For dense cores and clumps, it is reasonable to assume that they are optically thin at these wavelengths. Likewise, the linear variation of the flux is a good approximation, at least until the point that an H[ii]{} region forms. Although testing non-linear models is beyond the scope of this paper, if the flux variation does not vary linearly with angular radius, it is likely that the temperature would be overestimated and consequently the mass would be underestimated. After scaling the source fluxes in this way, we find that 224 sources out of the nearly 800 preliminary sources are well fit by a single-temperature grey body SED. Under the assumption that the data are accurately represented by the given SED function, we define a good fit such that there is a 95% chance that the calculated $\chi^{2}$ will be less than the $\chi^{2}$ value expected from random variations in the data (i.e. at the p-value $p = 0.05$ statistical significance level.) We fit a SED of the form
$$S(\nu) =
\frac{M_{\rm gas+dust}\kappa_{300}}{d^2}\left(\frac{\nu}{\nu_{300}}\right)^{\beta}
B(\nu,T_{\rm dust})~~,
\label{eq:gb}$$
where $M_{\rm gas+dust}$ is the total mass of the source, $d =
2.7$ kpc is the distance to the cloud, $\kappa_{300}$ is the the dust opacity per unit (gas + dust) mass at 300 [$\mu$m]{}, $\nu_{300}$ is the frequency corresponding to a wavelength of 300 [$\mu$m]{}, $\beta$ is the dust emissivity index, and $B(\nu,T_{dust})$ is the Planck function for dust temperature $T_{\rm
dust}$. We adopt a dust opacity of $\kappa_{300}$ = 0.1 cm$^{2}$ g$^{-1}$ and a fixed dust emissivity index of $\beta = 2.0$. These values are derived from @h83 and are consistent with those used in other HOBYS studies (e.g., @hobys) as well as other SPIRE Galactic key programmes. Note, however, that the dust opacity and emissivity index, $\beta$, are likely to vary with environment [@osse1994] which may imply non-systematic uncertainties in our measured dust masses (e.g., see Sadavoy et al. 2013, submitted). We assume a gas-to-dust ratio of 100. The total amount of mass associated with these 224 sources is on the order of 1$\times$10$^{4}$ [M$_{\odot}$]{}. The overwhelming majority of this mass is contained within sources having temperatures less than 20 K.
We do not include the 70 [$\mu$m]{} data in the SED fitting (see @hill2012 for further discussion). At wavelengths shorter than 100 [$\mu$m]{}, the emission may require an additional temperature component to properly include the contribution from the warmer material associated with the protostar. Also, the 70 [$\mu$m]{} emission may arise from very small dust grains (VSG) thereby changing the dust opacity law in this regime. Thus, in cases where the inner warm envelope material or VSG emission dominate the 70 [$\mu$m]{} flux associated with the bulk envelope (the material primarily traced by the $\lambda \ge$ 160 [$\mu$m]{} bands), the observed 70 [$\mu$m]{} emission is typically higher than the SED fits at that wavelength. With flux scaling applied to remove some of the influence of each source’s cold outer envelope, the SED fits are more representative of the interior parts of the sources where star formation, if present, would occur. For the purposes of assessing the evolutionary states of these sources, the conditions in their interiors are more important than those in their envelopes. For example, a cool interior is a more indicative sign of youth than a cool envelope.
In the future, we could leverage the information about each source’s emission on multiple spatial scales to construct more sophisticated, multi-temperature models, but for the present purpose of developing an overall picture of the star-formation activity in NGC 7538, we maintain this simple approach. Note that this procedure underestimates the source masses because it excludes flux from the outer envelope of each source as well as flux from wavelengths longer and shorter than those observed with *Herschel*. We deem this approach to be acceptable for our purposes since we are trying to identify conservatively sites of high-mass star formation.
Having derived each identified source’s temperature and mass from Equation \[eq:gb\], we compute their *Herschel* grey body luminosities as:
$$L = 4 \pi d^2 \int_{0}^{\infty} S(\nu) d\nu.$$
In Figure \[fig:lvsm\], we plot the luminosity of each source versus its total mass. The luminosity plotted here is that derived from the SED which, for cool, starless sources, should be similar to its bolometric luminosity. By taking the ratio of each source’s luminosity to its mass, we can assess the degree to which it is affected by internal heating, and therefore the likelihood that it is already forming stars [e.g. @andr2008; @hobys].
![ Herschel grey body luminosity vs. total mass (dust + gas) for compact sources in NGC 7538. Only the 224 sources whose SEDs were well-fit by a grey body function with $\beta =
2$ are included in the plot. The diagonal dashed lines represent loci of constant temperature, with the respective temperatures indicated on each line. The thick solid line represents $L/M$ = 1 [L$_{\odot}$]{}/[M$_{\odot}$]{}, equivalent to a temperature of about 15.7 K. The shaded region (M $\geq$ 40 [M$_{\odot}$]{} and L/M $\leq$ 1 [L$_{\odot}$]{}/[M$_{\odot}$]{}) highlights massive cold clumps which we identify as potential precursors of high-mass stars. \[fig:lvsm\]](fig-0.png){width="6in"}
The evolutionary state of a low-mass protostellar core is typically defined according to a scheme of classes, between Class 0 and Class III, using a combination of infrared and submillimeter observations [e.g. @lada1987; @andre]. This technique is generally less applicable to high-mass star formation for several reasons. First, the technique relies on positional coincidence to establish whether or not an infrared source is embedded within a given molecular cloud core. True physical association of infrared sources with dusty clumps is difficult to establish in high-mass star-forming regions, however, because these regions are typically distant and in clustered environments. Second, if a low-mass core contains an infrared point source, it is reasonable to assume that star formation is already underway and the star being formed is likely going to be the largest in the system. In a high-mass core, however, the high-mass star may not be the first star in the cluster to form. The presence of an infrared point source in a high-mass core may result from one or several low-mass protostars that formed before the most massive star in the cluster, and does not necessarily correspond to what will become the most massive star in the cluster.
For the above reasons, we employ the mass-luminosity diagram as a tool for characterizing the evolutionary states of the compact sources in NGC 7538. The mass-luminosity diagram helps separate sources according to their energetics and can be useful for separating Class I YSOs from Class 0 protostars and prestellar cores in the low mass regime and infrared-bright massive YSOs from infrared-quiet protostellar objects and starless dense cores in the high mass regime [@bon96; @mol08; @hennemann; @roy11]. Due to their high opacities, unevolved dusty clumps with no internal heat source ought to be a few degrees colder than the ambient temperature of their parent molecular clouds, having temperatures of about 15 K and luminosity-to-mass ratios $L/M
\lesssim$ 1 [L$_{\odot}$]{}/[M$_{\odot}$]{} [@roy11]. Indeed, Roy et al. divided high-mass sources in Cygnus X into two evolutionary categories: “Stage E,” denoting externally-heated sources, and “Stage A,” denoting sources at or above the ambient temperature due to heating by accretion. Stage E overlaps with the prestellar and Class 0 stages in the low-mass paradigm. In the mass-luminosity diagram, Stage E sources lie below $L/M \sim
1$ [L$_{\odot}$]{}/[M$_{\odot}$]{}, while Stage A sources lie strictly above $L/M =
1$ [L$_{\odot}$]{}/[M$_{\odot}$]{}.
To the Stage E classification, we add the additional requirement that a source must have a mass of at least 40 [M$_{\odot}$]{}. Given a star formation efficiency of 30% or greater [@lada], a core would require on the order of 20 [M$_{\odot}$]{} to form a high-mass star. In their study of Cygnus X, @m07 use a criterion of 40 [M$_{\odot}$]{} as the mass required for a core to form a high-mass star. In the study of several other HOBYS fields (Rosette, W48 and RCW 120), @hobys and @luong use a lower criterion of 20 [M$_{\odot}$]{} to enlarge the census to cores able to form an intermediate-mass star. Most of those objects are more compact (on the order of 0.1 pc compared to $\sim$0.7 pc for the objects in this study), however, so we choose a more conservative lower limit of 40 [M$_{\odot}$]{} in order to isolate the highest mass objects in the NGC 7538 region.
We find 27 compact objects in NGC 7538 that satisfy the criteria $M_{\rm gas+dust} \geq 40$ [M$_{\odot}$]{} and $L/M~\leq~1$ [L$_{\odot}$]{}/[M$_{\odot}$]{}. These high-mass precursor candidates occupy the shaded region in Figure \[fig:lvsm\]. Applying the further criteria that a source must be located within the region mapped by both PACS and SPIRE, that its temperature must be greater than 10 K, and that it be undetected at 24 [$\mu$m]{}, 13 sources remain. A list of properties for these 13 sources is given in Table \[tab:hmdc\]. As source extraction techniques evolve, this initial list of sources will need to be confirmed with a second detection method. These 13 sources range in diameter from [**0**.4 pc to 1.1 pc]{} in mass from [**4**0 [M$_{\odot}$]{} to a few hundred solar masses, ]{} and in density from $4\times10^{3}$ cm$^{-3}$ to $4\times10^{4}$ cm$^{-3}$. Their median diameters, masses, and densities are [**0**.7 pc, 80 [M$_{\odot}$]{}, and $9\times10^{3}$ cm$^{-3}$]{} respectively. We define the deconvolved diameter of a source using the major and minor axes measured at the shortest wavelength at which the source was resolved (which is 160 [$\mu$m]{} in all 13 cases. Its volume is defined as that of an ellipsoid having the same major and minor axes as the source, plus a third axis whose length is the arithmetic mean of the lengths of the other two.
Higher resolution data may reveal fragmentation within some of these sources, but we still present them here as a first cut of the potential intermediate and high mass star forming sites in NGC 7538. While many of these sites don’t fulfill the criteria set by @kauf2010, they still appear to be cold clumps of significant mass and are worthy of further investigation.
------ ------------ ------------ ----------------- --------------------------------- ------ ------ --------------------
HMDC RA Decl. Mass L/M Size Temp Density
h:m:s d:m:s [M$_{\odot}$]{} [L$_{\odot}$]{}/[M$_{\odot}$]{} pc K cm$^{-3}$
1 23:12:42.5 61:27:47.9 200 0.43 0.9 13 8 $\times$10$^{3}$
2 23:12:51 61:27:46.7 80 0.66 0.7 14 8 $\times$10$^{3}$
3 23:12:58.9 61:27:44.1 150 1.2 0.8 15 9 $\times$10$^{3}$
4 23:13:20.7 61:28:47.7 40 0.85 0.4 14 2 $\times$10$^{4}$
5 23:13:25 61:25:6.6 340 0.99 1.0 15 7 $\times$10$^{3}$
6 23:14:19.5 61:37:45.4 82 0.26 0.8 12 5 $\times$10$^{3}$
7 23:15:36.4 61:21:38.9 140 0.21 1.1 11 4 $\times$10$^{3}$
8 23:16:10.8 61:22:52.4 76 0.24 0.4 12 4 $\times$10$^{4}$
9 23:16:16.8 61:22:16.9 160 0.10 0.5 10 4 $\times$10$^{4}$
10 23:16:17.6 61:35:7.1 70 0.11 0.5 10 2 $\times$10$^{4}$
11 23:16:23 61:17:55.1 75 0.21 0.6 11 1 $\times$10$^{4}$
12 23:16:24.3 61:22:56.2 42 0.31 0.5 12 1 $\times$10$^{4}$
13 23:16:31.3 61:18:46.9 81 0.10 1.1 10 6 $\times$10$^{3}$
------ ------------ ------------ ----------------- --------------------------------- ------ ------ --------------------
: Properties of the 13 high-mass dense clump (HMDC) candidate objects. \[tab:hmdc\]
These 13 high-mass Stage E sources are distributed spatially throughout NGC 7538. They are generally visible as the reddest objects in Figure \[fig:threecolsources\] and are highlighted with white circles in that same figure. A more detailed analysis of the cloud structure and source distribution in NGC 7538 is beyond the scope of this first-look paper, but would be interesting to look into in a followup study.
Conclusions
===========
We have reported the first results of the *Herschel* HOBYS observations of the nearby high-mass star-forming region, NGC 7538. The thermal dust emission shows many compact sources distributed along filaments. We have detected nearly 800 compact sources and characterized the SEDs of 224 of them. Of these latter sources, we identify 13 as high-mass dense clump candidates, potential sites of future intermediate- to high-mass star formation. We present the characteristics of these select high-mass dense clump objects which require further follow-up observations to confirm that star formation is underway and determine the source kinematics.
We also report the discovery of a ring of cool thermal dust emission of as-yet unknown origin. With additional data from the JCMT, we further characterize the ring and determine properties of the ring such as its extent and energetics. We look into several possible origin scenarios for the ring, none of which provide a satisfactory explanation. We detect a large number of cold sources along the ring’s filamentary edge. We acknowledge the support of the Canadian Space Agency (CSA) via a Space Science Enhancement Program grant, the National Science and Engineering Rsearch Council (NSERC) via a Discovery grant, and the National Research Council of Canada (NRC).
SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including: Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA).
PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAF-IFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain).
The James Clerk Maxwell Telescope is operated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the United Kingdom, the National Research Council of Canada, and (until 31 March 2013) the Netherlands Organisation for Scientific Research.
This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France
KLJR acknowledges support by the Agencia Spaziale Italiana (ASI) under contract number I/005/11/0.
We would also like to thank the anonymous referee(s) for his/her/their comments which have significantly improved this manuscript.
[*Facilities:*]{}
Anderson, L. D., et al. 2012, , 542, 10 André, P., Ward-Thompson, D., & Barsony, M. 1993, , 406, 122 André, Ph., et al. 2008, , 490, L27 Beaumont, C. N., & Williams, J. P. 2010, , 709, 791 Bernard, J.-P., et al. 2010, , 518, L88 Bontemps, S., André, P., Terebey, S., & Cabrit, S. 1996, , 311, 858 Churchwell, E., et al. 2006, , 649, 759 Crampton, D., Georgelin, Y. M., & Georgelin, Y. P. 1978, , 66, 1 Davis, C. J., Moriarty-Schieven, G., Eisl[ö]{}ffel, J., Hoare, M. G., & Ray, T. P. 1998, , 115, 1118 Frieswijk, W. W. F., Spaans, M., Shipman, R. F., Teyssier, D., & Hily-Blant, P. 2007, , 475, 263 Frieswijk, W. F., et al. 2008, , 685, L51 Griffin, M., et al. 2010, , 518, L3 Hennemann, M., et al. 2010, , 518, L84 Hennemann, M., et al. 2012, , 543, L3 Hildebrand, R. H. 1983, , 24, 267 Hill, T., et al. 2011, , 533, 94 Hill, T., et al. 2012, , 542, 114 Kauffmann, J., & Pillai, T. 2010, , 723, L7 Lada, C. J. 1987, IAUS, 115, 1 Lada, C. J., & Lada, E. A. 2003, , 41, 57 Men’shchikov, A., et al. 2010, , 518, L103 Men’shchikov, A., et al. 2012, , 542, 81 Minier, V., et al. 2013, , 550, 50 Moffat, A. F. J., et al. 1998, , 331, 949 Molinari, S., et al. 2008, , 481, 345 Moscadelli L., et al. 2009, , 693, 406 Motte, F., André, P., & Neri, R. 1998, , 336, 150 Motte, F., et al. 2001, , 372, L41 Motte, F., Bontemps, S., Schilke, P., Schneider, N., Menten, K. M., & Broguière, D. 2007, , 476, 1243 Motte, F., et al. 2010, , 518, L77 Nguyen Luong, Q., et al. 2011, , 535, 76 Ossenkopf, V., & Henning, T. 1994, , 291, 943 Pickett, H. M., et al. 1998, JQSRT, 60, 883 Pilbratt, G. L., et al. 2010, , 518, L1 Poglitsch, A., et al. 2010, , 518, L2 Reed, B. C. 2005, , 130, 165 Reid, M. A. & Wilson, C. D. 2005, , 625, 891 Rivera-Ingraham, A., et al. 2013, , 766, 85 Roussel, H. 2012, arXiv:1205.2576v1 Roy, A., et al. 2011, , 727, 114 Schneider, N., et al. 2010, , 518, 83 Schneider, N., et al. 2012, , 540, L11 Ungerechts, H., Umbanhowar, P., & Thaddeus, P. 2000, , 537, 221 Wilson, J. R. 1985, in Numerical Astrophysics, ed. J. M. Centrella, J. M. Leblanc, & R. L. Bowers (Boston: Jones & Bartlett), 422 Wynn-Williams, C. G., Becklin, E. E., & Neugebauer, G. 1974, , 187, 473 Zavagno, A. et al., 2010, , 518, L81
[^1]: *Herschel* is an ESA space observatory that has science instruments provided by European-led Principal Investigator consortia with important participation from NASA.
[^2]: HIPE is a joint development software by the *Herschel* Science Ground Segment Consortium, consisting of ESA, the NASA *Herschel* Science Center, and the HIFI, PACS, and SPIRE consortia.
[^3]: The James Clerk Maxwell Telescope is operated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the United Kingdom, the National Research Council of Canada, and (until 31 March 2013) the Netherlands Organisation for Scientific Research.
[^4]: Note that small, less prominent bubbles were observed by @zava2010 in RCW120 and by @ande2012, and that @chur2006 provide a catalog of bubbles seen with *Spitzer* in the Galactic plane. However, as we do not see evidence for spherical symmetry in NGC 7538, the ring structure may be a different type of object. Some less clean rings are observed in M16 [@hill2012] and W3 [@rive2013].
[^5]: Vizier catalog V/125
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A study of kinematics of a 2-body system is used to show that the Mach principle, previously rejected by general relativity, can still serve as an alternative to the concept of absolute space, if one takes into account that the background of distant stars (galaxies) determines [*both*]{} the inertial and the gravitational masses of a body.'
author:
- 'Andrew E. Chubykalo'
title: 'Principle of Mach, the Equivalence Principle and concepts of inertial mass'
---
= 7.0in
[*Escuela de Física, Universidad Autónoma de Zacatacas\
98068 Zacatecas, ZAC., MÉXICO*]{}
$$$$
$$$$ $$$$ [**PACS**]{}: 04.20.-q,01.55.+b
It is well known that in classical mechanics there were two distinct concepts of inertial mass and absolute space: that of Newton ([*in which inertial mass is a property of a body with respect to absolute space*]{}), and that of Mach ([*in which inertial mass is the property determined by the masses of distant stars*]{}) \[1\]. It is also known that when construction general relativity, Einstein started with the Mach principle, but had to reject it thereafter (e.g.,see \[2\]) because of its unagreement with the Equivalence principle. However there still is no general agreement among scientists about the necessity of total rejection of the Mach principle (e.g., see \[3-7\]).
Let there be a two-body system with the inertial and gravitational masses $m_{\tt i},M_{\tt i}$ and $m_{\tt g},M_{\tt g}$, accordingly. Let it be surrounded by some collection of distant stars, at rest with respect to the center of inertia (CI) of the system. Let these two bodies rotate around their common CI with some angular velocity $\omega$. To be more specific, let us assume that the distance between the CI of the system and the nearest star is much greater than the distance $R$ between the bodies, which, in its turn, is much greater than the size of the bodies; this will allow us to consider them as material points. Let us also assume that their velocities are much smaller than the speed of light, which will allow us to use the equations of classical mechanics. We shall further assume that the angular velocity of rotation $\omega$ is such that the bodies can get closer to each other only because of the loss of energy due to gravitational radiation, i.e., in the absence of such radiation, the distance between the two bodies would remain constant.
Let us analyze the situation arising in the hypothetical case in which the distant stars [*disappear*]{}, assuming that at that instant the bodies rotated. In such setup, there are only [*two*]{} logically possible situations:
a)[*either the kinematics of the relative motion of the system changes, i.e., an observer located on one of the bodies perceives a picture different from the one he would see in the presence of fixed distant stars;*]{}
b\) [*or the observer does not perceive any difference in the kinematic pattern of the relative motion of the bodies.*]{} $$$$ The situation (a) can happen only if the Mach principle holds in the form in which it has been known so far \[1\], since in Newton description by definition there is no change in the state of the system under a removal of sufficiently distant stars (we set up that the [*gravitational*]{} influence of this stars is infinitesimal). A change in the state of the system would be possible only due to a change in the relationship between the inertial and gravitational masses (by virtue of the Mach principle, the absence of distant stars must lead to a strong decrease in the values of inertial masses, and, therefore, to sharp increase in the relative acceleration of their mutual approach). Such a situation would be contradict the principle of equality of inertial ($m_{\tt i}$) and gravitational ($m_{\tt g}$) masses but, generally speaking, we cannot consider the [ *principle of equivalence*]{} like one of [*underlying axioms*]{} of general relativity.[^1] The case (a) was studied already rather well (profound analysis of this case is given by P. Graneau \[9\] and A. Assis \[10,11\]). Following these works we have to infer that if the case (a) happens we can [*either*]{} say that the gravitational constant $\gamma$ [*or*]{} the inertial mass $m_{\tt
i}$ of the test body will be a function of the amount and distribution of distant bodies (stars and galaxies).
However, we cannot test neither the case (a) nor the case (b) (although a direct check-up of the Mach principle can be realized in the laboratory, the effect will be too small to be detected). Therefore, we still [*must*]{} consider the case (b) and case (a) like enjoying equal rights. In this paper we shall prove that if would be realized the case (b) the Mach principle can still remain an alternative to the Newton concept of absolute space, on the one hand, and allows for the equality of the inertial and gravitational masses, on the other hand (at least, in classical mechanics).
For detailed analysis of the case (b) we are going to consider two hypothetical situations: distant bodies (the rest of stars of the Universe) [*do not exist*]{} and the Universe only [*consists*]{} of two bodies $m$ and $M$; distant stars (galaxies) [*exist*]{}.
In the case (b) both concepts of inertial mass could be valid - the Newton and [*modified*]{}[^2] Mach ones (see below). Notice that in this reasoning we have to assume that aside from “local action" (in the Faraday terminology) and independently of it there is “action-at-a-distance" ([*instantaneous action*]{}) in nature.
So, in order to modify the Mach principle and still keep the kinematic equivalence of the two concepts of space for a circular motion, one has to assume that distant stars determine [*both*]{} inertial y gravitational properties of a body[^3]. We shall call such a concept “quasi-Mach". Notice that in quasi-Mach case of situation (b), the masses cannot be equal to zero, because each body serves as a background for the second one.
Notice that these bodies may turn around their common CI, following either elliptic or circular orbits. In our Gedanken experiment we chose the circular orbits. Such a selection of circular orbits may seem unfounded at the first sight, but in fact it can be easily explained: in order to obtain the relationships in which we are interested (see below), we have to choose such kind of motion whose relative kinematics does not depend on whether it is Newton or the quasi-Mach concept is true. In the case of elliptic motion, if the stars disappear, the observer will see either $(i)$ “oscillations" of bodies with respect to each other ([ *aphelion-perihelion*]{}), which would automatically signify the validity of Newton’s concept, or $(ii)$ these “oscillations" will cease, which would mean that there is an influence of stars on the masses of the bodies. Thus in both cases $(i)$ and$(ii)$ we would obtain a [ *unique*]{} answer in favor of one of the two concepts. In reality such an experiment is naturally impossible. However, in the case of circular orbits there is no unique choice of a valid concept for the observer, which will allow us to retain the assumption that the two concept are equally justified.
If we choose as the “true" concept the quasi-Mach one, then by equating kinematic properties of the same type in the [*presence*]{} and in the [*absence*]{} of distant stars, we can obtain a relationship between the “old" (in the [*presence*]{} of the rest of the matter) and the “new" (in its [*absence*]{}, accordingly) masses.
We sall use the equations of classical mechanics for the “new" masses, while for the “old" masses we shall use the equation for gravitational radiation of a system of two bodies rotating around their common CI.
In the case of “new" masses, we consider two bodies in an absolutely empty space, which come closer to each other under the influence of the gravitational force. Remember that speaking about a rotation of such a system has no meaning any more since both the stars and the notion of an absolute space are absent in this concept, so that the only “real" coordinate here is the [*distance*]{} between the two bodies. The second law of Newton and the inverse-square law lead to a relative acceleration in the two-body system: $$\ddot{x}=\gamma\frac{M_{\tt g}
m_{\tt g}}{M_{\tt i} m_{\tt i}}\biggl(\frac{M_{\tt i}+m_{\tt
i}}{x^2}\biggr),$$ where $x$ is the distance between the bodies; $M_{\tt i},m_{\tt i}$ and $M_{\tt g},m_{\tt g}$ are the inertial and gravitational masses of the bodies $M$ and $m$, accordingly.
In the case of “old" masses, the two bodies rotate around their common CI in the presence of stars (or in the absolute space, which in this case is the same). The potential energy of the system has the form $$\varepsilon_{\tt pot}=\gamma\frac{M_{\tt g} m_{\tt g}}{r},$$ where $r$ is the distance between bodies. From the condition of equality forces and the rotational frequencies, we can obtain the expressions for the linear velocities of the bodies: $$V^2_{\tt M}=\gamma\frac{m_{\tt
g}M_{\tt g}m_{\tt i}} {M_{\tt i}(M_{\tt i}+m_{\tt i})r}; \qquad V^2_{\tt
m}=\gamma\frac{m_{\tt g}M_{\tt g}M_{\tt i}} {m_{\tt i}(M_{\tt i}+m_{\tt
i})r}.$$ Substituting them into the equations for the kinetic energy of the bodies, we find $$\varepsilon^{\tt tot}_{\tt k}=\gamma\frac{M_{\tt
g}m_{\tt g}} {2r},$$ where $\varepsilon^{\tt tot}_{\tt k}$ is the total kinetic energy of the system (remember that we consider these bodies as material points). Notice that for a circular motion, the total kinetic energy of bodies depends [*only on their gravitational masses*]{}, rather than inertial ones. This fact allows us to prove following [**theorem**]{}: [*In the framework of*]{} [classical mechanics]{} [*the gravitational mass determines the “inertia" of material body*]{}.
[Proof]{}:
So, let some body $m$ (with inertial mass $m_i$) to move along a straight line with constant velocity $V$. Its kinetic energy is: $$K = \frac{m_i V^{2}}{2}$$
From kinematic point of view the movement with constant velocity along a straight line and (with constant [*linear*]{} velocity) along a circumference of infinity radius are equivalents.
We can consider also the movement of the similar body (with same inertial mass $m_i$) as circumference movement around the other body $M$. Now we can require that kinetic energy and [*linear*]{} velocity of body $m$ to be equal to that of the case (2). Nothing can forbid us to do it. Now from the equivalence force conditions we have: $$\frac{m_i
V^{2}}{R} = {\gamma}\frac{m_g M_g}{R^{2}}$$ where $m_g,M_g$ are gravitational masses of bodies $m$ and $M$, $R$ is distance between $m$ and $M$. Expressing “$V^{2}$” from (3) and substituting it into the formula of the kinetic energy of the body $m$ obtain: $$K
= {\gamma}\frac{m_g M_g}{2R}$$
Let us now to increase $M_g$ and $R$ conserving the same time value of $K$. In this case $M_g(R)$ and $R(M_g)$ are one-to-one functions. It is obvious that if $K$, $m_g$ and ${\gamma}$ are constants, $M_g(R)$ and $R$ will be [*linear dependent*]{} functions, i.e. $$M_g(R) = C\cdot R,$$ where $C$ is some suitable dimensional constant. After tending $R$ to infinity (conserving the same time values of $K,m_g$ and ${\gamma}$) we obtain from (4) $$K=m_g\frac{{\gamma} C}{2}$$ where “${\gamma} C$” has dimension of the “$V^{2}$”. It means that (5) can be rewritten as $$K=m_g \beta\frac{V^{2}}{2}$$ here $\beta$ is non-defined constant.
Recalling above-mentioned notice (equivalence between straight line movement and the same one along the circumference of the infinite radius), we conclude that kinetic energy of the body moving along the straight line with the constant velocity [*is proportional to the gravitational mass*]{}. Comparing (6) and (2) we obtain the equivalence $$m_i=m_g \beta,$$ where $\beta$ is a constant non-defined in frames of the above-mentioned considerations.
Now we can assumed that $\beta =1$ in (6) and as result, the total mechanical energy of the system (see above) is $$\varepsilon=-\gamma\frac{Mm}{2r}.$$ Here and below we shall skip the indices “i" and “g" according to the meaning of the problem.
The radiation rate of the gravitational energy during a circular motion has the form \[12\]: $$-\frac{d\varepsilon}{dt}=32\gamma\biggl(\frac
{mM}{m+M}\biggr)\frac{r^4\omega^6}{5c^2}.$$ From (8) we have $$\frac{d\varepsilon}{dt}=\gamma\frac{Mm}{2r^2}\frac
{dr}{dt}.$$ Then we obtain $$\omega^6=\gamma^3\frac{(m+M)^3}{r^9}.$$ Substituting (10) and (11) into (9) and differentiating the resulting expression with respect to time, we can find the relative acceleration of the mutual approach of the bodies: $$\frac{d^2r}{dt^2}=-3\gamma^6\biggl[\frac{64}{5c^5} mM(m+M)\biggr]^2r^{-7}.$$ Now, denoting in (1) both masses by the index “n" and in (12), by “o" (“[*new*]{}" and “[*old*]{}" masses), and comparing these two expressions, we obtain the desired relationship $$M_{\tt n}+m_{\tt n}=\alpha\frac{ m^2_{\tt o}M^2_{\tt o} (m_{\tt o}+M_{\tt
o})^2}{R^5}.$$ Here we denoted all constants coefficients by $\alpha$, while $R$ is the distance between the bodies.
Now let us consider another situation: distant stars (galaxies) exist “again" (remember that according to Mach, it is irrelevant whether it is the background or the bodies that rotate). Then $m_{\tt n}$ becomes $m_{\tt o}$, and $M_{\tt n}$ becomes $M_{\tt o}$. That is, $m_{\tt n}$ and $M_{\tt n}$ get multiplied by some factors $A$ and $B$ $(Am_{\tt
n}=m_{\tt o}; BM_{\tt n}=M_{\tt o})$ which are functions of th masses: $A=A(M_{\tt o},\Phi), B=B(m_{\tt o},\Phi)$, where $\Phi$ are masses of the rest of stars. Then (13) can be written in the form $$M_{\tt n}+m_{\tt n}=\alpha\frac {m^2_{\tt n}M^2_{\tt n}A^2B^2(Am_{\tt
n}+BM_{\tt n})^2} {R^5}.$$ Using the fact that $\Phi\gg
m_{\tt o},M_{\tt o}$, we can expand $A(M_{\tt o},\Phi),B(m_{\tt o},\Phi)$ in a power series in small parameters. In the zeroth approximation, $$A\cong A(0,\Phi);\qquad B\cong B(0,\Phi).$$ Since it is clear that the functional form of $A$ and $B$ must be same, we can write $$A(0,\Phi)\equiv B(0,\Phi)=A(\Phi).$$ We do not possess more information about the form of the function $A(\Phi)$, but we shall assume that when $\Phi\rightarrow\infty$, $A\rightarrow const$, i.e., we shall assume that $A$ is a constant specific for our Universe. Substituting (15) into (14), we obtain $$\alpha A^6 m^2_{\tt n}M^2_{\tt
n}(m_{\tt n}+M_{\tt n})=R^5.$$ The constant $A$ cannot be determined within the framework of this problem. We have only shown the necessity of its existence [*if the quasi-Mach concept is true*]{}. Now let us note that there exists such $p$ that $M_{\tt n}=pm_{\tt n}$. The Eq.(16) implies that $$[p^2(1+p)]^{1/5}m_{\tt n}=(\alpha
A^6)^{-1/5}R.$$
Thus we have shown that Newton solution of the problem of absolute space and inertial mass, taking into account the requirements of general relativity, is [*non-unique*]{} if the Mach principle is modified in a suitable way. We have also shown that if we choose the quasi-Mach concept as [*true*]{}, then in our approximation the masses of two bodies in an empty Universe [*are proportional to the distance between them*]{} (17). When the bodies approach each other, their masses tend to [*zero*]{} (we mast not forget that our computation are based on the bodies being pointlike, though this restriction is not a matter of principle). Notice that if the masses of the bodies vanish when they come close to each other, this would not signify that matter disappear, and therefore such vanishing of the masses would mean that there is some conserved property of matter in nature which, perhaps, is unrelated to the space-time structure of the Universe. In the framework of the problem under consideration we are unable to say anything more specific about this property.
[**Acknowledgments**]{}
I am grateful to Profs. Jayant V. Narlikar, A. Assis and V. Dvoeglazov for many stimulating discussions and critical comments.
E.Mach, [*Mechanics*]{} (St. Petersburg, 1906). A.Einstein, [*Autobiographical notes*]{} (Illinois, 1949). M. Sachs, Brit. J. Phil. Sci. [**23**]{}, 117 (1972). F.Hoyle, J.V.Narlikar, [*Action at a Distance in Physics and Cosmology*]{} (Freeman, San Francisco, 1974). A. Chubykalo and Kh. Granada, Sov. J. of Physics, [**6**]{}, p. 467 (1990). P. Graneau and N. Graneau, [*Newton versus Einstein - How matter interacts with matter*]{} (Carlton Press, New York, 1993). A. Assis, [*Weber’s Electrodynamics*]{} (Kluwer, 1994). M. Sachs, Brit. J. Phil. Sci. [**27**]{}, 225 (1976). P. Graneau, [*The riddle of inertia*]{} (Electronic World and World 96, 1990). A. Assis, Found. Phys. Lett. [**2**]{}, 301 (1989). A. Assis, In: [*Space and Time Problems in Modern Natural Sciences*]{}, p. 263 (Tomsk Scientific Center of the Russian Academy of Sciences, St. Petersburg, 1993). E.Amaldi and G.Pizzela: [*Astrofisica e Cosmologia Gravitazione Quanti e Relativita (Negli sviluppi del pensiero scientifico di Albert Einstein. “Centenario di Einstein” 1879-1979)*]{} (Giunti Barbera, Firenze, 1979).
[^1]: See, e.g., brilliant work of M. Sachs “On the Logical Status of Equivalence Principles in General Relativity Theory" \[8\]. That is, there is no $\alpha\ \pi\varrho\iota o\varrho\iota$ reason why it should necessarily follow that $m_{\tt i}=m_{\tt g}$, even though experimental observations confirm this equality to high accuracy.
[^2]: Generally accepted Mach principle can be realized in the case (a) only (see above and \[9-11\])
[^3]: Below we shall prove that this assumption has got rather strict arguments, at least, in classical mechanics (see [**Theorem**]{})
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{
"pile_set_name": "ArXiv"
}
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---
author:
- |
**[Eduard Mass[ó]{}]{}\
\
**
title: '**The Weight of Vacuum Fluctuations**'
---
**Abstract**
We examine the gravitational properties of Lamb shift energies. Using available experimental data we show that these energies have a standard gravitational behavior at the level of $\sim 10^{-5}$. We are motivated by the point of view that Lamb shift energies may be interpreted as a consequence of vacuum fluctuations of the electromagnetic field. If this is the case, our result is a test of the gravitational properties of quantum fluctuations. The result is of interest in relation to the problem of the zero-point energy contribution to the cosmological constant. Indeed, the problem presupposes that the zero-point energy gravitates as all other forms of energy, and this supposition is what we test.
———————————————————————
The Cosmological Constant and Vacuum Fluctuations
=================================================
Present cosmological data are well fitted by assuming the existence of a dark energy component that induces acceleration of the universe. This component is compatible with a simple cosmological constant term in the Friedmann equation, with an energy density $$\rho \simeq ( 2 \times 10^{-3} \ {\rm eV})^4
\label{rho}$$ in natural units, and pressure $p=-\rho$. With a few more parameters determined by data we have a sound description of all observations in the framework of the Standard Cosmological Model [@Frieman:2008sn].
The Standard Model of particles and their gauge electroweak and color interactions is also a well established model, able to account for all experimental particle physics data. In this model there are contributions to the cosmological constant. This is due to the fact that the model is based on quantum fields and, in vacuum, fields fluctuate around their minimal values giving a contribution to the vacuum energy. Provided we only measure energy differences, we can subtract this type of contributions and we do not need to worry when performing calculations (technically we call it normal ordering). However, this procedure is no longer possible in the presence of gravitation for in this case the absolute value of energy matters. One expects then a net cosmological constant from the zero-point field fluctuations [@Zel'dovich:1968zz]. It has been known for many years that these contributions exceed the observed value (\[rho\]) by many orders of magnitude. To solve this problem is one of the present challenges for Physics [@Weinberg:1988cp]. The problem presupposes that vacuum fluctuations have the same gravitational properties as all other forms of matter. While there is no a priori reason to doubt of such presupposition, it would be desirable to test it experimentally. Such a endeavor might seem fanciful, but in this Letter we will see that it could be possible.
It is often argued that experimental support for the presence of electromagnetic vacuum fluctuations is provided by the verification of the Casimir effect [@Casimir:1948dh] in the laboratory [@Lamoreaux:2005gf]. Indeed, the Casimir force can be deduced by considering the difference among the contribution to vacuum energies in two physical situations. It follows that it is not the absolute value of the vacuum energy what we measure but energy differences. Even so, it is interesting to study the gravitational properties of the Casimir energy. This has been done theoretically in [@Fulling:2007xa], with the conclusion that the Casimir energy gravitates according to the Equivalence Principle.
The measure and the theoretical prediction of the Lamb shift (LS) energy is historically considered a milestone in quantum field theory [@Kinoshita]. Along with the Casimir force, the LS energy can also be considered as a test of the vacuum fluctuations of the electromagnetic field. While perhaps this is not very obvious in Bethe’s original treatment [@Bethe], it is clear in the calculation by Welton [@Welton:1948zz]. He deduced the LS formula corresponding to the main contribution -see below, equation (\[LS\])- taking into consideration the effect of electromagnetic field fluctuations on the electron. Perhaps it is even more clear in the treatment given by Power [@Power], who followed suggestions by Feynman. In [@Power] the deduction of the LS is based on the consideration of the change in the zero-point field energy in a box containing atoms compared to the same box without atoms, and again one finds exactly the usual formula for the LS. The deduction in [@Power] has a striking similarity with the usual zero-point energy deduction of the Casimir effect. There are quantum field textbooks [@bdiz] reviewing all these topics. In the textbook [@Milonni] we can find a study with special emphasis in the connection with vacuum fluctuations.
The point of view that the Casimir effect is unambigous evidence for vacuum fluctuations has been disputed by Jaffe [@Jaffe:2005vp]. Indeed, he has calculated the effect with no reference whatsoever to vacuum. Although we are not aware of a similar calculation for the LS, a judicious attitude should be the following. Even if the LS can be calculated as originating from vacuum fluctuations, to take the point of view that the LS is evidence for quantum fluctuations has to be considered an assumption. Once this assumption is made, it is clearly worth to test the gravitational properties of the LS energy. In the next section we show how to test the Equivalence Principle (EP) for the LS energy.
Lamb Shift Energy and the Equivalence Principle
===============================================
The EP is a fundamental postulate in gravitational physics which has been experimentally tested to a high precision. We wish to test the principle for LS energies, or in other words, we would like to know to which extent this energy couples to external gravitational physics as all other types of matter and energy.
In order to proceed, we allow for a violation of the EP. We write the following relation between the gravitational mass $m_{gr}$ of an object and its inertial mass $m_{in}$, $$\label{g_i}
m_{gr}=m_{in} - \lambda\, \Delta E_{LS}$$ Here $ \Delta E_{LS}$ is the total LS energy contributions to the object and $\lambda$ is a parameter that would signal a violation of the EP: $\lambda=0$ corresponds to the case that the EP is obeyed, and $\lambda=1$ would correspond to a situation where the LS energies do no gravitate at all. Our main purpose is to get a bound on $\lambda$.
The main contribution to the LS energy of an electron in a level of principal quantum number $n$ in the H atom is given by $$\label{LS}
\Delta E_{LS} (H) = \frac{4 \alpha^2}{3} \frac{1}{m_e^2}\ \left(\log \frac{m_e}{\epsilon_n(H)} \right) | \Psi_n (0)|^2$$ Here, $\alpha$ is the fine structure constant, $m_e$ is the electron mass, and $\Psi_n(0)$ is the electron wave function at the origin. The parameter $\epsilon_n(H)$ is an average excitation energy, which has to be calculated numerically. For example, for the $2s$ level, $m_e/\epsilon\simeq 2 \times 10^3$. The 2s$_{1/2}$-2p$_{1/2}$ LS in H is very well measured, and one needs to consider subdominant contributions to the energy shift apart from the main contribution (\[LS\]); see ref. [@Kinoshita].
Actually, if we apply our EP test to the atomic LS we obtain poor limits on $\lambda$. To get an interesting bound, we will consider the LS of the proton energies due to the nuclear electromagnetic field. Of course this has not been measured, but we expect a tiny contribution to the total nuclear mass coming from it. We need to calculate a nuclear effect and therefore we know in advance that some approximations have to be done due to the intrincancies of nuclear physics.
In the nuclear shell model, the LS of the protons in a nucleus $N$ is $$\begin{aligned}
\Delta E_{LS} (N) &=& \frac{4 \alpha^2}{3} \frac{1}{m_p^2}\ \delta_N \nonumber \\
\delta_N &=& \sum_{i } \, Z_N^{(i)} \,
\left(\log \frac{m_p}{\epsilon_i(N)} \right) | \Psi_i (0)|^2
\label{EN}\end{aligned}$$ Here, $m_p$ is the proton mass. To get $\Psi_i (0) \neq 0$ one has to restrict to $i$ running over s-wave shells of protons in the nucleus $N$. We approximate the Coulomb field on a proton by a central field produced by the other protons in inner shells. Thus, $Z_N^{(i)}$ is the total number of protons in shells inner than $i$.
In the shell model one uses the Woods-Saxon $V_{WS}$ potential and a spin-orbit term $V_{LS}$, and one gets a fairly good understanding of the spectrum and other nuclear properties. The potential is $$\begin{aligned}
\label{WS}
V(r) &=&V_{WS} + V_{LS} \nonumber \\
V_{WS} &=& \frac{V_0}{1+\exp (r-R)/a} \end{aligned}$$ where $V_0 \simeq 50$ MeV, $R\simeq 1.3 A^{1/3}$ fm ($A$ is the atomic number), and $a\simeq 0.7$ fm. The spin-orbit coupling vanishes for the s-levels we are interested. No general analytical solution is obtained for the potential (\[WS\]), and one has to resort to numerical calculations. For our purposes it is enough to consider two simpler potentials where analytical results for the wave-function at the origin can be easily obtained. We start with the 3D harmonic (h) potential, $V^{\rm (h)}=(1/2)m_p\omega^2r^2 +$const, In fact, the s-wave functions near the origin for a harmonic potential with suitable chosen frequency $\omega$ and the numerically calculated wave functions corresponding to (\[WS\]) are practically indistinguishable [@bohr].
For the harmonic potential, if we numerate the s-waves with $n=0,1,\dots$ (corresponding to increasing energy) we find $$\label{psi_ho}
| \Psi_n^{\rm (h)} (0)|^2 = \left( \frac{m_p\, \omega}{\pi} \right)^{3/2}\, \frac{(2n+1)!!}{n!}\, 2^{-n}$$ We choose $\omega \simeq 9$ MeV, the value corresponding to $A=100$ [@bohr], and put $n=1$. It gives $$\label{50}
| \Psi_1^{\rm (h)} (0)|^2 \simeq (50\, {\rm MeV})^3$$ One can prove that the expression (\[psi\_ho\]) increases with $n$.
Let us find now $\Psi(0)$ for another simple potential. We shall consider a 3D square well (w) potential, which actually is the limit of $V_{WS}$ in (\[WS\]) when $R\gg a$, which is a reasonable limit even for not too large $A$. For simplicity, we shall consider the infinite well, which is a valid approximation for the lowest lying states. In this case, we obtain $$\label{psi_sw}
| \Psi_n^{\rm (w)} (0)|^2 = \frac{\pi}{2} \, \frac{n^2}{R^3}$$ where $n=1,2, \dots$ is the quantum number. Numerically, with the radius $R$ used before, we have $$\label{40}
| \Psi_n^{\rm (w)} (0)|^2 \simeq (40\, {\rm MeV})^3\, n^2 \, \left( \frac{100}{A} \right)^{1/3}$$
Let us now calculate the effect of a violation of the EP when $\lambda$ is not null. Given two elements 1 and 2, such a violation is signaled by a nonzero value of the parameter $$\label{eta}
\eta(1,2) = \frac{m_{gr}}{m_{in}} \bigg|_{2} - \frac{m_{gr}}{m_{in}} \bigg|_{1}$$ With the EP-violating assumption (\[g\_i\]) we have a contribution to this parameter given by $$\label{delta_m}
\eta(1,2) = \lambda\ \frac{4 \alpha^2}{3} \frac{1}{m_p^3}
\left( \frac{\delta_{1}}{{\rm A}_{1}} - \frac{\delta_{2}}{{\rm A}_{2}}\right)$$
To be conservative, we shall use the estimation (\[40\]), because (\[50\]) would lead to a stronger limit. Also, we point out that the log term in (\[EN\]) does not make a big difference between elements and shells. We shall factorize it in (\[delta\_m\]), and since we expect it to be larger than $O(1)$, we conservatively set $\log m_p/\epsilon=1$.
Let us consider aluminium ($Z=13$, $A=27$) and platinum ($Z=78$, $A=195$), the elements used in the Braginsky and Panov experiment [@Braginsky]. There are 2 s-shell protons in aluminium, $$\label{Al}
{\rm Al} = \{ (1\ s_{1/2})^{2p} \} + \dots$$ We employ the usual notation with the number and type of nucleons as a superindex, and the dots indicate protons in shells other than s and neutrons. There are 6 s-shell protons in platinum, $$\label{ }
{\rm Pt} ={\rm Al} + \{ (2\ s_{1/2})^{2p}, (3\ s_{1/2})^{2p} \} + \dots$$
We see that there are no inner protons for the 1s protons of aluminium. Thus, we get $\delta_{\rm Al}=0$. For platinum, for the 2s protons we shall consider the contribution of 1s protons, and for the 3s protons, we shall take into account the 1s and 2s protons. There are other protons contributing to the Coulomb field, but exactly how much is the contribution depends on the details of the potential. By ignoring them we get a conservative bound.
Numerically we have a contribution to (\[delta\_m\]) of about $\lambda \times 4 \times10^{-7}$. Using the experimental bound [@Braginsky] $$\label{EP_exp}
|\eta({\rm Al},{\rm Au}) | \lesssim 10^{-12}$$ we are led to $$\label{result}
\lambda \lesssim 3 \times 10^{-6}$$ Other experiments, done in laboratory or coming from lunar laser ranging data - see [@Fischbach] for a review- would give similar bounds. What can be done in the future in order to improve the limit (\[result\]) ? An obvious way is to get an experimental limit better than (\[EP\_exp\]). One could also think of using other elements than Al and Pt such that the nuclear shell structure would enhance our hypothetical EP-violating effects. However, this does not work because there is no known element with $4s$ protons. The best we can do is to choose a heavy element with $3s$ protons, which means $Z\geqslant 70$ (like Pt), and a lighter element with $1s$ protons and no other $s$-protons, namely, with $Z < 14$ (like Al).
We should mention that the modes our test probe are basically of wavelengths $\lesssim 2R \sim 10$ fm and thus of energies $\gtrsim 2\pi/R \sim 100$ MeV.
Lamb Shift as Active Gravitational Mass
=======================================
We can go a step further in the test of the gravitational properties of LS energies. One may distinguish the active gravitational mass $M_{gr}$ of an object, which is the source of gravitational fields, from the passive gravitational mass $m_{gr}$, which determines the force upon the object when placed in an external gravitational field [@Will]. Given two bodies, now the relevant parameter is $$\label{sigma}
\sigma(1,2) = \frac{m_{gr}}{M_{gr}} \bigg|_{2} - \frac{m_{gr}}{M_{gr}} \bigg|_{1}$$
Our test in the last section referred to the equality of inertial mass and passive gravitational mass for LS contributions. Now we would like to test the equality of the active and passive gravitational mass for LS contributions. Here laboratory experiments give poor limits and, fortunately, the authors of ref. [@Bartlett:1986zz] have been able to show how to place very tight limits to $\sigma$ using lunar laser ranging data. They find a limit concerning aluminium and iron, $$\label{Al_Fe}
|\sigma({\rm Al},{\rm Fe}) | \lesssim 4 \times 10^{-12}$$
In analogy with what we did in the last section, we allow for a difference in passive and active masses coming from an anomalous gravitational behavior of the contribution of Lamb shift energy, $$\label{a_p}
m_{gr}=M_{gr} - \lambda' \, \Delta E_{LS}$$ where now $\lambda' \neq 0$ would signal a violation of the equality among gravitational active and passive mass. To calculate the contribution to $\sigma$ coming from a potential non-vanishing $\lambda'$ we need the structure of s-wave protons in iron, $$\label{ }
{\rm Fe} ={\rm Al} + \{ (2\ s_{1/2})^{2p} \} + \dots$$ Proceeding as before, we find a contribution of about $\lambda' \times 10^{-7}$ so that using (\[Al\_Fe\]) we get the limit $$\label{result2}
\lambda' \lesssim 4 \times 10^{-5}$$
In [@Nordtvedt] it is shown that the limit on violations of active-passive gravitational masses can be improved with respect the limits presented in [@Bartlett:1986zz] by a factor $\sim 40$. However, there is not the extraction to elements as in equation (\[Al\_Fe\]) and because of this we do not employ [@Nordtvedt]. However, it means that our limit (\[result2\]) may be improved.
Conclusion
==========
Given the contribution of the zero-point field fluctuations to the cosmological constant, it is interesting to try to know whether fluctuating fields gravitate at all, or do so anomalously. With this motivation, we have worked out a test of the gravitational properties of the LS energies. We have shown that they obey the EP at the level of $3 \times 10^{-6}$. We have also established a limit on a violation of the equality of active and passive gravitational mass for the LS contributions; the limit is $4 \times 10^{-5}$. If we assume that LS energies are a consequence of electromagnetic vacuum fluctuations then what we are testing is the gravitational properties of these fluctuations. Our result might be of interest in such a case.
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank Enrique Alvarez and Matthias Jamin for bringing references [@Nordtvedt] and [@Milonni], respectively, to my attention. I also thank Subhendra Mohanty for discussions. I acknowledge support by the CICYT Research Project FPA 2008-01430 and the *Departament d’Universitats, Recerca i Societat de la Informaci[ó]{}* (DURSI), Project 2005SGR00916. This work was supported (in part) by the European Union through the Marie Curie Research and Training Network “UniverseNet” (MRTN-CT-2006-035863)."
[99]{}
For a recent review, see J. Frieman, M. Turner and D. Huterer, arXiv:0803.0982 \[astro-ph\].
Y. B. Zel’dovich, Sov. Phys. Usp. [**11**]{}, 381 (1968).
S. Weinberg, Rev. Mod. Phys. [**61**]{}, 1 (1989).
H. B. G. Casimir, Indag. Math. [**10**]{}, 261 (1948) \[Kon. Ned. Akad. Wetensch. Proc. [**51**]{}, 793 (1948 FRPHA,65,342-344.1987 KNAWA,100N3-4,61-63.1997)\].
For a review, see S. K. Lamoreaux, Rept. Prog. Phys. [**68**]{} (2005) 201. S. A. Fulling, K. A. Milton, P. Parashar, A. Romeo, K. V. Shajesh and J. Wagner, Phys. Rev. D [**76**]{}, 025004 (2007) \[arXiv:hep-th/0702091\];\
K. A. Milton, P. Parashar, K. V. Shajesh and J. Wagner, J. Phys. A [**40**]{}, 10935 (2007) \[arXiv:0705.2611 \[hep-th\]\].
For a review see “Quantum Electrodynamics”, edited by T. Kinoshita, Advanced Series on Directions in High Energy Physics, Vol. 25, World ScientiÞc (1990).
H. A. Bethe, Phys. Rev. [**72**]{}, 339 (1947).
T. A. Welton, Phys. Rev. [**74**]{}, 1157 (1948).
E. A. Power, Am. J. Phys. [**34**]{}, 516 (1966).
J.D. Bjorken, and S. D. Drell, Relativistic Quantum Fields, McGraw-Hill (1965);\
C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill (1980).
P. W. Milonni, The Quantum Vacuum, Academic Press, NY (1994).
R. L. Jaffe, Phys. Rev. D [**72**]{}, 021301 (2005) \[arXiv:hep-th/0503158\]. A. Bohr and B. R. Mottelson, Nuclear Structure, Vol. 1, Benjamin (1969).
V. B. Braginsky and V. I. Panov, V.I., Sov. Phys. JETP, [**34**]{}, 463 (1972).
E. Fischbach and C. L. Talmadge, The Search for Non-Newtonian Gravity, Springer (1999).
For the relation between a violation of active and passive masses and the Parametrized Post-Newtonian formalism see for example C. M. Will, Theory and Experiment in Gravitational Physics, Cambridge University (1993). D. F. Bartlett and D. Van Buren, Phys. Rev. Lett. [**57**]{}, 21 (1986).
K. Nordtvedt, Class. Quantum Grav. [**18**]{}, L133 (2001).
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**Sur une question de N. Chevallier liée à**
**l’approximation Diophantienne simultanée**
+0.5cm
par [**Nikolay Moshchevitin**]{}[^1]
+0.5cm
[**Résumé.**]{} Nous prouvons une conjecture proposée par Nicolas Chevallier qui concerne des matrices unimodulaires liées à l’approximation Diophantienne simultanée des nombres réels.
+0.5cm
[**1. Approximation Diophantienne simultanée.**]{}
Soit $n$ un entier naturel. Dans cet article nous considérons un vecteur réel $\xi$ de la forme $\xi = (1,\xi_1,...,\xi_n)$ dont les coordonnées sont linéairement indépendants sur $\mathbb{Z}$. Nous sommes intéressés à l’approximation du sous-espace vectoriel engendré par $\xi$ par des points entiers ${\bf z} =(q,a_1,...,a_n)$. On considère la fonction $$\psi_\xi (t) = \min_{q\in \mathbb{Z}, 1\le q \le t} \, \max_{1\le k \le n} ||q\xi_k||,$$ où $||\cdot ||$ désigne la distance à l’entier le plus proche. Cette fonction est décroissante et constante par morceaux. Soient $$q_0 =1\le q_1< q_2 <....<q_\nu<q_{\nu+1}
<...$$ les sauts de $\psi_\xi (t)$. Ils alors correspondent aux [*vecteurs meilleures approximations*]{} ${\bf g}_\nu = (q_\nu, a_{1,\nu},...,a_{n,\nu}) \in \mathbb{Z}^{n+1}$ qui sont définis par les conditions $$|| q_\nu \xi_k||
= |q_\nu \xi_k - a_{k,\nu}|,\,\,\,
1\le k \le n.$$ Notons que d’après le théorème de Minkowski sur les corps convexes on sait que $$\psi_\xi (t)\le t^{-\frac{1}{n}},$$ ou $$\label{apo}
\max_{1\le k \le n} ||q_\nu\xi_k|| \le q_{\nu+1}^{-\frac{1}{n}}.$$ Dans le cas $ n = 1$ d’après la théorie de fractions continues nous savons que pour les approximations (\[apo\]) nous avons une borne inférieure du même ordre, c’est $$(2q_{\nu+1})^{-1}<
||q_\nu \xi_1 || <q_{\nu+1}^{-1}.$$ Aussi nous savons que $$\left|
\begin{array}{cc}
q_\nu & q_{\nu+1}\cr
a_{\nu,1}&a_{\nu+1,1}
\end{array}
\right| = \pm 1.$$ Ces simples observations conduisent au corollaire suivant.
[**Proposition 1.**]{}[*Pour nombre irrationnel $\xi_1$ il existe une infinité de matrices unimodulaires $$\left(
\begin{array}{cc}
q' & q''\cr
a_{1}'&a_{1}''
\end{array}
\right)$$ telles que $$\max\left\{
q'{|q'\xi_1-a_1'|},q''{|q''\xi_1-a_1''|}
\right\}
\le 1.$$* ]{} La situation dans le cas $n \ge 2$ est tout à fait différent. Considérons une fonction décroissante $\varphi (t)$ telle que $$\label{tech}
\varphi (t) =o(1), \,\,\, t\to\infty.$$ Le résultat principal de cet article est le théorème suivant.
[**Théorème 1.**]{} [*Pour une foncion donnée $\varphi (t)$ qui deminue à zéro lorsque $ t\to \infty$, il existe un vecteur $(1,\xi_1,\xi_2) \in \mathbb{R}^3$ dont les composants sont linéairement indépendants sur $\mathbb{Z}$ et tel que pour toute matrice entière $$\label{meer}
M=
\left(
\begin{array}{ccc}
q'&q''&q'''\cr
a_{1}'& a_{1}''&a_{1}'''\cr
a_{2}'& a_{2}''&a_{2}'''
\end{array}
\right),\,\,\,\, {\rm det M} =\pm 1,\,\,\,\,\, q',q'',q'''\ge 1$$ on a $$\max \left\{ \frac{\max_{j=1,2} |q'\xi_j - a_j'|}{\varphi (q')},
\frac{\max_{j=1,2} |q''\xi_j - a_j''|}{\varphi (q'')},
\frac{\max_{j=1,2} |q'''\xi_j - a_j'''|}{\varphi (q''')}\right\}
\ge \varepsilon,$$ avec un certain nombre positif $\varepsilon $ qui dépend de la fonction $\varphi$.* ]{}
Notre Théorème 1 donne une réponse affirmative à la question posée par N. Chevallier dans [@che] dans le cas $s=2$. Bien sûr, un résultat similaire sera vrai pour tout $ n \ge 2$.
D’après l’argument ci-dessous, il est possible de prouver la déclaration suivante.
[**Proposition 2.**]{} [ *Il existe $\xi_1, \xi_2$ linéairement indépendants sur $\mathbb{Z}$ avec $1$ et tels que le déterminant d’une matrice de la forme $$\left(
\begin{array}{ccc}
q_{\nu_1}&q_{\nu_2}&q_{\nu_3}\cr
a_{1,\nu_1}& a_{1,\nu_2}&a_{1,\nu_3}\cr
a_{2,\nu_1}& a_{2,\nu_2}&a_{2,\nu_3}
\end{array}
\right),\,\,\,\,\, \nu_1<\nu_2<\nu_3$$ (ici $(q_\nu, a_{\nu,1},a_{\nu,2})$ sont les vecteurs meilleures approximations) n’est jamais égal à $\pm 1$.* ]{}
Bien sûr, ce résultat est aussi valable dans toute dimension $ n \ge 2$.
Nous tenons à rappeler deux résultats liés à l’approximation Diophantienne simultanée.
Le premier résultat va jusqu’à V. Jarník [@ja] (voir aussi [@D] et [@L]). Il déclare que pour $ n \ge 2$ il y a un nombre infini de triplets de [*linéairement indépendants*]{} vecteurs meilleures approximations consécutifs ${\bf g}_{\nu}, {\bf g}_{\nu+1}, {\bf g}_{\nu+2}$.
La seconde est due à l’auteur [@UMN]. Il déclare que pour tout $n\ge 2$ il existe $\xi_1,...,\xi_n$ linéairement indépendants sur $\mathbb{Z}$ avec $1$ et tels que pour tout $\nu$ la matrice $$\left(
\begin{array}{cccc}
q_{\nu}&q_{\nu +1}&...&q_{\nu+n}\cr
a_{1,\nu}& a_{1,\nu+1}&...&a_{1,\nu+n}\cr
....& ....&...&...\cr
a_{n,\nu}& a_{n,\nu+1}&...&a_{n,\nu+n}
\end{array}
\right)$$ de $n+1$ vecteurs meilleures approximations consécutifs est de rang $\le 3$. Ce résultat donne un contre-exemple à la conjecture de Lagarias [@L].
Pour plus d’informations concernant les vecteurs meilleures approximations nous nous référons à [@che; @M1; @M2]
Pour conclure cette section, nous formulons un résultat du type de Jarník qui est basé sur une observation simple par V. Jarník (voir [@TB], Satz 9 et Théorème 17 de [@M2]). Pour une matrice $M$ de la forme (\[meer\]) nous définissons $$R(M) =
\max \left\{ {\max_{j=1,2} |q'\xi_j - a_j'|},
{\max_{j=1,2} |q''\xi_j - a_j''|},
{\max_{j=1,2} |q'''\xi_j - a_j'''|}
\right\}.$$
[**Théorème 2.**]{} [ *Supposons que $1,\xi_1,\xi_2$ sont linéairement indépendants sur $\mathbb{Z}$. Il existe la suite $M_\nu$ des matrices de la forme (\[meer\]) telle que $R(M_\nu) \to 0$ lorsque $\nu \to \infty$.*]{} +0.2cm
Théorème 1 sera prouvée dans les sections 2 - 5. Nous donnons une preuve du Théorème 2 dans la section 6.
[**2. Lemmes.**]{}
Ici $|\cdot |$ signifie la norme Euclidienne, $ {\rm dist} \, ({\cal A}, {\cal B})$ désigne la distance Euclidienne entre les ensembles ${\cal A}, {\cal B}$. Par ${\rm angle} \, ({\bf u}, {\bf v})$ nous notons l’angle entre les vecteurs ${\bf u}$ et ${\bf v}$. Par ${\rm angle} \, (L, P)$ nous notons aussi l’angle entre les sous-espaces $L$ et $P$ de dimension un ou deux.
Pour deux vecteurs indépendants ${\bf z}', {\bf z}''\in \mathbb{R}^3$ nous définissons le sous-espace vectoriel $$L({\bf z}',{\bf z}'') = {\rm span} ({\bf z}',{\bf z}').$$ Pour ${\bf z}', {\bf z}''\in \mathbb{Z}^3$ nous considérons le réseau $$\Lambda ({\bf z}',{\bf z}'') = L({\bf z}',{\bf z}'') \cap \mathbb{Z}^3.$$ Notons que dans le cas où une paire ${\bf z}', {\bf z}''\in \mathbb{Z}^3$ peut être complétée à une base de tout $\mathbb{Z}^3$ on a $$\Lambda ({\bf z}',{\bf z}'') = \langle{\bf z}',{\bf z}''\rangle_\mathbb{Z}.$$ Pour deux points entiers ${\bf z}', {\bf z}''$ sous la condition $$\label{root}
L( {\bf z}', {\bf z}'') \cap \mathbb{Z}^3 = \langle{\bf z}', \bf {z}''\rangle_\mathbb{Z}$$ nous considérons un point $ {\bf y} ({\bf z}', {\bf z}'')$ qui complète la paire ${\bf z}', {\bf z}''$ à une base de $\mathbb{Z}^3$. Ensuite, nous définissons deux sous-espaces affines de dimension deux $$\label{recc}
L^{\pm} ({\bf z}', {\bf z}'' ) =
L({\bf z}',{\bf z}'')\pm {\bf y} ({\bf z}', {\bf z}'').$$ Nous devons d’introduire une quantité plus. Pour deux points entiers ${\bf z}', {\bf z}''$ sous la condition (\[root\]) nous considérons la valeur $$\label{vaal}
\eta ({\bf z}', {\bf z}'') = \min_{{\bf x}\in \Lambda ({\bf z}',{\bf z}'') \setminus {\rm span}({\bf z}')} {\rm dist} ({\bf x}, {\rm span} ({\bf z}'))> 0
.$$
[**Lemme 1.**]{}[*Soit ${\bf z} \in \mathbb{Z}^3$. Supposons que ${\bf z}', {\bf z}''\in \mathbb{Z}^3$ satisfait (\[root\]) et $ {\bf z} \not\in L( {\bf z}', {\bf z}'') $. Considérons le point ${\bf w} = L^+ ({\bf z}', {\bf z}'' )\cap {\rm span} \, ({\bf z})$. Soit $\delta$ l’angle $${\rm angle}\, ( L( {\bf z}', {\bf z}''), {\rm span} \, ({\bf z}))>\delta >0.$$ Supposons que $${\bf x}\in L^- ({\bf z}', {\bf z}'' )
\cup L^+ ({\bf z}', {\bf z}'' )$$ et $$\label{aar}
|{\bf x}|\ge 2|{\bf w}| .$$ Alors $${\rm dist }\, ({\bf x}, {\rm span} \, ({\bf z})) \ge \frac{|{\bf x}|}{2} \sin \delta .$$* ]{}
Preuve. Notons par ${\bf x}^*$ la projection orthogonale du point ${\bf x}$ sur le sous-espace ${\rm span} \, ({\bf z})$, qui est de dimension un. Alors, $${\rm dist }\, ({\bf x}, {\rm span} \, ({\bf z})) =
{\rm dist }\, ({\bf x}, {\bf x}^*) =
|{\bf x} \pm {\bf w}| \sin \theta,$$ où $\theta \ge \delta$ est l’angle entre ${\bf x} \pm {\bf w}$ et ${\bf z}$. Ainsi $${\rm dist }\, ({\bf x}, {\rm span} \, ({\bf z}))
\ge (|{\bf x}|-|{\bf w}|) \sin \delta \ge \frac{|{\bf x}|}{2} \sin \delta$$ (ici nous utilisons (\[aar\])). Lemme 1 est prouvé.$\Box$
+0.2cm
D’après Lemme 1 nous déduisons immédiatement
[**Corollaire 1.**]{}[*Soit ${\bf z} \in \mathbb{Z}^3$. Supposons que ${\bf z}', {\bf z}''\in \mathbb{Z}^3$ satisfait (\[root\]) et $ {\bf z} \not\in L( {\bf z}', {\bf z}'') $. Alors il existe des nombres positifs $\delta ({\bf z}, {\bf z}', {\bf z}'')$ et $T ({\bf z}, {\bf z}', {\bf z}'')$ tels que pour tous les vecteurs $\xi = (1,\xi_1,\xi_2)$ sous la condition $${\rm angle}\, (\xi , {\bf z}) \le \delta ({\bf z}, {\bf z}', {\bf z}'')$$ et pour tous les vecteurs entiers ${\bf x} = (q, a_1, a_2), q\ge 1$ sous la condition $${\bf x}\in L^- ({\bf z}', {\bf z}'' )
\cup L^+ ({\bf z}', {\bf z}'' )
\,\,\,\text{
et}
\,\,\,
|{\bf x}|\ge T ({\bf z}, {\bf z}', {\bf z}'')$$ on a $$\label{result}
\max_{j=1,2} |q\xi_j - a_j|\ge \varphi (q).$$* ]{} Il est clair que dans Corollaire 1 on peut considérer une collection finie de couples ${\bf z}', {\bf z}''$. Donc nous avons la déclaration suivante.
[**Corollaire 2.**]{}[*Soit ${\bf z} \in \mathbb{Z}^3$. Soit $\hbox{\got C}$ une collection finie de couples $({\bf z}', {\bf z}'')$ de points entiers tels que chacun d’eux satisfait (\[root\]) et $ {\bf z} \not\in L( {\bf z}', {\bf z}'') $. Alors il existe des nombres positifs $\delta ({\bf z}, \hbox{\got C})$ et $T ({\bf z}, \hbox{\got C})$ tels que pour tous les vecteurs $\xi = (1,\xi_1,\xi_2)$ sous la condition $${\rm angle}\, (\xi , {\bf z}) \le \delta ({\bf z},\hbox{\got C})$$ et pour tous les vecteurs entiers ${\bf x} = (q, a_1, a_2), q\ge 1$ sous la condition $${\bf x}\in
\bigcup_{
({\bf z}', {\bf z}'')\in \hbox{\got C}}
( L^- ({\bf z}', {\bf z}'' )
\cup L^+ ({\bf z}', {\bf z}'' )
)
\,\,\,\text{
et}
\,\,\,
|{\bf x}|\ge T ({\bf z}, \hbox{\got C})$$ on a (\[result\]).* ]{}
+0.2cm
Considérons un point entier $ {\bf z} \in \mathbb{Z}^3 \setminus \{{\bf 0}\}$. Soit $\Lambda\subset \mathbb{Z}^3$ un sous-réseau de dimension deux tel que $\Lambda\ni {\bf z}$ et $\varepsilon >0$, nous considérons l’ensemble des sous-réseaux $$\label{sub}
{\cal L} ={\cal L} ({\bf z},\Lambda,\varepsilon) =\{
\Lambda '\subset \mathbb{Z}^3: \,\, {\rm dim }\,\Lambda ' = 2,\,\, {\bf z } \in \Lambda',\,\,{\rm angle} \, ( {\rm span}\,\Lambda ', {\rm span}\, \Lambda)<\varepsilon\}.$$
[**Lemme 2.**]{}[*Considérons un ensemble ${\cal L} ({\bf z},\Lambda,\varepsilon) $ de la forme (\[sub\]). Soit $T$ un nombre positif. Alors, il existe un réseau $\Lambda \in {\cal L} ({\bf z},\Lambda,\varepsilon) $ tel que pour tout point ${\bf x}$ qui satisfait $${\bf x} \in \Lambda\,\,\,\text{et}\,\,\, |{\bf x} |\le T$$ on a $${\bf x }\in {\rm span}\,( {\bf z}).$$* ]{} Preuve. Le lemme résulte de l’observation que tout ensemble de la forme (\[sub\]) contient un nombre infini d’éléments.$\Box$
+0.2cm [**Lemme 3.**]{}[*Soit ${\bf z}$ un point entier et $\Lambda\ni {\bf z}$ un sous-réseau de dimension deux de $\mathbb{Z}^3$. Considérons un point entier ${\bf z}'$ indépendant de ${\bf z}$. Alors il existe positif $\varepsilon^*$ et un sous-réseau de dimension deux $ \Lambda_*\ni {\bf z}$ tels que $${\cal L}_*= {\cal L} ({\bf z}, \Lambda_*, \varepsilon_*)\subset {\cal L} ({\bf z},\Lambda,\varepsilon)$$ et pour tout $
\Lambda \in {\cal L}_*$ on a $$\Lambda \cap (L^-({\bf z}, {\bf z}') \cup L^+ ({\bf z}, {\bf z}') ) = \varnothing
.$$* ]{}
Preuve. Le sous-réseau affine $L^-({\bf z}, {\bf z}')\cap\mathbb{Z}^3$ (et le sous-réseau $L^+({\bf z}, {\bf z}')\cap\mathbb{Z}^3$) se divise en sous-réseaux affines (de dimension un) $\Gamma_i$ qui est parallèle à ${\rm span}, ({\bf z})$: $$L^-({\bf z}, {\bf z}')\cap\mathbb{Z}^3 = \bigsqcup_{i \in \mathbb{Z}} \Gamma_i.$$ Il suffit de traiter avec $L^-({\bf z}, {\bf z}')$, par l’argument de la symétrie. Nous considérons deux points différents $ {\bf w}_j\in L^-({\bf z}, {\bf z}'), j=1,2$ tels qu’ils appartiennent à deux sous-espaces affines voisins (de dimension un) $\Gamma_i$ et $\Gamma_{i+1}$, respectivement, et pour certains ${\bf w}$ de l’intervalle ouvert avec les bornes ${\bf w}_1, {\bf w}_2$ le sous-espace $ L ({\bf w}, {\bf z})$ contient un sous-réseau $$\Lambda_* = L ({\bf w}, {\bf z}) \cap \mathbb{Z}^3 \in {\cal L}.$$ Il est clair que pour un certain petit $\varepsilon_*$ l’ensemble ${\cal L}_*$ de la forme (\[sub\]) satisfait la propriété désirée.$\Box$
+0.2cm
D’après Lemme 3 on déduit immédiatement le suivant
[**Corollaire 3.**]{}
*Soit $\hbox{\got E}$ une collection finie de couples ${\bf z}'$ et chacun de ces points est indépendant de ${\bf z}$. Supposons que $ \Lambda \ni {\bf z}$ est un sous-réseau entier de dimension deux. Alors il existe un ensemble $${\cal L}_* ={\cal L} ({\bf z}, \Lambda_*, \varepsilon_*)$$ de la forme (\[sub\]) tel que pour tout $\Lambda \in {\cal L}_*$ on a $$\Lambda \cap \left( \bigcup_{{\bf z}'\in \hbox{\got E}} (L^-({\bf z}, {\bf z}') \cup L^+ ({\bf z}, {\bf z}') ) \right)= \varnothing
.$$*
+0.2cm
Pour deux points indépendants ${\bf z}$ et ${\bf z}^*$ nous considérons l’ensemble $${\cal P}({\bf z},{\bf z}^*)
=
\bigcup_P P,$$ où l’union est prise sur tous les sous-espaces vectoriels de dimension deux $P$ tels que $${\rm span}\, ({\bf z}) \subset P\,\,\,\text{et}\,\,\, {\rm angle} (P, L({\bf z},{\bf z}^*))\ge \frac{3\pi}{8}.$$ Nous avons aussi besoin d’un ensemble $$\overline{\cal P}({\bf z},{\bf z}^*)
=
\bigcup_P P,$$ où l’union est prise sur tous les sous-espaces linéaires de dimension deux $P$ tels que $${\rm span}\, ({\bf z}) \subset P\,\,\,\text{et}\,\,\, {\rm angle} (P, L({\bf z},{\bf z}^*))\ge \frac{\pi}{4}.$$ Il est clair que $$\overline{\cal P}({\bf z},{\bf z}^*)
\supset
{\cal P}({\bf z},{\bf z}^*).$$
[**Lemme 4.**]{} [*Supposons que les points entiers ${\bf z}$ et ${\bf z}^*$ peuvent être complétés à une base de $\mathbb{Z}^3$ Alors $${\rm dist} (\overline{\cal P}({\bf z},{\bf z}^*), \Lambda ({\bf z},{\bf z}^*)\setminus {\rm span}({\bf z}) ) \ge
\frac {\eta({\bf z},{\bf z}^*)}{\sqrt{2}},$$ où $\eta(\cdot,\cdot )$ est défini dans (\[vaal\]).* ]{}
Preuve. La distance entre $ {\bf x}\in \Lambda ({\bf z},{\bf z}^*)\setminus {\rm span}({\bf z})$ et $\overline{\cal P}({\bf z},{\bf z}^*)$ n’est pas inférieure à la distance entre $ {\bf x}$ et ${\rm span}({\bf z})$ multiplié par $\sin \frac{\pi}{4}$.$\Box$
+0.2cm [**3. Vecteurs ${\bf z}_\nu$.**]{}
Dans cette section nous construisons une suite de vecteurs d’entiers ${\bf z}_\nu$ par une certaine procédure inductive. On met $${\bf z}_1 = (1,0,0),\,\,\,\,\,\,{\bf z}_2 = (0,1,0).$$ Maintenant, nous supposons que les vecteurs $ {\bf z}_1,....,{\bf z}_\nu, \nu\ge 2$ sont déjà définis.
Pour une fonction décroissante $\varphi (t)$ nous définissons la fonction $\phi (t)$ qui est la fonction inverse de $\varphi (t)$. Nous définissons $$\label{hanu1}
H_\nu = \phi (2^{-3}\eta ( {\bf z}_{\nu-1}, {\bf z}_\nu)).$$ Considérons les ensembles $$\hbox{\got C}_\nu =
\bigcup_{\lambda=1}^\nu
\bigcup_{\mu=1}^{\nu-1}
\{ ({\bf z}', {\bf z}'') \,\,\text{satisfait (\ref{root})}:\, {\bf z}_\nu\not\in L({\bf z}', {\bf z}''),\,\,
{\bf z}' \in \Lambda_\lambda,\, |{\bf z}'| \le H_\lambda,\,
{\bf z}'' \in \Lambda_\mu ,\, |{\bf z}''| \le H_\mu\}$$ et $$\hbox{\got E}_\nu
=\bigcup_{\mu=1}^{\nu}
\{ {\bf z}': ({\bf z}_\nu, {\bf z}') \,\,\text{satisfait (\ref{root})},
\,{\bf z}' \in \Lambda_\mu \, |{\bf z}'| \le H_\mu\}$$ Il est claire que $
\hbox{\got C}_\nu
$ et $
\hbox{\got E}_\nu
$ sont des ensembles finis.
Mettons $$\delta_\nu = \delta ({\bf z}_\nu, \hbox{\got C}_\nu),\,\,\,\,
T_\nu = T ({\bf z}_\nu, \hbox{\got C}_\nu)
,\,\,\,\, \nu \ge 2,$$ où $\delta(\cdot,\cdot)$ et $T(\cdot,\cdot )$ sont définis dans Corollaire 2. Bien sûr, nous pouvons supposer que $ \delta_\nu <\delta_{\nu-1}/2$, où $\delta_{\nu-1}$ est défini à l’étape précédente de la construction (au début du processus on met $\delta_1=\delta_2 = \pi, T_1 = T_2 = 1$).
Nous supposons que $$\label{daba}
{\rm angle} ({\bf z}_\nu, {\bf z}_{\nu-1}) <\frac{\delta_{\nu-1}}{2}.$$
Nous devons définir maintenant le vecteur ${\bf z}_{\nu+1} $ de telle manière que le couple $ ({\bf z}_\nu, {\bf z}_{\nu+1})$ peut être complétée à une base de $\mathbb{Z}^3$ et définir le réseau correspondant $
\Lambda_{\nu+1} = \langle {\bf z}_\nu, {\bf z}_{\nu+1}\rangle_\mathbb{Z}.
$ Nous allons le faire de la manière suivante. [*Au début*]{} nous allons définir un réseau $
\Lambda_{\nu+1}\ni {\bf z}_\nu$ et [*ensuite*]{} nous allons choisir le vecteur $ {\bf z}_{\nu+1}$ pour compléter le couple $ ({\bf z}_\nu, {\bf z}_{\nu+1})$.
Nous prenons un réseau $\Lambda_{\nu+1}$ pour satisfaire les conditions
([**i**]{}) $ {\bf z}_\nu \in \Lambda_{\nu+1}$,
([**ii**]{}) $\mathbb{Z}^3 \cap {\rm span } \Lambda_{\nu+1} = \Lambda_{\nu+1}$,
([**iii**]{}) $\Lambda_{\nu+1} \subset {\cal P}({\bf z}_\nu, {\bf z}_{\nu-1})$,
([**iv**]{}) pour chaque $ {\bf x} \in \Lambda_{\nu+1}$ tel que $ |{\bf x}|\le T_\nu$ on a $ {\bf x}\in {\rm span} ({\bf z}_\nu)$,
([**v**]{}) $\Lambda_{\nu+1} \cap \left( \bigcup_{{\bf z}'\in \hbox{\got E}_\nu} (L^-({\bf z}, {\bf z}') \cup L^+ ({\bf z}, {\bf z}') ) \right)= \varnothing
.
$
L’existence d’un tel réseau $\Lambda_{\nu+1}$ résulte de Lemme 2 and Corollaire 3.
Maintenant, nous expliquons comment choisir le point entier $ {\bf z}_{\nu+1}=(q_{\nu+1},a_{1,\nu+1}, a_{2,\nu+1})\in \Lambda_{\nu+1}$. Comme $ {\bf z}_{\nu} \in \Lambda_{\nu+1}$ est un point primitif, le réseau $\Lambda_{\nu+1}$ se divise en réseaux affines parallèles (de dimension un) $\Gamma^k$ de telle manière que $
\Lambda_{\nu+1}=\bigcup_{k\in \mathbb{Z}}\Gamma^k$, $ \Gamma^0 = {\rm span } ({\bf z}_{\nu}) \, \cap \mathbb{Z}^3$. Ici $\Gamma^{\pm1}$ sont les réseaux les plus proches de $\Gamma^0$. Si nous prenons ${\bf z}_{\nu+1} \in \Gamma^1$, nous voyons que le couple $({\bf z}_\nu, {\bf z}_{\nu+1})$ peut être complétée à une base de $\mathbb{Z}^3$. Notons que si $|{\bf z}_{\nu+1}|$ est assez grand (et donc $q_{\nu+1}$ est grand) alors l’angle $ {\rm angle} ( {\bf z}_{\nu+1}, {\bf z}_\nu)$ est petit. Il est clair que cet angle tend vers zéro lorsque $|{\bf z}_{\nu+1}|$ tend vers l’infini. Il existe donc $$W_\nu^1 = W_\nu (\delta ({\bf z}_\nu, \hbox{\got C}_\nu))$$ tel que si $|{\bf z}_{\nu+1}|\ge W_\nu^1 $ alors $$\label{delta1}
{\rm angle} ( {\bf z}_{\nu+1}, {\bf z}_\nu)<\delta_\nu/2.$$ On peut supposer que $ \delta_\nu$ est assez petit, alors $$\label{delta2}
\{ {\bf x}\in \mathbb{R}^3: \,
{\rm angle} ( {\bf x}, {\bf z}_\nu)<\delta_\nu\} \subset
\{ {\bf x}\in \mathbb{R}^3: \,
{\rm angle} ( {\bf x}, {\bf z}_{\nu-1})<\delta_{\nu-1}\}
.$$
Si nous choisissons ${\bf z}_{\nu+1}$ nous pouvons considérer la distance Euclidienne $$\rho_\nu =
{\rm dist}\, ({\bf z}_\nu, {\rm span}({\bf z}_{\nu+1})).$$ Nous voyons d’après la définition que $\rho_\nu$ dépend du choix du point ${\bf z}_{\nu+1}$. Notons que pour tout choix de ${\bf z}_{\nu+1} \in \Gamma^{\pm1}$ la valeur $q_{\nu+1}\rho_\nu$ sera du même ordre que le volume fondamental du réseau de dimension deux $\Lambda_{\nu+1}$, qui est déjà définie. Plus précisément, la quantité $\frac{q_{\nu+1}\rho_\nu}{{\rm det}\Lambda_{\nu+1}}
$ est bornée par l’arrière et est séparé de zéro par une constante positive.
Mais $ \rho_{\nu}$ tend vers zéro lorsque $|{\bf z}_{\nu+1}|$ tend vers l’infini. Il existe donc $$W_\nu^2 = W_\nu ( {\bf z}_{\nu-1}, {\bf z}_\nu)$$ tel que pour tous ${\bf z}_{\nu+1}\in \Gamma^{\pm1}$ sous la condition $|{\bf z}_{\nu+1}|>W_\nu^2$ on a $\rho_{\nu} \le \rho_{\nu-1}/2$ et $$\label{ce}
\varphi \left(\frac{1}{64\rho_{\nu-1}\rho_\nu}\right) \le \frac{1}{16q_{\nu+1} \rho_{\nu}}$$ (ici $\rho_{\nu-1}$ supposé être défini par des moyens des points de ${\bf z}_{\nu-1}, {\bf z}_\nu$ à l’étape précédente de la construction).
Maintenant, nous fixons ${\bf z}_{\nu+1} $ avec $$|{\bf z}_{\nu+1}|>\max ( W_\nu^1, W_\nu^2)
.$$ Nous avons construit le point suivant ${\bf z}_{\nu+1}$. Pour le point construit les conditions (\[delta1\],\[delta2\],\[ce\]) sont valables. Et le réseau correspondant $\Lambda_{\nu+1}$ satisfait ([**i**]{}) - ([**v**]{}). De plus, notre construction donne (\[daba\]) avec $\nu$ remplacé par $\nu+1$.
Maintenant, nous mettons $$\xi_{j, \nu} = \frac{a_{i,\nu}}{q_\nu},\,\,\,j = 1,2$$ et $$\label{xi}
\xi_j = \lim_{\nu\to \infty} \xi_{j,\nu},\,\,\,\
\xi = (1,\xi_1,\xi_2).$$ Bien sûr, nous pouvons supposer que $ \xi_1,\xi_2 \in (0,1)$.
Nous voyons d’après (\[delta1\],\[delta2\]) que $$\label{dee}
\xi \in
\{ {\bf x}\in \mathbb{R}^3: \,
{\rm angle} ( {\bf x}, {\bf z}_\nu)<\delta_\nu\} \,\,\,\,\,
\forall \nu.$$ Notons que pour $\xi$ défini dans (\[xi\]) nous avons $${\rm dist }( {\bf z}_\nu, {\rm span }(\xi))
\le \sum_{k =\nu}^\infty {\rm dist} ({\bf z}_k, {\rm span} ({\bf z}_{k+1})) \le 2\rho_\nu,$$ et ainsi $$\label{uro}
\max_{j=1,2} |q_\nu \xi_j - a_{\nu, j} | \le 4\rho_\nu.$$ Par ailleurs il faut noter que d’après [**(iii)**]{}, il en résulte que $$\xi \in \overline{\cal P}({\bf z}_\nu, {\bf z}_{\nu-1}),\,\,\,\,\,\,\forall \,\nu$$ et donc d’après Lemme 4 et la definition de $H_\nu$ (égalité (\[hanu1\])) pour tout $ {\bf z} =(q,a_1,a_2) \in \Lambda_\nu \setminus {\rm span} ({\bf z}_\nu)$ avec $ |{\bf z}|\ge H_\nu$ on a $$\label{ha}
\max_{j=1,2}|q\xi_j - a_j | \ge \varphi (H_\nu)\ge \varphi(|{\bf z}|) \ge \varphi (q).$$
Dans le reste de l’article, nous montrons que le vecteur $\xi$ construit satisfait la conclusion de Théorème 1.
[**4. Inégalités.**]{}
Nous considérons l’intervalle $$\label{ii}
I_\nu =
\left[\phi\left( \frac{1}{16q_\nu\rho_{\nu-1}}\right),
\frac{1}{64\rho_{\nu-1}\rho_\nu}\right].$$
[**Lemme 5.**]{} [*Si $ z = (q,a_1,a_2)\in \mathbb{Z}^3$ est linéairement indépendant avec ${\bf z}_{\nu-1} $ et ${\bf z}_\nu$ et $$\label{quu}
q\in I_\nu$$ alors $$\label{relax}
\max_{j=1,2}||q\xi_j || \ge \varphi (q).$$* ]{}
Preuve.
Soit $
\rho = \max_{j=1,2}||q\xi_j ||.
$ D’après la condition d’indépendance, nous avons $$0\neq
\left|
\begin{array}{ccc}
q& a_1&a_2\cr
q_{\nu-1}& a_{1,\nu-1}&a_{2,\nu-1}\cr
q_{\nu}& a_{1,\nu}&a_{2,\nu}
\end{array}
\right|
=
\left|
\begin{array}{ccc}
q& a_1-q\xi_1&a_2-q\xi_1\cr
q_{\nu-1}& a_{1,\nu-1}-q_{\nu-1}\xi_1&a_{2,\nu-1}-q_{\nu-1}\xi_2\cr
q_{\nu}& a_{1,\nu}-q_{\nu}\xi_1&a_{2,\nu}-q_{\nu2}\xi_2
\end{array}
\right|.$$ Ainsi, selon (\[uro\]) nous avons $$1\le
32q\rho_{\nu-1}\rho_\nu+ 8q_\nu \rho\rho_{\nu-1} .$$ De la borne supérieure qui découle de (\[quu\]), nous avons $$\frac{1}{2}\le 8 q_\nu \rho\rho_{\nu-1}
.$$ Ainsi $$\max_{j=1,2}||q\xi_j ||= \rho \ge
\frac{1}{16q_\nu\rho_{\nu-1}}\ge \varphi (q)$$ (dans la dernière inégalité, nous utilisons la borne inférieure qui découle de (\[quu\])).$\Box$
[**5. Preuve du Théorème 1.**]{}
Notons que (\[ce\]) montre que l’union $\bigcup_\nu I_\nu$ couvre un certain rayon $ [I, +\infty)$. Alors, d’après Lemme 5, si $q$ est assez grand et le point $(q, a_1, a_2)$ est indépendent avec deux points quelconques $ {\bf z}_{\nu-1}, {\bf z}_\nu, \nu =1,2,3,...$ alors nous avons (\[relax\]) et ces points ne sont pas d’intérêt pour notre propos. Donc, si nous avons une matrice unimodulaire entier $$\label{maa}
\left(
\begin{array}{ccc}
q& q''& q'''\cr
a_1'&a_1''&a_1'''\cr
a_2'&a_2''&a_2'''
\end{array}
\right)$$ avec $ \min \{ |q'|. |q''|. |q'''|\} $ assez grand et $$\label{coco}
\max \left\{ \frac{\max_{j=1,2} |q'\xi_j - a_j'|}{\varphi (q')},
\frac{\max_{j=1,2} |q''\xi_j - a_j''|}{\varphi (q'')},
\frac{\max_{j=1,2} |q'''\xi_j - a_j'''|}{\varphi (q''')}\right\}
\le 1,$$ alors pour certains $\nu', \nu'', \nu'''$ on a $${\bf z}' = (q', a_1',a_2') \in \Lambda_{\nu'},\,\,\,
{\bf z}'' = (q'', a_1'',a_2'') \in \Lambda_{\nu''},\,\,\,
{\bf z}''' = (q''', a_1''',a_2''') \in \Lambda_{\nu'''} .$$ Nous prenons $\nu', \nu'', \nu'''$ être les quantités minimales à satisfaire cette propriété. Ainsi $${\bf z}' = (q', a_1',a_2') \in \Lambda_{\nu'}\setminus \Lambda_{\nu'-1} ,\,\,\,
{\bf z}'' = (q'', a_1'',a_2'') \in \Lambda_{\nu''}\setminus \Lambda_{\nu''-1} ,\,\,\,
{\bf z}''' = (q''', a_1''',a_2''') \in \Lambda_{\nu'''}\setminus \Lambda_{\nu'''-1} .$$ Nous supposons que $\nu'=\min \{ \nu', \nu'', \nu'''\} < \max \{ \nu', \nu'', \nu'''\} = \nu'''$ (sinon la matrice (\[maa\]) a déterminant zéro, et ce n’est pas possible).
Considérons le cas [**(A)**]{} où $\nu'\le \nu''<\nu'''$. Alors $ {\bf z}'''\in L^-({\bf z}'.{\bf z}'')\cup L^+({\bf z}'.{\bf z}'')$ par l’unimodularité de la matrice (\[maa\]). Cependant $ {\bf z}''' \in \Lambda_{\nu'''}$ et $ {\bf z}''' \not\in {\rm span} ({\bf z}_{\nu'''-1})$. Ainsi par [**(iv)**]{} nous avons $ |{\bf z}''' |\ge T_{\nu'''-1}$. Par (\[dee\]) nous voyons que $
{\rm angle} (\xi, {\bf z}_{\nu'''-1})<\delta_{\nu'''-1}
.
$
Supposons que $ ({\bf z}'.{\bf z}'')\in\hbox{\got C}_{\nu'''-1}$. Alors par Corollaire 2 nous avons (\[result\]) pour le point ${\bf z}'''$, c’est-à-dire $$\frac{\max_{j=1,2} |q'''\xi_j - a_j'''|}{\varphi (q''')}\ge 1
.$$ Nous avons donc une contradiction avec (\[coco\]).
Supposons que $ ({\bf z}'.{\bf z}'')\not\in\hbox{\got C}_{\nu'''-1}$. Ensuite, par la définition de $\hbox{\got C}_{\nu'''-1}$ soit $ |{\bf z }_{\nu'} |\ge H_{\nu'}$ ou $ |{\bf z }_{\nu''} |\ge H_{\nu''}$ Ainsi par (\[ha\]) avec $ \nu$ égal à $\nu'$ ou $\nu ''$, nous avons $$\max \left\{ \frac{\max_{j=1,2} |q'\xi_j - a_j'|}{\varphi (q')},
\frac{\max_{j=1,2} |q''\xi_j - a_j''|}{\varphi (q'') }\right\}
\ge 1.$$ Nous avons une contradiction avec (\[coco\]) à nouveau.
Donc, le cas [**(A)**]{} n’est pas possible.
Maintenant, nous considérons le cas [**(B**]{}) où $\nu'<\nu''=\nu'''$. Nous avons $${\bf z}' \in \Lambda_{\nu'},\,\,\,\,\,\, {\bf z}'',{\bf z}''' \in \Lambda_{\nu''}\setminus\Lambda_{\nu''-1}.$$ Comme la matrice (\[maa\]) est unimodulaire, le couple $({\bf z}'',{\bf z}''')$ forme une base de $\Lambda_{\nu''}$. Ainsi $${\bf z}' \in L^-({\bf z}_{\nu''}, {\bf z}_{\nu''-1})\cup L^+({\bf z}_{\nu''}, {\bf z}_{\nu''-1}).$$ Mais alors $${\bf z}_{\nu''} \in L^-({\bf z}_{\nu''-1}, {\bf z}')\cup L^+({\bf z}_{\nu''-1}, {\bf z}').$$ Comme $ {\bf z}_{\nu ''} \in \Lambda_{\nu''}$, par [**(v)**]{} avec $ \nu = \nu''-1$ nous voyons que $ {\bf z}' \not\in \hbox{\got E}_{\nu''-1}.$ Cela signifie que $|{\bf z}'|\ge H_{\nu'}$. Donc par (\[ha\]) nous avons $$\frac{\max_{j=1,2} |q'\xi_j - a_j'|}{\varphi (q')}
\ge 1.$$ Nous avons une contradiction avec (\[coco\]) à nouveau.
Donc, le cas [**(B)**]{} n’est pas possible.
Donc, dans le cas où $ \min \{ |q'|. |q''|. |q'''|\} \ge q_0 (\varphi ) $ nous avons $$\max \left\{ \frac{\max_{j=1,2} |q'\xi_j - a_j'|}{\varphi (q')},
\frac{\max_{j=1,2} |q''\xi_j - a_j''|}{\varphi (q'')},
\frac{\max_{j=1,2} |q'''\xi_j - a_j'''|}{\varphi (q''')}\right\}
\ge 1.$$ Maintenant, nous prenons $\varepsilon = \varepsilon (\varphi)>0$ assez peitit pour assurer l’inégalité nécessaire pour les vecteurs avec $ q\le q_0 (\varphi)$. Théorème 1 est prouvé. $\Box$
[**5. Preuve du Théorème 2.**]{}
Dans cette section, nous considérons les vecteurs meilleures approximations ${\bf z}_\nu = (q_\nu, a_{1,\nu}, a_{2,\nu})\in \mathbb{Z}^3$ dans le sens de l’approximation simultanée de $\xi_1,\xi_2$ (voir [@M2]). Nous savons que $$\max\left\{
\max_{j=1,2} ||q_\nu\xi_j - a_{j,\nu}||,
\max_{j=1,2} ||q_{\nu+1}\xi_j - a_{j,\nu+1}||\right\}
\le q_{\nu+1}^{-1/2} \to 0,\,\,\,\, \nu \to \infty.$$ En outre, le couple ${\bf z}_\nu, {\bf z}_{\nu+1}$ peut être étendue à une base de $\mathbb{Z}^3$. Le parallélogramme $$\Pi_\nu =\{ {\bf x}\in \mathbb{R}^3:\,\, {\bf x}= \lambda {\bf z}_\nu+\mu{\bf z}_{\nu+1},\, \lambda , \mu \in [0,1)\}.$$ forme un domaine fondamental du réseau de dimension deux $L({\bf z}_\nu, {\bf z}_{\nu+1})\cap \mathbb{Z}^3$. Considérons le sous-espace affine $L^+({\bf z}_\nu, {\bf z}_{\nu+1})$ et l’orthogonal projection $\Pi_\nu^*$ de $\Pi$ sur $L^+({\bf z}_\nu, {\bf z}_{\nu+1})$. Alors $$\Pi_\nu^* = \Pi_\nu +{\bf e}_\nu,$$ où le vecteur ${\bf e}_\nu\in \mathbb{R}^3$ a la longueur $$|{\bf e}_\nu| = ({\rm det} ( L({\bf z}_\nu, {\bf z}_{\nu+1})\cap \mathbb{Z}^3))^{-1} \asymp( q_{\nu+1} \cdot \max_{j=1,2} ||q_\nu\xi_j - a_{j,\nu}||)^{-1}
\to 0,\,\,\,\, \nu \to\infty$$ (la dernière déclaration ici est Théorème 17 de [@M2] qui est une minuscule généralisation de Satz 9 de [@TB]). Comme $\Pi_\nu^*$ est un domaine fondamental pour $ L({\bf z}_\nu, {\bf z}_{\nu+1})\cap \mathbb{Z}^3$, il existe un point entier ${\bf z}^* = (q^*, a_1^*, a_2^*)
\in \Pi_\nu^*$. Nous voyons que $$\max_{j=1,2} ||q^*\xi_j - a_{j}^*|| = O(
\max_{j=1,2} ||q_\nu\xi_j - a_{j,\nu}||+
\max_{j=1,2} ||q_{\nu+1}\xi_j - a_{j,\nu+1}||+ |{\bf e}_\nu|) \to 0,\,\,\,\nu \to \infty.$$ Mais la matrice $$\left(
\begin{array}{ccc}
q_\nu&q_{\nu+1}&q^*\cr
a_{1,\nu}& a_{1,\nu+1}&a_{1}^*\cr
a_{2,\nu}& a_{2,\nu+1}&a_{2}^*
\end{array}
\right)$$ est unimodulaire. Théorème 2 est prouvé. $\Box$
[1]{}
N. Chevallier, Best simultaneous Diophantine approximations and multidimensional continued fraction expansions, Mosc. J. Comb. Number Theory, vol. 3, iss. 1 (2013), pp. 3 - 56.
H. Davenport, W.M. Schmidt, Approximation to real numbers by algebraic integers, Acta Arithmetica, 15 (1969), 393 - 416.
V. Jarník, Zum Khintchineschen ’Übertragungssatz’, Acad. Sci. URSS, vol. 3, Travaux, Inst. Math., Tbilissi (1938), pp. 193-216.
V. Jarník, Contribution à la théorie des approximations diophantiennes linéaires et homogènes, Czechoslovak Math. J. 4 (1954), 330 - 353 (in Russian, French summary).
J.S. Lagarias, Best simultaneous Diophantine approximation II, Pac. J. Math., V. 102 (1982), No 1, p. 61 -88.
N.G. Moshchevitin, On best simultaneous approximations, Russian Mathematical Surveys, V. 51, No.6 (1996), p. 213 - 214.
, Best Diophantine approximations: the phenomenon of degenerate dimension, London Mathematical Society, Lecture Note Series, 338 (2007), p. 158 - 182.
N.G. Moshchevitin, Khintchine’s singular Diophantine systems and their applications, Russian Mathematical Survey 65:3 (2010), 43 - 126
[^1]: la recherche est financée par la subvention de RFBR No.12-01-00681-a et par la subvention de le Gouvernement Russe, projet 11. G34.31.0053.
|
{
"pile_set_name": "ArXiv"
}
|
1.0cm
.3in
Ghanashyam Date
.1in
The Institute of Mathematical Sciences, CIT Campus, Chennai 600 113, India
2.0in A School on Loop Quantum Gravity was held at the IMSc during Sept 8 – 18, 2009. In the first week a basic introduction to LQG was provided while in the second week the focus was on the two main application, to cosmology (LQC) and to the black hole entropy. These notes are an expanded written account of the lectures that I gave. These are primarily meant for beginning researchers. .3in
1.0cm
It has been felt for a while that our graduate students do not get an opportunity to get an exposure to the non-perturbative, background independent quantum theory of gravity at a pedagogic level. Although there are several excellent reviews and lecture notes available, an opportunity for complementing lectures by discussions is always an added bonus for the students. With this in mind and taking into account of the background preparation of the students, the [*School on Loop Quantum Gravity*]{} was organized at IMSc, for a period of 10 days. The first 5 days were devoted to the basics of connection formulation and loop quantization up to sketching steps involved in the quantization of the Hamiltonian constraint. The next 5 days were devoted to the applications to quantum cosmology and to the black hole entropy. In all 20 lectures and 10 tutorials were planned, however some tutorials ‘became’ additional lectures.
These notes are an expanded version of the topics that I covered. In particular the material of chapter 2, sections 4.2.1, 4.2.2, 4.2.3 and appendices 5.1, 5.3, 5.4 have been added. Email discussions on the sections of chapter 4 and appendix 5.4 with Abhay Ashtekar, Martin Bojowald and Madhavan Varadarajan have been very helpful and are gratefully acknowledged. There could still be some differences in the perceptions and formulations, what I have presented is my understanding of the issues.
The other main lecturers at the school were: Prof Amit Ghosh, Saha Institute of Nuclear Physics, Kolkata; Dr Alok Laddha, Raman Research Institute, Bengaluru; Parthasarathi Majumdar, SINP, Kolkata. In addition, Prof Romesh Kaul, IMSc, Dr Kinjal Banerjee, IUCAA, Pune and Ayan Chatterjee, SINP, Kolkata also gave a few lectures. Amit and Alok discussed the connection formulation and loop quantization up to the basic steps in the quantization of the Hamiltonian constraint. Partha, Amit and Ayan discussed classical formulation of isolated horizons and entropy associated with them. Romesh discussed the possibility of a ‘vacuum structure for gravity’ and Alok also briefly discussed the Brown-Kuchar dust model. It is a pleasure to acknowledge their contributions. At least some notes of the topics covered by these will be available in not-too-distant a future.
The funding for the school was provided by the Institute of Mathematical Sciences under the XI$^{\mathrm{th}}$ Plan Project entitled [*Numerical Quantum Gravity and Cosmology*]{}. It is envisaged that a more specialized workshop at an advanced level will be held in the summer of 2010 with the possibility of a similar School being repeated one more time.
March 17, 2010 Ghanashyam Date 0.3cm
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General Remarks:
================
Why a Quantum Theory of Gravity?
--------------------------------
This is not a rhetorical question but it is intended to [*identify physical context*]{} in which the classical theory of gravity, specifically the Einstein Theory of Gravity also referred to as General Relativity, is [*inadequate*]{} and calls for an extension. One has met with inadequacies of classical theories many times and has seen how their quantum versions have alleviated the inadequacies. For example, the classical theory of charges and electromagnetic fields was quite adequate until the hydrogen atom was found to have a central nucleus with an electron going around it. Classical theory predicts that since the electron is necessarily accelerated, it must radiate away its energy and spiral into the proton. Indeed in about $10^{-9}$ seconds (!) the classical trajectory of a (bound) electron would ‘end’ in the proton. We all know that this is physically wrong and the atoms are known to be stable for billions of years. We also know that the ‘fault’ lies not with the ‘Coulomb law’ (which does get modified) but with the classical [*framework*]{} of using well defined trajectories to describe the dynamical evolution for both the electron and the electromagnetic field. Inadequacies of classical theories are also revealed in the black body spectrum, specific heat of solids at low temperatures etc etc and the appropriate quantum theory of matter and electromagnetism cures these problems i.e. gives results consistent with experimental observations. The quantum nature of other interactions such as the strong and the weak is also verified in nuclear and particle physics. What about the gravitational interactions?
Gravitationally bound ‘atoms’ can also be considered. If gravity is described in the Newtonian manner, there is no gravitational radiation from an accelerated motion and the inward spiralling problem will not arise. But Einstein’s theory of gravity is very different and accelerated sources do radiate away energy and the stability issue re-surfaces. Of course the ‘weakness’ of gravitational interaction does not threaten the existence of such gravitationally bound atoms if the decay time is larger than the age of the universe, but in principle possibility exists. In fact Einstein did suggest a need for a quantum theory of gravity [@AbhayEinstein] almost immediately after GR was constructed[^1].
General relativity however uncovered two distinct contexts in which the theory calls for an extension – the context of (i) cosmological and black hole singularities and (ii) entropy of black hole horizons. Let us take a little closer look at these contexts.
[**The cosmological context:**]{} Under the assumption of [*homogeneity and isotropy*]{}, the space-time metric is described in terms of a single dynamical variable – the scale factor. As long as the energy density is positive and the pressure is not too negative (which is true for the properties of known matter), in an expanding universe (an observational fact), the scale factor vanishes at a finite time in the past. The universe has ‘beginning’, a finite age and the space-time curvature (or gravity) is infinitely large. Thus, a homogeneous, isotropic universe has [*singular beginning*]{}. If one relaxes isotropy but retains homogeneity, one has several [*types*]{} of space-times. There are now a maximum of [*six*]{} dynamical variables. The simplest of these, the vacuum Bianchi I space-time, has three ‘scale factors’ whose time dependence is given by the (exact) Kasner solution. This is also singular. As one evolves back in time, two of the scale factors vanish while the third one diverges. The most complex of these models, the vacuum Bianchi IX space, is also singular and the backward evolution has an [*oscillatory*]{} behaviour. Like the Kasner solution, two scale factors begin decreasing and the third one increasing. But after a while, the three scale factors change their behaviour and a different pair begins decreasing. This continues ad infinitum. If the [*non-diagonal*]{} metric components are included, then the directions along which contraction/expansion takes place are also ‘rotated’. If one relaxes homogeneity as well, then a beautiful analysis done by Belinskii-Khalatnikov-Lifschitz (BKL), shows that there exists a [*general solution*]{} of the vacuum Einstein equations which can be described as smaller and smaller portions of the spatial slice behaving as a homogeneous, Bianchi IX solution. The BKL analysis in particular shows that singular solutions found in the simpler situations are [*not*]{} due to high degree of symmetry (homogeneity and isotropy), but even without such symmetries, there exist general solutions which are singular (diverging curvatures) and the nature of singularities can be extremely complicated.
During the sixties Geroch-Penrose-Hawking used another approach to establish the [*Singularity theorems*]{} identifying conditions under which singularities are inevitable consequence of classical GR. For these theorems, singular space-times were defined as those [*in-extendible space-times which admit at least one causal (time-like or null) geodesic which is incomplete.*]{} Here incompleteness means that the geodesic cannot be defined for all real values of an affine parameter. There are three types of inputs in these theorems: (a) One restricts to a class of space-times which are [*causally well-behaved*]{} eg are free from closed causal curves. The space-times which are free of all causal pathologies and are fully deterministic are the so-called [*globally hyperbolic*]{} space-times. (b) The space-times are solutions of Einstein equation with the matter stress tensor satisfying suitable [*energy condition(s)*]{}. This incorporates that idea that gravity is attractive (for positive mass/energy). These two types of conditions are general requirements for a space-time model to be physically relevant. (c) the third input is a condition that distinguishes specific physical context such as an everywhere [*expanding universe*]{} or a gravitational collapse which has proceeded far enough to develop [*trapped surfaces*]{}. The presence of the last condition(s) shows that not every solution satisfying the first two conditions is singular (e.g. the Minkowski space-time). Thus, singularity theorems do [*not*]{} imply that gravitational interactions [*always*]{} produce singularities – the (c) type condition is necessary. While inclusion of (c) will imply singular space-times, it is [*not*]{} automatic that this condition is [*realized*]{} in the physical world. In our physical world however universe is expanding and it is widely believed that black holes also exist and hence condition (c) [*is* ]{} realized in nature. Thus, physical contexts exist wherein classical GR is inadequate.
The global hyperbolicity condition implies that the space-time is [*stably causal*]{} i.e. a global time function exists such that each hyper-surface of constant ‘time’ is space-like. Furthermore, a time function can be chosen such that the space-like hypersurfaces are [*Cauchy*]{} surfaces. The topology of such space-times is necessarily $\mathbb{R} \times \Sigma$. A Hamiltonian formulation makes sense only in such space-times. Thus, [*not every solution*]{} of Einstein equation yields a physically acceptable space-time (i.e. causally well behaved or globally hyperbolic). A Hamiltonian evolution however constructs such space-times [^2].
In specific solutions, one encounters singularities (regions of diverging curvatures) which are [*space-like*]{} (positive mass Schwarzschild solution, homogeneous cosmologies), [*time-like*]{} (negative mass Schwarzschild solution) or even [*null*]{} (some of the Weyl class of solutions). Singularities that [*arise*]{} in an evolution from [*non-singular*]{} initial conditions are the ones which strongly display inadequacy of the theory. Typically, these are the space-like singularities. Since the Hamiltonian formulation is an initial value formulation, it can “see” only such singularities.
[**The Black hole context:**]{} Black holes are objects whose ‘interiors’ are inaccessible to far away observers. More precisely, these are space-time geometries that have a [*horizon*]{} which leave some regions out of bounds for asymptotic observers. The special class of [*stationary*]{} black holes are characterized by a few parameters – mass ($M$), angular momentum ($J$) and electric charge (say) ($Q$). Associated with their horizons are some characteristic parameters – area of the horizon ($A$), surface gravity at the horizon ($\kappa$), angular velocity at the horizon ($\Omega$) and electromagnetic potential at the horizon ($\Phi$). In the seventies, a remarkable set of “laws” governing processes involving black holes were discovered. If in a process a black hole is disturbed (by accreting mass, say) and returns to a stationary state again, then the changes in the parameters obey: $$\delta M ~ = ~ \frac{\kappa}{8 \pi} \delta A + \Omega_H \delta J +
\Phi_H \delta Q ~ ~ , ~ ~ \delta A \ge 0\ .
$$ These are very temptingly analogous to the laws of thermodynamics! Especially after one also proves that $\kappa$ is constant over the surface of the horizon. Bekenstein in fact suggested that area of the horizon be identified with the entropy of a thermodynamic system. This suggests that the surface gravity be identified with a temperature. Hawking subsequently showed that when possibilities of quantum instabilities are taken into account, a black hole can be thought of a black body with temperature $T = {{\textstyle \frac{\kappa {\ell_{\mathrm P}}^2}{2 \pi}}}$ and hence $S = {{\textstyle \frac{1}{4}}} {{\textstyle \frac{A}{{\ell_{\mathrm P}}^2}}}, {\ell_{\mathrm P}}^2 := G\hbar$. For all other systems we know that thermodynamics is a manifestation of an underlying statistical mechanics of a large number of [*microscopic*]{} degrees of freedom. What are these micro-constituents of the black holes? Notice that from far away, only the exterior of a horizon is accessible and so also parameters such as mass and angular momentum. All detailed memory of what collapsed to form the black hole is lost. So these micro-constituents must be distinct from the matter degrees of freedom. They must represent “atoms” of [*geometry*]{}. But classical geometry is continuous so how does a particulate nature arise? Perhaps, not just the specific dynamics given by Einstein equation is inadequate but the [*framework of classical geometry*]{} itself is inadequate. Note that black hole horizons are [*not*]{} regions of high curvatures and geodesic incompleteness occurs in their interiors. Thus, black hole thermodynamics is a [*qualitatively different situation.*]{}
In summary, classical GR contains within its domain, physically realizable physical context where the theory is inadequate. At least one of its context involves [*highly dynamical*]{} geometries with high curvatures, matter densities etc. Because of these features, it is hard to imagine how any [*perturbative approach*]{} can be developed in these contexts. Since gravity (or space-time geometry) is dynamical and a perturbative approach is unlikely to be suitable, it is necessary to have a quantum theory of gravity which of [*does not use*]{} any fixed background space-time in its [*construction*]{}.
An Essential Feature of Classical Gravity
-----------------------------------------
Let us recall briefly that special relativity combines Newtonian notions of space and time into a single entity, the [*space-time*]{} (Minkowski space-time). The analysis of the geometry inferred in a rotating frame indicates that the geometry is non-Euclidean. Principle of equivalence, which identifies uniform acceleration with uniform gravity (in the Newtonian sense), then implies that gravity affects the space-time geometry and since matter affects gravity, it also affects geometry. In the final formulation of the relativistic theory of gravity, the space-time geometry, described by a metric tensor, is a dynamical (changeable) entity with the Newtonian gravity being a manifestation of the curvature. The law governing the matter-geometry interaction is encoded in the Einstein equation. That all observers are on equal footing to formulate the laws of physics implies that all quantities (and equations) be tensor fields (and equations) with respect to [*general coordinate transformations.*]{} Note that a general coordinate transformation corresponds to a [*change of chart*]{} in the framework of differentiable manifolds.
Such coordinate transformations however have another interpretation in terms of mapping of the manifold (or local regions thereof) into itself – the [*active diffeomorphisms*]{}. Under the action of such mappings, the pull-backs and push-forwards, generate “new” tensor fields from the old ones. That is, in a given neighbourhood, we will have the original tensor field and the one obtained via pull-back/push-forward. If $x \to y(x)$ represents the mapping in terms of local coordinates, then the pulled-back (pushed-forward) quantities are related to the original ones in precisely the same manner as general coordinate transformation [^3]. If the dynamical equations are covariant with respect to general coordinate transformations (coordinate transform of a solution is also a solution), they must also be covariant with respect to the active diffeomorphisms i.e. a configuration and its transform under active diffeomorphism are both solutions if any one of them is. This has far reaching implications.
[*The Einstein Hole Argument:*]{} The active diffeos can be chosen such that they map non-trivially only in some region (‘sub-manifold’) of the manifold. Choose a region which is free of any matter. Assume that the equations determining gravitational field and matter distribution are also tensor equations (i.e. generally covariant). Consider a solution which has certain curvature distribution inside our chosen ‘hole’. Make a diffeo which is non-trivial only inside the hole and change the curvature distribution. This will also be a solution by covariance. Thus we get a situation that even though matter distribution is unchanged, in a region where there is no matter, we can have two ‘different’ gravitational fields i.e. matter distribution does [*not*]{} determine the gravitational field. But in the non-relativistic limit Newtonian gravity is determined by matter. So [*either*]{} the equations should [*not*]{} be covariant [*or*]{} [*in the absence of any matter available for ‘marking’ points of a manifold, the ‘different’ distributions of curvature must be regarded as describing the ‘same’ gravitational field*]{}. It is the latter possibility that remains once the [*covariance of the equations*]{} is accepted. This in turn implies that [*it is the equivalence classes of solutions, with respect to the space-time diffeomorphisms, that correspond to physical reality.*]{}
[*Note*]{}: The size of the hole in the hole argument is unimportant. Also, the metric description itself does [*not*]{} play a role; one could repeat the argument for any other field. All that is used is that the fields are tensors under general coordinate transformations (chart change), field equations are covariant and the fields are inhomogeneous within a hole. Values of individual fields at any, manifold point are irrelevant but values of fields at points [*specified by other fields*]{} are invariant and thus physical.
Since ‘points’, not marked by any [*dynamical entity*]{}, have no physical meaning, the only, physically meaningful, questions are of relational nature. That is, it is physically meaningless to ask what is the curvature (or say electric field) “here and now”. The meaningful questions is what is the curvature where a certain field has a certain value. If we had [*any*]{} particular field to be fixed (non-dynamical), then with reference to that field we could ask the ‘here and now’ question. Such a field, constitutes a background. Note that the usual non-gravitational theories or in the perturbative treatment of gravity, the space-time geometry (metric) plays the role of a background. Since in the general relativistic theory [*all fields including the metric*]{} are fundamentally dynamical, such a theory is [*necessarily background free.*]{} The twin features of the framework namely [*all*]{} fields on a manifold being dynamical and the fundamental equation being [*generally covariant and deterministic*]{}, implies covariance with respect to active diffeomorphisms and physical characterization being in terms of 4-diffeo equivalence classes of fields [^4].
This is an essential feature of general relativity, much more fundamental than the particular Einstein equations themselves. The challenge is to construct a quantum theory which faithfully incorporates this feature i.e. a quantum theory of gravity must be background free (or at least recover background independence in the classical limit).
This also poses challenges, because we have to construct observables which are space-time diffeomorphism invariant. These alone could characterise specific equivalence classes of space-times and this problem is not understood even classically for spatially compact case and in absence of matter! Note that curvature invariants, although scalars, are local and [*not*]{} diffeo invariants. Hence these cannot be physical observables. Consequently identifying a physical state corresponding to (say) Minkowski space-time is much harder.
For a more detailed discussion of these conceptual issues, see [@Rovelli].
Towards the construction of a quantum theory
--------------------------------------------
Ultimately constructing a quantum theory of some phenomenon means specifying a state space - (projective) Hilbert space, identifying (self-adjoint) operators on it to correspond to physically observable quantities, a notion of evolution or dynamics such that in a suitable semi-classical approximation, the evolution of expectation values of a class of observables tracks the corresponding classical evolution with the quantum uncertainties less than the observational precision. Here the classical evolution is the one specified by the classical description of the phenomenon. Using such a framework, one can compute matrix elements of suitable observables or transition amplitudes etc. One familiar procedure is that of the [*canonical quantization*]{}.
In canonical quantization, typically we have a classical phase space which is cotangent bundle, ${\cal T}^*Q$ of some configuration manifold $Q$ and the Hilbert space is the space complex valued functions of $Q$ which are square integrable with respect to some suitable measure, $d\mu$. When $Q \sim \mathbb{R}^N$, we have the familiar $L_2(\mathbb{R}^N, d\mu_{\mathrm{Lebesgue}})$ which is unique thanks to the Stone-von Neumann theorem. This comfortable situation changes once $Q$ becomes topologically non-trivial and/or becomes infinite dimensional. The former can arise due to constraints while the latter arises in a field theory. In relativistic field theory, the [*classical configuration space*]{}, $Q$, is (say) the space of suitably smooth tensor fields which is inadequate to describe the corresponding quantum fields which can be arbitrarily non-smooth. Typically, $Q$ is [*extended*]{} to a [*quantum configuration space*]{}, $\bar{Q}$, which should admit a suitable measure.
For a quantum theory of gravity, there are two additional features - (i) we have a theory with first class constraints (i.e. a gauge theory) and (ii) we would like to have background independence.
In presence of constraints, the quantization procedure is a two step process. In the first step one constructs a [*kinematical Hilbert space*]{} on which are defined the constraint operators. The second step aims to ‘solve’ the constraints to get a quantum theory corresponding to physical degrees of freedom. Again, [*typically*]{}, there are no vectors in the kinematical Hilbert space which are annihilated by the constraint operators and one is forced to consider [*distributional solutions*]{}[^5]. The space of distributional solution however is [*not*]{} a Hilbert space and another inner product needs to be defined on this space to make it into the [*physical Hilbert space*]{}. The choice of this inner product is limited by demanding that a suitable class of [*Dirac observables*]{} – operators which leave the space of solutions invariant – be self-adjoint.
There are many choices to be made along the way. The requirement of background independence means that no non-dynamical fields should be used in any step. This poses a severe challenge to constructing even the kinematical Hilbert space. The connection formulation of gravity is of great help as the quantum configuration space of a gauge theory, $\overline{{\cal A}/{\cal G}}$ - space of generalized connections modulo generalized gauge transformations - admits several measures and the demand that the conjugate variables be represented by derivative operators essentially singles out a unique measure - the Ashtekar-Lewandowski measure, essentially constructed from the Haar measure on compact groups. One has a natural choice of $\Omega$ and (non-unique) definitions of constraint operators so that the kinematical set-up is well founded. We will see the details of these steps.
We will begin with the Hamiltonian formulation of GR in terms of the metric (ADM formulation). Discover the redundant variables and make a canonical transformation to a set of new variables (connection formulation) which are amenable to background independent presentation. These will lead us to the holonomy-flux variables and their Poisson bracket algebra whose representation theory will give us a Hilbert space. This will complete the [*first step*]{} in the construction of the quantum theory. Some of its novel features will be revealed through the properties of the geometrical operators. Our study of the basic formalism will conclude with the presentation of the constraint operators on the kinematical Hilbert space. The dynamical aspects will be studied through the simpler cases of homogeneous and isotropic cosmology leading to the Big Bang singularity resolution. The other application of quantum geometry, namely revealing the ‘atoms’ of geometry responsible for the black entropy will be discussed in the second week along with the loop quantum cosmology.
Classical Hamiltonian Formulation
=================================
We are familiar from usual spacial relativistic field theories (say a scalar field) that solutions of the field equations can be viewed as an evolution of fields, their spatial derivatives and their velocities from one spatial slice to another one (“a Cauchy evolution”). In the general relativistic case, one has to deal with space-times other than the Minkowski space-time and eventually make the space-time itself to be ‘dynamical’. However [*not all space-times support this notion of evolution.*]{}
To have a well defined, causal (no propagation faster than speed of light in vacuum) and deterministic (given certain data at one instance, the future data is [*uniquely*]{} determined), the space-time must be free of [*causal pathologies*]{} such as (i) no closed causal (i.e. time-like or null) curves (excludes by [*chronology condition*]{}); (ii) no closed causal curves but causal curves which return arbitrarily close to themselves (excluded by [*strong causality*]{}); (iii) strong causality holds but when the space-time metric is made slightly ‘wider’, it is violated (excluded by [*stable causality*]{}). All such pathologies are absent in space-time which are [*stably causal*]{} i.e. admit a differentiable function such that $\partial^{\mu} f$ is a time-like vector field. This alone is still not sufficient to guarantee the possibility of a [*Cauchy Problem*]{}. For this, one needs [*Globally Hyperbolic Space-times*]{}. These are space-times which are which admit a spatial hyper-surface such that events to the future (past) are completely determined by data specified on it. Such space-time admit a globally defined ‘time function’ whose equal time surfaces are Cauchy surfaces. It follows further that such space-times must have the topology: ${\cal M} = \mathbb{R} \times \Sigma_3$.
Thus, in order to have a well-posed causal theory of [*matter fields*]{}, the space-times must be globally hyperbolic.
It is a non-trivial result of analysis of Einstein equation that Einstein equation can be cast in a Hamiltonian form such that if initial conditions are chosen to satisfy certain [*constraints*]{}, then corresponding Hamiltonian evolution generates a solution (space-time) of the Einstein equation[^6]. Although a Hamiltonian formulations can be specified ab initio by giving a [*phase space*]{} which is a [*symplectic manifold*]{}, a [*Hamiltonian*]{} function and specifying evolution by the Hamilton’s equations of motion, it is more common that a theory is specified in terms an action which is a functional defined on a set of fields on a space-time manifold. Typically this is expressed as an integral of a Lagrangian density made up of finite order derivatives of a set of tensor (spinor) fields. In such a manifestly space-time covariant presentation, one needs to choose a “time” direction along which to ‘evolve’ data specified on a ‘equal time surface’. These data identify the [*configuration space variables and their velocities*]{}. The Hamiltonian formulations is then obtained from this [*Lagrangian formulation*]{} by passing through a Legendre transform. This identification of a time direction and a spatial slice on which data are to be specified is referred to as a “3 + 1 decomposition”.
The 3 + 1 decomposition
-----------------------
Let us assume that our would be space-time manifold is such as to admit a smooth function $T: {\cal M} \to \mathbb{R}$ such that the $T =
constant$ level sets, generate a foliation. Different possible $T$-functions will generate different foliations. For this to be possible, we must have ${\cal M} \sim \mathbb{R} \times \Sigma_3$.
Now choose a vector field $t^{\mu}\partial_{\mu}$ which is [*transversal*]{} to the foliation i.e. every integral curve of the vector field intersects each of the leaves, transversally. Furthermore, locally in the parameter of the curve, the leaves are intersected once and only once. [*Normalize*]{} the vector field so that $t^{\mu} \partial_{\mu} T
= 1$. This ensures that values of the $T-$functions can be taken as a “time” parameter which we denote as $t$.
Fix a leaf $\Sigma_{t_0}$ and introduce coordinates, $x^a, a = 1, 2, 3$ on it. Carry these along the integral curves of the vector fields, to the other leaves. This sets up a local coordinate system on ${\cal M}$ such that the normalized parametrization provides the coordinate $t$ while the integral curves themselves are labelled by the $\{x^a\}$. Note that there is no metric so ${\cal M}$ is not yet a space-time. We have only set up a coordinate system.
Choose tensors $g_{ab}, N^a, N$ on each of the leaves in a smooth manner ad [*define*]{} a space-time metric via the line element: $$ds^2 ~:=~ -N^2 dt^2 + \bar{g}_{ab}\left(dx^a + N^a dt\right)\left(dx^b +
N^b dt\right).
$$ Choosing $\bar{g}_{ab}$ to be [*positive definite*]{} and $N \ne 0$ ensures that the space-time metric $g_{\mu\nu}$ is [*invertible*]{}. Its inverse is given by, $$g^{tt} ~=~ -N^{-2} ~~,~~ g^{tb} ~=~ N^b N^{-2} ~~,~~ g^{ab} ~=~
\bar{g}^{ab} - N^{-2} N^a N^b ~~,~~ \bar{g}^{ac}\bar{g}_{cb} =
\delta^a_b ~ .
$$ We now have a space-time. The space-time metric is defined in terms of 10 independent functions and so there is no loss of generality. It is a convenient parametrization for reasons given below, but alternative parametrization are possible.
It follows that, (i) The induced metric on the leaves is the Riemannian metric $\bar{g}_{ab}$.
\(ii) $n_{\mu} := \partial_{\mu} T$ is normal to the leaves, since for any tangent vector $X^{\mu}\partial_{\mu}$, to $\Sigma_t$, $X^{\mu}
n_{\mu} = X^{\mu} \partial_{\mu} T = 0$. Thanks to the normalization of $t^{\mu}\partial_{\mu}$, we have $n_{\mu} = (1, 0, 0, 0)$.
\(iii) $n^{\mu} := g^{\mu\nu} n_{\nu} \Rightarrow n^{\mu} n_{\mu} =
g^{tt} = - N^{2} < 0$ and therefore the normal is [*time-like*]{} and hence the leaves are [*space-like*]{}. The $N n^{\mu}$ is a unit time-like vector.
\(iv) The original transversal vector field can be decomposed as $t^{\mu}
= a n^{\mu} + \tilde{N}^{\mu}$ where $\tilde{N}^{\mu} n_{\mu} = 0$ and hence $\tilde{N}^{\mu}$ is tangential to the leaves and $\tilde{N}^0 =
0$. This decomposition refers to $N^2$ as the [*lapse*]{} function and $\tilde{N}^{\mu}$ as the [*shift vector*]{}. Next, $t^{\mu}n_{\mu} = 1
\Rightarrow a = - N^2$. The integral curve equation, $d_t x^{\mu} = -
N^2 n^{\mu} + \tilde{N}^{\mu}$ implies for $\mu = a$, $\tilde{N}^a = N^2
n^a = N^2 g^{at} = N^2 ( N^{-2} N^a ) = N^a$. This identifies the $N^a$ with the shift vector (which is spatial).
The particular parametrization of the space-time metric can be said to be [*adapted*]{} to the pre-selected coordinate system. Since the coordinate system is defined [*without*]{} any reference to any metric, we can similarly parametrize other tensor fields, notably the [*co-tetrad*]{}, $e_{\mu}^I$.
We chose an arbitrary foliation (through an arbitrary choice of a “Time” function and then a transversal vector field to enable us to choose coordinate on the manifold. The foliation provides us with a normal $n_{\mu}$ and the transversal vector field can be parametrized in terms of this normal, a lapse function and a shift vector. Varying the lapse and shift varies the transversal vector field [*relative to the foliation*]{}. If we also change the foliation, then the normal changes and so must the shift vector. The changes induced by lapse and shift correspond to making a space-time diffeomorphism and [*every infinitesimal space-time diffeomorphism can be generated by infinitesimal changes in the lapse and shift.*]{}
Digression on tetrad formulation
--------------------------------
General relativity is formulated as theory consisting of tensorial fields on a manifold and a second rank, symmetric, non-degenerate (invertible) tensor field, $g_{\mu\nu}$ encoding gravitational phenomena. To do differential calculus on general tensor fields one also needs to define a covariant derivative, $\nabla_{\mu}$ which involves the introduction of an affine connection, $\Gamma^{\lambda}\,
_{\mu\nu}$, which is usually taken to symmetric and [*metric compatible*]{} i.e. $\nabla_{\lambda} g_{\mu\nu} = 0$. The non-abelian gauge theories already introduce quantities which are not just tensors with respect to general coordinate transformations but also transform under the action of “an internal” group, eg a Higgs field $\Phi^a$, a YM potential $A_{\mu}^a$, its corresponding field strength, $F^a_{\mu\nu}$ etc. The index $a$ indicates a response to the (adjoint) action of a group such as $SU(N)$. Developing calculus for such quantities, also needs a [*gauge*]{} covariant derivative and a corresponding [*gauge*]{} connection eg $A_{\mu}^a$.
Consider now a quantity, $e^I_{\mu}(x)$ where $\mu$ responds to a general coordinate transformation ($e^I_{\mu}$ transform as a covariant rank 1 tensor) and the index $I$ responds to the [*local*]{} action of the pseudo-orthogonal group, $SO(1,3)$ under the defining representation. This quantity can also be thought of as a $4 \times 4$ matrix and we will take it to be an invertible matrix. This is referred to as a [*co-tetrad*]{} while its inverse quantity, $e_I^{\mu}$ is referred to as a [*tetrad*]{}: $e^I_{\mu} e_J^{\mu} = \delta^I_J,
e^I_{\mu} e_I^{\nu} = \delta^{\nu}_{\mu}$. It is possible to formulate the theory of gravity in terms of a (co-)tetrad as follows.
[**(1)**]{} Let $\Lambda^I_J \in SO(1,3)$ i.e. $\Lambda^I_K \Lambda^J_L
\eta^{KL} = \eta^{IJ}$ holds where $\eta^{IJ} = \mathrm{diag}(-1, 1, 1,
1)$. Then the co-tetrad transforms as: $$(e')^{I}_{\mu}(x'(x)) ~ := ~ \Lambda^I_J(x) \frac{\partial
x'^{\nu}}{\partial x^{\mu}} e^J_{\nu}(x) \ .
$$ The $\Lambda-$ transformation is referred to as a [*Local Lorentz Transformation*]{} (LLT) while $x \to x'(x)$ is the [*General Coordinate Transformation*]{} (GCT). Clearly to have the derivatives of the co-tetrad to transform covariantly under both sets of transformations, we need two connections: an [*affine connection*]{} (not necessarily symmetric in the lower indices) and a [*Spin connection, $\omega_{\mu}\, ^{IJ}$*]{}. The spin connection is anti-symmetric in the $IJ$ indices and thus transforms as the [*adjoint representation*]{} of the pseudo-orthogonal group. The derivative covariant with respect to the LLT is denoted by $D_{\mu}$, that with respect to GCT is denoted by the usual $\nabla_{\mu}$ while the one with respect to both will be denoted by ${\cal D}_{\mu}$. The Lorentz indices will be raised/lowered using the [*Lorentz metric*]{} $\eta^{IJ}, \eta_{IJ}, \mathrm{where} ~
\eta^{IK}\eta_{KJ} = \delta^I_J$.
Armed with the tetrad, the spin connection and the Lorentz metric, [*define*]{} the following quantities: $
\begin{array}{|l|ccl|}
\hline
& & & \nonumber \\
\mathrm{Torsion:} & T^I(e, \omega) & := & d e^I + \omega^I\, _J \wedge
e^J \\
& T^I\, _{\mu\nu} & = &\partial_{\mu} e^I_{\nu} + \omega_{\mu}\, ^I\,_J
e^J_{\nu} - ({\mu \leftrightarrow \nu}) \nonumber \\
& & & \nonumber \\
\hline
& & & \nonumber \\
\mathrm{Curvature:} & R^{IJ}(\omega) & := & d \omega^{IJ} + \omega^I\,
_K \wedge \omega^{K J} \\
& R^{IJ}~_{\mu\nu} & = & \partial_{\mu}\omega_{\nu}\, ^{IJ} +
\omega_{\mu}\, ^{I}\,_{K} \omega_{\nu}\, ^{KJ} - ({\mu \leftrightarrow
\nu}) \nonumber \\
& & & \nonumber \\
\hline
& & & \nonumber \\
\mathrm{Bianchi~Identity:} & (D R)^I\,_J & = & 0 ~:=~ dR^I\,_J +
\omega^I\,_K\wedge R^K\,_J + \omega_J\,^K\wedge R^I\,_K \\
& & & \nonumber \\
\mathrm{Cyclic~Identity:} & (D T)^I & = & R^I\,_J\wedge e^J ~~:=~~d T^I
+ \omega^I\,_J\wedge T^J \\
& & & \nonumber \\
\hline \hline
& & & \nonumber \\
\mathrm{Metric:} & g_{\mu\nu}(e, \eta) & := & e_{\mu}^I e_{\nu}^J
\eta_{IJ} \\
& & & \nonumber \\
\hline
& & & \nonumber \\
\mathrm{Christoffel\,Connection:} & \left\{\begin{array}{c}
\lambda\\{\mu \nu}\end{array}\right\}(g(e)) & := & \frac{1}{2}
g^{\lambda\alpha}\left( g_{\alpha\mu, \nu} + g_{\alpha\nu, \mu} -
g_{\mu\nu, \alpha}\right) \\
& & & \nonumber \\
\hline
& & & \nonumber \\
\mathrm{Affine~connection:} & \Gamma^{\lambda}\, _{\mu\nu} (e, \omega) &
:= & \left\{\begin{array}{c} \lambda\\{\mu \nu}\end{array}\right\}(g(e))
+ \frac{1}{2} g^{\lambda\alpha}\left(T_{\alpha\mu\nu} - T_{\mu\nu\alpha}
- T_{\nu\mu\alpha}\right) \\
\hspace{2.2cm}\mathrm{where,} & {\bf T_{\alpha\mu\nu}(e, \omega)} & := &
{\bf e_{I\alpha} T^I\, _{\mu\nu}(e, \omega) }\nonumber \\
& & & \nonumber \\
\hline \hline
& & & \nonumber \\
\mathrm{These~imply} & & & \nonumber \\
& & & \nonumber \\
\mathrm{Matric~compatibility:} & \nabla_{\lambda} g_{\mu\nu} & = & 0 \\
\mathrm{tetrad~compatibility:} & {\cal D}_{\mu} e^I_{\nu} & = & 0 ~~ :=
~~ \nabla_{\mu} e^I_{\nu} + \omega_{\mu}\,^I\,_J e^J_{\nu} -
\Gamma^{\lambda}\,_{\mu\nu} e^I_{\lambda} \\
& & & \nonumber \\
\hline
& & & \nonumber \\
\mathrm{Compatibility} ~~ \Rightarrow &
\hspace{1.0cm}R^{\alpha}\,_{\lambda\mu\nu}(\Gamma) & = & e^{\alpha}_I
e_{\lambda\, J} R^{IJ}\,_{\mu\nu}(\omega) \\
& & & \nonumber \\
\hline
\end{array}
$ No assumption about the torsion tensor is made.
[**(2)**]{} It is possible to invert the torsion equation to ‘solve for’ the spin connection in terms of the tetrad, its derivatives and the torsion tensor. All one needs to do is manipulate the combination $T_{\lambda\mu\nu} + T_{\mu\nu\lambda} - T_{\nu\lambda\mu}$ and use the invertibility of the (co-)tetrad. The result is: $$\begin{aligned}
\omega_{\mu}\,^{IJ} & := & \hat{\omega}_{\mu}\,^{IJ}(e) +
K_{\mu}\,^{IJ}(e, T) \\
\hat{\omega}_{\mu}\,^{IJ} & := & \frac{1}{2}\left[ e^{\nu
I}\left(\partial_{\mu} e_{\nu}^J - \partial_{\nu} e_{\mu}^J\right)
- e^{\nu J}\left(\partial_{\mu} e_{\nu}^I - \partial_{\nu}
e_{\mu}^I\right)
- e^{\nu I} e^{\lambda J} \left(\partial_{\nu}e^K_{\lambda} -
\partial_{\lambda}e^K_{\nu} \right) e_{\mu K} \right] \\
K_{\mu}\,^{IJ} & := & - \frac{1}{2} e^{\nu I} e^{\lambda J} \left(
T_{\nu\lambda\mu} + T_{\lambda\mu\nu} - T_{\mu\nu\lambda} \right)
$$ The $K$ is called the [*con-torsion tensor*]{} and $\hat{\omega}$ is the [*torsion-free spin connection*]{} which is explicitly determined by the tetrad. The Affine connection equation is the corresponding inversion of the metric compatibility condition (covariant constancy of the metric) to express the general affine connection in terms of the torsion free Christoffel connection plus the torsion combinations.
Notice that a priori, we have two connections: the affine and the spin. Both define corresponding and independent torsions ($T^I\,_{\mu\nu}$ and the antisymmetric part of $\Gamma^{\lambda}\,_{\mu\nu}$). The introduction of the metric as the ‘square’ of the co-tetrad and the two compatibility conditions together identify the two torsions.
The last equation demonstrates that we can use the tetrad and the co-tetrad to convert the Lorentz indices and the general tensor indices into each other with the compatibility conditions ensuring the two distinct curvatures also going into each other.
[*Although we have referred to only 4 dimensions and Lorentz signature metric, the definitions generalise to any dimensions and any signature.*]{}
[**(3)**]{} Four dimensions have additional features available. One can define [*internal dual (Lorentz dual)*]{} for anti-symmetric rank-2 Lorentz tensors apart from the usual [*Hodge dual (space-time dual)*]{} for 2-forms.
Let, ${\cal E}^{IJKL}$ and ${\cal E}_{\mu\nu\alpha\beta}$ denote the Levi-Civita symbols; These are completely antisymmetric in their indices and we choose the conventions: ${\cal E}^{0123} = 1 = {\cal E}_{txyz}$. The indices on these are raised and lowered by the Lorentz and the space-time metric respectively. Using these we define: $$\begin{aligned}
\tilde{X}^{IJ} & := & \frac{1}{2} {\cal E}^{IJ}\,_{KL} X^{KL}
\hspace{1.0cm} (\mathrm{Internal~Dual}) \\
(*X)_{\mu\nu} & := & \frac{1}{2} {\cal E}_{\mu\nu}\,^{\alpha\beta}
X_{\alpha\beta} \hspace{1.3cm} (\mathrm{Hodge~Dual})
$$ [**(4)**]{} From the tetrad and the spin connection, the following Local Lorentz invariant four forms can be constructed whose integrals are candidate terms for an action.
1. [*Hilbert-Palatini:*]{} ${\cal L}_{HP}(e, \omega) ~:=~ \frac{1}{2}{\cal E}_{IJKL}
R^{IJ}(\omega)\wedge e^K\wedge e^L$.
The variational equations following from this are equivalent to the Einstein equations. The spin-connection equation implies that the torsion vanishes and the tetrad equation implies the vanishing of the Ricci tensor. The Hilbert-Palatini action is thus classically equivalent to the Einstein-Hilbert action of the metric formulation.
This term taken as an action with the tetrad and spin connection treated as independent variables is sometimes referred to as the tetrad formulation of gravity.
2. [*Cosmological Constant:*]{} ${\cal L}_{\Lambda}(e) ~ := ~
\frac{\Lambda}{4!}{\cal E}_{IJKL} e^I\wedge e^J\wedge e^K\wedge e^L$
This is the usual cosmological constant term, proportional to the volume form.
3. [*Euler Invariant:*]{} ${\cal L}_{E}(\omega) ~ := ~ \frac{1}{2} {\cal E}_{IJKL} R^{IJ} \wedge
R^{KL}$.
This 4-form is a [*topological term*]{} i.e. its variation under arbitrary infinitesimal changes in the spin connection, is an exact form and therefore the variation of its integral receives contributions only from the [*boundary values*]{}. Furthermore, explicitly, $${\cal L}_{E}(\omega) ~ = ~ - d\left\{ \frac{1}{2}{\cal E}^I\,_{JMN}
\omega^{MN}\wedge\left(d \omega^J\,_I + \frac{2}{3}
\omega^J\,_K\wedge\omega^K\,_I\right) \right\}
$$
4. [*Pontryagin Invariant:*]{} ${\cal L}_{P}(\omega) ~ := ~ R^{IJ} \wedge R^{IJ}$.
This 4-form is also a [*topological term*]{}. Furthermore, explicitly, $${\cal L}_{P}(\omega) ~ = ~ - d\left\{ \omega^I\,_J\wedge\left(d
\omega^J\,_I + \frac{2}{3} \omega^J\,_K\wedge\omega^K\,_I\right)
\right\}
$$ The terms enclosed within the braces is the [*Chern-Simmons*]{} 3-form.
5. [*Nieh-Yan Invariant:*]{} ${\cal L}_{NY}(e, \omega) ~ := ~ T^I\wedge T_I - R^{IJ} \wedge e^I\wedge
e^J$.
This 4-form is also a [*topological term*]{} which depends on both the tetrad and the spin connection. It vanishes if torsion is zero (for zero torsion, the second term vanishes by the cyclic identity.) Explicitly, $${\cal L}_{NY} ~ = ~ d\left\{ e_I\wedge T^I \right\}.
$$
Note that we have 5 different, Lorentz covariant 2-forms: $T^I,
\Sigma^{IJ} := e^I\wedge e^J, \tilde{\Sigma}^{IJ}, R^{IJ},
\tilde{R}^{IJ}$. From these, we can form the six Lorentz invariants: $T^2, \Sigma^2 (= 0 = - \tilde{\Sigma}^2), \Sigma\wedge\tilde{\Sigma},
R^2 (= - \tilde{R}^2), R\wedge\tilde{R}, R\wedge\Sigma,
R\wedge\tilde{\Sigma}$. If we are to get the Einstein equation (with a cosmological constant), then the $T^2$ and $R\wedge\Sigma$ must be combined into the Nieh-Yan combination,
[**(5)**]{} We will note a parametrization of the tetrad, adapted to the 3+1 decomposition, which leads to the corresponding metric decomposition. This can be derived from the identifications: $$\begin{aligned}
e^I_t e^J_t \eta_{IJ} & := & - N^2 + \bar{g}_{ab} N^a N^b \nonumber \\
e^I_t e^J_a \eta_{IJ} & := & \bar{g}_{ab} N^b \\
e^I_a e^J_b \eta_{IJ} & := & \bar{g}_{ab} \nonumber
$$ It follows,
$
\begin{array}{|ccl|l|}
\hline
& & & \\
\mathrm{Co-tetrad:} & & & ~~\mathrm{Introduces}~n_I \\
& & & \\
e^I_t & := & N n^I + N^a V^I_a ~~&~~ n^I n^J \eta_{IJ} := -1~~,~~ n^I V^J_a
\eta_{IJ} = 0 \\
& & & \\
e^I_a & := & V^I_a ~~~(\mathrm{free}) & \\
& & & \\
\bar{g}_{ab} & := & V^I_a V^J_b \eta_{IJ} ~~&~~ \mathrm{is~invertible;} \\
& & & \\
\hline
& & & \\
\mathrm{Tetrad:} & & & ~~\mathrm{Defines}~~V_I^a \\
& & & \\
e_I^t & := & - N^{-1} n_I ~~&~~ n^I V^a_I := 0 \\
& & & \\
e_I^a & := & N^{-1} n_I N^a + V_I^a & ~~ V^a_I V^I_b ~=~ \delta^a_b ~~,~~
V^a_I V^J_a ~=~ \delta_I^J + n_I n^J \\
& & & \\
\hline
\end{array}
$
In this parametrization, the 16 variables in the tetrad have been traded with $V_a^I (12), N, N^a (4)$ and $n^I (4)$ variables with 4 conditions: $n^2 = -1, n\cdot V_a = 0$. The conditions can be viewed as 4 conditions on $n^I$ given freely chosen $V^I_a$ [*or*]{} one condition on $n^I$ and 3 conditions on $V^I_a$ given freely chosen spatial vector $n^i$.
Notice that we have the [*normalized normal*]{}, $Nn_{\mu}$ defined by the foliation. From this we can construct an internal vector $\tilde{n}^I := e_I^{\mu}(Nn_{\mu})$. In the parametrization, we have also introduced an internal normalized time-like vector $n^I$, determined by the freely chosen $V^I_a$. These two are related by the parametrization of the tetrad as, $\tilde{n}^I = - n^I$.
We can view $n_{\mu}$ defined by the foliation and $n_I$ defined by a choice of $V^I_a$ as two time-like normalized vectors in the $T^*({\cal
M})$. These are not identical in general and in particular $n_I$ is not normal to the foliation. Demanding it to be so, puts a restriction on the $V^I_a$: $n_I \propto n_{\mu} ~\Rightarrow~ n_i = 0, ~ n_0n^0 = -1$ and $n\cdot V_a = 0 ~\Rightarrow V^0_a = 0$. This implies that $V^I_a$ are confined to $T^*(\Sigma)$. This choice is the so-called [*time gauge*]{}.
Finally, we reiterate that using the tetrad and co-tetrad we can freely convert the Lorentz and the general coordinate indices into each other. The normalized normal ($Nn_{\mu}$) can be used to define a [*projector*]{}, $P^{\mu}_{\nu} := \delta^{\mu}_{\nu} + N^2 n^{\mu} n_{\nu}$ which projects space-time tensors onto [*spatial tensors*]{}. This would lead to 3+1 decomposition (or parametrization) of all other tensorial quantities.
Symmetry Reduction
==================
There are different uses of the term ‘symmetry reduction’. Heuristically, if $S$ is a state space of a system, on which is specified an action of a group, $G$, which preserves the defining specification of the system (so that $G$ is its [*symmetry group*]{}), then the space $S$ gets “decomposed” into orbits of $G$. The space of orbits, $S/G$, is ‘smaller’ than $S$ and could constitute a simplification. $S/G$, is thought of as a [*symmetry reduction of $S$ by $G$*]{}. Alternatively, one could restrict to the subset of the so called [*invariant*]{} states which may be thought of as a collection of trivial orbits. In our context, we will be using the term in the latter sense. The system could be classical or quantum mechanical.
For example, if $S$ is the quantum mechanical state space of a particle with a rotationally invariant Hamiltonian, then the subspace of the invariant states would be all the states with zero angular momentum. If it is the phase space of a particle with a rotationally invariant dynamics, then the only invariant ‘state’ is the origin of the configuration space with zero momentum. If however, $S$ denotes the space of field configurations on a manifold, then the subset of invariant configurations is non-trivial. If the quantum mechanical state space of a system consists of [*distributions*]{} on a space of ‘test functions’, then invariant states could be defined as those distributions whose support consists of invariant test functions.
If one obtains a reduction by restricting to invariant states (and invariant observables) of a quantum system, one has followed the [*first quantize, then reduce*]{} route and the reduced system can be thought of as a [*symmetric sector*]{}. This is not always possible, since one does not have adequate explicit control over the quantum system. Alternatively, one can consider invariant subspace of a classical phase space and construct a corresponding quantum theory. This is the [*first reduce, then quantize*]{} route. In general, the relation between these two approaches is unclear.
While the former approach is more desirable, in practice, it is the latter approach which is followed commonly. We will also follow this approach. However, we will follow the methods – basic variables, construction of quantum Hilbert space etc – used in the full theory. The viability of these simplified models are then thought to constitute a test of the methods and premises of the full theory. The reduction of the classical theory is carried out by requiring certain [*symmetries*]{} to be exactly realized.
Symmetry Reduced Models
-----------------------
We are already familiar with use of symmetries to simplify a problem. For example, assuming spherical symmetry we choose coordinates and metric components to simplify the Einstein equation and obtain the Schwarzschild solution or using homogeneity and isotropy one obtains the FRW solutions. Thus symmetry groups (isometries) allow us to classify suitable ansatz for the basic variables of the theory. Note however that we are not interested in solving Einstein equations, but rather in obtaining a classical action with fewer degrees of freedom and constructing a corresponding quantum theory. In the context of spherical symmetry for example, this corresponds to restricting to only spherically symmetric form of 3-metrics: $ds^2 = \Lambda^2(t,r)dr^2 +
R^2(t,r)( d\theta^2 + \mathrm{sin}^2\theta d\phi^2)$ and reducing the Einstein-Hilbert action to get an action in terms of the two field degrees of freedoms – $\Lambda(r), R(r)$. Such reductions of degrees of freedom is termed [*mini-superspace*]{} model if the degrees of freedom is [*finite*]{} and a [*midi-superspace*]{} model, if the degrees of freedom is still infinite i.e. a lower dimensional field theory. The former occur in [*homogeneous cosmologies*]{} while examples of the latter include spherical symmetry, certain inhomogeneous cosmological models such as the Gowdy models, Einstein-Rosen waves etc. Needless to say that the midi-superspace models are still very complicated. We will concentrate on the mini-superspace models and specifically on (spatially) homogeneous cosmologies. We begin by defining spatially homogeneous space-times which are not necessarily solutions of Einstein equation.
### Spatially homogeneous models
A four dimensional space-time is said to be spatially homogeneous if (a) it can be foliated by a 1-parameter family of space-like hypersurfaces, $\Sigma_t$ and (b) possessing a (Lie) group of isometries such that for each $t$ and any two points $p, q \in \Sigma_t$ there exist an isometry of the space-time metric which maps $p$ to $q$. The isometry group $G$ is then said to act [*transitively*]{} on each of the $\Sigma_t$. If the group element connecting $p, q$ is unique, the group action is said to be [*simply transitive*]{} (otherwise multiply transitive). Spatially homogeneous space-times are further divided into two types.
A spatially homogeneous space-time is said to be of a [**Bianchi type**]{} if the group of isometries contains a subgroup (possibly itself), $G^*$, which acts simply transitively on $\Sigma_t$ otherwise it is said to be of the [**Kantowski-Sachs type**]{} (interior of Schwarzschild solution). It turns out that except for the special case of $\Sigma \sim S^2 \times
\mathbb{R}$ and $G = SO(3) \times \mathbb{R}$, in all other cases one has a Bianchi type space-time.
Transitive action implies that there must be at least three independent Killing vectors at each point of $\Sigma_t$ since $\Sigma_t$ is three dimensional. But there could be additional Killing vectors which vanish at a point. These Killing vectors generate the [*isotropy*]{} (or stability) subgroup, $H$ of $G$. Since $H$ will induce a transformation on the tangent spaces to the spatial slices, it must be a subgroup of $SO(3)$ and thus dimension of $G$ can be at most 6 and at least 3 since the dimension of $G^*$ is always 3. All 3 dimensional Lie groups have been classified by Bianchi into 9 types. The classification goes along the following lines[@LandauLifshitz].
A Lie algebra (or connected component of a Lie group) is characterised by structure constants $C^I_{~JK}$ with respect to a basis $X_I$, satisfying the antisymmetry and Jacobi identity namely, $$\left[X_J, X_K\right] ~ = ~ C^I_{~JK} X_I~~;~~ C^I_{~JK} = C^I_{~KJ}
~~;~~ \sum_{(IJK)} C^N_{~IL}C^L_{~JK} = 0 ~~,~~ I, J, K = 1, 2, 3\ .
$$ Using the availability of the Levi-Civita symbols, ${\cal E}_{IJK}, \
{\cal E}^{IJK}, ~ {\cal E}_{123} = 1 = {\cal E}^{123}$, we can write the structure constants as, $$\begin{aligned}
C^I_{~JK} & = & {\cal E}_{JKL}C^{LI} ~~,~~C^{IJ} := M^{IJ} + {\cal
E}^{IJK}A_K
$$ Thus, the 9 structure constants are traded for 6 $M^{IJ}$ (symmetric in $IJ$) and the 3 $A_K$. This has used only antisymmetry. The Jacobi identity implies, $M^{IJ}A_J = 0$.
Noting that the structure constants are subject to linear transformations induced by linear transformations, $X_I \to
S_I^{~J}X_J$, on the basis of the Lie algebra, the symmetric $M^{IJ}$ can be diagonalized by orthogonal transformations and the non-zero eigenvalues can be further scaled to $\pm 1$: $M^{IJ} = n^I\delta^{IJ}$. The condition $M^{IJ}A_J = 0$ implies that [*either*]{} $A_I = 0$ ([**Class A**]{}) [*or*]{} $A_I \neq 0$ ([**class B**]{}) in which case $M^{IJ}$ has a zero eigenvalue and we may take the non-zero eigenvector $A_I$ to be along the “1st” axis, i.e. $A_I = a\delta_{I, 1}$ and $n^1 = 0$. This leads to, $$\left[X_J, X_K\right] ~ = ~ n^I {\cal E}_{IJK} X_I + X_J A_K - X_K A_J
~.
$$ In the class A, there are precisely 6 possibilities organized by the [*rank of the matrix*]{} – 0, 1, 2, 3 and [*signature*]{} for ranks 2, 3 viz $(++, +-)$ and $(+++, ++-)$. The eigenvalues of $M^{IJ}$ can be taken to be $n^I = \pm 1, 0$.
In the class B, the rank of $M^{IJ}$ cannot be 3 and the possibilities are restricted to the ranks 0, 1, 2 and signatures $(++, +-)$ for rank 2. If the rank of $M$ is 0, all three eigenvalues are zero and scaling $X_1$, we can arrange $a = 1$. For rank 1, taking $n_3$ to be the non-zero eigenvalue, scaling $X_1, X_3$ ensures $a = 1$. For rank 2 however, ($n_2 = \pm 1, n_3 = \pm 1$), no scaling can preserve $n_2,
n_3$ and set $a = 1$ (of course $a = 1$ is possible).
Here is a table of the classification of Riemannian, homogeneous 3-geometries[@LandauLifshitz]:
Type a $n_1$ $n_2$ $n_3$ Remarks
----------------- ---- ------- ------- ------- -----------------------------------------
[**Class A**]{}
I 0 0 0 0 Euclidean space
(Leads to the Kasner space-time)
II 0 1 0 0
VII$_0$ 0 1 1 0
VI$_0$ 0 1 -1 0
IX 0 1 1 1 $S^3$ is a special case (with isotropy)
(Central to BKL Scenario)
VIII 0 1 1 -1
[**Class B**]{}
V 1 0 0 0 $H^3$ a special case (with isotropy)
IV 1 0 0 1
VII$_a$ a 0 1 1
III 1 0 1 -1 sub-case of type VI$_a$
VI$_a$ a 0 1 -1
Of interests to us are the so called [*class A*]{} models which are characterised by the structure constants satisfying $C^I\,_{I J} = 2 A_J
= 0$ [^7].
When $H = SO(3)$, one has homogeneity [*and*]{} isotropy i.e. FRW space-times. We know that these come in three varieties depending on the constant spatial curvature. The spatially flat case is of type Bianchi I while positively curved case is of type Bianchi IX. (The negatively curved case is in class B, type V).
The metrics of the general Bianchi type space-times have at the most 6 degrees of freedom thus constituting mini-superspaces. The spatial metrics [*can be put*]{} in the form: $ds^2 = g_{IJ}(t) e^I_i e^J_j dx^i
dx^j$, where $e^I_i dx^i$ are the so called Maurer-Cartan forms on the group manifold $G^*$, satisfying $d e^I = - {{\textstyle \frac{1}{2}}} C^I _{JK} e^J
\wedge e^K$. When one further restricts to [*diagonal*]{} $g_{IJ}$ one gets the so-called [*diagonal Bianchi models*]{}.
[*Remark:*]{} One should notice that restricting to a subclass of metrics amounts to introducing [*background structures*]{} from the perspective of the full theory. In the present case, these structures are the symmetry group and the coordinates adapted to the group action (which allowed the metric to be put in the specific form). This is unavoidable and constitutes a specification of the reduced model. From the perspective of a reduced model, these structures are [*non-dynamical, analogous to the manifold structure for the full theory*]{} and therefore do not automatically violate background independence. Instead, the background independence now means that quantization procedure should not depend the metric $g_{IJ}$ which is a dynamical variable.
Our basic variables however are not the 3-metric and the extrinsic curvatures. They are the $SU(2)$ connection and the densitized triad. In the metric variables, the natural notion of symmetry is isometry while in the connection formulation it is the [*group of automorphisms*]{} of the $SU(2)$ bundle. Thus, the cosmological models will now be understood to be characterised by groups of automorphisms of the $SU(2)$ bundle which acts on the base manifold $\Sigma$ transitively. The task is to characterise the connection and triad variables which are [*invariant*]{} under the group action (just as isometries mean invariant metrics). This requires more mathematical machinery and we will only state the conclusions[^8].
For the Bianchi models, the invariant connections and densitized triad are of the form: $$A_a^i(t, x) ~ := ~ \Phi^i_I(t) {\omega}^I_a(x)~ ~ , ~ ~ E^a_i(t,x) ~ :=
~ \sqrt{g_0}(x) p_i^I(t) X^a_I(x).
$$ In the above equation, $a$ refers to spatial coordinate index, $i$ refers to the adjoint representation of $SU(2)$ and $I$ refers to the adjoint index of the Lie algebra of the symmetry (sub) group $G^*$ (and hence takes 3 values). The $\omega^I_a dx^a$ are the Maurer-Cartan 1-forms (left-invariant 1-forms) on $\Sigma_t$ identified with the group manifold while $X_I^a{{\textstyle \frac{\partial}{\partial x^a}}}$ are the corresponding invariant vector fields dual to the 1-forms, i.e. $\omega^I(X_J) = \omega^I_a X_J^a = \delta^I_J$. The $g_0(x)$ is the determinant of the invariant metric on the symmetry group and provides the necessary density weight. It is regarded as a fiducial quantity and will drop out later. All the coordinate dependence resides in these forms, vector fields and the fiducial metric while the coefficients containing the $t$ dependence are the basic dynamical variables[^9].
If we have isotropy in addition, then the degrees of freedom are further reduced: $\Phi^i_I := c\Lambda^i_I, \ p_i^I := p \Lambda^I_i $ and there is only one degree of freedom left. Here the $\Lambda$’s are a set of orthonormal vectors satisfying, $\Lambda_I^i \Lambda_J^i = \delta_{IJ},
\Lambda_I^i \Lambda^j_J \Lambda^k_K \epsilon_{ijk} = \epsilon_{IJK}$. The phase space variables $c, p$ are gauge invariant.
The intermediate case of [*diagonal models*]{} arises from a [*choice*]{} $\Phi^i_I := c_I \Lambda^i_I, p_i^I := p^I \Lambda_i^I$ (no sum over I). The residual (SU(2)) gauge transformations act on the $\Lambda$’s and leaving the $c_I, p^I$ as the [*gauge invariant*]{} phase space variables thereby solving the Gauss constraint at the outset[^10]. Thus there are only 3 degrees of freedom [@MartinHomogeneous].
Having identified relevant degrees of freedom parameterising quantities invariant under symmetry transformation, the next task is to obtain the symplectic structure (basic Poisson brackets) and simplify the expressions for the constraints.
[*Symplectic form:*]{} In the full theory, this is given by $(8\pi G
\gamma)^{-1}\int_{\Sigma} d^3x \dot{A}^i_a(t,x)E_i^a(t,x)$. Direct substitution gives, $$\frac{1}{\kappa\gamma}\int_{\Sigma} d^3x \dot{A}^i_a(t,x)E_i^a(t,x) ~ =
~ \frac{1}{\kappa\gamma}\dot{\Phi}^i_I p_i^I\ \left\{\int_{\Sigma} d^3x
\sqrt{g_0}\right\} \ ,
~ \Rightarrow ~ \{\Phi^i_I, p^J_j\} = \frac{\kappa \gamma}{V_0}
\delta^i_j\delta^J_I\ .$$ The quantity in the braces is the [*fiducial*]{} volume, $V_0$, of $\Sigma_t$. For spatially flat, isotropic case, the slice is non-compact and the fiducial volume is infinite. This problem is addressed by restricting to a finite cell whose fiducial volume is finite. One has to ensure that the final results do [*not*]{} depend on the fiducial cell[^11]. The dependence on the fiducial volume is gotten rid off by redefining the basic variables as $\Phi \to \Phi
V_0^{-1/3}, \ p \to p V_0^{-2/3}$. If we have isotropy, the symplectic form would become ${{\textstyle \frac{3}{\kappa\gamma}}}V_0 \dot{c}p$ which leads to (after rescaling) to the Poisson bracket, $\{c, p\} =
{{\textstyle \frac{\kappa\gamma}{3}}}$. With this rescaling understood, we will now effectively put $V_0 = 1$.
[*Curvature:*]{} The curvature corresponding to the invariant connection above, is obtained as: $$\begin{aligned}
F^i & := & d A^i + \frac{1}{2} \epsilon^i\,_{jk} A^j \wedge A^k ~~:=~~
\frac{1}{2}F^i_{JK}\omega^J\wedge\omega^K \\
\therefore ~~ F^i_{JK} & = & -\Phi^i_I C^I\,_{JK} + \epsilon^i\,_{jk}
\Phi^j_J\Phi^k_K
$$
[*Gauss Constraint:*]{} The full theory expression is: $$\begin{aligned}
G(\Lambda) & := & \int_{\Sigma}
\Lambda^i\left\{\frac{1}{\kappa\gamma}\left(\partial_a E^a_i +
\epsilon_{ij}\,^k A^j_a E^a_k\right)\right\} \nonumber \\
& = & \frac{\Lambda^i}{\kappa\gamma}\left[ p^I_i
\underbrace{\int_{\Sigma} \partial_a(\sqrt{g_0} X^a_I)} ~ + ~
\epsilon_{ij}\,^k p^I_k \Phi^i_J\underbrace{\int_{\Sigma} \sqrt{g_0}
X^a_I \omega^J_a}\right] \nonumber \\
& & \hspace{1.7cm} - V_0 C^J\,_{IJ} \hspace{4.0cm} V_0 \\
\label{GaussOne}
\therefore ~~~ G_i & = & (\kappa\gamma)^{-1}\left\{-p^I_iC^J_{~IJ} +
\epsilon_{ij}^{~k}\Phi^j_Ip^I_k\right\} \ .
$$
Notice that for the class A models, the first term is zero and for the [*diagonal*]{} models the second term vanishes as well (since $\epsilon$ is antisymmetric in $j, k$ while the $\Lambda$ factors are symmetric in $j, k$). There are no continuous gauge invariances left. Note that the first term in eqn (\[GaussOne\]), is a surface term which [*could*]{} vanish if $\Sigma$ has no boundaries. But this would not be true for say spatially flat models which will have the $\Sigma$ as a cell on which the invariant vector fields need not vanish. The integrand however is proportional to $C^J_{~IJ}$ and these vanish for the class A models.
[*Diffeo Constraint:*]{} $$\begin{aligned}
C_{\mathrm{diff}}(\vec{N}) & := & \frac{1}{\kappa\gamma}\int_{\Sigma}
N^a(x) E^b_i(x) F^i_{ab}(x) - \int_{\Sigma} N^a(x) A^i_a(x) G_i(x)
\nonumber \\
N^a(t,x) & := & N^I(t)X^a_I(x) \\
\therefore~~N^I C_I & = &
\frac{N^I}{\kappa\gamma}\left[\left(C^K\,_{JK}\Phi^i_I +
C^K\,_{IJ}\Phi^i_K\right)p^J_i\right]
$$ This constraint again vanishes for [*diagonal*]{}, class A models.
[*Hamiltonian Constraint:*]{} The full theory Hamiltonian constraint is given by, $$C_{\mathrm{Ham}}(N) ~ := ~ \frac{1}{2\kappa}\int_{\Sigma} N
\frac{E^a_iE^b_j}{\sqrt|\mathrm{det}q|}\left[\epsilon^{ij}_{\,k}
F^k_{ab} - 2(1 + \gamma^2) K^i_{[a} K^j_{b]}\right]
$$ To carry out the integration, we need to note the expressions: $$\sqrt{g_0} ~ = ~ \frac{1}{3!}\epsilon_{IJK}\epsilon^{abc}
\omega_a^I\omega_b^J\omega_c^K ~~,~~ \frac{1}{\sqrt{g_0}} ~ =~
\epsilon_{abc}\epsilon^{IJK}X^a_I X^b_J X^c_K .
$$ This leads to, $$\mathrm{det} q ~ = ~ \mathrm{det}(E^a_i) ~ := ~ \frac{1}{3!}
\epsilon_{abc}\epsilon^{ijk} (g_0)^{3/2} X^a_I X^b_J X^c_K p^I_i p^J_j
p^K_k ~~=~~
\frac{1}{3!} g_0 \epsilon^{ijk}\epsilon_{IJK} p^I_i p^J_j p^K_k
$$ Now the integration can be carried out immediately to give, $$\begin{aligned}
H_{\mathrm{grav}} & = &
\frac{N}{2\kappa}\left[\frac{p^I_ip^J_j}{\sqrt{\frac{1}{6}|
\epsilon^{ijk}\epsilon_{IJK} p^I_i p^J_j p^K_k|}}
\left\{\epsilon^{ij}\,k F^k\,_{IJ} - 2 (1 + \gamma^2) K^{[i}_I
K^{j]}_J\right\} \right]
$$ In the above, $K^i_I = \gamma^{-1}( \Phi^i_I - \Gamma^i_I)$. These expressions are valid for general Bianchi models.
At this stage, we could in principle attempt to carry out the usual Schrodinger quantization with $\Phi^i_I$ being multiplicative operators and $P^I_i$ being the derivative operators. Both transform covariantly under the action of SU(2).
However, we can also imagine ‘specializing the holonomy-flux variables’ of the full theory, for these symmetric fields. It is natural to choose edges along the symmetry directions i.e. along integral curves of the $X^a_I$ vector fields. It follows that due to homogeneity, the path ordered exponentials, holonomies, become just the [*ordinary exponentials*]{}, $h_I(\Phi) := h_{e_I}(\Phi) := {\cal
P}\mathrm{exp}\{\int_{e_I}\Phi^i_I\tau_i\omega^I_a dx^a\} =
\mathrm{exp}\{\Phi_I^i(t)\tau_i\int_{e_I}\omega_a^Idx^a$}. There is no sum over $I$ in these expressions. These can be further expressed using the identity $e^{i\theta\hat{n}\cdot\vec{\sigma}} = \mathrm{cos}(\theta)
+ i \hat{n}\cdot\vec{\sigma}\mathrm{sin}\theta$. The holonomy is then given in terms of $\theta \sim \sqrt{\Phi^i_I\Phi^i_I}$ which is gauge invariant and two angular, gauge variant components corresponding to the direction $\hat{n} \sim $ unit vector in the direction of $\Phi^i_I$. A simplification occurs if we further restrict to the diagonal models: $\Phi^i_I := c_I\Lambda^i_I$ which makes the $\hat{n} = \vec{\Lambda}_I$ and now the matrix elements of these holonomies can be obtained from the elementary functions, $e^{\mu_{(I)}c_I/2}$. These have been termed as the [*point holonomies*]{}. The fluxes through surfaces perpendicular to the symmetry directions, likewise simplify to $E_{S_{JK}}(f) = p^I
\Lambda_i^I f^i\int_{S_{JK}} \sqrt{g_0(x)} \epsilon_{abc} X^a_IdS^{bc}
\propto p^I$. Unlike the flux operators in the full theory, these fluxes Poisson commute among themselves. Thus, in the diagonal models, we can extract gauge invariant phase space coordinates, with the holonomies and fluxes having the usual Poisson algebra. In quantum theory, a useful triad representation can then be set-up.
[*Point holonomies and commuting flux variables are new features*]{} which arise in the (diagonal) mini-superspace reduction. These are also responsible for the relative ease of analysis possible for these models. This will be discussed more below.
What about inhomogeneous models? There have fewer efforts regarding these. Among the inhomogeneous models, the reduction for the Gowdy model on 3-torus can be seen in [@GowdyClassical], while spherical symmetric model can be seen in [@Spherical]. Martin’s lattice model is briefly discussed in the appendix.
Singularity Resolution in Quantum Theory
========================================
The most detailed analysis of the singularity resolution is available for the homogeneous and isotropic geometry coupled to a massless scalar and this is the case that we discuss below. Prior to 2005, the singularity resolution was understood as the deterministic nature of fundamental equation (the Hamiltonian constraint) and in terms of an effective picture deduced either from the WKB approximation of by taking expectation values of the Hamiltonian. In this sense, resolution of singularities was seen for (i) FRW coupled to a scalar field with arbitrary positive semidefinite potential and (ii) diagonalised Bianchi class A (anisotropic) models. These resolutions were seen as an implication of the [*inverse triad quantum corrections*]{} which were present in the matter sector (and in the curvature for non-flat models). Post 2005, it was realized, at least for the FRW case, that [*the holonomy corrections by themselves could also resolve singularities*]{}. This required restriction to massless scalar and treating it as a clock variable, thereby paving the way for construction of physical states, Dirac observables and physical expectation values. Although restricted to special matter, it allows completion of the quantization program to the physical level and throws light on [*how*]{} a quantum singularity resolution may be viewed. For this reason, we this case is discussed in detail. Subsequently, Madhavan also showed another quantization for the same case, also completed to physical level, wherein holonomy corrections are absent and singularity resolution is achieved by inverse triad corrections only. There are also some issues which have been better understood in the past few years. These are briefly summarised and discussed in sections \[CellIndependence\] and \[LatticeView\] .
FRW, Classical Theory {#ClassicalFRW}
---------------------
[*Classical model:*]{} Using coordinates adapted to the spatially homogeneous slicing of the space-time, the metric and the extrinsic curvature are given by, $$\label{FRWMetric}
ds^2 := - dt^2 + a^2(t)\left\{dr^2 + r^2d\Omega^2\right\} ~~:= ~~ -dt^2
+ a^2(t) ds^2_{\mathrm{comoving}}\ .
$$ Starting from the usual Einstein-Hilbert action and scalar matter for definiteness, one can get to the Hamiltonian as, $$\begin{aligned}
S & := & \int dt \int_{\mathrm{cell}} dx^3 \sqrt{|det
g_{\mu\nu}|}\left\{ \frac{R(g)}{16 \pi G} + \frac{1}{2}\dot{\phi}^2 -
V(\phi) \right\} \nonumber\\
& = & V_0\int dt \left\{\frac{3}{8 \pi G}(- a\dot{a}^2) + \frac{1}{2}a^3
\dot{\phi}^2 - V(\phi) a^3 \right\} \\
p_a & = & - \frac{3 V_0}{4 \pi G} a \dot{a} ~ ~ , ~ ~ p_{\phi} ~ = ~ V_0
a^3 \dot{\phi} ~ ~,~ ~ V_0 ~ := ~ \int_{\mathrm{cell}}
d^3x\sqrt{g_{\mathrm{comoving}}} ~ ; \nonumber\\
H (a, p_a, \phi, p_{\phi}) & = & H_{\mathrm{grav}} + H_{\mathrm{matter}}
\nonumber \\
& = & \left[- \frac{2 \pi G}{3} \frac{p_a^2}{V_0 a} \right] +
\left[\frac{1}{2} \frac{p_{\phi}^2}{a^3 V_0} + a^3 V_0 V(\phi)\right] \\
& = & \left(\frac{3 V_0 a^3}{8 \pi G}\right)\left[ -
\frac{\dot{a}^2}{a^2} + \left(\frac{8 \pi G}{3}\right)
\left(\frac{H_{\mathrm{matter}}}{ V_0 a^3}\right)\right]
$$ Thus, $H = 0 \leftrightarrow $ Friedmann Equation. For the spatially flat model, one has to choose a fiducial cell whose fiducial volume is denoted by $V_0$.
In the connection formulation, instead of the metric one uses the densitized triad i.e. instead of the scale factor $a$ one has $\tilde{p}, |\tilde{p}| := a^2$ while the connection variable is related to the extrinsic curvature as: $\tilde{c} := \gamma \dot{a}$ (the spin connections is absent for the flat model). Their Poisson bracket is given by $\{\tilde{c}, \tilde{p}\} = (8\pi G \gamma)/(3
V_0)$. The arbitrary fiducial volume can be absorbed away by defining $c
:= V_0^{1/3} \tilde{c}, ~ p := V_0^{2/3}\tilde{p}$. Here, $\gamma$ is the Barbero-Immirzi parameter which is dimensionless and is determined from the Black hole entropy computations to be approximately $0.23$ [@BHEntropy]. From now on we put $8\pi G := \kappa$. The classical Hamiltonian is then given by, $$\label{ClassHam}
H ~ = ~ \left[- \frac{3}{\kappa}\left( \gamma^{-2}c^2 \sqrt{|p|}\right)
\right] + \left[\frac{1}{2}|p|^{-3/2} p_{\phi}^2 + |p|^{3/2}
V(\phi)\right] \ .
$$ For future comparison, we now take the potential for the scalar field, $V(\phi)$ to be zero as well.
One can obtain the Hamilton’s equations of motion and solve them easily. On the constrained surface ($H = 0$), eliminating $c$ in favour of $p$ and $ p_{\phi}$, one has, $$\begin{aligned}
c ~ = ~ \pm \gamma \sqrt{\frac{\kappa}{6}} \frac{|p_{\phi}|}{|p|} ~ & ,
& ~ \dot{p} ~ = ~ \pm 2 \sqrt{\frac{\kappa}{6}} |p_{\phi}||p|^{-1/2} \ .
\nonumber \\
\dot{\phi} ~ = ~ p_{\phi} |p|^{-3/2}~ & , & ~ \dot{p_{\phi}} ~ = ~ 0\ ,
\\
\frac{d p}{d \phi} ~ = ~ \pm \sqrt{\frac{2\kappa}{3}} |p| ~ &
\Rightarrow & ~
{\bf p(\phi) ~ = ~ p_* e^{\pm \sqrt{\frac{2\kappa}{3}}( \phi - \phi_*)}}
\label{ClassRelationalSoln}
$$ Since $\phi$ is a monotonic function of the synchronous time $t$, it can be taken as a new “time” variable. The solution is determined by $p(\phi)$ which is (i) independent of the constant $p_{\phi}$ and (ii) passes through $p = 0$ as $\phi \to \pm \infty$ (expanding/contracting solutions). It is immediate that, along these curves, $p(\phi)$, the energy density ($p^{-6}p_{\phi}^2/2$) and the extrinsic curvature diverge as $p \to 0$. Furthermore, the divergence of the density implies that $\phi(t)$ is [*incomplete*]{} i.e. $t$ ranges over a semi-infinite interval as $\phi$ ranges over the full real line[^12].
Thus a singularity is signalled by [*every*]{} solution $p(\phi)$ passing through $p = 0$ in [*finite*]{} synchronous time (or equivalently by the density diverging somewhere along any solution). A natural way to ensure that [*all*]{} solutions are non-singular is to ensure that either of the two terms in the Hamiltonian constraint is [*bounded*]{}. Question is: [*If and how does a quantum theory replace the Big Bang singularity by something non-singular?*]{}
There are at least two ways to explore this question. One can imagine computing corrections to the Hamiltonian constraint such that individual terms in the effective constraint are bounded. This approach presupposes the classical framework and thus will have a [*domain of validity of these corrections*]{}. Alternatively and more satisfactorily, one should be able to define suitable Dirac observables whose expectation values will generate the analogue of $p(\phi)$ curves along which physical quantities such as energy density, remain bounded. Both are discussed below.
FRW, Quantum Theory
-------------------
[**Schrodinger Quantization:**]{} In the standard Schrodinger quantization, one can introduce wave functions of $p, \phi$ and quantize the Hamiltonian operator by $c \to i\hbar \kappa\gamma/3 \partial_p ~,~
p_{\phi} \to -i \hbar \partial_{\phi}$, in equation (\[ClassHam\]). With a choice of operator ordering, $\hat{H}\Psi(p, \phi) = 0$ leads to the Wheeler-De Witt partial differential equation which has singular coefficients. We will return to this later.
[**Loop Quantization:**]{} The background independent quantization of Loop Quantum Gravity however suggest a different quantization of the isotropic model. One should look for a Hilbert space on which only exponentials of $c$ (holonomies of the connection) are well defined operators and not $\hat{c}$. Such a Hilbert space is obtained as the representation space of the C\* algebra of holonomies. In the present context this algebra is the algebra of [*almost periodic functions of $c$*]{}, finite linear combinations of functions of the form $e^{i\lambda_j
c}, \lambda_j \in \mathbb{R}$. Inner product (analogue of the Ashtekar-Lewandowski measure) on the space of the almost periodic functions is given by: $$(\Psi, \Phi) ~ := ~ \lim_{T \to \infty} ~ \frac{1}{2T} ~ \int_{-T}^{T}
dc ~ \Psi^{*}(c) \Phi(c) \ .
$$ The single exponentials form an orthonormal set. Let us denote it as, $\langle c|\mu\rangle := $exp$\{ {{\textstyle \frac{i}{2}}} \mu c \}, \mu \in
\mathbb{R}$. The holonomy-flux representation can now be made explicit as: $$\begin{aligned}
\label{HolonomyFluxRepren}
\hat{p}|\mu\rangle & = &
\frac{1}{6}\gamma{\ell_{\mathrm P}}^2\mu|\mu\rangle~~,~~\langle\mu|\mu'\rangle ~=~
\delta_{\mu, \mu'}~~,~~\mu \in \mathbb{R} \nonumber \\
\widehat{h_{\nu}}|\mu\rangle & := & \widehat{e^{{{\textstyle \frac{i}{2}}}\nu
c}}|\mu\rangle ~ = ~ |\mu + \nu\rangle
$$ Notice that that the triad operator has every real number as a [*proper eigenvalue*]{} (i.e. has a corresponding [*normalizable*]{} eigenvector, the spectrum is [*discrete*]{}). This implies that the holonomy operator, is [*not weakly continuous*]{} in the label $\nu$ i.e. arbitrary matrix elements of $\hat{h}_{\nu}$ are not continuous functions of $\nu$. Therefore one [*cannot*]{} define a $\hat{c}$ operator. Note that the [*volume operator*]{}, is given by $\hat{V} :=
|p|^{3/2}$.
[**Inverse Triad Operator:**]{} The fact that spectrum of the triad operator is [*discrete*]{}, has a major implication: [*inverses of positive powers of triad operators do not exist*]{} [^13]. These have to be defined by using alternative classical expressions and promoting them to quantum operators. This can be done with at least one parameter worth of freedom, eg. $$|p|^{-1} ~ = ~ \left[ \frac{3}{\kappa \gamma l}\{c, |p|^l\}\right]^{1/(1
-l)} ~,~ l \in (0, 1)\ .
$$ Only positive powers of $|p|$ appear now. However, this still cannot be used for quantization since there is no $\hat{c}$ operator. One must use holonomies: $h_j(c) ~ := ~ e^{\mu_0 c \Lambda^i\tau_i}\ ,$ where $\tau_i$ are anti-hermitian generators of $SU(2)$ in the $j^{th}$ representation satisfying $\mathrm{Tr}_j (\tau_i \tau_j) = - {{\textstyle \frac{1}{3}}}
j (j + 1) (2j + 1) \delta_{ij}$, $\Lambda^i$ is a unit vector specifying a direction in the Lie algebra of $SU(2)$ and $\mu_0$ is the coordinate length of the edge used in defining the holonomy. It is a fraction of $V_0^{1/3}$. Using the holonomies, $$\begin{aligned}
|p|^{-1} & = & (8 \pi G \mu_0\gamma l)^{\frac{1}{l - 1}} \left[
\frac{3}{j(j + 1)(2j + 1)} \mathrm{Tr}_j \Lambda\cdot\tau \
h_j\left\{h_j^{-1}, |p|^l\right\}\right]^{\frac{1}{1 - l}} \ ,
$$ which can be promoted to an operator. Two parameters, $\mu_0 \in
\mathbb{R}$ and $j \in \mathbb{N}/2$, have crept in and we have a three parameter family of inverse triad operators. The definitions are: $$\begin{aligned}
\widehat{|p|^{-1}_{(jl)}} |\mu\rangle & = &
\left(\frac{2j\mu_0}{6}\gamma{\ell_{\mathrm P}}^2\right)^{-1} (F_{l}(q))^{\frac{1}{1
-l}} |\mu\rangle ~ ~ , ~ ~ q := \frac{\mu}{2\mu_0j} ~ := ~
\frac{p}{2jp_0}~ ~, \label{InvTriad}\\
F_l(q) & := & \frac{3}{2l}\left[ ~ ~ \frac{1}{l + 2}\left\{ (q + 1)^{l +
2} - |q - 1|^{l + 2}\right\} \right. \nonumber \\ & & \hspace{0.7cm}
\left. - \frac{1}{l + 1} q \left\{ (q + 1)^{l + 1} - \mathrm{sgn}(q -1)
|q - 1|^{l + 1}\right\} ~ ~ \right] \nonumber \\
F_l( q \gg 1 ) & \approx & \left[q^{-1}\right]^{1 - l} \ ,
\label{Ffunction} \\
F_l( q \approx 0 ) & \approx & \left[\frac{3q}{l + 1}\right] \ .
\nonumber
$$ All these operators obviously commute with $\hat{p}$ and their eigenvalues are bounded above. This implies that the matter densities (and also intrinsic curvatures for more general homogeneous models), remain bounded over the classically singular region. Most of the phenomenological novelties are consequences of this particular feature predominantly anchored in the matter sector. In the effective Hamiltonian computations, this modification will imply the second term in the Hamiltonian constraint (\[ClassHam\]) is rendered bounded implying singularity avoidance.
We have also introduced two scales: $p_0 := {{\textstyle \frac{1}{6}}}\mu_0{\ell_{\mathrm P}}^2$ and $2jp_0 := {{\textstyle \frac{1}{6}}}\mu_0 (2j) {\ell_{\mathrm P}}^2$. The regime $|p| \ll p_0$ is termed the [*deep quantum regime*]{}, $p \gg 2jp_0$ is termed the [*classical regime*]{} and $p_0 \lesssim |p| \lesssim 2jp_0$ is termed the [*semi-classical regime*]{}. The modifications due to the inverse triad defined above are strong in the semi-classical and the deep quantum regimes. For $j = 1/2$ the semi-classical regime is absent. Note that such scales are not available for the Schrodinger quantization.
[**The Gravitational Constraint:**]{} Since $\hat{c}$ operator does not exist, the gravitational Hamiltonian (the first bracket in eq.(\[ClassHam\])), has to be expressed in an equivalent form using holonomies. For this, let us go back to the full theory Hamiltonian: $$\label{IsoGrav}
\frac{6}{\gamma^2} c^2\sqrt{p} ~ = ~ \gamma^{-2}\int_{\mathrm{cell}}d^3x
\frac{\epsilon_{ijk} E^{ai} E^{bj} F^k_{ab}}{\sqrt{|\mathrm{det}E|}}
$$ Now use the two identities: $$\begin{aligned}
\frac{\epsilon_{ijk} E^{ai} E^{bj}}{\sqrt{|\mathrm{det}E|}} & = & \sum_k
\frac{4 \ \mathrm{sgn} \ p}{\kappa \gamma \mu_0 V_0^{1/3}}~
\epsilon^{abc} ~ \omega^k_c \ \mathrm{Tr}\left( h^{(\mu_0)}_k \left\{
\left(h^{(\mu_0)}_k\right)^{-1}, V\right\} \tau_i \right) \\
F^k_{ab} & = & -2 \lim_{A_{\square} \to 0} \mathrm{Tr}
\left(\frac{h^{(\mu_0)}_{\square_{ij}} - 1}{\mu_0^2 V_0^{2/3}}\right)
\tau^{k} ~\omega_a^{i} ~ \omega_b^j \\
h^{(\mu_0)}_{\square_{ij}} & := & h^{(\mu_0)}_i h^{(\mu_0)}_j
\left(h^{(\mu_0)}_i\right)^{-1} \left(h^{(\mu_0)}_j\right)^{-1}
$$ In the above, the fiducial cell is thought to have been sub-divided into smaller cells of side $\mu_0\ell, \ell := V_0^{1/3}$. The area of a plaquette is $A_{\square} = \ell^2\mu_0^2$. The plaquette is to be shrunk such that its area goes to zero. The superscript $(\mu_0)$ on the holonomies is to remind that the length of the edge is $\mu_0$. The 1-forms $\omega^i_a$ are the fiducial 1-forms whose square gives the fiducial metric and the $\epsilon^{abc}$ is the (fiducial) metric dependent Levi-Civita density.
In quantum geometry however there is a gap in the spectrum of area operator and thus it is [*not*]{} appropriate to take the area to zero, but at the most to the smallest possible eigenvalue. Independently, if we force the limit, it will imply $\mu_0 \to 0$ which in turn amounts to defining $\hat{c}$ operator which does [*not*]{} exist on the Hilbert space.
Substituting these in the (\[IsoGrav\]) and carrying out the integration over the cell leads to (suppressing the $\mu_0$ superscript on the holonomies), $$\begin{aligned}
\label{RegularisedHam}
H_{\mathrm{grav}} & = & - \frac{4}{8 \pi G \gamma^3 \mu_0^3} \sum_{ijk}
\epsilon^{ijk}\mathrm{Tr}\left(h_i h_j h_i^{-1}h_j^{-1}h_k\{h_k^{-1},
V\}\right)
$$ In the above, we have used $j = 1/2$ representation for the holonomies and $V$ denotes the volume function. In the limit $\mu_0 \to 0$ one gets back the classical expression [^14].
If we promote this expression to a quantum operator (modulo ordering ambiguities) on the LQC Hilbert space constructed above, then we [*cannot*]{} take the limit $\mu_0 \to 0$ because it would imply that $\hat{c}$ exist which we have shown to be impossible. Thus, at the quantum level we should [*not*]{} take the limit $\mu_0 \to 0$. The best we can do is to take reduce $\mu_0$ such that the area reaches its smallest possible (and non-zero due to the gap) eigenvalue $\Delta :=
(2\sqrt{3} \pi \gamma) {\ell_{\mathrm P}}^2$. But which area do we consider, the fiducial or the physical? These are related by a factor of $|p|$. It seems appropriate to choose the physical area, which implies that we must take $\mu_0$ to be a function $\bar{\mu}$ of $p$ given by, $\bar{\mu}(p) := \sqrt{\Delta/|p|}$. Note that this is [*one*]{} prescription to interpret the limitation on shrinking of the plaquette. There are others which will be mentioned later. In the following we will continue to use the $\mu_0$ notation and replace it by $\bar{\mu}(p)$ when needed.
While promoting this expression to operators, there is a choice of factor ordering involved and many are possible. We will present two choices of ordering: the [*non-symmetric*]{} one which keeps the holonomies on the left as used in the existing choice for the full theory, and the particular [*symmetric*]{} one used in [@APSOne]. $$\begin{aligned}
\hat{H}^{\mathrm{non-sym}}_{\mathrm{grav}} & = & \frac{24 i }{\gamma^3
\mu_0^3 {\ell_{\mathrm P}}^2} \mathrm{sin}^2 \mu_0c \left( \mathrm{sin}\frac{\mu_0c}{2}
\hat{V} \mathrm{cos}\frac{\mu_0c}{2} - \mathrm{cos}\frac{\mu_0c}{2}
\hat{V} \mathrm{sin}\frac{\mu_0c}{2} \right) \\
\hat{H}^{\mathrm{sym}}_{\mathrm{grav}} & = & \frac{24 i
(\mathrm{sgn}(p))}{\gamma^3 \mu_0^3 {\ell_{\mathrm P}}^2} \mathrm{sin} \mu_0c \left(
\mathrm{sin}\frac{\mu_0c}{2} \hat{V} \mathrm{cos}\frac{\mu_0c}{2} -
\mathrm{cos}\frac{\mu_0c}{2} \hat{V} \mathrm{sin}\frac{\mu_0c}{2}
\right) \mathrm{sin} \mu_0c \end{aligned}$$ At the quantum level, $\mu_0$ cannot be taken to zero since $\hat{c}$ operator does not exist. The action of the Hamiltonian operators on $|\mu\rangle$ is obtained as, $$\begin{aligned}
\hat{H}^{\mathrm{non-sym}}_{\mathrm{grav}}|\mu\rangle & = &
\frac{3}{\mu_0^3\gamma^3{\ell_{\mathrm P}}^2}\left(V_{\mu + \mu_0} - V_{\mu -
\mu_0}\right) \left( |\mu + 4 \mu_0\rangle - 2 | \mu\rangle + |\mu -
4\mu_0\rangle\right) \\
\hat{H}^{\mathrm{sym}}_{\mathrm{grav}}|\mu\rangle & = &
\frac{3}{\mu_0^3\gamma^3{\ell_{\mathrm P}}^2}\left[ \left|V_{\mu + 3\mu_0} - V_{\mu +
\mu_0}\right||\mu + 4 \mu_0\rangle + \left|V_{\mu - \mu_0} - V_{\mu -
3\mu_0}\right||\mu - 4 \mu_0\rangle \right. \nonumber \\ & & \left.
\hspace{1.5cm} - \left\{\left|V_{\mu + 3\mu_0} - V_{\mu + \mu_0}\right|
+ \left|V_{\mu - \mu_0} - V_{\mu - 3\mu_0}\right|\right\}|\mu\rangle
\right]
$$ where $V_{\mu} := ({{\textstyle \frac{1}{6}}}\gamma {\ell_{\mathrm P}}^2 |\mu|)^{3/2}$ denotes the eigenvalue of $\hat{V}$.
We also have the Hilbert space for the matter degrees which for us is a single scalar, $\phi$ and the full kinematical Hilbert space is the tensor product of the $L_2(\mathbb{R}_{\mathrm{Bohr}},
d\mu_{\mathrm{Bohr}}) \otimes {\cal H}_{\mathrm{matter}}$.
[**Wheeler-DeWitt Difference Equation:**]{} Let $|\Psi\rangle :=
\sum_{\mu} \Psi(\mu, \phi) |\mu\rangle$, where the sum is over a countable subset of $\mathbb{R}$, the coefficients $\Psi(\mu, \phi)$ are valued in the matter Hilbert space and the argument $\phi$ is a reminder of that. The Hamiltonian constraint is imposed on these $|\Psi\rangle$ which leads to the [*Wheeler-DeWitt equation*]{} for the coefficients. Thanks to the presence of the trigonometric operators, this equation is a [*difference equation*]{}. In the Schrodinger quantization, this would be a differential equation.
For the non-symmetric operator we get, $$A(\mu + 4 \mu_0) \Psi(\mu + 4\mu_0, \phi) - 2 A(\mu) \Psi(\mu, \phi) +
A(\mu - 4 \mu_0) \Psi(\mu - 4\mu_0, \phi)
$$ $$~ = ~ -\frac{2 \kappa}{3}\mu_0^3 \gamma^3{\ell_{\mathrm P}}^2 H_{matter}(\mu)\Psi(\mu,
\phi)
$$ where, $A(\mu) := V_{\mu + \mu_0} - V_{\mu - \mu_0}$ and vanishes for $\mu = 0$.
For the symmetric operator one gets, $$f_+(\mu) \Psi(\mu + 4\mu_0, \phi) + f_0(\mu) \Psi(\mu, \phi) + f_-(\mu)
\Psi(\mu - 4\mu_0, \phi)
$$ $$~ = ~ -\frac{2 \kappa}{3}\mu_0^3 \gamma^3{\ell_{\mathrm P}}^2 H_{matter}(\mu)\Psi(\mu,
\phi) \hspace{3.0cm} \mbox{where,}
$$ $$f_+(\mu) ~ := ~ \left| V_{\mu + 3\mu_0} - V_{\mu + \mu_0} \right| ~ ,
~ f_-(\mu) ~ := ~ f_+(\mu - 4 \mu_0) ~, ~ f_0 ~ := ~ - f_+(\mu) -
f_-(\mu) \ .$$ Notice that $f_+(-2\mu_0) = 0 = f_-(2\mu_0)$, but $f_0(\mu)$ is never zero. The absolute values have entered due to the sgn($p$) factor.
The difference equations relate $\Psi(\mu)$’s only for $\mu$’s in a “[*lattice*]{}”, ${\cal L}_{\hat{\mu}} := \{\mu = \hat{\mu} + 4\mu_0
n, n \in \mathbb{Z}\}$ and the coefficients labelled by different lattices are completely independent. The $\hat{\mu} \in [0, 4\mu_0)$, label different [*superselected*]{} sectors.
The equations are effectively second order difference equations and the $\Psi(\mu, \phi)$ are determined by specifying $\Psi$ for two consecutive values of $\mu$ eg for $ \mu = \hat{\mu} + 4 \mu_0 N$ and $
\mu = \hat{\mu} + 4 \mu_0 (N + 1)$. Since the highest (lowest) order coefficients vanishes for some $\mu$, then the corresponding component $\Psi(\mu, \phi)$ is undetermined by the equation. Potentially this could introduce an arbitrariness in extending the $\Psi$ specified by data in the classical regime (eg $\mu \gg 2j $) to the negative $\mu$. Potentially, maintaining determinism of the quantum wave function, is one of the restrictive criteria for choosing the ordering.
For the non-symmetric case, the highest (lowest) $A$ coefficients vanish for their argument equal to zero thus leaving the corresponding $\Psi$ component undetermined. However, this undetermined component is decoupled from the others. Thus apart from admitting the trivial solution $\Psi(\mu, \phi) := \Phi(\phi)\delta_{\mu, 0}, ~ \forall \mu$, all other non-trivial solutions are completely determined by giving two consecutive components: $\Psi(\hat{\mu}, \phi), \Psi(\hat{\mu} + 4\mu_0,
\phi)$.
For the symmetric case, due to these properties of the $f_{\pm,0}(\mu)$, it looks as if the difference equation is [*non-deterministic*]{} if $\mu = 2\mu_0 + 4\mu_0 n, n \in \mathbb{Z}$. This is because for $\mu =
-2\mu_0$, $\Psi(2\mu_0, \phi)$ is undetermined by the lower order $\Psi$’s and this coefficient enters in the determination of $\Psi(2\mu_0, \phi)$. However, the symmetric operator also commutes with the parity operator: $(\Pi\Psi)(\mu, \phi) := \Psi(-\mu, \phi)$. Consequently, $\Psi(2\mu_0, \phi)$ is determined by $\Psi(-2\mu_0,
\phi)$. Thus, we can restrict to $\mu = 2\mu_0 + 4k\mu_0, k \ge 0$ where the equation [*is*]{} deterministic.
In both cases then, the space of solutions of the constraint equation, is completely determined by giving appropriate data for large $|\mu|$ i.e. in the classical regime. Such a deterministic nature of the constraint equation has been taken as a necessary condition for non-singularity at the quantum level [^15].
[**Effective Hamiltonian:**]{} By introducing an interpolating, slowly varying smooth function, $\Psi(p (\mu) := {{\textstyle \frac{1}{6}}}\gamma{\ell_{\mathrm P}}^2\mu)$, and keeping only the first non-vanishing terms, one deduces the Wheeler-De Witt differential equation (with a modified matter Hamiltonian) from the above difference equation. Making a WKB approximation, one infers an effective Hamiltonian which matches with the classical Hamiltonian for large volume ($\mu \gg \mu_0$) and small extrinsic curvature (derivative of the WKB phase is small). There are terms of $o(\hbar^0)$ which contain arbitrary powers of the first derivative of the phase which can all be summed up. The resulting effective Hamiltonian now contains modifications of the classical gravitational Hamiltonian, apart from the modifications in the matter Hamiltonian due to the inverse powers of the triad. The largest possible domain of validity of effective Hamiltonian so deduced must have $|p|
\gtrsim p_0$ [@SemiClass; @EffHam].
An effective Hamiltonian can alternatively obtained by computing expectation values of the Hamiltonian operator in semi-classical states peaked in classical regimes [@Willis]. The leading order effective Hamiltonian that one obtains is (spatially flat case): $$\begin{aligned}
H^{\mathrm{non-sym}}_{\mathrm{eff}} & = & - \frac{1}{16 \pi
G}\left(\frac{6}{\mu_0^3 \gamma^3 {\ell_{\mathrm P}}^2}\right) \left[ B_+(p)
\mathrm{sin}^2(\mu_0c) + \left( A(p) - \frac{1}{2}B_+(p) \right) \right]
+ H_{\mathrm{matter}} \ ; \nonumber\\
B_+(p) & := & A(p + 4 p_0) + A(p - 4 p_0) ~,~ A(p) ~ := ~ (|p +
p_0|^{3/2} - |p - p_0|^{3/2}) \ , \\
p & := & \frac{1}{6}\gamma {\ell_{\mathrm P}}^2 \mu ~ ~ , ~ ~ p_0 ~ := ~
\frac{1}{6}\gamma {\ell_{\mathrm P}}^2 \mu_0 \ . \nonumber
$$ For the symmetric operator, the effective Hamiltonian is the same as above except that $B_+(p) \to f_+(p) + f_-(p)$ and $2 A(p) \to f_+(p) +
f_-(p)$.
The second bracket in the square bracket, is the quantum geometry potential which is negative and higher order in ${\ell_{\mathrm P}}$ but is important in the small volume regime and plays a role in the genericness of bounce deduced from the effective Hamiltonian [@GenBounce]. This term is absent in the effective Hamiltonian deduced from the symmetric constraint. The matter Hamiltonian will typically have the eigenvalues of powers of inverse triad operator which depend on the ambiguity parameters $j, l$.
We already see that the quantum modifications are such that both the matter (due to inverse volume corrections) and the gravitational part (due to holonomy corrections) in the effective Hamiltonian, are rendered bounded and effective dynamics must be non-singular.
For large values of the triad, $p \gg p_0$, $B_+(p) \sim 6 p_0 \sqrt{p}
-~o(p^{-3/2})$ while $A(p) \sim 3 p_0 \sqrt{p} - o(p^{-3/2})$. In this regime, the effective Hamiltonians deduced from both symmetric and non-symmetric ordering are the same[^16]. The classical Hamiltonian is obtained for $\mu_0 \to 0$. From this, one can obtain the equations of motion and by computing the left hand side of the Friedmann equation, infer the effective energy density. For $p \gg p_0$ one obtains [^17], $$\label{EffDensity}
\frac{3}{8\pi G}\left(\frac{\dot{a}^2}{a^2}\right) ~ := ~
\rho_{\mathrm{eff}} ~ = ~
\left(\frac{H_{\mathrm{matter}}}{p^{3/2}}\right)\left\{1 - \frac{8 \pi G
\mu_0^2\gamma^2}{3} p
\left(\frac{H_{\mathrm{matter}}}{p^{3/2}}\right)\right\} ~ ~,~ ~ p :=
a^2/4\ .
$$
The effective density is quadratic in the classical density, $\rho_{cl}
:= H_\mathrm{matter} p^{-3/2}$. This modification is due to the quantum correction in the gravitational Hamiltonian (due to the sin$^2$ feature). This is over and above the corrections hidden in the matter Hamiltonian (due to the “inverse volume” modifications). As noted before, we have two scales: $p_0$ controlled by $\mu_0$ in the gravitational part and $2p_0 j$ in the matter part. For large $j$ it is possible that we can have $p_0 \ll p \ll 2p_0j $ in which case the above expressions will hold with $j$ dependent corrections in the matter Hamiltonian. In this semi-classical regime, the corrections from sin$^2$ term are smaller in comparison to those from inverse volume. If $p \gg
2p_0j$ then the matter Hamiltonian is also the classical expression. For $j = 1/2$, there is only the $p \gg p_0$ regime and $\rho_{cl}$ is genuinely the classical density.
To summarize:
\(1) The connection formulation, in the homogeneous and isotropic context, uses variables ($c, p \in \mathbb{R}$) in terms of which the classical singularity ($p = 0$) is in the [*interior*]{} of the phase space. By contrast, in the ADM variables ($a \ge 0, K$), in the same context, the classical singularity ($a = 0$) is on the [*boundary*]{}. This requires a boundary condition on the quantum wave functions to be specified in the deep quantum region where the classical framework is suspect. When the singularity is in the interior, only a continuation of the quantum wave function is required, given its specification in the semi-classical region.
\(2) The connection variables also strongly motivate the very different [*loop quantization*]{}. Its immediate implications are two types of corrections - the holonomy corrections and the inverse triad corrections. [*Either* ]{} of these is sufficient to indicate a bounce in the effective Hamiltonian picture. The same use of holonomies make the Wheeler-DeWitt equation, a [*difference equation*]{}.
\(3) The analysis at the level of effective Hamiltonian already indicates (i) replacement of big bang by big bounce; (ii) natural prediction of an inflationary (accelerated) phase; (iii) singularity resolution for more general homogeneous models with curvature.
\(4) There are at least three distinct ambiguity parameters: $\mu_0$ related to the fiducial length of the loop used in writing the holonomies; $j$ entering in the choice of $SU(2)$ representation which is chosen to be $1/2$ in the gravitational sector and some large value in the matter sector; $l$ entering in writing the inverse powers in terms of Poisson brackets. The first one was thought to be determined by the area gap from the full theory. The $j = 1/2$ in the gravitational Hamiltonian seems needed to avoid high order difference equation and larger $j$ values are hinted to be problematic in the study of a three dimensional model [@LargeJ]. Given this, the choice of a high value of $j$ in the matter Hamiltonian seems unnatural[^18]. Nevertheless the higher values of $j$ in the matter sector allow for a larger semi-classical regime. The $l$ does not play as significant a role.
\(5) LQC being a constrained theory, it would be more appropriate if singularity resolution is formulated and demonstrated in terms of physical expectation values of physical (Dirac) operators i.e. in terms of “gauge invariant quantities”. This can be done at present with self-adjoint constraint i.e. a symmetric ordering and for free, massless scalar matter.
[**Physical quantities and Singularity Resolution:**]{} When the Hamiltonian is a constraint, at the classical level itself, the notion of dynamics in terms of the ‘time translations’ generated by the Hamiltonian is devoid of any [*physical*]{} meaning. Furthermore, at the quantum level when one attempts to impose the constraint as $\hat{H}|\Psi\rangle = 0$, typically one finds that there are no solutions in the Hilbert space on which $\hat{H}$ is defined - the solutions are generically distributional. One then has to consider the space of all distributional solutions, define a new physical inner product to turn it into a Hilbert space (the physical Hilbert space), define operators on the space of solutions (which must thus act invariantly) which are self-adjoint (physical operators) and compute expectation values, uncertainties etc of these operators to make physical predictions. Clearly, the space of solutions depends on the quantization of the constraint and there is an arbitrariness in the choice of physical inner product. This is usually chosen so that a complete set of Dirac observables (as deduced from the classical theory) are self-adjoint. This is greatly simplified if the constraint has a [*separable*]{} form with respect to some degree of freedom[^19]. For LQC (and also for the Wheeler-De Witt quantum cosmology), such a simplification is available for a free, massless scalar matter: $H_{\mathrm{matter}}(\phi, p_{\phi}) := {{\textstyle \frac{1}{2}}} p_{\phi}^2
|p|^{-3/2}$. Let us sketch the steps schematically, focusing on the spatially flat model for simplicity [@APSOne; @APSTwo].
1. [*Fundamental constraint equation:*]{}
The classical constraint equations is: $$\label{ClassHamConstraint}
- \frac{6}{\gamma^2} c^2 \sqrt{|p|} + 8\pi G \ p_{\phi}^2 \ |p|^{-3/2}
~ = ~ 0 ~ = ~ C_{\mathrm{grav}} + C_{\mathrm{matter}}\ ;
$$ The corresponding quantum equation for the wave function, $\Psi(p,
\phi)$ is: $$8\pi G \hat{p}_{\phi}^2 \Psi(p, \phi) ~ = ~ [\tilde{B}(p)]^{-1}
\hat{C}_{\mathrm{grav}} \Psi(p, \phi)~ ~ , ~ ~ [\tilde{B}(p)] \mbox{ is
eigenvalue of } \widehat{|p|^{-3/2}} \ ;
$$ Putting $\hat{p}_{\phi} = - i \hbar \partial_{\phi}$, $p :=
{{\textstyle \frac{\gamma{\ell_{\mathrm P}}^2}{6}}}\mu$ and $\tilde{B}(p) :=
({{\textstyle \frac{\gamma{\ell_{\mathrm P}}^2}{6}}})^{-3/2} B(\mu)$, the equation can be written in a separated form as [^20], $$\frac{\partial^2 \Psi(\mu, \phi)}{\partial \phi^2} ~ = ~
[B(\mu)]^{-1}\left[8\pi G \left(\frac{\gamma}{6}\right)^{3/2}{\ell_{\mathrm P}}^{-1}
\hat{C}_{\mathrm{grav}}\right]\Psi(\mu, \phi) ~ := ~ -
\hat{\Theta}(\mu)\Psi(\mu, \phi).
$$ The $\hat{\Theta}$ operator for different quantizations is different. For Schrodinger quantization (Wheeler-De Witt), with a particular factor ordering suggested by the continuum limit of the difference equation, the operator $\hat{\Theta}(\mu)$ is given by, $$\hat{\Theta}_{\mathrm{Sch}}(\mu)\Psi(\mu, \phi) ~ = ~ - \frac{16\pi
G}{3}|\mu|^{3/2} \partial_{\mu} \left(\sqrt{\mu}\
\partial_{\mu}\Psi(\mu, \phi)\right)
$$ while for LQC, with symmetric ordering, it is given by, $$\begin{aligned}
\hat{\Theta}_{\mathrm{LQC}}(\mu)\Psi(\mu, \phi) & = & - [B(\mu)]^{-1}
\left\{ C^+(\mu) \Psi(\mu + 4 \mu_0, \phi) + C^0(\mu) \Psi(\mu, \phi) +
\right. \nonumber \\
& & \left. \hspace{5.0cm} C^-(\mu) \Psi(\mu - 4 \mu_0, \phi) \right\} \
, \nonumber \\
C^+(\mu) & := & \frac{\pi G}{9 \mu_0^3} \left| ~ |\mu + 3\mu_0|^{3/2} -
|\mu + \mu_0|^{3/2} \right| \ , \\
C^-(\mu) & := & C^+(\mu - 4 \mu_0) ~ ~ , ~ ~ C^0(\mu) ~ := ~ - C^+(\mu)
- C^-(\mu) \ . \nonumber
$$ Note that in the Schrodinger quantization, the $B_{\mathrm{Sch}}(\mu) =
|\mu|^{-3/2}$ diverges at $\mu = 0$ while in LQC, $B_{\mathrm{LQC}}(\mu)$ vanishes for all allowed choices of ambiguity parameters. In both cases, $B(\mu) \sim |\mu|^{-3/2}$ as $|\mu| \to
\infty$.
2. [*Inner product and General solution:*]{}
The operator $\hat{\Theta}$ turns out to be a self-adjoint, positive definite operator on the space of functions $\Psi(\mu, \phi)$ for each fixed $\phi$ with an inner product scaled by $B(\mu)$. That is, for the Schrodinger quantization, it is an operator on $L^2(\mathbb{R},
B_{\mathrm{Sch}}(\mu) d\mu)$ while for LQC it is an operator on $L^2(\mathbb{R}_{\mathrm{Bohr}},
B_{\mathrm{Bohr}}(\mu)d\mu_{\mathrm{Bohr}}).$ Because of this, the operator has a complete set of eigenvectors[^21]: $$\label{EigenEq}
\hat{\Theta}e_k(\mu) = \omega^2(k) e_k(\mu)~~,~~ \langle
e_k|e_{k'}\rangle = \delta(k, k')~~,~~ k,\, k' \in \mathbb{R}.
$$ Consequently, the general solution of the fundamental constraint equation can be expressed as $$\label{GeneralSolution}
\Psi(\mu, \phi) ~ = ~ \int dk ~ \tilde{\Psi}_+(k) e_k(\mu)
e^{i\omega(k)\phi} + \tilde{\Psi}_-(k) \bar{e}_k(\mu) e^{-i
\omega(k)\phi} \ .
$$ The orthonormality relations among the $e_k(\mu)$ are in the corresponding Hilbert spaces. Different quantizations differ in the form of the eigenfunctions, possibly the spectrum itself and of course $\omega(k)$. In general, these solutions are [*not*]{} normalizable in $L^2(\mathbb{R}_{\mathrm{Bohr}}\times\mathbb{R},
B_{\mathrm{Bohr}}(\mu)d\mu_{\mathrm{Bohr}}\times d\mu)$, i.e. these are distributional.
\(1) The $\hat{\Theta}$ operator acts in each of the superselected sector (thanks to the difference equation structure) and these are [*separable*]{}. Hence Dirac-$\delta$ appears in general when the label takes [*continuous*]{} values.
\(2) The group averaging can be seen as follows. Given any $f(\mu, \phi)$ one defines its [*group average*]{}, $$\Psi_f(\mu, \phi) ~ := ~ \int_{-\infty}^{\infty}d\lambda \
e^{i\lambda \hat{C}_{\mathrm{tot}} }\ f(\mu, \phi) ~~;~~
\hat{C}_{\mathrm{tot}} ~ := ~ \frac{\partial^2}{\partial \phi^2} +
\hat{\Theta} ~~,
$$ The eigenfunctions of the constraint operator are of a product form thanks to separability, $$\frac{\partial^2}{\partial \phi^2}g_{\sigma}(\phi) ~ = ~ - {\sigma}^2
g_{\sigma}(\phi) ~~,~~ \hat{\Theta}e_{k}(\mu) = \omega^2(k) e_{k}(\mu)
~~,~~ g_{\sigma}(\phi) = e^{i {\sigma}\phi}, ~~ k, {\sigma} ~ \in ~
\mathbb{R}
$$ Expanding the general function $f(\mu, \phi)$ in the eigenbasis of the constraint operator, $$f(\mu, \phi) ~ := ~ \int dk \int d\sigma \ A_{k, {\sigma}}\
g_{\sigma}(\phi)e_{k}(\mu) ~~,~~
$$ implies, $$\begin{aligned}
\Psi_f(\mu, \phi) & = & \int dk \int d\sigma \ A_{k, {\sigma}}
g_{\sigma}(\phi)e_{k}(\mu)\delta({\sigma}^2 - \omega^2(k)) \nonumber \\
& = & \int_{-\infty}^{\infty} \frac{dk}{2|\omega(k)|} \left[
\left\{A_{k, \omega(k)} e^{i\omega(k) \phi} + A_{k, - \omega(k)}
e^{-i\omega(k) \phi} \right\} e_{\omega}(\mu) \right]
$$ In the second equation above, we have carried out the integration over $\sigma$ using $\delta(\sigma^2 - \omega^2(k)) = {{\textstyle \frac{1}{2|\omega(k)|}}}
(\delta(\sigma - \omega(k)) + \delta(\sigma + \omega(k))$. Clearly the group average of a general function reproduces precisely the general solution given in equation (\[GeneralSolution\]).
3. [*Choice of Dirac observables:*]{} Since the classical kinematical phase space is 4 dimensional and we have a single first class constraint, the phase space of physical states (reduced phase space) is two dimensional and we need two functions to coordinatize this space. We should thus look for two (classical) Dirac observables: functions on the kinematical phase space whose Poisson bracket with the Hamiltonian constraint vanishes on the constraint surface. Specific values of these functions serve to label the physics states. Thus physical observables are values of the Dirac observables. Classically, the Dirac observables can be obtained as follows.
Our constraint is: $p_{\phi}^2/2 - {{\textstyle \frac{3}{\kappa}}}\gamma^{-2} c^2 p^2
\simeq 0$. A Dirac observable is a function $f(\phi, p_{\phi}, c, p)$ whose Poisson bracket with the constraint vanishes on the constraint surface. We can describe the constraint surface by solving for $c$ as: $c(\phi, p_{\phi}, p) := \pm \sqrt{\kappa/6} \gamma p_{\phi} p^{-1}$ (say). Consequently it should suffice to consider the Dirac observables to be a function of $(\phi, p_{\phi}, p)$ (we need only two independent Dirac observables). Then the Dirac observables are defined by the differential equation, $$\frac{\partial f}{\partial\phi} \pm \sqrt{2\kappa/3}\frac{\partial
f}{\partial\ell n p} = 0 ~~\Rightarrow~~ f = f(\zeta,
p_{\phi})~~,~~\zeta := \phi \mp \sqrt{\frac{3}{2\kappa}}\ell n p ~.
$$ Evidently, $f = p_{\phi}$ is a Dirac observable. For the second one, we can choose any function of $\zeta$. A particularly convenient choice is: $f(\zeta(\phi, p)) := \# \mathrm{exp}\{\mp \sqrt{2\kappa/3}\zeta\} = p \
\mathrm{exp}\{\mp \sqrt{2\kappa/3}(\phi - \phi_0)\}$. These Dirac observables taking a particular value, say $p_*$, define curves in the $(p, \phi)$ plane, $p(\phi) := p_*\mathrm{exp}\{\pm
\sqrt{2\kappa/3}(\phi - \phi_0)\}$ which are the classical solutions in (\[ClassRelationalSoln\]).
In the quantum theory, the notion of Dirac observable is that it is an operator which maps solutions of the constraint to (other) solutions. We already have the general solution in eq (\[GeneralSolution\]) which is obtained via unitary evolution (in $\phi$) from an initial $\Psi_{\pm}(\phi_0, \mu)$. Hence a Dirac observable is constructed by defining an operator on $\Psi_{\pm}(\phi_0, \mu)$ to generate a new ‘initial state’, and evolving the new state by the unitary operator, thereby constructing a new solution of the constraint. This procedure is followed for the two Dirac operators corresponding to $f_1(\phi,
p_{\phi}, p) := p_{\phi}$ and $f_2(\phi, p_{\phi}, p) := p \
\mathrm{exp}\{\mp \sqrt{2\kappa/3}(\phi - \phi_0)\}$. Notice that [*at*]{} $\phi = \phi_0$, these become the functions: $p_{\phi}$ and $p$ respectively and they act on the wavefunctions as the differential operator $-i\hbar\partial_{\phi}$ and as the multiplicative operator $p$, respectively. Explicit construction is as follows.
Consider initial data, $\Psi_{\pm}(\mu, \phi_0)$, with the corresponding solution is denoted by, $$\Psi(\mu, \phi) := e^{ i\sqrt{\hat{\Theta}}(\phi -
\phi_0)}\Psi_+(\mu, \phi_0) + e^{ - i\sqrt{\hat{\Theta}}(\phi -
\phi_0)}\Psi_-(\mu, \phi_0) \ .
$$ Generate [*new initial data*]{} via the actions of $\hat{p}_{\phi},
\widehat{|\mu|_{\phi_0}}$ as, $$\widehat{|\mu|_{\phi_0}}\Psi_{\pm}(\mu, \phi_0) ~ := ~
|\mu|\Psi_{\pm}(\mu, \phi_0) ~ ~ , ~ ~ \hat{p}_{\phi} \Psi_{\pm}(\mu,
\phi_0) ~ := ~ \hbar \sqrt{\hat{\Theta}}\Psi_{\pm}(\mu, \phi_0)\ .
$$ Evolve these respectively, by $e^{\pm i\sqrt{\hat{\Theta}}(\phi -
\phi_0)}$. By construction, these are solutions of the constraints, being of the form of (\[GeneralSolution\]). Explicitly, $$\begin{aligned}
\hat{p}_{\phi}\Psi(\mu, \phi) & := & e^{i\sqrt{\hat{\Theta}}(\phi -
\phi_0)} (\hbar\sqrt{\hat{\Theta}}) \Psi_+(\mu, \phi_0) + e^{-
i\sqrt{\hat{\Theta}}(\phi - \phi_0)} (- \hbar\sqrt{\hat{\Theta}})
\Psi_-(\mu, \phi_0) \nonumber \\
& = & -i \hbar\partial_{\phi}\Psi(\mu, \phi) \ , \\
\widehat{|\mu|_{\phi_0}}\Psi(\mu, \phi) & := &
e^{i\sqrt{\hat{\Theta}}(\phi - \phi_0)} |\mu| \Psi_+(\mu, \phi_0) + e^{-
i\sqrt{\hat{\Theta}}(\phi - \phi_0)} |\mu| \Psi_-(\mu, \phi_0)
$$ Expectation values and uncertainties of these operators are used to track the quantum ‘evolution’.
4. [*Physical inner product:*]{}
It follows that the Dirac operators defined on the space of solutions are self-adjoint if we define a [*physical inner product*]{} on the space of solutions as: $$\langle\Psi|\Psi'\rangle_{\mathrm{phys}} ~ := ~ ``\int_{\phi = \phi_0}
d\mu B(\mu)" ~ ~ \bar{\Psi}(\mu, \phi) \Psi'(\mu, \phi) \ .
$$ Thus the eigenvalues of the inverse volume operator crucially enter the definition of the physical inner product. For Schrodinger quantization, the integral is really an integral while for LQC it is actually a sum over $\mu$ taking values in a lattice. The inner product is independent of the choice of $\phi_0$.
A complete set of physical operators and physical inner product has now been specified and physical questions can be phrased in terms of (physical) expectation values of functions of these operators.
5. [*Semi-classical states:*]{}
To discuss semi-classical regime, typically one [*defines*]{} semi-classical states: physical states such that a chosen set of self-adjoint operators have specified expectation values with uncertainties bounded by specified tolerances. A natural choice of operators for us are the two Dirac operators defined above.
To be definite, let us consider the Wheeler-De Witt quantization. An arbitrary wavefunction at some given $\phi_0$ is expressed as an integral over $k$ of the eigenfunctions $e_k(\mu)$ multiplied by some function of $k$ and by a phase factor $e^{i\omega(k)\phi_0}$. The inner product involves integral over $\mu$. We can arrange to have a peak at some particular value $|\mu_*|$ by introducing a factor of $-
\sqrt{3/2\kappa}|\mu_*|$ so that for large $k$, the $\mu-$integral will be dominated by $\mu \sim \mu_*$. A large value of $k$ can be picked-up by choosing the function of $k$ to be a suitable Gaussian. This particular large value can be related to a desired $p_{\phi}^*$. Here are the expressions: $$\begin{aligned}
\Psi_{\mathrm{semi}}(\mu, \phi_0) & := & \int dk e^{- \frac{(k -
k^*)^2}{2 \sigma^2}} e_k(\mu) e^{i \omega (\phi_0 - \phi^*)} \\ k^* & =
& -\sqrt{3/2\kappa} \hbar^{-1} p^*_{\phi} ~ ~ , ~ ~ \phi^* = \phi_0
-\sqrt{3/2\kappa} \ell n|\mu^*| \ .
$$ The Gaussian allows the integrand to be approximated by $$e_{k^*}(\mu) e^{i\omega(k^*)(\phi_0 - \phi_*)} \sim e^{i\ell n|\mu| k^*
+ i(\omega(k^*)\sqrt{3/2\kappa})\ell n|\mu_*|} \sim e^{ik^*\ell
n|\mu/\mu_*|}
$$ where, in the last equality we used: $k^*$ is large and $\omega(k^*)
\simeq -\sqrt{2\kappa/3}\ k^*$ (see the footnote \[WDWEigenfunctions\]). The integral in the inner product will now pick-up contribution from near $\mu \simeq \mu_*$. With these observations, it is easy to verify that the ‘initial’ semiclassical wave function given above gives $\langle \hat{p}_{\phi}\rangle = p_{\phi}^*$ and $\langle \widehat{|\mu|_{\phi_0}}\rangle = \mu_*$. The initial semiclassical wavefunction evolves into $\Psi_{\mathrm{semi}}(\mu,
\phi)$ which is same as the initial wave function with $\phi_0 \to \phi$ [^22].
For LQC, the $e_k(\mu)$ functions are different [@APSTwo] and the physical expectation values are to be evaluated using the physical inner product defined in the LQC context.
6. [*Evolution of physical quantities:*]{}
We have now the physical wave function, evolved from $\Psi_{\mathrm{semin}}$. Since it retains the form of the initial wavefunction, the $k$ integral can be approximated as before and thus will lead to same expectation value for $\hat{p}_{\phi}$ for all $\phi$. For the expectation value of $\langle
\widehat{|\mu|_{\phi_0}}\rangle(\phi)$, the $\mu$ integral will be saturated by the new phase, $\phi - \phi_0 + \sqrt{3/2\kappa}\ \ell
n|\mu/\mu_*| \simeq 0$. And this we recognize as precisely the solution (\[ClassRelationalSoln\]).
Thus, in the WdW quantization, the classical relational evolution, $p =
p(\phi)$ is reproduced by the expectation values of the Dirac observables.
[*Exercise: compute/estimate the uncertainties of the Dirac observables in the semiclassical state given above.*]{}.
7. [*Resolution of Big Bang Singularity:*]{}
A classical solution is obtained as a curve in $(\mu, \phi)$ plane, different curves being labelled by the points ($\mu^*, \phi^*$) in the plane. The curves are independent of the constant value of $p^{*}_{\phi}$ These curves are already given in (\[ClassRelationalSoln\]).
Quantum mechanically, we first select a semi-classical solution, $\Psi_{\mathrm{semi}}(p_{\phi}^*, \mu^*: \phi)$ in which the expectation values of the Dirac operators, at $\phi = \phi_0$, are $p_{\phi}^*$ and $\mu^*$ respectively. These values serve as labels for the semi-classical solution. The former one continues to be $p_{\phi}^*$ for all $\phi$ whereas $\langle\widehat{|\mu|_{\phi_0}}\rangle(\phi) =:
|\mu|_{p_{\phi}^*, \mu^*}(\phi)$, determines a curve in the $(\mu,
\phi)$ plane. (We determined this curve above, using stationarity of the phase for the Schrodinger quantization). In general one expects this curve to be different from the classical curve in the region of small $\mu$ (small volume). This is what happens for the loop quantized theory.
The result of the computations is that Schrodinger quantization, the curve $ |\mu|_{p_{\phi}^*, \mu^*}(\phi)$, does approach the $\mu = 0$ axis asymptotically. However for LQC, the curve [*bounces away*]{} from the $\mu = 0$ axis. In this sense – and now inferred in terms of physical quantities – the Big Bang singularity is resolved in LQC. It also turns out that for large enough values of $p^*_{\phi}$, the quantum trajectories constructed by the above procedure are well approximated by the trajectories by the effective Hamiltonian. All these statements are for semi-classical solutions which are peaked at large $\mu_*$ at late times.
Two further features are noteworthy as they corroborate the suggestions from the effective Hamiltonian analysis.
First one is revealed by computing expectation value of the matter density operator, $\rho_{\mathrm{matter}} :=
{{\textstyle \frac{1}{2}}}\widehat{(p_{\phi}^*)^2 |p|^{-3}}$, at the bounce value of $|p|$. It turns out that this value is sensitive to the value of $p_{\phi}^*$ and can be made arbitrarily [*small*]{} by choosing $p_{\phi}^*$ to be [*large*]{}. Physically this is unsatisfactory as quantum effects are [*not*]{} expected to be significant for matter density very small compared to the Planck density. This is traced to the quantization of the gravitational Hamiltonian, in particular to the step which introduces the ambiguity parameter $\mu_0$. A novel solution proposed in the “improved quantization”, removes this undesirable feature.
The second one refers to the role of quantum modifications in the gravitational Hamiltonian compared to those in the matter Hamiltonian (the inverse volume modification or $B(\mu)$). The former is much more significant than the latter. So much so, that even if one uses the $B(\mu)$ from the Schrodinger quantization (i.e. switch-off the inverse volume modifications), one still obtains the bounce. So bounce is seen as the consequence of $\hat{\Theta}$ being different and as far as qualitative singularity resolution is concerned, the inverse volume modifications are [*un-important*]{}. As the effective picture (for symmetric constraint) showed, the bounce occurs in the classical region (for $j = 1/2$) where the inverse volume corrections can be neglected. For an exact model which seeks to understand why the bounces are seen, please see [@MartinExact].
[**Improved Quantization:**]{} The undesirable features of the bounce coming from the classical region, can be seen readily using the effective Hamiltonian, as remarked earlier. To see the effects of modifications from the gravitational Hamiltonian, choose $j = 1/2$ and consider the Friedmann equation derived from the effective Hamiltonian leading to the effective energy density (\[EffDensity\]), with matter Hamiltonian given by $H_{\mathrm{matter}} = {{\textstyle \frac{1}{2}}}p^2_{\phi} |p|^{-3/2}$. The positivity of the effective density implies that $p \ge p_*$ with $p_*$ determined by vanishing of the effective energy density: $\rho_* := \rho_{cl}(p_*)
= ({{\textstyle \frac{8\pi G \mu_0^2 \gamma^2}{3}}} p_*)^{-1}$. This leads to $|p_*| =
\sqrt{{{\textstyle \frac{4 \pi G \mu_0^2 \gamma^2}{3}}}} |p_{\phi}|$ and $\rho_* =
\sqrt{2} (\frac{8\pi G \mu_0^2 \gamma^2}{3})^{-3/2} |p_{\phi}|^{-1}$. One sees that for large $|p_{\phi}|$, the bounce scale $|p_*|$ can be large and the maximum density – density at bounce – could be small. Thus, [*within the model*]{}, there exist a possibility of seeing quantum effects (bounce) even when neither the energy density nor the bounce scale are comparable to the corresponding Planck quantities and this is an undesirable feature of the model. This feature is independent of factor ordering as long as the bounce occurs in the classical regime.
One may notice that [*if*]{} we replace $\mu_0 \to \bar{\mu}(p) :=
\sqrt{\Delta/|p|}$ where $\Delta$ is a constant, then the effective density vanishes when $\rho_{cl}$ equals the critical value $\rho_{\mathrm{crit}} := ({{\textstyle \frac{8\pi G \Delta \gamma^2}{3}}})^{-1}$, which is independent of matter Hamiltonian. The bounce scale $p_*$ is determined by $\rho_* = \rho_{\mathrm{crit}}$ which gives $|p_*| =
({{\textstyle \frac{p_{\phi}^2}{2\rho_{\mathrm{crit}}}}})^{1/3}$. Now although the bounce scale can again be large depending upon $p_{\phi}$, the density at bounce is always the universal value determined by $\Delta$. This is a rather nice feature in that quantum geometry effects are revealed when matter density (which couples to gravity) reaches a universal, critical value regardless of the dynamical variables describing matter. For a suitable choice of $\Delta$ one can ensure that a bounce always happens when [*the energy density*]{} becomes comparable to the Planck density. In this manner, one can retain the good feature (bounce) even for $j =
1/2$ thus “effectively fixing” an ambiguity parameter and also trade another ambiguity parameter $\mu_0$ for $\Delta$. This is precisely what is achieved by the “improved quantization” of the gravitational Hamiltonian [@APSThree].
The place where the quantization procedure is modified is when one expresses the curvature in terms of the holonomies along a loop around a “plaquette”. One shrinks the plaquette in the limiting procedure. One now makes an important departure: the plaquette should be shrunk only till the physical area (as distinct from a fiducial one) reaches its minimum possible value which is given by the area gap in the known spectrum of area operator in quantum geometry: $\Delta = 2\sqrt{3}\pi
\gamma G\hbar$. Since the plaquette is a square of fiducial length $\mu_0$, its physical area is $\mu_0^2|p|$ and this should set be to $\Delta$. Since $|p|$ is a dynamical variable, $\mu_0$ cannot be a constant and is to be thought of a function on the phase space, $\bar{\mu}(p) := \sqrt{\Delta/|p|}$. Thus we need to define an operator corresponding to the classical expression: $h_f :=
\mathrm{exp}(i{{\textstyle \frac{1}{2}}}f(p)c)$, we have taken a general function $f(p)$. This is little non-trivial since there is no $\hat{c}$ operator and $c, p$ are conjugate variables.
Observe that the usual holonomy operator effects a shift in the argument of eigenstates of the triad operator and formally the operator looks like exp$(\nu {{\textstyle \frac{d}{d\mu}}})$ i.e. it effects the action of a [*finite*]{} diffeomorphism generated by a vector field on the wavefunction. We will take this as a guiding principle.
Let $\Phi_f$ denote a diffeomorphism effecting a [*unit parameter shift*]{} along the integral curve of the vector field $f(\mu){{\textstyle \frac{d}{d\mu}}}$ and $\Phi_f^*$, the corresponding pull-back map. We define $\widehat{h_f}\Psi(\mu) := [\Phi^*_f(\Psi)] :=
\Psi(\Phi_f(\mu))$. As argued above, for a constant function, this reduces to the usual action (\[HolonomyFluxRepren\]). It can be checked directly that this action is also [*unitary*]{} in the kinematical Hilbert space: $(\Phi, \Psi) = \sum_{\mu} \Phi^*(\mu)
\Psi(\mu)$ where the sum is over a countable set (this follows from $|\Psi\rangle := \sum_{\mu \in \mathrm{countable~subset} \subset
\mathbb{R}} \Psi(\mu)|\mu\rangle$)[^23].
To compute a unit parameter shift due to the diffeomorphism generated by $f(\mu)$, solve the equation $$\label{AffineParameter}
\int_{\mu}^{\mu'} \frac{dx}{f(x)} ~=~ \int_v^{v + 1} dv = 1
$$ This will give $\mu' := \Phi_f(\mu)$.
For the specific choice of $f(p) := \bar{\mu}(p) := \sqrt{\Delta}
|p|^{-1/2},\ \Delta := \gamma\sqrt{3}{\ell_{\mathrm P}}^2/4,\ p(\mu) =
\gamma{\ell_{\mathrm P}}^2\mu/6$, one gets, $$\begin{aligned}
\sqrt{\frac{\Delta}{|p|}} & = & \sqrt{\frac{3\sqrt{3}}{2}}|\mu|^{-1/2}
~=: ~ f(\mu) \hspace{2.0cm} \Rightarrow \nonumber \\
\mathrm{sgn}(\mu')|\mu'|^{3/2} & = & \mathrm{sgn}(\mu)|\mu|^{3/2} +
K^{-1} ~~\hspace{2.0cm} K := \frac{2}{3}\sqrt{\frac{2}{3\sqrt{3}}} \\
\left[\widehat{e^{f(\mu)\frac{d}{d\mu}}}\Psi\right](\mu) & := &
\Psi(\mu'). \nonumber
$$ It is evident from the above that if we define $v := K
\mathrm{sgn}(\mu)|\mu|^{3/2}$, then the middle eqn reads: $v' = v + 1$. This suggests that we use $|v\rangle$ as a basis instead of $|\mu\rangle$. Apart from the constant $K$, and the sgn, $v$ is related to the eigenvalue of the volume operator $|p|^{3/2}$. Note that $v$ as a function of $\mu$ is one-to-one and on-to. Using the $h_{\bar{\mu}}$ operator and a basis labelled by volume eigenvalues, the Hamiltonian constraint is defined and difference equation is obtained as before. The relevant expressions are: $$\begin{aligned}
v & := & K \mathrm{sgn}(\mu) |\mu|^{3/2}~ ~; \label{VDefn}\\
\hat{V}|v\rangle & = &
\left(\frac{\gamma}{6}\right)^{3/2}\frac{{\ell_{\mathrm P}}^3}{K}\ |v|~ |v\rangle~,~ \\
\widehat{e^{ik{{\textstyle \frac{\bar{\mu}}{2}}}c}}\Psi(v) & := & \Psi(v + k)~, \\
\left.\widehat{|p|^{-1/2}}\right|_{j = 1/2, l = 3/4}\Psi(v) & = &
\frac{3}{2}\left(\frac{\gamma{\ell_{\mathrm P}}^2}{6}\right)^{-1/2}
K^{1/3}|v|^{1/3}\left| |v + 1|^{1/3} - |v - 1|^{1/3}\right| \Psi(v) \\
B(v) & = & \left(\frac{3}{2}\right)^{3/2} K |v| \left| |v + 1|^{1/3} -
|v - 1|^{1/3}\right|^3 \label{ImprovedInverse}\\
\hat{\Theta}_{\mathrm{Improved}}\Psi(v, \phi) & = & - [B(v)]^{-1}
\left\{ C^+(v) \Psi(v + 4, \phi) + C^0(v) \Psi(v, \phi) + \right.
\nonumber \\
& & \left. \hspace{5.0cm} C^-(v) \Psi(v - 4, \phi) \right\} \ , \\
C^+(v) & := & \frac{3 \pi K G}{8} |v + 2| \left| ~ |v + 1| - |v + 3|
\right| \ , \\
C^-(v) & := & C^+(v - 4) ~ ~ , ~ ~ C^0(v) ~ := ~ - C^+(v) - C^-(v) \ .
$$
Thus the main changes in the quantization of the Hamiltonian constraint are: (1) replace $\mu_0 \to \bar{\mu} := \sqrt{\Delta/|p|}$ in the holonomies; (2) [*choose*]{} symmetric ordering for the gravitational constraint; and (3) [*choose*]{} $j = 1/2$ in both gravitational Hamiltonian and the matter Hamiltonian (in the definition of inverse powers of triad operator). The “improvement” refers to the first point. This model is singularity free at the level of the fundamental constraint equation (even though the leading coefficients of the difference equation do vanish, because the parity symmetry again saves the day); the densities continue to be bounded above – and now with a bound independent of matter parameters; the effective picture continues to be singularity free and with undesirable features removed and the classical Big Bang being replaced by a quantum bounce is established in terms of [*physical*]{} quantities.
[*There is yet another spin on the story of singularity resolution!*]{}
### Madhavan Quantization [@Madhavan]:
The improved quantization scheme works primarily through the holonomy corrections, so much so that even if the inverse volume corrections in the matter are turned-off by hand, the singularity resolution continues. Madhavan works within the same kinematical Hilbert space of LQC but treats the Hamiltonian constraint differently, exploiting its specific, simple form for the massless scalar matter. In his quantization of the Hamiltonian constraint, it is the inverse volume corrections that are responsible for singularity resolution (also in terms of physical quantities) and holonomy corrections are by-passed completely.
He observes that the classical Hamiltonian constraint (\[ClassHamConstraint\]), is quadratic in $c$ and is of the form of difference of two squares. It can therefore be written as a product, $$C_{\mathrm{tot}} = - C_+C_- ~~,~~ C_{\pm} := - \sqrt{\frac{6}{\gamma}} c
|p|^{1/4} \pm \sqrt{\kappa} \frac{p_{\phi}}{|p|^{3/4}}
$$ The $C_{\pm}$ are linear in $c$. Since to define physical Hilbert space, a general procedure is to average over the group generated by the constraint, and this involves exponentiation of the constraint, one can directly define these operators which involve factors of the same form as the $h_f$ operators of the improved quantization!
The key differences in Madhavan quantization are: (1) The Hamiltonian constraint is regulated differently from the analogue of LQG and as a consequence, [*there are no holonomy corrections*]{} (sin$^2(c)$). However, the inverse triad corrections can be incorporated in the definition of $\hat{C}_{\pm}$ through the $\hat{p}_{\phi}$ term. One again uses the volume eigenvalues basis for the inverse triad definition; (2) the physical states are constructed directly by group averaging using the well-defined unitary operators, $e^{i\alpha
\hat{C}_{\pm}}$, consequently [*there is no difference/differential equation to be solved*]{}; (3) One of the Dirac observables, $p_{\phi}$ is the same but another one is somewhat different. Nevertheless, [*classical solutions*]{} can be derived from their expectation values; (4) The issue of independence from the fiducial cell (discussed in the next subsection) is also addressed differently.
The results are: (i) without inverse triad modifications, the classical (singular) solution is recovered; (ii) with inverse triad modifications, there [*is*]{} extension of the solution past the classical singularity with the energy density remaining bounded all through, making the extension non-singular. [*There is no bounce, but a regular extension!*]{}
Although Madhavan’s procedure of bypassing the holonomy corrections completely, is tied to the particular form of the constraint of the isotropic model (and hence may not extend to other models), it does demonstrate the possibility that there are inequivalent ways of constructing [*physical Hilbert space and observables*]{} starting from the [*same kinematical structures*]{}. Secondly, singularity resolution [*need not*]{} be seen only as a classical/quantum [*bounce*]{}, a regular extension is also a distinct possibility.
More details should be seen in [@Madhavan].
### Role of the Fiducial cell in spatially flat models {#CellIndependence}
Recall that in the description of spatially homogeneous and isotropic models one begins with a metric of the form (\[FRWMetric\]). The spatial metric is a metric with spatially constant (but possibly time dependent) curvature. This is conveniently taken to be a time dependent scaling of a fixed [*co-moving metric*]{} with corresponding [*co-moving coordinates*]{}. Although not strictly necessary, let us assign length dimension to the co-moving coordinates and take the scale factor to be dimensionless. For non-flat models the co-moving metric can be normalized to have the Ricci scalar to be $\pm 1$ in appropriate units (Ricci scalar has dimensions of (length)$^{-2}$). Note that this is a [*local*]{} condition, and by homogeneity, holds everywhere on the spatial manifold. It is independent of the [*size*]{} of the spatial manifold. For flat models, such a normalization of the co-moving metric is not possible. In this case, there is an arbitrariness in the [*definition*]{} of the scale factor. Clearly, by focusing only on those quantities which are invariant under constant scaling of the scale factor, eg $\dot{a}/a, \ddot{a}/a$, the energy density etc we can obviate the need for choosing a co-moving metric/coordinates. The equations of motion - the Friedmann equation and the Raychaudhuri/continuity equation - reflect this feature.
However, spatial flatness, homogeneity and isotropy also implies existence of (global) Cartesian coordinates with a metric $g^0_{ij} =
\delta_{ij}$ with the coordinate differences giving distances in the chosen unit of length. This unit is arbitrary, but also determines the unit of time by putting speed of light to be one. Change in this unit results in an overall scaling of the [*space-time metric*]{} but does not affect the scale factor. The scale factor is now unambiguously identified and co-moving coordinates and metric are also fixed.
Construction of a quantum theory of the scale factor degree of freedom (and matter homogeneous degrees of freedom) begins with a [*four dimensional action*]{} principle restricted to homogeneous modes of the fields. The action contains a spatial integration which is [*divergent*]{} for spatially flat models, thanks to homogeneity. To have a well defined phase space formulation, we need to regulate this divergence. This is done by introducing an [*arbitrarily chosen fiducial cell*]{}, specified by finite ranges of the co-moving coordinates (thus having a finite co-moving volume $V_0$) and restricting the integrations to this cell. Note that this [*need and the freedom*]{} in the choice of the fiducial cell arises strictly due to the need for an action formulation for the full theory and the assumption of spatial homogeneity[^24]. All subsequent computations will carry a dependence on this cell, either explicitly or implicitly. In the end, this dependence is to be removed by taking a suitable limit $V_0 \to \infty$[^25]. Precisely at what stage and how should one take the limit?
In the canonical formulation of the full theory, the fiducial volume, $V_0$, appears in the symplectic structure. This can however be absorbed away by redefining the canonical coordinates ($\tilde{c}, \tilde{p} \to
c, p$). This makes the canonical coordinate $p$ to have dimensions of (length)$^2$. Note that the physical volume of the cell is $a^3(t)\times
V_0$ is now directly given by $|p|^{3/2}$. [*Apparently*]{}, there is no reference to the fiducial cell any more in the model. However this is not so. The $|p|^{3/2}$ is the physical volume [*of the fiducial cell*]{}. All subsequent computations, whether classical or quantum, done using the ($c, p$) variables[^26] have [*no explicit reference*]{} to the cell. For example, the classical solution obtained in terms of phase space trajectory, (eqn. \[ClassRelationalSoln\]), does not depend on $V_0$.
As discussed above, the subsequent steps in the quantization, do not introduce any further dependence on the fiducial cell. It is no where in sight even in the computation of the phase space trajectory (expectation values of the Dirac observables). These trajectories of course differ from the corresponding classical trajectories. The problem of Big Bang singularity is however phrased in the framework of space-time geometry, specifically, in terms of backward evolution of the [*scale factor*]{}. So we need to [*transcribe*]{} the phase space trajectories (computed in terms of expectation values of Dirac observables) into evolution of the space-time geometry i.e. the scale factor. At this stage, a scale factor (and an explicit reference to the fiducial cell) is re-introduced via the triad variable as, $a := \xi
\sqrt{p}$ where $\xi$ has dimensions of (length)$^{-1}$ and can be identified with the fiducial volume: $\xi^{-1} = V_0^{1/3}$ (since $p^{3/2}$ is the physical volume [*of the fiducial cell*]{}). The phase space evolution then gives $a(t)$. The scale factor evolution so deduced could have some dependence on $\xi$. After taking the limit $\xi \to 0$ ($V_0 \to \infty$), the evolution that survives, is the prediction of the quantum theory, Whether or not this evolution is [*singularity free (i.e. all physical quantities remain bounded through out the evolution)*]{} is the central question of interest. Since the phase space curves are inferred from the expectation values, the states in which these are computed are also important to specify. These are expected to be computed in physical states peaked on large volume $p \gg {\ell_{\mathrm P}}^2$ and small energy density (say), corresponding to a classical regime. The singularity free evolution is required to hold for [*all*]{} such states.
The classical evolution given in eqn (\[ClassRelationalSoln\]) provides an example of this transcription. We see that $\xi^{-2}$ cancels out from both $p$ and $p_*$ and the classical evolution of the scale factor is [*independent*]{} of the fiducial cell as it should be. The LQC computed solution for $p$, always shows a bounce, is not very explicitly expressed and also contains an implicit dependence on the fiducial cell. It matches pretty closely with the classical solution in the large $p$ regime and therefore could be expected to be $V_0$ independent in these regimes. This removes the cell dependence in the initial condition and the question boils down to whether the bounce feature and value of $p$ at the bounce, is independent of $V_0$.
The APS investigations[@APSOne; @APSTwo] found that, in the $\mu_0-$scheme, a bounce can occur even for low values of energy density something which is not exhibited by the observed isotropic universe. Furthermore, the energy density at the bounce - which is a physical observable - has a $V_0-$dependence. So the quantization scheme has some problems. What exactly does this mean?
Note that this does not necessarily mean that there is any mathematical inconsistency in the process of quantization. However, the constructed quantum theory should agree with GR for low energy densities (i.e. have acceptable infra-red behaviour) and hopefully also [*imply a non-singular evolution*]{}. It is possible that it may fail this expectation. APS analysis concludes that the $\mu_0-$scheme with symmetric ordering fails this test.
[*In retrospect*]{}, this failure could have been inferred in the following way. The earlier methods of analysis were based on WKB approximation and effective Hamiltonians[@EffHam] derived from it. This allowed us to encode quantum modifications in terms of effective density and effective pressure, defined by computing the left hand sides of the Friedmann and the Raychaudhuri equations using the effective Hamiltonian. In these papers, the scale factor was introduced by setting $p := a^2/4$ (dimensionful) and therefore still refers (implicitly) to fiducial cell. To make the fiducial cell explicit, replace this scale factor as $a \to a\xi^{-1}$. The expressions for the energy density (say) can be transcribed in terms of the (dimensionless) scale factor and $\xi$. As explained above, prediction of the quantum theory for the scale factor evolution is obtained by taking the limit $V_0 \to \infty
(\xi \to 0)$. Note that this is now done at the level of equations as opposed to at the level of individual curves which need initial conditions to be chosen (which we have argued to be independent of $V_0$).
Recall the effective density given in eqn.\[\[EffDensity\]\], $$\begin{aligned}
\label{HolonomyCorrection}
\rho_{\mathrm{eff}} & = &
\left(\frac{H_{\mathrm{matter}}}{p^{3/2}}\right)\left[1 -
\frac{2\kappa\mu_0^2\gamma^2}{3}p
\left(\frac{H_{\mathrm{matter}}}{p^{3/2}}\right)\right] ~~~,
~~~\mathrm{where} \nonumber \\
H_{\mathrm{matter}} & = & \frac{1}{2} p_{\phi}^2 \left\{(2jp_0)^{-3/2}
\left(F_{\ell}(q)\right)^{{{\textstyle \frac{3}{2(1 - \ell)}}}} \right\} ~~,~~ q :=
\frac{p}{2jp_0}
$$ The square bracket contains the modification implied by [*holonomy corrections*]{}[^27]. The inverse volume corrections are contained in the matter Hamiltonian (see eqns.(\[InvTriad\], \[Ffunction\])).
Consider first the inverse volume corrections. Observe that with $p \to
a^2\xi^{-2}/4$ , $q \sim a^2\xi^{-2} \gg 1$ in the limit $\xi \to 0$. The limiting form of $F_{\ell}$ then implies that $p^{-3/2}
H_{\mathrm{matter}} \sim p_{\phi}^2\xi^6a^{-6}$. For massless scalar matter, the classical equation of state has $P/\rho = 1$ and hence the classical density behaves as $\sim a^{-6}$. Thus $p^2_{\phi}\xi^6$ must be a constant for any particular solution. For the leading term to give the classical evolution, we have to take the limit $\xi \to 0$ [*along with*]{} $p_{\phi} \to \infty$ keeping $p_{\phi}\xi^3$ a [*constant*]{} specifying a particular initial condition. This understood, the $p^{-3/2}H_{\mathrm{matter}}$ factors go over to the [*cell independent*]{} classical density plus corrections down by $q^{-2} \to 0$ in the limit. Thus, inverse volume [*corrections*]{}, simply vanish when $V_0 \to \infty$ is imposed.
Now consider the holonomy corrections. By the same logic as above, the holonomy corrections, second term in the square bracket in the effective density expression, goes as $\xi^{-2} \to \infty$!. This is clearly unacceptable. Thus, in the $\mu_0$ scheme of quantization, the inverse volume modifications do not survive the limit while the holonomy modifications give an inconsistency and neither shed any light on the singularity resolution issue.
True as these features are, they are not immediately conclusive to look for alternative quantizations because the fault may be with the WKB approximation and the corresponding effective Hamiltonians. For instance if one took the effective density from the first paper of [@EffHam], the holonomy corrections would also be down by inverse powers of $q$ and would vanish which is okay for the classical limit but the extrapolation to the quantum regime is unreliable since WKB is unreliable at turning points. Perhaps, physical level computations would clarify the issue. This is indeed the case. With the physical level computations, APS results show the unacceptability of the $\mu_0-$scheme while in Madhavan’s approach, with no holonomy corrections, the inverse volume corrections would simply vanish by the argument given above. Both APS and Madhavan have also suggested ways out.
The APS analysis discussed above shows that the $\mu_0 \to \bar{\mu}(p) =
\sqrt{\Delta}{\ell_{\mathrm P}}/\sqrt{p}$ substitution in the holonomies used in replacing the $c$ variable, suffices to obtain a non-singular evolution with good infra-red behaviour. It implies that the deviations from classical evolution (eg close to the bounce) occur when the energy density reaches a universal, maximal value. This substitution also renders the inverse triad correction from the matter sector highly suppressed[^28].
Madhavan suggests that along with the APS suggested substitution, one should also introduce a multiplicative parameter $\lambda$ as $\mu_0 \to
\lambda\bar{\mu}(p)$. Since only corrections that drive the quantum modifications are the inverse volume correction and these go as $(\lambda/v_1)^2 \sim (\lambda\xi^3{\ell_{\mathrm P}}^3/a^3)^2$ (see eqn \[Mu1GenVals\]), $\xi$ independence is achieved by choosing $\lambda
\sim (\bar{\xi}/\xi)^{3}$ where the new dimensionful parameter $\bar{\xi}$ is supposed to reflect a scale provided by an underlying LQG state supporting the homogeneity approximation. For details, please see [@Madhavan].
To summarize: For the spatially flat, isotropic models, let us choose the Cartesian coordinates with the standard Euclidean metric for the spatial slice and choose proper time as the time coordinate so that the space-time metric takes the form (\[FRWMetric\]). The scale factor is now specified unambiguously. For constructing a quantum theory, we need to choose a regulator cell with volume $V_0$. While the cell dependence can be hidden by choosing scaled variables, it manifests again because quantum computed evolution must be transcribed in terms of the scale factor evolution. This is necessary because the classical Big Bang singularity is understood as a singular evolution of the scale factor so its resolution lies in making the evolution non-singular.
The scale factor evolution can be cast in the form of the Friedmann equation with possible deviations from classical evolution, encoded in the effective density. A prediction of the quantum theory is the surviving correction terms after taking the limit $V_0 \to \infty$. A quantum theory could be understood to have resolved the Big Bang singularity if the surviving evolution is non-singular. Cell independence of quantum corrections automatically implies that non-trivial limit exists. Not every quantization scheme passes this test.
There are two types of corrections - the holonomy corrections and the inverse triad corrections. These have different properties in the limit. The APS quantization with $\mu_0-$scheme implies that holonomy corrections dominate and lead to unphysical implications. These are cured by the $\bar{\mu}-$scheme. The Madhavan quantization scheme, even with the $\bar{\mu}$ substitution in the inverse volume definition, these corrections again vanish unless additional $\lambda$ parameter is introduced. In either case, extra ingredients (scales) have to be ‘imported’ to get non-trivial results. Both the schemes have ingredients (role played by the area operator in the APS scheme[^29] and the specific form of constraint in the Madhavan scheme) which do not have counterparts in the full theory as it is understood at present.
So far the discussion has been within the context of full theory being classically reduced directly to a homogeneous and isotropic model. In the next subsection, we briefly discuss how homogeneity and isotropy can be viewed from within a particular quantized inhomogeneous model.
### A View from Inhomogeneity {#LatticeView}
To keep the flow of the in focus, basic details of the inhomogeneous model are given in the appendix 5.4.
The fundamental change in the way homogeneity is viewed, is that it is a property exhibited by a [*state*]{} of an inherently (spatially) inhomogeneous model. For definiteness, a lattice model with a lattice spacing $\ell_0$ is taken. This allows for [*states*]{} of the model which can be considered as [*homogeneous on a certain scale*]{}, eg $\ell_0 N^{1/3}$. This also allows the fields to be restricted to be periodic on this scale. Thus what is fundamental is the lattice spacing $\ell_0$ below which it makes no sense to consider inhomogeneity (or inhomogeneities are not probed) and a scale $N^{1/3} \ell_0$ provided by a state of the model. Let us call the former as the [*micro-*]{}scale and the latter as the [*macro-*]{}scale. The fundamentally isotropic model refers to the macro-scale. The fiducial cell of the isotropic model is determined by these two scales with fiducial volume given by, $V_0 := N \ell_0^3$. Notice that in this view, $V_0$ (or $N$) is a property of a quantum state and there is [*no reason*]{} to contemplate a limit $V_0 \to \infty$.
Now all quantum effects due to inverse volume and holonomies, arise at the micro-scale. To see how these translate or correspond to the quantum effects seen in the fundamentally isotropic model, let us begin by identifying variables.
In the lattice model, isotropic connection is defined by $\tilde{k}(x) =
\tilde{c}, \forall\ x, I$. Identifying this constant value with $\tilde{c}_{\mathrm{iso}}$ and comparing the basic link holonomies with the holonomies of the isotropic model implies, $$\label{CIdentification}
c_{\mathrm{iso}} ~:=~ V_0^{1/3}\tilde{c}_{\mathrm{iso}} ~=~ V_0^{1/3}
\tilde{k}_I ~:=~ V_0^{1/3} \ell_0^{-1} c_{\mathrm{lat}} ~=~
N^{1/3}c_{\mathrm{lat}}
$$ The first equality is the definition from the isotropic model, the second one identifies the isotropic connection with the lattice connection $\tilde{k}$, the third one defines $c_{\mathrm{lat}}$ and forth equality gives the final relation between the ‘$c$’ variables of the isotropic and the lattice models.
The state exhibiting isotropy on the macro-scale, may be characterised by stipulating that the $\tilde{p}_I(\vec{v})$ values are all mutually equal, and equal to $\tilde{p}_{\mathrm{lat}}$, for the vertices comprising the fiducial cell and that this value is identified with the isotropic variable, $\tilde{p}_{\mathrm{iso}}$. Recall that the isotropic operator is obtained by an averaging of the lattice operators. The stipulation says that the average value is realized at each of the lattice vertices. This leads to, $$\label{PIdentification}
p_{\mathrm{iso}} ~:=~ V_0^{2/3}\tilde{p}_{\mathrm{iso}} ~=~
V_0^{2/3}\tilde{p}_{\mathrm{lat}} ~=~
V_0^{2/3}p_{\mathrm{lat}}\ell_0^{-2} ~=~ p_{\mathrm{lat}} N^{2/3}
$$ These identification give a relation between isotropic variables and a state of lattice model with a scale parameter $N$. A dynamically evolving isotropic universe may be thought of as a family of lattice states with the scale $N$ being a function of the volume eg larger number of elementary cells get ‘homogeneised/isotropised’. As an example, if $N \propto p_{\mathrm{iso}}^{3/2}$, then $p_{\mathrm{lat}}$ will be a constant! The expressions seen in the context of isotropic models with fiducial cell of size $V_0$ are now to be applied to the elementary cell of size $\ell_0^3$.
In view of these identifications, let us consider the two specific corrections seen in the isotropic model, namely the the inverse volume corrections and the holonomy corrections. In the lattice model, these corrections arise in the same manner as in the isotropic model, but from the micro-cell. The inverse volume corrections are in powers of $p_*/p =
p_*/p_{\mathrm{lat}} = (p_*/p_{\mathrm{iso}})N^{2/3}$. For $N \propto
p_{\mathrm{iso}}^{3/2}$, these are independent of $p_{\mathrm{iso}}$. (ii) With the holonomy corrections included, the effective density (\[HolonomyCorrection\]), is of the form $\rho(1 -
\rho/\rho_{\mathrm{crit}})$ with $\rho^{-1}_{\mathrm{crit}} \sim
\kappa\gamma^2p_{\mathrm{lat}} =
\kappa\gamma^2p_{\mathrm{iso}}N^{-2/3}$. Again for $N \propto
p_{\mathrm{iso}}^{3/2}$, $\rho_{\mathrm{crit}}$ is independent of $p_{\mathrm{iso}}$. For a different dependence of $N$ on $p_{\mathrm{iso}}$, the corrections will have non-trivial dependence on $p_{\mathrm{iso}}$ and therefore on $V_0$, but $V_0$ is no longer a purely mathematical artifact but is dictated by the underlying inhomogeneous state.
In effect, a perspective from an underlying inhomogeneous model suggests that the fiducial cell of a homogeneous model is selected by a state of the inhomogeneous model and a dependence on the $V_0 := \ell_0^3 N$ is not necessarily unphysical. For a more detailed discussion of ramifications of these ideas, please see [@MartinLattice].
Appendix
========
Symmetric connections
---------------------
We assume that we have a manifold $M$ on which are defined connection $A_{\mu}^a(z)$ with $a$ taking values in the Lie algebra $\underline{G}$ of a [*gauge group*]{} $G$. Assume further that there is an action of a [*symmetry group*]{} $S$ on $M$ under which we want to have appropriate notion of invariance. The infinitesimal action of the symmetry group is generated by a set of vector fields $\xi_m^{\mu}\partial_{\mu}$ which represent the Lie algebra of $S$: $[\xi_m, \xi_n] = f^p_{~mn}X_p$. The Lie algebra of $G$ is generated by matrices $T_a$ satisfying: $[T_b,
T_c] = C^a_{~bc}T_c$.
When we have an ordinary tensor field, $T$, on a manifold, it is defined to be invariant under (or symmetric w.r.t.) the action of an infinitesimal diffeomorphism generated by a vector field $\xi$ if its Lie derivative with respect to $\xi$ vanishes: $L_{\xi} T = 0$. When the tensor fields also transform under the action of a gauge group, then the invariance condition allows the Lie derivative to be an infinitesimal gauge transform of the tensor field: $L_{\xi}T = \delta_{W(\xi)}T$, where $W(\xi)$ is valued in the Lie algebra $\underline{G}$. Notice that this associates a gauge transformation with a diffeomorphism.
This association has to satisfy two conditions: (a) If we took a gauge transform of the tensor field and then applied the diffeomorphism, the defining condition must be gauge covariant: $L_{\xi}(T^g) =
\delta_{W^g(\xi)} T$, where $W^g(\xi) = g^{-1}W(\xi)g +
g^{-1}L_{\xi}(g)$, and (b) The Lie derivatives represent the Lie algebra of the symmetry group: $[L_{\xi_m}, L_{\xi_n}] T = L_{[\xi_m, \xi_n]} T
= f^p_{~mn}L_{\xi_p} T$ and the $W_m$ must obey the consequent conditions. The task is to find those tensor fields which satisfy the invariance conditions subject to the allowed gauge transformations.
This is aided by another consequence of the symmetry action. The action of the the symmetry group $S$ on $M$ implies that $M$ can be expressed as a collection of [*orbits*]{} of $S$. We will assume the simpler case where [*all*]{} orbits are mutually diffeomorphic and are given by $S/F$ where $F$ is the stability subgroup of the $S-$action. Thus we obtain $M
\sim B + S/F$. Here $S/F$ is an orbit which is necessarily a coset space while $B$ is a manifold whose points label the orbits. Note that a non-trivial subgroup $F$ of $S$ means that a subset of vector fields $\xi_m$ vanish at some point. Corresponding to this structure of $M$, its tangent and cotangent spaces are also decomposed.
The solutions of the invariance conditions are constructed by using the available structure on the group manifold $S$ and projecting these onto $S/F$. Here are some details for the gauge connection[^30] [@ForgacsManton].
Let $A := A^a_{\mu}T_a dx^{\mu}$ denote the $\underline{G}$ valued connection 1-form (the gauge potential). Under a gauge transformation it transforms as: $A^g := g^{-1} A g + g^{-1} d g$ and infinitesimally, $g
= 1 + \epsilon W$, $\delta_{\epsilon W}A = \epsilon D(W) := \epsilon(d W
+ [A, W]), ~ W := W^a T_a.$ Under an infinitesimal diffeomorphism, $x^{'
\mu} := x^{\mu} + \epsilon\xi^{\mu}$, it transforms as $\delta_{\epsilon\xi}A := - \epsilon L_{\xi}A = -
\epsilon(\partial_{\mu}\xi^{\nu}A_{\nu} + \xi^{\nu}\partial_{\nu}
A_{\mu})dx^{\mu} ~= - \left\{i_{\xi}dA + d(i_{\xi}A)\right\}$.
The invariance conditions, when there are many symmetries, are: $$\label{InvCondition}
L_{\xi_m} A = D(A)\ W_m := d W_m + [A, W_m] ~\Leftrightarrow~
\partial_{\mu}\xi^{\nu}A_{\nu} + \xi^{\nu}\partial_{\nu} A_{\mu} =
\partial_{\mu} W_m + [A_{\mu}, W_m]\ .
$$ Here, $W_m$ are some $\underline{G}$ valued scalars on the manifold $M$ associated with $\xi_m$. We induce a gauge transformation on the $W_m$ by demanding that the above condition be [*gauge covariant*]{}: $$\label{GaugeTransform}
L_{\xi_m}A^g := D(A^g)\ W_m^g ~~~\Leftrightarrow~~~ W_m^g := g^{-1}W_m g
+ g^{-1} L_{\xi_m}g~~,~~L_{\xi_m}\ g := \xi^{\mu}_m\ \partial_{\mu}\ g
$$ The Lie algebra of the vector fields, $\xi_m$, implies that the $W_m$’s must satisfy: $$\begin{aligned}
\label{AlgebraCondition}
[L_{\xi_m}, L_{\xi_n}]A & = & L_{[\xi_m, \xi_n]}A ~=~f^p_{~mn}L_{\xi_p}A
\hspace{3.0cm}\Rightarrow \nonumber\\
D(W_{mn}) & = & f^p_{~mn}D(W_p) ~~~,~~~ W_{mn} := L_{\xi_m}W_n -
L_{\xi_n} W_m + [W_m, W_n]~,~ ~~ or \nonumber \\
0 & = & D(W_{mn} - f^p_{~mn}W_p)
$$ where $[W_m, W_n]$ is the bracket in the Lie algebra, $\underline{G}$.
[*Exercise:*]{} For the field strength $F_{\mu\nu}^aT_a$, verify that $L_{\xi_m} F = [F, W_m]$.
[*Exercise:*]{} Let $E$ be a vector field valued in $\underline{G}$ which transforms as $E^g = g^{-1} E g$. The condition for symmetric $E$ would be $L_{\xi_m} E = [E, W_m]$. Show that the $W_m$ transforms as before and the symmetry Lie algebra implies $[E, W_{mn} - f^p_{~mn}W_p]
= 0$.
Suppose that $\chi_{mn} := W_{mn} - f^p_{~mn}W_p \neq 0$. Then the above equations imply conditions on symmetric field strength (and indeed on all symmetric quantities). For example, $D\chi_{mn} = 0$ implies that $[F, \chi_{mn}] = 0$[^31]. This would mean that the field strengths must commute with the $\chi_{mn}$’s. We will assume that there does not exist any $\chi$ valued in $\underline{G}$ such that $D \chi = 0$. This implies that $\chi_{mn} =
0$ i.e. $$\label{Star}
L_{\xi_m}W_n - L_{\xi_n} W_m + [W_m, W_n] - f^p_{~mn}W_p ~=~ 0 \ .
$$ Note that this is a condition involving only the $W_n$’s and the symmetry generators $\xi_m$’s. Also observe that for vector fields corresponding to the stability subgroup $F$, the (\[Star\]) reduces to $[W_{m}, W_{n}] = f^p_{~mn} W_p$ at the points where these vector fields vanish.
The task is to characterise the symmetric connections satisfying eqn. (\[InvCondition\]) with $W_n$ satisfying eqn. (\[Star\]) modulo gauge transformations (\[GaugeTransform\]) on a manifold $M \sim B
\times S/F$. The strategy is to show that the gauge freedom allows $W_n
$ to be taken in appropriate form and then determine the form of symmetric connections in the same gauge.
Consider first the case where $F = \{e\}$ so that $S/F = S$ itself. Introduce local coordinates $(x^i, y^{\alpha})$ on $B \times S$. Without loss of generality, we can take the symmetry generators to be functions of $y$ and with zero components along $B$. Noting that the vector fields $\xi_m$ on $S$ are independent, it follows that the matrices $\xi^{~\alpha}_m(y)$ are invertible and therefore we can define new $\underline{G}$ valued 1-forms as: $W_m(x, y) := \xi^{~\alpha}_m
W_{\alpha}(x, y)$. It is easy to see that (i) $W_{\alpha}$ transform exactly as a $G-$connection and (ii) the condition (\[Star\]) is just the statement that this connection is flat. Therefore, [*locally*]{} it is always possible to choose $W_{\alpha}(x,y) = 0$ and hence $W_m(x,y) =
0$. Having chosen $W_m$’s to be zero, the gauge transformation freedom is restricted to $L_{\xi_m}(g) = \xi_m^{~\alpha}\partial_{\alpha}g = 0$ i.e. the gauge functions must depend only on the $x$ coordinates.
In this gauge, the invariance conditions can be written separately for $\mu = i$ and for $\mu = \alpha$ as: $$\xi_m^{~\alpha}\partial_{\alpha}A_i = 0 ~~\mathrm{and}~~
(L_{\xi_m}A)_{\alpha} = 0
$$ The first implies that $A_i$ depends only on $x$ and so do the gauge transformations. Hence $A_i$ is a $G-$connection on $B$. The solution for $A_{\alpha}$ is obtained as follows.
On a group manifold, there are left and right actions of the group onto itself which commute. Consequently, these generate left(right) invariant vector fields and 1-forms (the Maurer-Cartan forms). Apply these to the group manifold $S$. Assume that $\xi_m$ generate [*left*]{} action on the group manifold and use the [*left*]{} invariant 1-forms eg the unique, $\underline{S}-$valued Maurer-Cartan form $\Theta_{MC}$[^32]. It is immediate that $L_{\xi_m} \Theta_{MC} = 0$. To obtain a $G-$connection, we need a map $\Lambda: \underline{S} \to
\underline{G}$. Given such a map, we can define $A := \Lambda
(\Theta_{MC}) \leftrightarrow A_{\alpha}^a := \Phi^a_m
(\Theta_{MC})^m_{\alpha}$. Now $L_{\xi_m} A = 0 = L_{\xi_m}(\Phi)
\Theta_{MC} + 0$ implies that the “Higgs” fields, $\Phi_m$ are constants on $S$ i.e. are functions only of $x^i$.
Thus, for the case where the $S-$action is free ($F$ is trivial), the symmetric connections can be written as $A = {\cal A}_i(x)dx^i +
\Phi_m(x) \omega^m_{~\alpha}(y)dy^{\alpha}$ where ${\cal A}$ is an arbitrary $G-$connection on $B$ and $\Phi_m(x)$ are “Higgs” scalars valued in $\underline{G}$.
When the $S-$action is not free, the vector fields $\xi_m^{~\alpha}$ are tangent to $S/F$ and the above steps do not go through immediately. Nevertheless we can still find invariant, $\underline{S}-$valued 1-forms on $S/F$ and a suitable map $\Lambda$ to construct invariant connections. To see this, note that there is the natural projection map $\pi: S \to S/F$. Choose an [*embedding*]{} $i:S/F \to S$. As discussed above, on $S$ we have vector fields $\bar{\xi}_m$ generating left action and the corresponding Maurer-Cartan form $\Theta_{MC}$. Using $\pi_*$ we push-forward the vector field on to $S/F$ and using $i^*$ we pull-back $\Theta_{MC}$ on to $S/F$. The projected vector fields match with the $\xi_m$ (by definition of the symmetry action). Thus we get, $$\xi_n := \pi_*(\bar{\xi}_n) ~~,~~ \omega := i^*(\Theta_{MC}) ~~;~~
L_{\bar{\xi}_n}\Theta_{MC} = 0 ~\Rightarrow~ L_{\xi_n}\omega = 0 \ .
$$ As before, $\omega$ is valued in $\underline{S}$ since $\Theta_{MC}$ is. Introduce $\Phi^a_n$ as before and define $A^a_{\alpha} := \Phi^a_n
\omega^n_{\alpha}$. Using these definitions let us rewrite the defining equations as: $$\begin{aligned}
\mbox{Invariance condition} \hspace{0.9cm} & :~~ & \left(\xi^{\alpha}_n
\omega^k_{\alpha}\right)\left( L_{\xi_m} \Phi_k - [\Phi_k, W_m]\right) ~
= ~ L_{\xi_n} W_m \ ; \label{Inv} \\
\mbox{Lie Algebra condition} \hspace{0.65cm} & :~~ & L_{\xi_m} W_n -
L_{\xi_n} W_m - [W_m, W_n] ~ = ~ f^p_{~mn} W_p \ ; \label{AlgebraCond}\\
\mbox{Gauge transformations} \hspace{0.5cm} & :~~ & W^g_m ~ = ~ g^{-1}
W_m g + g^{-1} L_{\xi_m} g \ ; \label{GaugeW}\\
& :~~ & \left(\xi^{\alpha}_n \omega^k_{\alpha}\right)\left( \Phi^g_k -
g^{-1} \Phi_k g \right) ~ = ~ g^{-1} L_{\xi_n} g \label{GaugePhi}
$$ In writing the first equation we have used $L_{\xi_m}\omega^n = 0$ and also multiplied by $\xi^{\alpha}_n$.
The last equation implies that $\Phi_n$’s transform as the adjoint representation of $G$ [*iff*]{} the gauge transformations are constant over $S/F$. We can take $\Phi_n$ to transform by the adjoint representation of $G$, thereby restricting the gauge transformation to be constant over $S/F$. In such a case, $W_n$’s also transform the same way. For trivial $F$, we [*can*]{} transform away $W_n$ to [*zero*]{} and recover the previous case. For non-trivial $F$, this is not the case.
For non-trivial $F$, there exist a point, $y_0$ say, in $S/F$ at which the vector fields $\xi_{\underline{m}}, \underline{m} = 1, \cdots,
\mathrm{dim}(F)$ vanish. Then $L_{\xi_{\underline{m}}}$ terms drop out. Consider the equation (\[AlgebraCond\]) for $\underline{m},
\underline{n}$. Then, at $y_0$, we must have $[W_{\underline{m}},
W_{\underline{m}}] = f^{\underline{p}}_{~\underline{m}\underline{n}}
W_{\underline{p}}$. Since $F$ is a subgroup, the sum on the right hand side is restricted to $\underline{p}$. If there is a non-trivial homomorphism $\lambda: F \to G$, it will induce a corresponding homomorphism $\Lambda:\underline{F} \to \underline{G}$ on the Lie algebras and we can choose the $W_{\underline{m}}(y^0)$ to represent it.
Next, at $y_0$, consider the (\[Inv\],\[AlgebraCond\]) for $\underline{m}$. Eliminating $L_{\xi_n} W_{\underline{m}}$, and noting that $\Phi_k$ alone depends on $x$, we must have (a) $[W_{\underline{m}}, W_n] = f^p_{~\underline{m} n} W_p$ and (b) $[\Phi_n, W_{\underline{m}}] = 0$. Note that this implies that the residual gauge group is reduced to those elements of $G$ which commute with $W_{\underline{m}}$ i.e. to the [*centralizer of $\lambda(F)
\subset G$*]{}. The gauge transformations are already restricted to be functions of $x$ alone.
Considering the Jacobi identity for $\Phi_{\underline{k}}, \Phi_l,
W_{\underline{m}}$ it follows that $[\Phi_{\underline{k}}, \Phi_l] =
d^m_{~\underline{k} n}\Phi_m$ must hold for some $d$’s. This has the same form as the condition (a) on the $W$’s. Hence $d^m_{~\underline{k}l} = f^m_{~\underline{k}l}$ is obviously a solution.
In fact, it is a result that $S-$invariant connections, when they exist, are in on-to-one correspondence with homomorphisms of the groups $\lambda: S \to G$ and can be expressed as $A(x, y) = {\cal A}_i dx^i +
\Phi(x)_n i^*(\Theta_{MC})^n(y)$ where ${\cal A}$ is a connection on $B$ with the gauge group [*reduced to the centralizer of $\lambda(F)$ in G*]{} (i.e. group of all elements of $G$ which commute with the image of $F$ in $G$ under the homomorphism $\lambda$). Furthermore the Higgs fields have to satisfy the constraints: $[\Phi_{\underline{m}}, \Phi_n]
= f^p_{~\underline{m} n} \Phi_p$[^33].
What about other invariant fields, such as vector fields in the adjoint of the gauge group (eg the triad fields)? Now the invariance condition (\[Inv\]) will change and also the corresponding gauge transformations of the field. For $E^{\mu}_a\partial_{\mu}$, we will have $L_{\xi_m} E =
[E, W_m], E^g = g^{-1}E g$. Now use the projections $X_m$ of the [*left invariant vector fields*]{} $\bar{X}_m$ on $S$ (these generate the right action and are dual to the $\Theta_{MC}$) and write: $E^{\alpha}_a
:= \Psi^n_a X^{\alpha}_n$, $L_{\xi_m} X_n = 0$. The invariance condition then becomes $(\omega^k_{\alpha}X^{\alpha}_n)(L_{\xi_m}\Psi^n - [\Psi^n,
W_m]) = 0$ and $(\omega^k_{\alpha}X^{\alpha}_n)((\Psi^n)^g - g^{-1}
\Psi^n g ) = 0$. The gauge transformations imply $\Psi^n$ transforms by the adjoint representation and for $m = \underline{m}$, the invariance condition implies: $[\Psi_n, W_{\underline{m}}] = 0$. Exactly as before, the $\Psi^n$ must satisfy constraints analogous to the $\Phi_n$’s. Similar logic will hold for other tensor fields.
As a very simple illustration, consider the case of static magnetic field in three dimensions invariant under translations along the z-axix. We want to obtain the form of the vector potential $A_i$. In this case, the gauge group $G = U(1)$ and the symmetry group $S = \mathbb{R}$ which acts on $\mathbb{R}^3$ by translations. This action is free and therefore $F = \{e\}$. We have $\mathbb{R}^3 \sim \mathbb{R}^2 \times
\mathbb{R}$. The Maurer-Cartan form on $\mathbb{R}$ is just $dz$. The map from $\underline{\mathbb{R}} \to \underline{U(1)}$ is given by a single ‘Higgs’ scalar, $\Phi(x,y)$. The symmetric connection is then given by $A_i(x,y,z)dx^i = {\cal A}_x(x,y)dx + {\cal{A}}_y(x,y)dy +
\phi(x,y) dz$. This just says that all the three components of the vector potential depend [*only*]{} on $(x,y)$. Note that this is a statement in the gauge where “$W$” has been set to zero. This implies that the magnetic field is also independent of $z$. Note that since $A_z
:= \phi(x,y) \neq 0$, the magnetic field could be along any fixed direction.
[*Exercise:*]{} Work out spherically symmetric Yang-Mills fields in three dimensions. Now $G = SU(2), \ S = SO(3), \ F = U(1), \ \mathbb{R}^3 \sim
\mathbb{R}^+ \times S^2$.
Further examples may be seen in [@SymmetricConnections].
Schrodinger and Polymer Quantization
------------------------------------
We illustrate inequivalent quantization as well as the GNS procedure in a simple example.
### The Weyl-Heisenberg C\*-Algebra
Consider the usual Schrodinger quantization of a single degree of freedom. We have the usual Hilbert space $L^2(\mathbb{R}, dx)$, on which are defined the self-adjoint operators $x, p$, satisfying the canonical commutation relations: $[x, p] = i\hbar$.
Define the corresponding unitary operators: $$\begin{aligned}
U(\alpha) ~:=~ e^{i \alpha x} & , & V(\beta) ~:=~ e^{i\hbar^{-1}\beta
p}~~,~~\alpha, \beta \in \mathbb{R} \\
U^{\dagger}(\alpha) = e^{-i\alpha x} = U(-\alpha) = U(\alpha)^{-1} & , &
V^{\dagger}(\beta) = e^{-i\hbar^{-1}\beta p} = V(-\beta) = V(\beta)^{-1}
\nonumber
$$
Using the BCH formula, $$e^{A}\cdot e^{B} = e^{A + B + {{\textstyle \frac{1}{2}}}[A, B]
+ {{\textstyle \frac{1}{12}}}[A, [A, B]] - {{\textstyle \frac{1}{12}}}[B, [A, B]] + \cdots}
$$ it follows, $$U(\alpha)U(\alpha') ~=~ U(\alpha + \alpha')~~,~~ V(\beta)V(\beta') ~=~
V(\beta + \beta')~~,~~ U(\alpha)V(\beta) ~=~
e^{-i\alpha\beta}V(\beta)U)(\alpha)
$$
Define, for $z := (\alpha + i \beta)/\sqrt{2} \in \mathbb{C}$, the unitary operator, $$\begin{aligned}
W(z) & := & e^{i\frac{\alpha\beta}{2}} U(\alpha)V(\beta) \hspace{2.9cm}
= ~ e^{i(\alpha x + \hbar^{-1}\beta p)} \nonumber \\
& = & \mathrm{exp}\left[{i\left\{\frac{z + \bar{z}}{\sqrt{2}} x -i
\frac{z - \bar{z}}{\sqrt{2}} \frac{p}{\hbar}\right\}}\right] ~=~
\mathrm{exp}\left[{i\left\{z\frac{x - ip/\hbar}{\sqrt{2}} + \bar{z}
\frac{x + ip/\hbar}{\sqrt{2}}\right\}}\right] \nonumber \\
& = & e^{i\left(z a^{\dagger} + \bar{z} a\right)} \hspace{3.85cm} = ~
e^{-\frac{|z|^2}{2}}\ e^{iza^{\dagger}}\ e^{i\bar{z}a}~~~~
\mathrm{where,} \\
a & := & \frac{1}{\sqrt{2}}\left(x + i \frac{p}{\hbar}\right)
\hspace{3.05cm} \Rightarrow ~ [a, a^{\dagger} ] = 1 \ .
$$ It follows, $$\begin{aligned}
W(z_1) W(z_2) & = & e^{-\frac{i}{2}(\alpha_1\beta_2 -
\alpha_2\beta_1)}W(z_1 + z_2) \nonumber \\
& = & e^{\frac{1}{2}(z_1\bar{z}_2 - z_2\bar{z}_1)} W(z_1 + z_2)
\nonumber \\
& = & e^{\frac{1}{2}\mathrm{Im}(z_1\bar{z}_2)} W(z_1 + z_2)
\hspace{3.0cm} \mathrm{and,} \label{Algebra}\\
W(z)^{\dagger} & = & W(-z) ~=~W(z)^{-1} \ . \label{StarRelation}
$$
Taking finite linear combinations of products of the unitary operators $W(z)$, we get an algebra called the [*Weyl-Heisenberg algebra*]{}, ${\cal W}$. This is \*-algebra due to the Hermitian dagger defined for the operators. The unitary operators $W(z)$ are bounded and so are polynomials in them. With respect to the operator norm (which satisfies $||A^{\dagger}|| = ||A||, ||A^{\dagger}A|| = ||A||^2$), The Weyl-Heisenberg algebra is a C\*-algebra. Notice that the ${\cal W}$ C\*-algebra is [*non-commutative*]{} and has two [*commutative*]{} sub-algebras, namely those generated by the elements, $W({{\textstyle \frac{\alpha +
i0}{\sqrt{2}}}})$ and $W({{\textstyle \frac{0 + i\beta}{\sqrt{2}}}})$ respectively.
Thus at this stage we have constructed a C\*-algebra of bounded operators on the specific Hilbert space. We will now define a [*positive linear functional*]{} on the C\* algebra, ${\cal W}$, construct a unitary representation of the algebra and show its equivalence to that provided by the Schrodinger quantization. The same procedure will then be used to construct another representation, the [*Polymer Representation*]{}, of the same algebra.
### Re-construction of the Schrodinger Representation
In the Hilbert space, consider the wavefunction, $\langle x|0\rangle :=
\psi_0(x) := \pi^{-1/4} e^{-x^2/2}$ so that $\langle 0|0 \rangle := \int
dx |\psi_0(x)|^2 = 1$. Following results hold: $$\begin{aligned}
a |0\rangle & = & \frac{1}{\sqrt{2}}\left(x -
\frac{d}{dx}\right)\psi_0(x) ~~=~~0 \\
\left[W(z)\psi_0\right](x) & = & e^{{{\textstyle \frac{i}{2}}}\alpha\beta}
\left[U(\alpha) V(\beta) \psi_0\right](x) ~~=~~
e^{{{\textstyle \frac{i}{2}}}\alpha\beta}e^{i\alpha}\left[V(\beta)\psi_0\right](x)\nonumber
\\
& = & e^{{{\textstyle \frac{i}{2}}}\alpha\beta} e^{i\alpha x}\psi_0(x + \beta) \\
& = & \langle x|W(z)|0\rangle ~~=~~ e^{-\frac{|z|^2}{2}}\langle x|e^{i z
a^{\dagger}} |0\rangle ~~:=~~ \langle x|z\rangle \\
\therefore \int dx \psi_0^*(x) \left[W(z)\psi_0\right](x) & = &
\pi^{-1/2} \int dx \ e^{-{{\textstyle \frac{x^2}{2}}}} e^{i \alpha x - {{\textstyle \frac{(x +
\beta)^2}{2}}}} \nonumber \\
& = & e^{-{{\textstyle \frac{|z|^2}{2}}}} ~~:=~~ \langle 0|W(z)|0\rangle ~~=~~ \langle
0|z\rangle
$$
Define a linear functional $\Omega_{\mathrm{Sch}}$, on the Weyl-Heisenberg algebra $$\Omega_{\mathrm{Sch}}\left(\sum_i c_i W(z_i)\right) ~:=~ \sum_i c_i
\Omega_{\mathrm{Sch}}(W(z_i)) ~:=~ \sum_i c_i \langle 0|W(z_i)|0\rangle
~:=~ \sum_i c_i e^{-|z_i|^2/2} \ .
$$ The first definition ensures linearity and the third one completes the definition. The second definition (notational) makes it obvious that $\Omega_{\mathrm{Sch}}$ is a positive linear functional since $\Omega_{\mathrm{Sch}}(A^*A) = \langle 0|A^{\dagger}A|0\rangle =
||A|0\rangle||^2 \ge 0, \forall ~ A := c_i W(z_i) \in $ the C\*-algebra. The equality holds only if $A|0\rangle = c_i|z_i\rangle = 0$. Since $|z_i\rangle$ states are linearly independent, there are no non-trivial states in the Hilbert space which satisfy $A|0\rangle = 0$. The positive linear functional then defines an [*inner product*]{} on the algebra by, $$\langle W(z), W(z')\rangle ~:=~ \Omega_{\mathrm{Sch}}(W(z)^{\dagger}
W(z')) ~~=~~ \langle 0|W(-z)W(z')|0\rangle
~~=~~e^{-\frac{\mathrm{Im}(z\bar{z}')}{2}}\langle 0|z'
- z\rangle \ .
$$ and extended by linearity to the algebra. This turns the algebra into a Hilbert space (distinct from the original Hilbert space).
Next, define operators, $\hat{W}(z)$ acting on the algebra, by, $$\hat{W}(z) [W(z')] ~:=~ W(z)W(z') ~=~
e^{\frac{\mathrm{Im}(z\bar{z}')}{2}} W(z + z')
$$ and extended to the algebra by linearity. Similarly, one defines an operator for each element of the algebra, in an obvious manner.
[*Exercise:*]{} Show that $(\widehat{W(z)})^{\dagger} = \hat{W}(-z)$.
This implies that $\hat{W}(z)$ are unitary operators.
That $W(z) \to \hat{W}(z)$ provides a homomorphism of the ${\cal W}$ algebra is obvious from the action of the operators. Thus the algebra with the inner product defined, carries a representation of itself in which the $W(z)$ are represented unitarily.
Consider general matrix elements of the operators $\hat{W}(z)$: $$\begin{aligned}
\left\langle W(z_1), \hat{W}(z) \left[W(z_2)\right]\right\rangle & = &
\left\langle 0 |W(-z_1)W(z)W(z_2)|0\right\rangle \nonumber \\
& = & e^{\frac{1}{2}\left(z\bar{z}_2 - z_1\bar{z} - z_1\bar{z}_2\right)}
\left\langle 0|W(z + z_2 - z_1|0\right\rangle\nonumber \\
& = & e^{\frac{\left(z\bar{z}_2 - z_1\bar{z} - z_1\bar{z}_2\right)}{2}}
e^{- \frac{| z + z_2 - z_1 |^2}{2}}
$$
Observe that for $z = \alpha$ or $z = i\beta$, the above matrix elements are continuous in $\alpha, \beta$ respectively. General matrix elements are obtained from finite combinations of these and hence [*the $\hat{W}(\alpha)$ and $\hat{W}(i\beta)$*]{} are both weakly continuous families of unitary operators. Actually, these are also [*strongly*]{} continuous families i.e. w.r.t. vector space norm. The strong continuity can be checked by evaluating the norm $||(\hat{W}(z) -
\hat{W}(0))[W(z')]||$ and checking the limits for $z = \alpha/\sqrt{2}$ and $z = i \beta/\sqrt{2}$.
This allows us to define two self-adjoint operators (on the algebra) as, $$\begin{aligned}
\hat{X} & := & \lim_{\alpha \to 0} \frac{\hat{W}(\alpha/\sqrt{2}) -
\mathbb{I}}{i \alpha} ~~,~~ \hat{P} ~ := ~ \hbar ~ \lim_{\beta \to 0}
\frac{\hat{W}(i\beta/\sqrt{2}) - \mathbb{I}}{i \beta} \\
& \Longrightarrow & \left[ \hat{X} , \hat{P} \right] ~=~ i\hbar
\mathbb{I}
$$ The commutator can be evaluated directly using the definitions for $\alpha, \beta \ne 0$ and using the existence of the limits guaranteed by strong continuity.
### Another positive linear functional and the Polymer representation:
Now view the Weyl-Heisenberg algebra defined above as an abstract structure i.e. an algebra generated by elements $W(z), z \in
\mathbb{C}$, obeying the relations (\[Algebra\],\[StarRelation\]) with a norm defined by $||W(z)|| = 1 ~ \forall ~ z \in \mathbb{C}$ and extended by linearity. Define a linear functional by, $$\label{PolyPositive}
\Omega_{\mathrm{Poly}}(W(z)) ~:=~ \left\{ \begin{array}{ll}1 &
\mathrm{if ~ Im}(z) = 0 \\ 0 & \mathrm{otherwise} \end{array} \right.
$$ This is positive because $$\begin{aligned}
\Omega_{\mathrm{Ploy}}\left(\sum_i \left\{C_iW(z_i)\right\}^{\dagger}
\sum_j\left\{C_j W(z_j)\right\}\right) & = & \sum_{ij} C^*_iC_j
e^{-\frac{1}{2}\mathrm{Im}(z_i\bar{z}_j)}\Omega_{\mathrm{Ploy}}(W(z_j -
z_i)) \nonumber \\
& = & \sum_i |C_i|^2 + \sum_{i\ne j} C^*_iC_j
e^{\frac{i}{2}\beta_i(\alpha_j - \alpha_i)}\delta_{\beta_i, \beta_j}
\hspace{1.0cm}
$$ In the sum, only those pairs $(i,j)$ which have the same $\beta$, contribute. Group together all the terms whose $\beta_i$ are equal (eg $\{z_1, z_2, \ldots, z_m\}, \{z_{m+1}, \ldots z_{m+n}\} \ldots $), and consider one such group at a time. In each such group, the phases in the second term can be absorbed in the $C_i$’s (since the $\beta$ is common) and then combining with the first term gives $|\sum_i\{C_ie^{i\beta\alpha_i/2}\}|^2$ which is non-negative. This completes the proof. Note that positive linear functional must evaluate to 1 on the identity element of the algebra (namely, $\mathbb{I} :=
W(0)$) and therefore we [*cannot*]{} interchange $\beta \leftrightarrow
\alpha$ in the defining condition in (\[PolyPositive\]).
The $\Omega_{\mathrm{Poly}}$ defines a [*degenerate*]{} inner product on the algebra, $$\langle W(z'), W(z) \rangle ~:=~
\Omega_{\mathrm{Poly}}(W(z')^{\dagger}W(z)) ~=~
e^{-\frac{\mathrm{Im(z'\bar{z})}}{2}} \Omega_{\mathrm{Poly}}(W(z - z'))
$$ and extended by linearity. The elements whose norm w.r.t. this degenerate inner product is zero, forms a closed subspace ${\cal N}$, of the algebra and consists of elements of the form $\chi := \sum_i C_i
W({{\textstyle \frac{\alpha_i + i\beta}{\sqrt{2}}}}), \beta \in \mathbb{R} \,
\mathrm{such~that} \sum_i C_i e^{-i\beta\alpha_i/2} = 0$. Elements of ${\cal N}$ also satisfy the property: $\Omega_{\mathrm{Poly}}(A \chi) =
0 \ \forall \ A \ \in {\cal W}$, which is useful in the exercise below. The quotient space, ${\cal W}/{\cal N}$, is an inner product space and its Cauchy completion defines a Hilbert space of the [*Polymer Representation*]{}.
[*Exercise:*]{} Let $A$ denote a general element of the algebra and $\chi$ an element of ${\cal N}$. Define $[A] := \{B \in {\cal W}/B = A +
\chi\}$. Define $\langle [A], [B]\rangle := \langle A, B\rangle$ and $\hat{W}(z)\{ [A] \} := [ \hat{W}\{A\} ]$. Show that these definition are well defined and conclude that ${\cal W}/{\cal N}$ provides a [*unitary*]{} representation of the quotient algebra. From now on, we refer to the quotient representation without being explicit about it.
Observe that $\Omega_{\mathrm{Poly}}$ is continuous in $\alpha$ and [*discontinuous*]{} in $\beta$. This directly implies that $\hat{W}(i\beta/\sqrt{2})$ [*cannot be weakly continuous*]{} and therefore we cannot define the analogue of $\hat{P}$. This follows by noting that $\langle W(z'), \hat{W}(i\beta/\sqrt{2})\{W(z')\}\rangle
\sim \Omega_{\mathrm{Poly}}(W(i\beta/\sqrt{2} + z' - z'))$. The weak continuity (actually also [*strong continuity*]{}) in $\alpha$ however allows the definition of $\hat{X}$ self-adjoint operator. It remains to make the representation explicit.
In both the cases above, with the Schrodinger and the polymer functionals, we constructed a representation of the ${\cal W}$ in which the $W(z)$ are represented by unitary operators. This is the Gelfand-Naimark-Segal (GNS) construction. In the Schrodinger case, we obtained the $\hat{X}, \hat{P}$ operators satisfying the canonical commutation relations. In the Polymer case we obtained only $\hat{X}$. In both cases we get the following relations, directly by applying the definitions: $$\begin{aligned}
\hat{W}^{\dagger}\left(i\beta/\sqrt{2}\right)~\hat{X}~\hat{W}\left(i\beta/\sqrt{2}\right)
& = & \hat{X} - \beta\mathbb{I} ~~ \Rightarrow \label{Translation}\\
\hat{X}\hat{W}\left(i\beta/\sqrt{2}\right) -
\hat{W}\left(i\beta/\sqrt{2}\right) \hat{X} & = & -
\beta\hat{W}\left(i\beta/\sqrt{2}\right)
$$ From these relations follow an important result. Note that $\hat{X}$ is a self adjoint operator and therefore its spectrum is real. What can one say about its eigenvectors? The above relation implies that [*if $|x\rangle$ is an eigenvector of $\hat{X}$ with eigenvalues $x$, then $\hat{W}(i\beta/\sqrt{2})|x\rangle$ is also an eigenvector with eigenvalues $(x + \beta)$. Hence, either every real number is an eigenvalue or none is.*]{}
[*Suppose every $x$ is an eigenvalue*]{}. Then we have the orthogonality relation $\langle x|x'\rangle = \delta_{x,x'}$ - the Kronecker $\delta$. Taking expectation value of the second equation above, it follows that $\beta f(x, \beta) = 0, \forall \, x, \beta \in
\mathbb{R}$, where $f(x, \beta) := \langle
x|\hat{W}(i\beta/\sqrt{2})|x\rangle$. This implies that $f(x,\beta)$ is zero for $\beta \ne 0$ and $f(x, 0) = 1$ directly from the definition. Thus $f(x, \beta)$ [*cannot*]{} be continuous at $\beta = 0$, for any $x$. This means $\hat{W}(i\beta/\sqrt{2})$ cannot be weakly continuous at $\beta = 0$.
This also means that [*if*]{} $\hat{W}(i\beta/\sqrt{2})$ [*is*]{} weakly continuous (as for the Schrodinger representation), then $\hat{X}$ cannot have [*any eigenvector*]{}. Each $x \in \mathbb{R}$ is a [*generalised eigenvalue*]{} and hence, in the formal notation, $\langle
x|x'\rangle = \delta(x - x')$ - the Dirac $\delta$-function.
Thus, in the Schrodinger representation, $\hat{X}$ necessarily has only generalised eigenvalues, while in the polymer representation, it [*could*]{} have proper eigenvalues, but generalized eigenvalues is not ruled out.
However, in the polymer representation, we note: $$\begin{aligned}
\left\langle W(i\beta/\sqrt{2}), \, W(i\beta'/\sqrt{2}) \right\rangle &
= &
\Omega_{\mathrm{Poly}}\left(W(-i\beta/\sqrt{2})W(i\beta'/\sqrt{2})\right)
~ = ~ \delta_{\beta, \beta'} \ ; \\
\left\langle W(i\beta/\sqrt{2}), \,
\hat{X}\left\{W(i\beta'/\sqrt{2})\right\} \right\rangle & = &
- \beta\delta_{\beta, \beta'} \\
\left\|(\hat{X} - \lambda\mathbb{I})[W(i\beta/\sqrt{2})\right\| & = &
\lim_{\alpha \to 0} \left\| \frac{1}{i\alpha}\left[W({{\textstyle \frac{\alpha +
i\beta}{\sqrt{2}}}}) - (1 +
i\alpha\lambda)\right]W(i\beta/\sqrt{2})\right\| \nonumber \\ & = & 0
\hspace{2.0cm} \mathrm{for} ~ \lambda = -\beta \ .
$$ which show explicitly that $W(-i\beta/\sqrt{2})$ is a normalized eigenvector of $\hat{X}$ with eigenvalue $\beta$, for every $\beta \in
\mathbb{R}$. This means that the Polymer Hilbert space is [*non-separable*]{}.
This concludes the illustration of the GNS construction of representations of C\* algebras. In the next sub-section we see the analogue of the spin network construction.
### Polymer representation via ‘spin networks’
We begin by introducing ‘graphs’ in a ‘0-dimensional manifold’, define ‘holonomies’ and ‘spin network functions’, define an inner product and densely defined operators. More details may be seen in [@Shadows].
1. [*Graphs, holonomies, cylindrical functions:*]{} Any [*countable*]{} set of real numbers, $\{x_i\}$ represents a [*graph*]{} and is denoted by $\gamma$[^34]. Note that the ‘points’ $x_j$ on the real line correspond to [*edges*]{}. Associated to each edge, $x_j$, we define a [*point holonomy*]{}, $e^{ikx_j}$, $k \in \mathbb{R}$ plays the role of a [*connection*]{}. For each graph $\gamma$, define complex valued functions, $f_{\gamma}(k) := \sum_{j}f_j e^{ikx_j}$. Let Cyl$_{\gamma}$ denote the vector space of $f_{\gamma}(k)$. Elements of this vector space are said to be [*functions cylindrical with respect to the graph $\gamma$*]{}. Let Cyl := $\sum_{\oplus}$ Cyl$_{\gamma}$, where the sum is over all possible graphs.
Thus a general element of Cyl is a function of $k$ expressible as a countable linear combination of the [*elementary functions*]{} $f_{x_j}(k) := e^{ikx_j}$’s.
2. [*Inner Product on Cyl:*]{} Define $$\langle f, g \rangle_{\mathrm{Poly}} ~:=~ \lim_{L \to \infty}
\frac{1}{2L} \int_{-L}^{L} dk f^*(k)g(k)
$$ which for elementary functions $f_{x_j}(k), f_{x_l}(k)$ gives $\delta_{x_j, x_l}$. Introducing the notation, $|x_j\rangle
\leftrightarrow e^{ikx_j}$ this is expressed as $\langle x_j, x_l\rangle
:= \delta_{x_j, x_l}, \ \forall \ x_j, x_l \in \mathbb{R}$. Cauchy completion of Cyl with respect to this inner product defines a Hilbert space, ${\cal H}_{\mathrm{Poly}}$.
3. [*Action of ${\cal W}$:*]{} Define operators $\hat{W}(z)$ on Cyl by, $$\hat{W}\left({{\textstyle \frac{\alpha + i \beta}{\sqrt{2}}}}\right) f(k) ~:=~
e^{-i\alpha\beta/2}e^{i\beta k}f(k - \alpha) ~~,~~\forall ~ f \in
\mathrm{Cyl}, ~ \forall ~ \alpha, \beta \in \mathbb{R}
$$ These are densely defined and can be extended to bounded unitary operators on ${\cal H}_{\mathrm{Poly}}$. It is easily verified that this provides a representation of the abstract algebra defined in (\[Algebra\]).
[*Exercise:*]{} Show that the 1-parameter families of unitary operators, $\hat{W}(\alpha/\sqrt{2})$ and $\hat{W}(i\beta/\sqrt{2})$ are weakly continuous at $\alpha = 0$ and weakly discontinuous at $\beta = 0$ respectively. This implies that while $\hat{X}$ can be defined from the $\hat{W}(\alpha/\sqrt{2})$ family, there is no corresponding operator from the second family. Thus the holonomies - $h_{x_j}(k)$ are well defined but not the connection - $k$ itself.
### Harmonic Oscillator in the polymer representation
So we see two distinct representations of the same abstract algebra with the polymer representation being similar to the LQG representation. Are there observable quantities which would reveal which representation occurs in nature?
An obvious candidate is to consider the dynamics of the Harmonic oscillator, with the classical Hamiltonian, $H(x, p) := p^2/(2m) +
m\omega^2 x^2/2, \{x, p\} = 1$. In the quantum theory, the $x, p$ are expected to be replaced by the corresponding operators. However, in the polymer representation, there is no $\hat{p}$! So in proposing the quantum Hamiltonian we need to introduce a [*scale, $\mu_0$*]{} and [*define*]{} $\widehat{p^2} := \{ 2 - \hat{W}(i\mu_0/\sqrt{2}) +
\hat{W}(-i\mu_0/\sqrt{2}) \}/\mu_0^2$. We could of course define $\hat{p}$ first and then take its square. This is a quantization ambiguity not too important for our purposes here [@Shadows]. In the Schrodinger representation, we could use exactly the same definition, work out quantities of interest eg spectrum and then take the limit $\mu_0 \to 0$. In the polymer representation, we [*cannot*]{} take this limit.
Consider the eigenvalue equation for the Hamiltonian: $\hat{H}|\psi\rangle = E|\psi\rangle$. Writing $|\psi\rangle := \sum_{x
\ \in \ \mathrm{countable~set}} \psi(x)|x\rangle$, and noting that the unitary operators in $\widehat{p^2}$ shift the $|x\rangle \to |x \pm
\mu_0\rangle$, the eigenvalue equation becomes a [*difference*]{} equation, involving $\psi(x), \psi(x \pm \mu_0)$. This means that $\psi(x)$ with $x$ in a lattice $L_{x_0} := \{ x = x_0 + \mu_0 N, N \in
\mathbb{Z}\}$ constitute a solution while those belonging to different lattices are unrelated. Span of the vectors in any lattice form a [*separable*]{} subspace of the polymer Hilbert space. The spectrum can be determined for each lattice independentally. This is analyzed in detail in [@Shadows]. Suffice it to say that the spectrum differs from that in the Schrodinger representation by terms down by powers of $\mu_0/d$. Here, $d := \sqrt{\hbar/m\omega}$ is the length scale defined by the system while $\mu_0$ is the length scale introduced by the approximation for the momentum operator. For the physical systems modelled well by an oscillator (eg for vibrational spectra of molecules), $\mu_0/d$ is extremely small and so Schrodinger vs polymer representation cannot be resolved by observations.
Additional comments may be seen in [@Shadows].
Inverse Triad Operator(s)
-------------------------
As noted before, the discrete nature of the spectrum of the triad operator implies that its inverse is not densely defined. Consequently the counterpart of the classical function $p^{-1}$ needs to be defined indirectly, by a suitable prescription. Being a prescription, it introduces quantization ambiguities. We will consider a prescription which is sufficiently general.
We aim to define an operator $\widehat{\mathrm{sgn}(p) |p|^{-1}}$. Introduce the following notation: $n^i$ is a unit, 3-dimensional vector and $\tau_i$ are anti-hermitian generators of $SU(2)$ in the $J^{\mathrm{th}}$ representation, satisfying $$[\tau_i, \tau_j] = \epsilon_{ijk}\tau_k ~~,~~
\mathrm{Tr}_J(\tau_i\tau_j) = - \frac{1}{3}J(J + 1)(2J + 1)\delta_{ij}
~~~ := ~~~ - {\cal N}_J \delta_{ij}
$$ For $f_{\alpha}(p) := \mu_{\alpha}|p|^{-\alpha/2}$, and define $g_{\alpha}(p) := \int^p f_{\alpha}^{-1}(x)dx = \mu_{\alpha}^{-1}
\mathrm{sgn}(p)|p|^{1 + \alpha/2}(1 + \alpha/2)^{-1}$. For $\alpha = 0$ we have $f_0 = \mu_0$ while for $\alpha = 1$ we have $f_1 = \mu_1
|p|^{-1/2}$ and we choose $\mu_1 := {\ell_{\mathrm P}}\sqrt{\gamma\sqrt{3}/4}$. These recover the $\mu_0$ and the $\bar{\mu}$ schemes. We will suppress the $\alpha$ label. Define $h_f := e^{\lambda n^i\tau_i f(\alpha, p) c}$. This is matrix of order $(2J + 1)$. Classically, the following is true. $$\begin{aligned}
h_f\{h_f^{-1}, |g|^l(p)\} & = & - \frac{\kappa\gamma}{3} \left(\lambda
n^i\tau_i\right) l|g|^{l -1}\mathrm{sgn}(p)
~\hspace{0.0cm}\mathrm{where~we~used~~~}f\frac{d|g|}{dp} =
\mathrm{sgn}(p). \\
|p|^{(l -1)(1 + \alpha/2)} & = & \mathrm{sgn}(p)\left[
\frac{3}{\kappa\gamma l\lambda}\left(\mu_{\alpha}( 1 +
\alpha/2)\right)^{l - 1}{\cal N}_J^{-1} \right] \nonumber \\ & &
\hspace{4.0cm} \times \left[\mathrm{Tr}_J\left( (n^i\tau_i)
h_f\{h_f^{-1}, |g|^l\}\right)\right] \\ \label{Defn1}
& = & \mathrm{sgn}(p)\left[ \frac{\kappa\gamma
l\lambda}{3}\left(\mu_{\alpha}( 1 + \alpha/2)\right){\cal N}_J
\right]^{-1}\nonumber \\
& & \hspace{4.0cm}\times \left[\mathrm{Tr}_J\left( (n^i\tau_i)
h_f\{h_f^{-1}, |p|^{l(1 + \alpha/2)} \}\right)\right] \label{Defn2}
$$ Thus we have a classical expression for $|p|^{(l-1)(1 + \alpha/2)}$ which has [*four*]{} ambiguity parameters: $\alpha, J, l, \lambda$. $J$ is a positive half integer, $0 < l < 1, \alpha > -2$. The special cases would be: (a) $\mu_0-$scheme: $\alpha = 0, \lambda = 1$; (b) improved scheme: $\alpha = 1, \lambda = 1, j = 1/2$ and some special values of $l$ explored; (c) Madhavan scheme: similar to the improved scheme except $\lambda$ is correlated with the fiducial volume $V_0$ (more on this later). We could [*choose*]{} $l -1 = (1 + \alpha/2)^{-1}$ to define inverse triad, but we will postpone such choices.
The corresponding quantum operator is obtained by replacing the Poisson bracket by $-i\hbar^{-1}$ times the commutator. The $-i$ is combined with $n\cdot\tau$ to make the generators Hermitian and the $\hbar^{-1}$ combines with $\kappa$ to replace $\kappa$ by ${\ell_{\mathrm P}}^2$. The commutator is expanded as: $\hat{h}_f [ \hat{h}^{-1}_f, \widehat{|g|}^l ] =
\mathrm{I}. \widehat{|g|}^l - \hat{h}_f \widehat{|g|}^l
\hat{h}^{-1}_f$.
Observe that $n\cdot(-i\tau)$ can be diagonalised with diagonal elements being $-J, -J + 1, \ldots J -1, J$. So the $h_f$ becomes the diagonal matrix $e^{(i\lambda f_{\alpha}c)(J, J -1, ~, , ,~ -J + 1, -J)}$. So the commutator terms are diagonal matrices.
The computations simplify if we label the basis states by [^35] $$v_{\alpha} ~ := ~
\left(\frac{1}{6}\gamma{\ell_{\mathrm P}}^2\right)^{\alpha/2}\left(\frac{1}{\mu_{\alpha}(1
+ \alpha/2)}\right) \mathrm{sgn}(\mu) |\mu|^{1 + \alpha/2} ~~,~~
\hat{g}_{\alpha}|v_{\alpha}\rangle =
\frac{\gamma{\ell_{\mathrm P}}^2}{6}v_{\alpha}|v_{\alpha}\rangle
$$ So that the $h_f$ shifts the $v_{\alpha}$ labels simply as, $$\widehat{e^{i \lambda k f_{\alpha}c}}|v_{\alpha}\rangle ~ = ~
|v_{\alpha} + 2 k \lambda\rangle ~~~~ \because ~~ e^{ifc/2}|v\rangle =
|v + 1\rangle \ .
$$
With these, acting on a basis state $|v_{\alpha}\rangle$, the Tr$_J$ evaluates to, $$\begin{aligned}
\left[\mathrm{Tr}_J\{ \cdots \}\right] & = &
\left(\frac{\gamma{\ell_{\mathrm P}}^2}{6}\right)^{l}\sum_{k = -J}^J k\left\{
|v_{\alpha}|^l
- |v_{\alpha} - 2 k\lambda|^l \right\} \nonumber \\
& = & \left(\frac{\gamma{\ell_{\mathrm P}}^2}{6}\right)^{l}\sum_{k = -J}^J k
|v_{\alpha} + 2 k\lambda|^l \mathrm{~~~and~defining~} v_{\alpha} := 2
J\lambda q_{\alpha} \ , \nonumber \\
& = & \left(\frac{\gamma{\ell_{\mathrm P}}^2\ \lambda}{6}\right)^{l} 2^l \sum_{k =
-J}^J k |J q_{\alpha} + k|^l
~~:=~~\left(\frac{\gamma{\ell_{\mathrm P}}^2\lambda}{3}\right)^{l}
\mathrm{sgn}(q_{\alpha})G_{J,l}(q_{\alpha}) \\
G_{J,l}(q_{\alpha}) & := & \mathrm{sgn}(q_{\alpha}) \sum_{k = -J}^J k |J
q_{\alpha} + k|^l \label{GjlDefn}
$$ The eigenvalues of $\widehat{|p|^{(l -1)(1 + \alpha/2)}}$ are then given by $$\begin{aligned}
\widehat{|p|^{(l -1)(1 + \alpha/2)}}|v_{\alpha}\rangle & := &
\mathrm{sgn}(v_{\alpha})\Lambda_{J, l,
\alpha}(v_{\alpha})|v_{\alpha}\rangle \\
\Lambda_{J, l, \alpha}(v_{\alpha}) & = & \left(\frac{\mu_{\alpha}
\gamma{\ell_{\mathrm P}}^2\lambda}{3}\right)^{l -1}\frac{(1 + \alpha/2)^{l -
1}}{l}{\cal N}_J^{-1}\left[ G_{J,l}(q_{\alpha})\right] \nonumber
$$ The first bracket takes care of the dimensions and the remaining factors are dimensionless. It remains to calculate the last square bracket which is a $\lambda, \alpha$ independent, universal function of its argument and depends only on $J, l$.
From its definition, it is easy to see that $G_{J,l}(0) = 0$ and $G_{J,l}(-q_{\alpha}) = G_{J,l}(q_{\alpha})$, the $\mathrm{sgn}(q_{\alpha})$ factor is crucial for this. Thus it suffices to consider only $q_{\alpha} > 0$. $$\begin{aligned}
G_{j,l}(q_{\alpha} > 0) & = & \sum_{k = -J}^{J} k |k + Jq_{\alpha}|^l
\nonumber \\
& = & \sum_{k = -J}^{J} \left\{ (k + Jq_{\alpha})|k + Jq_{\alpha}|^l -
Jq_{\alpha}|k + Jq_{\alpha}|^l \right\} \nonumber \\
& = & \sum_{k = J(q_{\alpha} -1)}^{J(q_{\alpha} +1)}
\left\{\mathrm{sgn}(k)|k|^{l + 1} - Jq_{\alpha}|k|^l\right\} \\
G_{j,l}(q_{\alpha} \ge 1) & = & \sum_{k = J(q_{\alpha}
-1)}^{J(q_{\alpha} +1)} \left\{|k|^{l + 1} - Jq_{\alpha}|k|^l\right\}
\label{QGreater1}\\
G_{j,l}(0 < q_{\alpha} < 1) & = & - \sum_{k = J(q_{\alpha} -1)}^{0_-}
\left\{|k|^{l + 1} + Jq_{\alpha}|k|^l\right\} + \sum_{0_+}^{k =
J(q_{\alpha} +1)} \left\{|k|^{l + 1} - Jq_{\alpha}|k|^l\right\}
\label{QLess1}
$$ In the second step, we have shifted $k \to k - Jq_{\alpha}$. $k$ is no longer integral but still changes in steps of 1. Clearly, for $q_{\alpha} \ge 1$, $k$ is positive and the sgn as well as the absolute value are redundant (the $k = 0$ term for $q_{\alpha} = 1$ gives zero and hence the sum is confined to positive $k$ only), as in (\[QGreater1\]). For $q_{\alpha} < 1$, the sum splits in two groups as in (\[QLess1\]), and the $0_{\pm}$ denote the respective limits on the values of $k$ which must match with the other limits and shift in steps of 1.
We will not simplify/approximate this further but consider the special cases (i) $\alpha = 0, \lambda = 1$ and (ii) $\alpha = 1, J = 1/2, l =
2/3$.
\(i) : (eigenvalues of $|p|^{l -1}$) $$f_0(p) = \mu_0~~ , ~~ g_0(p) = \mu_0^{-1}\mathrm{sgn}(p)|p|~~ , ~~ V_0 =
\mu_0^{-1} \mathrm{sgn}(\mu)|\mu|
$$ $$\begin{aligned}
\Lambda_{J,l,0}(V_0) & = & \left(\frac{\mu_0\gamma{\ell_{\mathrm P}}^2}{3}\right)^{l -
1} (l {\cal N}_J)^{-1} G_{J,l}(|\mu|/(2J\mu_0)) \\
\Lambda_{{{\textstyle \frac{1}{2}}},{{\textstyle \frac{1}{2}}},0}(V_0) & = &
\left(\frac{\gamma{\ell_{\mathrm P}}^2}{6}\right)^{-1/2} \frac{1}{\sqrt{\mu_0}} \left(
\left|\frac{\mu}{\mu_0} + 1 \right|^{1/2}
- \left|\frac{\mu}{\mu_0} - 1 \right|^{1/2}\right) \label{Mu0Vals}
$$
\(ii) : (eigenvalues of $|p|^{{{\textstyle \frac{3}{2}}}(l - 1)}$) $$f_1(p) = \mu_1 |\mu|^{-1/2} ~~ , ~~ \mu_1 :=
\sqrt{\frac{\gamma{\ell_{\mathrm P}}^2}{6}\frac{3\sqrt{3}}{2}} ~~,~~
g_0(p) = \mu_1^{-1}\mathrm{sgn}(p)|p|^{3/2} (3/2)^{-1} ~~ ,
$$ $$v_1 = \left(\frac{\gamma{\ell_{\mathrm P}}^2}{6}\right)^{1/2} \mu_1^{-1} (3/2)^{-1}
\mathrm{sgn}(\mu)|\mu|^{3/2} ~~ = ~~ K \mathrm{sgn}(\mu)|\mu|^{3/2}
$$ $$\Lambda_{{{\textstyle \frac{1}{2}}},l,1}(v_1) =
\left(\frac{\mu_1\gamma{\ell_{\mathrm P}}^2}{3}\frac{3}{2}\lambda\right)^{l - 1} (l
{\cal N}_{{{\textstyle \frac{1}{2}}}})^{-1} G_{1/2,l}(|v_1|/(2{{\textstyle \frac{1}{2}}}\lambda))
$$ The $G$ can be computed directly from the (\[GjlDefn\]), for $v_1 >
0$, as $$\label{Mu1GenVals}
G_{{{\textstyle \frac{1}{2}}}, l}(v_1/\lambda) = \left(\frac{1}{2}\right)^{l + 1}
\left( \left| \frac{v_1}{\lambda} + 1 \right|^{l} - \left|
\frac{v_1}{\lambda} - 1 \right|^{l} \right)
$$ For $l = 2/3$ the operator becomes $|p|^{-1/2}$ and the eigenvalue becomes, $$\label{Mu1Vals}
\Lambda_{{{\textstyle \frac{1}{2}}},{{\textstyle \frac{2}{3}}},1}(v_1) ~=~
\left(\frac{\gamma{\ell_{\mathrm P}}^2}{6}\right)^{-1/2}
\frac{3}{4}\left(\frac{K}{\lambda}\right)^{{{\textstyle \frac{1}{3}}}}\left[
\left|\frac{v_1}{\lambda} + 1\right|^\frac{2}{3} -
\left|\frac{v_1}{\lambda} - 1\right|^\frac{2}{3} \right]
$$
For large $v_1$ and $\lambda = 1$, this matches with the eigenvalue given by [@APSThree][^36]. The difference arises because the APS prescription takes the $\alpha = 0$ expression and replaces $\mu_0$ by $\bar{\mu}$ in equation (\[Defn2\]).
For large $J$ the sum can be approximated using, $$\int_0^1 dx x^{r} ~=~ \frac{1}{r + 1} \approx \sum_{i = 1}^N
\left(\frac{i}{N}\right)^r \frac{1}{N} ~~~~ \Rightarrow
\sum_{i = 1}^N i^r \approx \frac{N^{r+1}}{(r + 1)}
$$ and applying it to the sums in the definition of the $G_{J,l}$.
Inhomogeneous Lattice Models
----------------------------
In the main body, we focused on quantization of [*symmetry reduced models*]{} which are based on a homogeneous and isotropic background. From a perspective of the full theory, this background presumably corresponds to a state of full theory. Generic states of the full theory would be inhomogeneous. One way in which symmetric states of the full theory have been understood in the LQC context is that the symmetric states are those distributions in Cyl$^*$ of the full theory which have support on the invariant connections [@SymmetricConnections]. In the same spirit, we may stipulate certain kinds of inhomogeneous states as those distributions which have support on certain form of ‘inhomogeneous connections’. Specific models can be then constructed using similar strategies as used in LQC constructions. Such models can shed some light on how homogeneous and isotropic models could be viewed from an inhomogeneous perspective. These so-called lattices models are briefly summarised below. Details should be seen in [@MartinLattice].
For definiteness, let us continue to work with homogeneous (not necessarily isotropic), spatially flat, diagonalised model with a fiducial cell of co-moving volume $V_0$ as before. The spatial isometries provide directions (of the Killing vectors) and the fiducial metric provides coordinates as background structures which are to be kept fixed. Using these background structures, Bojowald constructs another model as follows.
Choose a cubical lattice (say) aligned with the isometry directions and with a spacing $\ell_0 := (V_0/N)^{1/3}$. Let the vertices of the lattices be denoted by $\vec{v}$ and the three oriented links be denoted by $\vec{e}_{I,\vec{v}}(t) := \vec{v} + t \hat{e}_I, ~ t \in [0,
\ell_0]$ and $\hat{e}_I$ is the unit vector in the I$^{\mathrm{th}}$ direction. For future reference, let $S_{I, \vec{v}}$ denote the elementary surface perpendicular to the elementary link $\vec{e}_{I,
\vec{v}}$ and passing through its mid-point.
Restrict the connections and triad variables to be of the form, $$\label{LatticeFields}
A^i_a(x) := \tilde{k}_I(x)\delta_{(I)}^{i}\delta^I_a ~~,~~ E_i^a(x) :=
\tilde{p}^I(x)\delta^{(I)}_{i}\delta_I^a \ .
$$ These are the local versions of the diagonalised homogeneous models. The diagonal form of the connections implies that the holonomies - path ordered exponentials - become ordinary exponentials of line integrals.
The $\tilde{k}_I, \tilde{p}^I$ are further taken to be spatially periodic with period $V_0^{1/3}$, $$\label{Periodicity}
\tilde{k}_I(x) ~=~ \sum_{\vec{m}} \tilde{k}_I(\vec{m})e^{i
\vec{m}\cdot\vec{x}} ~~,~~
\tilde{p}^I(x) ~=~ \sum_{\vec{m}} \tilde{p}^I(\vec{m})e^{i
\vec{m}\cdot\vec{x}} ~~,~~\vec{m} ~=~ 2\pi V_0^{-1/3}\vec{n}, ~~ \vec{n}
\in \mathbb{Z}^3 \ .
$$ The Poisson brackets between the connection and the triad lead to, $$\{\tilde{k}_I(x), \tilde{p}^J(y)\} ~=~ \kappa\gamma\delta^J_I\delta^3(x,
y) \ .
~~\Rightarrow~~ \{\tilde{k}_I(\vec{m}), \tilde{p}^J(\vec{m'})\} ~=~
\kappa\gamma V_0^{-1}\delta^J_I\delta^3(\vec{m},-\vec{m'}) \ .
$$
In loop quantization, basic variables of the model will be holonomies of the lattice connection along the three elementary links at each vertex and the three fluxes of the lattice triad variables along the elementary surfaces through the mid-points of the elementary links and perpendicular to the link. As noted above, these holonomies will be ordinary exponentials thanks to the diagonal form of the connection.
The line integrals of the connection along elementary links of the lattice are given by, $$\begin{aligned}
{\cal I}_{I,\vec{v}} & := & \int_{\vec{e}_{I, \vec{v}}} dt \
\tilde{k}_I(\vec{e}_I(t)) ~=~ \int_{\vec{e}_{I, \vec{v}}} dt
\sum_{\vec{m}}\tilde{k}_I(\vec{m})e^{i\vec{m}\cdot\vec{e}_{I,\vec{v}}(t)}
~=~ \sum_{\vec{m}}
\tilde{k}_I(\vec{m})e^{i\vec{m}\cdot\vec{v}}\int_0^{\ell_0} dt e^{i t
\vec{m}\cdot\hat{e}_I}
\nonumber \\
& = & \sum_{\vec{m}} \tilde{k}_I(\vec{m})\left\{2\
e^{i\vec{m}\cdot\vec{v}}e^{im_I\ell_0/2}
\left(\frac{\mathrm{sin}(m_I\ell_0/2)}{m_I}\right)\right\} ~~~
\mathrm{where,}~ m_I := \vec{m}\cdot\hat{e}_I \ .
\label{HolonomyIntegrand} \\
& \approx & \tilde{k}_I(\vec{v})\ell_0 ~~\because ~~~ m_I\ell_0 \ll 1
\mathrm{~dominates\ the\ sum.} \nonumber \\
h_{I, \vec{v}} & := & e^{{{\textstyle \frac{i}{2}}} {\cal I}_{I,\vec{v}}} ~~ \approx ~~
e^{{{\textstyle \frac{i}{2}}}\tilde{k}_I(\vec{v})\ell_0} ~~ := ~~
e^{{{\textstyle \frac{i}{2}}}{k}_I(\vec{v})} \hspace{2.0cm}
(\mathrm{elementary~holonomies}) \label{LatticeHolonomy}
$$ Likewise, the fluxes of the lattice triad along elementary surfaces are given by, $$\begin{aligned}
{\cal F}^J_{\vec{v}} & := & \int_{S_{J,\vec{v}}}
\sum_{\vec{m}}\tilde{p}^J(\vec{m})e^{\vec{m}\cdot\vec{y}}
~=~\sum_{\vec{m}}\tilde{p}^J(\vec{m})e^{\vec{m}\cdot\vec{v}}e^{im_J\ell_0/2}\int_{-\ell_0/2}^{\ell_0/2}
e^{i t m_K}dt \int_{-\ell_0/2}^{\ell_0/2} e^{i t m_L}dt \nonumber \\
& = & \sum_{\vec{m}}\tilde{p}^J(\vec{m})\left\{4
e^{i\vec{m}\cdot\vec{v}}e^{im_J\ell_0/2}\left(\frac{\mathrm{sin}(m_K\ell_0/2)\mathrm{sin}(m_L\ell_0/2)}{m_K
m_L}\right)\right\} \label{Fluxes} \\
& \approx & \tilde{p}_I(\vec{v})\ell_0^2 \hspace{6.7cm}
(\mathrm{elementary \ fluxes}) \label{ElementaryFlux}
$$ The $J,K,L$ indices are chosen such that $\epsilon_{JKL} = 1$. This takes care of the orientations. There are no smearing functions above because the $\tilde{p}^I$ variables are (U(1)) gauge invariant thanks to diagonalised form.
These variables satisfy the Poisson brackets, $$\begin{aligned}
\{{\cal I}_{I,\vec{v}}, {\cal F}^J_{\vec{v}'}\} & = & \kappa\gamma
\delta^J_I \left[8 V_0^{-1}\sum_{\vec{m}}e^{i\vec{m}\cdot(\vec{v} -
\vec{v}')}\frac{\mathrm{sin}(m_I\ell_0/2)\mathrm{sin}(m_K\ell_0/2)\mathrm{sin}(m_L\ell_0/2)}{m_I
m_K m_L}\right] \nonumber \\
& = & \kappa\gamma\delta^J_I \left[\chi_{\ell_0}(\vec{v} -
\vec{v}')\right] ~~=~~ \kappa\gamma\delta^J_I\delta_{\vec{v}, \vec{v}'}
$$ The square brackets above is the characteristic function of width $\ell_0$ and centered at $(\vec{v} - \vec{v}')$ which is just the Kronecker delta.
The kinematical Hilbert space is then described in terms of the flux representation as: $$\begin{aligned}
\hat{{\cal F}}^I_{\vec{v}} |\ldots, \ \mu_{I, \vec{v}}\ , \
\ldots\rangle & = & \left(\frac{\gamma{\ell_{\mathrm P}}^2}{2} \mu_{I, \vec{v}}\right)
|\ldots, \ \mu_{I, \vec{v}}\ , \ \ldots\rangle ~~~,~~~\mu_{I,\vec{v}} \
\in \ \mathbb{Z} \label{FluxBasis} \\
\hat{h}_{I,\vec{v}}|\ldots, \ \mu_{I, \vec{v}}\ , \ \ldots\rangle & = &
|\ldots, \ \mu_{I, \vec{v}} ~ + ~ 1\ , \ \ldots\rangle
\label{HolonomyAction}
$$ The flux eigenvalues are in [*integer*]{} steps of $\gamma{\ell_{\mathrm P}}^2/2$ because the elementary holonomies suffice to separate the [*lattice connections*]{} (periodic) and thus only their integer powers appear.
Subsequent steps are similar to what is done in the homogeneous models. In particular, the volume corresponding to the cell with $N^3$ lattice sites, can be expressed as $$\begin{aligned}
V & = & \int d^3x \sqrt{|\tilde{p}^1\tilde{p}^2\tilde{p}^3|} \approx
\sum_{\vec{v}}\ell_0^3\sqrt{|\tilde{p}^1(\vec{v})\tilde{p}^2(\vec{v})\tilde{p}^3(\vec{v})|}
= \sum_{\vec{v}}\sqrt{|{p}^1(\vec{v}){p}^2(\vec{v}){p}^3(\vec{v})|}
\nonumber \\ & \approx & \sum_{\vec{v}}\sqrt{|{\cal F}^1_{\vec{v}}{\cal
F}^2_{\vec{v}}{\cal F}^3_{\vec{v}}|}
$$ leading to the corresponding operator expression.
There is no diffeomorphism constraint since the background coordinates are fixed in defining the lattice, the SU(2) gauge invariance is first reduced to the U(1)$^3$ due to restriction to diagonal connection and triad and by the form of these variables, the $\tilde{k}_I, \tilde{p}^I$ are gauge invariant variables. Hamiltonian constraint remains as in the case of homogeneous models. In essence, we have $N^3$ ‘homogeneous models’ (labelled by the Fourier label $\vec{m}$) at the level of basic variables and the kinematical Hilbert space. The role that $V_0$ played in the homogeneous model is now played by $\ell_0$. The [*inhomogeneity*]{} is reflected by basis states having [*different values*]{} of $\mu_{I,\vec{v}}$ variables.
How do we relate this set-up to the isotropic one discussed before?
Observe that a generic basis state in the lattice model will be, $$\psi_{\{\mu_{I,\vec{v}}\}}[h_{I,\vec{v}}] = \prod_{I,\vec{v}}
(h_{I,\vec{v}})^{\mu_{I,\vec{v}}} ~:=~ \langle k_J(x)|\ldots, \ \mu_{I,
\vec{v}}\ , \ \ldots\rangle ~~~,~~~k_J(x) := \tilde{k}_J(x)\ell_0 \ .
$$ If we choose $\tilde{k}(x) := \tilde{c} := V_0^{-1/3}c ~ \forall~ x, I$, then the basis function becomes a function of a single variable $c$ (which is independent of $x$), and is of the form: $$\psi_{\mu}(c) ~ = ~ e^{i\mu c/2} ~~,~~ \mu := N^{-1/3}\sum_{I, \vec{v}}
\mu_{I, \vec{v}} ~~ \in \mathbb{Q} \ ;
$$ which can be viewed as a basis element of Cyl$_{\mathrm{isotropic}}$. Thus we can define a map $\pi:$ Cyl$_{\mathrm{lattice}} \to $ Cyl$_{\mathrm{isotropic}}$, $$\begin{aligned}
\pi: |\ldots,\ \mu_{I,\vec{v}}\ ,\ \ldots\rangle ~ \to ~ |\mu\rangle &
\Leftrightarrow &
\langle c|\mu\rangle ~ := ~ \langle k_J(x)|\ldots,\ \mu_{I,\vec{v}}\ ,\
\ldots\rangle\left. \right|_{\tilde{k}(x) = \tilde{c}} \nonumber \\
\mathrm{with}~~ \mu & := & N^{-1/3}\sum_{I, \vec{v}} \mu_{I, \vec{v}}
$$ Note that the image of $\pi$-map is a separable subspace of Cyl$_{\mathrm{isotropic}}$, spanned by $|\mu\rangle, \mu \in
\mathbb{Q}$.
Clearly we cannot [*uniquely*]{} identify a cylindrical state of the lattice model, given a cylindrical state of the isotropic model. However, we can define a map $\sigma:$ Cyl$_{\mathrm{isotropic}}$ $\to$ Cyl$^*_{\mathrm{lattice}}~$, $~\sigma: |\mu\rangle \to (\mu|, \ \mu \in
\mathbb{R}$, such that, $$(\mu|\ldots, \ \nu_{I, \vec{v}}\ , \ \ldots\rangle ~ = ~
\langle\mu|\pi\left(|\ldots, \ \nu_{I, \vec{v}}\ , \
\ldots\rangle\right) ~~ = ~~\delta_{\mu,\nu}~~,~~ \nu :=
N^{-1/3}\sum_{I, \vec{v}} \nu_{I, \vec{v}} \ .
$$ In the second equality, we have used the inner product of the isotropic model. This map embeds cylindrical states of the isotropic model into the distributional states of the lattice model[^37].
Now, we have Operators $A^*$ acting on Cyl$^*_{\mathrm{lattice}}$ corresponding to operators $A$ acting on the Cyl$_{\mathrm{lattice}}$, defined in the usual manner. Those of these operators which act [*invariantly*]{} on the image of $\sigma$ in Cyl$^*_{\mathrm{lattice}}$, can be identified with operators of the isotropic model. For these operators, we can define $A_{\mathrm{isotropic}}$ via the equation: $\sigma( A_{\mathrm{isotropic}}|\mu\rangle ) := (\sigma
|\mu\rangle)A^*_{\mathrm{lattice}}$. Since we have embedded isotropic states in the distributions of the lattice model and also have correspondence between operators, matrix elements computed in the isotropic model can be understood as actions of lattice distributions on lattice cylindrical states. Consider an operator $A_{\mathrm{lattice}}$ on Cyl$_{\mathrm{lattice}}$. This defines an operator $A^*_{\mathrm{lattice}}$ on Cyl$^*_{\mathrm{lattice}}$: $(A^*_{\mathrm{lattice}} \phi|\ldots,\
\nu_{I,\vec{v}}\ ,\ \ldots\rangle := (\phi |\left\{A_{\mathrm{lattice}}
|\ldots,\ \nu_{I,\vec{v}}\ ,\ \ldots\rangle\right\}$. [*If*]{}, for every distribution $(\phi| = (\mu| =: \sigma(|\mu\rangle)$, the operator $A^*_{\mathrm{lattice}}$ gives another distribution $(\mu'| =:
\sigma|\mu'\rangle$, [*then*]{} we get an operator on Cyl$_{\mathrm{isotropic}}\ $: $A_{\mathrm{isotropic}}|\mu\rangle :=
|\mu'\rangle$.
It is easy to see that the multiplicative operators on Cyl$_{\mathrm{lattice}}$, give multiplicative operators on Cyl$_{\mathrm{isotropic}}$. For example, taking $A_{\mathrm{lattice}} =
h_{J,\vec{v}'}$, the lattice state $|\ldots,\ \nu_{I,\vec{v}}\ ,\
\ldots\rangle$ will have the $\nu_{J,\vec{v}'}$ incremented by 1. The action of the $(\mu|$ will give $\delta_{\mu, \nu + 1}$. This can be understood as the action of $(\mu - 1|$ on the original lattice state. Thus $(A^*_{\mathrm{lattice}}\mu| = (\mu - 1|$ which implies the a multiplicative action $A_{\mathrm{isotropic}}|\mu\rangle := |\mu -
1\rangle$.
For elementary flux operators, little more work is needed. For example, action of $\hat{{\cal F}}^I_{\vec{v}}$ on a basis state, $|\ldots,\
\nu_{J,\vec{v}'}\ ,\ \ldots\rangle$ is [*zero*]{} unless $\nu_{J,\vec{v}'} \neq 0$ for some $J = I$ and at some $\vec{v}' =
\vec{v}$ i.e. $$\begin{aligned}
\hat{{\cal F}}^I_{\vec{v}}|\nu_{J,\vec{v}'}\rangle & = &
{{\textstyle \frac{1}{2}}}\gamma{\ell_{\mathrm P}}^2\nu_{J,\vec{v}'}\delta^I_J\delta_{\vec{v},
\vec{v}'}|\nu_{J,\vec{v}'}\rangle ~~~~~~~or \nonumber \\
(\mu|\hat{{\cal F}}^I_{\vec{v}}|\nu_{J,\vec{v}'}\rangle & = &
\left[\frac{1}{2}\gamma{\ell_{\mathrm P}}^2\nu_{J,\vec{v}'}\right]\delta^I_J\delta_{\vec{v},
\vec{v}'}\delta_{\mu,\nu}~~~,~~~\nu := N^{-1/3}\nu_{J,\vec{v}'}
~~~However, \\
(\mu'|\nu_{I,\vec{v}'}\rangle & = & (\mu'|\nu_{I,\vec{v}''}\rangle
~~~~~\forall ~~~~~ (\mu'|,\ \vec{v}', \ \vec{v}'' \ .
$$ Thus $(\hat{{\cal F}}^I_{\vec{v}})^*$ [*cannot*]{} act invariantly on the image of $\sigma$ in Cyl$^*_{\mathrm{lattice}}$. It is clear though that if we [*sum*]{} the elementary flux operators (with the directional index $I$) over [*all*]{} the lattice sites (this is a finite sum due to the cell), then the sum will act invariantly. By averaging over the directions as well, we can construct an operator corresponding to a ‘flux’ operator on Cyl$_{\mathrm{isotropic}}$. In equations, $$\begin{aligned}
\hat{p}^I_{\mathrm{lattice}} & := & N^{-1/3} \sum_{\vec{v}} \hat{{\cal
F}}^I_{\vec{v}} ~~,~~ \hat{p}_{\mathrm{lattice}} ~ := ~
\frac{1}{3}\sum_I \hat{p}^I_{\mathrm{lattice}} ~~ \Rightarrow \\
(\mu| \left\{\hat{p}_{\mathrm{lattice}}|\ldots,\ \nu_{J,\vec{v}'}\ ,\
\ldots\rangle\right\} & = & \frac{1}{6}\gamma{\ell_{\mathrm P}}^2 N^{-1/3}\sum_{J,
\vec{v}} \nu_{J, \vec{v}}(\mu|\ldots,\ \nu_{J,\vec{v}'}\ ,\
\ldots\rangle \nonumber \\
& = & \frac{1}{6}\gamma{\ell_{\mathrm P}}^2\ \nu\ \delta_{\mu,\nu} ~~,~~ \nu :=
N^{-1/3} \sum_{I, \vec{v}}\nu_{J, \vec{v}}
$$ The last expression is exactly the matrix element of the $\hat{p}$ operator defined in the isotropic model. As noted in the footnote, the identification of the matrix elements is restricted to $\mu, \nu \in
\mathbb{Q}$.
This completes our summary of the lattice model and how isotropic model is ‘embedded’ in the lattice model. This ‘embedding’ refers to embedding of the particular separable subspace of Cyl$_{\mathrm{isotropic}}$.
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[^1]: At that time, the universe was supposed to be eternal and hence, however small the gravitational instability, existence of stable atoms would threaten GR unless gravitational radiation is also terminated at certain stage.
[^2]: One could [*analytically extend*]{} such maximally Cauchy evolved solutions further. These will have Cauchy Horizons, an example being Reissner-Nordstrom space-time.
[^3]: Let $\phi: p \to \phi(p)$ be a smooth map of a manifold onto itself. Given a function $f':M \to \mathbb{R}$, define another function $f:M \to \mathbb{R}$ as: $f(p) := f'(\phi(p))$. This function is the [*pull-back*]{} of $f'$ and is also denoted as $f :=
\phi^*(f')$. Likewise, given a vector field $X$ on $M$ define a new vector field $X'$ as: $X'(f')|_{\phi(p)} := X(f)|_{p}$. The new vector field is called the [*push-forward*]{} of $X$ and is also denoted as: $X' = \phi_*(X)$. Now introduce local coordinates $x^i$ around $p$ and $y^i$ around $\phi(p)$. It is easy to see that the components of tensors relative to these coordinates are related in exactly the same manner as though $x \to y(x)$ is a change of chart.
[^4]: Since we have taken (differential) equations as specifying a presentation of a theory, the manifold [*cannot*]{} be thought of as a background, but rather part of the specification of the theory. If we take some transition amplitudes (among topological spaces, sets, ...) as specifying a theory, then the choice of a particular differential structure [*will*]{} constitute a background since it is also a dynamical entity. We will restrict to manifold category. This also explains why metric by itself is not essential for background independence of gravity, a dynamical tetrad with compatible spin connection would do just as well.
[^5]: Let $\Omega \subset H_{\mathrm{kin}}$ be a dense subspace of the kinematical Hilbert space. Let $\Omega^*$ denote its algebraic dual (space of linear functions on $\Omega$) so that $\Omega
\subset H_{\mathrm{kin}} \subset \Omega^*$. $\Omega$ is chosen so that it contains the domains of the constraint operators as well as of other operators of interest. Distributional solutions of constraints are those elements of $\Omega^*$ which evaluate to zero on all elements of $\Omega$ of the form $\hat{C}|\psi\rangle, \forall\ \psi \in \Omega$.
[^6]: This is local existence and uniqueness theorem for the Einstein equation. Since these are short time evolutions, one cannot guarantee that largest possible space-time constructed will be globally hyperbolic. However, if a globally hyperbolic solution is to exist, one can perform a time + space decomposition to put the equations in a Hamiltonian form. That Einstein equation admit a well-posed initial value problem is a necessary condition for globally hyperbolic solutions. That the equations are of Hamiltonian form is an additional, non-trivial property. This follows most directly via the Einstein-Hilbert action formulation.
[^7]: The remaining, Class B models are thought not have a Hamiltonian formulation and hence are not amenable to analysis by canonical methods [@AshtekarSamuel].
[^8]: A few essential details from Forgacs and Manton are summarised in the appendix.
[^9]: Similar decomposition is made for all quantities with spatial and the Lorentz indices. The [*contravariant*]{} spatial index is expressed using the invariant vector fields and the covariant one using the invariant 1-forms. In particular, $K_a^i := K^i_I \omega_a^I
~,~ \Gamma^i_a := \gamma^i_I \omega_a^I$.
[^10]: There is still a discrete invariance remaining and involves changing the sign of two of the triad and connection components.
[^11]: This is discussed in more details in section \[CellIndependence\]
[^12]: For the FRW metric, integral curves of $\partial_t$ are time-like geodesics and hence incompleteness with respect to $t$ is synonymous with geodesic incompleteness.
[^13]: The domain of the inverse power operator(s) will have to exclude the subspace corresponding to the zero eigenvalue of the triad operator. But this makes the domain [*non-dense*]{} and its adjoint cannot be defined [@ABL].
[^14]: The expression for the Hamiltonian constraint follows exactly from the full theory procedure. Starting from the equation (\[IsoGrav\]), the integral will be replaced by a sum over smaller cells of a triangulation. The size parameter of the cells will drop out thanks to density weight 1 of the Hamiltonian. Due to homogeneity, contribution from each cell will be the same and hence the total sum will be number of cells of the triangulation times the contribution of one cell. There are exactly $\mu_0^{-3}$ cubical (say) cells with side of length $\mu_0 V^{1/3}_0$ and this produces the factor of $\mu_0^{-3}$ in equation (\[RegularisedHam\]). The $V_0$ of course disappears as in the full theory \[SubCells\]
[^15]: For contrast, if one just symmetrizes the non-symmetric operator (without the sgn factor), one gets a difference equation which [*is non-deterministic*]{}. Note that this issue arises only in [*one*]{} superselection sector so may not really be an issue. However, requiring deterministic equation in [*all*]{} sectors could be invoked as a criterion to discriminate between different factor ordering.
[^16]: The effective Hamiltonian then reduces to $-{{\textstyle \frac{3}{\kappa}}}\gamma^{-2}\sqrt{p} \left[\mu_0^{-2}
\mathrm{sin}^2(\mu_0 c)\right]$. This is also the Hamiltonian in eq. (\[RegularisedHam\]) for non-zero $\mu_0$.
[^17]: For $p$ in the semi-classical regime, one should include the contribution of the quantum geometry potential present in the non-symmetric ordering, especially for examining the bounce possibility [@EffHam].
[^18]: For an alternative view on using large values of $j$, see reference [@MartinLattice].
[^19]: A general abstract procedure using group averaging is also available.
[^20]: Our primary goal here is to compare the classical geometry (Wheeler-DeWitt quantization) and quantum geometry (loop quantization). Consequently, the gravitational constraint is quantized in two different ways but [*for simplicity*]{}, the matter sector is quantized in the usual Schrodinger way. In both quantizations, $p_{\phi} = -i\hbar\partial_{\phi}$ and there is no $\phi$ dependence in the matter Hamiltonian, so the two quantum Hamiltonian will have identical expressions. However, using $\phi$ as labelling an ‘emergent time’ would be questionable.
[^21]: For the Schrodinger quantization, the explicit eigenfunctions are: $e_k(\mu) :=
{{\textstyle \frac{|\mu|^{1/4}}{4\pi}}} e^{ik\ell n|\mu|}$ and the eigenvalues are: $\omega^2(k) := {{\textstyle \frac{2\kappa}{3}}}(k^2 + 1/16)$ [@APSTwo].\[WDWEigenfunctions\]
[^22]: In the above heuristic reasoning for the form of the semiclassical state, we have glossed over some technical issues such as whether the states exhibited are in the domain of the $\hat{\Theta}$ operator which requires carefully stipulating conditions on the function of $k$ (which has been taken to be a Gaussian). These are too technical to go into here and are not expected to affect the corresponding discussion for LQC. A discussion of these issues may be seen in [@Madhavan].
[^23]: For a comparison in Schrodinger quantization, see remarks in [@APSTwo].
[^24]: The action formulation (Lagrangian or Hamiltonian) in turn is required for a quantum theory. The classical theory needs only equations of motion which are independent of any cell.
[^25]: The limit $V_0 \to
\infty$ can be viewed as a convenient way to pick-out $V_0-$independent terms and/or could also be heuristically motivated by noting that the definition of homogeneity identifies the spatial manifold with the group manifold and this group manifold is $\mathbb{R}^3$ for the present case.
[^26]: During the process of [*loop quantization*]{}, fiducial scales could appear again eg through the holonomies along edges. However as explained in the footnote \[SubCells\], the $V_0$ disappear.
[^27]: In the first paper of [@EffHam], the effective density contained only the leading terms of the holonomy corrections which have been summed up in the second paper.
[^28]: From eqn. \[Mu1GenVals\], with $\lambda = 1$, one sees that the corrections go as $v_1^{-2} \sim {\ell_{\mathrm P}}^6 p^{-3} \sim {\ell_{\mathrm P}}^6
(\xi)^6 a^{-6} \to 0$ as $\xi \to 0$.
[^29]: The logic used to motivate the role of area operator, is also extended to other Bianchi models such that when isotropy is imposed, the $\bar{\mu}$-scheme of isotropic model is recovered back[@AWE].
[^30]: In the context of Kaluza-Klein approach to unification, the analysis of invariant quantities was carried out to construct suitable ansatz. There, the space-time is taken to be of the product form $M_4 \times B$ with $B$ a compact manifold. The isometry groups of the compact manifold played the role of symmetries and the forms of the fields on the space-times were obtained.
[^31]: Note that $D\chi = 0$ follows only for the symmetric connections. If it were to hold for all connections, then $[F,
\chi] = 0$ would hold for all field strengths and this would correspond to a reduction of the gauge group to the little group of $\chi_{mn}$. This is analogous to the non-trivial Higgs vacua situation.
[^32]: For classical matrix groups, these are given by $g^{-1}dg$ and $dg g^{-1}$ and are left and right invariant respectively.
[^33]: The classification of connections invariant under some group of automorphisms of appropriate bundles is given by [*generalised Wang theorem*]{}. There are many mathematical fine prints in the above discussion which should be seen in the references in [@SymmetricConnections].
[^34]: For precise technical conditions, please refer to [@Shadows]
[^35]: The functions $f_{\alpha}(p)$ are taken to be dimensionless. This makes the $\mu_{\alpha}$ to have dimensions of ${\ell_{\mathrm P}}^{\alpha}$ and $g_{\alpha}(p)$ to have dimensions of ${\ell_{\mathrm P}}^2$. The $v_{\alpha}$ is defined to be dimensionless.
[^36]: Actually, for large volume, the leading term is [*independent*]{} of $\lambda$. The sub-leading (correction) terms, do depend on $\lambda$.
[^37]: Notice that $\mu, \nu$ defined above are [*rationals with a common denominator $N^{1/3}$*]{}. Therefore the distributions $(\mu|$ are non-trivial only for $\mu \in \mathbb{Q}$ with the same denominator.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate whether current data on the distribution of observed flux densities of Fast Radio Bursts (FRBs) are consistent with a constant source density in Euclidean space. We use the number of FRBs detected in two surveys with different characteristics along with the observed signal-to-noise ratios of the detected FRBs in a formalism similar to a $V/V_\mathrm{max}$-test to constrain the distribution of flux densities. We find consistency between the data and a Euclidean distribution. Any extension of this model is therefore not data-driven and needs to be motivated separately. As a byproduct we also obtain new improved limits for the FRB rate at 1.4GHz, which had not been constrained in this way before.'
author:
- |
Niels Oppermann,$^{1,2}$[^1] Liam D. Connor,$^{1,2,3}$ Ue-Li Pen$^{1,2,4,5}$\
$^{1}$Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto ON, M5S 3H8, Canada\
$^{2}$Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto ON, M5S 3H4, Canada\
$^{3}$Department of Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto ON, M5S 3H4, Canada\
$^{4}$Canadian Institute for Advanced Research, 180 Dundas St. West, Toronto ON, M5G 1Z8, Canada\
$^{5}$Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo ON, N2L 2Y5, Canada
bibliography:
- 'alpha.bib'
date: 'Accepted XXX. Received YYY; in orignal form ZZZ'
title: The Euclidean distribution of Fast Radio Bursts
---
\[firstpage\]
methods: statistical – pulsars: general
Introduction
============
Fast Radio Bursts (FRBs) are bright, short radio pulses that are highly dispersed. Most FRBs observed so far have dispersion measures that are much larger than what is expected due to the Milky Way’s interstellar medium. Thus, it is likely that the sources producing these bursts are extragalactic. If most of the dispersion is due to the ionised intergalactic medium, the sources have to be at redshifts of the order of one [e.g., @champion-2015]. In this case, they may soon become useful as cosmological probes [e.g., @masui-2015a]. However, the large observed dispersion may also be due to dense material surrounding the source [e.g., @connor-2016a], putting them at potentially much smaller distances.
An observable distinction between cosmological models and more local models for the FRB population is given by the distribution of flux densities. This is independent of the pulse dispersion. The simplest model is one in which FRBs are uniformly distributed and their location does not influence any of their observable properties other than that farther bursts appear dimmer. In Euclidean space this leads to an unambiguous prediction for their flux densities, namely $$\mathrm{d}N \propto S^{-(\alpha + 1)} \, \mathrm{d}S
\label{eq:powerlaw}$$ with $\alpha = 3/2$, where $S$ is the flux density and $\mathrm{d}N$ is the number of FRBs with a flux density in an infinitesimal interval $\mathrm{d}S$. This relation holds independently from their intrinsic luminosity distribution and does not require them to be standard candles. Deviations from an index 3/2 are for example expected if the number density of FRBs is not constant and/or if their distances are large enough that deviations from Euclidean geometry become important. If FRBs are extragalactic, but local, then the default expectation should be $\alpha = 3/2$. It is worth noting that a constant-density Euclidean distribution is the only model that leads to a clear prediction for the observed flux densities. If FRBs are at large enough distances that the expansion history of the Universe is important, then the population of FRBs can be expected to also have undergone some evolution, thus introducing functional degrees of freedom that are at present completely unconstrained. An additional effect that can in principle influence the measured distribution of flux densities is dispersion smearing, i.e., the dispersion of the pulses’ arrival times within a frequency bin of a survey, resulting in a reduction of the signal-to-noise ratio for very high dispersion measures. If dispersion measures were correlated with distance, this could potentially lead to a systematic flattening of the distribution, but will not affect its shape if the dispersion measure and distance of an FRB are uncorrelated.
In this note we investigate whether the constant-density Euclidean model is consistent with current observations. We do this by studying the one-parametric class of models in which $\alpha$ is a free parameter and which includes the constant-density Euclidean model as a special case. Deriving the observational constraints on the parameter $\alpha$ has the potential of showing that $\alpha=3/2$ is disfavoured by the data without the need to make any more complicated model assumptions. We derive constraints from a combination of source counts of different surveys and the observed $V/V_\mathrm{max}$-values [@schmidt-1968].
In the simple constant-density Euclidean model, the power-law behaviour of Eq. is not only valid for the flux densities, but also for any other observable that depends linearly on flux density. One example would be the fluence $$\begin{aligned}
F &= \int\mathrm{d}t \, S(t)\nonumber\\
&= S \, \tau,\end{aligned}$$ where $\tau$ is the duration of a burst and $S$ is the average flux density within the interval of length $\tau$. Another example is the observed signal-to-noise ratio, which depends on the flux density of the burst, its duration, and of course on properties of the telescope and the survey. In many cases it can be approximated as $$s \approx K \, S \, \tau^{1/2},$$ where we include all instrumental properties in the constant $K$ [e.g., @caleb-2016]. The detection of an FRB is always subject to a sensitivity cutoff in signal-to-noise, and not in flux density or fluence. This makes the statistics of flux density and fluence more complicated than the statistics of the signal-to-noise ratio $s$ and we choose to cast all equations in terms of $s$. Note that, as is common in the field, we use the term signal-to-noise ratio to mean the amplitude of the FRB signal plus noise, divided by the standard deviation of the noise, which can be approximately determined empirically for each search window length $\tau$.
We derive the necessary statistical methodology in the following section, discuss the data that we use and show our results in Sect. \[sec:results\]. We conclude with a discussion in Sect. \[sec:discussion\].
Methodology {#sec:methods}
===========
Likelihood for the observed signal-to-noise ratios {#sec:snr}
--------------------------------------------------
Clearly some information on the parameter $\alpha$ is contained in the distribution of observed signal-to-noise ratios. If we assume a value for $\alpha$ and suppose that a survey with a given signal-to-noise threshold $s_\mathrm{min}$ detects an FRB, then the likelihood for its signal-to-noise ratio to be $s$ is, according to Eq. , $$\mathcal{P}(s|s_\mathrm{min},\alpha) = \left\{\begin{array}{cc}\frac{\alpha}{s_\mathrm{min}} \, \left(\frac{s}{s_\mathrm{min}}\right)^{-(\alpha + 1)} & \textnormal{if } s \geq s_\mathrm{min}\\0&\textnormal{else}\end{array}\right..$$ If the survey detects $n$ independent FRBs then the joint likelihood for their signal-to-noise ratios is simply the product of the individual ones.
For $N$ different surveys, each detecting $n_1,\dots, n_{N}$ FRBs, the situation is the same, except that we have to take into account that each survey has a different detection threshold $s_\mathrm{min}$. If we denote the observed signal-to-noise value of the $i$-th FRB in the $I$-th survey as $s_{I,i}$, the $(n_1+\cdots+n_N)$-dimensional vector of all these observed values as $\vec{s}$, the $N$-dimensional vector of all threshold signal-to-noise values as $\vec{s}_\mathrm{min}$, and the $N$-dimensional vector of the numbers of detections in each survey as $\vec{n}$, the complete likelihood becomes $$\mathcal{P}(\vec{s}|\vec{n},\vec{s}_\mathrm{min},\alpha) = \prod_{I=1}^N \prod_{i=1}^{n_I} \mathcal{P}(s_{I,i}|s_{\mathrm{min},I},\alpha).$$ This is easily calculated and we will do so in Sect. \[sec:results\]. In the following we will refrain from mentioning $\vec{s}_\mathrm{min}$ explicitly in the notation of probabilities and imply that all survey properties are always fixed.
Note that the combination $(s/s_\mathrm{min})^{-\alpha}$ for $\alpha = 3/2$ corresponds to the ratio of the volume interior to the FRB and the volume in which this particular FRB could have been detected by the survey, $V/V_\mathrm{max}$, for a constant source density in three-dimensional Euclidean space. The likelihood we are using here to constrain $\alpha$ is thus closely related to the $V/V_\mathrm{max}$-test used in many contexts to check for deviations from a constant density for a source population [e.g., @schmidt-1968].
Likelihood for the number of observed FRBs {#sec:number}
------------------------------------------
In addition to the information contained in the signal-to-noise ratios of the observed bursts, some information is also contained in the numbers of bursts detected by different surveys. For any one survey, the number of detected FRBs puts constraints on the rate of FRBs occurring above the detection threshold of that survey. This rate can be rescaled to a different survey with a different detection threshold and confronted with the observed number of bursts for that survey. However, the rescaling depends on the parameter $\alpha$ and thus the number of bursts detected by two or more surveys puts constraints on $\alpha$.
To include these constraints in our analysis we introduce the FRB rate explicitly as an unknown parameter. Since the rate observable by a given survey depends on various properties of the survey, we define the rate $r_0$ occurring above the detection threshold of a hypothetical survey described by a system temperature $T_{\mathrm{sys},0} = 1\,\mathrm{K}$, a gain $G_0 = 1\,\mathrm{K}\,\mathrm{Jy}^{-1}$, $n_{\mathrm{p},0}=2$ observed polarizations, a bandwidth $B_0 = 1\,\mathrm{MHz}$, and a signal-to-noise threshold $s_{\mathrm{min},0} = 1$. As explained by @connor-2016b, the FRB rate above the detection threshold of the $I$-th survey is then a rescaled version of this rate, namely $$\begin{aligned}
r_I &= r_0 \, \left(\frac{T_{\mathrm{sys},I}}{T_{\mathrm{sys},0}} \, \frac{G_0}{G_I} \, \sqrt{\frac{n_{\mathrm{p},0} \, B_0}{n_{\mathrm{p},I} \, B_I}} \, \frac{s_{\mathrm{min},I}}{s_{\mathrm{min},0}}\right)^{-\alpha}\nonumber\\
&= r_0 \, \left(\frac{T_{\mathrm{sys},I}}{G_I} \, \sqrt{\frac{2\,\mathrm{MHz}}{n_{\mathrm{p},I} \, B_I}} \, s_{\mathrm{min},I} \, \mathrm{Jy}\right)^{-\alpha}.
\label{eq:rescaling}\end{aligned}$$
The expected number of FRBs detected by the $I$-th survey will then be $$M_I = r_I \, \Omega_I \, T_I,$$ where $\Omega_I$ is the angular size of the survey’s field of view and $T_I$ is the time spent surveying. The likelihood for the actual number of FRBs observed in this survey is then a Poissonian distribution with this expectation value, $$P(n_I|r_0,\alpha) = \frac{{M_I}^{n_I}}{n_I!} \, \mathrm{e}^{-M_I}.$$ For $N$ surveys the complete likelihood again becomes a product of the likelihoods for the individual surveys, $$P(\vec{n}|r_0,\alpha) = \prod_{I=1}^N P(n_I|r_0,\alpha),$$ and can be used to put constraints on the distribution of flux densities via the parameter $\alpha$, as well as on the overall rate of FRBs, here parameterized as the rate above the detection threshold of our hypothetical survey, $r_0$.
Posterior {#sec:posterior}
---------
To get the complete set of constraints on the distribution of flux densities, both from the observed signal-to-noise ratios and from the detection numbers of different surveys, we combine the results of Sects. \[sec:snr\] and \[sec:number\]. We write the joint likelihood for the number of observed FRBs and their signal-to-noise ratios as $$\begin{aligned}
\mathcal{P}(\vec{s},\vec{n}|r_0,\alpha) &= \mathcal{P}(\vec{s}|\vec{n},r_0,\alpha) \, P(\vec{n}|r_0,\alpha)\nonumber\\
&= \mathcal{P}(\vec{s}|\vec{n},\alpha) \, P(\vec{n}|r_0,\alpha).\end{aligned}$$ If we assume flat priors for $r_0$ and for $\alpha > 0$, this likelihood is proportional to the joint posterior for the parameter $\alpha$ and the rate $r_0$.
The likelihood for observed fluxes within a survey obviously only gives us constraints if the survey has in fact detected at least one FRB. Note, however, that we can in principle include surveys without FRB detection by setting $$\mathcal{P}(\vec{s}|n=0,s_\mathrm{min},\alpha) = 1$$ and thus still use them to constrain the parameter $\alpha$ via their implications on the FRB rate above their detection thresholds. Similarly, the numbers of FRBs detected by different surveys only have implications for the parameter $\alpha$ if we assume that the surveys observe the same source population, described by the same rate $r_0$. This assumption will in general be violated if different surveys have different frequency coverage or different observational strategies. Specifically, the observations of a deep and narrow survey will in general not be described by the same statistics as those of a shallow and wide survey, as explained by @connor-2016c. Care is thus warranted when comparing detection numbers of qualitatively different surveys. Such an attempt will require more parameters or simply setting $$P(n|r_0,\alpha) = 1$$ for all surveys that are not expected to be described by $r_0$.
Data and results {#sec:results}
================
We make use of 15 observed FRBs from seven surveys. For definiteness, we list all values used in our calculation in Tables \[tab:surveys\] and \[tab:FRBs\]. For the likelihood of the numbers of detected FRBs, we only make use of two dedicated pulsar surveys with well-defined characteristics, namely the High Time Resolution Universe Pulsar Survey (HTRU; @keith-2010) at the Parkes telescope and the Pulsar ALFA survey (PALFA; @cordes-2006) at the Arecibo Observatory. We choose these two surveys because for most other discovered FRBs it is hard to estimate the surveying period $T$ that has been searched for FRBs, especially in the case of non-detections. The survey of @masui-2015b is similarly well-defined, but sensitive to different frequencies. We assume our parameter $r_0$ to describe the rate at frequencies around 1.4GHz and do not want to make any assumption about the relation between this rate and the rate at 800MHz, which is the central frequency of @masui-2015b.
Even for HTRU and PALFA, the parameters needed in Eq. are defined somewhat ambiguously. To avoid building complicated models of the telescopes and surveys, we generally opt for simple choices that can be made consistently for both surveys. Specifically, this means that we do not include any estimate of the sky temperature due to the Milky Way in the values we assume for the system temperature $T_\mathrm{sys}$. For the gain $G$ we use the arithmetic mean of the gains corresponding to the beam centres of the multibeam receivers. The bandwidth $B$ does not include frequencies deemed unusable by the surveying team and the angular size of the field of view $\Omega$ is intended to approximate the area within the half-maximum beam power. The resulting numerical values are listed in Table \[tab:surveys\]. Other reasonable choices for these parameters will typically lead to deviations on the order of 10%. Note that the exact definition of each parameter does not impact the results as long as the same definition is used for all surveys that are being compared.
For the constraint on $\alpha$ coming from the likelihood for the observed signal-to-noise ratios, we can use all detected FRBs, as long as there is a well-defined signal-to-noise cutoff $s_\mathrm{min}$. Since we are investigating the population of sources, we are not including repeated bursts from the same object [@spitler-2016]. We also exclude the single burst detections by @lorimer-2007 and @keane-2011, since no definitive value of $s_\mathrm{min}$ can be determined. We list the values of $s$ and $s_\mathrm{min}$ that we use in Table \[tab:FRBs\].
After calculating the two-dimensional posterior for $\alpha$ and $r_0$, we derive the final constraint on $\alpha$ by marginalising over $r_0$ and vice versa. These posterior distributions are shown in Fig. \[fig:2dpost\].
0= =0 ‘\*=
[ccccccccc]{} survey$^1$& $s_\mathrm{min}$& $T_\mathrm{sys}/\mathrm{K}$& $G/(\mathrm{K/Jy})$& $n_\mathrm{p}$& $B/\mathrm{MHz}$& $\Omega/(\mathrm{deg}^2)$& $T/\mathrm{h}$& $n$ HTRU \[1\]& 10& 23& 0.64& 2& 340& $13\times0.043*$& 3650& 9 PALFA \[2\]& \*7& 30& 8.5\*& 2& 300& $*7\times0.0027$& \*886& 1
$^1$ \[1\] @champion-2015 [@thornton-2013; @keith-2010]; \[2\] @spitler-2014 [@cordes-2006]
0= =0 ‘\*=
[lccc]{} name& $s$& $s_\mathrm{min}$& survey$^2$ FRB090625& 30& 10& \[1\] FRB110220& 49& 10& \[1\] FRB110626& 11& 10& \[1\] FRB110703& 16& 10& \[1\] FRB120127& 11& 10& \[1\] FRB121002& 16& 10& \[1\] FRB130626& 21& 10& \[1\] FRB130628& 29& 10& \[1\] FRB130729& 14& 10& \[1\] FRB121102& 14& \*7& \[2\] FRB010125& 17& \*7& \[3\] FRB131104& 30& \*8& \[4\] FRB140514& 16& 10& \[5\] FRB150418& 39& 10& \[6\] FRB110523& 42& \*8& \[7\]
$^1$ <http://www.astronomy.swin.edu.au/pulsar/frbcat/>\
$^2$ \[1\] @champion-2015 [@thornton-2013; @keith-2010]; \[2\] @spitler-2014 [@scholz-2016]; \[3\] @burke-spolaor-2014; \[4\] @ravi-2015; \[5\] @petroff-2015; \[6\] @keane-2016; \[7\] @masui-2015b [@connor-2016b]
![\[fig:2dpost\]Posterior distribution for the parameter $\alpha$ describing the distribution of FRB flux densities and the FRB rate $r_0$. The bottom left panel shows the two-dimensional posterior for both parameters, the smaller panels show the marginalised posteriors for each parameter individually. We show separate curves and contours for the constraints coming from the signal-to-noise ratios $\vec{s}$ (green dashed), from the detection numbers $\vec{n}$ (blue dashed), and their combination (orange solid). The contour lines show the 68%, 95%, and 99% confidence regions.](figure1.pdf)
Discussion {#sec:discussion}
==========
Figure \[fig:2dpost\] shows a strong correlation between the FRB rate and the slope parameter $\alpha$. This can be understood in terms of regions of parameter space that are in tension with the data. If the rate of FRBs is high, a shallow flux density distribution will overpredict the number of FRBs occurring at high signal-to-noise ratios. If the rate is low, on the other hand, a steep distribution will underpredict the number of FRBs occurring above the detection threshold of current surveys, especially at high signal-to-noise values.
The figure also shows the posterior distributions that are obtained if only the detection numbers or the signal-to-noise ratios are used, instead of their combination. Obviously, the signal-to-noise ratios alone do not constrain the FRB rate at all. Thus, the corresponding contours appear as horizontal lines in the main panel of the figure. And even for the parameter $\alpha$, the main constraint comes from the comparison of the detection numbers for the two surveys HTRU and PALFA. Using the signal-to-noise ratios in addition does, however, add some information, in that it rules out close-to-flat flux distributions and very low rates.
The full posterior for the parameter $\alpha$ still allows a wide range of values. The 95% confidence interval is $$0.8 \leq \alpha \leq 1.7.$$ This is to be contrasted with recent results from the literature. @caleb-2016, for example, find $\alpha = 0.9 \pm 0.3$ and @li-2016 claim $\alpha = 0.14 \pm 0.20$. We stress that our constraints are model-independent in the sense that we have not assumed any specific relation between flux density, burst duration, and dispersion measure. An important conclusion of our analysis is that the simplest possible model for the distribution of FRBs, constant density in Euclidean space, is consistent with current data. Of course this does not mean that it is proven to be correct, but it does mean that any extension to this model is not data-driven but has to be motivated independently. This finding is consistent with the qualitative conclusions of @katz-2016a and @katz-2016b.
As a byproduct of our attempt to constrain $\alpha$, we also obtain constraints on the rate of FRBs at 1.4GHz. The 95% confidence interval for our parameter $r_0$ is $$4.8\times10^4\,\mathrm{sky}^{-1}\,\mathrm{day}^{-1} \leq r_0 \leq 5.3\times10^5\,\mathrm{sky}^{-1}\,\mathrm{day}^{-1}.$$
It may be worth noting that the constraints on the FRB rate tighten up somewhat if the parameter $\alpha$ is fixed. In the constant-density Euclidean model, for example, the 95% confidence limit on the rate is $$1.6\times10^5\,\mathrm{sky}^{-1}\,\mathrm{day}^{-1} \leq r_{0,\alpha=3/2} \leq 5.4\times10^5\,\mathrm{sky}^{-1}\,\mathrm{day}^{-1}.$$ For any specific survey this rate has to be rescaled according to Eq. . As an example we calculate the FRB rate above the detection threshold of the HTRU survey, again for $\alpha=3/2$, $$1.9\times10^3\,\mathrm{sky}^{-1}\,\mathrm{day}^{-1} \leq r_{\mathrm{HTRU},\alpha=3/2} \leq 6.3\times10^3\,\mathrm{sky}^{-1}\,\mathrm{day}^{-1}.$$ These are slightly lower values than the range derived by @champion-2015. We stress again that all rates we calculate are subject to a signal-to-noise cutoff. Converting them to rates above a given fluence is impossible without making further assumptions.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Vikram Ravi and Kiyo Masui for helpful discussions and Jonathan Katz, Evan Keane, and an anonymous referee for useful comments on the manuscript. This research has made use of NASA’s Astrophysics Data System. The figure was produced using the `matplotlib` library [@hunter-2007] and we acknowledge use of the <span style="font-variant:small-caps;">FRBcat</span> database [@petroff-2016]. We acknowledge NSERC support.
\[lastpage\]
[^1]: E-mail: [email protected] (NO)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In a previous paper, we have constructed, for an arbitrary Lie group $G$ and any of the fields $F=\R$ or $\C$, a good equivariant cohomology theory $KF_G^*(-)$ on the category of proper $G$-CW-complex and have justified why it deserved the label “equivariant K-theory". It was shown in particular how this theory was a logical extension of the construction of Lück and Oliver for discrete groups and coincided with Segal’s classical K-theory when $G$ is a compact group and only finite $G$-CW-complexes are considered. Here, we compare our new equivariant K-theory with that of N.C. Phillips: it is shown how a natural transformation from ours to his may be constructed which gives rises to an isomorphism when $G$ is second-countable and only finite proper $G$-CW-complexes are considered. This solves the long-standing issue of the existence of a classifying space for Phillips’ equivariant K-theory.'
author:
- 'Clément de Seguins Pazzis [^1] [^2]'
title: 'A classifying space for Phillips’ equivariant K-theory'
---
Introduction
============
The problem
-----------
In this paper, $G$ will denote a second-countable Lie group, $\mu$ a right Haar measure on $G$, and $F$ one of the fields $\R$ or $\C$. We wish to compare our equivariant K-theory [@Ktheo1], which is defined on the category of proper $G$-CW-complexes, with Phillips’ equivariant K-theory $KF_G^{\text{Ph}}(-)$ which is defined on the category of proper locally compact Hausdorff $G$-spaces (see [@Phillips]).
To do this, we will construct a natural transformation $\eta : KF_G(-) \longrightarrow KF_G^{\text{Ph}}(-)$ on the category of finite proper $G$-CW-complexes, such that the diagram $$\xymatrix{
KF_G(-) \ar[rr]^{\eta} & & KF_G^{\text{Ph}}(-) \\
& \mathbb{K}F_G(-) \ar[ul]^{\gamma} \ar[ur]
}$$ commutes (the natural transformation $\mathbb{K}F_G(-) \longrightarrow KF_G^{\text{Ph}}(-)$ is the one defined by Phillips p.40 of [@Phillips]). Once we will have done so, it will be easy to check that $\eta$ is an isomorphism on coefficients (i.e. for spaces of the type $G/H \times Y$ where $Y$ is a finite CW-complex with trivial action of $G$), and it will easily follow that $\eta_X$ is an isomorphism for every finite proper $G$-CW-complex $X$. Before explaining the construction, we will start with a quick recollection of the definition of our version of equivariant K-theory.
A review of equivariant K-theory for proper $G$-CW-complexes
------------------------------------------------------------
### $\Gamma$-spaces
The simplicial category is denoted by $\Delta$ (cf. [@G-Z]). Recall that the category $\Gamma$ (see [@Segal-cat]) has the finite sets as objects, a morphism from $S$ to $T$ being a map from $\mathcal{P}(S)$ to $\mathcal{P}(T)$ which preserves disjoint unions (with obvious composition of morphisms); this is equivalent to having a map $f$ from $S$ to $\mathcal{P}(T)$ such that $f(s) \cap f(s') =\emptyset$ whenever $s \neq s'$.
For every $n \in \N$, we set $\mathbf{n}:=\{1,\dots,n\}$ and $[n]:=\{0,\dots,n\}$. Recall the canonical functor $\Delta \rightarrow \Gamma$ obtained by mapping $[n]$ to $\mathbf{n}$ and the morphism $\delta: [n] \rightarrow [m]$ to $$\begin{cases}
\mathbf{n} & \longrightarrow \mathcal{P}(\mathbf{m}) \\
k & \longmapsto \{j \in \N: \bigl\{\delta(k-1) < j \leq \delta(k)\bigr\}.
\end{cases}$$
By a $\Gamma$-space, we mean a *contravariant* functor $\underline{A}: \Gamma \rightarrow \text{CG}$ to the category of k-spaces such that $\underline{A}(\mathbf{0})$ is a well-pointed contractible space. The space $\underline{A}(\mathbf{1})$ is then simply denoted by $A$. We say that $\underline{A}$ is a $\Gamma$-space when, in addition, for all $n \in \mathbb{N}^*$, the continuous map $\underline{A}(\mathbf{n}) \rightarrow \underset{i=1}{\overset{n}{\prod}} A$, induced by all morphisms $\mathbf{1} \rightarrow \mathbf{n}$ which map $1$ to $\{i\}$, is a homotopy equivalence. From now on, when we talk of $\Gamma$-spaces, we will actually mean good $\Gamma$-spaces.
When $\underline{A}$ is a $\Gamma$-space, composition with the previously defined functor $\Delta \rightarrow \Gamma$ yields a simplicial space, which we still write $\underline{A}$, and we can take its thick geometric realization (as defined in appendix A of [@Segal-cat]), which we write $BA$. Since $\underline{A}(\mathbf{0})$ is well-pointed and contractible, we have a map $A \rightarrow \Omega B\underline{A}$ that is “canonical up to homotopy”. Recall that we have an H-space structure on $A$ by composing the map $\underline{A}(\mathbf{2}) \rightarrow A$ induced by $\begin{cases}
\{1\} & \rightarrow \mathcal{P}(\mathbf{2}) \\
1 & \mapsto \{1,2\}
\end{cases}$ and a homotopy inverse of the map $\underline{A}(\mathbf{2}) \rightarrow A \times A$ mentioned earlier.
Given a topological group $G$, a is a contravariant functor $\underline{A}: \Gamma \rightarrow \text{CG}_G$ such that:
(i) $\underline{A}(\mathbf{0})$ is equivariantly well-pointed and equivariantly contractible;
(ii) For any $n \in \N^*$, the canonical map $\underline{A}(\mathbf{n}) \rightarrow \underset{i=1}{\overset{n}{\prod}} A$ is an equivariant homotopy equivalence.
When $\underline{A}$ is a $\Gamma-G$-space, we may define as before a $G$-map $A \rightarrow \Omega BA$.
### k-categories {#struct}
If $\mathcal{C}$ is a small category, then:
- ${\operatorname{Ob}}(\mathcal{C})$ (resp. ${\operatorname{Hom}}(\mathcal{C})$) will denote its set of objects (resp. of morphisms).
- The structural maps of $\calC$ i.e. the initial, final, identity and composition maps are respectively denoted by $${\operatorname{In}}_{\mathcal{C}}: {\operatorname{Hom}}(\mathcal{C}) \rightarrow {\operatorname{Ob}}(\mathcal{C}) \quad ; \quad
{\operatorname{Fin}}_{\mathcal{C}}: {\operatorname{Hom}}(\mathcal{C}) \rightarrow {\operatorname{Ob}}(\mathcal{C});$$ $${\operatorname{Id}}_{\mathcal{C}}: {\operatorname{Ob}}(\mathcal{C}) \rightarrow {\operatorname{Hom}}(\mathcal{C}) \quad \text{and} \quad
{\operatorname{Comp}}_{\mathcal{C}}: {\operatorname{Hom}}(\mathcal{C}) \underset{\vartriangle}{\times} {\operatorname{Hom}}(\mathcal{C}) \rightarrow {\operatorname{Hom}}(\mathcal{C}).$$
- The nerve of $\calC$ is denoted by $\mathcal{N}(\calC)$, whilst $\mathcal{N}(\calC)_m$ will denote its $m$-th component for any $m\in \N$.
By a , we mean a small category with k-space topologies on the sets of objects and spaces, such that the structural maps induce continuous maps in the category of k-spaces. To every topological category $\calC$, we assign a k-category whose space of objects and space of morphisms are respectively ${\operatorname{Ob}}(\calC)_{(k)}$ and ${\operatorname{Hom}}(\calC)_{(k)}$.
To a k-category, we may assign its *nerve* in the category of k-spaces, and then take one of the two geometric realizations $\| \quad \|$ (the “thick realization") or $| \quad |$ (the “thin realization") of it in the category of k-spaces (see [@Segal-cat]). When $\calC$ and $\calD$ are two k-categories, we may define another k-category, denoted by ${\operatorname{Func}}(\calC,\calD)$, whose objects are the topological functors from $\calC$ to $\calD$, and whose morphisms are the continuous natural transformations between continuous functors from $\calC$ to $\calD$. The structural maps are obvious, as are the topologies on the sets of objects and morphisms (see [@Ktheo1] for more details).
Given a $k$-space $X$, we define a $k$-category $\calE X$ with $X$ as space of objects, $X {\operatorname{\underset{k}{\times}}}X$ as space of morphisms, and $(x,y)$ as the only morphism from $x$ to $y$ in $\calE X$.
The category of k-categories is denoted by $\text{kCat}$.
### Proper $G$-CW-complexes
A $G$-space $X$ is called a $G$ when it is obtained as the direct limit of a sequence $(X_{(n)})_{n \in \mathbb{N}}$ of subspaces for which there exists, for every $n \in \mathbb{N}$, a set $I_n$, a family $(H_i)_{i \in I_n}$ of closed subgroups of $G$ and a push-out square $$\begin{CD}
\underset{i \in I_n}{\coprod} (G/H_i) \times S^{n-1} @>>> \underset{i \in I_n}{\coprod} (G/H_i) \times D^n \\
@VVV @VVV \\
X_{(n-1)} @>>> X_{(n)}
\end{CD}$$ in the category of $G$-spaces (where we have a trivial action of $G$ on both the $(n-1)$-sphere $S^{n-1}$ and the closed $n$-disk $D^n$), with the convention that $X_{-1}=\emptyset$. The spaces $(G/H_i) \times \overset{\circ}{D^n}$ are called the equivariant cells (or $G$-cells) of $X$. A $G$-CW-complex is when all its isotropy subgroups are compact, i.e. all the groups $H_i$ in the preceding description are compact.
Relative $G$-CW-complexes are defined accordingly. A is a relative $G$-CW-complex $(X,*)$, with $*$ a point, such that the $G$-space $X {\smallsetminus}*$ is proper. Notice that whenever $(X,A)$ is a relative $G$-CW-complex such that $X {\smallsetminus}A$ is proper, the $G$-space $X/A$ inherits a natural structure of pointed proper $G$-CW-complex.
### The category of compactly-generated $G$-spaces
Let $G$ be a topological group. A consists of a $G$-space which is a k-space together with a point in it which is fixed by the action of $G$.
The category $CG_G^{\bullet}$ is the one whose objects are the $G$-pointed k-spaces and whose morphisms are the pointed $G$-maps. The category $CG_G^{h\bullet}$ is the category with the same objects as $CG_G^{\bullet}$, and whose morphisms are the equivariant pointed homotopy classes of $G$-maps between objects (i.e. $CG_G^{h\bullet}$ it is the homotopy category of $CG_G^{\bullet})$. Given two $G$-spaces (resp. two pointed $G$-spaces) $X$ and $Y$, we let $[X,Y]_G$ (resp. $[X,Y]_G^\bullet$) denote the set of equivariant homotopy classes of $G$-maps (resp. pointed $G$-maps) from $X$ to $Y$.
A morphism $f: X \rightarrow Y$ is called a when the restriction $f^H:X^H \rightarrow Y^H$ is a weak equivalence for every compact subgroup $H$ of $G$ (here, $X^H$ denotes the subspace $\bigl\{x \in X : \; \forall h \in H, \; h.x=x\bigr\}$ of $X$). We finally define $W_G$ as the class of morphisms in $CG_G^{h\bullet}$ which have $G$-weak equivalences as representative maps. We may then consider the category of fractions $CG_G^{h\bullet}[W_G^{-1}]$, with its usual universal property. The crucial property in this paper is the following one:
\[pointedweakeq\] Let $Y \overset{f}{\rightarrow} Y'$ be a $G$-weak equivalence between pointed $G$-spaces.\
Then, for every proper pointed $G$-CW-complex $X$, the map $f$ induces a bijection $$f_*: [X,Y]_G^\bullet \longrightarrow [X,Y']_{G.}^\bullet$$
### $G$-fibre bundles
Let $G$ be a topological group. Given a $G$-space $X$, we call (resp. $G$-) over $X$ the data consisting of a pseudo-vector bundle (resp. a vector bundle) $p : E \rightarrow X$ over $X$ and of a (left) $G$-action on $E$, such that $p$ is a $G$-map, and, for all $g \in G$ and $x \in X$, the map $E_x \rightarrow E_{g.x}$ induced by the $G$-action on $E$ is a linear isomorphism.
Given an integer $n \in \N$ and a $G$-space $X$, $\mathbb{V}\text{ect}_G^{F,n}(X)$ will denote the set of isomorphism classes of $n$-dimensional $G$-vector bundles over $X$. Accordingly, $\mathbb{V}\text{ect}_G^{F}(X)$ will denote the abelian monoid of isomorphism classes of finite-dimensional $G$-vector bundles over $X$. \[Gvectorbundle\]
Given another topological group $H$, a is an $H$-principal bundle $\pi : E \rightarrow X$ with structures of $G$-spaces on $E$ and $X$ for which $\pi$ is a $G$-map and $\forall (g,h,x)\in G \times H \times E, \; g.(x.h)=(g.x).h$.
### Topological categories attached to simi-Hilbert bundles {#orthogroups}
We denote by $U_n(F)$ (resp. ${\operatorname{GU}}_n(F)$) the group of orthogonal automorphisms (resp. of similarities) of the vector space $F^n$.
A is a finite-dimensional vector space $V$ (with ground field $F$) with a linear family $(\lambda \langle -,- \rangle)_{\lambda \in \mathbb{R}_+^*}$ of inner products on $V$.
The relevant notion of isomorphisms between two simi-Hilbert spaces is that of similarities. We do have a notion of orthogonality, but no notion of orthonormal families. The relevant notion is that of families: a family will be said to be simi-orthonormal when it is orthogonal and all its vectors share the same positive norm (for any inner product in the linear family). Equivalently, a family of vectors is simi-orthonormal if and only if it is orthonormal for some inner product in the linear family.
Let $G$ be a topological group.\
For $n \in \mathbb{N}$, an $n$-dimensional simi-$G$-Hilbert bundle is a $G$-vector bundle with fiber $F^n$ and structural group ${\operatorname{GU}}_n(F)$.\
A disjoint union of $k$-dimensional $G$-simi-Hilbert bundles, for $k\in \mathbb{N}$, is called a $G$-**simi-Hilbert bundle.**
We fix an integer $n \in \mathbb{N}$ for the rest of the paragraph. Let $\varphi: E \rightarrow X$ be an $n$-dimensional simi-Hilbert over a locally-countable CW-complex, and $\tilde{\varphi}: \tilde{E} \rightarrow X$ the ${\operatorname{GU}}_n(F)$-principal bundle canonically associated to it (by considering $\tilde{E}$ as a subspace of $E^{\oplus n}$).\
\[defsframe\]\[defsmod\]\[defsBdl\]We define:
- $\varphi {\operatorname{\text{-}sframe}}$ as the category $\mathcal{E}\tilde{E}$, with a natural right-action of ${\operatorname{GU}}_n(F)$;
- $\varphi{\operatorname{\text{-}smod}}$ as $\varphi{\operatorname{\text{-}sframe}}/{\operatorname{GU}}_n(F)$: an object of $\varphi{\operatorname{\text{-}smod}}$ corresponds to a point of $X$, and a morphism from $x$ to $y$ (with $(x,y) \in X^2$) corresponds to a similarity $E_x \overset{\cong}{\rightarrow} E_y$;
- $\varphi{\operatorname{\text{-}sBdl}}$ as the category whose space of objects is $E$, and whose space of morphisms is the (closed) subspace of $E {\operatorname{\underset{k}{\times}}}E {\operatorname{\underset{k}{\times}}}{\operatorname{Hom}}(\varphi{\operatorname{\text{-}smod}})$ consisting of those triples $(e,e',f)$ such that $\varphi(e) \underset{f}{\longrightarrow} \varphi(e')$ and $f(e)=e'$.
### The category $\Gamma \text{-Fib}_F$ and the ${\operatorname{\text{-}smod}}$ functor {#4.1.1}
We define the category $\Gamma \text{-Fib}_F$ as follows:
- An object of $\Gamma \text{-Fib}_F$ consists of a finite set $S$, a locally-countable CW-complex $X$, and, for every $s \in S$, of a Hilbert bundle $p_s:E_s \rightarrow X$ with ground field $F$. Such an object is called an **$S$-object** over $X$. If $S=\mathbf{n}$ for some $n \in \mathbb{N}$, an $S$-object will be called an $n$-object.
- A morphism $f:(S,X,(p_s)_{s\in S}) \longrightarrow
(T,Y,(q_t)_{t\in T})$ consists of a morphism $\gamma: T \rightarrow S$ in the category $\Gamma$, a continuous map $\bar{f}:X \rightarrow Y$, and, for every $t \in T$, a strong morphism of Hilbert bundles $$\begin{CD}
\underset{s \in \gamma(t)}{\oplus}E_s @>{f_t}>> E'_t \\
@V{\underset{s \in \gamma(t)}{\oplus}p_s}VV @VV{q_t}V \\
X @>>{\bar{f}}> Y.
\end{CD}$$
If $f:(S,X,(p_s)_{s\in S}) \rightarrow (T,Y,(q_t)_{t\in T})$ is the morphism in $\Gamma \text{-Fib}_F$ corresponding to $(\gamma,\bar{f},(f_t)_{t\in T})$, and $g:(T,Y,(q_t)_{t\in T}) \rightarrow (U,Z,(r_u)_{u\in U})$ is the morphism in $\Gamma \text{-Fib}_F$ corresponding to $(\gamma',\bar{g},(g_u)_{u\in U})$, then the composite morphism $g \circ f: (S,X,(p_s)_{s\in S}) \rightarrow (U,Z,(r_u)_{u\in U})$ is the one which corresponds to the triple consisting of $\gamma \circ \gamma'$, $\bar{g} \circ \bar{f}$, and the family $\left(g_u \circ \left[\underset{t\in \gamma'(u)}{\oplus}f_t\right]\right)_{u \in U.}$
\[4.2.2\] Let $(X,p:E \rightarrow X)$ be a $1$-object of $\Gamma \text{-Fib}_F$. The natural map $\dim_p:\begin{cases}
X & \longrightarrow \mathbb{N} \\
x & \longmapsto \dim(E_x)
\end{cases}$ is continuous because $X$ is a CW-complex. Setting $X_{n}:=\dim_p^{-1}\{n\}$, $E_n:=p^{-1}(X_n)$, and $p_n=p_{|E_n}: E_n \rightarrow X_{n}$, we then have $p=\underset{n \in \mathbb{N}}{\coprod}p_n.$\
Set $$p {\operatorname{\text{-}smod}}:=\underset{n \in \mathbb{N}}{\coprod}(p_n {\operatorname{\text{-}smod}}).$$ We obtain a functor $p {\operatorname{\text{-}smod}}\rightarrow \mathcal{E}X \times \mathcal{B}\mathbb{R}_+^*$ by assigning $(x,y,\|\varphi\|)$ to every morphism $\varphi:E_x \rightarrow E_y$ (here, $\|\varphi\|$ denotes the norm of the similarity $\varphi$ with respect to the respective inner product structures on $E_x$ and $E_y$): this is compatible with the composition of morphisms, since we are dealing with similarities here.\
For any $S$-object $\varphi=(S,X,(p_s)_{s \in S})$, $\varphi {\operatorname{\text{-}smod}}$ is defined as the fiber product of the categories $p_s {\operatorname{\text{-}smod}}$ over $\mathcal{E}X \times \mathcal{B}\mathbb{R}_+^*$ for all $s\in S$. For any $S$-object $\varphi=(S,X,(p_s:E_s \rightarrow X)_{s \in S})$, an object of $\varphi {\operatorname{\text{-}smod}}$ simply corresponds to a point $x\in X$, while a morphism $x \rightarrow y$ in $\varphi {\operatorname{\text{-}smod}}$ is a family $(\varphi_s)_{s\in S}$ of similarities $\varphi_s: (E_s)_x \overset{\cong}{\rightarrow} (E_s)_y$ *which share the same norm*. It is then easy to extend this construction to obtain a functor $${\operatorname{\text{-}smod}}: \Gamma \text{-Fib}_F \longrightarrow \text{kCat}.$$
### Hilbert $\Gamma$-bundles {#4.4}
A is a contravariant functor $\varphi:\Gamma \longrightarrow \Gamma \text{-Fib}_F$ which satisfies the following conditions:
(i) $\mathcal{O}^F_\Gamma \circ \varphi={\operatorname{id}}_{\Gamma}$;
(ii) $\varphi(\mathbf{0})=(\mathbf{0},*,\emptyset)$;
(iii) For every $n \in \mathbb{N}^*$, there exists a morphism $f_n:n.\varphi
(\mathbf{1}) \rightarrow \varphi(\mathbf{n})$ in $\Gamma \text{-Fib}_F$ such that $\mathcal{O}^F_\Gamma(f_n)={\operatorname{id}}_{\mathbf{n}}$.
\[gammaspaces\]
Let $\varphi$ be an object of $\Gamma-\text{Fib}_F$ and $G$ be a Lie group. We define $$s{\operatorname{Vec}}_G^\varphi:=\left|{\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})\right|.$$
In what follows, we let $\varphi$ be a Hilbert $\Gamma$-bundle and $G$ be a Lie group. We define a functor from $\Gamma$ to $\text{CG}_G$: $$s\underline{{\operatorname{Vec}}}_G^\varphi: S \longmapsto s{\operatorname{Vec}}_G^{\varphi(S)}.$$ This is actually a $\Gamma-G$-space.
For every finite set $S$, we let $\Gamma (S)$ denote the set of maps $f:\mathcal{P}(S)
\rightarrow \mathcal{P}(\mathbb{N})$ which respect disjoint unions (and in particular $f(\emptyset)=\emptyset$), and such that $f(S)$ is finite. We will write $S \overset{f}{\rightarrow}
\mathbb{N}$ when $f\in \Gamma(S)$.
For an inner product space $\mathcal{H}$ (with underlying field $F$) of finite dimension or isomorphic to $F^{(\infty)}$, and for a finite subset $A$ of $\mathbb{N}$, we may consider the inner product space $\mathcal{H}^A$ as embedded in the Hilbert space $\mathcal{H}^\infty$. We then define $G_A(\mathcal{H})$ as the set of subspaces of dimension $\# A$ of $\mathcal{H}^A$, with the limit topology for the inclusion of $G_{\#A}(E)$, where $E$ ranges over the finite dimensional subspaces of $\mathcal{H}$. When $A$ is empty, we set $G_\emptyset(\mathcal{H})=*$.
We let $p_A(\mathcal{H}): E_A(\mathcal{H}) \longrightarrow G_A(\mathcal{H})$ denote the canonical Hilbert bundle of dimension $\#A$ over $G_A(\mathcal{H})$. The set $E_A(\mathcal{H})$ is constructed as a subspace of the product of $\mathcal{H}^A$ (with the limit topology described above) with $G_A(\mathcal{H})$.
\[4.5\]For every finite set $S$, we define the following object of $\Gamma \text{-Fib}_F$: $$\text{Fib}^{\mathcal{H}}(S):=
\left(S,X^\mathcal{H}(S),(p^\mathcal{H}(s))_{s \in S}\right),$$ where $$X^\mathcal{H}(S):=\underset{f \in \Gamma(S)}{\coprod}\left[\underset{s \in S}{\prod}
G_{f(s)}(\mathcal{H})\right]$$ and, for every $s \in S$, $$p^\mathcal{H}(s):=
\underset{f \in \Gamma(S)}{\coprod}\left[p_{f(s)}(\mathcal{H})
\times \underset{s' \in S{\smallsetminus}\{s\}}{\prod}
{\operatorname{id}}_{G_{f(s')}(\mathcal{H})}\right].$$ Let $\gamma: S \rightarrow T$ be a morphism in $\Gamma$. We define a morphism $\text{Fib}^{\mathcal{H}}(\gamma):\text{Fib}^{\mathcal{H}}(T) \rightarrow
\text{Fib}^{\mathcal{H}}(S)$ of $\Gamma \text{-Fib}_F$ for which $\mathcal{O}^F_\Gamma(\text{Fib}^{\mathcal{H}}(\gamma))=\gamma$, in the following way: for every $f \in \Gamma(S)$, we consider the map $$\begin{cases}
\underset{t \in \gamma(s)}{\prod}
G_{f(t)}(\mathcal{H}) & \longrightarrow
G_{f \circ \gamma (s)}(\mathcal{H}) \\
(E_t)_{t \in \gamma(s)} & \longmapsto \underset{t \in \gamma(s)}{\overset{\bot}{\oplus}}
E_t,
\end{cases}$$ and, for every $s \in S$ and $f \in \Gamma(S)$, we have a commutative square: $$\begin{CD}
\left[\underset{t \in \gamma(s)}{\oplus}
E_{f(t)}(\mathcal{H})\right]
\times \underset{t \in T {\smallsetminus}\gamma(s)}{\prod}
G_{f(t)}(\mathcal{H}) @>>>
E_{f\circ \gamma(s)}(\mathcal{H})
\times \underset{s' \in S {\smallsetminus}\{s\}}{\prod}
G_{f(s')}(\mathcal{H}) \\
@VVV @VVV \\
\underset{t \in T}{\prod}
G_{f(t)}(\mathcal{H}) @>>>
\underset{s_1 \in S}{\prod}
G_{f\circ \gamma(s_1)}(\mathcal{H}),
\end{CD}$$ where the upper morphism is given by the previous map and the following one: $$\begin{cases}
\underset{t \in \gamma(s)}{\oplus}
E_{f(t)}(\mathcal{H}) & \longrightarrow
E_{f \circ \gamma (s)}(\mathcal{H}) \\
(x_t)_{t \in \gamma(s)} & \longmapsto
\underset{t \in \gamma(s)}{\sum} x_t.
\end{cases}$$
\[defFoo\]Denoting by $F^{(\infty)}$ the direct limit of the sequence $(F^k)_{k \geq 0}$ for the standard inclusion of inner product spaces, we have the following result:
Let $\mathcal{H}$ be an inner product space with ground field $F$. Assume that $\mathcal{H}$ is finite-dimensional or isomorphic to $F^{(\infty)}$. Then $\text{Fib}^{\mathcal{H}}$ is a Hilbert $\Gamma$-bundle.
\[deftypique\]We may now set: $$s\underline{{\operatorname{Vec}}}_G^{F,\infty}:=
s\underline{{\operatorname{Vec}}}_G^{\text{Fib}^{F^{(\infty)}}} \quad; \quad
s{\operatorname{Vec}}_G^{F,\infty}:=
s{\operatorname{Vec}}_G^{\text{Fib}^{F^{(\infty)}}(\mathbf{1})}
\quad; \quad
Es{\operatorname{Vec}}_G^{F,\infty}:=
Es{\operatorname{Vec}}_G^{\text{Fib}^{F^{(\infty)}}(\mathbf{1})}.$$ The $\Gamma$-space structure of $\underline{{\operatorname{Vec}}}_G^{F,\infty}$ induces a structure of equivariant H-space on ${\operatorname{Vec}}_G^{F,\infty}=\underline{{\operatorname{Vec}}}_G^{F,\infty}(\mathbf{1})$. The following result was established in [@Ktheo1]:
Let $G$ be a second-countable Lie group Then $Es{\operatorname{Vec}}_G^{F,\infty} \rightarrow s{\operatorname{Vec}}_G^{F,\infty}$ is universal for finite-dimensional $G$-simi-Hilbert bundles over $G$-CW-complexes, and, for every $G$-CW-complex $X$, the induced bijection $$\Phi: [X,{\operatorname{Vec}}_G^{F,\infty}]_G \overset{\cong}{\longrightarrow}
{\operatorname{\mathbb{V}ect}}_G^F(X)$$ is a homomorphism of abelian monoids.
### A definition of equivariant K-theory {#6}
For any Lie group $G$, we set: $$sKF_G^{[\infty]}:=\Omega Bs{\operatorname{Vec}}_G^{F,\infty}.$$
Let $G$ be a Lie group, $F=\mathbb{R}$ or $\mathbb{C}$, $(X,A)$ a proper $G$-CW-pair and $n \in \mathbb{N}$. We set: $$KF_G^{-n}(X,A):=\bigl[\Sigma ^n(X/A),sKF_G^{[\infty]}\bigr]_G^\bullet,$$ and $$KF_G^{-n}(X):=KF_G^{-n}(X\cup \{*\},\{*\}).$$ In particular, for every proper $G$-CW-complex $X$, $$KF_G(X):=KF_G^0(X)=\bigl[X,sKF_G^{[\infty]}\bigr]_{G.}$$
It was shown in [@Ktheo1] that $KF_G(-)$ may be extended to positive degrees in order to recover a good equivariant cohomology theory. Given a $G$-CW-complex $X$, the canonical map $i:s{\operatorname{Vec}}_G^{F,\infty} = \underline{s{\operatorname{Vec}}}_G^{F,\infty}(\mathbf{1}) \rightarrow sKF_G^{[\infty]}$ induces a natural homomorphism $[X,s{\operatorname{Vec}}_G^{F,\infty}]_G \overset{i_*}{\rightarrow} [X,sKF_G^{[\infty]}]_G$ of abelian monoids, which, pre-composed with the inverse of the natural isomorphism $[X,s{\operatorname{Vec}}_G^{F,\infty}]_G \overset{\cong}{\rightarrow} \mathbb{V}\text{ect}_G^{F}(X)$, yields a natural homomorphism of abelian monoids $$\mathbb{V}\text{ect}_G^{F}(X) \longrightarrow [X,sKF_G^{[\infty]}]_G.$$ By the universal property of the Grothendieck construction, this yields a natural homomorphism of abelian groups $$\gamma _X: \mathbb{K}F_G(X) \longrightarrow KF_G(X).$$ This clearly defines a natural transformation $\gamma: \mathbb{K}F_G(-) \longrightarrow KF_G(-)$ on the category of proper $G$-CW-complexes. It was shown in [@Ktheo1] how this natural transformation may be extended in arbitrary degrees so as to be compatible with long exact sequences. Finally, the following property of $\gamma$ was established in [@Ktheo1]:
For every compact subgroup $H$ of $G$, every integer $n$ and every finite CW-complex $Y$ on which $G$ acts trivially, the map $$\gamma _{(G/H) \times Y}^{-n}: \mathbb{K}F_G^{-n}((G/H) \times Y)
\overset{\cong}{\longrightarrow} KF_G^{-n}((G/H) \times Y)$$ is an isomorphism.
Main ideas and structure of the paper
-------------------------------------
The basic idea is to start from the “quasi”-classifying space that naturally arises in the study of Phillips’ equivariant K-theory, i.e. the space ${\operatorname{Fred}}(L^2(G)^\infty)$ of Fredholm operators on the Hilbert $G$-module $L^2(G)^\infty$, with the norm topology. Indeed, any $G$-map $f : X \longrightarrow {\operatorname{Fred}}(L^2(G)^\infty)$, where $X$ is a proper locally-compact Hausdorff $G$-space, yields a morphism $$\begin{cases}
X \times L^2(G)^\infty & \longrightarrow X \times L^2(G)^\infty \\
(x,y) & \longmapsto (x,f(x)[y])
\end{cases}$$ of $G$-Hilbert bundle over $X$, which in turns yields an element of $KF_G^{\text{Ph}}(X)$ (cf. example 3.5 p.41 of [@Phillips]). This gives rise to a natural transformation $[-,{\operatorname{Fred}}(L^2(G)^\infty)]_G \longrightarrow KF_G^{\text{Ph}}(-)$ on the category of proper locally compact Hausdorff $G$-spaces, and it is quite easy to check that, if we consider the structure of topological monoid on ${\operatorname{Fred}}(L^2(G)^\infty)$ induced by the composition of Fredholm morphisms, this is actually a natural transformation between monoid-valued functors.
However, ${\operatorname{Fred}}(L^2(G)^\infty)$ is not always a classifying space for $KF_G^{\text{Ph}}(-)$: unless $G$ is a discrete group, it may be shown indeed that ${\operatorname{Fred}}(L^2(G)^\infty)$ is not a $G$-space (since the usual $G$-action on this topological space is not continuous). Classically, this is handled by replacing ${\operatorname{Fred}}(L^2(G)^\infty)$ with its subspace $\underline{{\operatorname{Fred}}}(L^2(G)^\infty)$ consisting of those elements $x$ such that the action of $G$ on the orbit of $x$ is continuous. Classically, $\underline{{\operatorname{Fred}}}(L^2(G)^\infty)$ is a $G$-space and every continuous equivariant map $X \longrightarrow {\operatorname{Fred}}(L^2(G)^\infty)$ has its range in $\underline{{\operatorname{Fred}}}(L^2(G)^\infty)$ when $X$ is a $G$-space. Hence we really have a natural transformation $$[-,\underline{{\operatorname{Fred}}}(L^2(G)^\infty)]_G \longrightarrow KF_G^{\text{Ph}}(-)$$ which is not known in general to be an isomorphism (the compact case however has been solved, see [@Twisted]).
![Going from $sKF_G^{[\infty]}$ to $\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H})^\infty)$](structure.eps)
####
Our basic idea is to construct a reasonable morphism $sKF_G^{[\infty]} \longrightarrow \underline{{\operatorname{Fred}}}(L^2(G)^\infty)$ in the category $CG_G^{h\bullet}[W_G^{-1}]$, and then compose the natural transformation $KF_G(-) \longrightarrow [-,\underline{{\operatorname{Fred}}}(L^2(G)^\infty)]_G$ derived from it with the one discussed earlier. The construction of this morphism is now briefly explained:
- In Section \[10.1\], we will define $L^2$ functors from $\mathcal{E}G$ to $\varphi {\operatorname{\text{-}smod}}$ (where $\varphi$ is a Hilbert bundle), we define the space $s{\operatorname{Vec}}_{G,L^2}^{\varphi}$ from those functors very much like $s{\operatorname{Vec}}_G^{\varphi}$, and will construct a canonical $G$-weak equivalence $s{\operatorname{Vec}}_{G,L^2}^{\varphi} \rightarrow s{\operatorname{Vec}}_G^{\varphi}$. This construction will be extended to a $\Gamma-G$-space $s\underline{{\operatorname{Vec}}}_{G,L^2}^{F,\infty}$, and a canonical $G$-weak equivalence $\Omega B s {\operatorname{Vec}}_{G,L^2}^{F,\infty} \longrightarrow \Omega B s {\operatorname{Vec}}_{G}^{F,\infty}$ will be produced.
- In Section \[10.2\], we will define the simplicial $G$-space $s\underline{{\operatorname{Vec}}}_{G,L^2}^{F,\infty *}$ by only considering functors that are $L^2$ and “contiguous” label-wise, and natural transformations that are non-decreasing label-wise. There will then be a canonical morphism $s\underline{{\operatorname{Vec}}}_{G,L^2}^{F,\infty*} \longrightarrow s\underline{{\operatorname{Vec}}}_{G,L^2}^{F,\infty}$ which will induce a $G$-weak equivalence $\Omega B s{\operatorname{Vec}}_{G,L^2}^{F,\infty*} \longrightarrow \Omega B s {\operatorname{Vec}}_{G,L^2}^{F,\infty}$.
- In Section \[10.3\], we will briefly introduce the $G$-space $\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}))$ together with the simplicial $G$-space ${\operatorname{Fred}}(G,\mathcal{H})$ associated to this monoid, and prove that the canonical map $\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H})) \longrightarrow \Omega B \underline{{\operatorname{Fred}}} (L^2(G,\mathcal{H}))$ is a $G$-weak equivalence. We will also briefly recall the definition of the shift map ${\operatorname{Sh}}: \underset{n\in \mathbb{N}}{\bigcup}{\operatorname{sub}}_n(L^2(G,\mathcal{H})) \longrightarrow \underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H})^\infty)$ and its basic properties.
- Section \[10.4\] is devoted to the construction of a morphism of simplicial $G$-spaces $s{\operatorname{Vec}}_{G,L^2}^{F,\infty} \longrightarrow \underline{{\operatorname{Fred}}}(G,\mathcal{H}^\infty)$: this part is very technical so we will start with a lengthy explanation of the main ideas before going into more details. Note that this construction uses an unresolved - yet very reasonable - conjecture on relative triangulations of smooth compact manifolds, see Section \[conjecturesection\].
- Finally, we wrap up the construction in Section \[10.5\], where we obtain a natural transformation $KF_G(-) \longrightarrow KF_G^{\text{Ph}}(-)$ (that is independent of some choices made during the construction), and we finish by proving that it is an isomorphism on the category of finite proper $G$-CW-complexes.
Additional notations and definitions
------------------------------------
Given an integer $n \in \mathbb{N}$, and $G$-space $X$, $\mathbb{V}\text{ect}_G^{F,n}(X)$ will denote the set of isomorphism classes of $n$-dimensional $G$-vector bundles over $X$. Accordingly, $\mathbb{V}\text{ect}_G^{F}(X)$ will denote the abelian monoid of isomorphism classes of finite-dimensional $G$-vector bundles over $X$.
We denote by $\Delta^*$ the hemi-simplicial category, i.e. the subcategory of $\Delta$ with the same objects, and with the monomorphisms of $\Delta$ as morphisms. A $G$-space is a contravariant functor from $\Delta^*$ to the category of $G$-spaces.
\[hilbertspace\]When $\mathcal{H}$ is an inner product space, and $n \in \mathbb{N}^*$, $B_n(\mathcal{H}) \subset \mathcal{H}^n$ will denote the space of linearly independent $n$-tuples of elements of $\mathcal{H}$ (with the convention that $B_0(\mathcal{H})=*$), while $V_n(\mathcal{H}) \subset \mathcal{H}^n$ will denote the space of orthonormal $n$-tuples of elements of $\mathcal{H}$ (with the convention that $V_0(\mathcal{H})=*$). We also let $B_{\mathcal{H}}$ denote the unit ball of $\mathcal{H}$, ${\operatorname{sub}}(\mathcal{H})$ denote the set of closed linear subspaces of $\mathcal{H}$, and, for $n \in \mathbb{N}, {\operatorname{sub}}_n(\mathcal{H})$ the set of $n$-dimensional linear subspaces of $\mathcal{H}$. Finally, $\mathcal{B}(H)$ will denote the space of bounded linear operators on $H$.
For every smooth manifold $N$, and every set $x$, we will let $T_x N$ denote the tangent space of $N$ at $x$ when $x \in N$, and we set $T_x N=\emptyset$ when $x \not\in N$.
\[undercat\]Given a category $\mathcal{C}$ and an object $X$ of $\mathcal{C}$, we denote by $\mathcal{C} \downarrow X$ will denote the category whose objects are the morphisms $Y \rightarrow X$ of $\mathcal{C}$, and the morphisms from $Y \overset{f}{\rightarrow} X$ to $Y' \overset{g}{\rightarrow} X$ are the morphisms $Y \overset{\varphi}{\rightarrow} Y'$ of $\mathcal{C}$ such that $g \circ \varphi=f$.
#### List of important notation
${}$
The equivariant $\Gamma$-space $s{\operatorname{Vec}}_{G,L^2}^\varphi$ {#10.1}
======================================================================
In this section, $L^2_*(G)$ will denote $L^2(G,\mathbb{R}_+^*) \coprod \{0\}$ (with the $L^2$-topology on $L^2(G,\mathbb{R}_+^*)$), and $C_*(G)$ will denote $C(G,\mathbb{R}_+^*)\coprod \{0\}$ (with the compact-open topology on $C(G,\mathbb{R}_+^*)$). Finally, $C_*(G)\cap L^2_*(G)$ is equipped with the initial topology for the diagonal map $C_*(G)\cap L^2_*(G) \longrightarrow C_*(G)\times L^2_*(G)$, i.e. the topology with the fewest open sets so that the diagonal map is continuous.
The topological category ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$ {#10.1.1}
----------------------------------------------------------------------------------------------------------
### The topological category induced by a functor
Let $\mathcal{C}$ be a small category, $\mathcal{D}$ a topological category, and $F: \mathcal{C} \rightarrow \mathcal{D}$ a functor. We define ${\operatorname{Ob}}(\mathcal{C}_F)$ as the topological space with underlying set ${\operatorname{Ob}}(\mathcal{C})$ and the initial topology with respect to $F_{|{\operatorname{Ob}}} : {\operatorname{Ob}}(\mathcal{C}) \rightarrow {\operatorname{Ob}}(\mathcal{D})$ (i.e. the topology with the fewest open sets such that $F_{|{\operatorname{Ob}}}$ is continuous). We define ${\operatorname{Hom}}(\mathcal{C}_F)$ as the topological space with underlying set ${\operatorname{Hom}}(\mathcal{C})$ and the initial topology with respect to $F_{|{\operatorname{Hom}}} : {\operatorname{Hom}}(\mathcal{C}) \rightarrow {\operatorname{Hom}}(\mathcal{D})$. Since the squares $$\begin{CD}
{\operatorname{Ob}}(\mathcal{C}) @>{F_{|{\operatorname{Ob}}}}>> {\operatorname{Ob}}(\mathcal{D}) \\
@V{{\operatorname{Id}}_\mathcal{C}}VV @V{{\operatorname{Id}}_\mathcal{D}}VV \\
{\operatorname{Hom}}(\mathcal{C}) @>{F_{|{\operatorname{Hom}}}}>> {\operatorname{Hom}}(\mathcal{D})
\end{CD} \quad , \quad
\begin{CD}
{\operatorname{Hom}}(\mathcal{C}) @>{F_{|{\operatorname{Hom}}}}>> {\operatorname{Hom}}(\mathcal{D}) \\
@V{{\operatorname{Fin}}_\mathcal{C}}VV @V{{\operatorname{Fin}}_\mathcal{D}}VV \\
{\operatorname{Ob}}(\mathcal{C}) @>{F_{|{\operatorname{Ob}}}}>> {\operatorname{Ob}}(\mathcal{D})
\end{CD} \quad , \quad
\begin{CD}
{\operatorname{Hom}}(\mathcal{C}) @>{F_{|{\operatorname{Hom}}}}>> {\operatorname{Hom}}(\mathcal{D}) \\
@V{{\operatorname{In}}_\mathcal{C}}VV @V{{\operatorname{In}}_\mathcal{D}}VV \\
{\operatorname{Ob}}(\mathcal{C}) @>{F_{|{\operatorname{Ob}}}}>> {\operatorname{Ob}}(\mathcal{D})
\end{CD}$$ and $$\begin{CD}
{\operatorname{Hom}}(\mathcal{C}) \underset{\vartriangle}{\times} {\operatorname{Hom}}(\mathcal{C}) @>{F_{|{\operatorname{Hom}}} \times F_{|{\operatorname{Hom}}}}>> {\operatorname{Hom}}(\mathcal{D}) \underset{\vartriangle}{\times} {\operatorname{Hom}}(\mathcal{D}) \\
@V{{\operatorname{Comp}}_\mathcal{C}}VV @V{{\operatorname{Comp}}_\mathcal{D}}VV \\
{\operatorname{Ob}}(\mathcal{C}) @>{F_{|{\operatorname{Ob}}}}>> {\operatorname{Ob}}(\mathcal{D})
\end{CD}$$ are all commutative, $\mathcal{C}_F$ is a topological category, and we call it the category $\mathcal{C}$ *with the topology induced by* $F$.
\[L2nerve\] For every $n \in \mathbb{N}$, $\mathcal{N}(\mathcal{C}_F)_n$ has the initial topology for $\mathcal{N}(F)_n: \mathcal{N}(\mathcal{C}_F)_n \rightarrow \mathcal{N}(\mathcal{D})_n$.
This is true by definition for $n\in \{0,1\}$. Let $n\in \mathbb{N} {\smallsetminus}\{0,1\}$. Then $\mathcal{N}(\mathcal{C}_F)_n \rightarrow \mathcal{N}(\mathcal{D})_n$ is continuous as a restriction of the continuous map ${\operatorname{Hom}}(\mathcal{C}_F)^n \rightarrow {\operatorname{Hom}}(\mathcal{D})^n$. Let $Z$ be a topological space, and $\alpha : Z \rightarrow \mathcal{N}(\mathcal{C}_F)_n$ be a map such that the composite map $\beta : Z \rightarrow \mathcal{N}(\mathcal{C}_F)_n \rightarrow \mathcal{N}(\mathcal{D})_n$ is continuous. Then all the corresponding maps $\beta_1,\dots,\beta_n$ from $Z$ to ${\operatorname{Hom}}(\mathcal{D})$ are continuous, and it follows that the maps $\alpha_1,\dots,\alpha_n$ from $Z$ to ${\operatorname{Hom}}(\mathcal{C}_F)$ which correspond to $\alpha$ are continuous. Hence $\alpha$ is continuous. This proves that $\mathcal{N}(\mathcal{C}_F)_n$ has the initial topology for $\mathcal{N}(F)_n$.
### The topological category ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$ {#the-topological-category-operatornamefunc_l2mathcalegvarphi-operatornametext-smod}
Let $S$ be a finite set, and $\varphi$ be an $S$-object of $\Gamma \text{-Fib}_F$.
We define $$\alpha_\varphi :
\begin{cases}
{\operatorname{Ob}}({\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})) & \longrightarrow C_*(G) \times \mathbb{N}^S \\
\mathbf{F}: \mathcal{E}G \rightarrow \varphi {\operatorname{\text{-}smod}}& \longmapsto \begin{cases}
\left([g \mapsto \|\underset{s \in S}{\oplus}\mathbf{F}(1,g)_s\|],(\dim(\mathbf{F}(1)_s))_{s \in S}\right)
& \text{if} \quad
\exists s \in S: \dim(\mathbf{F}(1)_s)\neq 0 \\
(0,(0,\dots,0)) & \text{otherwise}.
\end{cases}
\end{cases}$$ The map $\alpha_\varphi$ is obviously continuous. We then define ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$ as the full subcategory of ${\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$ whose set of objects is $\alpha_\varphi^{-1}((L^2_*(G)\cap C_*(G)) \times \mathbb{N}^S)$.
We will now let $\alpha_\varphi : {\operatorname{Ob}}({\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}}))
\longrightarrow (C_*(G)\cap L^2_*(G)) \times \mathbb{N}^S$ denote the restriction of $\alpha_\varphi$. Finally, we equip ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$ with the topology induced by the functor $${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})
\longrightarrow {\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}}) \times \mathcal{E}\bigl(
(C_*(G)\cap L^2_*(G)) \times \mathbb{N}^S
\bigr).$$ The topology on ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$ is also induced by the functor $${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})
\longrightarrow {\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}}) \times \mathcal{E}\bigl(
C_*(G) \times L^2_*(G)
\bigr).$$ We finally set: $$s{\operatorname{Vec}}_{G,L^2}^\varphi:=\Big|{\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}}) \Big|.$$
### The action of $G$ on ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$
Here, we prove that the action of $G$ on ${\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$ induces a continuous action of $G$ on ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$.
We first need to check that ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$ is a stable subcategory for the action of $G$ on ${\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$. The left-action of $G$ on ${\operatorname{Ob}}({\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}}))$ gives rise to a commutative square $$\begin{CD}
G \times {\operatorname{Ob}}({\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})) @>{{\operatorname{id}}_G \times \alpha_\varphi}>>
G \times C_*(G) \\
@VVV @VVV \\
{\operatorname{Ob}}({\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})) @>{\alpha_\varphi}>> C_*(G),
\end{CD}$$ in which the right-hand vertical map is $$\begin{cases}
G \times C_*(G) & \longrightarrow C_*(G) \\
(g,f) & \longmapsto
\begin{cases}
0 & \quad \text{if} \quad f=0 \\
\left[h \longmapsto \cfrac{f(hg)}{f(g)}\right] & \quad \text{otherwise}.
\end{cases}
\end{cases}$$ Since we are working with a right-invariant Haar measure on $G$, the image of $G \times (C_*(G) \cap L^2_*(G))$ by this map is included in $C_*(G) \cap L^2_*(G)$. It follows that ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$ is a stable subcategory for the action of $G$ on ${\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$.
Moreover, the induced map $G \times (C_*(G) \cap L^2_*(G)) \longrightarrow C_*(G) \cap L^2_*(G)$ is continuous since the action of $G$ on $L^2(G)$ is continuous, and it follows that the induced action of $G$ on ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$ is continuous.
### The functor ${\operatorname{Func}}_{L^2}(\mathcal{E}G,- {\operatorname{\text{-}smod}})$
Given a morphism $f: \varphi \rightarrow \psi$ in $\Gamma \text{-Fib}_F$, we want to check that the functor $f^*:{\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}}) \longrightarrow {\operatorname{Func}}(\mathcal{E}G,\psi {\operatorname{\text{-}smod}})$ induces a continuous functor $f_{L^2}^*: {\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}}) \longrightarrow
{\operatorname{Func}}_{L^2}(\mathcal{E}G,\psi {\operatorname{\text{-}smod}})$.
Let $\gamma=\mathcal{O}_\Gamma^F(f): S \rightarrow T$. Then $\gamma$ induces a map $$\gamma^{\mathbb{N}}_*: \begin{cases}
\mathbb{N}^T & \rightarrow \mathbb{N}^S \\
(n_t)_{t \in T} & \mapsto \left(\underset{i \in \gamma(s)}{\sum}n_i\right)_{s \in S,}
\end{cases}$$ and finally a map $$\gamma_* : \begin{cases}
C_*(G) \times \mathbb{N}^T & \longrightarrow C_*(G) \times \mathbb{N}^S \\
(f,(n_t)_{t\in T}) & \longmapsto \begin{cases}
(0,0) & \text{if} \quad \gamma^{\mathbb{N}}_*((n_t)_{t\in T})=0 \\
(f,\gamma^{\mathbb{N}}_*((n_t)_{t\in T})) & \text{otherwise}.
\end{cases}
\end{cases}$$ Clearly $\gamma_*$ maps $(L^2_*(G) \cap C_*(G)) \times \mathbb{N}^T$ into $(L^2_*(G) \cap C_*(G)) \times \mathbb{N}^S$, and its restriction $(L^2_*(G) \cap C_*(G)) \times \mathbb{N}^T \longrightarrow (L^2_*(G) \cap C_*(G)) \times \mathbb{N}^S$ is obviously continuous.
Since the square $$\begin{CD}
{\operatorname{Ob}}({\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})) @>{\alpha_\varphi}>> C_*(G) \times \mathbb{N}^T \\
@V{f^*}VV @VV{\gamma_*}V \\
{\operatorname{Ob}}({\operatorname{Func}}(\mathcal{E}G,\psi {\operatorname{\text{-}smod}})) @>{\alpha_\psi}>> C_*(G) \times \mathbb{N}^S
\end{CD}$$ is commutative, it follows that $f^*$ restricts to a continuous functor $$f^*_{L^2} : {\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}}) \longrightarrow {\operatorname{Func}}_{L^2}(\mathcal{E}G,\psi {\operatorname{\text{-}smod}}).$$ We have just constructed a functor ${\operatorname{Func}}_{L^2}(\mathcal{E}G, - {\operatorname{\text{-}smod}})$ from $\Gamma \text{-Fib}_F$ to the category of topological categories. By composing it with the canonical functor $\text{TopCat} \rightarrow \text{kCat}$, we recover a functor from $\Gamma \text{-Fib}_F$ to the category of k-categories, and we still write it ${\operatorname{Func}}_{L^2}(\mathcal{E}G, - {\operatorname{\text{-}smod}})$.
**Remark :** If $G$ is compact, then all continuous functions on $G$ are square integrable and $C(G) \hookrightarrow
L^2(G)$ is continuous, therefore ${\operatorname{Func}}_{L^2}(\mathcal{E}G, - {\operatorname{\text{-}smod}})={\operatorname{Func}}(\mathcal{E}G, - {\operatorname{\text{-}smod}})$.
The $\Gamma-G$-space $s{\operatorname{Vec}}_{G,L^2}^\varphi$ {#10.1.2}
------------------------------------------------------------
Let $\varphi$ denote a Hilbert $\Gamma$-bundle. We define a contravariant functor from $\Gamma$ to $CG_G$ by: $$s\underline{{\operatorname{Vec}}}_{G,L^2}^\varphi:
S \longmapsto |{\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi(S) {\operatorname{\text{-}smod}})|=s{\operatorname{Vec}}_{G,L^2}^{\varphi(S)}.$$
We wish to prove that $s\underline{{\operatorname{Vec}}}_{G,L^2}^\varphi$ is a $\Gamma-G$-space. In order to do this, we need the following lemma.
\[L2transformations\] Let $S$ be a finite set, $\varphi$ and $\psi$ be two $S$-objects in $\Gamma \text{-Fib}_F$ together with two morphisms $f: \varphi \rightarrow \psi$ and $g:\psi \rightarrow \varphi$ such that $\mathcal{O}_\Gamma^F(f)=\mathcal{O}_\Gamma^F(g)={\operatorname{id}}_S$.\
Then the natural transformations $$\eta : {\operatorname{id}}_{\varphi {\operatorname{\text{-}smod}}} \longrightarrow (g {\operatorname{\text{-}smod}})\circ (f {\operatorname{\text{-}smod}}) \quad \text{and} \quad
{\varepsilon}: {\operatorname{id}}_{\psi {\operatorname{\text{-}smod}}} \longrightarrow (f {\operatorname{\text{-}smod}})\circ (g {\operatorname{\text{-}smod}})$$ from Corollary 3.3 of [@Ktheo1] induce, for every Lie group $G$, continuous natural transformations $$\eta^*: {\operatorname{id}}_{{\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})} \longrightarrow
g^*_{L^2} \circ f^*_{L^2}$$ and $${\varepsilon}^*: {\operatorname{id}}_{{\operatorname{Func}}_{L^2}(\mathcal{E}G,\psi {\operatorname{\text{-}smod}})} \longrightarrow
f^*_{L^2} \circ g^*_{L^2}.$$
Since ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$ is a full subcategory of ${\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$, it suffices to check that the natural transformations $\eta^*$ and ${\varepsilon}^*$ are continuous. In the case of $\eta^*$, we only need to prove that the composite map $${\operatorname{Ob}}({\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})) \overset{\eta^*}{\longrightarrow}
{\operatorname{Hom}}({\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})) \overset{({\operatorname{Fin}},{\operatorname{In}})}{\longrightarrow}
{\operatorname{Ob}}({\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}}))^2$$ is continuous. This is a simple consequence of the fact that $g_{L^2}^* \circ f_{L^2}^*$ is continuous. The case of ${\varepsilon}^*$ may be treated similarly.
For any Hilbert $\Gamma$-bundle $\varphi$, $s\underline{{\operatorname{Vec}}}_{G,L^2}^\varphi$ is a $\Gamma-G$-space.
It is clear that $s\underline{{\operatorname{Vec}}}_{G,L^2}^\varphi(\mathbf{0})=*$. Let $n$ be a positive integer, and $\varphi$ be a Hilbert $\Gamma$-bundle. We set $p:=\varphi(\mathbf{1})$. The same strategy as in the proof of Proposition 3.1 of [@Ktheo1] yields three functors $F^p: n.p {\operatorname{\text{-}smod}}\longrightarrow (p {\operatorname{\text{-}smod}})^n$, $G^p: (p {\operatorname{\text{-}smod}})^n \longrightarrow n.p {\operatorname{\text{-}smod}}$ and $H^p: (p {\operatorname{\text{-}smod}})^n \times I \longrightarrow (p {\operatorname{\text{-}smod}})^n$, which, in turn, induce three functors $$F^p_{L^2}: {\operatorname{Func}}_{L^2}(\mathcal{E}G,n.p {\operatorname{\text{-}smod}}) \longrightarrow ({\operatorname{Func}}(\mathcal{E}G,p {\operatorname{\text{-}smod}}))^n,$$ $$G^p_{L^2}: ({\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}}))^n \longrightarrow {\operatorname{Func}}(\mathcal{E}G,n.p {\operatorname{\text{-}smod}}),$$ and $$H^p_{L^2} : ({\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}}))^n \times I \longrightarrow
({\operatorname{Func}}(\mathcal{E}G,p {\operatorname{\text{-}smod}}))^n.$$ We need to prove that $F^p_{L^2}$ and $H^p_{L^2}$ (resp. $G^p_{L^2}$) take values in $({\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}}))^n$ (resp. in ${\operatorname{Func}}_{L^2}(\mathcal{E}G,n.p {\operatorname{\text{-}smod}})$) and that they induce continuous functors $$F^p_{L^2}: {\operatorname{Func}}_{L^2}(\mathcal{E}G,n.p {\operatorname{\text{-}smod}})
\longrightarrow ({\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}}))^n,$$ $$G^p_{L^2}: ({\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}}))^n \longrightarrow
{\operatorname{Func}}_{L^2}(\mathcal{E}G,n.p {\operatorname{\text{-}smod}}),$$ and $$H^p_{L^2} : ({\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}}))^n \times I \longrightarrow
({\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}}))^n.$$ This is however quite easy: for example, in the case of $F^p_{L^2}$, it suffices to consider the commutative square: $$\begin{CD}
{\operatorname{Ob}}({\operatorname{Func}}_{L^2}(\mathcal{E}G,n.p {\operatorname{\text{-}smod}})) @>{\alpha_{n.p}}>> C_*(G) \\
@V{F^p_{L^2}}VV @VVV \\
{\operatorname{Ob}}({\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}}))^n @>{(\alpha_p)^n}>> C_*(G)^n,
\end{CD}$$ where the right-hand vertical map is the diagonal map.
We then deduce that $|F^p_{L^2}|$ and $|G^p_{L^2}|$ are inverse one to the other up to equivariant homotopy, and it follows that the canonical map $$|{\operatorname{Func}}_{L^2}(\mathcal{E}G,n.p {\operatorname{\text{-}smod}})| \longrightarrow
|{\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}})|^n$$ is an equivariant homotopy equivalence.
Using Lemma \[L2transformations\] and the definition of a Hilbert $\Gamma$-bundle, it is then easy to check that the canonical map $$|{\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi(\mathbf{n}) {\operatorname{\text{-}smod}})| \longrightarrow
|{\operatorname{Func}}_{L^2}(\mathcal{E}G,n.\varphi(\mathbf{1}) {\operatorname{\text{-}smod}})|$$ is an equivariant homotopy equivalence.
We conclude that the canonical map $$|{\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi(\mathbf{n}) {\operatorname{\text{-}smod}})| \longrightarrow
|{\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi(\mathbf{1}) {\operatorname{\text{-}smod}})|^n$$ is an equivariant homotopy equivalence.
Given a Hilbert $\Gamma$-bundle $\varphi$ and a second countable Lie group $G$, we define $$\boxed{sKF_{G,L^2}^\varphi:=\Omega Bs\underline{{\operatorname{Vec}}}_{G,L^2}^\varphi.}$$
The morphism $s{\operatorname{Vec}}_{G,L^2}^\varphi \rightarrow
s{\operatorname{Vec}}_G^\varphi$ and its properties {#10.1.3}
---------------------------------------------------------------
### Main statements
For every object $\psi$ in the category $\Gamma \text{-Fib}_F$, the inclusion of categories defines a canonical continuous functor ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\psi {\operatorname{\text{-}smod}}) \longrightarrow
{\operatorname{Func}}(\mathcal{E}G,\psi {\operatorname{\text{-}smod}})$. Thus, for every Hilbert-$\Gamma$-bundle $\varphi$, we recover a morphism of equivariant $\Gamma$-spaces $$s\underline{{\operatorname{Vec}}}_{G,L^2}^\varphi \longrightarrow s\underline{{\operatorname{Vec}}}_{G.}^\varphi$$
When $G$ is compact, this morphism is simply the identity.
\[L2=indif\] Let $p$ be a $1$-object of $\Gamma \text{-Fib}_F$. Then, for any compact subgroup $H$ of $G$, $$(s{\operatorname{Vec}}_{G,L^2}^p)^H \longrightarrow
(s{\operatorname{Vec}}_G^p)^H$$ is a homotopy equivalence.
\[L2=indiff\] For every Hilbert $\Gamma$-bundle $\varphi$, the morphism $s\underline{{\operatorname{Vec}}}_{G,L^2}^\varphi
\longrightarrow s\underline{{\operatorname{Vec}}}_G^\varphi$ induces a $G$-weak equivalence: $$sKF_{G,L^2}^\varphi \longrightarrow sKF_G^\varphi.$$
Let $H$ be a compact subgroup of $G$ and $n$ be any non-negative integer. The canonical commutative diagram $$\begin{CD}
s\underline{{\operatorname{Vec}}}_{G,L^2}^\varphi(\mathbf{n}) @>>> s\underline{{\operatorname{Vec}}}_G^\varphi(\mathbf{n}) \\
@VVV @VVV \\
\Bigl(s\underline{{\operatorname{Vec}}}_{G,L^2}^\varphi(\mathbf{1})\Bigr)^n @>>> \Bigl(s\underline{{\operatorname{Vec}}}_G^\varphi(\mathbf{1})\Bigr)^n
\end{CD}$$ gives rise, by restriction to sets of points fixed by $H$, to a commutative diagram $$\begin{CD}
\bigl(s\underline{{\operatorname{Vec}}}_{G,L^2}^\varphi(\mathbf{n})\bigr)^H @>>> \bigl(s\underline{{\operatorname{Vec}}}_G^\varphi(\mathbf{n})\bigr)^H \\
@VVV @VVV \\
\Bigl(\bigl(s\underline{{\operatorname{Vec}}}_{G,L^2}^\varphi(\mathbf{1})\bigr)^H\Bigr)^n @>>>
\Bigl(\bigl(s\underline{{\operatorname{Vec}}}_G^\varphi(\mathbf{1})\bigr)^H\Bigr)^n.
\end{CD}$$ By Proposition \[L2=indif\] applied to $p=\varphi(\mathbf{1})$, the canonical map $\bigl(s{\operatorname{Vec}}_{G,L^2}^{\varphi(\mathbf{1})}\bigr)^H
\longrightarrow \bigl(s{\operatorname{Vec}}_G^{\varphi(\mathbf{1})}\bigr)^H$ is a homotopy equivalence. Hence $\Bigl(\bigl(s\underline{{\operatorname{Vec}}}_{G,L^2}^\varphi(\mathbf{1})\bigr)^H\Bigr)^n \longrightarrow
\Bigl(\bigl(s\underline{{\operatorname{Vec}}}_G^\varphi(\mathbf{1})\bigr)^H\Bigr)^n$ is a homotopy equivalence. Since $s\underline{{\operatorname{Vec}}}_{G,L^2}^\varphi$ and $s\underline{{\operatorname{Vec}}}_G^\varphi$ are both $\Gamma-G$-spaces, we use the previous commutative diagram to deduce that the canonical map $\bigl(s\underline{{\operatorname{Vec}}}_{G,L^2}^\varphi(\mathbf{n})\bigr)^H \longrightarrow
\bigl(s\underline{{\operatorname{Vec}}}_G^\varphi(\mathbf{n})\bigr)^H$ is a homotopy equivalence.
Since this is true for any non-negative integer $n$, it follows (by Proposition A.1 of [@Segal-cat]), that the canonical map $$\bigl(Bs\underline{{\operatorname{Vec}}}_{G,L^2}^\varphi\bigr)^H \longrightarrow
\bigl(Bs\underline{{\operatorname{Vec}}}_G^\varphi\bigr)^H$$ is a homotopy equivalence. Finally, taking loop spaces shows that the map $$\bigl(sKF_{G,L^2}^\varphi\bigr)^H \longrightarrow \bigl(sKF_G^\varphi\bigr)^H$$ is a homotopy equivalence.
### The proof of Proposition \[L2=indif\]
We may assume that $G$ is non-compact. Let $H$ denote a compact subgroup of $G$. In order to prove that $(s{\operatorname{Vec}}_{G,L^2}^p)^H \longrightarrow
(s{\operatorname{Vec}}_G^p)^H$ is a homotopy equivalence, we will construct a particular continuous functor $${\operatorname{Func}}(\mathcal{E}G,p {\operatorname{\text{-}smod}})
\longrightarrow {\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}})$$ that maps $H$-invariant functors to $H$-invariant functors.
Here is the basic idea: given a continuous functor $F: \mathcal{E}G \rightarrow p {\operatorname{\text{-}smod}}$, we want to build a continuous functor $F': \mathcal{E}G \rightarrow p {\operatorname{\text{-}smod}}$ such that $g \mapsto \|F'(1_G,g)\|$ is square integrable. Since only the norm of $F'(1_G,g)$ matters, we only have to multiply the map $g \mapsto F(1_G,g)$ by some “pit map” $\psi$ from $G$ to $\mathbb{R}_+^*$ so that $g \mapsto \psi(g)\|F(1_G,g)\|$ is square integrable. Of course, $\psi$ should depend continuously on $F$, and should be $H$-invariant.
We start by defining the functor on the sets of objects. When $F:\mathcal{E}G \rightarrow p {\operatorname{\text{-}smod}}$ is a functor, we define the *dimension* of $F$ as $\dim F:=\dim(F(1_G))$. We define ${\operatorname{Func}}^0(\mathcal{E}G,p {\operatorname{\text{-}smod}})$ (resp. ${\operatorname{Func}}^{>0}(\mathcal{E}G,p {\operatorname{\text{-}smod}})$) as the full subcategory of ${\operatorname{Func}}(\mathcal{E}G,p {\operatorname{\text{-}smod}})$ whose space of objects consists of those functors of dimension $0$ (resp. of positive dimension). Similarly, we define ${\operatorname{Func}}^0_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}})$ (resp. ${\operatorname{Func}}^{>0}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}})$) as the full subcategory of ${\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}})$ whose space of objects consists of those functors of dimension $0$ (resp. of positive dimension).
We easily see that $${\operatorname{Func}}^0_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}})={\operatorname{Func}}^0(\mathcal{E}G,p {\operatorname{\text{-}smod}}),$$ $${\operatorname{Func}}(\mathcal{E}G,p {\operatorname{\text{-}smod}}) ={\operatorname{Func}}^0(\mathcal{E}G,p {\operatorname{\text{-}smod}}) \coprod {\operatorname{Func}}^{>0}(\mathcal{E}G,p {\operatorname{\text{-}smod}})$$ and $${\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}}) ={\operatorname{Func}}^0_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}}) \coprod {\operatorname{Func}}^{>0}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}}).$$ We then set $\Phi(\mathbf{F}):=\mathbf{F}$ for every $\mathbf{F} \in {\operatorname{Func}}^0(\mathcal{E}G,p {\operatorname{\text{-}smod}})$.
The Haar measure $\mu$ induces a measure on $G/H$ which is finite on compact subsets (i.e. $\sigma$-compact). Since $G/H$ is locally compact, there exists a map $\lambda : G/H \rightarrow \mathbb{R}_+^*$ which is continuous and square integrable. We choose such a map. The following lemma is then straightforward:
The map $$\beta : \begin{cases}
{\operatorname{Ob}}({\operatorname{Func}}_{L^2}^{>0}(\mathcal{E}G,p {\operatorname{\text{-}smod}})) & \longrightarrow C(G,\mathbb{R}_+^*) \\
\mathbf{F} & \longmapsto \left[g \longmapsto \cfrac{\lambda(gH)}{\underset{h \in H}{\max} \|F(1_G,gh)\|}\right]
\end{cases}$$ is well defined and continuous.
Finally, we define a continuous map: $$\gamma : \begin{cases}
C(G,\mathbb{R}_+^*) \times
{\operatorname{Ob}}\left({\operatorname{Func}}^{>0}(\mathcal{E}G,p {\operatorname{\text{-}smod}})\right) & \longrightarrow {\operatorname{Ob}}\left({\operatorname{Func}}^{>0}(\mathcal{E}G,p {\operatorname{\text{-}smod}})\right) \\
(\psi,\mathbf{F}) & \longmapsto
\begin{cases}
g & \longmapsto \mathbf{F}(g) \\
(g,g') & \longmapsto \frac{\psi(g')}{\psi(g)}\cdot \mathbf{F}(g,g')
\end{cases}
\end{cases}$$ and consider the composite map: $$\begin{gathered}
\Phi : {\operatorname{Ob}}\left({\operatorname{Func}}^{>0}(\mathcal{E}G,p {\operatorname{\text{-}smod}})\right) {\overset{(\beta,{\operatorname{id}})}{{\;{\count255=0 \loop \relbar\mathrel{\mkern-6mu} \advance\count255 by1\ifnum\count255<5\repeat\rightarrow}\;}}}
C(G,\mathbb{R}_+^*) \times {\operatorname{Ob}}\left({\operatorname{Func}}^{>0}(\mathcal{E}G,p {\operatorname{\text{-}smod}})\right) \\
\overset{\gamma}{\longrightarrow} {\operatorname{Ob}}\left({\operatorname{Func}}(\mathcal{E}G,p {\operatorname{\text{-}smod}})\right),\end{gathered}$$ which is clearly continuous.
Given a functor $\mathbf{F}:\mathcal{E}G \rightarrow p {\operatorname{\text{-}smod}}$ of positive dimension, $\Phi(\mathbf{F})$ is an object of ${\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}})$. Indeed, for every $g \in G$, $$\|\Phi(\mathbf{F})(1_G,g)\|=
\frac{1}{\beta(\mathbf{F})(1_G)}\cfrac{\|\mathbf{F}(1_G,g)\|}{\underset{h \in H}{\max}\,\|\mathbf{F}(1_G,gh)\|}\,\lambda(gH)
\leq \frac{1}{\beta(\mathbf{F})(1_G)}\, \lambda(gH)$$ Since $\lambda$ is square integrable, $g \mapsto \|\Phi(\mathbf{F})(1_G,g)\|$ is square integrable. Hence $\Phi(\mathbf{F}) \in {\operatorname{Ob}}({\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}}))$.
We have thus constructed a map $$\Phi : {\operatorname{Ob}}\bigl({\operatorname{Func}}(\mathcal{E}G,p {\operatorname{\text{-}smod}})\bigr)
\longrightarrow {\operatorname{Ob}}\bigl({\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}})\bigr).$$
The map $\Phi$ is continuous.
It suffices to prove that the composite map $${\operatorname{Ob}}\left({\operatorname{Func}}^{>0}(\mathcal{E}G,p {\operatorname{\text{-}smod}})\right)
\overset{\Phi}{\longrightarrow} {\operatorname{Ob}}\left({\operatorname{Func}}_{L^2}^{>0}(\mathcal{E}G,p {\operatorname{\text{-}smod}})\right) \overset{\alpha_p}{\longrightarrow} L^2(G)$$ is continuous. It thus suffices to establish the continuity of $$\Phi_1 :\begin{cases}
C(G,\mathbb{R}_+^*) & \longrightarrow L^2(G,\mathbb{R}_+^*) \\
f & \longmapsto \left[g \mapsto \cfrac{f(g)}{\underset{h \in H}{\max}\,f(gh)}\,\lambda(gH)\right].
\end{cases}$$ Let $f \in C(G,\mathbb{R}_+^*)$ and ${\varepsilon}>0$. Since $G/H$ is $\sigma$-compact, we may choose a compact subset $K$ of $G$ such that $\int_{G {\smallsetminus}K} \lambda(gH)\diff g \leq {\varepsilon}$. We already know that the map $$\begin{cases}
C(G,\mathbb{R}_+^*) & \longrightarrow C(G,\mathbb{R}_+^*) \\
f & \longmapsto \left[g \mapsto \frac{f(g)}{\underset{h \in H}{\max}f(gh)} \lambda(gH)\right]
\end{cases}$$ Hence $$O_{\varepsilon}:=\left\{f_1 \in C(G,\mathbb{R}_+^*):
\|\Phi_1(f_1)-\Phi_1(f)\|_\infty^K <\sqrt{\frac{{\varepsilon}}{\mu(K)}} \right\}$$ is an open subset of $C(G,\mathbb{R}_+^*)$ which contains $f$.\
For every $f_1\in O_{\varepsilon}$, $$\int_G \big|\Phi_1(f_1)-\Phi_1(f)\big|^2\diff \mu
\leq 2\left(\int_{G {\smallsetminus}K}(\big|\Phi_1(f_1)\big|^2 +\big|\Phi_1(f)\big|^2)\diff \mu \right)
+ \int_K \big|\Phi_1(f_1)-\Phi_1(f)\big|^2 \diff \mu
\leq 5{\varepsilon}.$$ It follows that $\Phi_1$ is continuous, hence $\Phi$ also is.
We may now define $\Phi$ on morphisms by: $$\Phi: \begin{cases}
{\operatorname{Hom}}({\operatorname{Func}}(\mathcal{E}G,p {\operatorname{\text{-}smod}})) & \longrightarrow {\operatorname{Hom}}({\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}})) \\
\mathbf{F} \overset{\eta}{\longrightarrow} \mathbf{F}' & \longmapsto
\begin{cases}
\left[g \longmapsto \cfrac{\beta(\mathbf{F}')[g]}{\beta(\mathbf{F})[g]}\cdot \eta_g \right] &
\text{if} \quad \dim \mathbf{F}>0 \\
[g \mapsto \eta_g] & \text{otherwise},
\end{cases}
\end{cases}$$ which yields a continuous functor $$\Phi : {\operatorname{Func}}(\mathcal{E}G,p {\operatorname{\text{-}smod}}) \longrightarrow
{\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}}).$$
Since $\tilde{\beta}(x)$ is $H$-invariant for every $x\in \left]0,+\infty\right[^\mathbb{N}$, $\Phi$ induces a functor $$\Phi^H: {\operatorname{Func}}(\mathcal{E}G,p {\operatorname{\text{-}smod}})^H
\longrightarrow {\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}})^H.$$ We define $i^H: {\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}})^H
\longrightarrow {\operatorname{Func}}(\mathcal{E}G,p {\operatorname{\text{-}smod}})^H$ as the functor induced by inclusion of categories.
Finally, we define a continuous natural transformation $\eta : {\operatorname{id}}_{{\operatorname{Func}}_{L^2}(\mathcal{E}G,p {\operatorname{\text{-}smod}})^H} \longrightarrow
\Phi^H \circ i^H$ by: $$\eta: \mathbf{F} \longmapsto
\begin{cases}
\left[g \mapsto \beta(\mathbf{F})[g].{\operatorname{id}}_{\mathbf{F}(g)} \right] & \text{if} \quad \dim(\mathbf{F})>0 \\
\left[g \mapsto {\operatorname{id}}_{\mathbf{F}(g)}\right] & \text{otherwise},
\end{cases}$$ and a continuous natural transformation ${\varepsilon}: {\operatorname{id}}_{{\operatorname{Func}}(\mathcal{E}G,p {\operatorname{\text{-}smod}})^H} \longrightarrow
i^H \circ \Phi^H$ by $${\varepsilon}: \mathbf{F} \longmapsto
\begin{cases}
\left[g \mapsto \beta(\mathbf{F})[g].{\operatorname{id}}_{\mathbf{F}(g)}\right] & \text{if} \quad \dim(\mathbf{F})>0 \\
\left[g \mapsto {\operatorname{id}}_{\mathbf{F}(g)}\right] & \text{otherwise}.
\end{cases}$$ We conclude that $|\Phi^H|$ is a homotopy-inverse of $|i^H|$. This finishes the proof of Proposition \[L2=indif\].
A universal bundle over $|{\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})|$ {#10.1.4}
-----------------------------------------------------------------------------------------------------------
Let $\varphi$ be a 1-object of $\Gamma \text{-Fib}_F$. We define ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}sBdl}})$ as the full-subcategory of ${\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}sBdl}})$ whose set of objects is the inverse image of ${\operatorname{Ob}}({\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}}))$ by the canonical functor ${\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}sBdl}}) \longrightarrow
{\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}}).$\
We then equip ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}sBdl}})$ with the topological category structure induced by the canonical functor $${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}sBdl}})
\longrightarrow {\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})
\times {\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}sBdl}}).$$
Similarly, if $n$ is a non-negative integer and $\varphi$ an $n$-dimensional $1$-object of $\Gamma \text{-Fib}_F$, we define ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}sframe}})$ as the full subcategory of ${\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}sframe}})$ whose set of objects is the inverse image of ${\operatorname{Ob}}({\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}}))$ by the canonical functor ${\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}sframe}}) \longrightarrow {\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})$.
We then equip ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}sframe}})$ with the topology induced by the canonical functor $${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}sframe}})
\longrightarrow {\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}smod}})
\times {\operatorname{Func}}(\mathcal{E}G,\varphi {\operatorname{\text{-}sframe}}).$$ For any $1$-object $\varphi$ of $\Gamma \text{-Fib}_F$, set: $$Es{\operatorname{Vec}}_{G,L^2}^\varphi:=|{\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}sBdl}})|.$$ If $\varphi$ has dimension $n\in \mathbb{N}$, set: $$s\widetilde{{\operatorname{Vec}}}_{G,L^2}^\varphi:=|{\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi {\operatorname{\text{-}sframe}})|.$$
With those definitions, we find (compare with Theorem 2.6 of [@Ktheo1]):
\[theoL2\] Let $n\in \mathbb{N}$ and $\varphi$ be an $n$-dimensional $1$-object of $\Gamma \text{-Fib}_F$. Then:
- $s\widetilde{{\operatorname{Vec}}}_{G,L^2}^\varphi \rightarrow
s{\operatorname{Vec}}_{G,L^2}^\varphi$ is a $(G,{\operatorname{GU}}_n(F))$-principal bundle;
- $Es{\operatorname{Vec}}_{G,L^2}^\varphi \rightarrow s{\operatorname{Vec}}_{G,L^2}^\varphi$ is an $n$-dimensional $G$-simi-Hilbert bundle;
- The morphism $s\widetilde{{\operatorname{Vec}}}_{G,L^2}^\varphi \times _{{\operatorname{GU}}_n(F)}F^n \rightarrow
Es{\operatorname{Vec}}_{G,L^2}^\varphi$ is an isomorphism of $G$-simi-Hilbert bundles.
One needs to go back to the details of the proof of Theorem 2.6 of [@Ktheo1] adapted to the case of simi-Hilbert bundles. The careful reader will then easily check that the corresponding maps $\nu_m$, ${\varepsilon}_m$, $\chi_m$, et $\psi_{g,m}$ are continuous in the present situation (the proof of this claim relies upon Lemma \[L2nerve\]). The rest of the proof is identical to that of Theorem 2.6 of [@Ktheo1].
For $\varphi=\text{Fib}^{F^{(\infty)}}$, set now $$s{\operatorname{Vec}}_{G,L^2}^{F,\infty}:=s{\operatorname{Vec}}_{G,L^2}^{\varphi(\mathbf{1})} \quad \text{and} \quad
Es{\operatorname{Vec}}_{G,L^2}^{F,\infty}:=Es{\operatorname{Vec}}_{G,L^2.}^{\varphi(\mathbf{1})}$$ The canonical injection ${\operatorname{Func}}_{L^2}(\mathcal{E}G,\varphi(\mathbf{1}) {\operatorname{\text{-}sBdl}})
\rightarrow {\operatorname{Func}}(\mathcal{E}G,\varphi(\mathbf{1}) {\operatorname{\text{-}sBdl}})$ then induces a strong morphism of $G$-simi-Hilbert bundles $$\begin{CD}
Es{\operatorname{Vec}}_{G,L^2}^{F,\infty} @>>> Es{\operatorname{Vec}}_G^{F,\infty} \\
@VVV @VVV \\
s{\operatorname{Vec}}_{G,L^2}^{F,\infty} @>>> s{\operatorname{Vec}}_G^{F,\infty}.
\end{CD}$$
It follows that, for every proper $G$-CW-complex $X$, the composite map
$$[X,s{\operatorname{Vec}}_{G,L^2}^{F,\infty}]_G \overset{\cong}{\longrightarrow}
[X,s{\operatorname{Vec}}_G^{F,\infty}]_G
\overset{\cong}{\longrightarrow} s{\operatorname{\mathbb{V}ect}}_G^F(X)$$ is the one obtained by pulling-back the $G$-simi-Hilbert bundle $Es{\operatorname{Vec}}_{G,L^2}^{F,\infty} \rightarrow s{\operatorname{Vec}}_{G,L^2}^{F,\infty}$. This last morphism is thus systematically an isomorphism of monoids, which yields:
The $G$-simi-Hilbert bundle $Es{\operatorname{Vec}}_{G,L^2}^{F,\infty} \rightarrow s{\operatorname{Vec}}_{G,L^2}^{F,\infty}$ is universal amongst $G$-simi-Hilbert bundles over proper $G$-CW-complexes.
The simplicial $G$-space $s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}$ {#10.2}
=====================================================================
Categorical background {#10.2.1}
----------------------
Let $S$ be a totally ordered set. A subset $I$ of $S$ is said to be an **interval** of $S$ when $$\forall (x,y,z)\in I^2 \times S,\quad x <z<y \Rightarrow z \in I.$$ For $(S_1,S_2)\in \mathcal{P}(S)^2$, we will write $S_1 <S_2$ when $\forall (x,y)\in S_1 \times S_2,\; x<y$, and $S_1 \leq S_2$ when $\forall y \in S_2, \exists x \in S_1 : \, x \leq y$.
Let $S$ and $T$ be two totally ordered sets, and $f:\mathcal{P}(S) \rightarrow \mathcal{P}(T)$ be a map.\
We say that $f$ is **increasing** when $$\forall (S_1,S_2) \in \mathcal{P}(S)^2, \; S_1<S_2 \Rightarrow f(S_1) < f(S_2).$$ We say that $f$ is **hole-free** when $f(I)$ is an interval of $T$ for every interval $I$ of $S$.
**Remarks :**
(i) If $S_1, S_2, S_3$ are totally ordered sets, and $f:\mathcal{P}(S_1) \rightarrow \mathcal{P}(S_2)$ and $g:\mathcal{P}(S_2) \rightarrow \mathcal{P}(S_3)$ are two maps, we easily see that $g \circ f$ is increasing if $f$ and $g$ are both increasing, and $g \circ f$ is hole-free if $f$ and $g$ are both hole-free.
(ii) If $S$ is a totally ordered finite set, $T$ is a totally ordered set, and $f: \mathcal{P}(S) \rightarrow \mathcal{P}(T)$ is a map which respects disjoint unions and such that $f(S)$ is finite, then
- $f$ is increasing if and only if $\forall (k,k')\in S^2, \; k<k' \Rightarrow f(\{k\}) < f(\{k'\})$;
- in the case $f$ is increasing, it is hole-free if and only if $f(S)=\emptyset$ or $f(S)=\{k\in T : \inf f(S) \leq k \leq \sup f(S)\}$.
\[gammastar\] We define $\Gamma^*$ as the category whose objects are the totally ordered finite sets, and whose morphisms are the morphisms of $\Gamma$ that are both increasing and hole-free. Forgetting the order structures gives rise to a “forgetful” functor $\Gamma^* \rightarrow \Gamma$.
The previous remarks make it then easy to check that the canonical functor: $\Delta \rightarrow \Gamma$ induces a functor $\Delta \rightarrow \Gamma^*$.
The simplicial $G$-space $s{\operatorname{Vec}}_{G,L^2}^{F,m*}$ {#10.2.2}
---------------------------------------------------------------
For every totally ordered finite set $S$, we define $\Gamma^*(S)$ as the subset of $\Gamma(S)$ consisting of those maps $\mathcal{P}(S) \rightarrow
\mathcal{P}(\mathbb{N})$ which are increasing and hole-free. We define a structure of poset on $\Gamma^*(S)$ by $$\forall (f,g)\in \Gamma^*(S)^2, \; f \leq g \underset{\text{def}}{\Longleftrightarrow} \bigl(\forall X \in \mathcal{P}(S),\; f(X) \leq g(X)\bigr).$$
Every morphism $f:S \rightarrow T$ in $\Gamma^*$ induces a non-decreasing map $f^*:\Gamma^*(T) \rightarrow \Gamma^*(S)$ by precomposition.
Let $\mathcal{H}$ be an inner product space that is either finite-dimensional or isomorphic to $F^{(\infty)}$. For every object $S$ of $\Gamma^*$, we set $$\text{Fib}^{\mathcal{H}*}(S):=
\left(S,\underset{f \in \Gamma^*(S)}{\coprod}\underset{s \in S}{\prod}
G_{f(s)}(\mathcal{H}),
\left(\underset{f \in \Gamma^*(S)}{\coprod}\left[p_{f(s)}
(\mathcal{H}) \times \underset{s' \in S{\smallsetminus}\{s\}}{\prod}
{\operatorname{id}}_{G_{f(s')}(\mathcal{H})}\right]\right)_{s \in S}\right).$$ Since $\Gamma^*(S) \subset \Gamma(S)$, we may view $\text{Fib}^{\mathcal{H}*}(S)$ as a sub-object of $\text{Fib}^{\mathcal{H}}(S)$ in the category $\Gamma \text{-Fib}_F$. The morphism $\text{Fib}^{\mathcal{H}}(T) \longrightarrow \text{Fib}^{\mathcal{H}}(S)$ induced by a morphism $f : S \rightarrow T$ in $\Gamma^*$ then yields a morphism $\text{Fib}^{\mathcal{H}*}(T) \longrightarrow \text{Fib}^{\mathcal{H}*}(S)$. This defines a contravariant functor $\text{Fib}^{\mathcal{H}*}:\Gamma^* \longrightarrow \Gamma \text{-Fib}_F$. The canonical morphisms $\text{Fib}^{\mathcal{H}*}(S) \longrightarrow \text{Fib}^{\mathcal{H}}(S)$ then induce a natural transformation from $\text{Fib}^{\mathcal{H}*}$ to the composite of $\text{Fib}^{\mathcal{H}}:\Gamma \rightarrow \Gamma \text{-Fib}_F$ with the forgetful functor $\Gamma^* \rightarrow \Gamma$.
For every object $S$ of $\Gamma^*$, we equip the set of objects of $\text{Fib}^{\mathcal{H}*}(S) {\operatorname{\text{-}smod}}$ with the following preorder: $$\forall (f,g)\in \Gamma^*(S)^2, \;\forall (x,y)\in \left(\underset{s \in S}{\prod}
G_{f(s)}(\mathcal{H})\right) \times \left(\underset{s \in S}{\prod} G_{g(s)}(\mathcal{H})\right), \quad x \leq y
\underset{\text{def}}{\Longleftrightarrow} f\leq g.$$
A morphism $x \overset{\varphi}{\rightarrow} y$ in $\text{Fib}^{\mathcal{H}*}(S) {\operatorname{\text{-}smod}}$ is called **non-decreasing** when $x \leq y$.
It is then obvious that, for every morphism $\gamma : S \rightarrow T$ in $\Gamma^*$, $\text{Fib}^{\mathcal{H}*}(\gamma) {\operatorname{\text{-}smod}}$ maps the set of non-decreasing morphisms of $\text{Fib}^{\mathcal{H}*}(T) {\operatorname{\text{-}smod}}$ into the set of non-decreasing morphisms of $\text{Fib}^{\mathcal{H}*}(S) {\operatorname{\text{-}smod}}$. \[increasingfunc\] Finally, we let ${\operatorname{Func}}_{\uparrow L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}*}(S) {\operatorname{\text{-}smod}}\right)$ denote the subcategory of ${\operatorname{Func}}_{L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}*}(S) {\operatorname{\text{-}smod}}\right)$ which has the same space of objects and whose morphisms are the natural transformations that map $G$ into the set of non-decreasing morphisms of $\text{Fib}^{\mathcal{H}*}(S){\operatorname{\text{-}smod}}$.
Composing $\text{Fib}^{\mathcal{H}*}$ with the canonical functor $\Delta \rightarrow \Gamma^*$, we obtain a functor $\Delta \rightarrow \Gamma \text{-Fib}_F$ which we will still denote by $\text{Fib}^{\mathcal{H}*}$.
Finally, we recover a simplicial $G$-space: $$s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^{\mathcal{H}*}}: [n] \longmapsto
\left|{\operatorname{Func}}_{\uparrow L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}*}([n]) {\operatorname{\text{-}smod}}\right)\right|.$$ With a slight abuse of notation, we will also denote by $s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^\mathcal{H}}$ the simplicial $G$-space deduced from the equivariant $\Gamma$-space $s\underline{{\operatorname{Vec}}}_{G,L^2}^{\text{Fib}^\mathcal{H}}$. The previous natural transformation yields a morphism of simplicial $G$-spaces: $s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^{\mathcal{H}*}} \longrightarrow s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^\mathcal{H}}$. Our main result follows:
\[increasindif\] For every $n \in \mathbb{N}$, $s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^{\mathcal{H}*}}[n]
\rightarrow s{\operatorname{Vec}}_{G,L^2}^{\mathcal{H}}[n]$ is an equivariant homotopy equivalence.
Let $n\in \mathbb{N}$ and $f\in \Gamma(\mathbf{n})$. If $f(\mathbf{n})=\emptyset$, we set $r_n(f):=f$. Otherwise, we define $r_n(f)$ as the unique element of $\Gamma^*(\mathbf{n})$ such that $0 \in r_n(f)(\mathbf{n})$ and $\# r_n(f)(k)=\# f(k)$ for every $k\in \mathbf{n}$. For every $k \in \mathbf{n}$, we then let $i_k^f$ denote the unique increasing map from $f(k)$ to $r_n(f)(k)$. For every $k\in \mathbb{N}$, $i_k^f$ yields an isometry $\mathcal{H}^{f(k)} \longrightarrow \mathcal{H}^{r_n(f)(k)}$, and these isometries give rise to a strong morphism of Hilbert bundles $$\begin{CD}
E_{f(i)}(\mathcal{H}) \times \underset{k \in \mathbf{n}{\smallsetminus}\{i\}}{\prod} G_{f(k)}(\mathcal{H})
@>>> E_{r_n(f)(i)}(\mathcal{H}) \times \underset{k \in \mathbf{n}{\smallsetminus}\{i\}}{\prod} G_{r_n(f)(k)}(\mathcal{H}) \\
@VVV @VVV \\
\underset{k \in \mathbf{n}}{\prod} G_{f(k)}(\mathcal{H}) @>>> \underset{k \in \mathbf{n}}{\prod} G_{r_n(f)(k)}(\mathcal{H}) \\
\end{CD}$$ for every $i\in \mathbf{n}$. This defines a morphism $r_n^{\mathcal{H}} : \text{Fib}^{\mathcal{H}}(\mathbf{n})
\rightarrow \text{Fib}^{\mathcal{H}*}(\mathbf{n})$ such that $\mathcal{O}_\Gamma^F(r_n^{\mathcal{H}})={\operatorname{id}}_{\mathbf{n}}$ and $r_n^{\mathcal{H}} {\operatorname{\text{-}smod}}$ maps any morphism of $\text{Fib}^{\mathcal{H}}(\mathbf{n}) {\operatorname{\text{-}smod}}$ to a non-decreasing morphism of $\text{Fib}^{\mathcal{H}*}(\mathbf{n}) {\operatorname{\text{-}smod}}$. Hence, $r_n^{\mathcal{H}}$ induces a $G$-map $s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^\mathcal{H}}[n]
\rightarrow s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^{\mathcal{H}*}}[n]$, and we now prove that it is an equivariant homotopy inverse of the canonical map $s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^{\mathcal{H}*}}[n] \rightarrow s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^\mathcal{H}}[n]$.
Since the canonical map is induced by the inclusion of $\Gamma^*(\mathbf{n})$ into $\Gamma(\mathbf{n})$, it is induced by a morphism $\text{Fib}^{\mathcal{H}*}(\mathbf{n}) \rightarrow \text{Fib}^{\mathcal{H}}(\mathbf{n})$ over ${\operatorname{id}}_\mathbf{n}$, and we then deduce from Lemma \[L2transformations\] that the composite map $s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^\mathcal{H}}[n]
\rightarrow s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^{\mathcal{H}*}}[n] \rightarrow s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^\mathcal{H}}[n]$ is $G$-homotopic to the identity.
By construction, one has $r_n(f) \leq f$ for every $f \in \Gamma^*([n])$. We deduce that the inverse of the natural transformation from ${\operatorname{id}}_{{\operatorname{Func}}_{L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}*}([n]) {\operatorname{\text{-}smod}}\right)}$ to the composite functor $${\operatorname{Func}}_{L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}*}([n]) {\operatorname{\text{-}smod}}\right)
\longrightarrow {\operatorname{Func}}_{L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}}([n]) {\operatorname{\text{-}smod}}\right)
\longrightarrow {\operatorname{Func}}_{L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}}([n]) {\operatorname{\text{-}smod}}\right)$$ obtained in Lemma \[L2transformations\] and the proof of Corollary 3.3 in [@Ktheo1] is really a natural transformation from $${\operatorname{Func}}_{\uparrow L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}*}([n]) {\operatorname{\text{-}smod}}\right)
\longrightarrow {\operatorname{Func}}_{L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}}([n]) {\operatorname{\text{-}smod}}\right)
\longrightarrow {\operatorname{Func}}_{\uparrow L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}}([n]) {\operatorname{\text{-}smod}}\right)$$ to ${\operatorname{id}}_{{\operatorname{Func}}_{\uparrow L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}*}([n]) {\operatorname{\text{-}smod}}\right)}$. Hence the composite map $s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^{\mathcal{H}*}}[n] \rightarrow s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^\mathcal{H}}[n] \rightarrow
s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^{\mathcal{H}*}}[n]$ is $G$-homotopic to the identity map.
\[ultimatevec\] For $m\in \mathbb{N}^* \cup \{\infty\}$, we now set $s{\operatorname{Vec}}_{G,L^2}^{F,m *}:=s{\operatorname{Vec}}_{G,L^2}^{\text{Fib}^{F^{m}*}}$.
The canonical map $\Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,m*} \longrightarrow
\Omega Bs\underline{{\operatorname{Vec}}}_{G,L^2}^{F,m}$ is a $G$-weak equivalence.
We deduce from Proposition \[increasindif\] that, for every compact subgroup $H$ of $G$,\
$\bigl(s{\operatorname{Vec}}_{G,L^2}^{F,m *}[n]\bigr)^H \longrightarrow \bigl(s{\operatorname{Vec}}_{G,L^2}^{F,m}[n]\bigr)^H$ is a homotopy equivalence for every non negative integer $n$. It follows then from proposition A.1 of [@Segal-cat] that $\bigl(\Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,m*}\bigr)^H \longrightarrow \bigl(\Omega Bs\underline{{\operatorname{Vec}}}_{G,L^2}^{F,m}\bigr)^H$ is a homotopy equivalence for every compact subgroup $H$ of $G$.
The $G$-space $\Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,\infty*}$ is a classifying space for $KF^*_G(-)$ on the category of proper $G$-CW-complexes.
The universal bundle $Es{\operatorname{Vec}}_{G,L^2}^{F,\infty*} \longrightarrow s{\operatorname{Vec}}_{G,L^2}^{F,\infty*}[1]$ {#10.2.3}
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We denote by ${\operatorname{Func}}_{\uparrow L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}*}([1]) {\operatorname{\text{-}sBdl}}\right)$ the subcategory of ${\operatorname{Func}}_{L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}*}([1]) {\operatorname{\text{-}sBdl}}\right)$ which has the same space of objects, and whose morphisms are the natural transformations which, after composition with the canonical functor $\text{Fib}^{\mathcal{H}*}([1]) {\operatorname{\text{-}sBdl}}\longrightarrow \text{Fib}^{\mathcal{H}*}([1]) {\operatorname{\text{-}smod}}$, map $G$ into the set of non-decreasing morphisms of $\text{Fib}^{\mathcal{H}*}(\mathbf{1}){\operatorname{\text{-}smod}}$.
\[ultimatebdl\] We define $$Es{\operatorname{Vec}}_{G,L^2}^{F,\infty*}:=
\left|{\operatorname{Func}}_{\uparrow L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}*}([1]) {\operatorname{\text{-}sBdl}}\right)\right|.$$ Then $Es{\operatorname{Vec}}_{G,L^2}^{F,\infty*}$ is simply the inverse image of $s{\operatorname{Vec}}_{G,L^2}^{F,\infty*}[1]$ by the canonical map $Es{\operatorname{Vec}}_{G,L^2}^{F,\infty} \longrightarrow s{\operatorname{Vec}}_{G,L^2}^{F,\infty}[1]$. The canonical morphism $\text{Fib}^{F^{(\infty)}*}([1])
\rightarrow \text{Fib}^{F^{(\infty)}}([1])$ then induces a strong morphism of $G$-simi-Hilbert bundles: $$\begin{CD}
Es{\operatorname{Vec}}_{G,L^2}^{F,\infty*} @>>> Es{\operatorname{Vec}}_{G,L^2}^{F,\infty} \\
@VVV @VVV \\
s{\operatorname{Vec}}_{G,L^2}^{F,\infty*}[1] @>>> s{\operatorname{Vec}}_{G,L^2}^{F,\infty}.
\end{CD}$$ The following result follows readily:
For every proper $G$-CW-complex $X$, the map $[X,s{\operatorname{Vec}}_{G,L^2}^{F,\infty*}[1]]_G \rightarrow s{\operatorname{\mathbb{V}ect}}_G^F(X)$ induced by pulling back the $G$-simi-Hilbert bundle $Es{\operatorname{Vec}}_{G,L^2}^{F,\infty*} \rightarrow s{\operatorname{Vec}}_{G,L^2}^{F,\infty*}[1]$ is the composite map $$\bigl[X,s{\operatorname{Vec}}_{G,L^2}^{F,\infty*}[1]\bigr]_G \overset{\cong}{\longrightarrow}
\bigl[X,s{\operatorname{Vec}}_{G,L^2}^{F,\infty}\bigr]_G \overset{\cong}{\longrightarrow} s{\operatorname{\mathbb{V}ect}}_G^F(X).$$
Finally, denote by $\varphi_n$ the $n$-dimensional part of the Hilbert bundle $\text{Fib}^{F^{(\infty)}*}([1])$, and define ${\operatorname{Func}}_{\uparrow L^2}\left(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}}\right)$ as the subcategory of ${\operatorname{Func}}_{L^2}\left(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}}\right)$ which has the same space of objects and whose morphisms are the natural transformations which, after composition with the canonical functor $\varphi_n {\operatorname{\text{-}sframe}}\longrightarrow \text{Fib}^{\mathcal{H}*}([1]) {\operatorname{\text{-}smod}}$, map $G$ into the set of non-decreasing morphisms of $\text{Fib}^{\mathcal{H}*}([1]){\operatorname{\text{-}smod}}$.
\[ultimateframe\] Set $s\widetilde{{\operatorname{Vec}}}_{G,L^2}^{n,F,\infty*}:=|{\operatorname{Func}}_{\uparrow L^2}\left(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}}\right)|$, and notice that $s\widetilde{{\operatorname{Vec}}}_{G,L^2}^{n,F,\infty*}$ is simply the inverse image of $\left|{\operatorname{Func}}_{\uparrow L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}*}([1]){\operatorname{\text{-}smod}}\right)\right|
\cap s{\operatorname{Vec}}_{G,L^2}^{\varphi_n}$ under the identification map $s\widetilde{{\operatorname{Vec}}}_{G,L^2}^{\varphi_n} \longrightarrow s{\operatorname{Vec}}_{G,L^2}^{\varphi_n}$. We conclude that $\underset{n=0}{\overset{\infty}{\coprod}}s\widetilde{{\operatorname{Vec}}}_{G,L^2}^{n,F,\infty*} \longrightarrow
s{\operatorname{Vec}}_{G,L^2}^{F,\infty*}[1]$ is an identification map.
Denote finally by $Es{\operatorname{Vec}}_{G,L^2}^{n,F,\infty*}$ the inverse image of $\big|{\operatorname{Func}}_{\uparrow L^2}\left(\mathcal{E}G,\text{Fib}^{\mathcal{H}*}([1]){\operatorname{\text{-}smod}}\right)\big| \cap s{\operatorname{Vec}}_{G,L^2}^{\varphi_n}$ under the $G$-vector bundle map $E{\operatorname{Vec}}_{G,L^2}^{\varphi_n} \longrightarrow s{\operatorname{Vec}}_{G,L^2}^{\varphi_n}$. Proposition \[theoL2\] then yields that the canonical morphism $s\widetilde{{\operatorname{Vec}}}_{G,L^2}^{n,F,\infty*} \times _{{\operatorname{GU}}_n(F)}F^n \rightarrow Es{\operatorname{Vec}}_{G,L^2}^{n,F,\infty*}$ is an isomorphism of $G$-simi-Hilbert bundles.
The simplicial $G$-space $({\operatorname{Fred}}(G,\mathcal{H})[n])_{n \in \mathbb{N}}$ {#10.3}
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The $G$-space ${\operatorname{Fred}}(L^2(G,\mathcal{H}))$ {#10.3.1}
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Let $\mathcal{H}$ be a separable Hilbert space. The map $$(g,f) \longmapsto \left[x \mapsto f(xg)\right]$$ classically defines a structure of Hilbert $G$-module on $L^2(G,\mathcal{H})$. It follows that $$\begin{cases}
G \times \mathcal{B}(L^2(G,\mathcal{H})) & \longrightarrow \mathcal{B}(L^2(G,\mathcal{H})) \\
(g,f) & \longmapsto \left[x \mapsto g.f(g^{-1}.x) \right]
\end{cases}$$ defines an action of the group $G$ on the space $\mathcal{B}(L^2(G,\mathcal{H}))$ of bounded operators on $L^2(G,\mathcal{H})$, but this action is not continuous *a priori* (if $G$ is non-discrete and $\mathcal{H} \neq \{0\}$, it may actually be shown that this action is non-continuous).
For every $g \in G$ and $f \in \mathcal{B}(L^2(G,\mathcal{H}))$, notice that $${\operatorname{Ker}}(g.f)=g.{\operatorname{Ker}}(f) \quad \text{and} \quad {\operatorname{Im}}(g.f)=g.{\operatorname{Im}}(f).$$ The previous action thus induces an action of the group $G$ on the space ${\operatorname{Fred}}(L^2(G,\mathcal{H}))$ of Fredholm operators on $L^2(G,\mathcal{H})$ (again, a non-continuous action *a priori*). Denote by $\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}))$ the subset of ${\operatorname{Fred}}(L^2(G,\mathcal{H}))$ consisting of the operators $F$ such that $g \mapsto g.F$ is a continuous map. Since $F \mapsto g.F$ is an isometry of ${\operatorname{Fred}}(L^2(G,\mathcal{H}))$ for every $g \in G$, it may easily be shown that $\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}))$ is a $G$-space.
The simplicial $G$-space $({\operatorname{Fred}}(G,\mathcal{H})[n])_{n \in \mathbb{N}}$ {#10.3.2}
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Let $g\in G$ and $(f,f')\in \mathcal{B}(L^2(G,\mathcal{H}))$. Then $g.(f\circ f')=(g.f)\circ (g.f')$. It follows that the composition of operators in $\mathcal{B}(L^2(G,\mathcal{H}))$ induces a (continuous) composition law in $\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}))$. Therefore, $\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}))$ has a structure of (non-abelian) topological monoid for this composition law, and $G$ acts by morphisms on $\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}))$ .
We then set $$({\operatorname{Fred}}(G,\mathcal{H})[n])_{n \in \mathbb{N}}:=\mathcal{N}\left(\mathcal{B}
\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}))\right),$$ where the nerve is constructed in the category of k-spaces.
Thus, for every $n \in \mathbb{N}$, we have $${\operatorname{Fred}}(G,\mathcal{H})[n]=\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}))^n,$$ with the usual face and degeneracy maps: this is obviously a k-space.
\[fredweak\] The canonical map $$\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H})) \longrightarrow
\Omega B{\operatorname{Fred}}(G,\mathcal{H})$$ is a $G$-weak equivalence.
Let $H$ denote a compact subgroup of $G$. For every $g \in G$ and $f \in \mathcal{B}(L^2(G,\mathcal{H}))$, one has $(g.f)^*=g.f^*$. It follows that the adjunction map induces a continuous self isometry on $\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}))$. The maps $$h_1: (t,f) \longmapsto t.{\operatorname{id}}_{L^2(G,\mathcal{H})}+(1-t)f\circ f^* \quad \text{and} \quad
h_2: (t,f) \longmapsto t.{\operatorname{id}}_{L^2(G,\mathcal{H})}+(1-t)f^*\circ f.$$ then yield equivariant homotopies respectively from ${\operatorname{id}}_{\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}))}$ to $f\circ f^*$, and from ${\operatorname{id}}_{\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}))}$ to $f^*\circ f$. We deduce that the topological monoid $\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}))^H$ has an inverse up to homotopy.
Since ${\operatorname{Fred}}(G,\mathcal{H})^H$ is a simplicial k-space such that $$\forall n \in \mathbb{N}, \; {\operatorname{Fred}}(G,\mathcal{H})[n]^H
\overset{\cong}{\longrightarrow} \left({\operatorname{Fred}}(G,\mathcal{H})[1]^H\right)^n,$$ and ${\operatorname{Fred}}(G,\mathcal{H})[1]^H$ is an H-space with an inverse up to homotopy, we deduce from proposition 1.5 of [@Segal-cat] that the canonical map ${\operatorname{Fred}}(G,\mathcal{H})[1]^H \longrightarrow \Omega B{\operatorname{Fred}}(G,\mathcal{H})^H$ is a homotopy equivalence. Hence $\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H})) \longrightarrow \Omega B{\operatorname{Fred}}(G,\mathcal{H})$ is a $G$-weak equivalence.
From finite dimensional subspaces of $L^2(G,\mathcal{H})$ to Fredholm operators on $L^2(G,\mathcal{H}^\infty)$: the shift map {#10.3.3}
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Recall that when $\mathcal{H}$ is a Hilbert space, $\mathcal{H}^\infty$ is defined as the Hilbert space completion of the inner product space $\mathcal{H}^{(\infty)}$. Recall also the canonical isomorphism $L^2(G,\mathcal{H}^\infty) \cong L^2(G,\mathcal{H})^\infty$.
#####
We will now let $H$ denote a Hilbert $G$-module. The case of interest is $H=L^2(G,\mathcal{H})$.
We write $H^\infty=\overline{\underset{i\in \mathbb{N}}{\oplus} (H\times \{i\})}$. We define the shift operator on $H^\infty$ as $$S_H : (x_i,i)_{i\in \mathbb{N}} \longmapsto (x_{i+1},i)_{i\in \mathbb{N}.}$$ Clearly, $S_H$ is a bounded linear operator of norm $1$, it is onto and its kernel is $H \times \{0\}$.\
Let $V$ be a closed linear subspace of $H$. Then $H^\infty=V^\infty \overset{\bot}{\oplus} (V^\bot)^\infty$, and we can define ${\operatorname{Sh}}(V)$ as the linear operator on $H^\infty$ such that $$\begin{cases}
\forall x \in V^\infty, & {\operatorname{Sh}}(V)[x]=S_V(x) \\
\forall x \in (V^\bot)^\infty, & {\operatorname{Sh}}(V)[x]=x
\end{cases}$$ i.e. ${\operatorname{Sh}}(V)$ should be thought of as the shift alongside $V^\infty$ in $H^\infty$. Obviously, ${\operatorname{Sh}}(V)$ is a bounded operator of norm $1$. Also, ${\operatorname{Sh}}(V)$ is onto and ${\operatorname{Ker}}({\operatorname{Sh}}(V))=V \times \{0\}$. We have thus defined a map $${\operatorname{Sh}}: \begin{cases}
{\operatorname{sub}}(H) & \longrightarrow \mathcal{B}(H^\infty) \\
V & \longmapsto {\operatorname{Sh}}(V).
\end{cases}$$ The following properties are then classical and easily checked:
\[propertiesofshift\] ${}$\
(i) The map ${\operatorname{Sh}}$ is a $G$-map.
(ii) For every pair $(V,V')\in {\operatorname{sub}}(H)^2$ such that $V \bot V'$, $${\operatorname{Sh}}(V \oplus V')={\operatorname{Sh}}(V) \circ {\operatorname{Sh}}(V')={\operatorname{Sh}}(V') \circ {\operatorname{Sh}}(V).$$
The map ${\operatorname{Sh}}$ induces a $G$-map $${\operatorname{Sh}}: \underset{n\in \mathbb{N}}{\bigcup}{\operatorname{sub}}_n(H) \longrightarrow \underline{{\operatorname{Fred}}}(H^\infty).$$
Let $V \in \underset{n\in \mathbb{N}}{\bigcup}{\operatorname{sub}}_n(H)$. Then ${\operatorname{Sh}}(V)$ is surjective and its kernel is $V \times \{0\}$, which is a finite-dimensional space. It follows that ${\operatorname{Sh}}(V)$ is a Fredholm operator on $H^\infty$. Hence ${\operatorname{Sh}}$ induces a continuous equivariant map $\underset{n\in \mathbb{N}}{\bigcup}{\operatorname{sub}}_n(H) \longrightarrow {\operatorname{Fred}}(H^\infty)$. However $\underset{n\in \mathbb{N}}{\bigcup}{\operatorname{sub}}_n(H)$ is a $G$-space, hence, for every $V \in \underset{n\in \mathbb{N}}{\bigcup}{\operatorname{sub}}_n(H)$, one recovers that $g \mapsto g.{\operatorname{Sh}}(V)$ is continuous on $G$, which shows that ${\operatorname{Sh}}$ maps $\underset{n\in \mathbb{N}}{\bigcup}{\operatorname{sub}}_n(H)$ into $\underline{{\operatorname{Fred}}}(H^\infty)$.
In particular, for every separable Hilbert space $\mathcal{H}$, we have constructed a $G$-map $${\operatorname{Sh}}: \underset{n\in \mathbb{N}}{\bigcup}{\operatorname{sub}}_n(L^2(G,\mathcal{H})) \longrightarrow \underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}^\infty)).$$
Construction of a morphism $sK_{G,L^2}^{F,\infty} \rightarrow \underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H})^\infty)$
===========================================================================================================================
Main ideas {#10.4}
----------
The purpose of this short section is to help motivate the very technical constructions that the reader will have to face in our construction of a “good" morphism $sK_{G,L^2}^{F,\infty} \rightarrow \underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}^\infty))$ in the category $CG_G^{h\bullet}[W_G^{-1}]$ (where $\mathcal{H}$ is a separable Hilbert space which contains $F^{(\infty)}$, to be defined later on).
First of all, we know that the canonical maps $\Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,\infty*} \rightarrow sKF_G^{[\infty]}$ and $\underline{{\operatorname{Fred}}}(G,\mathcal{H}^\infty) \longrightarrow
\Omega B{\operatorname{Fred}}(G,\mathcal{H}^\infty)$ are $G$-weak equivalences, so all we need is “good" $G$-map from $\Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,\infty*}$ to $\Omega B{\operatorname{Fred}}(G,\mathcal{H}^\infty)$. Such a map will of course be obtained by constructing a morphism of hemi-simplicial $G$-spaces from $s\underline{{\operatorname{Vec}}}_{G,L^2}^{F,\infty*}$ to $({\operatorname{Fred}}(G,\mathcal{H}^\infty)[n])_{n \in \mathbb{N}}$, i.e. a collection of $G$-maps $s{\operatorname{Vec}}_{G,L^2}^{F,\infty*}(\mathbf{n}) \longrightarrow \underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}^\infty))^n$ which are compatible with the face maps[^3]. Assume that a $G$-map $$\alpha_G : s{\operatorname{Vec}}_{G,L^2}^{F,\infty*}(\mathbf{1}) \longrightarrow \underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}^\infty))$$ has been obtained at the level $1$. For every $n \in \mathbb{N}$, we have a canonical $G$-map $s\underline{{\operatorname{Vec}}}_{G,L^2}^{F,\infty}(\mathbf{n})
\longrightarrow (s\underline{{\operatorname{Vec}}}_{G,L^2}^{F,\infty}(\mathbf{1}))^n$. Composing it with $\alpha_G^n : (s\underline{{\operatorname{Vec}}}_{G,L^2}^{F,\infty}(\mathbf{1}))^n \longrightarrow
(\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}^\infty)))^n$ yields a $G$-map $$s\underline{{\operatorname{Vec}}}_{G,L^2}^{F,\infty}(\mathbf{n}) \longrightarrow
(\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}^\infty)))^n.$$ Elementary considerations on simplicial sets show that those maps yield a morphism of hemi-simplicial $G$-spaces if and only if the diagram $$\label{comdiagcond}
\xymatrix{
s\underline{{\operatorname{Vec}}}_{G,L^2}^{F,\infty}(\mathbf{2}) \ar[d]_{d_1^2} \ar[r] &
(s{\operatorname{Vec}}_{G,L^2}^{F,\infty})^2 \ar[r]^(0.45){(\alpha_G)^2} & (\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H})^\infty))^2
\ar[d]^{\circ} \\
s{\operatorname{Vec}}_{G,L^2}^{F,\infty} \ar[rr]^{\alpha_G} & & \underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H})^\infty)
}$$ is commutative.
Let us now see how $\alpha_G$ should be constructed. Let $\varphi:=\text{Fib}^{F^{(\infty)}*}(\mathbf{1})$. Recall that the objects of the category $\varphi {\operatorname{\text{-}smod}}$ are the pairs consisting of a finite *interval* $A$ of $\mathbb{N}$ (the “label”), and a $\#A$-dimensional subspace of $(F^{(\infty)})^A$, which may of course be seen as a subspace of $\mathcal{H}:=(F^{(\infty)})^{(\mathbb{N})}$. Let $\mathbf{F} : \mathcal{E}G \longrightarrow \text{Fib}^{F^{(\infty)}}(\mathbf{1}) {\operatorname{\text{-}smod}}$ be a square integrable continuous functor, i.e. an object of ${\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\text{Fib}^{F^{(\infty)}}(\mathbf{1}){\operatorname{\text{-}smod}})$. If we choose a basis $(e_1,\dots,e_n)$ of $\mathbf{F}(1_G)$, then the maps $\begin{cases}
G & \longrightarrow (F^{(\infty)})^{(\mathbb{N})} \\
g & \longmapsto \mathbf{F}(1_G,g)[e_i]
\end{cases}$, for $i \in \{1,\dots,n\}$, define an $n$-tuple of linearly independent elements of $L^2(G,\mathcal{H})$. In turn, this $n$-tuple defines an $n$-dimensional subspace of $L^2(G,\mathcal{H})$ which does not depend on the choice of $(e_1,\dots,e_n)$, and this subspace yields a Fredholm operator on $L^2(G,\mathcal{H})^\infty$ by the usual shift construction (cf. Section \[10.3.3\]). This procedure defines a map from the set of $0$-simplices in $s{\operatorname{Vec}}_{G,L^2}^{F,\infty*}=\big|{\operatorname{Func}}(\mathcal{E}G,\text{Fib}^{F^{(\infty)}}(\mathbf{1}){\operatorname{\text{-}smod}})\big|$ to the $G$-space $\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H})^\infty)$.
So far, we have associated to every $0$-simplex in the geometric realization $s{\operatorname{Vec}}_{G,L^2}^{F,\infty*}$ a finite-dimensional subspace of $L^2(G,\mathcal{H})$, which itself defines a Fredholm operator on $L^2(G,\mathcal{H})^\infty$. We now want to build a map: $${\operatorname{Hom}}\left({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\text{Fib}^{F^{(\infty)}}(\mathbf{1}) {\operatorname{\text{-}smod}})\right)
\times [0,1] \longrightarrow \underset{n=0}{\overset{\infty}{\bigcup}}{\operatorname{sub}}_n(L^2(G,\mathcal{H}))$$ which is compatible with the preceding one. Assume first that $G$ is trivial. Let $x$ be a morphism of ${\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}\{1\},\text{Fib}^{F^{(\infty)}}(\mathbf{1}) {\operatorname{\text{-}smod}})=\text{Fib}^{F^{(\infty)}}(\mathbf{1}) {\operatorname{\text{-}smod}}$. Then $x$ is a $5$-tuple consisting of two finite intervals $A$ and $B$ of $\mathbb{N}$, a $\#A$-dimensional subspace $x_0$ of $(F^{(\infty)})^A$, a $\#B$-dimensional subspace $x_1$ of $(F^{(\infty)})^B$, and a similarity $\varphi : x_0 \overset{\cong}{\rightarrow} x_1$. We set $n:=\#A=\#B$.
Let $\tilde{x}$ be a morphism of $\varphi_n {\operatorname{\text{-}sframe}}$ in the fiber of $x$. Then $\tilde{x}=(\mathbf{B}_0,\mathbf{B}_1)$, where $\mathbf{B}_0$ is a simi-orthonormal basis of $x_0$, and $\mathbf{B}_1$ is a simi-orthonormal basis of $x_1$. We may write $\mathbf{B}_0=\lambda_0.\mathbf{B}'_0$ and $\mathbf{B}_1=\lambda_1.\mathbf{B}'_1$, where $(\lambda_0,\lambda_1) \in (\mathbb{R}_+^*)^2$, $\mathbf{B}'_0$ is an orthonormal basis of $x_0$, and $\mathbf{B}'_1$ is an orthonormal basis of $x_1$. The basic idea is to create a path from $\mathbf{B}'_0$ to $\mathbf{B}'_1$ in the space of orthonormal $n$-tuples of elements of $\mathcal{H}$.
Once we have a such a path, we may multiply it with the affine path from $\lambda_0$ to $\lambda_1$ in $\R_+^*$ in order to recover a path from $\mathbf{B}_0$ to $\mathbf{B}_1$ in the space of simi-orthonormal $n$-tuples of elements of $\mathcal{H}$. This path would yield a continuous map $\{x\} \times [0,1] \longrightarrow {\operatorname{sub}}_n(\mathcal{H})$. Of course, this map should not depend on the choice of $\tilde{x}$ in the fiber of $x$. It should also be constant in the case $A=B$ and $\mathbf{B}_0'=\mathbf{B}_1'$ (since we need compatibility with the degeneracy maps), it should remain in $(F^{(\infty)})^A$ when $A=B$, and it should be continuous with respect to $x$.
The construction is no more complicated in the case $G$ is non trivial. In this case, a morphism of ${\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}})$ is a family of morphisms of $\varphi_n {\operatorname{\text{-}sframe}}$ indexed over $G$: each of these morphisms will yield a path in the space of simi-orthonormal $n$-tuples of elements of $\mathcal{H}$: we recover a family indexed over $G$ of paths in the space of simi-orthonormal $n$-tuples of elements of $\mathcal{H}$; this can be seen as a path in the space of maps from $G$ to the space of simi-orthonormal $n$-tuples of elements of $\mathcal{H}$. Finally, a map from $G$ to the space of simi-orthonormal $n$-tuples of elements of $\mathcal{H}$ defines, under certain conditions, an $n$-dimensional subspace of $L^2(G,\mathcal{H})$.
With similar ideas, it is quite easy to see how one can expect to construct a reasonable map $$\mathcal{N}\left(
{\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\text{Fib}^{F^{(\infty)}}(\mathbf{1}) {\operatorname{\text{-}smod}})\right)_m
\times \Delta^m \longrightarrow \underset{n=0}{\overset{\infty}{\bigcup}}{\operatorname{sub}}_n(L^2(G,\mathcal{H}))$$ for every $m \in \mathbb{N}$.
Assume now that we have a $G$-map $s{\operatorname{Vec}}_{G,L^2}^{F,\infty *} \longrightarrow
\underset{n\in \mathbb{N}}{\bigcup}{\operatorname{sub}}_n(L^2(G,\mathcal{H}))$ which extends the previous construction already done on the $0$-simplices, and sketched on the $1$-skeleton of $s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}$. By composition with the shift map ${\operatorname{Sh}}: \underset{n\in \mathbb{N}}{\bigcup}{\operatorname{sub}}_n(L^2(G,\mathcal{H})) \longrightarrow
\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H})^\infty)$ (cf. Section \[10.3\]), we recover the $G$-map $\alpha_G : s{\operatorname{Vec}}_{G,L^2}^{F,\infty *} \longrightarrow \underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H})^\infty)$ we were looking for.
Let us now look at the condition on $\alpha_G$ imposed by the commutativity of diagram . Assume $G$ is trivial for sake of simplicity. Let $(n,m) \in \mathbb{N}^2$ and $(x_1,x_2,y_1,y_2)\in (F^{(\infty)})^{\{n\}} \times (F^{(\infty)})^{\{n+1\}} \times
(F^{(\infty)})^{\{m\}} \times (F^{(\infty)})^{\{m+1\}}$ be a $4$-tuple of unit vectors. By the previous construction, we have a path from $(n,x_1)$ to $(m,y_1)$ and from $(n+1,x_2)$ to $(m+1,y_2)$. For the previous square to be commutative, it is necessary that the path from $(\{n,n+1\},(x_1,x_2))$ to $(\{m,m+1\},(y_1,y_2))$ in the previous construction be precisely obtained by juxtaposing the two previous paths. In particular, this means that the two paths should be “orthogonal” at every step. Even taking those conditions as granted, the preceding requirement will not be fulfilled by every choice of paths. The key idea now is that, when $m>n$, we create a new inner product space, labeled $\mathcal{H}_{n,m}$, which is isomorphic to $(F^{(\infty)})^{\{n\}} \oplus (F^{(\infty)})^{\{m\}}$, and we choose an isomorphism $\varphi : (F^{(\infty)})^{\{n\}} \oplus (F^{(\infty)})^{\{m\}} \overset{\cong}{\longrightarrow}
\mathcal{H}_{n,m}$. Instead of creating a path from $x_1$ to $y_1$ in $(F^{(\infty)})^{\{n,m\}}$, we create a path in the orthogonal direct sum $(F^{(\infty)})^{\{n,m\}} \overset{\bot}{\oplus} \mathcal{H}_{n,m}$ as follows:
- First of all, we use a rotation to go from $x_1$ to $\varphi(x_1)$ in the orthogonal direct sum $(F^{(\infty)})^{\{n\}} \overset{\bot}{\oplus} \mathcal{H}_{n,m}$.
- Then we go from $\varphi(x_1)$ to $\varphi(y_1)$ in $\mathcal{H}_{n,m}$.
- Finally, we go from $\varphi(y_1)$ to $y_1$ by a rotation in the orthogonal direct sum $(F^{(\infty)})^{\{m\}} \overset{\bot}{\oplus} \mathcal{H}_{n,m}$.
This will solve the problem of orthogonality. Of course, by doing so, we need to define $\mathcal{H}$ as much larger than $(F^{(\infty)})^{(\N)}$: this will be done in Section \[10.4.3\].
Finally let $n<m$ be non-negative integers and $x \in (F^{(\infty)})^{\{n\}}$, $y \in (F^{(\infty)})^{\{m\}}$ and $z \in (F^{(\infty)})^{\{m\}}$ be unit vectors. Assume we already have constructed a path $f$ from $y$ to $z$ in $(F^{(\infty)})^{\{m\}}$, and, for every $y \in (F^{(\infty)})^{\{m\}}$, a path from $x$ to $y'$. Then for every $t \in [0,1]$, we have a path from $x$ to $f(t)$ in the orthogonal direct sum $(F^{(\infty)})^{\{n\}} \overset{\bot}{\oplus} \mathcal{H}_{n,m}$, and those paths may be used directly to create a map from $\Delta^2$ to the orthogonal direct sum $(F^{(\infty)})^{\{n\}} \overset{\bot}{\oplus} \mathcal{H}_{n,m}$, using the “associativity of barycenters” in the $2$-simplex (cf. next figure).
**Structure of the rest of the section:**\
In Section \[10.4.1\], we develop an elementary construction of subdivisions of the $n$-simplex that generalizes the subdivision of the interval into three parts which is the basis of the path construction that we have just discussed. Section \[10.4.2\] is devoted to the combinatorial material that we need in order to generalize the “associativity of barycenters” argument which was mentioned earlier. In Section \[10.4.3\], we turn to the definition of the inner-product space $\mathcal{H}$ we will work with. The detailed construction of “paths” is the topic of Section \[10.4.4\], in which we formulate the conjecture that a certain construction of paths exists. It is then proved in Section \[10.4.5\] that this conjecture is a consequence of a simple conjecture on the triangulation of smooth manifolds (see Section \[conjecturesection\]). Since the construction involves choices of extensions, we also use the conjecture on smooth manifolds to establish that two possible constructions will necessarily be “homotopic”. In Section \[10.4.6\], we use the previous construction to obtain a $G$-map $s{\operatorname{Vec}}_{G,L^2}^{F,\infty*}[1] \longrightarrow \underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H})^\infty)$, and we finally recover a morphism $s{\operatorname{Vec}}_{G,L^2}^{F,\infty*} \longrightarrow \underline{{\operatorname{Fred}}}(G,\mathcal{H}^\infty)$ in Section \[10.4.7\], as explained earlier.
Additional definitions {#10.4.0}
----------------------
Recall that when $(A_n)_{n \in \mathbb{N}}$ is a simplicial set, an element $x \in A_n$ is called **degenerate** when $x \in \sigma_*(A_k)$ for some $\sigma : [n] \twoheadrightarrow [k]$ with $k<n$.
For every $n\in \mathbb{N}$ and every $x \in A_n$, there is a unique pair $(\sigma,y)$ such that $\sigma : [n] \twoheadrightarrow [k]$ is an epimorphism of $\Delta$, the element $y$ is a non-degenerate element of $A_k$, and $x=\sigma_*(y)$. We will call $y$ the **root** of $x$, and $\sigma$ its **reduction**.
For every $n \in \mathbb{N}$, we denote by ${\operatorname{Hom}}_\uparrow([n],\mathbb{N})$ the set of non-decreasing maps from $[n]$ to $\mathbb{N}$. The structure of cosimplicial set on $\Delta$ thus induces a simplicial set ${\operatorname{Hom}}_\uparrow(\Delta,\mathbb{N})$ so that, for every morphism $\tau
: [k] \rightarrow [n]$ in $\Delta$, $$\tau_*: \begin{cases}
{\operatorname{Hom}}_\uparrow([n],\mathbb{N}) & \longrightarrow {\operatorname{Hom}}_\uparrow([k],\mathbb{N}) \\
f & \longmapsto f \circ \tau.
\end{cases}$$ Of course, $f\in {\operatorname{Hom}}_\uparrow([n],\mathbb{N})$ is non-degenerate if and only if it is an increasing map. Finally, for every $N \in \mathbb{N}$ and $f\in {\operatorname{Hom}}_\uparrow([n],\mathbb{N})$, we set $f+N : \begin{cases}
[n] & \longrightarrow \mathbb{N} \\
i & \longmapsto f(i)+N.
\end{cases}$
Subdivisions of the $n$-simplex {#10.4.1}
-------------------------------
### The relation $<$
Set $$E:=\underset{(k,n)\in \mathbb{N}^2}{\coprod}{\operatorname{Hom}}_{\Delta^*}([k],[n]).$$ We define a binary relation $<$ on $E$ as follows: for every $\delta: [k] \hookrightarrow [n]$ and $\delta' : [k'] \hookrightarrow [n']$, $$\delta< \delta' \; \underset{\text{def}}{\Longleftrightarrow} \;
\begin{cases}
k<n \\
n=k'
\end{cases}$$ (i.e. $\delta<\delta'$ if and only if $\delta' \circ \delta$ exists and $\delta$ is not bijective).\
A finite sequence $(\delta_1,\delta_2,\dots,\delta_m)$ in $E$ is called **increasing** when
- $\forall i \in \{1,\dots,m-1\},\; \delta_i <\delta_{i+1}$;
- $\delta_m={\operatorname{id}}_{[n]}$ for some $n \in \mathbb{N}$.
If $u$ is an increasing sequence, we simply write $u=\delta_1<\delta_2<\dots <\delta_{m-1}<[n]$ if $\delta_m={\operatorname{id}}_{[n]}$, we will say that $u$ ends at $[n]$, and we will call $m$ the **length** of $u$. We say that $u$ is **trivial** when its length is $1$. Finally, we define $${\operatorname{Lat}}(u):=
\bigl\{\delta_m \circ \delta_{m-1} \circ \dots \circ \delta_i \mid i \in \{1,\dots,m\} \bigr\}.$$ Let $u=\delta_1<\dots<\delta_{m-1}<[n]$ be a non trivial increasing sequence. We define the **differential** of $u$ as $$u':=\delta_1<\dots<\delta_{m-2}<{\operatorname{id}}_{[k]},$$ where $[k]$ is the domain of $\delta_{m-1}$. More generally, we define, for every $i \in \{1,\dots,m-1\}$, the sequence $$u^{(i)}:=\delta_1<\dots<\delta_{m-i-1}<{\operatorname{id}}_{[k_i]},$$ where $[k_i]$ is the domain of $\delta_{m-i}$. Obviously, ${\operatorname{Lat}}(u)=\{{\operatorname{id}}_{[n]}\} \cup \bigl\{\delta_{m-1} \circ \delta\; |\; \delta \in {\operatorname{Lat}}(u')\bigr\}$.\
For every $n \in \mathbb{N}$, we denote by $S(n)$ the set of increasing sequences in $E$ which end at $[n]$, and we define an order relation $\subset$ on $S(n)$ by $$\forall (u,v)\in S(n)^2, \quad u \subset v \; \underset{\text{def}}{\Longleftrightarrow} \; {\operatorname{Lat}}(u) \subset {\operatorname{Lat}}(v).$$ Finally, for every $n \in \mathbb{N}$ and every pair $(u,v)\in S(n)^2$, we define $$S_{u,v}:=\bigl\{w \in S(n) : \quad w \subset u \quad \text{and} \quad w \subset v\bigr\}.$$
### Subdivisions of the $n$-simplex {#core}
For $n \in \mathbb{N}$, we consider the $n$-simplex $$\Delta^n=\bigl\{(t_0,\dots,t_n) \in (\mathbb{R}_+)^{n+1}:t_0+t_1+\dots+t_n=1\bigr\} \subset
\mathbb{R}^{n+1}.$$
For any $i \in \{0,\dots,n\}$, we set $$\Delta_i^n:=\left\{(t_0,\dots,t_n) \in \Delta^n: \quad
t_i\leq \frac{1}{2(n+1)} \quad \text{and} \quad \forall
j \in \{0,\dots,n\}, \; t_i \leq t_j\right\}$$ and $$C(\Delta^n):=\left\{(t_0,\dots,t_n) \in \Delta^n: \quad
\forall i \in \{0,\dots,n\}, \; t_i \geq \frac{1}{2(n+1)}\right\}.$$
Notice the homeomorphism $$\alpha^n : \begin{cases}
C(\Delta^n) & \overset{\cong}{\longrightarrow} \Delta^n \\
(t_0,\dots,t_n) & \longmapsto \left(2(t_0-\frac{1}{2(n+1)}),
2(t_1-\frac{1}{2(n+1)}),\dots, 2(t_n-\frac{1}{2(n+1)})\right).
\end{cases}$$ Furthermore, for every $i\in \{0,\dots,n\}$, we define a homeomorphism $$\alpha_i^n : \begin{cases}
\Delta_i^n & \overset{\cong}{\longrightarrow} \Delta^{n-1} \times I \\
(t_0,\dots,t_n) & \longmapsto \left[(\frac{t_0}{1-t_i},\dots,\frac{t_{i-1}}{1-t_i}
,\frac{t_{i+1}}{1-t_i},\dots\frac{t_n}{1-t_i}), 2(n+1)t_i \right].
\end{cases}$$ Gluing the maps $(\alpha_0^n)^{-1},(\alpha_1^n)^{-1},\dots,(\alpha_n^n)^{-1}$ together yields a homeomorphism $$r_n : \partial \Delta^n \times [0,1] \overset{\cong}{\longrightarrow}
\underset{0\leq i\leq n}{\bigcup}\Delta_i^n.$$
\[deltau\] The following recursive definition yields, for every $n \in \mathbb{N}$ and every increasing sequence $u=\delta_1<\dots<\delta_{m-1}<n$ in $S(n)$, a compact subset $\Delta_u$ of $\Delta^n$:
- If $u$ is trivial, then $\Delta_u:=C(\Delta^n)$.
- Otherwise, $\Delta_u:=r_n(\delta_{m-1}^*(\Delta_{u'}) \times [0,1])$.
For every $n \in \mathbb{N}$, we obtain a subdivision of the $n$-simplex in this manner.
**Example :** There are thirteen increasing sequences that end at $[2]$. They are: $$\begin{aligned}
[2], & \quad u_1:=(0,1) <[2], & \quad u_2:=(0,2) <[2], \\
u_3:=(1,2) <[2] & \quad u_4:= 0<(0,1) <[2], & \quad u_5:=1<(0,1) <[2], \\
u_6:= 0<(0,2) <[2], & \quad u_7:= 1<(0,2) <[2], & \quad u_8:=0<(1,2) <[2], \\
u_9:=1<(1,2) <[2], & \quad u_{10}:= 0<[2], & \quad u_{11}:=1 <[2], \quad u_{12}:=2<[2].\end{aligned}$$ The corresponding subdivision of $\Delta^2$ is detailed in the following figure:
### Some properties of the subdivisions
In the rest of the construction, we will need the following two results:
\[glue\] Let $n \in \mathbb{N}$. Then:
- $\underset{u \in S(n)}{\bigcup}\Delta_u=\Delta^n$;
- $\underset{u \in S(n){\smallsetminus}\{[n]\}}{\bigcup}\Delta_u=\underset{0\leq i \leq N}{\bigcup}\Delta_i^n$;
- the space $\underset{0\leq i \leq N}{\bigcup}\Delta_i^n$ has the final topology for the inclusions $\Delta_u \subset
\underset{0\leq i \leq N}{\cup}\Delta_i^n$, with non-trivial $u\in S(n)$.
\[intersection\] Let $n \in \mathbb{N}$, and $(u,v)\in S(n)^2$ be a pair of non-trivial sequences. Then $$\Delta_u \cap \Delta_v \subset \underset {w \in S_{u,v}{\smallsetminus}\{[n]\}}{\bigcup}\Delta_w.$$
We prove (i) and (ii) by induction on $n$. When $n=0$, $\Delta^0=C(\Delta^0)=\Delta_{[0]}$. Let $n \in \mathbb{N}^*$, and assume the result is true for $n-1$. It suffices to prove that $\Delta^n \subset \underset{u \in S(n)}{\bigcup}\Delta_u$ and $\underset{0\leq i \leq N}{\bigcup}\Delta_i^n \subset \underset{u \in S(n){\smallsetminus}\{[n]\}}{\bigcup}\Delta_u$.\
Let $x \in \Delta^n$. If $x \in C(\Delta^n)$, then $x \in \Delta_{[n]}$. Otherwise, we choose $i \in [n]$ such that $x \in \Delta_i^n$, and we set $(y,t):=r_n^{-1}(x)$, and $z \in \Delta^{n-1}$ such that $y=(\delta_i^n)^*(z)$. Then, by the induction hypothesis, there is some $u \in S(n-1)$ such that $z \in \Delta^{n-1}$, and we write $u=\delta_1<\dots<\delta_{m-1}<[n-1]$. We then set $v:=\delta_1<\dots<\delta_{m-1}<\delta_i^n<[n]$. Then $v$ is non trivial, $v \in S(n)$, $v'=u$, and we deduce that $\Delta_v=r_n((\delta_i^n)^*(\Delta_u)\times [0,1])$. Since $x=r_n((\delta_i^m)^*(z),t))$ and $z \in \Delta_u$, it follows that $x \in \Delta_v$.\
We conclude that $\Delta^n \subset \underset{u \in S(n)}{\bigcup} \Delta_u$ and $\underset{0\leq i \leq N}{\bigcup}\Delta_i^n \subset \underset{u \in S(n){\smallsetminus}\{[n]\}}{\bigcup} \Delta_u$, and this proves (i) and (ii) by induction on $n$.
Finally, let $n \in \mathbb{N}$. Since $S(n)$ is finite, $\underset{u \in S(n){\smallsetminus}\{[n]\}}{\coprod}\Delta_u$ is a compact space. Also, $\underset{0\leq i \leq n}{\bigcup}\Delta_i^n$ is compact. Since the canonical map $\underset{u \in S(n){\smallsetminus}\{[n]\}}{\coprod}\Delta_u \longrightarrow \underset{0\leq i \leq n}{\cup}\Delta_i^n$ is continuous and onto, it follows that it is an identification map.
In order to prove Proposition \[intersection\], we will need the following technical lemma:
\[faceintersectionlemma\] Let $n \in \mathbb{N}$, $u \in S(f)$, and $\delta: [k] \hookrightarrow [n]$.\
If $\delta^*(\Delta^k) \cap \Delta_u \not\subset \delta^*(\partial \Delta^k)$, then $\delta \in {\operatorname{Lat}}(u)$ and there exists an $i \in \mathbb{N}$ such that $$\delta^*(\Delta^k) \cap \Delta_u=\delta^*(\Delta_{u^{(i)}}).$$
We will use the following simple fact of simplicial geometry:
for every $n \in \mathbb{N}$, $\delta : [k] \hookrightarrow [n]$ and $\delta' : [k'] \hookrightarrow [n]$, one has $\delta^*(\Delta^k) \cap (\delta')^*(\Delta^{k'}) \not \subset (\delta')^*(\partial \Delta^{k'})$ if and only if there is some $\delta'': [k'] \hookrightarrow [k]$ such that $\delta' =\delta \circ \delta''$.
Now, we prove the claimed result by induction on $n$. It is obvious when $n=0$.\
Let $n\in \mathbb{N}^*$, and assume the result holds for every $n'<n$, every $u \in S(n')$ and $\delta : [k'] \hookrightarrow [n']$.\
Let $u \in S(n)$ and $\delta : [k] \hookrightarrow [n]$. We assume that $\delta^*(\Delta^k) \cap \Delta_u \not\subset \delta^*(\partial \Delta^k)$. If $\delta={\operatorname{id}}_{[n]}$, then $\delta \in {\operatorname{Lat}}(u)$ and $\delta^*(\Delta^k) \cap \Delta_u=\Delta^n \cap \Delta_u=\Delta_u$.
Assume now that $k<n$. If $u$ is trivial, then $\partial \Delta^n \cap \Delta_u =\emptyset$, and this contradicts $\delta^*(\Delta^k) \cap \Delta_u \neq \emptyset$. Hence $u$ is non-trivial. We now write $u=\delta_1<\dots<\delta_{m-1}<n$, with $\delta_{m-1} : [n'] \hookrightarrow [n]$. By construction, $\Delta_u \cap \partial \Delta^n =\delta_{m-1}^*(\Delta_{u'})$. It follows that $\delta_{m-1}^*(\Delta^{n'}) \cap \delta^*(\Delta^k) \not\subset \delta^*(\partial\Delta^k)$, and we deduce that $\delta=\delta_{m-1} \circ \delta''$ for some $\delta'' : [k'] \hookrightarrow [n']$.
It follows that $$\begin{aligned}
\delta^*(\Delta^k) \cap \Delta_u & =\delta^*(\Delta^k) \cap (\partial \Delta^n \cap \Delta_u) \\
& =\delta^*(\Delta^k) \cap \delta_{m-1}^*(\Delta_{u'}) \\
\delta^*(\Delta^k) \cap \Delta_u & =\delta_{m-1}^*((\delta'')^*(\Delta^{k}) \cap \Delta_{u'}).\end{aligned}$$ Moreover $\delta^*(\Delta^k)=(\delta_{m-1})^*((\delta'')^*(\Delta^{k}))$, hence $(\delta'')^*(\Delta^k) \cap \Delta_{u'} \not\subset (\delta'')^*(\partial \Delta^k)$.
It follows from the induction hypothesis that $\delta''\in {\operatorname{Lat}}(u')$ and $(\delta'')^*(\Delta^k) \cap \Delta_{u'}=(\delta'')^*(\Delta_{(u')^{(i)}})$ for some integer $i$. Hence $\delta=\delta_{m-1} \circ \delta''\in {\operatorname{Lat}}(u)$, and $$\delta^*(\Delta^k) \cap \Delta_u=\delta_{m-1}^*((\delta'')^*(\Delta_{u^{(i+1)}}))=\delta^*(\Delta_{u^{(i+1)}}).$$ This proves lemma \[faceintersectionlemma\].
We prove the result by induction on $n$. For $n=0$, the result is obvious. Let $n\in \mathbb{N}^*$, and assume the result holds for every $n'<n$ and every pair $(u,v)\in S(n')^2$ of non-trivial sequences.
Let $(u,v)\in S(n)^2$ be a pair of non-trivial sequences. Let $x \in \Delta_u \cap \Delta_v$. Then $x \in \underset{i \in [n]}{\bigcup} \Delta_i^n$ and we may therefore set $(y,t):=(r_n)^{-1}(x)$. By construction of $\Delta_u$ and $\Delta_v$, we deduce that $y \in \Delta_u \cap \Delta_v$. Then $y \in \partial \Delta^n$ and there is a unique lattice $\delta : [k] \hookrightarrow [n]$, with $k<n$, such that $y \in \delta^*(\Delta^k {\smallsetminus}\partial \Delta^k)$. Since $y \in \Delta_u \cap \Delta_v$, we deduce that $\delta^*(\Delta^k) \cap \Delta_u \not\subset \delta^*(\partial \Delta^k)$ and $\delta^*(\Delta^k) \cap \Delta_v \not\subset \delta^*(\partial \Delta^k)$. It then follows from Lemma \[faceintersectionlemma\] that $\delta \in {\operatorname{Lat}}(u) \cap {\operatorname{Lat}}(v)$ and that there exists a pair $(i,j)$ of integers such that $\delta^*(\Delta^k) \cap \Delta_u=\delta^*(\Delta_{u^{(i)}})$ and $\delta^*(\Delta^k) \cap \Delta_v=\delta^*(\Delta_{v^{(j)}})$. Then $y \in \delta^*(\Delta_{u^{(i)}} \cap \Delta_{v^{(j)}})$.
Let now $z\in \Delta^k$ be such that $y=\delta^*(z)$. Hence $z \in \Delta_{u^{(i)}} \cap \Delta_{v^{(j)}}$. If $u^{(i)}$ or $v^{(j)}$ is trivial, then $w \in \Delta_{[k]}$. Otherwise, we deduce from the induction hypothesis that there exists $w \in S_{u^{(i)},v^{(j)}} {\smallsetminus}\{[k]\}$ such that $z \in \Delta_w$. In any case, we have $w \in S_{u^{(i)},v^{(j)}}$ such that $z \in \Delta_w$. We write $w=\delta_1<\dots<\delta_{m-1} < [k]$, and set $w_1:=\delta_1<\dots<\delta_{m-1} < \delta < [n]$. Then $$\begin{aligned}
{\operatorname{Lat}}(w_1) & =\{[n]\} \cup \{\delta \circ \delta', \delta' \in {\operatorname{Lat}}(w)\} \\
& \subset \{[n]\} \cup \{\delta \circ \delta', \delta' \in {\operatorname{Lat}}(u^{(i)})\} \\
{\operatorname{Lat}}(w_1)& \subset {\operatorname{Lat}}(u),\end{aligned}$$ and similarly ${\operatorname{Lat}}(w_1) \subset {\operatorname{Lat}}(v)$. We deduce that $w_1 \in S_{u,v}{\smallsetminus}\{[n]\}$.\
Finally, $w_1'=w$, and it follows that $x=r_n(\delta^*(z),t)) \in \Delta_w$.\
We conclude that $\Delta_u \cap \Delta_v \subset \underset{w \in S_{u,v}{\smallsetminus}\{[n]\}}{\bigcup}\Delta_w$.
Associativity of barycenters in the $n$-simplex {#10.4.2}
-----------------------------------------------
### Cartesian squares in the simplicial category
In the category $\Delta$, every pair of morphisms $(\sigma,\delta)$, with $\sigma : [n] \twoheadrightarrow [k]$, and $\delta : [k'] \hookrightarrow [k]$, gives rise to a cartesian square: $$\xymatrix{
[m] \ar@{^{(}->}[d]^{\sigma \natural \delta} \ar@{>>}[r]^{\delta \natural \sigma} & [k'] \ar@{^{(}->}[d]^\delta \\
[n] \ar@{>>}[r]^{\sigma} & [k]
}$$ With $\delta : [k'] \hookrightarrow [k]$ fixed, every commutative triangle $\xymatrix{
[n] \ar@{>>}[dr]^{\sigma} \ar[rr]^{\tau} & & [n'] \ar@{>>}[dl]^{\sigma'} \\
& [k]
}$ yields a commutative diagram $$\xymatrix{
[m]
\ar@{>>}[dr]^{\delta \natural \sigma} \ar[rr]^{\tau'}
\ar[drrrr]^(0.7){\sigma \natural \delta}
& & [m'] \ar@{>>}[dl]^{\delta \natural \sigma'}
\ar[drrrr]^{\sigma' \natural \delta} \\
& [k'] \ar@{^{(}->}[drrrr]_\delta & & & [n] \ar[rr]^\tau \ar[dr]_\sigma & & [n'] \ar[dl]^{\sigma'} \\
& & & & & [k]
}.$$ For a fixed epimorphism $\sigma : [n] \twoheadrightarrow [k]$, we obtain a functor $\sigma\natural :\; \Delta^* \downarrow [k] \longrightarrow \Delta^* \downarrow [n]$. For every $i \in [k]$, we also set $\sigma \natural i:=\sigma \natural \delta_i$, where $\delta_i : [0] \hookrightarrow [k]$ maps $0$ to $i$.
### The maps defined by the associativity of barycenters
Recall that $(\mathbb{R}^{[n]})_{n\in \mathbb{R}}$ is equipped with both a canonical structure of cosimplicial space and a canonical structure of simplicial set.
Let $\sigma : [n] \twoheadrightarrow [k]$ be an epimorphism in $\Delta$, and set $n_i:=\# \sigma^{-1}\{i\}$ for every $i\in [k]$. The canonical ring structure on $\mathbb{R}^{[n]}$ yields a continuous map $$\lambda_\sigma : \begin{cases}
\mathbb{R}^{[n]} \times \mathbb{R}^{[k]} & \longrightarrow \mathbb{R}^{[n]} \\
(x,t) & \longmapsto \sigma_*(t) \times x
\end{cases}$$ where $\times$ is the standard $(n+1)$-fold product on $\R^{[n]}$ (the set of functions from $[n]$ to $\R$). For every $i \in [k]$, we also set $$\lambda^{(i)}_\sigma : \begin{cases}
\mathbb{R}^{\sigma^{-1}\{i\}} \times \mathbb{R}^{\{i\}} & \longrightarrow \mathbb{R}^{\sigma^{-1}\{i\}} \\
(x,t) & \longmapsto t.x
\end{cases}$$ and notice that $\lambda_\sigma=\underset{i=0}{\overset{k}{\prod}}\lambda^{(i)}_\sigma$.
For every non-empty finite subset $I$ of $\mathbb{N}$, we denote by $\Delta(\mathbb{R}^I)$ the subset of $\mathbb{R}^I$ consisting of those families $(t_i)_{i \in I}$ such that $\underset{i \in I}{\sum} t_i=1$. The previous decomposition of $\lambda_\sigma$ helps us see that $\lambda_\sigma$ induces a continuous map $$\left(\underset{i=0}{\overset{k}{\prod}}\Delta\bigl(\mathbb{R}^{\sigma^{-1}\{i\}}\bigr)\right) \times \Delta^k
\longrightarrow \Delta^n,$$ which, composed with $$\left(\underset{i=0}{\overset{k}{\sum}}(\sigma \natural i)^*\right)\times {\operatorname{id}}:
\left(\underset{i=0}{\overset{k}{\prod}}\Delta^{n_i-1}\right) \times \Delta^k
\longrightarrow
\left(\underset{i=0}{\overset{k}{\prod}}\Delta(\mathbb{R}^{\sigma^{-1}\{i\}})\right) \times \Delta^k,$$ yields a map $$\left(\underset{i=0}{\overset{k}{\prod}}\Delta^{n_i-1}\right) \times \Delta^k
\longrightarrow \Delta^n$$ which we still write $\lambda_\sigma$.
Note that $\lambda_\sigma$ is onto. Indeed, let $t=(t_i)_{0 \leq i \leq n}$: set $t':=\sigma^*(t)$ and define $(x_i)_{0 \leq i \leq k}$ as: $x_i=(1,0,0,\dots,0)$ if $t'_i= 0$, and $x_i:=\left(\frac{t_{(\sigma \natural i)(j)}}{t_i}\right)_{j \in [n_i-1]}$ otherwise. It is then easily checked that $\lambda_\sigma(x,t')=t$.
If $k=0$ or $k=n$, then $\lambda_\sigma$ is the identity map. In any other case, it may easily be proven that $\lambda_\sigma$ is *not* one-to-one. This lack of injectivity is a problem: in the following paragraphs, we explain what needs to be done in order to construct a homeomorphism from $\lambda_\sigma$.
### The functors $F_\sigma$ and $H_\sigma$
By composing $\sigma \natural -$ with the forgetful functor $\Delta^* \downarrow [n] \longrightarrow \Delta$ and the functor defined by the canonical structure of simplicial space on $(\mathbb{R}^{[n]})_{n\in \mathbb{N}}$, we recover a contravariant functor $F_\sigma : \Delta^* \downarrow [k] \longrightarrow \text{Top}$.
By composing the forgetful functor $\Delta^* \downarrow [k] \longrightarrow \Delta$ with the canonical structure of cosimplicial space on $(\Delta^n)_{n\in \mathbb{N}}$, we recover a functor $H_\sigma : \Delta^* \downarrow [k] \longrightarrow \text{Top}$ which maps $\delta : [m] \hookrightarrow [n]$ to $\Delta^m$ and $\xymatrix{
[m] \ar@{^{(}->}[dr]^{\delta} \ar@{^{(}->}[rr]^{\delta''} & & [m'] \ar@{^{(}->}[dl]^{\delta'} \\
& [k]
}$ to $(\delta'')^*: \Delta^m \longrightarrow \Delta^{m'}$.
For every pair of morphisms $(\sigma,\delta)$, with $\sigma : [n] \twoheadrightarrow [k]$, $\delta : [k'] \hookrightarrow [k]$, and $\sigma \natural \delta : [m] \hookrightarrow [k']$, the square $$\begin{CD}
\mathbb{R}^{[k']} @>{(\delta \natural \sigma)_*}>> \mathbb{R}^{[m]} \\
@VV{\delta^*}V @VV{(\sigma \natural \delta)^*}V \\
\mathbb{R}^{[k]} @>{\sigma_*}>> \mathbb{R}^{[n]}
\end{CD}$$ is easily shown to be commutative.\
It follows that, for every morphism $\varphi=\xymatrix{
[m] \ar@{^{(}->}[dr]^{\delta} \ar@{^{(}->}[rr]^{\delta''} & & [m'] \ar@{^{(}->}[dl]^{\delta'} \\
& [k]
}$ in $\Delta^*\downarrow [k]$, the square $$\begin{CD}
F_\sigma(\delta') \times H_\sigma(\delta) @>>{{\operatorname{id}}\times H_\sigma(\varphi)}> F_\sigma(\delta') \times H_\sigma(\delta') \\
@VV{F_\sigma(\varphi)\times {\operatorname{id}}}V @VV{(\sigma \natural \delta')^* \circ \lambda_{\delta' \natural \sigma}}V \\
F_\sigma(\delta) \times H_\sigma(\delta) @>>{(\sigma \natural \delta)^* \circ \lambda_{\delta \natural \sigma}}> \Delta^n
\end{CD}$$ is commutative.
### The functor $G_\sigma$ {#gsigma}
Let $\sigma : [n] \twoheadrightarrow [k]$ be an epimorphism in $\Delta$. For every monomorphism $\delta : [m] \hookrightarrow [k]$, set $G_\sigma(\delta):= \underset{i=0}{\overset{m}{\prod}}\Delta^{n_{\delta(i)}-1}$ and $$\alpha_\sigma(\delta):=\underset{i=0}{\overset{m}{\sum}}((\delta \natural \sigma)\natural i)^*:
G_\sigma(\delta) \longrightarrow F_\sigma(\delta).$$ For every morphism $\varphi=\xymatrix{
[m] \ar@{^{(}->}[dr]^{\delta} \ar@{^{(}->}[rr]^{\delta''} & & [m'] \ar@{_{(}->}[dl]^{\delta'} \\
& [k]}$ in $\Delta^*\downarrow [k]$, set $$G_\sigma(\varphi): \begin{cases}
\underset{i=0}{\overset{m'}{\prod}}\Delta^{n_{\delta'(i)}-1}
& \longrightarrow \underset{j=0}{\overset{m}{\prod}}\Delta^{n_{\delta(j)}-1} \\
(x_i)_{0 \leq i \leq m'} & \longmapsto (x_{\delta''(j)})_{0 \leq j \leq m.}
\end{cases}$$ This yields a contravariant functor $G_\sigma : \Delta^*\downarrow [k] \longrightarrow \text{Top}$. It is then easy to check that $\alpha_\sigma : G_\sigma \longrightarrow F_\sigma$ is a natural transformation.\
We deduce that for every morphism $\varphi=\xymatrix{
[m] \ar@{^{(}->}[dr]^{\delta} \ar@{^{(}->}[rr]^{\delta''} & & [m'] \ar@{_{(}->}[dl]^{\delta'} \\
& [k]}$ in $\Delta^*\downarrow [k]$, the square $$\begin{CD}
G_\sigma(\delta') \times H_\sigma(\delta) @>>{{\operatorname{id}}\times H_\sigma(\varphi)}> G_\sigma(\delta') \times H_\sigma(\delta') \\
@VV{G_\sigma(\varphi)\times {\operatorname{id}}}V @VV{(\sigma \natural \delta')^* \circ \lambda_{\delta' \natural \sigma}}V \\
G_\sigma(\delta) \times H_\sigma(\delta) @>>{(\sigma \natural \delta)^* \circ \lambda_{\delta \natural \sigma}}> \Delta^n
\end{CD}$$ is commutative.
\[nusigma\] It follows that for every object $\delta : [m] \hookrightarrow [k]$ of $\Delta^* \downarrow [k]$, the maps $(\sigma \natural \delta)^* \circ \lambda_{\delta \natural \sigma} : G_\sigma(\delta)\times \Delta^m
\longrightarrow \Delta^n$ yield a continuous map $$\nu_\sigma : \left(\underset{ \delta \in {\operatorname{Ob}}(\Delta^* \downarrow [k])}{\coprod} G_\sigma(\delta) \times H_\sigma(\delta)\right)/_\sim
\longrightarrow \Delta^n,$$ where $\sim$ is the equivalence relation on $\underset{\delta \in {\operatorname{Ob}}(\Delta^* \downarrow [k])}{\coprod} G_\sigma(\delta) \times H_\sigma(\delta)$ generated by the collection of elementary relations $$\forall \varphi : \delta \rightarrow \delta', \; \forall (x,t) \in
G_\sigma(\delta') \times H_\sigma(\delta),\quad (x,\varphi^*(t)) \sim (\varphi_*(x),t).$$
Of course, since $G_\sigma({\operatorname{id}}_{[k]}) \times H_\sigma({\operatorname{id}}_{[k]})=\left(\underset{i=0}{\overset{k}{\prod}}
\Delta^{n_i-1}\right) \times \Delta^k$ and $\sigma \natural {\operatorname{id}}_{[k]}={\operatorname{id}}_{[n]}$, the map $\lambda_\sigma$ is the composite of $\nu_\sigma$ and of the canonical map $$\left(\underset{i=0}{\overset{k}{\prod}}
\Delta^{n_i-1}\right) \times \Delta^k \longrightarrow
\left(\underset{\delta \in {\operatorname{Ob}}(\Delta^* \downarrow [k])}{\coprod} G_\sigma(\delta) \times H_\sigma(\delta)\right)/_{\sim.}$$
\[superglue\] The map $$\nu_\sigma : \left(\underset{ \delta \in
{\operatorname{Ob}}(\Delta^* \downarrow [k])}{\coprod} G_\sigma(\delta) \times H_\sigma(\delta)\right)/_\sim
\overset{\cong}{\longrightarrow} \Delta^n$$ is a homeomorphism.
Since $\lambda_\sigma$ is onto, $\nu_\sigma$ is also onto. Since ${\operatorname{Ob}}(\Delta^* \downarrow [k])$ is finite, $G_\sigma (\delta) \times H_\sigma(\delta)$ is compact for every $\delta \in {\operatorname{Ob}}(\Delta^* \downarrow [k]$, and $\Delta^n$ is compact, we deduce that $\nu_\sigma$ is an identification map.
In order to prove that $\nu_\sigma$ is one-to-one, it suffices to construct a retraction of $\nu_\sigma$. Let $t \in \Delta^n$, and denote by $\delta : [m] \hookrightarrow [k]$ the unique monomorphism such that $\sigma^*(t) \in \delta^*(\partial \Delta^i)$. Let $t' \in \partial \Delta^i$ such that $\delta^*(t')=\sigma^*(t)$, and set $x:=\underset{i=0}{\overset{m}{\sum}}((\delta \natural \sigma)\natural i)^*(\frac{t'_j}{t'_i})
_{j \in [n_{\delta(i)}-1]}$. Finally, we consider the class of $(x,t')$. This construction yields a map which is easily seen to be a retraction of $\nu_\sigma$, and we deduce that $\nu_\sigma$ is one-to-one, which finishes the proof.
### Compatibility with the structure of cosimplicial space on $(\Delta^n)_{n\in \mathbb{N}}$
Let $\tau : [n'] \rightarrow [n]$ be a morphism in $\Delta$. We then have a unique decomposition of $\sigma \circ \tau$ into $$\xymatrix{
[n'] \ar[rr]^{\sigma \circ \tau}
\ar@{>>}[dr]^{\sigma'} & & [k] \\
& [k'] \ar@{^{(}->}[ur]_{\delta'}
}.$$ The morphism $\delta'$ is the root of $\sigma \circ \tau$, whilst $\sigma'$ is its reduction.\
Let $\delta' \natural \sigma : [m] \rightarrow [k']$. The universal property of cartesian squares yields a morphism $\tau': [n'] \rightarrow [m]$ which renders commutative the following diagram: $$\xymatrix{
[n'] \ar@/_/[ddr]_\tau \ar@/^/[drr]^{\sigma'} \ar@{.>}[dr]|-{\tau'} \\
& [m] \ar@{>>}[r]^{\delta' \natural \sigma} \ar@{^{(}->}[d]^{\sigma \natural \delta'} & [k'] \ar@{^{(}->}[d]^{\delta'} \\
& [n] \ar@{>>}[r]^{\sigma} & [k].
}$$ For every $i \in [k']$, set $n'_i:=\#(\sigma')^{-1}\{i\}$: then the commutative triangle $\xymatrix{
[n'] \ar@{>>}[dr]^{\sigma'} \ar[rr]^{\tau'} & & [m] \ar@{>>}[dl]^{\delta' \natural \sigma} \\
& [k']
}$ yields a commutative diagram $$\xymatrix{
[n'_i-1]
\ar@{>>}[dr] \ar[rr]^{\tau_i}
\ar[drrrr]^(0.7){\sigma' \natural i}
& & [n_{\delta'(i)}-1] \ar@{>>}[dl]
\ar[drrrr]^{(\delta' \natural \sigma) \natural i} \\
& [0] \ar@{^{(}->}[drrrr]_i & & & [n'] \ar[rr]^\tau \ar[dr]_{\sigma'} & & [m]. \ar[dl]^{\delta' \natural \sigma} \\
& & & & & [k']
}$$ The family $(\tau_i)_{0 \leq i \leq k'}$ will be called the **decomposition of $\tau$ over** $\sigma$.
\[splitting\] The square $$\begin{CD}
G_{\sigma'}({\operatorname{id}}_{[n']}) \times \Delta^{k'} @>{\nu_{\sigma'}}>> \Delta^{n'} \\
@V{\left(\underset{i=0}{\overset{k'}{\prod}}\tau_i^*\right) \times {\operatorname{id}}}VV @V{\tau^*}VV \\
G_\sigma(\delta') \times \Delta^{k'} @>{\nu_\sigma}>> \Delta^{n}
\end{CD}$$ is commutative.
For every $i \in [k']$, one has $\tau'((\sigma')^{-1}\{i\}) \subset (\delta' \natural \sigma)^{-1}\{i\}$. We deduce that the square $$\begin{CD}
\mathbb{R}^{(\sigma')^{-1}\{i\}} \times \mathbb{R}^{[k']} @>{\lambda^{(i)}_{\sigma'}}>> \mathbb{R}^{(\sigma')^{-1}\{i\}} \\
@V{(\tau')^* \times {\operatorname{id}}}VV @V{(\tau')^*}VV \\
\mathbb{R}^{(\delta' \natural \sigma)^{-1}\{i\}} \times \mathbb{R}^{[k']} @>{\lambda^{(i)}_{\delta' \natural \sigma}}>>
\mathbb{R}^{(\delta' \natural \sigma)^{-1}\{i\}}
\end{CD}$$ is well defined and commutative for every $i \in [k']$, and it follows that the square
$$\begin{CD}
\mathbb{R}^{[n']} \times \mathbb{R}^{[k']} @>{\lambda_{\sigma'}}>> \mathbb{R}^{[n']} \\
@V{(\tau')^* \times {\operatorname{id}}}VV @V{(\tau')^*}VV \\
\mathbb{R}^{[m]} \times \mathbb{R}^{[k']} @>{\lambda_{\delta' \natural \sigma}}>> \mathbb{R}^{[m]}
\end{CD}$$ is commutative.\
Also, the square $$\begin{CD}
\mathbb{R}^{[n'_i-1]} @>>> \mathbb{R}^{(\sigma')^{-1}\{i\}} \\
@V{(\tau_i)^*}VV @V{(\tau')^*}VV \\
\mathbb{R}^{[n_{\delta'(i)}-1]} @>>> \mathbb{R}^{(\delta' \natural \sigma)^{-1}\{i\}}
\end{CD}$$ is commutative for every $i\in [k']$. We deduce that the square $$\begin{CD}
\underset{i=0}{\overset{k'}{\prod}}\Delta^{n'_i-1} @>{\alpha_{\sigma'}({\operatorname{id}}_{[n']})}>>
\mathbb{R}^{[n']} \\
@V{\underset{i=0}{\overset{k'}{\prod}}\tau_i^*}VV @V{(\tau')^*}VV \\
\underset{i=0}{\overset{k'}{\prod}}\Delta^{n_{\delta'(i)}-1} @>{\alpha_{\sigma}(\delta')}>>
\mathbb{R}^{[n]}
\end{CD}$$ is commutative, and it follows that the square $$\begin{CD}
\underset{i=0}{\overset{k'}{\prod}}\Delta^{n'_i-1} \times \Delta^{n'} @>{\lambda_{\sigma'}}>>
\Delta^{n'} \\
@V{\left(\underset{i=0}{\overset{k'}{\prod}}\tau_i^*\right) \times {\operatorname{id}}}VV @V{(\tau')^*}VV \\
\underset{i=0}{\overset{k'}{\prod}}\Delta^{n_{\delta'(i)}-1} \times \Delta^{n'}
@>{\lambda_{\delta' \natural \sigma}}>> \Delta^{n'}
\end{CD}$$ is commutative.\
Since $\tau=(\sigma \natural \delta')\circ \tau'$, we conclude that the square $$\begin{CD}
G_{\sigma'}({\operatorname{id}}_{[n']}) \times \Delta^{k'} @>{\nu_{\sigma'}}>> \Delta^{n'} \\
@V{\left(\underset{i=0}{\overset{k'}{\prod}}\tau_i^*\right) \times {\operatorname{id}}}VV @V{\tau^*}VV \\
G_\sigma(\delta') \times \Delta^{k'} @>{\nu_\sigma}>> \Delta^{n}
\end{CD}$$ is commutative.
The definition of $\mathcal{H}$ {#10.4.3}
-------------------------------
We now set $B:=\underset{n \in \mathbb{N}}{\coprod} {\operatorname{Hom}}_\uparrow([n],\mathbb{N})$ and $C:=\{f \in B : f \quad \text{is non degenerate}\}$.\
For every non-negative integer $n$, and every non-degenerate $f : [n] \rightarrow \mathbb{N}$, we set $$\partial(f):=\left\{\delta_*(f),
\delta \in \underset{k<n}{\coprod}{\operatorname{Hom}}_{\Delta^*}([k],[n])\right\}$$ and $$\mathcal{H}_f:=(F^{(\infty)})^{\{f\}} \subset (F^{(\infty)})^{(B)}.$$ Let $f: [n] \rightarrow \mathbb{N}$ be a non-degenerate map. Then $\mathcal{H}_f$ has a countable dimension (as a real vector space), and $\forall g \in C, \; f \neq g \Rightarrow \mathcal{H}_f \bot \mathcal{H}_g$. If $n>0$, we may therefore choose a bijective isometry $$\varphi_f: \underset{g \in \partial(f)}{\overset{\bot}{\oplus}}\mathcal{H}_g
\overset{\cong}{\longrightarrow} \mathcal{H}_{f.}$$ We finally set $$\mathcal{H}:=\overline{(F^{(\infty)})^{(C)}}=
\overline{\underset{f \in C}{\overset{\bot}{\oplus}}\mathcal{H}_f}$$ which is a separable Hilbert space since $C$ is countable.
By identifying every $f: [0] \rightarrow \mathbb{N}$ with $f(0)$, we may view $(F^{(\infty)})^{(\mathbb{N})}$ as a subspace of $\mathcal{H}$. For every $n \in \mathbb{N}$, we also set $\mathcal{H}_n:=(F^{(\infty)})^{\{n\}}$, seen as a subspace of $\mathcal{H}$.
The filtration of $F^{(\infty)}$ by the sequence $$F^1\hookrightarrow F^2 \hookrightarrow \cdots \hookrightarrow F^l \hookrightarrow F^{l+1} \cdots$$ gives rise, for every $k\in \mathbb{N}$, to a filtration of $\mathcal{H}_k$ by an increasing sequence of finite dimensional subspaces $$\mathcal{H}_k^{(1)} \subset \mathcal{H}_k^{(2)} \subset \cdots \subset \mathcal{H}_k^{(l)} \subset \mathcal{H}_k^{(l+1)} \cdots$$ By induction on $n$, we recover, for every non degenerate $f : [n] \rightarrow \mathbb{N}$, a filtration of $\mathcal{H}_f$ by an increasing sequence of finite dimensional subspaces $$\mathcal{H}_f^{(1)} \subset \mathcal{H}_f^{(2)} \subset \cdots \subset \mathcal{H}_f^{(l)} \subset \mathcal{H}_f^{(l+1)} \cdots$$ which is identical to the preceding one when $n=0$, and such that $\varphi_f\left(\underset{g \in \partial(f)}{\bigoplus}\mathcal{H}_g^{(l)}\right)=\mathcal{H}_f^{(l)}$ for every positive integer $l$, when $n>0$.\
For every positive integer $l$, we finally set $$\mathcal{H}^{(l)} :=\overline{\underset{f \in C}{\overset{\bot}{\oplus}}\mathcal{H}^{(l)}_f}.$$ This defines a filtration of $\mathcal{H}$ by an increasing sequence of subspaces $$\mathcal{H}^{(1)} \subset \mathcal{H}^{(2)} \subset \cdots \subset \mathcal{H}^{(l)} \subset \mathcal{H}^{(l+1)} \cdots$$
\[hnuf\] Let $m\in \mathbb{N}$, $f: [m] \rightarrow \mathbb{N}$ be an increasing map, $u\in S(m)$, and $n\in \mathbb{N}$. Set $$\mathcal{H}_{n,u,f}:= \underset{(i,\delta)\in [n-1] \times {\operatorname{Lat}}(u)}{\bigoplus} \mathcal{H}_{\delta_*(f)+i} \subset \mathcal{H},$$ and, for every $l \in \mathbb{N}^*$, $\mathcal{H}^{(l)}_{n,u,f}:= \mathcal{H}_{n,u,f} \cap \mathcal{H}^{(l)}$.\
Assume $u$ is non-trivial, and write $u=\delta_1<\cdots<\delta_{k-1}<[m]$. Then $$\mathcal{H}_{n,u',f} \subset \underset{i=0}{\overset{n-1}{\bigoplus}}\left[\underset{g \in \partial (f+i)}{\bigoplus} \mathcal{H}_g\right],$$ and it follows that the condition $$\forall (i,g)\in [n-1] \times {\operatorname{Lat}}(u'), \; \forall x \in \mathcal{H}_{\delta_*((\delta_{k-1})_*(f))+i},
\quad \varphi_{n,u,f}(x)=\varphi_{f+i}(x)$$ defines an isometry $$\varphi_{n,u,f} : \mathcal{H}_{n,u',(\delta_{k-1})_*(f)} \hookrightarrow \underset{i=0}{\overset{n-1}{\oplus}}\mathcal{H}_{f+i}=\mathcal{H}_{n,[m],f}$$ which is compatible with the respective filtrations of $\mathcal{H}_{n,u',(\delta_{k-1})_*(f)}$ and $\mathcal{H}_{n,[m],f.}$\
Obviously, if $v$ is another non-trivial class in $S(m)$ such that $u \subset v$, where, $v=\delta'_1<\cdots<\delta'_{k'-1}<[m]$, then $\mathcal{H}_{n,u',(\delta_{k-1})_*(f)} \subset \mathcal{H}_{n,v',(\delta'_{k'-1})_*(f)}$, and $(\varphi_{n,v,f})_{|\mathcal{H}_{n,u',(\delta_{k-1})_*(f)}}=\varphi_{n,u,f}$.\
Finally, if $n'$ is another non-negative integer, then $$\mathcal{H}_{n,u,f} \overset{\bot}{\oplus} \mathcal{H}_{n',u,f+n}=\mathcal{H}_{n+n',u,f},$$ $$\mathcal{H}_{n,u',(\delta_{m-1})_*(f)} \overset{\bot}{\oplus} \mathcal{H}_{n',u',(\delta_{m-1})_*(f)+n}=\mathcal{H}_{n+n',u',(\delta_{m-1})_*(f),}$$ and $$\forall (x,y) \in \mathcal{H}_{n,u',(\delta_{m-1})_*(f)} \times \mathcal{H}_{n',u',(\delta_{m-1})_*(f)+n},
\quad \varphi_{n+n',u,f}(x+y)=\varphi_{n,u,f}(x)+\varphi_{n',u,f+n}(y).$$
Universal paths between orthonormal bases {#10.4.4}
-----------------------------------------
### The simplicial space $(V_n(m))_{m\in \mathbb{N}}$
For every $l \in \mathbb{N}^*$, every linear subspace $\mathcal{H'}$ of $\mathcal{H}^{(l)}$, and every non-negative integer $n$, we denote by $V_n(\mathcal{H'})$ the subspace of $(\mathcal{H}')^n$ consisting of orthonormal $n$-tuples, which we consider as a $U_n(F)$-space with the canonical right-action of $U_n(F)$.
For any linear subspace $\mathcal{H'}$ of $\mathcal{H}$ and any integer $n$, we denote by $V_n(\mathcal{H}')$ the subset of $(\mathcal{H}')^n$ consisting of orthonormal $n$-tuples, equipped with the final topology for the canonical map $$\underset{l \in \mathbb{N}^*}{\coprod}V_n(\mathcal{H}'\cap \mathcal{H}^{(l)}) \longrightarrow V_n(\mathcal{H}').$$ This definition is clearly compatible with the preceding one, and the canonical right-action of $U_n(F)$ on $V_n(\mathcal{H'})$ is clearly continuous. Moreover, given two subspaces $\mathcal{H}'$ and $\mathcal{H}''$ of $\mathcal{H}$ such that $\mathcal{H}' \subset \mathcal{H}''$, then the inclusion $V_n(\mathcal{H}') \subset V_n(\mathcal{H}'')$ is an immersion.
For every $n \in \mathbb{N}$, $k \in \mathbb{N}$ and $l \in \mathbb{N}$, we define $V_{n,k}^{(l)}$ as $V_n\left(\underset{j=k}{\overset{k+n-1}{\bigoplus}}\mathcal{H}^{(l)}_j \right)$. For every $n \in \mathbb{N}$ and $k \in \mathbb{N}$, we define $V_{n,k}$ as $V_n\left(\underset{j=k}{\overset{k+n-1}{\bigoplus}}\mathcal{H}_j \right)$, so that $V_{n,k}=\underset{l \in \mathbb{N}^*}{{\operatorname{\underset{\longrightarrow}{{\operatorname{colim}}}}}}\, V_{n,k}^{(l)}$.\
For every $f:[m] \rightarrow \mathbb{N}$ and $n \in \mathbb{N}$ set $$V_n(f):=\underset{0 \leq i \leq n}{\prod} V_{n,f(i)}$$ which is $U_n(F)$-space for the diagonal action.\
For every $l \in \mathbb{N}^*$, set $V_n^{(l)}(f):=\underset{0 \leq i \leq n}{\prod}V^{(l)}_{n,f(i)}$, so that $V_n(f)=\underset{l \in \mathbb{N}^*}{{\operatorname{\underset{\longrightarrow}{{\operatorname{colim}}}}}}\, V_n^{(l)}(f)$.\
If $n$ and $n'$ are non-negative integers, and $f: [m] \rightarrow \mathbb{N}$ is a map, then $\underset{j=f(i)}{\overset{f(i)+n-1}{\bigoplus}}\mathcal{H}^{(l)}_j$ and $\underset{j=f(i)+n}{\overset{f(i)+n+n'-1}{\bigoplus}}\mathcal{H}^{(l)}_j$ are orthogonal for every $i \in [m]$ in the case $n>0$; it follows that the juxtaposition of families yields a canonical injection: $$V_n(f) \times V_{n'}(f+n) \hookrightarrow V_{n+n'}(f).$$
For every $m\in \mathbb{N}$, we define $$V_n(m):=\underset{f\in {\operatorname{Hom}}_\uparrow([m],\mathbb{N})}{\coprod} V_n(f)$$ as a $U_n(F)$-space, with the previous action on every component.\
For every $n\in \mathbb{N}$, and every $f : [m] \rightarrow \mathbb{N}$, every morphism $\tau : [m'] \rightarrow [m]$ in the category $\Delta$ induces a $U_n(F)$-map: $$\tau_* : \begin{cases}
V_n(f) & \longrightarrow V_n(f \circ \tau) \\
(x_i)_{0 \leq i \leq m} & \longmapsto (x_{\tau(i)})_{0 \leq i \leq m'.}
\end{cases}$$ This defines a structure of simplicial $U_n(F)$-space on $(V_n(m))_{m\in \mathbb{N}}$, for every $n \in \mathbb{N}$.
For every $n\in \mathbb{N}$ and every compact space $K$, the structure of cosimplicial space of $(\Delta^m)_{m \in \mathbb{N}}$ induces a structure of simplicial $U_n(F)$-space on $({\operatorname{Hom}}(\Delta^m \times K,V_n(\mathcal{H})))_{m \in \mathbb{N}}$ where, for every $m \in \mathbb{N}$, ${\operatorname{Hom}}(\Delta^m \times K,V_n(\mathcal{H}))$ denotes the space of continuous maps from $\Delta^m \times K$ to $V_n(\mathcal{H})$ with the compact-open topology.
### The main goal
Our main goal in the rest of the section is to construct, for every $n \in \mathbb{N}$, a morphism of simplicial $U_n(F)$-spaces $$\psi_n: (V_n(m))_{m \in \mathbb{N}} \longrightarrow ({\operatorname{Hom}}(\Delta^m,V_n(\mathcal{H})))_{m \in \mathbb{N}}$$
which fullfills a certain list of requirements.
Such a morphism is simply the data of a family $(\psi_{n,m})_{m\in \mathbb{N}}$ of continuous maps, with $\psi_{n,m} : V_n(m) \times \Delta^m \longrightarrow V_n(\mathcal{H})$ for every $m \in \mathbb{N}$, which satisfies a set of compatibility conditions.
Assuming that we have built two such families of morphisms $\psi=(\psi_n)_{n\in \mathbb{N}}$ and $\psi'=(\psi'_n)_{n\in \mathbb{N}}$, we also want to build a “homotopy” from $\psi$ to $\psi'$, i.e. a family $\Psi=(\Psi_n)_{n\in \mathbb{N}}$ of morphisms, such that $$\Psi_n: (V_n(m))_{m \in \mathbb{N}} \longrightarrow ({\operatorname{Hom}}(\Delta^m \times I,V_n(\mathcal{H})))_{m \in \mathbb{N}}$$ is a morphism of simplicial $U_n(F)$-spaces for every $n \in \mathbb{N}$, which satisfies a set of compatibility conditions, and such that, for every $(n,m) \in \mathbb{N}^2$, $(\Psi_{n,m})_{|V_n(m) \times \{0\}}=\psi_{n,m}$ and $(\Psi_{n,m})_{|V_n(m) \times \{1\}}=\psi'_{n,m}$.
We will actually describe a construction that will fulfill both needs at once. We fix $p \in \mathbb{N}$, and we assume that we have a family $(\psi_n^\partial)_{n\in \mathbb{N}}$, such that $$\psi_n^\partial : (V_n(m))_{m \in \mathbb{N}} \longrightarrow ({\operatorname{Hom}}(\Delta^m \times \partial \Delta^p,V_n(\mathcal{H})))_{m \in \mathbb{N}}$$ is a morphism of simplicial $U_n(F)$-spaces for every $n \in \mathbb{N}$, which satisfies compatibility conditions (i) to (vi) detailed in the next paragraph. We want to construct a family $(\psi_n)_{n\in \mathbb{N}}$, such that $$\psi_n : (V_n(m))_{m \in \mathbb{N}} \longrightarrow ({\operatorname{Hom}}(\Delta^m \times \Delta^p,V_n(\mathcal{H})))_{m \in \mathbb{N}}$$ is a morphism of simplicial $U_n(F)$-spaces for every $n \in \mathbb{N}$, which satisfies compatibility conditions (i) to (vi) detailed in the next paragraph, and, for every $(n,m)\in \mathbb{N}^2$, $(\psi_{n,m})_{|V_n(m) \times \Delta^m \times \partial \Delta^p}=\psi_{n,m}^\partial$.
In the case $p=0$, $\psi_n^\partial$ is trivial, and the construction will yield the family $(\psi_n)_{n \in \mathbb{N}}$ we are looking for. In the case $p=1$, if we have two sequences of morphisms $\psi$ and $\psi'$, they define a family of morphisms $(\Psi_n^\partial)_{n\in \mathbb{N}}$, with $$\Psi_n^\partial : (V_n(m))_{m \in \mathbb{N}} \longrightarrow
({\operatorname{Hom}}(\Delta^m \times \{0,1\},V_n(\mathcal{H})))_{m \in \mathbb{N}},$$ and the construction of $\Psi$ from $\Psi^\partial$ will yield a homotopy from $\psi$ to $\psi'$.
### The compatibility conditions
Let $K$ be a compact space, and $\psi=(\psi_n)_{n\in \mathbb{N}}$ be a family such that $\psi_n : (V_n(m))_{m \in \mathbb{N}} \longrightarrow {\operatorname{Hom}}(\Delta^n \times K,V_n(\mathcal{H}))$ is a morphism of simplicial $U_n(F)$-spaces for every $n\in \mathbb{N}$. We define the following conditions on $\psi$, some of which depend on three integers $n$, $n'$ and $m$, and a non-decreasing map $f : [m] \rightarrow \mathbb{N}$.
(i) $\psi_{n,0}: V_n(0) \times \Delta^0 \times K \rightarrow V_n(\mathcal{H})$ is the composite of the projection on the first factor and the map $V_n(0) \rightarrow V_n(\mathcal{H})$ induced by the inclusion $\underset{i \in \mathbb{N}}{\oplus} \mathcal{H}_i \subset \mathcal{H}$.
(ii) $$\forall (\mathbf{B},\mathbf{B}',t)\in V_n(f) \times V_{n'}(f+n) \times (\Delta^m \times K), \quad
\psi_{n,m}(\mathbf{B},t) \,\bot\, \psi_{n',m}(\mathbf{B}',t).$$
(iii) The diagram $$\xymatrix{
V_n(f) \times V_{n'}(f+n) \ar[dd] \ar[drr]^{\psi_{n,m} \times \psi_{n',m}} \\
& & {\operatorname{Hom}}(\Delta^m \times K,\mathcal{H}^n\times \mathcal{H}^{n'}) \\
V_{n+n'}(f) \ar[urr]_{\psi_{n+n',m}}
}$$ is commutative.
(iv) If $g : [k] \rightarrow \mathbb{N}$ denotes the root of $f$ and $\sigma : [m] \twoheadrightarrow [k]$ its reduction, then, for every $u\in S(k)$, $$\psi_{n,m}\bigl(V_n(f) \times (\sigma^*)^{-1}(\Delta_u) \times K\bigr) \subset V_n(\mathcal{H}_{n,u,g}).$$
(v) If $g : [k] \rightarrow \mathbb{N}$ denotes the root of $f$ and $\sigma:[m] \twoheadrightarrow [k]$ its reduction, and $\forall i \in [k], n_i:=\# \sigma^{-1}\{i\}$, then, $$\forall \mathbf{B} \in V_n(f),
\forall i \in [k],
\forall t \in \Delta^{n_i-1} \times K, \;
\psi_{n,n_i-1}((\sigma \natural i)_*(\mathbf{B}),t) \in V_n(\mathcal{H}_{g(i)}),$$ and, for every $\delta : [k'] \hookrightarrow [k]$, the composite of $\psi_{n,m}$ with\
$\nu_\sigma : V_n(f) \times (G_\sigma(\delta) \times \Delta^{k'}) \times K
\overset{\delta_* \times \nu_\sigma \times {\operatorname{id}}_K}{\longrightarrow} V_k(f) \times \Delta^m \times K$ is the composite map $$\begin{gathered}
V_n(f) \times (G_\sigma(\delta) \times \Delta^{k'}) \times K
\overset{(\sigma \natural \delta)_*\times {\operatorname{id}}}{\longrightarrow}
\left(\underset{i=0}{\overset{k'}{\prod}}(V_n(\mathcal{H}_{g(i)}))^{n_{\delta(i)}}
\times \Delta^{n_{\delta(i)}-1} \times K\right)
\times \Delta^{k'} \times K \\
\overset{\underset{i=0}{\overset{k'}{\prod}}\psi_{n,n_{\delta(i)}-1}}{\longrightarrow}
V_n(\delta_*(g)) \times \Delta^{k'} \times K
\overset{\psi_{n,k'}}{\longrightarrow} V_n(\mathcal{H}).\end{gathered}$$
(vi) In the case $f$ is non-degenerate: for every non-trivial increasing sequence\
$u=\delta_1<\dots<\delta_{k-1}<[m]$ in $S(m)$, with $\delta_{k-1} : [m'] \hookrightarrow [m]$, and every quadruple $(\mathbf{B},y,t,z)\in V_n(f) \times \Delta_{u'} \times [0,1] \times K$: if we set $\mathbf{B}':=\psi_{n,m'}((\delta_{k-1})_*(\mathbf{B}),y,z)$ and $x=r_m(\delta_{k-1}^*(y),t)$, then $\mathbf{B}'\in V_n(\mathcal{H}_{n,u',(\delta_{k-1})_*(f)})$ and $$\psi_{n,m}(\mathbf{B},x,z)=\cos\left(\frac{\pi}{2}t\right).\mathbf{B}'
+\sin\left(\frac{\pi}{2}t\right).\varphi_{n,u,f}(\mathbf{B}') .$$
**Remarks:**
- Conditions (ii) and (iii) hold for all $m$ and $f: [m] \rightarrow \mathbb{N}$ when $n=0$ or $n'=0$.
- When $f$ is constant, condition (iv) simply means that $\psi_{n,m}(V_n(f) \times \Delta^m \times K) \subset V_{n,f(0).}$
- Conditions (iv), (v) and (vi) are only there so that we can carry out the construction, and they will be useless when the construction is over.
### Relationships between the conditions
- In condition (v), the first requirement holds if and only if condition (iv) holds for every triple $(n,n_i-1,f \circ (\sigma \natural i))$ with $i\in [k]$.
- In the case $f$ is constant, condition (v) for $(n,m,f)$ is logically equivalent to condition (iv) for the same triple.
- In the case $f$ is non-degenerate and (i) holds for $n$, condition (v) holds if and only if the square $$\begin{CD}
V_n(f) @>{\psi_{n,k}}>> {\operatorname{Hom}}(\Delta^m \times K,V_n(\mathcal{H})) \\
@V{\delta_*}VV @V{\delta_*}VV \\
V_n(\delta_*(f)) @>{\psi_{n,k'}}>> {\operatorname{Hom}}(\Delta^k \times K,V_n(\mathcal{H}))
\end{CD}$$ is commutative for every $\delta : [k] \hookrightarrow [m]$.
- In condition (vi), only the second requirement is interesting since the first one is obviously true when condition (iv) holds for $(n,m',f\circ \delta_{k-1})$.
- Assume that condition (iv) holds for the two triples $(n,m,f)$ and $(n',m,f)$.\
Let $g$ be the root of $f$ and $\sigma : [m] \twoheadrightarrow [k]$ its reduction. Let $(\mathbf{B},\mathbf{B}') \in V_n(f) \times V_{n'}(f+n)$, $t \in \Delta^m$ and $z\in K$. By Proposition \[glue\], we may choose $u\in S(k)$ such that $\sigma^*(t) \in \Delta_u$.
Since condition (iv) is true for both triples $(n,m,f)$ and $(n',m,f)$, we deduce that $\psi_{n,m}(\mathbf{B},x,z) \in V_n(\mathcal{H}_{n,u,g})$ and $\psi_{n',m}(\mathbf{B}',x,z) \in V_n'(\mathcal{H}_{n',u,g+n})$. Since $\mathcal{H}_{n,u,g} \bot \mathcal{H}_{n',u,g+n}$, we deduce that $\psi_{n,m}(\mathbf{B},x,z) \bot \psi_{n',m}(\mathbf{B}',x,z)$. Hence condition (ii) holds for $(n,n',m,f)$.
### Main result
We may now state our main result:
\[mighty\] Let $p \in \mathbb{N}$, and $\psi^\partial=(\psi_n^\partial)_{n\in \mathbb{N}}$ be a family such that:
- For every $n \in \mathbb{N}$, $\psi_n^\partial : (V_n(m))_{m\in \mathbb{N}} \longrightarrow
({\operatorname{Hom}}(\Delta^m \times \partial \Delta ^p,V_n(\mathcal{H})))_{m\in \mathbb{N}}$ is a morphism of simplicial $U_n(F)$-spaces;
- Conditions (i) to (vi) are satisfied by $\psi^\partial$ for every compatible $4$-tuple $(n,n',m,f)$.
Then there exists a family $\psi=(\psi_n)_{n\in \mathbb{N}}$ such that
- For every $n \in \mathbb{N}$, $\psi_n : (V_n(m))_{m\in \mathbb{N}} \longrightarrow
({\operatorname{Hom}}(\Delta^m \times \Delta ^p,V_n(\mathcal{H})))_{m\in \mathbb{N}}$ is a morphism of simplicial $U_n(F)$-spaces.
- Conditions (i) to (vi) are satisfied by $\psi$ for every compatible $4$-tuple $(n,n',m,f)$.
- On has $(\psi_{n,m})_{|V_n(m) \times \Delta^m \times \partial \Delta^p}=\psi_{n,m}^\partial$ for every $(n,m)\in \mathbb{N}^2$.
The proof of Proposition \[mighty\] {#10.4.5}
-----------------------------------
Our proof of Proposition \[mighty\] will be done by induction. An essential is played by a general conjecture on relative triangulations that we were not able to prove: we will begin by stating it and drawing the consequences that will be necessary in our proof.
### A conjecture on relative triangulations, and some consequences {#conjecturesection}
Let $M$ be an $n$-dimensional smooth manifold, and $(M_i)_{i \in I}$ be a finite family of closed subspaces of $M$. For every $x \in M$, set $I_x:=\{i\in I :\; x \in M_i\}$. We say that the family $(M_i)_{i \in I}$ **intersects cleanly** if, for every $x \in M$, there is an open neighborhood $U_x$ of $x$ in $M$, an open neighborhood $V$ of $0$ in $\mathbb{R}^n$, a family $(F_i)_{i\in I_x}$ of linear subspaces of $\mathbb{R}^n$, and a smooth diffeomorphism $\varphi : U_x \overset{\cong}{\longrightarrow} V$ such that $$\forall i \in I {\smallsetminus}I_x, \quad U_x \cap M_i =\emptyset$$ and $$\forall J \in \mathcal{P}(I_x), \quad \varphi\left(U \cap \left(\underset{j \in J}{\cap}M_j\right)\right) =
V \cap\left(\underset{j \in J}{\cap}F_j\right).$$
In this case, $(M_j)_{j \in J}$ obviously intersects cleanly for every $J \subset I$, and the $M_j$’s are all smooth submanifolds of $M$.
**Remarks:**
- Let $M$ be a smooth manifold, $I$ be a finite set, and $(M_i)_{i \in I}$ be a family of closed smooth submanifolds of $M$ indexed over $I$. Assume that, for every $x \in M$, there is a smooth manifold $N_x$, a family $(N_{x,i})_{i \in I}$ of closed smooth submanifolds of $N_x$ which intersects cleanly, an open neighborhood $U_x$ of $x$ in $M$, an open subset $V_x$ of $N$, and a smooth diffeomorphism $\varphi_x : U_x \overset{\cong}{\longrightarrow} V_x$ such that $U_x \cap (\underset{i \in I {\smallsetminus}I_x}{\bigcup}M_i)=\emptyset$ and $\forall i \in I_x,\; \varphi_x(U_x \cap M_i)=V_x \cap N_{i,x}$. Then $(M_i)_{i \in I}$ intersects cleanly.
- Let $M$ and $N$ be two smooth manifolds, $I$ a finite set, and $(M_i)_{i\in I}$ (resp. $(N_i)_{i\in I}$) a family of closed smooth submanifolds of $M$ (resp. of $N$) indexed over $I$ which intersects cleanly. Then $(M_i \times N_i)_{i \in I}$ intersects cleanly in $M \times N$.
**Example 1:** Let $E$ be an affine variety, and $(E_i)_{i \in I}$ be a finite family of affine subvarieties of $E$. Then $(E_i)_{i \in I}$ intersects cleanly.
**Example 2:** Let $G$ be a Lie group and $(H_i)_{i \in I}$ be a finite family of closed subgroups of $G$. Then $(H_i)_{i \in I}$ intersects cleanly.
Let $x \in G$, $I_x:=\{i\in I : x \in H_i\}$, and let $V$ be an open neighborhood of $x$ in $G {\smallsetminus}\underset{i \in I {\smallsetminus}I_x}{\bigcup}H_i$. We set $\varphi_x : \begin{cases}
G & \longrightarrow G \\
g & \longmapsto g.x^{-1}
\end{cases}$ and choose an open neigborhood $V_1$ of $0$ in $LG$ and an open neighborhood $V_2$ of $1_G$ in $G$ such that $\exp_{|V_1} : V_1 \overset{\cong}{\longrightarrow} V_2$ is a diffeomorphism. We set $V':=(\varphi_x)^{-1}(V_2) \cap V$ and $\varphi : \begin{cases}
V' & \overset{\cong}{\longrightarrow} V_1 \\
z & \longmapsto (\exp_{|V_2})^{-1}(\varphi_x(z)).
\end{cases}$\
Then, for every $J \subset I_x$, one has $$\varphi\left(V' \cap \left(\underset{j\in J}{\cap}H_j\right) \right)=V_1 \cap\left(\underset{j\in J}{\cap}LH_j\right).$$ Then Example 1 and the previous remarks show that $(H_i)_{i\in I}$ intersects cleanly.
We may now state our conjecture:
\[unionofsubmanifolds\] Let $M$ be a smooth compact manifold and $(M_i)_{i \in I}$ be a finite family of closed subspaces of $M$ which intersects cleanly. Then the pair $\left(M, \underset{i \in I}{\bigcup}M_i\right)$ is a finite relative CW-complex.
Let us now draw some important consequences of this conjecture.\
Let $m$ and $n$ be two positive integers. Let $(E_0,\dots,E_m)$ be an $(m+1)$-tuple of finite dimensional inner product spaces (with ground field $F$). For every $k \in [m-1]$, we consider an isometry $\varphi_k : E_k \overset{\cong}{\longrightarrow} E_{k+1}$ For every $k \in [m]$, we consider a linear subspace $F_k$ of $E_k$, and a decomposition $E_k=\underset{1\leq i \leq n}{\overset{\bot}{\bigoplus}}E_k^{(i)}$ such that:
- for all $k \in [m-1], \; \varphi_k(F_k)=F_{k+1}$.
- for all $k \in [m-1], \; \forall i \in \{1,\dots,n\}, \varphi_k(E_k^{(i)})=E_{k+1}^{(i)}$.
- for all $k \in [m], \; F_k= \underset{1\leq i \leq n}{\overset{\bot}{\bigoplus}}F_k^{(i)}$, where $F_k^{(i)}:=F_k\cap E_k^{(i)}$ for all $i\in \{1,\dots,n\}$.
For every $k \in [m]$, every subspace $V$ of $E_k$, and every subset $A$ of $\{1,\dots,n\}$, we define $$V^{[A]}:=V \cap \bigl(\underset{i \in A}{\oplus}E_k^{(i)}\bigr).$$ Set then $$\mathcal{M}:= \left(\bigl(E_k,F_k,(E_k^{(i)})_{1 \leq i \leq n}\bigr)_{0 \leq k \leq m},
(\varphi_k)_{0 \leq k \leq m-1} \right).$$
We define $V_N(\mathcal{M})$ as the product space $\underset{k=0}{\overset{m}{\prod}}V_n(E_k)$ with the diagonal right-action of $U_n(F)$. For every $(p,q) \in (\mathbb{N}^*)^2$ such that $p+q=n$, we define $$V^{(p,q)}_n(\mathcal{M}) :=
\left(\underset{k=0}{\overset{m}{\prod}}\left[V_p\left(\underset{1\leq i \leq p}{\overset{\bot}{\bigoplus}}E_k^{(i)}\right)
\times V_q\left(\underset{p+1\leq j \leq n}{\overset{\bot}{\bigoplus}}E_k^{(j)}\right)\right]\right).U_n(F) \subset
V_n(\mathcal{M}),$$ and we set $$V_n^{\text{prod}}(\mathcal{M}):=\underset{p \geq 1,q \geq 1, p+q=n}{\bigcup}
V^{(p,q)}_n(\mathcal{M}) \subset V_n(\mathcal{M}).$$ For every $k \in [M-1]$, set $$V_n^k(\mathcal{M}):=\bigl\{(\mathbf{B}_i)_{0 \leq i \leq m} \in V_n(\mathcal{M}) :
\varphi_k(\mathbf{B}_k)=\mathbf{B}_{k+1}\bigr\},$$ $$V_n^{\text{deg}}(\mathcal{M}):=\underset{0 \leq k \leq m-1}{\bigcup}V_n^k(\mathcal{M}) \quad \text{and}
\quad V'_n(\mathcal{M}):=\underset{k=0}{\overset{m}{\prod}}V_n(F_k).$$ We now define $G_n(\mathcal{M})$ as the quotient space $V_n(\mathcal{M})/U_n(F)$. Since $V_n(\mathcal{M})$ is a smooth compact manifold with a smooth free action of $U_n(F)$, we deduce that there is a unique structure of smooth manifold on $G_n(\mathcal{M})$ such that the canonical projection $V_n(\mathcal{M}) \longrightarrow G_n(\mathcal{M})$ is a smooth principal $U_n(F)$-bundle. For every $(p,q)\in (\mathbb{N}^*)^2$ such that $p+q=n$ (resp. for every $k \in [m-1]$), we define $G_N^{(p,q)}(\mathcal{M})$ as the direct image of $V^{(p,q)}_N(\mathcal{M})$ (resp. of $V_N^k(\mathcal{M})$) by the canonical projection $V_n(\mathcal{M}) \rightarrow G_n(\mathcal{M})$. Finally, we define $G'_n(\mathcal{M})$ as the direct image of $V'_n(\mathcal{M})$ by the canonical projection. Obviously, all those subspaces of $G_n(\mathcal{M})$ are compact, and therefore closed.
**Remark :** An element $x$ of $G_n(\mathcal{M})$ may be identified with a diagram $x_0 \overset{f_0}{\rightarrow} x_1 \overset{f_1}{\rightarrow}
\dots \overset{f_{m-1}}{\rightarrow} x_m$, where, for every $k \in [m]$, $x_k$ is an $n$-dimensional subspace of $E_k$, and, for every $k \in [m-1]$, $f_k$ is a linear isometry from $x_k$ to $x_{k+1}$.\
Let $(p,q)\in (\mathbb{N}^*)^2$ such that $p+q=n$. Then $x \in G_n^{(p,q)}(\mathcal{M})$ if and only if, for every $k \in [m]$, $x_k=x_k^{[\{1,\dots,p\}]} \oplus x_k^{[\{p+1,\dots,n\}]}$, and, for every $k \in [m-1]$, $f_k\left(x_k^{[\{1,\dots,p\}]}\right)=x_{k+1}^{[\{1,\dots,p\}]}$ and $f_k\left(x_k^{[\{p+1,\dots,n\}]}\right)=x_{k+1}^{[\{p+1,\dots,n\}]}$.\
Let $k \in [m-1]$. Then $x \in G_n^k(\mathcal{M})$ if and only if $f_k=\varphi_k\big|_{x_k.}^{x_{k+1}}$\
Finally, $x \in G'_n(\mathcal{M})$ if and only if $x_k \subset F_k$ for every $k \in [m]$.
We are now ready to state the two consequences of conjecture \[unionofsubmanifolds\] that will be used in the proof of Proposition \[mighty\].
\[clean\] The family $\left((G_n^{(p,q)}(\mathcal{M}))_{p+q=n},\,(G^k_n(\mathcal{M}))_{0 \leq k \leq m},\, G'_n(\mathcal{M})\right)$ intersects cleanly in $G_n(\mathcal{M})$.
Obviously, the subspaces considered here are all closed in $G_n(\mathcal{M})$ since they are compact.\
Let $x=x_0 \overset{f_0}{\rightarrow} x_1 \overset{f_1}{\rightarrow}
\dots \overset{f_{m-1}}{\rightarrow} x_m$ in $G_n(\mathcal{M})$, and set: $$I_x:=\{(p,q) \in (\mathbb{N}^*)^2 : x \in G_n^{(p,q)}(\mathcal{M})\} \cup \{k \in [m-1] : x \in G_n^k(\mathcal{M})\}$$ and $$J_x:=\{(p,q) \in (\mathbb{N}^*)^2 : x \not\in G_n^{(p,q)}(\mathcal{M})\} \cup \{k \in [m-1] : x \not\in G_n^k
(\mathcal{M})\}.$$ Denote by $U_x$ the subset of $G_n(\mathcal{M})$ consisting of those elements $x'=x'_0 \overset{f'_0}{\rightarrow} x'_1 \overset{f'_1}{\rightarrow} \dots \overset{f'_{m-1}}{\rightarrow} x'_m$ such that $x'_k \cap x_k^\bot =\{0\}$ for all $k\in [m]$. If $x'$ is such an element, we set $\psi_{x_k,x'_k}:= \pi_{x'_k} \circ (\pi_{x_k}^{x'_k} \circ \pi_{x'_k}^{x_k})^{-\frac{1}{2}}$ for every $k \in [m]$ (this is a well-defined isometry from $x_k$ to $x'_k$). Obviously, $U_x$ is an open neighborhood of $x$ in $G_N(\mathcal{M})$.\
We finally set $$\overline{M}:=\underset{k=0}{\overset{m}{\prod}}L(x_k,x^\bot_k) \quad ,
\quad \overline{N}:=\underset{k=0}{\overset{m-1}{\prod}}U(x_k)$$ and $$\varphi_x : \begin{cases}
U_x & \longrightarrow \overline{M} \times \overline{N} \\
x'_0 \overset{f'_0}{\rightarrow} x'_1 \overset{f'_1}{\rightarrow} \dots \overset{f'_{m-1}}{\rightarrow} x'_m
& \longmapsto
\left(\left(\pi_{x_k^\bot} \circ (\pi_{x_k}^{x'_k})^{-1}\right)_{k \in [m]},
\left(f_k^{-1} \circ (\psi_{x_{k+1},x'_{k+1}})^{-1}
\circ f'_k \circ \psi_{x_k,x'_k}\right)_{k \in [m-1]}\right).
\end{cases}$$ For every $(p,q)\in I_x$, we set $$M_{(p,q)}:=\underset{k=0}{\overset{m}{\prod}}
\left(L\left(x^{[\{1,\dots,p\}]}_k,(x^\bot_k)^{[\{1,\dots,p\}]}\right)\oplus
L\left(x^{[\{p+1,\dots,n\}]}_k,(x^\bot_k)^{[\{p+1,\dots,n\}]}\right)\right)$$ and $$N_{(p,q)}:=\underset{k=0}{\overset{m-1}{\prod}}\left(U(x_k^{[\{1,\dots,p\}]})\times
U(x_k^{[\{p+1,\dots,n\}]})\right).$$ For every $k \in I_x$, we set $$M_k:=\left\{(\alpha_i)_{0\leq i \leq m}\in \overline{M} : \;\alpha_k=\alpha_{k+1}\right\},$$ and $$N_k:=\left\{(\beta_i)_{0\leq i \leq m-1}\in \overline{N} : \; \beta_k={\operatorname{id}}_{x_k}\right\}.$$ Finally, we set $M':=\underset{k=0}{\overset{m}{\prod}}L(x_k,x^\bot_k \cap F_k)$. By Example 1, $((M_i)_{i \in I_x},M')$ intersects cleanly in $\overline{M}$. By Example 2, $((N_i)_{i \in I_x},N)$ intersects cleanly in $\overline{N}$. Therefore $((M_i \times N_i)_{i \in I_x},M' \times \overline{N})$ is a family of subsets of $\overline{M} \times \overline{N}$ which intersects cleanly. It is then a straightforward task to check that $\varphi_x(U_x \cap G_n^i(\mathcal{M}))=M_i \times N_i$ for every $i \in I_x$, and $\varphi_x(U_x \cap G_n'(\mathcal{M}))=M' \times \overline{N}$ if $x \in G_n'(\mathcal{M})$.\
Set finally $$V_x:=
\begin{cases}
G_N(\mathcal{M}) {\smallsetminus}\underset{j \in J_x}{\bigcup}G_N^j(\mathcal{M}) & \text{if} \quad x \in G'_N(\mathcal{M}) \\
G_N(\mathcal{M}) {\smallsetminus}\left(G'_N(\mathcal{M}) \cup \underset{j \in J_x}{\bigcup}G_N^j(\mathcal{M})\right) &
\text{otherwise}.
\end{cases}$$ By restricting $\varphi_x$ to the open neighborhood $U_x \cap V_x$ of $x$ in $G_n(\mathcal{M})$, we deduce from a previous remark that the family $\left((G_n^{(p,q)}(\mathcal{M}))_{p+q=n},(G^k_n(\mathcal{M}))_{0 \leq k \leq m}, G'_n(\mathcal{M})\right)$ of subsets of $G_n(\mathcal{M})$ intersects cleanly.
\[CWconjecture\] $(V_n(\mathcal{M}),V_n^{\text{prod}}(\mathcal{M}) \cup V_n^{\text{deg}}(\mathcal{M}) \cup V_n'(\mathcal{M}))$ and $(V_n(\mathcal{M}),V_n^{\text{prod}}(\mathcal{M}) \cup V_n'(\mathcal{M}))$ are finite relative $U_n(F)$-CW-complexes.
By Corollary \[clean\], each pair $\bigl(G_n(\mathcal{M}),G_n^{\text{prod}}(\mathcal{M}) \cup G_n^{\text{deg}}(\mathcal{M}) \cup G_n'(\mathcal{M})\bigr)$ and $\bigl(G_n(\mathcal{M}),G_n^{\text{prod}}(\mathcal{M}) \cup G_n'(\mathcal{M})\bigr)$ is a finite relative CW-complex. Since the respective inverse images of $G_n^{\text{prod}}(\mathcal{M}) \cup G_n^{\text{deg}}(\mathcal{M}) \cup
G_n'(\mathcal{M})$ and $G_n^{\text{prod}}(\mathcal{M}) \cup G_n'(\mathcal{M})$ by the canonical projection $V_n(\mathcal{M}) \rightarrow G_n(\mathcal{M})$ are $V_n^{\text{prod}}(\mathcal{M}) \cup V_n^{\text{deg}}(\mathcal{M}) \cup V_n'(\mathcal{M})$ and $V_n^{\text{prod}}(\mathcal{M}) \cup V_n'(\mathcal{M})$, and since the projection is a $U_n(F)$-principal bundle with a compact total space, there exists some barycentric subdivision of each relative CW-complex structure that lifts, and we can thus define a structure of finite relative $U_n(F)$-CW-complex for each pair $\bigl(V_n(\mathcal{M}),V_n^{\text{prod}}(\mathcal{M}) \cup V_n^{\text{deg}}(\mathcal{M}) \cup V_n'(\mathcal{M})\bigr)$ and $\bigl(V_n(\mathcal{M}),V_n^{\text{prod}}(\mathcal{M}) \cup V_n'(\mathcal{M})\bigr)$.
We finish with another technical result. Let $(p,q,r)\in (\mathbb{N}^*)^3$ such that $p+q+r=n$. We define $$V_N^{(p,q,r)}(\mathcal{M}) :=
\left(\underset{k=0}{\overset{M}{\prod}}\left[V_p\left(\underset{1\leq i \leq p}{\overset{\bot}{\bigoplus}}E_k^{(i)}\right)
\times V_q\left(\underset{p+1\leq i \leq p+q}{\overset{\bot}{\bigoplus}}E_k^{(i)}\right)
\times V_r\left(\underset{p+q+1\leq i \leq n}{\overset{\bot}{\bigoplus}}E_k^{(i)}\right)
\right]\right).U_n(F).$$
\[productintersection\] Let $(p,q) \in (\mathbb{N}^*)^2$ and $(p',q') \in (\mathbb{N}^*)^2$ such that $n=p+q=p'+q'$ and $p<p'$. Then: $V_n^{(p,q)}(\mathcal{M}) \cap V_n^{(p',q')}(\mathcal{M}) = V_n^{(p,p'-p,q')}(\mathcal{M})$.
We let $G_n^{(p,p'-p,q')}(\mathcal{M})$ denote the image of $V_n^{(p,p'-p,q')}(\mathcal{M})$ by the canonical projection $\pi : V_n(\mathcal{M}) \rightarrow G_n(\mathcal{M})$. It suffices to check that $G_n^{(p,q)}(\mathcal{M}) \,\cap\, G_n^{(p',q')}(\mathcal{M})
=G_n^{(p,p'-p,q')}(\mathcal{M})$.
Let $x=x_0 \overset{f_0}{\rightarrow} x_1 \overset{f_1}{\rightarrow}
\dots \overset{f_{m-1}}{\rightarrow} x_m$ be an element of $G_n(\mathcal{M})$. Then $x \in G_n^{(p,p'-p,q')}$ if and only if, $x_k=x_k^{[\{1,\dots,p\}]} \oplus x_k^{[\{p+1,\dots,p'\}]} \oplus x_k^{[\{p'+1,\dots,n\}]}$ for every $k \in [m]$, and, for every $k \in [m-1]$, one has $$f_k\bigl(x_k^{[\{1,\dots,p\}]}\bigr) =x_{k+1}^{[\{1,\dots,p\}]}, \quad
f_k\bigl(x_k^{[\{p+1,\dots,p'\}]}\bigr) =x_{k+1}^{[\{p+1,\dots,p'\}]} \quad \text{and} \quad
f_k\bigl(x_k^{[\{p'+1,\dots,n\}]}\bigr) =x_{k+1}^{[\{p'+1,\dots,n\}]}.$$ We are thus reduced to the following easy lemma.
Let $E$ be a vector space, and $E=E_1 \oplus E_2 \oplus E_3$ be a decomposition of $E$. Let $F$ be a linear subspace of $E$ such that $F=(F\cap E_1)\oplus (F\cap(E_2 \oplus E_3))$ and $F= (F\cap(E_1 \oplus E_2))\oplus (F\cap E_3)$. Then $F=(F\cap E_1) \oplus (F\cap E_2) \oplus (F\cap E_3)$.
We denote by $\pi_1$ (resp. $\pi_2$, resp. $\pi_3$) the projection on $E_1$ alongside $E_2 \oplus E_3$ (resp. on $E_2$ alongside $E_1 \oplus E_3$, resp. on $E_3$ alongside $E_1\oplus E_2$).\
Then $\pi_1+\pi_2$ is the projection on $E_1\oplus E_2$ alongside $E_3$, and $\pi_2+\pi_3$ is the projection on $E_2\oplus E_3$ alongside $E_1$. The assumptions on $F$ show that, for any $x \in F$, all the vectors $\pi_1(x)$, $(\pi_2+\pi_3)(x)$, $\pi_3(x)$ and $(\pi_1+\pi_2)(x)$ belong to $F$. It follows that $\forall x \in F,\; (\pi_1(x),\pi_2(x),\pi_3(x))\in F^3$, which yields the claimed result.
### Starting the induction
We let $p \in \mathbb{N}$, and $\psi^\partial=(\psi_n^\partial)_{n\in \mathbb{N}}$ be a family which satisfies the conditions of conjecture \[mighty\].
We are going to construct a family $(\psi_n)_{n \in \mathbb{N}}$ by a double induction process on both $m$ and $n$. Since $V_0(\mathcal{H})=*$, we define $\psi_{0,m}$ for every $m \in \mathbb{N}$ as the trivial map . This yields a morphism $\psi_0$ of simplicial $U_0(F)$-spaces which clearly satisfies conditions (i) to (vi), and of course $\psi_0^\partial$ is a restriction of $\psi_0$.
For every non negative integer $n \in \mathbb{N}$, we define $\psi_{n,0}$ by condition (i), then conditions (iii), (iv), (v) and (vi) are easily seen to be true, and, since $\psi^\partial_{n,0}$ also satisfies condition (i), $\psi^\partial_{n,0}$ is a restriction of $\psi_{n,0}$.
### The induction hypothesis
We now fix a pair $(N,M) \in (\mathbb{N}^*)^2$, we define $I(N,M):=\bigl([N-1] \times \mathbb{N}\bigr) \cup \bigl(\{N\} \times [M-1]\bigr)$, and we assume that we have a family $(\psi_{n,m})_{(n,m)\in I(N,M)}$ such that:
- $\psi_{n,m} : V_n(m) \times \Delta^m \times \Delta^p \longrightarrow V_n(\mathcal{H})$ is a $U_n(F)$-map for every $(n,m)\in I(N,M)$;
- $\psi_n$ is a morphism of simplicial sets for every $n<N$;
- For every $(m,m')\in [M-1]^2$ and every morphism $\tau : [m'] \rightarrow [m]$ in $\Delta$, the square $$\begin{CD}
V_n(m) @>{\psi_{n,m}}>> {\operatorname{Hom}}(\Delta^m \times \Delta^p,V_n(\mathcal{H})) \\
@VV{\tau_*}V @VV{\tau_*}V \\
V_n(m') @>{\psi_{n,m'}}>> {\operatorname{Hom}}(\Delta^{m'} \times \Delta^p,V_n(\mathcal{H}))
\end{CD}$$ is commutative;
- The restriction of $\psi_{n,m}$ to $V_n(m) \times \Delta^m \times \partial\Delta^p$ is $\psi_{n,m}^\partial$ for every $(n,m)\in I(N,M)$;
- Condition (i) is satisfied;
- Condition (ii) is satisfied for every $4$-tuple $(n,n',m,f)$ such that $(n,m) \in I(N,M)$ and $(n',m) \in I(N,M)$;
- Condition (iii) is satisfied for every $4$-tuple $(n,n',m,f)$ such that $(n,m) \in I(N,M)$, $(n',m) \in I(N,M)$ and $n+n'\leq N$;
- Conditions (iv), (v) and (vi) are satisfied for every triple $(n,m,f)$ such that $(n,m) \in I(N,M)$.
### The requirements
We need to build a $U_N(F)$-map $\psi_{N,M}: V_N(M) \times \Delta^M \times \Delta^p \longrightarrow V_N(\mathcal{H})$ such that
- The restriction of $\psi_{N,M}$ to $V_N(M) \times \Delta^M \times \partial\Delta^p$ is $\psi_{N,M}^\partial$;
- For every $i \in [M-1]$, the square $$\begin{CD}
V_N(M-1) @>{\psi_{N,M-1}}>> {\operatorname{Hom}}(\Delta^{M-1} \times \Delta^p,V_N(\mathcal{H})) \\
@VV{s_i^M}V @VV{s_i^M}V \\
V_N(M) @>{\psi_{N,M}}>> {\operatorname{Hom}}(\Delta^M \times \Delta^p,V_N(\mathcal{H}))
\end{CD}$$ is commutative;
- For every $i \in [M]$, the square $$\begin{CD}
V_N(M) @>{\psi_{N,M}}>> {\operatorname{Hom}}(\Delta^M \times \Delta^p,V_N(\mathcal{H})) \\
@VV{d_i^{M-1}}V @VV{d_i^{M-1}}V \\
V_N(M-1) @>{\psi_{N,M-1}}>> {\operatorname{Hom}}(\Delta^{M-1} \times \Delta^p,V_N(\mathcal{H}))
\end{CD}$$ is commutative.
- Condition (iii) is satisfied for every $4$-tuple $(n,n',m,f)$ such that $m=M$, $n \leq N$, $n' \leq N$ and $n+n'= N$.
- Conditions (iv), (v) and (vi) are satisfied by the family $(\psi_{n,m})_{(n,m)\in I(N,M+1)}$ for every triple $(n,m,f)$ with $(n,m)=(N,M)$.
### The basic strategy to construct $\psi_{N,M}$
We fix a map $f : [M] \rightarrow \mathbb{N}$, and we build $\psi_{N,M}$ on $V_N(f)\times \Delta^M \times \Delta^p$. We have to distinguish between three cases, whether $f$ is degenerate but non-constant, $f$ is constant, or $f$ is non-degenerate.
In the first case, the definition is completely forced by condition (v), and all that needs to be done is check that the requirements are satisfied by this definition. In the other two cases, the requirements force the definition on subsets of $V_N(f)\times \Delta^M \times \Delta^p$. What we first do is check that those definitions are compatible. We then find that they help define $\psi_{N,M}$ on a subset of $V_N(f)\times \Delta^M \times \Delta^p$. In order to complete the definition, we use an extension argument which relies on practical consequences of our conjecture on triangulations.
### The case $f$ is degenerate and non-constant
We assume here that $f$ is degenerate and non-constant. We denote by $g : [k] \rightarrow \mathbb{N}$ its root and by $\sigma : [M] \twoheadrightarrow [k]$ its reduction, and for all $i \in [k]$, we set $n_i:=\# \sigma^{-1}\{i\}$. Notice, since $0<k<n$, that the definition of $\psi_{N,M}$ on $V_N(f)\times \Delta^M \times \Delta^p$ is forced by condition (v) for $(N,M,f)$.
Since $k>0$, we have $n_i-1 <M$ for all $i\in [k]$. Since $(\sigma \natural i)_*(f)$ is constant and its value is $g(i)$ for every $i \in [k]$, we deduce from condition (iv) applied to $(N,n_i,(\sigma \natural i)_*(f))$ that the first requirement in condition (v) holds for $(N,M,f)$.\
It follows that we may define a map $$\begin{gathered}
V_N(f) \times (G_\sigma(\delta) \times \Delta^{k'}) \times \Delta^p
{\overset{(\sigma \natural \delta)_*\times {\operatorname{id}}}{{\;{\count255=0 \loop \relbar\mathrel{\mkern-6mu} \advance\count255 by1\ifnum\count255<5\repeat\rightarrow}\;}}}
\left(\underset{i=0}{\overset{k'}{\prod}}(V_N(\mathcal{H}_{g(\delta(i))}))^{n_{\delta(i)}}
\times \Delta^{n_{\delta(i)}-1} \times \Delta^p \right)
\times \Delta^{k'} \times \Delta^p \\
{\overset{\underset{i=0}{\overset{k'}{\prod}}\psi_{N,n_{\delta(i)}-1}}{{\;{\count255=0 \loop \relbar\mathrel{\mkern-6mu} \advance\count255 by1\ifnum\count255<5\repeat\rightarrow}\;}}}
V_N(\delta_*(g)) \times \Delta^{k'} \times \Delta^p
\overset{\psi_{N,k'}}{\longrightarrow} V_N(\mathcal{H})\end{gathered}$$ for every $\delta : [k'] \hookrightarrow [k]$. By the induction hypothesis applied to $\psi_{N,k'}$ and $\psi_{N,n_i-1}$ for every $i \in [k']$, this map is a $U_N(F)$-map.\
Let $\varphi=\xymatrix{
[k'] \ar@{^{(}->}[dr]^{\delta} \ar@{^{(}->}[rr]^{\delta''} & & [k''] \ar@{_{(}->}[dl]^{\delta'} \\
& [k]}$ be a morphism in $\Delta^*\downarrow [k]$. Let then $\mathbf{B} \in V_N(f)$, and for every $i \in [k]$, set $\mathbf{B}_i:=(\sigma \natural i) _*(\mathbf{B})$. Let also $(t_0,\dots,t_{k'})\in \Delta^{k'}$, $\mathbf{t}=(\mathbf{t}_0,\dots,\mathbf{t}_{k''}) \in G_\sigma(\delta')$ and $x \in \Delta^p$. Then $(\delta'')_*(\mathbf{t})=(\mathbf{t}_{\delta''(i)})_{0 \leq i \leq k'}$.\
By the induction hypothesis, the square $$\begin{CD}
V_N(k'') @>{\psi_{N,k''}}>> {\operatorname{Hom}}(\Delta^{k''} \times \Delta^p,V_N(\mathcal{H})) \\
@VV{\delta''_*}V @VV{\delta_*''}V \\
V_N(k') @>{\psi_{N,k'}}>> {\operatorname{Hom}}(\Delta^{k'} \times \Delta^p,V_N(\mathcal{H}))
\end{CD}$$ is commutative, and we deduce that $$\begin{aligned}
\psi_{N,k'}\left(\left(\psi_{N,n_{\delta(i)}-1}\left(\mathbf{B}_{\delta(i)},\mathbf{t}_{\delta''(i)},x\right)\right)_{0 \leq i \leq k'}
,t,x \right) & = \psi_{N,k'}\left((\delta'')_*\left((\psi_{N,n_{\delta'(i)}-1}
\left(\mathbf{B}_{\delta'(i)},\mathbf{t}_i,x)\right)_{0 \leq i \leq k''}
,t,x \right)\right) \\
& = \psi_{N,k''}\left(\left(\psi_{N,n_{\delta'(i)}-1}
(\mathbf{B}_{\delta'(i)},\mathbf{t}_i,x)\right)_{0 \leq i \leq k''}
,(\delta'')^*(t),x \right).\end{aligned}$$ Hence the previous maps are compatible, and it follows from Proposition \[superglue\] that they yield an equivariant map $\psi_{N,M} : V_N(f) \times \Delta^M \times \Delta^p \longrightarrow V_N(\mathcal{H})$. Since $V_N(f)$ is filtered by an increasing sequence of compact spaces, we also deduce from Proposition \[superglue\] that $\psi_{N,M}$ is actually a continuous map.
Since condition (v) is now checked for $(n,M,f)$ for any $n\leq N$, it follows from the induction hypothesis[^4] that condition (ii) holds for $(n,n',M,f)$, for every $(n,n')$ such that $n+n'=N$.
Since condition (v) is satisfied by both $\psi_{N,M}$ and $\psi_{N,M}^\partial$, it follows from the induction hypothesis[^5] that the restriction of $\psi_{N,M}$ to $V_N(f) \times \Delta^M \times \partial \Delta^p$ is $\psi^\partial_{N,M}$.
Let $u \in S(k)$, $\mathbf{B} \in V_N(f)$, $t \in (\sigma^*)^{-1}(\Delta_u)$ and $x \in \Delta^p$. Let $\mathbf{t}=(\mathbf{t}_i)_{0\leq i \leq n} \in \underset{i=0}{\overset{k}{\prod}}\Delta^{n_i-1}$ such that $t=\lambda_\sigma(\mathbf{t},\sigma^*(t))$. By the induction hypothesis, condition (iv) holds for the triple $(N,k,g)$; we deduce that $$\psi_{N,k}\left(\left(\psi_{N,n_i-1}\left(\mathbf{B}_i,\mathbf{t}_i,x\right)\right)_{0 \leq i \leq k},\sigma^*(t),x \right)
\in V_N(\mathcal{H}_{n,u,g})$$ and we conclude that $\psi_{N,M}(\mathbf{B},t,x) \in V_N(\mathcal{H}_{n,u,g})$. This proves that condition (iv) holds for the triple $(N,M,f)$.
We finish by checking the compatibility with face and degeneracy maps. Let $\tau : [m] \rightarrow [M]$ be any morphism in $\Delta$, with $m<M$. Let $\tau=\delta' \circ \sigma'$ be the decomposition of $\sigma \circ \tau$ into the composite of an epimorphism and a monomorphism. $$\xymatrix{
[k'] \ar@{^{(}->}[r]^{\delta'} & [k] \ar@{^{(}->}[r]^{g} & \mathbb{N}. \\
[m] \ar@{>>}[u]^{\sigma'} \ar[r]^{\tau} & [M] \ar@{>>}[u]^{\sigma} \ar[ur]_f
}$$ Then $g \circ \delta'$ is the root of $f \circ \tau$, and $\sigma'$ is its reduction. We set $n'_i:=\#(\sigma')^{-1}\{i\}$ for every $i\in [k']$. Let $(\tau_i)_{0 \leq i \leq k'}$ denote the decomposition of $\tau$ over $\sigma$. Let $\mathbf{B} \in V_N(f)$. For every $i \in [k]$, set $\mathbf{B}_i:=(\sigma \natural i) _*(\mathbf{B})$, and for every $i \in [k']$, set $\mathbf{B}'_i:=(\sigma' \natural i) _*(\tau_*(\mathbf{B}))=(\tau_i)_*((\delta'\natural \sigma)_*(\mathbf{B}))$. Let also $(t_0,\dots,t_m)\in \Delta^m$, $\mathbf{t}=(\mathbf{t}_0,\dots,\mathbf{t}_{k'}) \in G_{\sigma'}({\operatorname{id}}_{[k']})$ and $x \in \Delta^p$.\
By the induction hypothesis, the square $$\begin{CD}
V_N(n_{\delta'(i)}-1) @>{\psi_{N,n_{\delta(i)}-1}}>> {\operatorname{Hom}}(\Delta^{n_{\delta'(i)}-1} \times \Delta^p,V_N(\mathcal{H})) \\
@VV{(\tau_i)_*}V @VV{(\tau_i)_*}V \\
V_N(n'_i-1) @>{\psi_{N,n'_i-1}}>> {\operatorname{Hom}}(\Delta^{n'_i-1} \times \Delta^p,V_N(\mathcal{H}))
\end{CD}$$ is commutative for every $i \in [k']$.\
It follows that $$\psi_{N,k'}\left(\left(\psi_{N,n'_i-1}\left(\mathbf{B}'_i,\mathbf{t}_i,x\right)\right)_{0 \leq i \leq k'},t,x \right)
= \psi_{N,k'}\left(\left(\psi_{N,n_{\delta'(i)}-1}\left(((\delta' \natural \sigma)\natural i)_*(\mathbf{B}),\tau_i^*(\mathbf{t}_i)
,x\right)\right)_{0 \leq i \leq k'},t,x \right),$$ and we deduce from Proposition \[splitting\] and condition (v) that $\psi_{N,m}(\tau_*(\mathbf{B}),t',x)=\psi_{N,M}(\mathbf{B},\tau^*(t'),x)$, where $t'=\lambda_{\sigma'}(\mathbf{t},t) \in \Delta^m$. This proves that the square $$\begin{CD}
V_N(f) @>{\psi_{N,M}}>> {\operatorname{Hom}}(\Delta^M \times \Delta^p,V_N(\mathcal{H})) \\
@VV{\tau_*}V @VV{\tau_*}V \\
V_N(\tau_*(f)) @>{\psi_{N,m}}>> {\operatorname{Hom}}(\Delta^m \times \Delta^p,V_N(\mathcal{H}))
\end{CD}$$ is commutative.
Finally, let $\tau : [M] \twoheadrightarrow [m]$ and $f_1 : [m] \rightarrow \mathbb{N}$ such that $f =f_1 \circ \tau$. Then $\sigma=\sigma' \circ \tau$, where $\sigma' : [m] \twoheadrightarrow [k]$ is the reduction of $f_1$ (and $g$ is its root):
$$\xymatrix{
[M] \ar@{>>}[r]^{\tau} \ar[rd]_f & [m] \ar@{>>}[r]^{\sigma'} \ar[d]_{f_1} & [k] \ar[dl]_g \\
& \mathbb{N}.
}$$ For every $i\in [k]$, set $n'_i:=\#(\sigma')^{-1}\{i\}$. Let $(\tau_i)_{0 \leq i \leq k}$ denote the decomposition of $\tau$ over $\sigma'$. Let $\mathbf{B} \in V_N(f_1)$. For every $i \in [k]$, set $\mathbf{B}_i:=(\sigma' \natural i) _*(\mathbf{B})$ and $\mathbf{B}'_i:=(\sigma \natural i) _*(\tau_*(\mathbf{B}))=(\tau_i)_*(\sigma_*(\mathbf{B}))$. Let also $(t_0,\dots,t_M)\in \Delta^M$, $\mathbf{t}=(\mathbf{t}_0,\dots,\mathbf{t}_{k}) \in G_{\sigma}({\operatorname{id}}_{[k]})$ and $x \in \Delta^p$.\
By the induction hypothesis, the square $$\begin{CD}
V_N(n'_i-1) @>{\psi_{N,n'_i-1}}>> {\operatorname{Hom}}(\Delta^{n'_i-1} \times \Delta^p,V_N(\mathcal{H})) \\
@VV{(\tau_i)_*}V @VV{(\tau_i)_*}V \\
V_N(n_i-1) @>{\psi_{N,n_i-1}}>> {\operatorname{Hom}}(\Delta^{n_i-1} \times \Delta^p,V_N(\mathcal{H}))
\end{CD}$$ is commutative for every $i \in [k]$. It follows that $$\psi_{N,k}\left(\left(\psi_{N,n_i-1}\left(\mathbf{B}'_i,\mathbf{t}_i,x\right)\right)_{0 \leq i \leq k},t,x \right)
= \psi_{N,k}\left(\left(\psi_{N,n'_i-1}\left(\mathbf{B}_i,\tau_i^*(\mathbf{t}_i),x\right)\right)_{0 \leq i \leq k},t,x \right),$$ and we deduce from Proposition \[splitting\] and condition (v) that $\psi_{N,M}(\tau_*(\mathbf{B}),t',x)=\psi_{N,m}(\mathbf{B},\tau^*(t'),x)$, where $t'=\lambda_\sigma(\mathbf{t},t) \in \Delta^M$. This proves that the square $$\begin{CD}
V_N(f_1) @>{\psi_{N,m}}>> {\operatorname{Hom}}(\Delta^m \times \Delta^p,V_N(\mathcal{H})) \\
@VV{\tau_*}V @VV{\tau_*}V \\
V_N(f) @>{\psi_{N,M}}>> {\operatorname{Hom}}(\Delta^M \times \Delta^p,V_N(\mathcal{H}))
\end{CD}$$ is commutative.
### The case $f$ is constant
We now assume $f$ is constant and we set $n_0:=f(0)$. The root of $f$ is the map $[0] \rightarrow \mathbb{N}$ whose image is $\{n_0\}$, and its reduction is the canonical map $[M] \twoheadrightarrow [0]$. For conditions (iv) and (v) to be satisfied, it suffices to build $\psi_{N,M} : V_N(f) \times \Delta^M \times \Delta^p \longrightarrow V_{N,n_0}$. Notice that condition (vi) is irrelevant here.
Let $i\in [M]$. We define $\psi_{N,M}$ on $V_N(f) \times (\delta_i^M)^*(\Delta^{M-1}) \times \Delta^p$ by $$\forall (\mathbf{B},t,x) \in V_N(f) \times \Delta^{M-1} \times \Delta^p, \quad
\psi_{N,M}(\mathbf{B},(\delta_i^{M-1})^*(t),x):=\psi_{N,M-1}(d_i^{M-1}(\mathbf{B}),t,x).$$ By compatibility with the face maps at lower levels and condition (iv) at lower levels, these definitions are compatible, and they yield a $U_N(F)$-map $$\boxed{\psi_{N,M} : V_N(f) \times \partial \Delta^M \times \Delta^p \longrightarrow V_{N,n_0}}.$$
Let $i \in [M-1]$. We define $\psi_{N,M}$ on $s_i^M((V_{N,n_0})^{[M-1]}) \times \Delta^M \times \Delta^p$ by $$\forall (\mathbf{B},t,x) \in s_i^M((V_{N,n_0})^{[M-1]}) \times \Delta^M \times \Delta^p,
\psi_{N,M}(\mathbf{B},t,x):=\psi_{N,M-1}(d_i^{M-1}(\mathbf{B}),(\sigma_i^M)^*(t),x).$$ By the induction hypothesis, this is a $U_N(F)$-map. Those maps are compatible. Let $(i,j)\in [M-1]^2$ be such that $i<j$, and let $(\mathbf{B},t,x) \in \left(s_i^{M-1}((V_{N,n_0})^{[M-1]}) \cap s_j^{M-1}((V_{N,n_0})^{[M-1]})\right)
\times \Delta^M \times \Delta^p$.\
Let $\mathbf{B'}\in (V_{N,n_0})^{[M-2]}$ be such that $\mathbf{B}=s_i^{M-1}(s_{j-1}^{M-2}(\mathbf{B'}))=s_j^{M-1}(s_i^{M-2}(\mathbf{B'}))$. Then, by the induction hypothesis, $$\begin{aligned}
\psi_{N,M-1}(d_i^M(\mathbf{B}),(\sigma_i^{M-1})^*(t),x) & = \psi_{N,M-1}(s_{j-1}^{M-2}(\mathbf{B}'),(\sigma_i^{M-1})^*(t),x) \\
& = \psi_{N,M-1}(\mathbf{B}',(\sigma_{j-1}^{M-2} \circ \sigma_i^{M-1})^*(t),x) \\
& = \psi_{N,M-1}(\mathbf{B}',(\sigma_{i}^{M-2} \circ \sigma_j^{M-1})^*(t),x) \\
& = \psi_{N,M-1}(s_i^{M-2}(\mathbf{B}'),(\sigma_j^{M-1})^*(t),x) \\
& = \psi_{N,M-1}(d_j^M(\mathbf{B}),(\sigma_j^{M-1})^*(t),x).\end{aligned}$$ We deduce that the preceding maps yield an equivariant map $$\boxed{\psi_{N,M} : V_N^{\text{deg}}(f) \times \Delta^M \times \Delta^p \longrightarrow V_{N,n_0},}$$ where $V_N^{\text{deg}}(f):=\underset{i \in [M-1]}{\prod}s_i^M((V_{N,n_0})^{[M-1]})$ (the continuity of this map will be checked later on).
We now check that this map is compatible with the preceding one.\
Let $(i,j)\in [M-1]\times [M]$, and $(\mathbf{B},t,x) \in s_i^M((V_{N,n_0})^{[M-1]}) \times (\delta_j^{M-1})^*(\Delta^{M-1}) \times \Delta^p$, $t'\in \Delta^{M-1}$ such that $t=(\delta_j^{M-1})^*(t')$, and $\mathbf{B}'\in (V_{N,n_0})^{[M-1]}$ such that $\mathbf{B}=s_i^M(\mathbf{B'})$. Then $d_i^{M-1}(\mathbf{B})=\mathbf{B}'$, and we deduce from the induction hypothesis that $$\begin{aligned}
\psi_{N,M-1}(d_i^{M-1}(\mathbf{B}),(\sigma_i^M)^*(t),x) & =
\psi_{N,M-1}(\mathbf{B}',(\sigma_i^M\circ \delta_j^{M-1})^*(t'),x) \\
& = \psi_{N,M-1}((d_j^{M-1} \circ s_i^M)(\mathbf{B}'),t',x) \\
& = \psi_{N,M-1}(d_j^{M-1}(\mathbf{B}),t',x).\end{aligned}$$ This proves that the two previous definitions are compatible.
Now, let $p \in \{1,\dots,N-1\}$ and set $q:=N-p$. Should we identify $V_p(f) \times V_q(f+p)$ with a subspace of $V_N(f)$, we may use the induction hypothesis concerning condition (ii) to define $\psi_{N,M}$ on $V_p(f) \times V_q(f+p) \times (\Delta^M \times \Delta^p)$ by: $$\forall (\mathbf{B},\mathbf{B}',t)\in V_p(f) \times V_q(f+p) \times (\Delta^M \times \Delta^p), \quad
\psi_{N,M}(\mathbf{B},\mathbf{B}',t):=\bigl(\psi_{p,M}(\mathbf{B},t),\psi_{q,M}(\mathbf{B}',t)\bigr).$$ Since $\psi_p$ (resp. $\psi_q$) is a morphism of simplicial $U_p(F)$-spaces (resp. of simplicial $U_q(F)$-spaces), we deduce that we have just defined a $(U_p(F) \times U_q(F))$-map. By the induction hypothesis, it is compatible with the preceding ones. Set $$V_N^{(p,q)}(f):=(V_p(f) \times V_q(f+p)).U_N(F) \subset V_N(f).$$ The canonical $U_N(F)$-map $$(V_p(f) \times V_q(f+p)) \times U_N(F) \longrightarrow V_N^{(p,q)}(f)$$ is an identification map (indeed, for every $l\in \mathbb{N}^*$, $(V_p^{(l)}(f) \times V_q^{(l)}(f+p)) \times U_N(F) \longrightarrow
V_N^{(p,q)}(f)\cap V_N^{(l)}(f)$ is a continuous surjection between compact spaces). We deduce that the canonical map $$(V_p(f) \times V_q(f+p)) \times_{U_p(F) \times U_q(F)} U_N(F) \longrightarrow V_N^{(p,q)}(f)$$ is an equivariant homeomorphism. It follows that the following definition extends $\psi_{N,M}$ as a $U_N(F)$-map on $V_N^{(p,q)}(f) \times \Delta^M \times \Delta^p$: $$\forall (\mathbf{B},\mathbf{B}',t,\mathbf{M}) \in V_p(f) \times V_q(f+p)
\times (\Delta^M \times \Delta^p) \times U_N(F), \quad
\psi_{N,M}((\mathbf{B},\mathbf{B}').\mathbf{M},t)=\psi_{N,M}((\mathbf{B},\mathbf{B}'),t).\mathbf{M}.$$
Let $p'\in \{1,\dots,N-1\}$ such that $p<p'$, and set $q':=N-q$. We have to check that the respective definitions of $\psi_{N,M}$ on $V_N^{(p,q)}(f) \times \Delta^M \times \Delta^p$ and $V_N^{(p',q')}(f) \times \Delta^M \times \Delta^p$ are compatible.\
Let $(\mathbf{B},t,x)\in (V_N^{(p,q)}(f) \cap V_N^{(p',q')}(f))\times \Delta^M \times \Delta^p$. Proposition \[productintersection\] then yields a triple $(\mathbf{B}_1,\mathbf{B}_2,\mathbf{B}_3)\in V_p(f) \times V_{p'-p}(f+p) \times V_{q'}(f+p')$ and $\mathbf{M} \in U_N(F)$ such that $\mathbf{B}=(\mathbf{B}_1,\mathbf{B}_2,\mathbf{B}_3).\mathbf{M}$.
We deduce from the induction hypothesis that $$\begin{aligned}
(\psi_p(\mathbf{B}_1,t,x),\psi_q((\mathbf{B}_2,\mathbf{B}_3),t,x)) & = (\psi_p(\mathbf{B}_1,t,x),\psi_{p'-p}(\mathbf{B}_2,t,x),
\psi_{q'}(\mathbf{B}_3,t,x)) \\
(\psi_p(\mathbf{B}_1,t,x),\psi_q((\mathbf{B}_2,\mathbf{B}_3),t,x)) & =
(\psi_{p'}((\mathbf{B}_1,\mathbf{B}_2),t,x),\psi_{q'}(\mathbf{B}_3,t,x))\end{aligned}$$ This proves that the previous maps are all compatible, and that they therefore yield a $U_N(F)$-map: $$\boxed{\psi_{N,M} : V_N^{\text{prod}}(f) \times \Delta^M \times \Delta^p \longrightarrow V_{N,n_0},}$$ where $V_N^{\text{prod}}(f):= \underset{p \in \{1,\dots,N-1\}}{\bigcup} V_N^{(p,N-p)}(f)$. This map is compatible with the preceding ones.
Finally, we may define $\psi_{N,M}$ on $V_N(f)\times \Delta^M \times \partial\Delta^p$ as $\psi_{N,M}^\partial$. Since $\psi^\partial$ satisfies conditions (i) to (iv) and is a family of morphisms of simplicial spaces, we deduce from the induction hypothesis that this definition is compatible with the preceding ones.\
We have just constructed an equivariant map $$\boxed{\psi_{N,M} : \left(V_N(f) \times \partial(\Delta^M \times \Delta^p)\right) \cup
\left((V_N^{\text{prod}}(f) \cup V_N^{\text{deg}}(f)) \times \Delta^M \times \Delta^p\right) \longrightarrow
V_{N,n_0}}.$$ For every $l \in \mathbb{N}^*$, its restriction to the intersection of $V_N^{(l)}(f) \times \Delta^M \times \Delta^p$ and its domain is obtained by gluing together a finite family of continuous maps, where each map is defined on a compact subspace of $V_N^{(l)}(f) \times \Delta^M \times \Delta^p$, and is therefore continuous. Since $V_N(f)=\underset{l\in \mathbb{N}^*}{{\operatorname{\underset{\longrightarrow}{{\operatorname{colim}}}}}}\, V_N^{(l)}$, we deduce that $\psi_{N,M}$ is continuous.
By construction, it suffices to extend $\psi_{N,M}$ to a $U_N(F)$-map from $V_N(f) \times \Delta^M \times \Delta^p$ to $V_{N,n_0}$, and such an extension will fulfill all the expected conditions: indeed, the compatibility with face maps follows from the definition of $\psi_{N,M}$ on $V_N(f) \times \partial \Delta^M \times \Delta^p$; the compatibility with degeneracy maps follows from the definition of $\psi_{N,M}$ on $V_N^{\text{deg}}(f) \times \Delta^M \times \Delta^p$; it follows from the definition of $\psi_{N,M}$ on $V_N^{\text{prod}}(f) \times \Delta^M \times \Delta^p$ that condition (iii) holds; finally, the compatibility with $\psi^\partial$ comes from the definition of $\psi_{N,M}$ on $V_N(f) \times \Delta^M \times \partial\Delta^p$.
The extension is based upon the following result:
The pair $$\left(V_N(f)\times \Delta^M \times \Delta^p,\left(V_N(f) \times \partial(\Delta^M \times \Delta^p)\right) \cup
\left((V_N^{\text{prod}}(f) \cup V_N^{\text{deg}}(f)) \times (\Delta^M \times \Delta^p)\right)
\right)$$ is filtered by a sequence of relative $U_N(F)$-CW-complexes.
To start with, $(\Delta^M \times \Delta^p,\partial(\Delta^M \times \Delta^p))$ is a relative CW-complex. Also, for every $l \in \mathbb{N}^*$, we set $V_N^{\text{prod} (l)}(f):=V_N^{\text{prod}}(f)\cap V_N^{(l)}(f)$, $V_N^{\text{deg} (l)}(f):=V_N^{\text{deg}}(f)\cap V_N^{(l)}(f)$ and $$\mathcal{M}_l:=\left(\left(\underset{i=0}{\overset{N-1}{\oplus}}\mathcal{H}_{n_0+i}^{(l+1)},
\underset{i=0}{\overset{N-1}{\oplus}}\mathcal{H}_{n_0+i}^{(l)},(\mathcal{H}_{n_0+i-1}^{(l)})_{1 \leq i \leq N}\right)_{0 \leq k \leq M}
,({\operatorname{id}})_{0 \leq k \leq M-1}\right).$$ We then deduce from corollary \[CWconjecture\] - applied to $\mathcal{M}_l$ - that the pair\
$(V_N^{(l+1)}(f), V_N^{(l)}(f) \cup V_N^{\text{prod}(l+1)}(f) \cup
V_N^{\text{deg}(l+1)}(f))$ is a relative $U_N(F)$-CW-complex for every positive integer $l$. Since $V_N^{(1)}(f)=V_N^{\text{prod}(1)}(f)$, we deduce that the pair $(V_N(f), V_N^{\text{prod}}(f) \cup V_N^{\text{deg}}(f))$ is filtered by a sequence of $U_N(F)$-CW-complexes, which proves the claimed result.
Finally, $V_{N,n_0}$ is contractible, and $U_N(F)$ acts freely on $V_N(f) \times \Delta^M \times \Delta^p$; we may thus extend $\psi_{N,M}$ to obtain a $U_N(F)$-map $\psi_{N,M} : V_N(f) \times \Delta^M \times \Delta^p \longrightarrow V_{N,n_0}$. We choose such an extension, and this finishes the construction in the case $f$ is constant.
### The case $f$ is non-degenerate
We finally assume that $f:[M] \rightarrow \mathbb{N}$ is non-degenerate.
Let $i\in [M]$. We define $\psi_{N,M}$ on $V_N(f) \times (\delta_i^{M-1})^*(\Delta^{M-1}) \times \Delta^p$ by: $$\forall (\mathbf{B},t,x) \in V_N(f) \times \Delta^{M-1} \times \Delta^p, \quad
\psi_{N,M}(\mathbf{B},(\delta_i^{M-1})^*(t),x):=\psi_{N,M-1}(d_i^{M-1}(\mathbf{B}),t,x).$$ By compatibility with the face maps at lower levels and since condition (iv) is satisfied at lower levels, those definitions are compatible and, since condition (iv) is satisfied at the lower levels, yield a $U_N(F)$-map: $$\boxed{\psi_{N,M} : V_N(f) \times \partial \Delta^M \times \Delta^p \longrightarrow V_N(\mathcal{H}).}$$ Let $u=\delta_1<\dots<\delta_{k-1}<[M]$ be a non-trivial sequence in $S(M)$, where $\delta_{k-1} : [m] \hookrightarrow [M]$. Let $\mathbf{B} \in V_N(f)$, $x \in \Delta_u$, $z \in \Delta^p$, and $(y,t)\in \Delta^m \times [0,1]$ such that $x=r_M(\delta_{k-1}^*(y),t)$. Then $y \in \Delta_{u'}$ and we deduce from the induction hypothesis that $\psi_{N,m}((\delta_{k-1})_*(\mathbf{B}),y,z) \in V_N(\mathcal{H}_{N,u',(\delta_{k-1})_*(f)})$. We deduce from the previous definition and the compatibility with face maps at lower levels that $\mathbf{B}':=
\psi_{N,M}(\mathbf{B},(\delta_{k-1})^*(y),z)=\psi_{N,m}((\delta_{k-1})_*(\mathbf{B}),y,z) \in V_N(\mathcal{H}_{N,u',(\delta_{k-1})_*(f)})$. It follows that we may define $$\psi_{N,M}(\mathbf{B},x,z):=\cos\left(\frac{\pi t}{2}\right)\mathbf{B'}+\sin\left(\frac{\pi t}{2}\right).\varphi_{N,u,f}(\mathbf{B'})
\in V_N(\mathcal{H}_{N,u,f}),$$ and this yields a $U_N(F)$-map $V_N(f) \times \Delta_u \times \Delta^p \longrightarrow V_N(\mathcal{H}_{N,u,f})$ which is compatible with the previous one.
Let $v=\delta'_1<\dots<\delta_{k'-1}<[M]\in S(M)$ be another non-trivial class. We need to check that the respective definitions of $\psi_{N,M}$ on $V_N(f) \times \Delta_u \times \Delta^p$ and $V_N(f) \times \Delta_v \times \Delta^p$ agree. By Proposition \[intersection\], it suffices to do this when $u \subset v$. Let $\mathbf{B} \in V_N(f)$, $x \in \Delta_u$, and $(y,t)\in \partial \Delta^M \times \Delta^p$ such that $x=r_M(y,t)$, and $\mathbf{B}' :=\psi_{N,M}(\mathbf{B},y,z)$. Then $y \in \Delta_u$, and we deduce from the preceding remarks that $\mathbf{B}' \in V_N(\mathcal{H}_{N,u',(\delta_{k'-1})_*(f)})$. From the remarks at the end of Section \[10.4.3\], we derive that $\varphi_{N,u,f}(\mathbf{B'})=\varphi_{N,v,f}(\mathbf{B'})$, and it follows that $$\cos\left(\frac{\pi t}{2}\right)\mathbf{B'}+\sin\left(\frac{\pi t}{2}\right).\varphi_{N,u,f}(\mathbf{B'})
=\cos\left(\frac{\pi t}{2}\right)\mathbf{B'}+\sin\left(\frac{\pi t}{2}\right).\varphi_{N,v,f}(\mathbf{B'}).$$ We deduce that all the definitions are compatible so far, and it follows from Proposition \[glue\] that they yield a $U_N(F)$-map: $$\boxed{\psi_{N,M} : V_N(f) \times \left(\underset{0 \leq i \leq M}{\cup}\Delta_i^M\right)\times \Delta^p \longrightarrow V_N(\mathcal{H})
.}$$ By construction, this map is compatible with face maps (and therefore condition (v) holds), and condition (vi) also holds. Moreover, condition (iv) holds for every non-trivial sequence $u \in S(M)$. The compatibility with degeneracy maps is not relevant here.
Let now $p \in \{1,\dots,N-1\}$ and $q:=N-p$. Identifying $V_p(f) \times V_q(f+p)$ with a subspace of $V_N(f)$, we may use the induction hypothesis concerning condition (ii) to define $\psi_{N,M}$ on $(V_p(f) \times V_q(f+p)) \times (\Delta^M \times \Delta^p)$ by: $$\forall (\mathbf{B},\mathbf{B}',t)\in V_p(f) \times V_q(f+p) \times (\Delta^M \times \Delta^p), \;
\psi_{N,M}(\mathbf{B},\mathbf{B}',t)=(\psi_{p,M}(\mathbf{B},t),\psi_{q,M}(\mathbf{B}',t)).$$ Since $\psi_p$ (resp. $\psi_q$) is a morphism of simplicial $U_p(F)$-spaces (resp. of simplicial $U_q(F)$-spaces), we deduce that we have just defined a $(U_p(F) \times U_q(F))$-map.\
Set $$V_N^{(p,q)}(f):=(V_p(f) \times V_q(f+p)).U_N(F) \subset V_N(f).$$ The canonical $U_N(F)$-map $$(V_p(f) \times V_q(f+p)) \times U_N(F) \longrightarrow V_N^{(p,q)}(f)$$ is an identification map (since for every positive integer $l$, $(V_p^{(l)}(f) \times V_q^{(l)}(f+p)) \times U_N(F) \longrightarrow
V_N^{(p,q)}(f)\,\cap\, V_N^{(l)}(f)$ is a continuous surjection between compact spaces). We deduce that the canonical map $$(V_p(f) \times V_q(f+p)) \times_{U_p(F) \times U_q(F)} U_N(F) \longrightarrow V_N^{(p,q)}(f)$$ is an equivariant homeomorphism. It follows that we can extend $\psi_{N,M}$ as a $U_N(F)$-map on $V_N^{(p,q)}(f) \times \Delta^M \times \Delta^p$ by: $$\forall (\mathbf{B},\mathbf{B}',t,\mathbf{M}) \in V_p(f) \times V_q(f+p) \times (\Delta^M \times \Delta^p) \times U_N(F),
\; \psi_{N,M}((\mathbf{B},\mathbf{B}').\mathbf{M},t)=\psi_{N,M}((\mathbf{B},\mathbf{B}'),t).\mathbf{M}.$$ With the same line of reasoning as in the case $f$ is constant, those definitions are compatible and yield a $U_N(F)$-map: $$\boxed{\psi_{N,M} : V_N^{\text{prod}}(f) \times \Delta^M \times \Delta^p \longrightarrow V_N(\mathcal{H}),}$$ where $$V_N^{\text{prod}}(f):= \underset{1 \leq p \leq N-1}{\bigcup} V_N^{(p,N-p)}(f).$$ We now check that this map is compatible with the previous one. The compatibility with the definition on $V_n(F) \times \partial \Delta^M \times \Delta^p$ is a straightforward consequence of the induction hypothesis. Let $u \in S(M)$ be a non trivial class, $p \in \{1,\dots,N-1\}$ and $q:=N-p$. Let $\mathbf{B} \in V_N(f)$, $x \in \Delta_u$, $z \in \Delta^p$, and $(y,t)\in \partial\Delta^M \times [0,1]$ such that $x=r_M(y,t)$.
Let $(\mathbf{B}',\mathbf{B}'') \in V_p(f)\times V_q(f+p)$ and $\mathbf{M} \in U_N(F)$ such that $\mathbf{B}=(\mathbf{B}',\mathbf{B}'').\mathbf{M}$. By the induction hypothesis, $$\psi_{p,M}(\mathbf{B}',x,z)=\cos \left(\frac{\pi t}{2}\right)\psi_{p,M}(\mathbf{B}',y,z)
+\sin \left(\frac{\pi t}{2}\right)\varphi_{p,u,f}(\psi_{p,M}(\mathbf{B}',y,z)),$$ $$\psi_{q,M}(\mathbf{B}'',x,z)=\cos \left(\frac{\pi t}{2}\right)\psi_{q,M}(\mathbf{B}'',y,z)+
\sin \left(\frac{\pi t}{2}\right)\varphi_{q,u,f+p}(\psi_{q,M}(\mathbf{B}'',y,z)),$$ and we deduce from the last remark in Section \[10.4.3\] that $$\varphi_{N,u,f}(\psi_{p,M}(\mathbf{B}',y,z),\psi_{q,M}(\mathbf{B}'',y,z))=
\left(\varphi_{p,u,f}(\psi_{p,M}(\mathbf{B}',y,z)),\varphi_{q,u,f+p}(\psi_{q,M}(\mathbf{B}'',y,z))\right).$$ We then deduce from the compatibility of the definitions on $V_N(f) \times \partial \Delta^M \times \Delta^p$ that $$\bigl(\psi_{p,M}(\mathbf{B}',x,z),\psi_{q,M}(\mathbf{B}'',x,z)\bigr).\mathbf{M}=
\cos \left(\frac{\pi}{2}t\right)\psi_{N,M}(\mathbf{B},y,z)+
\sin \left(\frac{\pi}{2}t\right)\varphi_{N,u,f}(\psi_{N,M}(\mathbf{B},y,z)),$$ where $\psi_{N,M}(\mathbf{B},y,z)$ is given by the earlier definition.
Finally, we may define $\psi_{N,M}$ on $V_N(f)\times \Delta^M \times \partial\Delta^p$ as $\psi_{N,M}^\partial$. Since $\psi^\partial$ satisfies condition (i) to (vi) and is a family of morphisms of simplicial spaces, we deduce from the induction hypothesis that this definition is compatible with the above ones.
Proposition \[glue\] then shows that we have just constructed an equivariant map $$\boxed{\psi_{N,M} : \left(V_N(f) \times \left((\Delta^M \times \partial \Delta^p)\cup\left((\underset{0\leq i \leq M}{\bigcup}\Delta_i^M) \times \Delta^p
\right)\right)\right) \bigcup\left(V_N^{\text{prod}}(f) \times \Delta^M \times \Delta^p\right)
\longrightarrow V_N(\mathcal{H}).}$$ For every positive integer $l$ and every non-trivial $u \in S(M)$, the restriction of $\psi_{N,M}$ to the intersection of $V_N^{(l)}(f) \times \Delta_u \times \Delta^p$ and its domain is obtained by gluing together a finite family of continuous maps, where each map is defined on a compact subspace of $V_N^{(l)}(f) \times \Delta_u \times \Delta^p$: therefore, this restriction is continuous. Since $V_N(f)=\underset{l\in \mathbb{N}^*}{{\operatorname{\underset{\longrightarrow}{{\operatorname{colim}}}}}}\, V_N^{(l)}(f)$, we deduce from Proposition \[glue\] that $\psi_{N,M}$ is continuous.
Notice that $\psi_{N,M}$ maps $\Bigl(V_N(f) \times \partial (C(\Delta^M) \times \Delta^p)\Bigr)
\bigcup \left(V_N^{\text{prod}}(f) \times C(\Delta^M) \times \Delta^p\right)$ into $V_N(\mathcal{H}_{N,[M],f})$. If we can extend this map to a $U_N(F)$-map $V_N(f)\times C(\Delta^M) \times \Delta^p \longrightarrow V_N(\mathcal{H}_{N,[M],f})$, condition (iv) will be checked, and we will recover a $U_N(F)$-map: $V_N(f) \times \Delta^M \times \Delta^p \longrightarrow V_N(\mathcal{H})$ which fulfills all the requirements. That $\psi_{N,M}$ may be extended in this manner is a consequence of the following result:
The pair $$\left(V_N(f)\times C(\Delta^M) \times \Delta^p,\left(V_N(f) \times \partial(C(\Delta^M) \times \Delta^p)\right) \cup
\left(V_N^{\text{prod}}(f)\times C(\Delta^M) \times \Delta^p\right)
\right)$$ is filtered by a sequence of relative $U_N(F)$-CW-complexes.
First of all, $(C(\Delta^M) \times \Delta^p,\partial(C(\Delta^M) \times \Delta^p))$ is a relative CW-complex. For every positive integer $l$, set $V_N^{\text{prod} (l)}(f):=V_N^{\text{prod}}(f)\cap V_N^{(l)}(f)$, and $$\mathcal{N}_l:=\left(\left(\underset{i=0}{\overset{N-1}{\oplus}}\mathcal{H}_{f(k)+i}^{(l+1)},
\underset{i=0}{\overset{N-1}{\oplus}}\mathcal{H}_{f(k)+i}^{(l)},(\mathcal{H}_{f(k)+i-1}^{(l)})_{1 \leq i \leq N})\right)
_{0 \leq k \leq M},(f_k)_{0 \leq k \leq M-1}\right),$$ where, for every $k \in [M-1]$, we have chosen a linear isometry $f_k$ from $\underset{i=0}{\overset{N-1}{\oplus}}\mathcal{H}_{f(k)+i}^{(l+1)}$ to $\underset{i=0}{\overset{N-1}{\oplus}}\mathcal{H}_{f(k+1)+i}^{(l+1)}$ which maps $\underset{i=0}{\overset{N-1}{\oplus}}\mathcal{H}_{f(k)+i}^{(l)}$ to $\underset{i=0}{\overset{N-1}{\oplus}}\mathcal{H}_{f(k+1)+i.}^{(l)}$ We then deduce from conjecture \[CWconjecture\], applied to $\mathcal{N}_l$, that the pair $(V_N^{(l+1)}(f), V_N^{(l)}(f) \cup V_N^{\text{prod}(l+1)}(f))$ is a relative $U_N(F)$-CW-complex for every positive integer $l$. Since $V_N^{(1)}(f)=V_N^{\text{prod}(1)}(f)$, we deduce that the pair $(V_N(f), V_N^{\text{prod}}(f))$ is filtered by a sequence of $U_N(F)$-CW-complexes: this proves the claimed result.
Finally, $V_N(\mathcal{H}_{N,[M],f})$ is contractible, and $U_N(F)$ acts freely on $V_N(f) \times C(\Delta^M) \times \Delta^p$. It follows that we may extend $\psi_{N,M}$ to recover a $U_N(F)$-map $$\psi_{N,M} : V_N(f) \times C(\Delta^M) \times \Delta^p \longrightarrow V_N(\mathcal{H}_{N,[M],f}).$$ This finishes the construction in the case $f$ is non-degenerate (up to a choice of extension, of course), and we have thus completed our proof of Proposition \[mighty\].
The map $s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[1] \longrightarrow {\operatorname{sub}}(L^2(G,\mathcal{H}))$ {#10.4.6}
----------------------------------------------------------------------------------------------------------------
For every $n \in \mathbb{N}$, set $$\varphi_n : \underset{k \in \mathbb{N}}{\coprod}E_n\bigl((F^{(\infty)})^{\{k,\dots,k+n-1\}}\bigr) \longrightarrow
\underset{k \in \mathbb{N}}{\coprod}G_n\bigl((F^{(\infty)})^{\{k,\dots,k+n-1\}}\bigr),$$ so that $\text{Fib}^{F^{(\infty)} *}(\mathbf{1})=\underset{n\in \mathbb{N}}{\coprod} \varphi_n$.\
For every object $x$ in $\varphi_n {\operatorname{\text{-}sframe}}$, we let $k_x$ denote the only integer such that $x \in V_n((F^{(\infty)})^{\{k_x,\dots,k_x+n-1\}})$.
Let $n \in \mathbb{N}^*$. For every object $\mathbf{F}$ of ${\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}})$, we define $$\alpha(\mathbf{F}) : \begin{cases}
G & \longrightarrow \mathbb{R}_*^+ \\
g & \longmapsto \|\mathbf{F}(g)\|
\end{cases} \quad ; \quad
\beta(\mathbf{F}) : \begin{cases}
G & \longrightarrow V_n(\mathcal{H}) \\
g & \longmapsto \frac{1}{\|\mathbf{F}(g)\|}\mathbf{F}(g)
\end{cases}
\quad \text{and} \quad
k(\mathbf{F}) : \begin{cases}
G & \longrightarrow \mathbb{N} \\
g & \longmapsto k_{\beta(\mathbf{F})[g]}.
\end{cases}$$ Clearly, those maps are continuous and yield continuous maps $$\alpha : {\operatorname{Ob}}({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}})) \rightarrow C_*(G) \cap L^2_*(G),$$ $$\beta : {\operatorname{Ob}}({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}})) \rightarrow \mathcal{H}^n \quad \text{and} \quad
k : {\operatorname{Ob}}({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}})) \rightarrow \mathbb{N}^G.$$ Let $m \in \mathbb{N}$, and $\mathbf{F}:F_0 \rightarrow \dots \rightarrow F_m$ be an element of $\mathcal{N}({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}}))_m$. The definition of ${\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}})$ shows that the list $k(\mathbf{F})[g]:=(k(F_i)[g])_{0 \leq i \leq m}$ is a non-decreasing map from $[m]$ to $\mathbb{N}$ for every $g \in G$. We then define $$\beta(\mathbf{F}) :
\begin{cases}
G & \longrightarrow V_n(m) \\
g & \longmapsto (\beta(F_i)[g])_{0\leq i \leq m},
\end{cases}$$ where, for every $g \in G$, $(\beta(F_i)[g])_{0\leq i \leq m}$ is considered as an element of $V_n(k(\mathbf{F})[g])$. Notice that $\beta(\mathbf{F})$ is continuous. This yields a continuous map $$\beta : \mathcal{N}({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}}))_m \longrightarrow V_n(m).$$ Let now $K$ be a compact space, and assume that we have a family $\psi=(\psi_n)_{n\in \mathbb{N}}$ such that $\psi_n : (V_n(m))_{m \in \mathbb{N}} \longrightarrow ({\operatorname{Hom}}(\Delta^n \times K,V_n(\mathcal{H})))_{m \in \mathbb{N}}$ is a morphism of simplicial $U_n(F)$-spaces for every $n\in \mathbb{N}$, which satisfies conditions (i), (ii) and (iii).
Let $n$ be a positive integer. Let $\mathbf{F}:F_0 \rightarrow \dots \rightarrow F_m$ be an element of $\mathcal{N}({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}}))_m$, $t=(t_0,\dots,t_m) \in \Delta^m$ and $z \in K$. Set $$\chi_{n,m}(\mathbf{F},t,z):
\begin{cases}
G & \longrightarrow \mathcal{H}^n \\
g & \longmapsto \left(\underset{0 \leq i \leq m}{\sum}t_i.\alpha(F_i)[g]\right).
\psi_{n,m}(\beta(\mathbf{F})[g],t,z).
\end{cases}$$
For every $g \in G$, $\chi_{n,m}(\mathbf{F},t,z)[g]$ is the product of a positive real number with an orthonormal $n$-tuple. Also $\underset{0 \leq i \leq m}{\sum}t_i.\alpha(F_i) \in L^2(G)$ since $L^2(G)$ is convex. It follows that the factors $\pi_i\circ \chi_{n,m}(\mathbf{F},t,z) :G \rightarrow \mathcal{H}$ of $\chi_{n,m}(\mathbf{F},t,z)$ (where $\pi_i : \mathcal{H}^n \rightarrow \mathcal{H}$ denotes the projection on the $i$-th factor) form a (linearly independent) orthogonal $n$-tuple of elements of $L^2(G,\mathcal{H})$, and we conclude that $\chi_{n,m}(\mathbf{F},t,z)$ defines an element of $B_n(L^2(G,\mathcal{H}))$.
Using the fact that $\psi_{n,m}$ is continuous and the respective definitions of the various left $G$-actions considered here, we find that we have just defined a $G$-map: $$\chi_{n,m} : \mathcal{N}({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}}))_m \times \Delta^m \times K \longrightarrow
B_n(L^2(G,\mathcal{H})).$$ Note that $\chi_{n,m}$ is also a ${\operatorname{GU}}_n(F)$-map. Indeed, let $\mathbf{F}:F_0 \rightarrow \dots \rightarrow F_m$ be an element of $\mathcal{N}({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}}))_m$, $t=(t_0,\dots,t_m) \in \Delta^m$ and $z \in K$. Let also $(\lambda,\mathbf{M}) \in \mathbb{R}_+^* \times U_n(F)$. It then follows from the $U_n(F)$-equivariance of $\psi_{n,m}$ that $$\begin{aligned}
\chi_{n,m}(\mathbf{F}.(\lambda\mathbf{M}),t,z)[g]
& =\left(\underset{0 \leq i \leq m}{\sum}t_i.\alpha(F_i.(\lambda\mathbf{M}))[g]\right).\psi_{n,m}(\beta(\mathbf{F}.(\lambda\mathbf{M})),t,z) \\
& = \left(\underset{0 \leq i \leq m}{\sum}t_i\lambda \,\alpha(F_i)[g]\right).
\psi_{n,m}(\beta(\mathbf{F}).\mathbf{M},t,z) \\
& = \chi_{n,m}(\mathbf{F},t,z)[g].(\lambda\mathbf{M}).\end{aligned}$$ We conclude that $\chi_{n,m}$ is a $(G \times {\operatorname{GU}}_n(F))$-map.
We now check that the maps $\chi_{n,0},\dots,\chi_{n,m},\dots$ are compatible with the simplicial structures. Let $\tau : [m'] \rightarrow [m]$ be a morphism in $\Delta$. Let $\mathbf{F}:F_0 \rightarrow \dots \rightarrow F_m$ be an element of $\mathcal{N}({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}}))_m$. Let $t=(t_0,\dots,t_m) \in \Delta^m$ and $z \in K$. Clearly $\beta(\tau_*(\mathbf{F}))=\tau_*(\beta(\mathbf{F}))$, so we deduce from the compatibility of $\psi_n$ with the simplicial structure that, for any $g \in G$, $$\begin{aligned}
\chi_{n,m}(\mathbf{F},\tau^*(t),z)[g]
& = \left(\underset{0 \leq i \leq m}{\sum}\left(\underset{j \in \tau^{-1}(\{i\})}{\sum}t_j\right).\alpha(F_i)[g]\right).
\psi_{n,m}(\beta(\mathbf{F})[g],\tau^*(t),z) \\
& = \left(\underset{0 \leq j \leq m'}{\sum}t_j.\alpha(F_{\tau(j)})[g]\right).\psi_{n,m'}(\tau_*(\beta(\mathbf{F})[g]),t,z) \\
& = \chi_{n,m}(\tau_*(\mathbf{F}),t,z)[g].\end{aligned}$$ We deduce that the family $(\chi_{n,m})_{m\in \mathbb{N}}$ yields a $(G\times {\operatorname{GU}}_n(F))$-map: $$\chi_n : s\widetilde{{\operatorname{Vec}}}_{G,L^2}^{n,F,\infty *} \times K \longrightarrow B_n(L^2(G,\mathcal{H})).$$ We define $\chi_0$ as the trivial map.
Since the canonical map $\underset{n \in \mathbb{N}}{\coprod} s\widetilde{{\operatorname{Vec}}}_{G,L^2}^{n,F,\infty *} \longrightarrow
s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[1]$ is an identification map and $K$ is compact, we may finally define $\chi$ as the unique $G$-map which renders commutative the diagram $$\begin{CD}
\left(\underset{n \in \mathbb{N}}{\coprod} s\widetilde{{\operatorname{Vec}}}_{G,L^2}^{n,F,\infty *}\right) \times K
@>{\underset{n\in \mathbb{N}}{\coprod}\chi_n}>>
\underset{n \in \mathbb{N}}{\coprod} B_n(L^2(G,\mathcal{H})) \\
@VVV @VVV \\
s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[1] \times K @>{\chi}>> \underset{n \in \mathbb{N}}{\bigcup}{\operatorname{sub}}_n(L^2(G,\mathcal{H})).
\end{CD}$$
The morphism $s{\operatorname{Vec}}_{G,L^2}^{F,\infty *} \longrightarrow {\operatorname{Fred}}(G,\mathcal{H}^\infty)$ {#10.4.7}
---------------------------------------------------------------------------------------------------------------------
We define $$\alpha_G^\psi(0) : s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[0] \times K \longrightarrow {\operatorname{Fred}}(G,\mathcal{H}^\infty)[0]$$ as the trivial map.\
For every positive integer $n$, we define $\alpha_G^\psi(n)$ as the composite $G$-map $$s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[n] \times K
\longrightarrow \left(s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[1]\times K\right)^n
\overset{\chi^n}{\longrightarrow}
\left(\underset{k \in \mathbb{N}}{\bigcup}{\operatorname{sub}}_k(L^2(G,\mathcal{H}))\right)^n \overset{{\operatorname{Sh}}^n}{\longrightarrow}
{\operatorname{Fred}}(G,\mathcal{H}^\infty)[n],$$ where the first map is the product of the simplicial maps $(\alpha_i^n)_* : s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[n] \rightarrow
s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[1]$ for $i \in \{1,\dots,n\}$ (where $\alpha_i^n : [1] \rightarrow [n]$ maps $0$ to $i-1$ and $1$ to $i$). The compact space $K$ is considered here as the simplicial space with $K$ as the space at every level and all morphisms equal to ${\operatorname{id}}_K$.
$\alpha_G^\psi : s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}\times K \longrightarrow
{\operatorname{Fred}}(G,\mathcal{H}^\infty)$ is a morphism of hemi-simplicial $G$-spaces.
**Remark :** In fact, $\alpha_G^\psi$ is even a morphism of simplicial $G$-spaces, but this is not relevant here since we are only interested in thick geometric realizations.
Obviously, $\alpha_G^\psi(n)$ is a $G$-map for every $n\in \mathbb{N}$. In order to prove that $\alpha_G^\psi$ is compatible with degeneracy maps, it suffices to check that the square $$\begin{CD}
s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[2] \times K @>{\alpha_G^\psi(2)}>> {\operatorname{Fred}}(L^2(G,\mathcal{H})^\infty)^2 \\
@VV{d_1^2 \times {\operatorname{id}}_K}V @VVV \\
s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[1]\times K @>{\alpha_G^\psi(1)}>> {\operatorname{Fred}}(L^2(G,\mathcal{H})^\infty)
\end{CD}$$ is commutative.
Let $x \in s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[2]$ and $a \in K$. Let $(y,t) \in \mathcal{N}({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\text{Fib}^{F^{(\infty)}*}(\mathbf{2}) {\operatorname{\text{-}smod}}))_m \times
\Delta^m$ such that $x=[y,t]$. We write $t=(t_0,\dots,t_m)\in \Delta^m$. We set $y_i:= (\alpha_i^2)_*(y)$ for every $i \in \{1,2\}$. Set $(n,n')\in \mathbb{N}^2$ such that $y_1 \in \mathcal{N}({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_n {\operatorname{\text{-}smod}}))_m$ and $y_2 \in \mathcal{N}({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_{n'} {\operatorname{\text{-}smod}}))_m$.
We choose $z_1 \in \mathcal{N}({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_n {\operatorname{\text{-}sframe}}))_m$ and $z_2 \in \mathcal{N}({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_{n'} {\operatorname{\text{-}sframe}}))_m$ in the respective fibers of $y_1$ and $y_2$, and we write $z_1=F_0 \rightarrow \dots \rightarrow F_m$ and $z_2=F'_0 \rightarrow \dots \rightarrow F'_m$. Set $z:=(F_0,F'_0) \rightarrow \dots \rightarrow (F_m,F'_m)$, and define $y' \in \mathcal{N}({\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\varphi_{n+n'} {\operatorname{\text{-}smod}}))_m$ as the image of $z$ by the canonical map. Then $[y',t]=d_1^2(x)$.
The definition of ${\operatorname{Func}}_{\uparrow L^2}(\mathcal{E}G,\text{Fib}^{F^{(\infty)}*}(\mathbf{2}) {\operatorname{\text{-}smod}})$, shows that $\|F_i(g)\|=\|F'_i(g)\|$ and $k(F'_i)[g]=k(F_i)[g]+n$ for any $i \in [m]$ and any $g \in G$. Also, $\beta(z)=(\beta(z_1),\beta(z_2))$. Hence $\alpha(F_i)=\alpha(F'_i)=\alpha(F_i,F'_i)$ for every $i\in [m]$, and $k(z_2)=k(z_1)+n=k(z)+n$. It follows that $(\beta(z_1)[g],\beta(z_2)[g]) \in V_n(k(z_1)[g]) \times V_{n'}(k(z_1)[g]+n)$ for every $g \in G$.
Since condition (iii) holds for $\psi$, we deduce that, for any $g \in G$, $$\begin{aligned}
\chi_{n+n',m}(z,t,a)[g] & = \left(\underset{0 \leq i \leq n}{\sum} t_i.\alpha((F_i,F'_i))[g]\right).
\psi_{n+n',m}(\beta(z)[g],t,a) \\
& = \left(\underset{0 \leq i \leq n}{\sum} t_i.\alpha((F_i,F'_i))[g]\right).
\left(\psi_{n,m}(\beta(z_1)[g],t,a),\psi_{n',m}(\beta(z_2)[g],t,a)\right) \\
& = \left(\left(\underset{0 \leq i \leq n}{\sum} t_i.\alpha(F_i)[g]\right).
\psi_{n,m}(\beta(z_1)[g],t,a), \left(\underset{0 \leq i \leq n}{\sum} t_i.\alpha(F'_i)[g]\right).\psi_{n',m}(\beta(z_2)[g],t,a)\right) \\
& = \bigl(\chi_{n,m}(z_1,t,a)[g],\chi_{n',m}(z_2,t,a)[g]\bigr).\end{aligned}$$ It follows that the subspaces of $L^2(G,\mathcal{H})$ respectively generated by $\chi_{n,m}(z_1,t,a)$ and $\chi_{n',m}(z_2,t,a)$ are orthogonal and that their direct sum is the subspace generated by $\chi_{n+n',m}(z,t,a)$.\
We deduce that $\chi([y_2,t],a) \overset{\bot}{\oplus} \chi([y_1,t],a)=\chi([y',t],a)$, i.e. $\chi((\alpha_{2,2})_*(x),a) \overset{\bot}{\oplus} \chi((\alpha_{1,2})_*(x),a)
=\chi(d_1^2(x),a)$. We finally derive from Proposition \[propertiesofshift\] that ${\operatorname{Sh}}(\chi(d_1^2(x),a))={\operatorname{Sh}}(\chi((\alpha_{2,2})_*(x),a)) \circ {\operatorname{Sh}}(\chi((\alpha_{1,2})_*(x),a))$.
Since $\alpha_G^\psi : s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}\times K \longrightarrow
{\operatorname{Fred}}(G,\mathcal{H}^\infty)$ is a morphism of hemi-simplicial $G$-spaces, it yields a $G$-map: $Bs{\operatorname{Vec}}_{G,L^2}^{F,\infty *}\times K \longrightarrow B{\operatorname{Fred}}(G,\mathcal{H}^\infty)$, and, furthermore, a $G$-map $$\Omega B\alpha_G^\psi : \bigl(\Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,\infty *}\bigr)\times K \longrightarrow \Omega B{\operatorname{Fred}}(G,\mathcal{H}^\infty).$$ Apply now the result of Proposition \[mighty\]. For $p=0$, we recover a family $(\psi_n)_{n\in \mathbb{N}}$ such that $\psi_n : (V_n(m))_{m \in \mathbb{N}} \longrightarrow
({\operatorname{Hom}}(\Delta^m,V_n(\mathcal{H})))_{m \in \mathbb{N}}$ is a morphism of simplicial $U_n(F)$-spaces for every $n \in \mathbb{N}$, and which satisfies conditions (i) to (vi). This family yields a $G$-map $$\Omega B\alpha_G^\psi : \Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,\infty *} \longrightarrow \Omega B{\operatorname{Fred}}(G,\mathcal{H}^\infty).$$ If $\psi'$ is another such family, it yields another $G$-map $$\Omega B\alpha_G^{\psi'} : \Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,\infty *} \longrightarrow \Omega B{\operatorname{Fred}}(G,\mathcal{H}^\infty).$$ Those two maps are $G$-homotopic. Indeed, by Proposition \[mighty\] applied in the case $p=1$ to $\psi \coprod \psi'$, we recover a family $(\Psi_n)_{n\in \mathbb{N}}$ such that $\Psi_n : (V_n(m))_{m \in \mathbb{N}} \longrightarrow
({\operatorname{Hom}}(\Delta^m \times [0,1],V_n(\mathcal{H})))_{m \in \mathbb{N}}$ is a morphism of simplicial $U_n(F)$-spaces for every $n \in \mathbb{N}$, which satisfies conditions (i) to (vi), and such that $\Psi_{|\{0\}}=\psi$ and $\Psi_{|\{1\}}=\psi'$. This family then yields an equivariant homotopy $$\Omega B\alpha_G^\Psi : \bigl(\Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,\infty *}\bigr) \times I \longrightarrow \Omega B{\operatorname{Fred}}(G,\mathcal{H}^\infty)$$ from $\Omega B\alpha_G^\psi$ to $\Omega B\alpha_G^{\psi'}$.
The natural transformation $KF_G(-) \rightarrow KF_G^{\text{Ph}}(-)$ {#10.5}
====================================================================
The definition of $\eta$
------------------------
We denote by $KF_G^{\text{Ph}}(-)$ the equivariant K-theory of Phillips (as defined and studied in [@Phillips]) on the category of finite proper $G$-CW-complexes.
Since $\mathcal{H}^\infty$ is a separable Hilbert space, we may consider, for every finite proper $G$-CW-complex $X$, the map: $$\begin{cases}
[X,\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}^\infty))]_G & \longrightarrow KF_G^{\text{Ph}}(X)
\\
\left[f:X \rightarrow \underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}^\infty))\right]
& \longmapsto \left[(X \times L^2(G,\mathcal{H}^\infty),
X \times L^2(G,\mathcal{H}^\infty),f)\right].
\end{cases}$$ This yields a natural transformation of group-valued functors: $$[-,\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}^\infty))]_G \longrightarrow KF_G^{\text{Ph}}(-).$$ Since we know from Proposition \[fredweak\] that the canonical map $\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}^\infty)) \longrightarrow \Omega B{\operatorname{Fred}}(G,\mathcal{H}^\infty)$ is a $G$-weak equivalence, it yields a natural equivalence $$[-,\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}^\infty))]_G
\overset{\cong}{\longrightarrow} [-,\Omega B{\operatorname{Fred}}(G,\mathcal{H}^\infty)]_G$$ on the category of proper $G$-CW-complexes.\
The $G$-map $\alpha_G^\psi : \Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,\infty *} \longrightarrow \Omega B{\operatorname{Fred}}(G,\mathcal{H}^\infty)$ constructed earlier then induces a uniquely defined natural transformation $$[-,\Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,\infty *}]_G
\overset{\alpha_G^\psi \circ -}{\longrightarrow} [-,\Omega B{\operatorname{Fred}}(G,\mathcal{H}^\infty)]_G$$ on the category of $G$-spaces.
Moreover, the composite $\Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,\infty *} \longrightarrow
\Omega Bs\underline{{\operatorname{Vec}}}_{G,L^2}^{F,\infty} \longrightarrow KF_G^{[\infty]}$ of the previously discussed $G$-weak equivalences is a $G$-weak equivalence and yields a natural equivalence $$[-,\Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,\infty *}]_G \overset{\cong}{\longrightarrow}
[-,KF_G^{[\infty]}]_G=KF_G(-)$$ on the category of proper $G$-CW-complexes.
We may now define a natural transformation $\eta : KF_G(-) \longrightarrow KF_G^{\text{Ph}}(-)$ on the category of finite proper $G$-CW-complexes by composing from left to right: $$KF_G(-) \overset{\cong}{\longleftarrow}
[-,\Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,\infty *}]_G
\overset{\alpha_G^\psi \circ -}{\longrightarrow}
[-,\Omega B{\operatorname{Fred}}(G,\mathcal{H}^\infty)]_G
\overset{\cong}{\longleftarrow}
[-,\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}^\infty))]_G \longrightarrow KF_G^{\text{Ph}}(-).$$ Obviously, $\eta_X$ is a homomorphism of abelian groups for every finite proper $G$-CW-complex $X$.
The definition of Phillips’ K-theory in negative degrees [@Phillips] pp 80-81 makes it clear that $\eta$ may be extended to negative degrees as a natural transformation $KF_G^*(-) \longrightarrow KF_G^{*\text{Ph}}(-)$ which preserves the long exact sequence of a finite proper $G$-CW-pair.
Properties of $\eta$
--------------------
Recall from [@Phillips] example 3.4 page 40 that there is a natural transformation $\mathbb{K}F_G(-) \longrightarrow KF_G^{\text{Ph}}(-)$ on the category of locally compact $G$-spaces, defined by mapping every finite dimensional $G$-Hilbert bundle $E \rightarrow X$ to the class of the cocycle $(E,0,0)$ in $KF_G^{Ph}(X)$.
\[transformationcommute\] The diagram $$\xymatrix{
KF_G(-) \ar[rr]^{\eta} & & KF_G^{\text{Ph}}(-) \\
& \mathbb{K}F_G(-) \ar[ul]^{\gamma} \ar[ur]
}$$ is commutative on the category of finite proper $G$-CW-complexes.
It suffices to prove that the triangle $$\xymatrix{
KF_G(-) \ar[rr]^{\eta} & & KF_G^{\text{Ph}}(-) \\
& \mathbb{V}\text{ect}_G^F(-) \ar[ul]^{\gamma} \ar[ur]
}$$ is commutative on the category of finite proper $G$-CW-complexes.\
However $\gamma$ is the composite of the natural transformations: $$\mathbb{V}\text{ect}_G^F(-) \overset{\cong}{\longrightarrow} [-,s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}]_G
\longrightarrow [-,\Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,\infty *}]_G
\longrightarrow [-,KF_G^{[\infty]}]_G$$ where the first morphism is the one induced by pulling back the universal bundle $Es{\operatorname{Vec}}_{G,L^2}^{F,\infty *} \longrightarrow s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[1]$.\
Since the square $$\begin{CD}
\Omega Bs{\operatorname{Vec}}_{G,L^2}^{F,\infty *} @>{\alpha_G^\psi}>> \Omega B{\operatorname{Fred}}(G,\mathcal{H}^\infty) \\
@AAA @AAA \\
s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[1] @>{\chi}>> \underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}^\infty))
\end{CD}$$ is commutative, the composite natural transformation $\mathbb{V}\text{ect}_G^F(-) \longrightarrow KF_G(-) \overset{\eta}{\longrightarrow} KF_G^{\text{Ph}}(-)$ is simply the composite natural transformation $$\mathbb{V}\text{ect}_G^F(-) \overset{\cong}{\longrightarrow} [-,s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}]_G
\overset{\chi \circ -}{\longrightarrow}
[-,\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}^\infty))]_G \longrightarrow KF_G^{\text{Ph}}(-).$$ Moreover, $\chi$ defines a morphism of Hilbert $G$-bundles $$\overline{\chi} : \begin{cases}
s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[1] \times L^2(G,\mathcal{H}^\infty) & \longrightarrow
s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[1] \times L^2(G,\mathcal{H}^\infty) \\
(x,y) & \longmapsto \bigl(x,\chi(x)[y]\bigr).
\end{cases}$$ and the maps $$\begin{cases}
s\widetilde{{\operatorname{Vec}}}_{G,L^2}^{n,F,\infty*} \times {\operatorname{GU}}_n(F) & \longrightarrow L^2(G,\mathcal{H}) \\
(x,M) & \longmapsto \tilde{\chi_n}(x).M,
\end{cases}$$ for $n \in \mathbb{N}$, yield a morphism of $G$-simi-Hilbert bundles $$\xymatrix{
Es{\operatorname{Vec}}_{G,L^2}^{F,\infty*} \ar[dr] \ar[rr]^{\underline{\chi}} & &
s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[1] \times L^2(G,\mathcal{H}^\infty)
\ar[dl] \\
&
s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[1]
}$$ which maps $Es{\operatorname{Vec}}_{G,L^2}^{F,\infty*}$ onto the kernel of $\overline{\chi}$.
Let now $X$ be a finite proper $G$-CW-complex, and $X \overset{f}{\longrightarrow} s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}$ be a $G$-map.\
Pulling back the diagram of bundle morphisms $$\xymatrix{
Es{\operatorname{Vec}}_{G,L^2}^{F,\infty*} \ar[dr] \ar[r]^(0.35){\underline{\chi}} &
s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[1] \times L^2(G,\mathcal{H}^\infty) \ar[r]^{\overline{\chi}}
\ar[d] & s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[1] \times L^2(G,\mathcal{H}^\infty) \ar[dl] \\
& s{\operatorname{Vec}}_{G,L^2}^{F,\infty *}[1]
}$$ by $f$ yields a diagram of bundle morphisms $$\xymatrix{
E \ar[dr] \ar[r]^(0.3){f_*(\underline{\chi})} &
X \times L^2(G,\mathcal{H}^\infty) \ar[r]^{f_*(\overline{\chi})} \ar[d] & X \times L^2(G,\mathcal{H}^\infty), \ar[dl] \\
& X
}$$ where $f_*(\underline{\chi})$ maps $E$ onto the kernel of $f_*(\overline{\chi})$.
Thus $\eta(\gamma([f]))$ is the class of $f_*(\overline{\chi})$ in $KF_G^{\text{Ph}}(X)$, and we need to prove that it is also the class $[E \rightarrow 0]$.\
Since the dimension of the kernel of $f_*(\overline{\chi})$ is locally constant, it follows that ${\operatorname{Ker}}f_*(\overline{\chi})$ is a finite-dimensional $G$-vector bundle over $X$ (see [@Janich]). The above diagram thus induces a strong morphism of $G$-vector bundles: $$\xymatrix{
E \ar[rr]^{\cong} \ar[dr] & & {\operatorname{Ker}}f_*(\overline{\chi}). \ar[dl] \\
& X
}$$ Hence ${\operatorname{Ker}}f_*(\overline{\chi})$ and $E$ are isomorphic.
Moreover $({\operatorname{Ker}}f_*(\overline{\chi}))^\bot$ is a sub-$G$-Hilbert bundle of $X \times L^2(G,\mathcal{H}^\infty)$, and $f_*(\overline{\chi})$ induces an isomorphism of $G$-Hilbert bundles $${\operatorname{Ker}}f_*(\overline{\chi}) ^\bot \overset{\cong}{\longrightarrow} X \times L^2(G, \mathcal{H}^\infty).$$ We deduce that the class of ${\operatorname{Ker}}f_*(\overline{\chi})^\bot \longrightarrow X \times L^2(G, \mathcal{H}^\infty)$ in $KF_G^{\text{Ph}}(X)$ is $0$.\
Finally $$[f_*(\overline{\chi})]=\left[{\operatorname{Ker}}f_*(\overline{\chi}) \rightarrow 0\right] +
\left[{\operatorname{Ker}}f_*(\overline{\chi})^\bot \rightarrow X \times L^2(G,
\mathcal{H}^\infty) \right]=\left[E \rightarrow 0\right]$$ in $KF_G^{\text{Ph}}(X)$.
For every finite proper $G$-CW-complex $X$ $$\eta_X : KF_G(X) \overset{\cong}{\longrightarrow} KF_G^{\text{Ph}}(X)$$ is an isomorphism of abelian groups.
Since $\eta$ is a natural transformation between good equivariant cohomology theories in negative degrees, it suffices to establish the result in the case $X=(G/H) \times Y$, where $H$ is a compact subgroup of $G$, and $Y$ is a finite CW-complex on which $G$ acts trivially (by an argument that is similar to that of the proof of Proposition 1.5 in [@Bob2]). In this case, we deduce from Proposition \[transformationcommute\] that the diagram $$\xymatrix{
KF_G((G/H) \times Y) \ar[rr]_{\cong}^{\eta_{(G/H) \times Y}} & & KF_G^{\text{Ph}}((G/H) \times Y) \\
& \mathbb{K}F_G((G/H) \times Y) \ar[ul]_{\gamma_{(G/H) \times Y}}^{\cong} \ar[ur]_{\cong}
}$$ is commutative. By [@Phillips], the map $\mathbb{K}F_G((G/H) \times Y) \longrightarrow KF_G^{\text{Ph}}((G/H) \times Y)$ is an isomorphism. By Proposition 4.4 of [@Ktheo1], $\gamma_{(G/H) \times Y}$ is an isomorphism. Hence $\eta_{(G/H) \times Y}$ is an isomorphism.
**An open problem :** there remains essentially one problem to be solved here: is $\eta$ compatible with the product maps (or with Bott-periodicity maps)? This has proved out of our reach, for two main reasons:
1. it is not clear at all how the multiplicative structure on $KF_G^{\text{Ph}}(-)$ may be understood in terms of the space $\underline{{\operatorname{Fred}}}(L^2(G,\mathcal{H}^\infty))$;
2. in constructing $\eta$, we have dumped the very categorical structures that helped constructed the product maps, i.e. the $\Gamma-G$-space structure.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank Bob Oliver for his constant support and the countless good advice he gave me during my research on this topic.
[10]{}
M. Atiyah, G. Segal, Twisted K-theory, ,
P. Gabriel, M. Zisman,
K. Jänich, Vektorraumbundel und der Raum der Fredholm-Operatoren. ,
W. Lück, B. Oliver, Chern characters for equivariant K-theory for proper $G$-cw-complexes, in
A.T. Lundell, S. Weingram,
N.C. Phillips,
G. Segal, Categories and cohomology theories. ,
C. de Seguins Pazzis, Equivariant K-theory for proper actions of non-compact Lie groups ,
N.E. Steenrod, A convenient category of topological spaces.
[^1]: e-mail adress: [email protected]
[^2]: This work is part of the author’s PhD thesis that he completed at the Institut Galilée in Université Paris Nord, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, FRANCE
[^3]: Note that only the hemi-simplicial space structure is taken into account since the thick realization only uses face maps.
[^4]: and, more specifically, from the part concerning condition (iii), applied to $(n,n',k,g)$, and $(n,n',n_i-1,(\sigma \natural i)_*(f))$, for every $i\in [k]$ and every $(n,n')$ such that $n+n'=N$.
[^5]: and, more specifically, the part concerning the compatibility of $\psi$ with $\psi^\partial$ for $(N,k)$, and $(N,n_i-1)$ for every $i\in [k]$.
|
{
"pile_set_name": "ArXiv"
}
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abstract: 'We study an upper bound of ranks of $n$-tensors with size $2\times\cdots\times2$ over the complex and real number field. We characterize a $2\times 2\times 2$ tensor with rank $3$ by using the Cayley’s hyperdeterminant and some function. Then we see another proof of Brylinski’s result that the maximal rank of $2\times2\times2\times2$ complex tensors is $4$. We state supporting evidence of the claim that $5$ is a typical rank of $2\times2\times2\times2$ real tensors. Recall that Kong and Jiang show that the maximal rank of $2\times2\times2\times2$ real tensors is less than or equal to $5$. The maximal rank of $2\times2\times2\times2$ complex (resp. real) tensors gives an upper bound of the maximal rank of $2\times\cdots\times 2$ complex (resp. real) tensors.'
author:
- 'Toshio Sumi, Toshio Sakata and Mitsuhiro Miyazaki'
title: 'Rank of tensors with size $2\times \cdots\times 2$'
---
Introduction
============
Let $\FFF$ be the real number field $\RRR$ or the complex number field $\CCC$. For a positive integer $n$, an $n$-tensor $T$ over $\FFF$ with size $2\times \cdots\times 2$ is $$(t_{i_1,i_2,\ldots,i_n})$$ consisting of $2^n$ elements where $i_1,i_2,\ldots,i_n$ are taken $1$ and $2$, and $t_{i_1,i_2,\ldots,i_n}\in \FFF$ for $i_1,i_2,\ldots,i_n=1,2$. Let $(\FFF^{2})^{\otimes n}$ be the set of all $n$-tensors over $\FFF$ with size $2\times \cdots\times 2$. This set is closed by sum operation and scalar multiplication: $$(t_{i_1,i_2,\ldots,i_n})+(s_{i_1,i_2,\ldots,i_n})=(t_{i_1,i_2,\ldots,i_n}+s_{i_1,i_2,\ldots,i_n})$$ $$c(t_{i_1,i_2,\ldots,i_n})=(ct_{i_1,i_2,\ldots,i_n})$$ And ${\mathrm{GL}}(2,\FFF)^n$ acts on the set $(\FFF^{2})^{\otimes n}$. We call $T$ is a rank one tensor if $T$ is irreducible, that is, $T=T_1+T_2$ for some nonzero tensors $T_1,T_2$ implies that $T_1=sT_2$ for some $s\in \FFF$. The rank of $T$, denoted by ${\mathrm{rank}}_{\FFF}(T)$ is the smallest integer $s\geq0$ such that $T$ is expressed as the sum of $s$ rank one tensors. The rank of the zero tensor is zero. Rank is invariant under the ${\mathrm{GL}}(2,\FFF)^n$-action. In general the determination of the rank of a tensor is hard.
The rank and classification of $2\times 2\times 2$ tensors are well-known, for example, see [@JaJa:1979b] or [@Sumi-etal:2009]. The maximal rank of $2\times 2\times 2$ real tensors is equal to one of $2\times 2\times 2$ complex tensors.
For a real $2\times 2\times 2\times 2$ tensor, Kong et al. [@Kong-etal:2012] show that ${\mathrm{rank}}_{\RRR}(T)\leq 5$.
\[thm:Kong-etal\] Any real $2\times 2\times 2\times 2$ tensor has rank less than or equal to $5$.
Brylinski gave the maximal rank of $2\times2\times2\times 2$ tensors over $\CCC$.
\[thm:Brylinski\] Any complex $2\times2\times2\times 2$ tensor has rank less than or equal to $4$.
$2\times 2\times 2\times 2$ tensors over $\FFF$ are used to represent the entanglement of four quantum bits (qubits). Verstraete et al [@Verstraete-etal:2002] gave a classification of $2\times 2\times 2\times 2$ rank one tensors. We also show that ${\mathrm{rank}}_{\FFF}(A_1;A_2)\leq 2$ then ${\mathrm{rank}}_{\FFF}(T)\leq 4$ (see Propositions \[prop:leq2implies4a\]). This was obtained by Kong et al. [@Kong-etal:2012] over $\RRR$. By a numerical analysis, it seems that there are tensors over $\RRR$ with rank $5$. There are tensors over the finite field $\mathbb{F}_3$ with rank $5$ (cf. [@Bremner-Stavrou:2012]). The purpose of this paper is to give an upper bound of rank of tensors with size $2\times\cdots\times2$ by using the maximal rank of $2\times 2\times 2\times2$ real tensors (see Theorem \[thm:higher2x...x2\]). Our main tool is a matrix theory.
$2\times2\times 2$ tensors
==========================
The maximal rank of $2\times2\times2$ tensors over $\FFF$ is equal to $3$. In this section, we clarify a condition for a $2\times2\times2$ tensor to have rank three.
We denote the $2\times2$ identity matrix by $E_2$ or simply $E$. Let $A=(a_{ij})=(\aaa_1,\aaa_2)$ and $B=(b_{ij})=(\bbb_1,\bbb_2)$ be $2\times 2$ matrices. ${\mathrm{GL}}(2,\FFF)^3$ acts on the set of $2\times 2\times 2$ tensors by $$(P,Q,R)\cdot (A;B)=(r_{11}PAQ^\top+r_{12}PBQ^\top;r_{21}PAQ^\top+r_{22}PBQ^\top),$$ where $R=\begin{pmatrix} r_{11}&r_{12}\\ r_{21}&r_{22}\end{pmatrix}$. For a subgroup $G$ of ${\mathrm{GL}}(2,\FFF)^3$, two $2\times 2\times 2$ tensors $T_1$ and $T_2$ are $G$-equivalent if $g\cdot T_1=T_2$ for some $g\in G$.
Suppose that $xA+yB$ are nonzero for any $(x,y)\ne(0,0)$ in $\FFF^2$. If $(A;B)$ has rank two, then there is a tensor $X=(X_1;X_2)$ such that $X$ is $\{E_2\}^2\times {\mathrm{GL}}(2,\FFF)$-equivalent to $(A;B)$ and $\det(X_1)=\det(X_2)=0$.
Suppose that $(A;B)$ has rank two. There are two rank one matrices $C_1$ and $C_2$ such that $A=pC_1+qC_2$ and $B=rC_1+sC_2$. By the assumption, $P:=\begin{pmatrix} p&q\\ r&s\end{pmatrix}$ is nonsingular. Let $X=(E_2,E_2,P^{-1})\cdot (A;B)$. Then $X=(C_1;C_2)$.
Remark that if $xA+yB$ are nonzero for any $(x,y)\ne(0,0)$, then ${\mathrm{rank}}(A;B)\geq 2$.
Define $$\begin{split}
\Delta(A;B)&=(\det(A+B)-\det(A-B))^2/4-4\det(A)\det(B) \\
&=(\det(\aaa_1,\bbb_2)+\det(\bbb_1,\aaa_2))^2-4\det(\aaa_1,\aaa_2)\det(\bbb_1,\bbb_2).
\end{split}$$ This number is called Cayley’s hyperdeterminant up to sign. The discriminant of the polynomial $\det(xA+B)$ (resp. $\det(A+xB)$) on $x$ is equal to $\Delta(A;B)$ if $A$ (resp. $B$) is nonsingular. Thus, over $\RRR$, if $A$ (resp. $B$) is nonsingular and $\Delta(A;B)>0$ then there are $x_1$ and $x_2$ in $\RRR$ such that $x_1\ne x_2$ and $\det(x_1A+B)=\det(x_2A+B)=0$ (resp. $\det(A+x_1B)=\det(A+x_2B)=0$), and ${\mathrm{rank}}(A;B)\leq2$. If $\det(A)=\det(B)=0$ then ${\mathrm{rank}}(A;B)\leq {\mathrm{rank}}(A)+{\mathrm{rank}}(B)\leq 2$. A $2\times 2\times 2$ tensor $T$ over a field with characteristic not $2$ is nonsingular if and only if $\Delta(T)$ is not a square in the field (see [@Coolsaet:2012]).
We show the following property straightforwardly.
\[prop:Silva\] $$\Delta(A;B)=\Delta(B;A).$$ $$\Delta(A+xB;yB)=y^2\Delta(A;B)$$ for any $x$. $$\Delta((P,Q,R)\cdot (A;B))=\Delta(A;B)\det(P)^2\det(Q)^2\det(R)^2$$ for any matrices $P$, $Q$ and $R$.
If $A$ is nonsingular, ${\mathrm{rank}}(A;B)=2$ if and only if $A^{-1}B$ is diagonalizable. If $A^{-1}B$ has distinct eigenvalues, the descriminant of the characteristic polynomial of $A^{-1}B$ which is equal to $$\begin{split}
(a_{11}b_{22}-a_{22}b_{11}+a_{12}b_{21}-a_{21}b_{12})^2+4(a_{22}b_{12}-a_{12}b_{22})(-a_{21}b_{11}+a_{11}b_{21}) \\
=(\det(\aaa_1,\bbb_2)-\det(\bbb_1,\aaa_2))^2-4\det(\aaa_1,\bbb_1)(\aaa_2,\bbb_2)
\end{split}$$ is positive in $\RRR$ and nonzero in $\CCC$. Note that $$\Delta(A;B)=(\det(\aaa_1,\bbb_2)-\det(\bbb_1,\aaa_2))^2-4\det(\aaa_1,\bbb_1)(\aaa_2,\bbb_2).$$ Thus we have the following proposition and theorem.
\[prop:Silva-etal\] Let $T$ be a $2\times2\times 2$ real tensor.
1. The sign of $\Delta$ is invariant under the ${\mathrm{GL}}(2,\RRR)^{\times 3}$-action.
2. If $\Delta(T)>0$ then ${\mathrm{rank}}(T)\leq 2$.
3. If $\Delta(T)<0$ then ${\mathrm{rank}}(T)=3$.
4. If ${\mathrm{rank}}(T)\leq 2$ then $\Delta(T)\geq 0$.
Put $S=\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}$ and $R=\begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$. Note that $\Delta(E;S)=0$ and $\Delta(E;R)=-4$.
\[thm:2x2x2\] Let $A=(\aaa_1,\aaa_2)$ and $B=(\bbb_1,\bbb_2)$ be $2\times2$ real (resp. complex) matices and let $T=(A;B)$ be a $2\times 2\times 2$ tensor. ${\mathrm{rank}}_{\FFF}(T)\leq 2$ if and only if at least one of the following holds:
1. \[thm:2x2x2:parallel1\] $\alpha A+\beta B=O$ for some $(\alpha,\beta)\ne(0,0)$,
2. \[thm:2x2x2:parallel2\] $\alpha (\aaa_1,\bbb_1)+\beta(\aaa_2,\bbb_2)=O$ for some $(\alpha,\beta)\ne(0,0)$,
3. \[thm:2x2x2:same\] $\Delta(A;B)=0$ and $\det(\aaa_1,\bbb_1)+\det(\aaa_2,\bbb_2)=0$, or
4. \[thm:2x2x2:different\] $\Delta(A;B)$ is positive (resp. nonzero).
First suppose that $|x_1A+y_1B|\ne0$ for some $x_1$ and $y_1$. There are $x_2$ and $y_2$ such that $\left|\begin{matrix} x_1&x_2\\ y_1&y_2\end{matrix}\right|\ne 0$ and $(A;B)$ is equivalent to $(x_1A+y_1B;x_2A+y_2B)$. ${\mathrm{rank}}(A;B)\leq 2$ is equivalent to that $(x_1A+y_1B)^{-1}(x_2A+y_2B)$ is diagonalizable. It is equivalent to
- $x_2A+y_2B=\alpha (x_1A+y_1B)$ for some $\alpha$, or
- all eigenvalues of $(x_1A+y_1B)^{-1}(x_2A+y_2B)$ lie in $\FFF$ and are distinct.
We have (i) $\Leftrightarrow$ , and (ii) $\Leftrightarrow$ $\Delta(x_1A+y_1B;x_2A+y_2B)$ is positive (resp. nonzero) $\Leftrightarrow$ $\Delta(A;B)$ is positive (resp. nonzero).
Next suppose that $|xA+yB|=0$ for any $x$ and $y$. Then we have $${\mathrm{rank}}(A;B)\leq {\mathrm{rank}}(A)+{\mathrm{rank}}(B)\leq 1+1=2.$$ We see that $|xA+yB|=0$ for any $x$ and $y$ if and only if $|A|=|B|=|\aaa_1,\bbb_2|+|\bbb_1,\aaa_2|=0$, since $$|x\aaa_1+y\bbb_1,x\aaa_2+y\bbb_2|=x^2|\aaa_1,\aaa_2|+xy(|\aaa_1,\bbb_2|+|\bbb_1,\aaa_2|)+y^2|\bbb_1,\bbb_2|.$$ Thus, $|xA+yB|=0$ for any $x$ and $y$ if and only if $|A|=|B|=\Delta(A;B)=0$. We divide into four cases:
- $\aaa_2=\alpha\aaa_1$, $\bbb_2=\beta\bbb_1$ for some $\alpha$ and $\beta$;
- $\aaa_1\ne\zerovec$, $\aaa_2=\alpha\aaa_1$ for some $\alpha$, $\bbb_1=\zerovec$, $\bbb_2\ne\zerovec$;
- $\aaa_1=\zerovec$, $\aaa_2\ne\zerovec$, $\bbb_1\ne\zerovec$, $\bbb_2=\beta\bbb_1$ for some $\beta$; and
- $\aaa_1=\zerovec$, $\aaa_2\ne\zerovec$, $\bbb_1=\zerovec$, $\bbb_2\ne\zerovec$.
\(a) Since $|\aaa_1,\bbb_2|+|\bbb_1,\aaa_2|=(\beta-\alpha)|\aaa_1,\bbb_1|$, if $|\aaa_1,\bbb_1|=0$ implies and otherwise $(\aaa_2,\bbb_2)=\alpha(\aaa_1,\bbb_1)$. (b) Since $|\aaa_1,\bbb_2|+|\bbb_1,\aaa_2|=|\aaa_1,\bbb_2|$, we have $|\aaa_1,\bbb_1|+|\aaa_2,\bbb_2|=\alpha|\aaa_1,\bbb_2|=0$ which implies . (c) Since $|\aaa_1,\bbb_2|+|\bbb_1,\aaa_2|=|\bbb_1,\aaa_2|$, we have $|\aaa_1,\bbb_1|+|\aaa_2,\bbb_2|=\beta|\bbb_1,\aaa_2|=0$ which implies . (d) If $|\aaa_2,\bbb_2|=0$ implies and otherwise $(\aaa_1,\bbb_1)=O$ which implies .
We define a function $\Theta\colon \FFF^{2\times2\times2} \to \FFF$ by $\Theta((\aaa_1,\aaa_2);(\bbb_1,\bbb_2))=|\aaa_1,\bbb_1|+|\aaa_2,\bbb_2|$. We have the following corollary by Theorem \[thm:2x2x2\].
\[cor:2x2x2\] Let $T=((\aaa_1,\aaa_2);(\bbb_1,\bbb_2))$ be a $2\times 2\times 2$ tensor.
1. \[cor:2x2x2:complex\] A complex tensor $T$ has rank three if and only if $\dim\langle \begin{pmatrix} \aaa_1\\ \aaa_2\end{pmatrix}, \begin{pmatrix} \bbb_1\\ \bbb_2\end{pmatrix}\rangle=\dim\langle \begin{pmatrix} \aaa_1\\ \bbb_1\end{pmatrix}, \begin{pmatrix} \aaa_2\\ \bbb_2\end{pmatrix}\rangle=2$, $\Delta(T)=0$ and $\Theta(T)\ne0$.
2. \[cor:2x2x2:real\] A real tensor $T$ has rank three if and only if $\Delta(T)<0$, or $\dim\langle \begin{pmatrix} \aaa_1\\ \aaa_2\end{pmatrix}, \begin{pmatrix} \bbb_1\\ \bbb_2\end{pmatrix}\rangle=\dim\langle \begin{pmatrix} \aaa_1\\ \bbb_1\end{pmatrix}, \begin{pmatrix} \aaa_2\\ \bbb_2\end{pmatrix}\rangle=2$, $\Delta(T)=0$ and $\Theta(T)\ne0$.
We put $A\cdot B=\det(A+B)-\det(A)-\det(B)$.
Let $\mathbb{K}$ be a field and $A, B \in \mathbb{K}^{2\times 2}$. Then $(A ; B)$ is nonsingular if and only if $\det(A)\ne 0$ and the quadratic equation $(\det A)x^2 - (A \cdot B)x + (\det B) = 0$, has no solutions for $x$ in $\mathbb{K}$. Equivalently, $(A ; B)$ is nonsingular if and only if $A$ is nonsingular and the eigenvalues of $BA^{-1}$ do not belong to $\mathbb{K}$.
Theoretical results
===================
We refer to the paper [@Friedland:2008] by Friedland. He wrote properties for $3$-tensors but almost all properties with respect to the map $f_k$ canonically hold for $n$-tensors in general.
\[typical rank by Friedland\] The space $\mathcal{T}:=\RRR^{m_1\times\cdots\times m_n}$, $m_1,\ldots,,m_n \in \NNN$, contains a finite number of open connected disjoint semi-algebraic sets $O_1, \ldots, O_M$ satisfying the following properties.
1. $\mathcal{T} \smallsetminus \cup_{i=1}^M O_i$ is a closed semi-algebraic set of $\mathcal{T}$ of dimension strictly less than $m_1\cdots m_n=\dim\mathcal{T}$.
2. Each $T \in O_i$ has rank $r_i$ for $i = 1,\ldots, M$.
3. $\min(r_1, \ldots, r_M) = grank(m_1,\ldots,m_n)$.
4. $mtrank(m_1,\ldots,m_n) := \max(r_1,\ldots, r_M)$ is the minimal $k \in \NNN$ such that the closure of $f_k((\RRR^{m_1}\times\cdots\times \RRR^{m_n})^k)$ is equal to $\mathcal{T}$.
5. For each integer $r \in [{\mathrm{grank}}(m_1,\ldots,m_n), {\mathrm{mtrank}}(m_1,\ldots,m_n)]$ there exists $r_i = r$ for some integer $i \in [1,M]$.
We call the number $r_i$, $i\in [1,M]$ a typical rank of $\mathcal{T}$.
We denote by $\mathcal{T}_{\leq p}$ the subset of all tensors with rank less than or equal to $p$ of $\mathcal{T}$.
\[thm:judge typical rank\] Let $\mathcal{T}:=\RRR^{m_1\times\cdots\times m_n}$, $p\in \NNN$ and $L$ a closed semi-algebraic set of dimension less than $\dim\mathcal{T}$. Let $f\colon \mathcal{T}\smallsetminus L \to \RRR$ be a continuous map such that $f(T)\geq 0$ for any $T$ and $f(T)=0$ for $T\in \mathcal{T}_{\leq p}$. If $f$ is not zero, then there exists a typical rank $q$ of $\mathcal{T}$ with $q>p$.
Suppose that there does not exist a typical rank $q$ of $\mathcal{T}$ with $q>p$. Then $p$ is greater than or equal to the maximal typical rank of $\mathcal{T}$. By Theorem \[typical rank by Friedland\], there is an open dense semi-algebraic set $O$ of $\mathcal{T}$ such that $O\subset \mathcal{T}_{\leq p}$ and then $O\smallsetminus L$ is also an open dense semi-algebraic set of $\mathcal{T}$. Since $f(T)=0$ for any $T\in O\smallsetminus L$ and $f$ is continuous, we have $f$ must be a constant zero map.
Let $p$ be a typical rank of $\mathcal{T}$. If there is a nonzero map in Theorem \[thm:judge typical rank\], then $p+1$ is a typical rank of $\mathcal{T}$.
$2\times 2\times 2\times 2$ tensors
===================================
Let $\underline{T}=(t_{ijk\ell})$ be a $2\times 2\times2\times 2$ tensor. Put $T_{k\ell}=\begin{pmatrix} t_{11k\ell}& t_{12k\ell}\\
t_{21k\ell} & t_{22k\ell}\end{pmatrix}$ for $k,\ell=1,2$. We describe $\underline{T}$ as $$\begin{array}{c|c}
T_{11} & T_{12}\vrule width0pt depth8pt\\
\hline
T_{21} & T_{22}\vrule width0pt height12pt
\end{array},\quad \begin{array}{c|c}
T_{\cdot1} & T_{\cdot2}
\end{array},\quad\text{or}\quad \begin{array}{c}
T_{1\cdot} \vrule width0pt depth8pt\\
\hline
T_{2\cdot} \vrule width0pt height12pt
\end{array}.$$ Let $G_1,\ldots,G_4={\mathrm{GL}}(2,\CCC)$. The action of ${\mathrm{GL}}(2,\CCC)^4$ is as follows. $(P_1,P_2,E,E)\cdot \underline{T}$ is given by $$\begin{array}{c|c}
P_1T_{11}P_2^\top & P_1T_{12}P_2^\top\vrule width0pt depth8pt\\
\hline
P_1T_{21}P_2^\top & P_1T_{22}P_2^\top\vrule width0pt height12pt
\end{array},$$ $(E,E,\begin{pmatrix} p_{11}&p_{12}\\ p_{21}&p_{22}\end{pmatrix},E)\cdot (T_{\cdot1}|T_{\cdot2})$ is given by $$\begin{array}{c|c}
p_{11}T_{\cdot1}+p_{12}T_{\cdot2}&p_{21}T_{\cdot1}+p_{22}T_{\cdot2}\vrule width0pt depth8pt
\end{array},$$ and $(E,E,E,\begin{pmatrix} q_{11}&q_{12}\\ q_{21}&q_{22}\end{pmatrix})\cdot \left(\begin{array}{c} T_{1\cdot}\vrule width0pt depth5pt\\ \hline T_{2\cdot}\vrule width0pt height12pt\end{array}\right)$ is given by $$\begin{array}{c} q_{11}T_{1\cdot}+q_{12}T_{2\cdot}\vrule width0pt depth8pt\\ \hline q_{21}T_{1\cdot}+q_{22}T_{2\cdot}\vrule width0pt height12pt\end{array}.$$ We also denote $\underline{T}$ by $((T_{11};T_{12});(T_{21};T_{22}))$.
\[prop:action\] Let $g\in {\mathrm{GL}}(2,\FFF)^4$ and $\underline{T}^\prime=g\cdot \underline{T}$. Suppose that ${\mathrm{rank}}_{\FFF}(T_{1\cdot})=3$. If ${\mathrm{rank}}_{\FFF}(T^\prime_{1\cdot})\leq2$ then there is $x\in\FFF$ such that ${\mathrm{rank}}_{\FFF}(xT_{1\cdot}+T_{2\cdot})\leq 2$.
Suppose that ${\mathrm{rank}}_{\FFF}(T^\prime_{1\cdot})\leq2$ for $g=(g_1,g_2,g_3,g_4)$. Then ${\mathrm{rank}}(T^{\prime\prime}_{1\cdot})={\mathrm{rank}}_{\FFF}(T^\prime_{1\cdot})$, where $\underline{T}^{\prime\prime}=(E,E,E,g_4)\cdot \underline{T}$. Thus there is $(x_1,x_2)\ne(0,0)$ such that ${\mathrm{rank}}(x_1T_{1\cdot}+x_2T_{2\cdot})={\mathrm{rank}}(T^\prime_{1\cdot})$. Note that ${\mathrm{rank}}(yS_{1\cdot})={\mathrm{rank}}(S_{1\cdot})$ for any $y\ne 0$, where $\underline{S}=(g_1,g_2,g_3,E)\cdot \underline{T}$. Since ${\mathrm{rank}}(T^{\prime\prime}_{1\cdot})\leq 2$, we have $x_2\ne 0$ and then ${\mathrm{rank}}(\frac{x_1}{x_2}T_{1\cdot}+T_{2\cdot})={\mathrm{rank}}(T^\prime_{1\cdot})\leq2$.
\[lem:onewayseparate\] Let $A$ and $B$ be real (resp. complex) $2\times 2\times 2$ tensors. Suppose that $A=T_1+T_2$ for some rank one tensors $T_1$ and $T_2$. If $\Delta(B+xT_1)$ is positive (resp. nonzero) for some $x$, then ${\mathrm{rank}}(A;B)\leq 4$.
We replace the values of $\Delta$ are nonzero instead of positive over $\CCC$ in the following argument. So, we assume that the base field is $\RRR$. Suppose that $\Delta(B+xT_1)>0$. Then ${\mathrm{rank}}(B+xT_1)=2$. Therefore we have $$\begin{split}
{\mathrm{rank}}((A;B)+(-T_1;xT_1)) &={\mathrm{rank}}(T_2;B+xT_1) \\
&\leq {\mathrm{rank}}(T_2)+{\mathrm{rank}}(B+xT_1) \\
&=1+2=3
\end{split}$$ and $$\begin{split}
{\mathrm{rank}}(A;B) &\leq {\mathrm{rank}}(T_1;-xT_1)+{\mathrm{rank}}(Y+(-T_1;xT_1)) \\
&\leq 1+3=4.
\end{split}$$
\[lem;onepartiszero\] Let $\underline{Y}=((E;O);(B_1;B_2))$ be a $2\times 2\times 2\times 2$ tensor. Then ${\mathrm{rank}}_{\FFF}(\underline{Y})\leq 4$ holds.
Put $$S_1={\mathrm{diag}}(1,0), S_2={\mathrm{diag}}(0,1), S_3=\frac{1}{2}\begin{pmatrix} 1&1\\ 1&1\end{pmatrix}
\text{ and }
S_4=\frac{1}{2}\begin{pmatrix} 1&-1\\ -1&1\end{pmatrix}.$$ Note that $(E;O)=(S_1;O)+(S_2;O)=(S_3;O)+(S_4;O)$ and $$\begin{array}{l}
\Delta((B_1;B_2)+x(S_1;O))=b_{222}^2x^2+(\text{lower term}), \\
\Delta((B_1;B_2)+x(S_2;O))=b_{112}^2x^2+(\text{lower term}), \\
\Delta((B_1;B_2)+x(S_3;O))=(b_{112} - b_{122} - b_{212} + b_{222})^2x^2/4+(\text{lower term}), \\
\Delta((B_1;B_2)+x(S_4;O))=(b_{112} + b_{122} - b_{212} - b_{222})^2x^2/4+(\text{lower term}).
\end{array}$$ If $b_{222}\ne 0$ then ${\mathrm{rank}}(B_1;B_2)+x_0(S_1;O))=2$ for some $x_0$ and ${\mathrm{rank}}(\underline{Y})\leq 4$ by Lemma \[lem:onewayseparate\]. Similarly, if $b_{112}$, $b_{112} - b_{122} - b_{212} + b_{222}$ or $b_{112} + b_{122} - b_{212} - b_{222}$ is nonzero then ${\mathrm{rank}}(\underline{Y})\leq 4$. If $b_{222}=b_{112}=b_{112} - b_{122} - b_{212} + b_{222}=b_{112} + b_{122} - b_{212} - b_{222}=0$ then $B_2=0$ and thus ${\mathrm{rank}}(\underline{Y})\leq 3$, since ${\mathrm{rank}}(\underline{Y})={\mathrm{rank}}(E;B_1)\leq 3$.
\[prop:leq2implies4a\] Let $\underline{Y}=((A_1;A_2);(B_1;B_2))$ be a $2\times 2\times 2\times 2$ tensor. Suppose that ${\mathrm{rank}}_{\FFF}(A_1;A_2)\leq 2$. Then ${\mathrm{rank}}_{\FFF}(\underline{Y})\leq 4$ holds.
If ${\mathrm{rank}}(B_1;B_2)\leq 2$ then we have $${\mathrm{rank}}(\underline{Y})\leq {\mathrm{rank}}(A_1;A_2)+{\mathrm{rank}}(B_1;B_2)\leq 2+2=4$$ and similarly if ${\mathrm{rank}}(A_1;A_2)\leq 1$ then $${\mathrm{rank}}(\underline{Y})\leq {\mathrm{rank}}(A_1;A_2)+{\mathrm{rank}}(B_1;B_2)\leq 1+3=4.$$ Suppose that ${\mathrm{rank}}(A_1;A_2)=2$ and ${\mathrm{rank}}(B_1;B_2)=3$. Then there is $\mu\in {\mathrm{GL}}(2,\FFF)^{\times 3}$ such that $\mu\cdot(A_1;A_2)=(E;{\mathrm{diag}}(a,b))$ for some $a,b\in\mathbb{R}$. Putting $(C_1;C_2)=\mu\cdot(B_1;B_2)$, we have $$\Delta((C_1;C_2)+x(E;{\mathrm{diag}}(a,b)))=(a-b)^2x^4+(\text{lower term}).$$ Then if $a\ne b$ then $\Delta((B_1;B_2)+x_0(A_1;A_2))$ is positive (resp. nonzero) and thus ${\mathrm{rank}}((B_1;B_2)+x_0(A_1;A_2))\leq 2$ for some $x_0$. We have $$\begin{split}
{\mathrm{rank}}(\underline{Y})&\leq {\mathrm{rank}}(A_1;A_2)+{\mathrm{rank}}((B_1;B_2)+x_0(A_1;A_2)) \\
&\leq 2+2=4.
\end{split}$$ Finally, suppose that $a=b$. Since $\underline{Y}$ is equivalent to $((E;O);(C_1;C_2-aC_1))$, by Lemma \[lem;onepartiszero\], we have ${\mathrm{rank}}(\underline{Y})\leq 4$.
Over $\RRR$, the following proposition has been obtained (see [@Kong-etal:2012 Proposition 4.1]). For the reader’s convenience, we show the proof of Theorem \[thm:Kong-etal\].
Let $\underline{Y}=(A;B)$ be a $2\times2\times2\times2$ real tensor. If ${\mathrm{rank}}(A)\leq 2$, then ${\mathrm{rank}}(\underline{Y})\leq 4$ by Proposition \[prop:leq2implies4a\]. Suppose that ${\mathrm{rank}}(A)>2$. Then ${\mathrm{rank}}(A)=3$, since the maximal rank of $2\times2\times2$ tensors is $3$. Take a $2\times2\times2$ rank one tensor $C$ such that ${\mathrm{rank}}(A-C)=2$. Again by Proposition \[prop:leq2implies4a\], we have ${\mathrm{rank}}(\underline{Y}-(C;O))\leq 4$. Therefore, ${\mathrm{rank}}(\underline{Y})\leq {\mathrm{rank}}(\underline{Y}-(C;O))+{\mathrm{rank}}(C;O)\leq 4+1=5$. This completes the proof.
\[prop:exceptnone\] Let $\underline{T}$ be a $2\times 2\times2\times 2$ complex tensor. There is a tensor $\underline{A}$ such that $\underline{A}$ is equivalent to $\underline{T}$ and $(A_{11};A_{12})$ has rank less than or equal to $2$.
We may suppose that ${\mathrm{rank}}(T_{11};T_{12})=3$. There is $\alpha\in {\mathrm{GL}}(2,\FFF)^3$ such that $S_{11}=E$, $S_{12}=\begin{pmatrix} 0&1\\ 0&0\end{pmatrix}$ for $\underline{S}=(\alpha,E)\cdot \underline{T}$. Since $\Theta(xS_{1\cdot}+S_{2\cdot})$ is a polynomial of $x$ with degree two, there is $x_0\in\CCC$ such that $\Theta(x_0S_{1\cdot}+S_{2\cdot})=0$. Thus by Corollary \[cor:2x2x2\] , we have ${\mathrm{rank}}(x_0S_{1\cdot}+S_{2\cdot})\leq 2$. Let $P=\begin{pmatrix} x_0&1\\ 1&0 \end{pmatrix}$ and $\underline{A}=(\alpha,P)\cdot\underline{T}$. Then ${\mathrm{rank}}(A_{11};A_{12})\leq2$.
The maximal and typical rank of $\CCC^{2\times2\times2\times2}$ is equal to $4$.
By Propositions \[prop:leq2implies4a\] and \[prop:exceptnone\], the maximal rank of $\CCC^{2\times2\times2\times2}$ is equal to $4$. For $\underline{A}=((A_{11};A_{12});(A_{21};A_{22}))\in \CCC^{2\times2\times2\times2}$, since $${\mathrm{rank}}\underline{A}\geq
{\mathrm{rank}}\begin{pmatrix} A_{11}&A_{12} \\ A_{21}&A_{22}\end{pmatrix},$$ a typical rank of $\CCC^{2\times2\times2\times 2}$ is greater than or equal to $4$. Therefore a typical rank of $\CCC^{2\times2\times2\times 2}$ is equal to $4$.
Let $$\label{eq:exampletensor}
\underline{X}=\begin{array}{cc|cc}
1&0&0&1\\
0&1&-1&0\vrule width0pt depth4pt\\
\hline
0&-1&1&0\vrule width0pt height12pt\\
2&0&0&2\vrule width0pt depth4pt\\
\end{array}.$$
We do not proceed in the real number field as in the complex number field:
\[prop:notConverttorank2slice\] There is a tensor $\underline{T}$ in $\RRR^{2\times2\times2\times2}$ such that ${\mathrm{rank}}(S_{11};S_{12})={\mathrm{rank}}(S_{11};S_{21})={\mathrm{rank}}(S_{21};S_{22})={\mathrm{rank}}(S_{12};S_{22})=3$ for any $g\in {\mathrm{GL}}(2,\RRR)^4$, where $\underline{S}=\begin{array}{c|c} S_{11}&S_{12}\vrule width0pt depth4pt\\
\hline
S_{21}&S_{22}\vrule width0pt height12pt
\end{array}=g\cdot \underline{T}$.
We show $\underline{T}=\underline{X}$ satisfies the assertion. It suffices to show that $\Delta(S_{11};S_{12})$, $\Delta(S_{11};S_{21})$, $\Delta(S_{21};S_{22})$, $\Delta(S_{12};S_{22})$ are all negative for any $g\in {\mathrm{GL}}(2,\RRR)^4$. Let $g\in {\mathrm{GL}}(2,\RRR)^4$. By Proposition \[prop:Silva\], we may suppose that $g=(E,E,P,Q)$. For $\Delta(S_{11};S_{12})$, we may further suppose that $P=E$ by Proposition \[prop:Silva\]. We straightforwardly see that $\Delta(S_{11};S_{12})=-4(q_{11}^2 + 2q_{12}^2)^2<0$. Similarly, for $\Delta(S_{11};S_{21})$, we may suppose that $Q=E$ and see that $\Delta(S_{11};S_{21})=-8 (p_{11}^2 + p_{12}^2)^2<0$.
Now we give a condition for a tensor $\underline{T}\in \RRR^{2\times2\times2\times2}$ to have rank $4$.
Let $\underline{T}=(t_{ijkl})$ be a $2\times 2\times2\times 2$ real tensor. We consider the following condition (E): $$\begin{aligned}
\left|\begin{matrix}
t_{1111}&t_{1211}&t_{2111}&t_{2211} \\
c_{12}d_{12} & c_{12}d_{22} & c_{22}d_{12} & c_{22}d_{22} \\
c_{13}d_{13} & c_{13}d_{23} & c_{23}d_{13} & c_{23}d_{23} \\
c_{14}d_{14} & c_{14}d_{24} & c_{24}d_{14} & c_{24}d_{24} \end{matrix}\right|
\left|\begin{matrix}
t_{1122}&t_{1222}&t_{2122}&t_{2222} \\
c_{12}d_{12} & c_{12}d_{22} & c_{22}d_{12} & c_{22}d_{22} \\
c_{13}d_{13} & c_{13}d_{23} & c_{23}d_{13} & c_{23}d_{23} \\
c_{14}d_{14} & c_{14}d_{24} & c_{24}d_{14} & c_{24}d_{24} \end{matrix}\right|
\hbox to 30mm{}\notag \\
-\left|\begin{matrix}
t_{1112}&t_{1212}&t_{2112}&t_{2212} \\
c_{12}d_{12} & c_{12}d_{22} & c_{22}d_{12} & c_{22}d_{22} \\
c_{13}d_{13} & c_{13}d_{23} & c_{23}d_{13} & c_{23}d_{23} \\
c_{14}d_{14} & c_{14}d_{24} & c_{24}d_{14} & c_{24}d_{24} \end{matrix}\right|
\left|\begin{matrix}
t_{1121}&t_{1221}&t_{2121}&t_{2221} \\
c_{12}d_{12} & c_{12}d_{22} & c_{22}d_{12} & c_{22}d_{22} \\
c_{13}d_{13} & c_{13}d_{23} & c_{23}d_{13} & c_{23}d_{23} \\
c_{14}d_{14} & c_{14}d_{24} & c_{24}d_{14} & c_{24}d_{24} \end{matrix}\right|=0,
\\
\left|\begin{matrix}
c_{11}d_{11} & c_{11}d_{21} & c_{21}d_{11} & c_{21}d_{21} \\
t_{1111}&t_{1211}&t_{2111}&t_{2211} \\
c_{13}d_{13} & c_{13}d_{23} & c_{23}d_{13} & c_{23}d_{23} \\
c_{14}d_{14} & c_{14}d_{24} & c_{24}d_{14} & c_{24}d_{24} \end{matrix}\right|
\left|\begin{matrix}
c_{11}d_{11} & c_{11}d_{21} & c_{21}d_{11} & c_{21}d_{21} \\
t_{1122}&t_{1222}&t_{2122}&t_{2222} \\
c_{13}d_{13} & c_{13}d_{23} & c_{23}d_{13} & c_{23}d_{23} \\
c_{14}d_{14} & c_{14}d_{24} & c_{24}d_{14} & c_{24}d_{24} \end{matrix}\right|
\hbox to 30mm{}\notag \\
-\left|\begin{matrix}
c_{11}d_{11} & c_{11}d_{21} & c_{21}d_{11} & c_{21}d_{21} \\
t_{1112}&t_{1212}&t_{2112}&t_{2212} \\
c_{13}d_{13} & c_{13}d_{23} & c_{23}d_{13} & c_{23}d_{23} \\
c_{14}d_{14} & c_{14}d_{24} & c_{24}d_{14} & c_{24}d_{24} \end{matrix}\right|
\left|\begin{matrix}
c_{11}d_{11} & c_{11}d_{21} & c_{21}d_{11} & c_{21}d_{21} \\
t_{1121}&t_{1221}&t_{2121}&t_{2221} \\
c_{13}d_{13} & c_{13}d_{23} & c_{23}d_{13} & c_{23}d_{23} \\
c_{14}d_{14} & c_{14}d_{24} & c_{24}d_{14} & c_{24}d_{24} \end{matrix}\right|=0,
\\
\left|\begin{matrix}
c_{11}d_{11} & c_{11}d_{21} & c_{21}d_{11} & c_{21}d_{21} \\
c_{12}d_{12} & c_{12}d_{22} & c_{22}d_{12} & c_{22}d_{22} \\
t_{1111}&t_{1211}&t_{2111}&t_{2211} \\
c_{14}d_{14} & c_{14}d_{24} & c_{24}d_{14} & c_{24}d_{24} \end{matrix}\right|
\left|\begin{matrix}
c_{11}d_{11} & c_{11}d_{21} & c_{21}d_{11} & c_{21}d_{21} \\
c_{12}d_{12} & c_{12}d_{22} & c_{22}d_{12} & c_{22}d_{22} \\
t_{1122}&t_{1222}&t_{2122}&t_{2222} \\
c_{14}d_{14} & c_{14}d_{24} & c_{24}d_{14} & c_{24}d_{24} \end{matrix}\right|
\hbox to 30mm{}\notag \\
-\left|\begin{matrix}
c_{11}d_{11} & c_{11}d_{21} & c_{21}d_{11} & c_{21}d_{21} \\
c_{12}d_{12} & c_{12}d_{22} & c_{22}d_{12} & c_{22}d_{22} \\
t_{1112}&t_{1212}&t_{2112}&t_{2212} \\
c_{14}d_{14} & c_{14}d_{24} & c_{24}d_{14} & c_{24}d_{24} \end{matrix}\right|
\left|\begin{matrix}
c_{11}d_{11} & c_{11}d_{21} & c_{21}d_{11} & c_{21}d_{21} \\
c_{12}d_{12} & c_{12}d_{22} & c_{22}d_{12} & c_{22}d_{22} \\
t_{1121}&t_{1221}&t_{2121}&t_{2221} \\
c_{14}d_{14} & c_{14}d_{24} & c_{24}d_{14} & c_{24}d_{24} \end{matrix}\right|=0,
\\
\left|\begin{matrix}
c_{11}d_{11} & c_{11}d_{21} & c_{21}d_{11} & c_{21}d_{21} \\
c_{12}d_{12} & c_{12}d_{22} & c_{22}d_{12} & c_{22}d_{22} \\
c_{13}d_{13} & c_{13}d_{23} & c_{23}d_{13} & c_{23}d_{23} \\
t_{1111}&t_{1211}&t_{2111}&t_{2211} \end{matrix}\right|
\left|\begin{matrix}
c_{11}d_{11} & c_{11}d_{21} & c_{21}d_{11} & c_{21}d_{21} \\
c_{12}d_{12} & c_{12}d_{22} & c_{22}d_{12} & c_{22}d_{22} \\
c_{13}d_{13} & c_{13}d_{23} & c_{23}d_{13} & c_{23}d_{23} \\
t_{1122}&t_{1222}&t_{2122}&t_{2222} \end{matrix}\right|
\hbox to 30mm{}\notag \\
-\left|\begin{matrix}
c_{11}d_{11} & c_{11}d_{21} & c_{21}d_{11} & c_{21}d_{21} \\
c_{12}d_{12} & c_{12}d_{22} & c_{22}d_{12} & c_{22}d_{22} \\
c_{13}d_{13} & c_{13}d_{23} & c_{23}d_{13} & c_{23}d_{23} \\
t_{1112}&t_{1212}&t_{2112}&t_{2212} \end{matrix}\right|
\left|\begin{matrix}
c_{11}d_{11} & c_{11}d_{21} & c_{21}d_{11} & c_{21}d_{21} \\
c_{12}d_{12} & c_{12}d_{22} & c_{22}d_{12} & c_{22}d_{22} \\
c_{13}d_{13} & c_{13}d_{23} & c_{23}d_{13} & c_{23}d_{23} \\
t_{1121}&t_{1221}&t_{2121}&t_{2221} \end{matrix}\right|=0,
\\
\det(M)\ne 0,
$$ where $$\label{eq:MatrixM}
M=\begin{pmatrix}
c_{11}d_{11} & c_{11}d_{21} & c_{21}d_{11} & c_{21}d_{21} \\
c_{12}d_{12} & c_{12}d_{22} & c_{22}d_{12} & c_{22}d_{22} \\
c_{13}d_{13} & c_{13}d_{23} & c_{23}d_{13} & c_{23}d_{23} \\
c_{14}d_{14} & c_{14}d_{24} & c_{24}d_{14} & c_{24}d_{24} \end{pmatrix}$$ is a $4\times 4$ matrix.
For a $2\times 2$ matrix $X=(\xxx_1,\xxx_2)$, put ${\mathrm{vec}}(X)=\begin{pmatrix} \xxx_1\\ \xxx_2\end{pmatrix}$.
\[thm:suffcond\] Suppose that $T_{11},T_{12},T_{21},T_{22}$ are linearly independent. There are $c_{ik}$, $d_{ik}\in \KKK$ ($i=1,2$, $k=1,2,3,4$) such that the condition [(E)]{} holds, if and only if ${\mathrm{rank}}_{\KKK}(\underline{T})=4$.
Recall that ${\mathrm{rank}}_{\KKK}(\underline{T})\leq r$ if and only if there are $x_{ik},y_{ik}\in \KKK$ and rank one matrices $A_k$ for $i=1,2$ and $k=1,2,\ldots,r$ such that $T_{ij}=\sum_{k=1}^r x_{ik}y_{jk}A_k$ for $i,j=1,2$. Since $T_{11},T_{12},T_{21},T_{22}$ lie in the vector space generated by $A_k$’s, we have ${\mathrm{rank}}_{\KKK}(\underline{T})\geq 4$.
Consider the problem $T_{ij}=\sum_{k=1}^4 c_{ik}d_{jk}A_k$ for $i,j=1,2$. Then we have $$\label{eq:B}
({\mathrm{vec}}(T_{11}),{\mathrm{vec}}(T_{12}),{\mathrm{vec}}(T_{21}),{\mathrm{vec}}(T_{22}))
=({\mathrm{vec}}(A_{1}),{\mathrm{vec}}(A_{2}),{\mathrm{vec}}(A_{3}),{\mathrm{vec}}(A_{4}))M.$$ Since $({\mathrm{vec}}(T_{11}),{\mathrm{vec}}(T_{12}),{\mathrm{vec}}(T_{21}),{\mathrm{vec}}(T_{22}))$ is nonsingular, $M$ is nonsingular, and $$\label{eq:A}
({\mathrm{vec}}(A_{1}),{\mathrm{vec}}(A_{2}),{\mathrm{vec}}(A_{3}),{\mathrm{vec}}(A_{4}))=
({\mathrm{vec}}(T_{11}),{\mathrm{vec}}(T_{12}),{\mathrm{vec}}(T_{21}),{\mathrm{vec}}(T_{22}))M^{-1}.$$ Therefore, for each $k=1,2,3,4$, the equation ($k+1$) is zero if and only if ${\mathrm{rank}}(A_k)\leq1$. Conversely, for $c_{ik}$, $d_{ik}\in \KKK$ such that the condition [(E)]{} holds, determining $A_k$ by , $A_k$ is a rank one matrix and $$\underline{T}=\sum_{k=1}^4 A_k\odot \begin{pmatrix} c_{1k}\\ c_{2k}\end{pmatrix}\odot
\begin{pmatrix} d_{1k}\\ d_{2k}\end{pmatrix},$$ which implies that ${\mathrm{rank}}_{\KKK}(\underline{T})=4$.
Numerical approach
==================
We compute $\Delta(A;B)$, $\Delta(C;D)$, $\Delta(A;C)$, $\Delta(B;D)$, $\Delta(A+xC;B+xD)$, $\Delta(A+xB;C+xD)$, and $\Delta(A+xB+z(C+xD);yA+B+z(yC+D))$ for $\begin{array}{c|c}
A& B\vrule width0pt depth4pt\\
\hline
C&D\vrule width0pt height12pt
\end{array}.$
There are $24$ flattening pattern corresponding to permutations $(i,j,k,\ell)$. The following $4$ patterns are essentially same, since they correspond to the transpose of matrices.
$$(t_{ijk\ell})=\begin{array}{cc|cc}
t_{1111}&t_{1211} &t_{1112} &t_{1212}\\
t_{2111}&t_{2211}&t_{2112}&t_{2212}\vrule width0pt depth4pt\\
\hline
t_{1121}&t_{1221} &t_{1122} &t_{1222}\vrule width0pt height12pt\\
t_{2121}&t_{2221}&t_{2122}&t_{2222}\vrule width0pt depth4pt\\
\end{array}, \quad
(t_{jik\ell})=\begin{array}{cc|cc}
t_{1111}&t_{2111} &t_{1112} &t_{2112}\\
t_{1211}&t_{2211}&t_{1212}&t_{2212}\vrule width0pt depth4pt\\
\hline
t_{1121}&t_{2121} &t_{1122} &t_{2122}\vrule width0pt height12pt\\
t_{1221}&t_{2221}&t_{1222}&t_{2222}\vrule width0pt depth4pt\\
\end{array},$$
$$(t_{ij\ell k})=\begin{array}{cc|cc}
t_{1111}&t_{1211} &t_{1121} &t_{1221}\\
t_{2111}&t_{2211}&t_{2121}&t_{2221}\vrule width0pt depth4pt\\
\hline
t_{1112}&t_{1212} &t_{1122} &t_{1222}\vrule width0pt height12pt\\
t_{2112}&t_{2212}&t_{2122}&t_{2222}\vrule width0pt depth4pt\\
\end{array},\quad
(t_{ji\ell k})=\begin{array}{cc|cc}
t_{1111}&t_{2111} &t_{1121} &t_{2121}\\
t_{1211}&t_{2211}&t_{1221}&t_{2221}\vrule width0pt depth4pt\\
\hline
t_{1112}&t_{2112} &t_{1122} &t_{2122}\vrule width0pt height12pt\\
t_{1212}&t_{2212}&t_{1222}&t_{2222}\vrule width0pt depth4pt\\
\end{array}$$
Thus there are essentially $6$ patterns:
$$(t_{ijk\ell})=\begin{array}{cc|cc}
t_{1111}&t_{1211} &t_{1112} &t_{1212}\\
t_{2111}&t_{2211}&t_{2112}&t_{2212}\vrule width0pt depth4pt\\
\hline
t_{1121}&t_{1221} &t_{1122} &t_{1222}\vrule width0pt height12pt\\
t_{2121}&t_{2221}&t_{2122}&t_{2222}\vrule width0pt depth4pt\\
\end{array},\quad
(t_{kji\ell})=\begin{array}{cc|cc}
t_{1111}&t_{1211} &t_{1112} &t_{1212}\\
t_{1121}&t_{1221}&t_{1122}&t_{1222}\vrule width0pt depth4pt\\
\hline
t_{2111}&t_{2211} &t_{2112} &t_{2212}\vrule width0pt height12pt\\
t_{2121}&t_{2221}&t_{2122}&t_{2222}\vrule width0pt depth4pt\\
\end{array},$$
$$(t_{ikj\ell})=\begin{array}{cc|cc}
t_{1111}&t_{1121} &t_{1112} &t_{1122}\\
t_{2111}&t_{2121}&t_{2112}&t_{2122}\vrule width0pt depth4pt\\
\hline
t_{1211}&t_{1221} &t_{1212} &t_{1222}\vrule width0pt height12pt\\
t_{2211}&t_{2221}&t_{2212}&t_{2222}\vrule width0pt depth4pt\\
\end{array}, \quad
(t_{i\ell kj})=\begin{array}{cc|cc}
t_{1111}&t_{1112} &t_{1211} &t_{1212}\\
t_{2111}&t_{2112}&t_{2211}&t_{2212}\vrule width0pt depth4pt\\
\hline
t_{1121}&t_{1122} &t_{1221} &t_{1222}\vrule width0pt height12pt\\
t_{2121}&t_{2122}&t_{2221}&t_{2222}\vrule width0pt depth4pt
\end{array},$$
$$(t_{k\ell ij})=\begin{array}{cc|cc}
t_{1111}&t_{1112} &t_{1211} &t_{1212}\\
t_{1121}&t_{1122}&t_{1221}&t_{1222}\vrule width0pt depth4pt\\
\hline
t_{2111}&t_{2112} &t_{2211} &t_{2212}\vrule width0pt height12pt\\
t_{2121}&t_{2122}&t_{2221}&t_{2222}\vrule width0pt depth4pt\\
\end{array}, \quad
(t_{j\ell ki})=\begin{array}{cc|cc}
t_{1111}&t_{2111} &t_{1211} &t_{2211}\\
t_{1112}&t_{2112}&t_{1212}&t_{2212}\vrule width0pt depth4pt\\
\hline
t_{1121}&t_{2121} &t_{1221} &t_{2221}\vrule width0pt height12pt\\
t_{1122}&t_{2122}&t_{1222}&t_{2222}\vrule width0pt depth4pt
\end{array}$$
The tensor $\underline{X}$ given in implies the following result.
\[prop:notConverttorank2sliceforanyflatten\] There is a tensor $\underline{T}$ in $\RRR^{2\times2\times2\times2}$ such that ${\mathrm{rank}}(S_{11};S_{12})={\mathrm{rank}}(S_{11};S_{21})={\mathrm{rank}}(S_{21};S_{22})={\mathrm{rank}}(S_{12};S_{22})=3$ for any flattening pattern $\underline{T}^\prime$ of $\underline{T}$ and any $g\in {\mathrm{GL}}(2,\RRR)^4$, where $\underline{S}=\begin{array}{c|c} S_{11}&S_{12}\vrule width0pt depth4pt\\
\hline
S_{21}&S_{22}\vrule width0pt height12pt
\end{array}=g\cdot \underline{T}^\prime$.
Therefore, there is a tensor in $\RRR^{2\times2\times2\times2}$ which does not apply Proposition \[prop:leq2implies4a\].
\[thm:estimate\] Let $\underline{T}$ be a $2\times 2\times2\times2$ real tensor. Suppose that the $4\times4$ matrix obtained from $\underline{T}$ is nonsingular. Put $$\label{eq:function}
f((c_{1k},d_{1k},c_{2k},d_{2k})^\top_{1\leq k\leq 4})=\frac{\displaystyle\sum_{j=1}^4 (n_{1j}n_{4j}-n_{2j}n_{3j})^2}{\displaystyle(\sum_{i,j=1}^4 n_{ij}^2)^2}.
$$ where $$(n_{ij})=({\mathrm{vec}}(T_{11}),{\mathrm{vec}}(T_{12}),{\mathrm{vec}}(T_{21}),{\mathrm{vec}}(T_{22}))M^{-1}.$$ If ${\mathrm{rank}}_{\RRR}(\underline{T})=4$ then $f=0$ and $\det(M)\ne0$ at some $(c_{1k},d_{1k}c_{2k},d_{2k})^\top_{1\leq k\leq 4}\in\RRR^{16}$.
Consider the equation . By the assumption, the matrix $M$ is nonsingular. By Theorem \[thm:suffcond\], if ${\mathrm{rank}}_{\RRR}(\underline{T})=4$ then there is $(c_{ik},d_{jk})^\top\in\RRR^{16}$ such that $n_{1k}n_{4k}=n_{2k}n_{3k}$ for $1\leq k\leq 4$.
Let $M$ be a matrix in , and put $\underline{T}=\underline{X}$ in . Note that a $4\times4$ matrix $$({\mathrm{vec}}(T_{11}),{\mathrm{vec}}(T_{12}),{\mathrm{vec}}(T_{21}),{\mathrm{vec}}(T_{22}))=\begin{pmatrix}
1&0&0&1\\
0&1&-1&0\\
0&-1&2&0\\
1&0&0&2\\
\end{pmatrix}$$ is nonsingular. Thus ${\mathrm{rank}}_{\RRR}(\underline{T})\geq 4$ and $M$ must be nonsingular. By we consider the function $f$ from an open subset $S:=\{ (c_{ik},d_{jk})^\top\mid \det(M)\ne 0\}$ of $\RRR^{16}$ to $\RRR$ defined as . Although $f$ might have no minimum value in general, Theorem \[thm:estimate\] yields us that if ${\mathrm{rank}}_{\RRR}(T)=4$ then $f$ must take the value zero, and $\inf f>0$ says that $5$ is a typical rank of $\RRR^{2\times2\times2\times2}$. So we consider the problem:\
------------ ---------------
infimize $f$
subject to $\det(M)\ne0$
------------ ---------------
We estimate by using the command FindMinimum in the software Mathematica [@Wolfram-Mathematica] and obtains the minimum value $0.04$ in the $10000$ times iterations.
T = {{1, 0, 0, 1}, {0, 1, -1, 0}, {0, -1, 1, 0}, {2, 0, 0, 2}};
myf[T_, x11_, x12_, x13_, x14_, x21_, x22_, x23_, x24_, y11_, y12_,
y13_, y14_, y21_, y22_, y23_, y24_] := Block[{U, V, mx},
U = {{x11*y11, x11*y21, x21*y11, x21*y21},
{x12*y12, x12*y22, x22*y12, x22*y22},
{x13*y13, x13*y23, x23*y13, x23*y23},
{x14*y14, x14*y24, x24*y14, x24*y24}};
V = Simplify[T.Inverse[U]]; mx := Sum[V[[i, j]]^2, {i, 1, 4}, {j, 1, 4}];
Sum[(V[[1, k]]*V[[4, k]] - V[[2, k]]*V[[3, k]])^2, {k, 1, 4}]/mx^2];
tryfind[T_] := Block[{vars, UU, U0, det, a, U, V, f}, a = 0;
While[a == 0, vars = Table[Random[Real, {-1, 1}], {16}];
U0 = {{vars[[1]]*vars[[9]], vars[[1]]*vars[[13]], vars[[5]]*vars[[9]],
vars[[5]]*vars[[13]]}, {vars[[2]]*vars[[10]], vars[[2]]*vars[[14]],
vars[[6]]*vars[[10]], vars[[6]]*vars[[14]]}, {vars[[3]]*vars[[11]],
vars[[3]]*vars[[15]], vars[[7]]*vars[[11]], vars[[7]]*vars[[15]]},
{vars[[4]]*vars[[12]], vars[[4]]*vars[[16]], vars[[8]]*vars[[12]],
vars[[8]]*vars[[16]]}};
a = Det[U0]];
FindMinimum[
myf[T,x11,x12,x13,x14,x21,x22,x23,x24,y11,y12,y13,y14,y21,y22,y23,y24],
{{x11, vars[[1]]}, {x12, vars[[2]]}, {x13, vars[[3]]}, {x14, vars[[4]]},
{x21, vars[[5]]}, {x22, vars[[6]]}, {x23, vars[[7]]}, {x24, vars[[8]]},
{y11, vars[[9]]}, {y12, vars[[10]]},{y13, vars[[11]]},{y14, vars[[12]]},
{y21, vars[[13]]},{y22, vars[[14]]},{y23, vars[[15]]},{y24, vars[[16]]}}]]
res = Table[tryfind[T], {10000}];
val = Table[res[[k, 1]], {k, 1, Length[res]}];
Min[val]
Thus we have
The maximal rank of $\RRR^{2\times2\times2\times2}$ is $5$ and the typical rank of $\RRR^{2\times2\times2\times2}$ is $\{4,5\}$.
High dimensional tensors
========================
A lower bound of the maximal rank of $n$-tensors with size $2\times\cdots\times 2$ is $$\frac{2^n}{2n-n+1}=\frac{2^n}{n+1}$$ (cf. [@Brylinski:2002 Proposition 1.2]) and a canonical upper bound of those is $2^n$. We give an upper bound by using the maximal rank of $\FFF^{2\times2\times2\times2}$ tensors.
\[prop:highertensorrank\] Let $1\leq s< n$. $${\mathrm{mrank}}_{\KKK}(m_1,m_2,\ldots,m_n)\leq {\mathrm{mrank}}_{\KKK}(m_1,\ldots,m_s)\prod_{t=s+1}^n m_t$$
Let $\underline{A}=(a_{i_1i_2\ldots i_n})$ be an $n$-tensor with size $m_1\times m_2\times \cdots \times m_n$.
Let $e_{i_1,\ldots,i_k}$, $1\leq i_t\leq a_t, 1\leq t\leq n$ be a standard basis of $\KKK^{m_1\times\cdots\times m_n}$, that is, $\eee_{i_1,\ldots,i_n}$ has $1$ at the $(i_1,\ldots,i_n)$-element and otherwise $0$.
The tensor $\underline{A}$ is described as $$\sum_{i_1,\ldots,i_n} a_{i_1,\ldots,i_n}\eee_{i_1,\ldots,i_n}
=\sum_{i_{s+1},\ldots,i_n} (\sum_{i_1,\ldots,i_s} a_{i_1,\ldots,i_n}\eee_{i_1,\ldots,i_s})\odot\eee_{i_{s+1},\ldots,i_n}.$$
For each $(j_{s+1},\ldots,j_n)$, the $s$-tensor $(a_{i_1\ldots i_sj_{s+1}\ldots j_n})$ is described as $$\sum_{k=1}^{{\mathrm{mrank}}(m_1,\ldots,m_s)} C_{j_{s+1},\ldots,j_n}^{(k)},$$ where $C_{j_{s+1},\ldots,j_n}^{(k)}$ are rank one tensors. Then $$\underline{A}
=\sum_{i_{s+1},\ldots,i_n} \sum_{k=1}^{{\mathrm{mrank}}(m_1,\ldots,m_s)} C_{i_{s+1},\ldots,i_n}^{(k)} \odot\eee_{i_{s+1},\ldots,i_n}$$ which is a sum of ${\mathrm{mrank}}_{\KKK}(m_1,\ldots,m_s)\prod_{t=s+1}^n m_t$ rank one tensors.
For $n\geq 4$, The maximal rank of $n$-tensors with size $2\times\cdots\times 2$ over the complex number field is less than or equal to $2^{n-2}$.
Theorem \[thm:Brylinski\] covers the case where $n=4$. Suppose $n>4$. The maximal rank of complex $2\times 2\times 2\times 2$ tensors is equal to $4$. By applying Proposition \[prop:highertensorrank\] with $s=4$, we have $${\mathrm{mrank}}_{\CCC}(2,2,\ldots,2)\leq {\mathrm{mrank}}_{\CCC}(2,2,2,2)\prod_{t=5}^n 2
=4\cdot 2^{n-4}=2^{n-2}.$$
\[lem:essentialrank2\] Let $n$ be a positive integer and let $A_j$ and $B_j$, $1\leq j\leq n$ be $2\times 2$ real matrices. There is a rank one real matrix $C$ such that ${\mathrm{rank}}_{\RRR}(A_j;B_j+C)\leq 2$ for any $1\leq j\leq n$.
Put $A_j=\begin{pmatrix} a_j&b_j\\ c_j& d_j\end{pmatrix}$ and $C=\begin{pmatrix} su&sv\\ tu& tv\end{pmatrix}$. Since $$\Delta(A_j;C)=(s(ud_j-vc_j)-t(ub_j-va_j))^2,$$ there exists a rank one matrix $C_0$ such that $\Delta(A_j;C_0)>0$ for any $j\in S_2$. Let $C=\gamma C_0$. Since $(A_j;B_j+C)$ is $\{E\}^2\times{\mathrm{GL}}(2,\RRR)$-equivalent to $(A_j; \gamma^{-1}B_j+C_0)$, The continuity of $\Delta$ implies that for each $j$, there is $h_j>0$ such that $\Delta(A_j;B_j+C)>0$ for any $\gamma\geq h_j$ by Proposition \[prop:Silva-etal\] (1). For $C=(\max_j h_j)C_0$, we have ${\mathrm{rank}}(A_j;B_j+C)\leq 2$ by Proposition \[prop:Silva-etal\] (2).
\[thm:higher2x...x2\] Let $k\geq 2$. The maximal rank of real $k$-tensors with size $2\times\cdots\times 2$ is less than or equal to $2^{k-2}+1$.
The assertion is true for $k=2,3$. Then suppose that $k\geq 4$. Let $e_{i_1,\ldots,i_k}$, $i_1,\ldots,i_k=1,2$ be a standard basis of $(\RRR^{2})^{\otimes k}$, that is, $\eee_{i_1,\ldots,i_k}$ has $1$ at the $(i_1,\ldots,i_k)$-element and otherwise $0$. Any tensor $\underline{A}$ of $(\RRR^{2})^{\otimes k}$ is written by $$\sum_{i_1,\ldots,i_k} a_{i_1,\ldots,i_k}\eee_{i_1,\ldots,i_k}.$$ This is described as $$\sum_{i_4,\ldots,i_k} B(i_4,\ldots,i_k)\odot \eee_{i_4,\ldots,i_k},$$ where $B(i_4,\ldots,i_k)=\sum_{i_1,i_2,i_3}
a_{i_1,\ldots,i_k}\eee_{i_1,i_2,i_3}$ is a $2\times 2\times 2$ tensor. By Lemma \[lem:essentialrank2\], there is a rank one $2\times2$ matrix $C$ such that $B(i_4,\ldots,i_k)+(O;C)$ has rank less than or equal to $2$ for any $i_4,\ldots,i_k$. We have $$\begin{split}
\underline{A}&=\sum_{i_4,\ldots,i_k} (B(i_4,\ldots,i_k)+(O;C))\odot \eee_{i_4,\ldots,i_k}
-\sum_{i_4,\ldots,i_k} (O;C)\odot \eee_{i_4,\ldots,i_k} \\
&=\sum_{i_4,\ldots,i_k} (B(i_4,\ldots,i_k)+(O;C))\odot \eee_{i_4,\ldots,i_k}
-C\odot\eee_2\odot \uuu\odot\cdots\odot\uuu,
\end{split}$$ where $\uuu=\begin{pmatrix} 1\\ 1\end{pmatrix}$ and $\eee_2=\begin{pmatrix} 0\\ 1\end{pmatrix}$, and then $${\mathrm{rank}}\underline{A}\leq \sum_{i_4,\ldots,i_k} {\mathrm{rank}}(B(i_4,\ldots,i_k)+(O;C))+1=2^{k-2}+1.$$
[1]{}
M. R. Bremner and S. G. Stavrou. Canonical forms of $2\times 2\times 2$ and $2\times 2\times 2\times
2$ arrays over $\mathbb{F}_2$ and $\mathbb{F}_3$. , published online: 28 Sep 2012.
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K. Coolsaet. On the classification of nonsingular $2\times 2\times 2\times 2$ hypercubes. , published online: 15 August 2012.
V. de Silva and L.-H. Lim. Tensor rank and the ill-posedness of the best low-rank approximation problem. , 30(3):1084–1127, 2008.
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\times 2$ tensors. , first published: 11 Dec 2012.
T. Sumi, M. Miyazaki, and T. Sakata. Rank of $3$-tensors with $2$ slices and [K]{}ronecker canonical forms. , 431(10):1858–1868, 2009.
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|
{
"pile_set_name": "ArXiv"
}
|
---
bibliography:
- 'Planck\_bib.bib'
- 'diffuse\_compsep.bib'
title: '*Planck* 2018 results. IV. Diffuse component separation'
---
Introduction {#sec:introduction}
============
This paper, one of a set associated with the 2018 release of data from the [^1] mission [@planck2014-a01], describes the cosmological and astrophysical component maps derived from the full set of observations [@planck2016-l01], and compares these to earlier versions of the corresponding products. was launched on 14 May 2009, and observed the sky nearly without interruption for four years. The raw, time-ordered observations were released to the public in their entirety in February2015 as part of the second data release (PR2), together with associated frequency and component sky maps and higher-level science data products, including cosmic microwave background (CMB) power spectra and cosmological parameters. These observations represent a cornerstone of modern cosmology, and they severely constrain the history of the early Universe.
The time-ordered data selection adopted for the current (third, PR3) release is similar to that used in the second release (@planck2016-l02; @planck2016-l03); the second and third product deliveries therefore have nearly identical scientific constraining power, as measured in terms of raw integration time and instrumental noise levels. The difference between the two releases lies in their overall levels of instrumental systematic uncertainties. A substantial fraction of the second-release papers was dedicated to identifying, quantifying, and characterizing residual uncertainties due to a wide range of instrumental effects, including effective gain variations, analogue-to-digital converter (ADC) nonlinearities, residual temporal transfer functions, and foreground bandpass leakage. Indeed, these residuals were sufficiently large to prohibit extraction of a robust polarization signal on large angular scales from the High Frequency Instrument (HFI) observations, significantly limiting the science scope of the polarization observations as a whole. Fortunately, as discussed extensively in @planck2016-l03, these residuals are now not only better understood and modelled, but also greatly reduced in the final dataset, particularly through the use of improved end-to-end processing techniques.
In this paper, we present updated full-sky CMB maps in both temperature and polarization, as well as new synchrotron and thermal dust emission maps in polarization, and compare these to previous versions [@planck2013-p06; @planck2014-a11; @planck2014-a12]. In terms of temperature foreground products, we provide an update of the Generalized Needlet Internal Linear Combination (;[@Remazeilles2011b]) thermal dust model, to be used in conjunction with the updated 2018 polarization map, but no new [@eriksen2008] foreground products. The reason for this is one of necessity: as described in @planck2016-l03, the latest HFI processing exploits the full information content of each frequency in order to suppress large-scale polarization systematics, and the processing has thus been tuned to optimize the polarization solution. The cost of this choice, however, is that individual single-bolometer maps are no longer available; see section 3.1.2 of [@planck2016-l03] for details. Specifically, some of the single-bolometer maps only contain part of the sky signal and thus cannot be used for component separation. This, in turn, has an impact on the ability of the algorithm to resolve individual foreground components in temperature. The single most important effect is on our ability to constrain CO line emission, which benefits particularly strongly from intra-frequency measurements. Because each bolometer has a different bandpass amplitude at the CO-line centre frequency of 115.27GHz (and multiples thereof), each bolometer observes the true CO signal with different effective responses, and these differences provide a strong handle on the true intensity of the CO signal. Furthermore, both thermal dust and free-free emission correlate strongly with CO emission, and are therefore also negatively affected by the lack of single-bolometer maps. In turn, free-free emission is strongly correlated with both synchrotron and anomalous microwave emission. In summary, we believe that the 2015 Commander-based temperature (i.e., Stokes $I$) foreground model represents a more accurate description of the true temperature sky than what can be extracted from the current (2018) data set. To avoid confusion, we therefore do not release the latest version publicly, although we compare the two models in Sect. \[sec:foregrounds\]. For the CMB component, we find that the latest processing produces results that are fully consistent with the previous incarnation, while for polarization the new results represent a major improvement, both in terms of CMB and foregrounds.
The methodologies adopted in this paper mirror those used in earlier releases, with only minor algorithmic updates and improvements. In particular, for CMB extraction we adopt the same four component-separation implementations used in earlier releases, namely , , , and , each of which was initially selected as a representative of a particular class of algorithms(blind versus non-blind methods and pixel-based versus harmonic-based methods). In combination, they represent most approaches proposed in the literature. In the current release, all four CMB methods adopt the same data selection, based only on full-frequency maps, in order to facilitate a direct comparison of the results. As in previous releases, we strongly suggest considering all four CMB maps in any higher-level map-based CMB analysis, in order to assess robustness with respect to algorithmic choices. We also provide again cleaned CMB maps at individual frequencies constructed by . More specifically, in this release, intensity and polarization CMB maps are produced at four different frequencies from 70 to 217 GHz. These maps are particularly useful to test, for example, the robustness of results versus the presence of foregrounds and/or systematics. In addition, one fundamentally new data product is delivered in this release, namely a CMB temperature map generated by from which Sunyaev-Zeldovich(SZ) sources have been projected out. This can be used, for instance, in lensing studies [@planck2016-l08].
For astrophysical component separation, which depends inherently on explicit parametric modelling, we adopt as our primary computational engine, mirroring the processing adopted in the two previous releases. However, since the last release the internal mechanics of this code have been significantly re-written. now allows for analysis of data sets with different angular resolutions at each frequency, and thereby allows for production of frequency maps at the full angular resolution of the data [@seljebotn2017]. In addition, we employ both and for foreground reconstruction in the new release.
The rest of the paper is organized as follows. Section \[sec:methods\] reviews the algorithms and methods used in the analysis, focusing primarily on updates and improvements made since the 2015 release. Section \[sec:inputs\] describes the data selection and pre-processing steps applied to the data before analysis. Section \[sec:cmb\] presents the 2018 CMB maps in both temperature and polarization, and characterizes their properties in terms of residuals with respect to earlier versions, along with angular power spectra, cosmological parameters, and simple higher-order statistics. Section \[sec:foregrounds\] discusses the updated polarization foreground products. Section \[sec:conclusions\] gives conclusions. The various algorithms are reviewed in Appendices \[app:commander\]–\[app:gnilc\]. A brief summary of temperature foregrounds derived from the 2018 frequency maps is provided in Appendix \[app:foregrounds\] and, finally, additional CMB figures are provided in Appendices \[app:splits\] and \[app:npt\_functions\].
Component-separation methods {#sec:methods}
============================
Earlier publications give detailed descriptions of the four main component-separation methods used in this paper [@planck2013-p06; @planck2014-a11; @planck2014-a12]. For some methods, notable improvements have been implemented since the last release, and these are described below. Further technical details may be found in the Appendices.
We also employ the algorithm for thermal dust extraction. This method and corresponding results are described in detail in @Remazeilles2011b, @planck2016-XLVIII, and @planck2016-l11B. A detailed comparison of the foreground products derived with and is presented in the current paper.
{#sec:commander}
[@eriksen2004; @eriksen2008; @planck2014-a12] has undergone the most significant changes since the previous release. is a Bayesian approach employing a Monte Carlo method called Gibbs sampling as its central computational engine. Within this Bayesian framework, a parametric model is fitted to the the data set in question with standard posterior sampling or maximization techniques, including both cosmological, astrophysical, and instrumental parameters.
We start by writing down a generic model on the form, $${\vec{d}}_{\nu}(p) = g_{\nu} \sum_{c=1}^{N_{\rm c}} {\tens{F}}_{\nu}(\beta_c) {\tens{T}}(p) {\vec{a}}_{c} + {\vec{n}}_{\nu}(p).$$ Here ${\vec{d}}_{\nu}(p)$ denotes the observed data at frequency $\nu$ and pixel $p$. The sum runs over $N_{\rm c}$ components, each with an amplitude vector ${\vec{a}}_c$, a map projection operator ${\tens{T}}(p)$, and frequency scaling operator ${\tens{F}}_{\nu}(\beta_c)$ that depends on astrophysical spectral parameters $\beta_c$. The quantity $g(\nu)$ denotes an overall instrumental calibration factor per frequency channel, and $n_{\nu}(p)$ indicates instrumental noise. With this notation, the component sum runs over both astrophysical components (CMB, synchrotron, CO, thermal dust emission etc.) and possible spurious monopole and dipole terms. The projection operator ${\tens{T}}$ indicates any step required in going from a general amplitude vector (such as a pixelized sky map, a set of spherical harmonic coefficients, or a template amplitude) to a map as observed by the current detector. Thus, this matrix encodes both the choice of basis vectors (pixels, spherical harmonics, templates) and higher-level operations such as beam convolution. Given this data model, samples are drawn from the full posterior as described in @eriksen2004 [@eriksen2008] and @seljebotn2017.
In previous releases the above model was fitted to the combination of and external data using the implementation described by @eriksen2008. This implementation adopted map-space pixels as its basis set for astrophysical foregrounds, for coding efficiency reasons. Although computationally fast, this approach has a significant limitation in that it requires all data sets under consideration to have the same angular resolution. Specifically, this implies that the angular resolution of the final output maps are limited to that of the lowest resolution frequency channel under consideration, which typically is $1^{\circ}$ FWHM for the combination of , , and Haslam 408MHz, which formed the basis of the previous astrophysically oriented foreground analysis. Higher-resolution products could then only be derived by dropping lower-resolution channels, which in turn carried a significant cost in terms of model fidelity.
In the current release, we implement the algorithm described by @seljebotn2017, which we refer to as . This approach, which models the foreground amplitude maps in terms of spherical harmonics instead of pixels, offers three important improvements over the pixel-based approach.
First, since amplitudes are modelled in harmonic space, it is computationally trivial to convolve with a separate instrumental beam transfer function at separate frequencies, so that for the first time we can solve for full-resolution signal models with multi-resolution data sets. is thus able to produce a foreground model at native resolution, limited only by the effective signal-to-noise ratio of each component. The computational cost is greater; however, as shown by @seljebotn2017, this is manageable with modern computers, even for -sized data sets.
Second, the new approach offers the option of imposing a prior on the foreground signal amplitudes in the form of an angular power spectrum. This can be used to regularize the foreground solution at small angular scales, and thereby reduce degeneracies between different components at high multipoles.
Third, the improvements allow for joint fitting of compact or unresolved objects and diffuse components. This improves the reconstruction of the diffuse components themselves, including the CMB, and also allows production of a new catalogue of compact objects. The details of this procedure are described in Appendix \[app:commander\].
Overall, from an algorithmic point of view the implementation used in the current data release is more powerful than in previous releases. At the same time, there is also one important aspect of the 2018 data release that limits our ability to perform a component separation as detailed as that in the 2015 analysis. As mentioned in Sect. \[sec:introduction\], the 2018 data set includes only full-frequency maps, not single-bolometer maps. For the temperature analysis, this implies that a simpler foreground model must be employed than in the corresponding 2015 analysis. In the previous analysis we considered seven different physical components, namely CMB, synchrotron, free-free, spinning and thermal dust emission, a general line emission component at 95 and 100GHz, and CO with individual components at 100, 217, and 353GHz. Single-detector maps played a central part in constraining this rich model, in particular with respect to CO line emission. With the new and more limited data set, we instead adopt a similar model as employed in the 2013 analysis, which includes only four diffuse signal components in temperature, namely CMB, a single general low-frequency power-law component, thermal dust, and a single CO component with spatially constant line ratios between 100, 217, and 353GHz. For polarization the model remains the same as in 2015, and includes only CMB, synchrotron, and thermal dust emission. The latter two components are as usual modelled in terms of simple power-law and modified blackbody SEDs, respectively.
The above general specification provides a basic summary of the framework used for parametric fitting. However, there are still some free choices that must be made, the two most important of which are: (1) the angular resolution of the foreground spectral indices; and (2) the spatial priors imposed on the foreground amplitudes. For the spectral indices, we are primarily driven by signal-to-noise considerations, as adopting too high resolution for such parameters leads to an undesirable increase in noise in all components. In the temperature case, we adopt a smoothing scale of $40\arcm$ FWHM for low-frequency foregrounds, slightly larger than the 30-GHz instrumental beam. For the dust spectral index, we adopt $10\arcm$ FWHM, which is slightly larger than the 100-GHz beam. The dust temperature is fitted at the full resolution of $5\arcm$ FWHM, since this parameter is well supported by all frequencies between 217 and 857GHz. For polarization, we fit only a spatially-constant spectral index for synchrotron,[^2] while for thermal dust emission, we fit the dust spectral index at $3\deg$ FWHM. The dust temperature for the polarization model is fixed at the values derived in the intensity analysis, as the 545- and 857-GHz frequency channels are unpolarized, and the observations therefore do not constrain the thermal dust temperature in polarization.
Finally, for spatial priors, we adopt minimally informative power-spectrum priors, defined simply as flat spectra in units of $C_\ell \ell(\ell+1)/2\pi$ for all components, with an amplitude that is larger than that observed in the high signal-to-noise regime. In addition, this flat spectrum is smoothly apodized at high multipoles in order to suppress ringing around bright compact objects. For the low-frequency temperature foreground and the CO line-emission components, the apodization is performed with a Gaussian beam with a FWHM roughly matching the dominant frequency for the respective component, while for thermal dust only a mild apodization is applied in the form of an exponentially-falling cut-off between $\ell=5000$ and 6000. For polarization, we apodize with Gaussian smoothing kernels, as in the low-frequency foreground and CO case.[^3] Full details regarding these choices are summarized in Appendix \[app:commander\].
{#sec:nilc}
(Needlet Internal Linear Combination) is described by @2012MNRAS.419.1163B [@2013MNRAS.435...18B]. The overall goal of is to extract the CMB component from multi-frequency observations while minimizing the contamination from Galactic and extragalactic foregrounds and instrumental noise. This is done by computing the linear combination of input maps that minimizes the variance in a basis spanned by a particular class of spherical wavelets called needlets [@narcowich06localizedtight]. Needlets allow localized filtering in both pixel space and harmonic space. Localization in pixel space allows the weights of the linear combination to adapt to local conditions of foreground contamination and noise, whereas localization in harmonic space allows the method to favour foreground rejection on large scales and noise rejection on small scales. Needlets permit the weights to vary smoothly on large scales and rapidly on small scales, which is not possible by cutting the sky into zones prior to processing . The pipeline is applicable to scalar fields on the sphere, hence we work separately on maps of temperature and the $E$ and $B$ modes of polarization. The decomposition of input polarization maps into $E$ and $B$ is done on the full sky. At the end, the CMB $Q$ and $U$ maps are reconstructed from the $E$ and $B$ maps. Further details of the method are provided in Appendix \[app:nilc\].
The pipeline employed in the 2018 analysis is essentially unchanged from that employed in the 2015 analysis; we therefore refer to @planck2014-a11 and references therein for full details.
{#sec:sevem}
[@leach2008; @fernandez-cobos2012] is an implementation of an internal template-cleaning approach in real space. It has been used in the previous releases to produce clean CMB maps in both intensity and polarization, and has been demonstrated to provide robust results. A detailed description of the pipeline can be found in Appendix \[app:sevem\].
The starting point for is a set of internal templates typically constructed as difference maps between two neighboring channels convolved to the same resolution, ensuring that the CMB signal vanishes. These templates trace the foreground contaminants at the corresponding frequency ranges. Next, a linear combination of such templates is then subtracted from some set of CMB-dominated frequency maps, typically 70 to 217GHz for . The coefficients of the linear fit are derived by minimizing the variance of the clean map outside a given mask. A final, co-added CMB map is obtained by combining individually-cleaned frequency maps in harmonic space.
is also able to produce cleaned CMB maps at specific channels. Individually-cleaned frequency CMB maps are useful to test the robustness of results versus the presence of foregrounds and/or systematics, for instance for isotropy and statistics estimators [@planck2014-a16] or the integrated Sachs-Wolfe effect stacking analysis [@planck2014-a26]. They are also valuable to construct cross-frequency estimators, which allow one to minimize the impact of certain types of systematic effects (e.g., possible correlated noise in data splits). In addition, they can be used to search for frequency-dependent effects in the CMB itself, such as those arising from relativistic boosting [@planck2013-pipaberration] or the Sunyaev-Zeldovich effect [@sunyaev:1970], although for this type of analysis the contribution of the templates (which would contain a certain level of any effect that is not constant with frequency) to the cleaned maps should be taken into account.
Since the 2015 release, we have introduced two significant improvements to the pipeline for polarization. First, in the previous release we produced cleaned maps at three frequencies, 70, 100, and 143GHz, and the final map was produced by combining the cleaned 100 and 143-GHz maps. However, given the improvements in the new polarization data, we are now also able to robustly clean the 217-GHz channel map, and this is now included in the final combination. As a result, the signal-to-noise ratio of the cleaned CMB polarization map is significantly improved with respect to the previous version. Second, in the updated pipeline, we now produce polarization maps at full resolution ($N_{\rm side}=2048$), whereas in the last release all polarization maps were constructed at N$_{\rm side}$=1024. However, recognizing the fact that the 217-GHz channel is likely to be somewhat more susceptible to large-scale systematic residuals and calibration uncertainties due its higher foreground levels than the two lower frequencies [@planck2016-l03], we introduce at the same time a relative down-weighting of the 217-GHz channel on the largest scales. In summary, these modifications yield significantly improved polarization maps, both in terms of the combined CMB map and individually cleaned frequency maps. Regarding intensity, the pipeline is essentially identical to that used in the previous release; however, we now also provide a cleaned 70-GHz map in intensity. In addition to the final CMB map, therefore now provides the complete set of $\{T,Q,U\}$ CMB maps for each of the four frequency channels between 70 and 217GHz.
{#sec:smica}
(Spectral Matching Independent Component Analysis) is described in @cardoso2008, and details regarding the actual implementation used in the following analysis (pre-processing, masking and mask correction, beam correction, binning, possible re-calibration, etc.) are provided in Appendix \[app:smica\].
synthesizes CMB $\{T, E, B\}$ maps from spherical harmonic coefficients $\hat s_\lm$ obtained by combining the coefficients of $\ncha$ frequency maps with an $\ell$-dependent $\ncha\times 1$ vector of weights $\vec w_\ell$, $$\label{eq:smica:ilc}
\hat s_\lm = \vec{w}_\ell \adj \vec{x}_\lm
\quad\text{where }\quad
\vec{w}_\ell = \frac{\bC_\ell\inv \ba}{\ba\adj \bC_\ell\inv \ba}.$$ Here the $\ncha\times 1 $ vector $\vec a$ describes the emission law of the CMB, and the $\ncha\times \ncha$ spectral covariance matrix $\bC_\ell$ contains (estimates of) all auto- and cross-spectra of the $\ncha$ input maps. On small angular scales, where a large number of harmonic coefficients are available, $\bC_\ell$ may be accurately estimated as $$\label{eq:smica:scm}
\widehat\bC_\ell = \frac1{2\ell+1}\sum_m \vec x_\lm^{\vphantom{\dagger}}\vec x_\lm\adj ,$$ which is used “as is,” in Eq. (\[eq:smica:ilc\]). On large angular scales, we resort to a parametric model $\bC_\ell(\theta)$ of the spectral covariance matrices in order to reduce the estimation variance and mitigate the effects of chance correlation between the CMB field and the foregrounds. The model is adjusted to the data by selecting best-fit parameters $\theta$ obtained as $$\label{eq:smica:mmcrit}
\hat\theta
=
\arg\min_\theta \sum_\ell (2\ell+1)
\left[
\mathrm{Tr}\left(\widehat\bC_\ell \bC_\ell (\theta)\inv \right)
+
\log\det \bC_\ell (\theta)
\right]
.$$ The minimization in Eq. (\[eq:smica:mmcrit\]) is equivalent to maximizing the joint likelihood of the $\ncha$ input maps assuming that they follow a Gaussian isotropic distribution characterized by the spectra and cross-spectra collected in the spectral covariance matrices $\bC_\ell(\theta)$. For a motivation of this likelihood, see @LVAICA2017.
The spectral model fitted by , ${\tens{C}}_\ell(\theta)$, is agnostic, as it assumes only that the foreground emission can be described by an unconstrained $\nfg$-dimensional component with a covariance matrix on the form $$\label{eq:smica:model}
\bC_\ell(\theta)
=
\left[\begin{array}{cc} \vec a & \tens F \end{array} \right]
\left[\begin{array}{cc} C_\ell^\mathrm{cmb} & 0 \\ 0 & \tens P_\ell \end{array} \right]
\left[\begin{array}{cc} \vec a & \tens F \end{array} \right]\adj
+
\tens N_\ell.$$ Here the $\ncha\times\nfg$ matrix $\tens F$ represents the foreground emissivities, which are $\ell$-independent, and the $\nfg\times\nfg$ matrix $\tens P_\ell$ contains the foreground auto- and cross-spectra. The diagonal matrix $\tens N_\ell$ represents the noise contribution, and $\theta$ contains whatever parameters are needed to determine the quantities $C_\ell^\mathrm{cmb}$, $\vec a$, $\tens F$, $\tens P_\ell$, and $\mathrm{diag}(\tens N_\ell)$. In most cases, a fit is conducted with $\vec a$ fixed to assumed known values (i.e., assuming perfect calibration) and leaving all other parameters free. $\tens P_\ell$ is only constrained to be positive. In other words, foreground spectra (emissivities and angular spectral behaviour) and their correlations are freely fitted by . In this release, however, we also consider two variations that include constraints on foreground emissions. The first of these is used to produce an SZ-free CMB map in intensity (see Appendix \[app:smica\]), and the second results in thermal dust and synchrotron maps in polarization (see Sect. \[sec:foregrounds\]). No attempt is made to reconstruct temperature foregrounds, since the combination of synchrotron, free-free, spinning and thermal dust, and CO emission is intrinsically much more tightly coupled and difficult to disentangle than synchrotron and thermal dust emission in polarization.
Since the last release, changes have been introduced for both intensity and polarization maps. Starting with the temperature case, the most important change in this release is the introduction of hybrid CMB rendering, merging two different CMB maps produced independently by the pipeline. The first CMB map, $X_\text{high}$, is designed to describe the cleanest region of the sky and intermediate-to-small angular scales. It is obtained from all six HFI channels using a foreground dimension of $\nfg=4$. The second CMB map, $X_\text{full}$, is designed to describe the full sky and all harmonic scales. It includes all nine frequency channels using a maximal foreground dimension of $\nfg=8$. The final hybrid CMB map $X$ is then computed by merging $X_\text{high}$ and $X_\text{full}$ according to $$\label{eq:smica:merge}
X
= \mathcal{P} X_\text{high} + (\mathcal{I}-\mathcal{P})\, X_\text{full}
\ =\
X_\text{full} + \mathcal{P} (X_\text{high}-X_\text{full}),$$ where $\mathcal{P}$ is a linear operator that smoothly removes large harmonic scales, and masks out an area close to the Galactic plane. Hence, in the resulting hybridized map, the multipoles of highest degree and the areas of highest Galactic latitude are provided by $X_\text{high}$, while the remaining information is provided by $X_\text{full}$. In practice, the hybridization operator $\mathcal{P}$ is implemented by high-pass filtering in the harmonic domain (with a transfer function that smoothly transitions from $0$ to $1$ according to an arc-cosine function over the multipole range $50\leq \ell\leq150$), followed by multiplication by an apodized Galactic mask that is similar to the mask used at 100GHz in the 2018 likelihood () [see @planck2016-l05 for details].
Hybridization of two CMB renderings has several benefits compared to using a single set of harmonic weights over all areas of the sky. First, the data suggest it: the weights are quite different if they are based on spectral statistics computed over the full sky rather than over a region with much lower foreground contamination. This is the rationale behind , which extends the idea to many more than the two sky regions considered regions considered by . Second, the reason for leaving out the LFI channels in producing $X_\text{high}$ except at large angular scales is that would put very small weights on those channels (this is not the case when the weights are based on statistics computed for $X_\text{high}$, as seen on Fig. \[fig:smica\_T\_weights\] which shows a significant contribution from the 70-GHz channel). We could still include those channels and let automatically down-weight them, but by excluding the channels with the lowest resolution, we avoid large, ‘low-resolution’ holes in the common point source mask, and therefore in the final CMB map. Finally, hybridization matches well the high-$\ell$ TT likelihood function in , uses a clean fraction of the sky, does not include LFI channels, and only involves high frequency foregrounds.
adopts its own relative calibration between frequency channels. In 2015, this process was applied to frequency channels from 44 to 353GHz; however, since then we have found that the uncertainty in the 44-GHz channel was larger than expected, and that the previously reported value was inaccurate (see Fig. \[fig:smica\_diff2015\]). In the new release, we adopt a more conservative approach, and limit re-calibration to 70, 100, and 217GHz, taking the 143-GHz channel as a reference; see Appendix \[app:smica:temp\] for further details.
For polarization, we have introduced two changes since the previous release. First, the CMB polarization maps are now generated by independently processing $E$ and $B$ modes, while in 2015 they were jointly fitted and filtered. Second, we run two independent fits, one targeted at CMB extraction, the other at foreground separation.
For CMB extraction, we conduct a fit using a maximal foreground dimension of $\nfg=7-1=6$, which makes $[\vec a \ \tens F ]$ a square matrix. This is the largest dimension supported blindly (i.e., without any constraint on the foreground contribution) by , given the number of available polarized channels.
For foreground separation, we conduct a separate fit using a foreground model of dimension $\nfg=2$, implicitly targeting synchrotron and dust emissions. The degeneracy of the foreground model (Eq. \[eq:smica:model\]) can then be fixed by requesting that synchrotron (thermal dust) emission should be negligible at 353GHz (30GHz); Appendix \[app:smica\] describes the implementation details. This analysis yields, without any other prior information, the angular spectra and emissivities of both foreground components and the corresponding synchrotron and dust maps. The results are summarized in Sect. \[sec:foregrounds\]. Note that in 2015, a foreground model at $\nfg=2$ dimensions for capturing synchrotron and thermal dust emissions was already explored, but no maps were released (although a dust comparison appeared in [@planck2014-a12]) because additional “foreground dimensions” were clearly needed to accommodate the systematic errors. In 2018, we use the same dimensions as in 2015 (a fit with maximal dimension for CMB cleaning and a fit with $\nfg=2$ for dust and synchrotron maps); however, contrary to 2015, the $\nfg=2$ fit yields a clean CMB reconstruction, almost as clean as when using the maximal foreground dimension. For that reason, this release includes -derived synchrotron and dust polarized maps.
{#sec:gnilc}
The above four methods were the standard CMB extraction algorithms in each of the three data releases. In this release, we also consider the Generalized Needlet Internal Linear Combination (; @Remazeilles2011b) method as a foreground extraction algorithm. is not designed to extract CMB information from the data.[^4] is a wavelet-based component-separation method that generalizes the method by exploiting not only the *spectral* information (SED) but also the *spatial* information (angular power spectra) from non-astrophysical components (cosmic infrared background, CIB, CMB, and instrumental noise) to extract clean estimates of the correlated emission from Galactic foregrounds, with reduced contamination from CIB, CMB, and noise. This additional spatial discriminator adopted by enables in particular disentanglement of emission components that suffer from spectral degeneracies, such as modified blackbody emissions like the CIB and Galactic dust. has been successfully applied to 2015 intensity data to disentangle Galactic thermal dust emission and CIB anisotropies over the entire sky [@planck2016-XLVIII]. In this paper, CMB and instrumental noise were also filtered out from the dust intensity map by using the same strategy as for CIB removal.
In this work, we apply to the 2018 polarization data in order to extract the Stokes parameters $Q$ and $U$ of Galactic thermal dust polarization, while removing the contamination from CMB polarization and instrumental noise over the entire sky. $I$, $Q$, and $U$ dust maps have been produced in a self-consistent way by processing the seven polarized channels (30 to 353GHz). The reason for discarding the 545- and 857-GHz channels is as follows. The main characteristic of the method is to estimate the local number of independent foreground degrees of freedom over the sky and over angular scales. The estimated dimension of the foreground subspace depends on the local signal-to-noise ratio in the $9\times 9$ (intensity) or $7\times 7$ (polarization) observation space of the frequency-by-frequency data covariance matrix. In some parts of sky where the data are found by to be fully compatible with CIB, CMB, and noise at small angular scales, the dimension of the Galactic foreground subspace can go down to zero. The result of this is that the dust products have a variable resolution over the sky, with the local FWHM fully determined and publicly released [@planck2016-XLVIII]. However, because of decorrelation effects, the local dimension of the foreground subspace found by will be larger in a 9-dimensional space of observations (30–857GHz) than in a 7-dimensional space of observations (30–353GHz), so that the effective local resolution of the dust products will be different over the sky for intensity and polarization. For the purpose of polarization fraction studies in the 2018 release [@planck2016-l11B], we prefer to have the same local resolution over the sky both for intensity and polarization, hence our choice of processing with the same data set for $I$, $Q$, and $U$, namely the seven polarized channels (30–353GHz).
Omission of the 545- and 857-GHz channels limits the ability of to clean CIB anisotropies in the 2018 dust intensity map compared to the 2015 dust intensity map [@planck2016-XLVIII], for which the full set of unpolarized channels (30–857GHz) and the IRAS map were used in the component-separation pipeline. For analyses of dust intensity (e.g., dust optical depth, emissivity, and temperature), we recommend use of the 2015 dust intensity map, which has reduced CIB contamination. Conversely, for analysis of dust polarization (e.g., polarization fraction) we recommend use of 2018 $I$, $Q$, and $U$ maps.
Data selection, preprocessing, splits, and simulations {#sec:inputs}
======================================================
Frequency maps
--------------
The low-level data processing and mapmaking algorithms adopted for the current release are described in detail in @planck2016-l02 and @planck2016-l03. For the LFI maps at 30, 44, and 70GHz, there are only minor changes compared to the previous release, the most important of which is a better calibration procedure that explicitly accounts for polarized foregrounds in the calibration sources. For HFI, more significant changes have been implemented, all designed to suppress instrumental systematics at various scales. These include better ADC and transfer-function corrections, and explicit bandpass corrections employing a detailed foreground model.
A particularly important problem for both LFI and HFI with respect to polarization reconstruction is bandpass mismatch between multiple detectors within a single frequency channel. The issue may be summarized as follows. In order to solve for both temperature and linear polarization in each pixel on the sky, a total of three parameters per pixel, it is necessary to include information from at least three polarization-sensitive detectors in any given mapmaking operation. The polarization signal is estimated by taking pairwise differences between the signals observed by these detectors, while accounting for the relative orientation of their polarization detector angles at any given time. However, there are other effects in addition to true sky polarization signals that may induce effective signal differences between detectors. The largest of these is different effective bandpasses. Since each detector has a slightly different frequency response function, each detector observes a slightly different foreground signal. Unless explicitly accounted for during mapmaking, these differences create a spurious polarization signal in the maps.
In the LFI mapmaking procedure, this effect is accounted for in two different ways, as described in @planck2016-l02. First, for gain estimation, an iterative scheme is established, in which a proper foreground model is derived jointly with the sky maps using . Each iteration of this procedure consists of three individual steps. First, a gain model is established for each radiometer, accounting for the orbital and Solar dipoles as well as astrophysical foregrounds as estimated by . Second, frequency maps are derived based on this gain model using [MADAM]{} [@keihanen2005; @planck2014-a07], a well-established destriper. Third, these frequency maps are used by to derive a new foreground model. A total of four such iterations are used to derive the final LFI maps; however, even after these iterations there may be non-negligible large-scale residuals present in the 70-GHz sky map, as described by @planck2016-l02. To account for this, a gain correction template, based on differences between consecutive iterations, is subtracted from the final LFI 70-GHz map, with an amplitude derived from a low-resolution likelihood fit [@planck2016-l05]. This procedure accounts for biases in the time-variable gain solutions, but it does not remove direct temperature-to-polarization leakage from bandpass mismatch. That effect, which is stationary on the sky, is corrected through use of a static template, as described in detail in @planck2014-a03. The same procedure is applied to the LFI sky maps in the current release with an updated foreground model [@planck2016-l02].
For HFI a different but related approach is adopted. The 2015 temperature model is used to explicitly adjust the effective bandpass response of all bolometers within a frequency channel, by subtracting a small fraction of each foreground signal (thermal dust, free-free, and CO emission, but not synchrotron or spinning dust emission) from the individual bolometer timestreams. These “foreground-equalized” timestreams are then combined into a single frequency map by standard destriping. Since only a spin-0 temperature signal is subtracted in this procedure, the resulting polarization maps are unbiased with respect to foreground leakage, to the extent that the foreground model is accurate. However, the resulting temperature maps will be very slightly biased, in the sense that the predicted bandpass response of a given map does not perfectly match the observed signal, and this causes complications for any method that explicitly employs such information. In the current paper, this applies to and . The three remaining methods (, and ) do not explicitly use such information.
An additional complication arises from the updated 2018 HFI mapmaking procedure, due to the fact that the single-bolometer maps produced by the latest processing are not reliable for component-separation purposes [@planck2016-l03]. Since the CO emission lines are very narrow, their measured amplitudes are very sensitive to small variations in bandpass shape between individual detectors. In 2015, this sensitivity was exploited to extract line-emission maps at each of the affected frequencies. However, since single-bolometer maps are not available in 2018, this is no longer possible. The new processing represents a conscious choice of optimizing the polarization extraction at non-negligible expense in terms of our ability to perform high-fidelity astrophysical foreground reconstruction with temperature maps. For individual foreground components in temperature, we therefore recommend continued usage of the 2015 data products.
To summarize the overall data selection, all diffuse component separation codes employ all nine frequency maps between 30 and 857GHz in temperature, and all seven frequency maps between 30 and 353GHz in polarization, for the 2018 analysis. For the LFI polarization maps, we apply a set of template corrections that account for bandpass mismatch and gain corrections, as described in @planck2016-l02, while no additional corrections are applied to the HFI maps. All maps are defined by the [^5] pixelization [@gorski2005].
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Instrument characterization
---------------------------
In addition to the raw frequency maps, each method requires various degrees of knowledge about the instrument itself. The most important characterization is the beam response of the individual frequency channels. These have been updated to reflect the latest changes in the data processing pipelines, and are described in @planck2016-l02 and @planck2016-l03. We note that in the 2015 data release, CMB polarization maps for two of the methods ( and ) were given at 10 FWHM, compared to 5 FWHM for the temperature maps; however, in this release all CMB maps in both polarization and temperature are provided at the maximum angular resolution of 5 FWHM.
Each CMB map must also be associated with a statistical characterization of the instrumental noise. For this purpose, we compute and analyse null maps derived from subsets of the full data set, as done in earlier releases. In the previous release, we focused on half-mission splits, yearly splits, and half-ring splits [@planck2014-a11]. In the current release, we drop the yearly split, since this behaves similarly to the half-mission split, and we replace the half-ring split with a so-called “odd-even” split, in which scanning rings from HFI are grouped according to odd or even pointing IDs. The odd-even split nullifies long-time-correlated signals, similarly to the half-ring split, but suffers less from inter-ring correlations. For LFI, we still adopt the same half-ring split as in 2015, but nevertheless refer to this split as “odd-even,” recognizing the different signal-to-noise ratios of the LFI and HFI maps. We consider this to be our best instrumental noise tracer among the splits, whereas the half-mission split represents the best instrumental systematics tracer. Simulations including either pure CMB signal or the sum of instrumental noise and residual systematics are individually propagated through each analysis pipeline, and these simulations form the basis of all subsequent goodness-of-fit tests.
As described in @planck2016-l03, the HFI polarization frequency maps are associated with a significant uncertainty regarding polarization efficiencies, corresponding in effect to an uncertainty in the overall calibration of the Stokes $Q$ and $U$ maps. Ideally, such polarization efficiencies would be perfectly accounted for during mapmaking. However, as reported by @planck2016-l05, a cosmological analysis of power spectra of the individual frequency maps suggests that small but notable residual calibration uncertainties may remain in a few channels. The reported best-fit correction values are $+0.7 \pm 1.0$% (100GHz), $-1.7 \pm 1.0$% (143GHz), and $+1.9 \pm 1.0$% (217GHz). For 353GHz, the foreground contribution is too large to allow a robust CMB-based measurement. These corrections are only marginally statistically significant, therefore we do not apply them by default in this paper. Instead, we compute results with and without the corrections, and report the difference between the two solutions as a known systematic error. For the CMB, we find that the differences due to polarization efficiency uncertainties are small, while for polarized foregrounds, we find that the inclusion of polarization efficiencies changes the spectral index of thermal dust by $\Delta\beta_{\mathrm{d}}=-0.03$. See Sect. \[sec:foregrounds\] for details.
Treatment of unobserved pixels {#sec:misspix}
------------------------------
As described in @planck2016-l03, the HFI split maps contain a non-negligible number of unobserved pixels at the full $N_{\textrm{side}}=2048$ resolution. These are pixels that were either never seen by any bolometer at a given frequency, or for which the polarization angle coverage is too poor to support a reliable decomposition into the three Stokes parameters. For most methods considered in this paper,[^6] such unobserved pixels represent a notable algorithmic problem, and must be treated before analysis. For these methods, we simply replace all unobserved pixels in a given frequency map by the same pixels in a corresponding map downgraded to a resolution of $\nside=64$, corresponding to a pixel size of 55. Of course, this procedure introduces correlations between neighbouring unobserved pixels, and we therefore mask all high-resolution pixels after the analysis; separate masks for each data split are provided to account for this effect. The details of how the unobserved pixel mask has been generated are described in Sect. \[sec:masks\]. Finally, to account for possible leakage from unobserved to observed pixels during inter-analysis smoothing operations, we apply the same procedure to the reference simulations described below.
Comparison between 2015 and 2018 frequency maps
-----------------------------------------------
It is useful to compare the new 2018 frequency maps to the previous 2015 frequency maps. Structures seen in these difference maps should be expected to propagate into the corresponding CMB differences at some level. Starting with the temperature case, the left columns of Figs. \[fig:dx11\_vs\_dx12\_lfi\] and \[fig:dx11\_vs\_dx12\_hfi\] show the differences between each 2018 frequency map and the 2018 CMB solution.[^7] Overall, the behaviour is consistent with what has been found in earlier releases, with: an absolute foreground minimum around 70GHz; LFI monopoles of 10–20; increasing HFI monopoles with frequency, corresponding to the expected offset due to the cosmic infrared background (CIB), which is manually introduced into the HFI frequency maps [@planck2016-l03]; and overall morphologies consistent with some combination of synchrotron, free-free, CO, and dust emission.
More interesting are the second and third columns in each figure, which show the raw the fractional differences between the 2018 and 2015 frequency maps, respectively. In the latter we have removed the best-fit offset and dipole outside a Galactic mask, defining the fractional difference, $f$, as $${\vec{f}}= \frac{{\vec{m}}^{2018} - {\vec{m}}^{2015} - \Delta M - \Delta D }{{\vec{m}}^{2015}-{\vec{m}}^{\mathrm{CMB}}},$$ where ${\vec{m}}^{2018}$ is the new 2018 frequency map, ${\vec{m}}^{2015}$ is the 2015 map, $\Delta M$ and $\Delta D$ are the monopole and dipole differences between these maps, and ${\vec{m}}^{\mathrm{CMB}}$ is the 2015 CMB temperature map.
Starting with the LFI 30-GHz difference maps, two effects stand out. At high latitudes, we see broad stripes following the scanning pattern. These are due to an improved time-varying gain calibration procedure in the 2018 analysis that takes into account astrophysical foregrounds as computed by , in an iterative gain-estimation$\rightarrow$mapmaking$\rightarrow$component-separation procedure. This new iterative scheme is one of the main new features of the LFI 2018 processing pipeline [@planck2016-l02]. A second effect is seen in the Galactic plane, where the 2018 amplitude is lower by about 0.2% compared to 2015. This is due to re-estimation of the overall absolute calibration, due to a new estimate of the Solar CMB dipole [@planck2016-l01].
Similar considerations hold for the 44-GHz channel, although with a significantly lower striping level. In fact, in this case the striping is sufficiently low to reveal a small residual dipole of about 1 in the raw difference map, directly showing the effect of the new Solar dipole estimate. Even smaller differences are seen in the 70-GHz channel, but in this case the iterative foreground estimation process was not used, because the foreground level of this channel near the foreground minimum is too low to allow robust foreground estimation [@planck2016-l02].
The HFI frequencies (Fig. \[fig:dx11\_vs\_dx12\_hfi\]) show many qualitatively similar structures, in addition to a few unique HFI-type features. First, in the 100-GHz channel we see a fairly large dipole of 2–3. In the new HFI processing, thermal dust emission is explicitly included in the dipole estimation model, resulting in improved consistency in the dipole estimates among the various frequency channels. As a result of this process, the best-fit 2018 dipole estimate changed by 2.4 relative to 2015, and this is visually apparent in the 100-GHz raw difference map. In addition, we see significant striping in the fractional difference map, with an amplitude of more than 3% of the foreground level at high latitudes. As is the case for LFI, these stripes are due to improved time-variable gain estimation, which in turn is responsible for the overall improvement in the large-scale polarization reconstruction. Of course, for this channel the absolute foreground levels are low at high Galactic latitudes, and a 3% relative difference corresponds only to 1–2 in absolute value. For temperature this is small, while for polarization it is highly relevant, as we discuss below.
Qualitatively speaking, similar considerations hold for the 143 and 217-GHz channels as well. However, in these cases we see an additional effect, namely a significantly blue Galactic plane in the fractional difference map, indicating relative absolute differences of about 1% in the high signal-to-noise regime. At first sight, this may appear puzzling, since the absolute CMB calibration between the 2018 and 2015 has changed by less than 0.1% [@planck2016-l05]. The explanation is the new HFI treatment of bandpass differences among individual bolometers. As described in Sect. \[fig:dx11\_vs\_dx12\_hfi\], each frequency map is now generated as the sum over all bolometer timestreams within that frequency channel, each of which has been *bandpass equalized* prior to co-addition. This equalization is implemented by fitting foreground templates of thermal dust, CO, and free-free emission jointly with other instrumental parameters, with the goal of minimizing inter-bolometer bandpass differences that otherwise generate spurious polarization contamination.
For component-separation purposes, this implies that the overall bandpass profile of each HFI frequency channel has changed. Furthermore, this process also leads to a complicated bandpass definition overall, in which the bandpass in principle is component dependent. While thermal dust, free-free, and CO emission are associated with bandpasses given as straight averages of the individual bolometer bandpasses (due to their inclusion in the bandpass equalization procedure), synchrotron, spinning dust, and thermal Sunyaev-Zeldovich signals are associated with inverse noise-variance-weighted bandpasses as in earlier releases. In practice, though, we adopt the straight averaged bandpasses for all HFI channels in the current release, since the affected non-equalized components are sub-dominant at HFI frequencies, and implementing multi-bandpass integration would require significant algorithm re-structuring. However, this is also one of the reasons why we do not release new individual synchrotron and spinning dust products in temperature in the current release.
Turning to the 353-GHz frequency channel, two additional effects are seen. First, at high latitudes one can see a weak imprint of zodiacal light emission [@planck2013-pip88] in the fractional difference map, taking the form of a blue band along the Ecliptic plane with an amplitude of 1%. Second, we also see two deep blue bands on either side of the Galactic plane with amplitudes of 2%; these are due to changes in the 353-GHz transfer function. From such difference maps alone, it is of course impossible to conclude whether the additional residuals are due to defects in the 2015 or 2018 maps. On the other hand, such structures tend to stand out quite prominently in maps of foreground spectral indices, which in essence measure small differentials between frequencies. Thus, through subsequent -type astrophysical analyses, we find that these two 353-GHz effects are indeed present in the 2018 maps, and not in the corresponding 2015 maps. These residual effects, along with the lack of single-bolometer maps, are thus part of the cost of producing as clean polarization maps as possible, which is the primary goal of the current data release.
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At 545 and 857GHz, most of the effects are similar to those described above, with one additional effect for the 857-GHz channel, where residual sidelobe contamination dominates the high-latitude residuals, with amplitudes of 2–3% of the full foreground signal. In this case, the 2018 processing represents an absolute improvement over the 2015 processing, in the sense that the full 2018 frequency map has lower sidelobe contamination than the corresponding 2015 frequency map. At the same time, it is worth noting that single-bolometer maps are available in the 2015 release, and the 857-2 bolometer map has significantly lower sidelobe contamination than any of the other three [@planck2014-a12]. Thus, if a given scientific analysis does not require the signal-to-noise ratio of the full 857-GHz channel, the 2015 857-2 bolometer channel may be an even better choice than the full 857-GHz 2018 frequency map. However, in the current paper, which is dedicated to the 2018 release itself, we adopt the 2018 full-frequency map in all analyses.
Figure \[fig:dx11\_vs\_dx12\_pol\] shows the corresponding plots for polarization. Here we do not subtract any CMB component (since it is small), and we also do not show fractional difference maps (since polarized foreground amplitudes can go both positive and negative). The two leftmost columns show the raw 2018 frequency maps in Stokes $Q$ and $U$, and the two rightmost columns shows the straight differences between the 2018 and 2015 frequency maps.
As expected, the various features seen in the polarization difference maps trace those observed in the corresponding temperature differences. For 30 and 44GHz, the main features at high latitudes are due to bandpass mismatch and time-variable gain corrections, achieved by iterating between gain estimation, mapmaking and component separation. For 70GHz, only very small differences are seen, since the gain estimation procedure is unchanged from 2015; however, it is important to note that a separate residual gain template has been produced for this channel, and this is applied in the scientific processing (see @planck2016-l02).
For the HFI channels, we see similar effects of improved effective gain estimation at high latitudes, as well as improved bandpass corrections at low latitudes, in particular for 100, 217, and 353GHz, which are strongly affected by CO emission. At 353GHz, we additionally see the residual effect of transfer-function convolution near the Galactic plane in Stokes $U$. Thus, caution is warranted when studying polarized thermal dust emission near the Galactic plane with this frequency map.
![Differences between 2015 and 2018 CMB $I$ maps at 80 resolution. From top to bottom, rows show results for , , , and . Monopoles and dipoles have been subtracted with parameters fitted outside a $|b|<30^{\circ}$ mask.[]{data-label="fig:cmb_diff_release_maps"}](figs/diff_dx11_v2_dx12_v3_commander_cmb_080a_0128_I_10uK_compress.pdf "fig:"){width="\columnwidth"}\
![Differences between 2015 and 2018 CMB $I$ maps at 80 resolution. From top to bottom, rows show results for , , , and . Monopoles and dipoles have been subtracted with parameters fitted outside a $|b|<30^{\circ}$ mask.[]{data-label="fig:cmb_diff_release_maps"}](figs/diff_dx11_v2_dx12_v3_nilc_cmb_080a_0128_I_10uK_compress.pdf "fig:"){width="\columnwidth"}\
![Differences between 2015 and 2018 CMB $I$ maps at 80 resolution. From top to bottom, rows show results for , , , and . Monopoles and dipoles have been subtracted with parameters fitted outside a $|b|<30^{\circ}$ mask.[]{data-label="fig:cmb_diff_release_maps"}](figs/diff_dx11_v2_dx12_v3_sevem_cmb_080a_0128_I_10uK_compress.pdf "fig:"){width="\columnwidth"}\
![Differences between 2015 and 2018 CMB $I$ maps at 80 resolution. From top to bottom, rows show results for , , , and . Monopoles and dipoles have been subtracted with parameters fitted outside a $|b|<30^{\circ}$ mask.[]{data-label="fig:cmb_diff_release_maps"}](figs/diff_dx11_v2_dx12_v3_smica_cmb_080a_0128_I_10uK_compress.pdf "fig:"){width="\columnwidth"}\
![Differences between 2015 and 2018 CMB $I$ maps at 80 resolution. From top to bottom, rows show results for , , , and . Monopoles and dipoles have been subtracted with parameters fitted outside a $|b|<30^{\circ}$ mask.[]{data-label="fig:cmb_diff_release_maps"}](figs/colourbar_10uK "fig:"){width="0.7\columnwidth"}
To summarize, we observe typically (at most) 2–3% differences between the 2015 and 2018 frequency maps at high latitudes, as measured in units of foreground signal. In most cases, these differences are directly due to improvements in the updated processing, although with a few notable exceptions, in particular for the 353-GHz channel. It is important to note, however, that the design philosophy of the 2018 release has been to optimize the quality of the polarization products, which in some cases comes at the expense of temperature analysis. In particular, the non-availability of single-bolometer maps represents a limiting factor for astrophysical component separation in temperature. For this reason, we expect both 2015 and 2018 temperature products to be in common use in the future, depending on the needs of a particular application, whereas for polarization we strongly recommend usage of the 2018 products.
Simulations {#sec:simulations}
-----------
The instrumental noise characteristics of the observations are complex, and a simple white-noise approximation is inadequate for high-precision analyses of these data. The only realistic approach to handling both instrumental noise and residual systematics is through end-to-end simulations. As part of the 2018 data release, we therefore provide a set of 300 independent noise-plus-systematics simulations for each frequency band and for each of the data splits described above, as well as 999 CMB-only simulations that include the effects of satellite scanning and asymmetric beams; see @planck2016-l02 and @planck2016-l03 for full details. These simulations are available through the Planck Legacy Archive.[^8]
These simulations are propagated through each of the pipelines; we adopt the same frequency weights (mixing matrices, spectral indices etc.) as for the real data. The two main advantages of fixing the weights are, first, that the noise properties actually correspond to the real final maps; and, second, that the system becomes linear, and CMB and noise may be propagated independently through each pipeline. In the following, we will employ CMB-only, noise-only, and CMB-plus-noise combinations for various applications.
Standardization of simulations and data {#sec:standardization}
---------------------------------------
Each of the four pipelines processes both the data and simulations somewhat differently with respect to harmonic space truncation ($\ell_{\mathrm{max}}$) and high-$\ell$ regularization. In order to facilitate meaningful direct comparisons between the various maps, we convolve all four data sets to a common effective resolution prior to analysis, as described below. We emphasize, however, that the released data products are provided at their native resolution, in order to allow external users to exploit the full resolution of each data set, if so desired.
For temperature, the most aggressive smoothing applied by any of the four pipelines is defined by , for which the effective high-$\ell$ apodization kernel reads $$B(\ell) =
\begin{cases}
1, & \ell\leq \ell_{\mathrm{peak}}, \\
\cos^2\left[(\pi/2)(\ell-\ell_{\mathrm{peak}})/(\ell_{\mathrm{max}}-\ell_{\mathrm{peak}})\right],
& \ell > \ell_{\mathrm{peak}},
\end{cases}$$ where $\ell_{\mathrm{peak}} = 3400$ and $\ell_{\mathrm{max}}=3999$. We therefore apply this kernel to each of the three other pipelines, on top of their intrinsic $5\arcm$ FWHM smoothing kernels. For we additionally apply the pixel window for $N_{\mathrm{side}}=2048$, which is not by default applied for this code.
For polarization, the most aggressive high-$\ell$ truncation is applied by , which enforces a hard harmonic space truncation at $\ell_{\mathrm{max}}=3000$. This same truncation is applied to each of the three other codes in polarization as a post-processing step.
CMB maps {#sec:cmb}
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The CMB maps and associated products obtained by the various pipelines as applied to the 2018 data are presented in this section; astrophysical foreground results are presented in the next section. For a detailed analysis of the higher-order statistical properties of these maps, see @planck2016-l07.
Full-mission maps and comparison with 2015 release {#sec:fullmission}
--------------------------------------------------
![Standard deviation of the CMB maps between the four component-separation methods, at 80 resolution. Temperature is shown in the top panel and polarization in the bottom panel. The polarization standard deviation is defined as $\sqrt{\mathrm{var}(Q) + \mathrm{var}(U)}$.[]{data-label="fig:cmb_maps_stddev"}](figs/dx12_v3_stddev_cmb_080a_0128_T_v2_compress.pdf "fig:"){width="\linewidth"}\
![Standard deviation of the CMB maps between the four component-separation methods, at 80 resolution. Temperature is shown in the top panel and polarization in the bottom panel. The polarization standard deviation is defined as $\sqrt{\mathrm{var}(Q) + \mathrm{var}(U)}$.[]{data-label="fig:cmb_maps_stddev"}](figs/dx12_v3_stddev_cmb_080a_0128_P_v2_compress.pdf "fig:"){width="\linewidth"}
Figure \[fig:cmb\_maps\] shows the final full-mission 2018 CMB component-separated maps derived by each of the four pipelines,[^9] both in intensity (left column) and polarization (middle and right columns). Only [SMICA]{} has been inpainted within a Galactic mask (see Appendix \[app:smica\]). All maps are smoothed to a common resolution of $80\arcm$ FWHM for visualization purposes.
At first sight, the consistency among the various pipeline maps appears to be reasonable outside the central Galactic plane, and, as expected, more so in temperature than in polarization.
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![Mask used for inpainting the cleaned CMB temperature maps.[]{data-label="fig:inpaint"}](figs/mask_inpainting_n512_compress.pdf){width="\columnwidth"}
In the polarization maps, however, we can identify several notable artefacts already at this stage, which prospective future users of these maps need to be aware of. The visually most striking features are of course residual foreground contamination in the Galactic plane. In particular, the alternating sign along the plane is a classic signature of temperature-to-polarization leakage, and the data set is particularly sensitive to residual CO emission in this respect. These features are extremely difficult to suppress to the level of the CMB fluctuations during processing, and must in practice be removed by standard Galactic masking.
The second most striking feature in the polarization maps is a blue stripe in the upper right quadrant of the Stokes $U$ map. This stripe corresponds to a few bad scanning rings that ideally should have been removed by flagging during mapmaking. Unfortunately, this issue was not caught at a sufficiently early stage of the processing, and remains in the final maps. We therefore mask this stripe in the same way that we mask Galactic residuals.
Third, and somewhat less obvious, we observe broad large-scale structures in both Stokes $Q$ and $U$ that are aligned with the scanning strategy. These structures are effectively due to gain-modelling uncertainties coupled to monopole and dipole leakage, and corresponding features are present in the associated simulations. In principle, therefore, these need not be removed prior to subsequent analyses, as long as the appropriate simulations are used to quantify all relevant uncertainties. In practice, however, we note that these modes are associated with significant additional systematic uncertainties, and we therefore caution against over-interpretation of the very largest scales in these maps. In particular, we warn against employing these maps for auto-correlation type analysis, unless the statistic of choice is explicitly shown to be robust against this type of systematic effect, based on end-to-end simulations.
Figure \[fig:cmb\_diff\_release\_maps\] shows maps of temperature differences between each of the 2018 pipeline maps and the corresponding 2015 pipeline maps. Corresponding maps of polarization differences are not shown, since the high level of large-scale systematics in the 2015 maps renders a direct difference-map comparison non-informative. In Fig. \[fig:cmb\_diff\_release\_maps\], we recognize many of the features seen in the raw input frequency difference maps shown in Figs. \[fig:dx11\_vs\_dx12\_lfi\] and \[fig:dx11\_vs\_dx12\_hfi\], and discussed in Sect. \[sec:inputs\].
Starting with , the most striking difference is a dark blue Galactic plane residual with a clear CO-like morphology. This reflects the fact that it is more difficult for the parametric pipeline to estimate CO emission from co-added frequency maps (as in the 2018 processing) than with individual bolometer maps (as in the 2015 processing). For this reason, the map adopts a larger Galactic mask in the new release than in the previous one, specifically targeting CO emission; see Appendix \[app:commander\] for further details. The second most notable feature in the difference map is a $\lesssim2\muK$ blue signal at intermediate latitude with a thermal dust imprint, and this is due to the changes in bandpass modelling in the high-frequency channels.
Only small differences are observed for , for which very few pipeline modifications have been introduced since 2015. already used full-frequency maps in the previous release. The most significant change is a large-scale quadrupolar structure at high latitudes, which directly reflects the effective gain changes at 100, 143, and 217GHz seen in Fig. \[fig:dx11\_vs\_dx12\_hfi\]. Likewise, also used full-frequency maps in 2015, and only minor pipeline modifications have been introduced, and consequently, only minor differences are observed in temperature from 2015 to 2018.
For , three qualitatively different types of differences are seen. First, the weak large-scale background pattern is similar to that observed in , and simply reflects the slight changes in input data discussed above. In addition, we see significant changes in the compact sources that can be explained by the change of masking strategy described in Sect. \[sec:methods\]. Third and finally, the strong near-Galactic-plane differences that include free-free sources (e.g., the Gum Nebula and Rho Ophucius) are explained by the miscalibration of the 44-GHz channel in the 2015 released map (as recalled in Sect. \[sec:methods\]). The impact of this issue is assessed in Appendix \[app:smica\].
Figure \[fig:cmb\_diff\_pipe\_maps\] shows all pairwise difference maps between each of the pipeline CMB maps. The structures seen in these plots correspond closely to those already discussed above. Finally, Fig. \[fig:cmb\_maps\_stddev\] shows the standard deviation evaluated from the four cleaned CMB maps, smoothed to $80\arcm$ FWHM angular scales; the polarization standard deviation is here defined as $\sqrt{\mathrm{var}{Q} + \mathrm{var}{U}}$.
Confidence masks {#sec:masks}
----------------
From the above discussion, it is clear that significant residuals are present in the CMB maps, in particular close to the Galactic plane. Therefore, appropriate masking is required for scientific exploration of the 2018 maps in both temperature and polarization, as in earlier releases. For this purpose, we adopt a conservative strategy similar to that of 2015, and we construct a common confidence mask for all maps, even if the various maps may have different levels of residuals.
In previous releases, a common mask was generated simply as the product of the individual confidence masks derived for each pipeline. However, the pipeline masks were established using qualitatively different criteria in each case, and a direct comparison was therefore non-trivial. In the current analysis, we adopt a more direct route, starting with the inter-pipeline standard deviation maps shown in Fig. \[fig:cmb\_maps\_stddev\]. Specifically, for temperature we first threshold at 3 the standard deviation map evaluated at 80 FWHM smoothing scale, and adopt this as our primary mask. The specific smoothing scale of 80 represents a compromise between suppressing noise while still retaining small features, while the threshold of 3 is defined by the region at high Galactic latitude in the top panel of Fig. \[fig:cmb\_maps\_stddev\]. Second, we smooth this binary mask, consisting of 0 and 1s, with a $10^\circ$ FWHM Gaussian beam, and remove any pixels with a value lower than 0.5; this is to remove isolated small “islands” within the main Galactic plane. Third, we threshold at 10 a corresponding standard deviation map evaluated at 10 FWHM smoothing scale in order to remove compact objects.
The resulting mask ensures that only pixels for which the four pipelines agree in their CMB solutions to better than 3 in standard deviation on large scales (10 on small scales) are allowed in the final analysis. However, quantitative agreement among codes is only a necessary criterion; it is not sufficient. We therefore augment this mask by the absolute individual confidence masks of and (see Appendices \[app:commander\] and \[app:sevem\] for details), by the point-source masks used for inpainting by and , and by the processing mask employed by . The first two of these employ $\chi^2$ and difference maps to define their acceptable regions, and thereby correspond to standard absolute goodness-of-fit statistics, while the latter two correspond to basic processing masks. The inpainted point-source mask is constructed from point sources detected in the 143- and 217-GHz channels, and is described in detail in Appendix \[app:sevem\] (see also Fig. \[fig:dx12\_masks\_inpainting\_sevem\]). We find no evidence for significant artefacts in the and maps outside the and masks defined above, and we therefore do not apply any special measure for these maps.
We adopt a similar procedure for polarization, but with a few notable additions. First, the inter-pipeline standard deviation map evaluated at 80 FWHM is thresholded at 1. The resulting mask is smoothed to 5 FWHM, and thresholded at a value of 0.9, effectively expanding the original mask by a few degrees in all directions. This mask is then multiplied with a corresponding mask derived by thresholding at 0.6 the original standard deviation map at 80 FWHM, to remove sharper features. We then exclude all pixels flagged by the and confidence masks. Next, we remove the region contaminated by cosmic rays discussed in Sect. \[sec:fullmission\], as defined in Ecliptic coordinates following ’s scanning path. Third, as an additional guard against temperature-to-polarization leakage from CO emission, we exclude any pixels for which the CO emission at 100GHz (see Sect. \[sec:foregrounds\]) is brighter than $20\muK$, evaluated at a smoothing scale of $5^{\circ}$ FWHM. Isolated “islands” in the main Galactic plane are then removed with the same procedure as for temperature. Finally, we also add the point-source masks used for inpainting by and . For polarization, the inpainted point-source mask is constructed from point sources detected in the 100-, 143-, and 217-GHz channels and is shown in Fig. \[fig:dx12\_masks\_inpainting\_sevem\].
The resulting common masks are shown in the top row of Fig. \[fig:commonmask\] for temperature (left panel) and polarization (right panel). The final accepted sky fractions are $f_{\mathrm{T}} = 0.780$ and $f_{\mathrm{P}} = 0.782$. These sky fractions are similar to those reported in 2015, namely $f_{\mathrm{T}}^{2015} = 0.77$ and $f_{\mathrm{P}}^{2015}
= 0.78$.
As discussed in Sect. \[sec:misspix\], the half-mission and odd-even split maps contain a number of unobserved or poorly conditioned pixels. For split-map analysis, we therefore recommend additional unobserved pixel masks. These are produced by thresholding the $3\times3$ Stokes parameter condition number hit-count maps produced during mapmaking [@planck2016-l02; @planck2016-l03]. The resulting unobserved pixel mask is further extended in a three-step iterative process in which the neighbours of each unobserved pixel have been masked. The bottom row in Fig. \[fig:commonmask\] shows the products of the temperature and polarization unobserved pixel masks for both the half-mission (left panel) and odd-even (right panel) splits.
As a final mask-related issue, we note that the 2018 product delivery includes Wiener-filtered versions of each pipeline map, in which all high-foreground regions are replaced with a Gaussian constrained realization. For temperature, these regions are defined simply by thresholding the maximum difference between any of the four cleaned CMB maps at $100\muK$, and additionally removing all pixels excluded by the processing mask. This mask is shown in Fig. \[fig:inpaint\], and excludes 2% of the sky. For polarization inpainting we conservatively adopt the common confidence mask defined above. In either case, we note that the inpainted CMB maps are primarily intended for publication and presentation purposes, rather than scientific analysis. For full scientific analysis purposes, we recommend corresponding processing of end-to-end simulations; however, these are not provided in the current release due to large data volume and processing costs.
Effective transfer functions
----------------------------
As noted in Sect. \[sec:methods\], all 2018 CMB maps have a common nominal target resolution of $5\arcm$ FWHM, as output by each of the respective pipelines. However, this resolution is not exact, as it does not take into account the effect of spatially-varying asymmetric beams on the sky. The nominal $5\arcm$ beam kernel must therefore be corrected by an effective transfer function for each pipeline prior to any harmonic space analysis of these maps, including cosmological power spectrum and parameter estimation.
![Effective transfer functions $f_{\ell}$ for each of the four pipeline CMB maps, after deconvolving a $5\arcm$ FWHM Gaussian beam and $\nside=2048$ pixel window. From top to bottom, the panels show results for $T$, $E$, and $B$. In the two bottom panels, the dotted lines show the effective residual transfer functions for the three CMB-dominated HFI frequencies between 100 and 217GHz, after deconvolving the azimuthally-symmetric [QuickBeam]{}-based transfer function and pixel window in each case.[]{data-label="fig:transferfunctions"}](figs/compsep_transfunc_v5_compress.pdf){width="\columnwidth"}
We estimate the effective transfer functions from the CMB signal-only simulations discussed in Sect. \[sec:simulations\] through the following expression, $$f_{\ell} = \frac{1}{b_{\ell}^{5\arcm} p_{\ell}^{2048}}
\sqrt{\left<\frac{C_{\ell}^{\mathrm{out}}}{C_{\ell}^{\mathrm{in}}}\right>},
\label{eq:transfunc}$$ where $C_{\ell}^{\mathrm{out}}$ and $C_{\ell}^{\mathrm{in}}$ denote the simulated output and input power spectra of each CMB signal realization. The former includes both instrumental beam and pixel window convolution, while the latter includes neither. The quantity $b_{\ell}^{5\arcm}$ denotes the beam transfer function of a 5 FWHM Gaussian beam, $p_{\ell}^{2048}$ is the $N_{\mathrm{side}}=2048$ pixel window [@gorski2005], and brackets indicate an average over 50 simulations. Equation \[eq:transfunc\] is evaluated independently for temperature and both $E$- and $B$-mode polarization. Finally, each transfer function is smoothed with a third-order Savitzky-Golay filter with a window size of $\Delta\ell=51$ to reduce residual uncertainty from the finite number of Monte Carlo simulations. The examples shown in this paper correspond to full-sky transfer functions; these functions should in principle be re-evaluated for each sky fraction used in a given analysis. [^10]
The resulting transfer functions are shown in Fig. \[fig:transferfunctions\]. Starting with the temperature case, we first note that the range spanned by the four curves is well within $\pm0.5\,$%, and, therefore, these effects are quite minor for all the considered multipoles. Overall, qualitatively similar behaviour is observed for the four codes, with showing a slightly larger deviation. In particular, for , we see that the effective residual transfer function is very close to unity up to $\ell\approx700$, after which it starts to fall off, eventually reaching an amplitude of about 0.3% at $\ell\approx2000$, before it begins to rise sharply. The small excess of $\lesssim0.1\%$ around $\ell=500$ is associated with the effective cut-off of the LFI-dominated low-frequency signal component employed by . These general trends are due to small mismatches between the full asymmetric beams, as implemented through pixel-space [FEBeCoP]{} [@mitra2010] convolutions, and the azimuthally symmetric effective beam transfer functions, as implemented with [QuickBeam]{} [@hivon2017]. For instance, a fall-off of 0.3% at $\ell\approx2000$ corresponds to a mismatch of about $0.05'$ FWHM in the two models.
Turning our attention to the $E$-mode transfer functions, the most striking new feature is a pattern of systematic wiggles. These are seen both in transfer functions derived from each frequency alone (shown as dotted lines) and in the component-separated maps. These wiggles are due to temperature-to-polarization leakage through the asymmetric beam shapes, and the positions of the peaks coincide with the peaks in the CMB temperature power spectrum.
Similar considerations apply to the $B$-mode transfer functions, although in this case the wiggles are largely dominated by an increasing trend caused by a wide range of both temperature-to-polarization and polarization-to-polarization leakage effects. Overall, however, the net sum of all these effects is smaller than 10% of the underlying (lensing-induced) $B$-mode signal up to $\ell\lesssim1600$. We also see that the component-separated map is strongly dominated by the 217-GHz channel for $\ell\gtrsim500$.
Noise characterization and consistency with simulations {#sec:noise_consistency}
-------------------------------------------------------
We now characterize the statistical properties of the component-separated CMB map, and we start with a description of instrumental noise and residual systematic effects. We adopt three main measures for this purpose, each designed to highlight different aspects of the effective noise properties; these are designed for different applications.
Our first noise measure is defined in terms of the so-called odd-even half-difference (OEHD) maps, in which the full time-ordered data volume is divided according to odd and even ring numbers. This is a fine-grained time split, and as such, the OEHD map tends to cancel most systematic effects. This noise measure is thus our cleanest probe of pure instrumental (white and correlated) noise. OEHD maps are plotted in Appendix \[fig:cmb\_oehd\_maps\] for each pipeline and for each of the three Stokes parameters. Overall, we see that these difference maps exhibit very few visually-apparent systematic effects at high latitudes, and the only significant residuals occur in the Galactic centre, where the overall signal amplitude is very larget.
Our second noise measure is defined in terms of the half-mission half-difference (HMHD) maps, in which the time-ordered data are split according to long time periods, defined by years [@planck2016-l02; @planck2016-l03]. This measure is thus a coarse-grained time split, and more sensitive to systematic effects that vary on long time scales, such as gain variations or sidelobe contamination. This is our preferred estimate for the combined impact of instrumental noise and systematic effects. HMHD maps are shown in Appendix \[fig:cmb\_hmhd\_maps\] for each pipeline and for each of the three Stokes parameters. In these maps, we clearly see the imprint of the Galactic plane, which is largely caused by calibration uncertainties, as well as more pronounced scan-aligned structures at high latitudes.
The third noise measure comprises the full-blown end-to-end simulations, in which all known systematics have been modelled to the best of our ability (see @planck2016-l02 and @planck2016-l03 for full details). These simulations are generated as raw time-ordered data, and processed through each step of the analysis pipeline, including map making and component separation. Unfortunately, this process is computationally very expensive, and only a limited set of 300 realizations has been produced for the current release. However, for each realization a full set of results are produced, including full mission maps, half-mission and odd-even splits. Combined, these form the basis of most goodness-of-fit statistics presented in the following sections.
### Power spectrum analysis
In Fig. \[fig:simspec\] we compare the power spectra of the cleaned CMB maps with the simulations. All spectra are evaluated outside the common mask described in Sect. \[sec:masks\] using [PolSpice]{} [@chon2004]. Furthermore, all spectra have been normalized relative to the mean of the simulated ensemble, and plotted in terms of the fractional deviation, $$\eta_{\ell} \equiv \frac{D_{\ell}^{\mathrm{data}} - \left< D_{\ell}^{\mathrm{sim}}\right>}{\left< D_{\ell}^{\mathrm{sim}}\right>}.$$ This function thus measures the fractional difference of the observed power spectrum from the mean of the simulations, plotted as a percentage in Fig. \[fig:simspec\]. This function is evaluated both for temperature and polarization ($TT$, $EE$, and $BB$), as well as for full-mission, HMHD, and OEHD data splits. For clarity, each function has been binned with $\Delta\ell=25$ after computing the above single-$\ell$ quantity.
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For full-mission temperature data, we find that the CMB-plus-noise simulations agree well with the data in terms of angular power up to $\ell\lesssim 750$. At higher multipoles, we see a slow increase in power up to $\ell\approx2000$, corresponding to a positive contribution from point sources not included in the simulations. The level of point-source residuals is highest in , which does not apply any inpainting of sources during pre-processing, and lowest in . At high multipoles, $\ell \gtrsim 2000$, the spectra turn over. As described in @planck2016-l03, the power in the HFI simulations for the 100–217GHz channels underestimates the true noise in the real data by a few percent (with variations depending both on angular scale and frequency), and this translates into a negative bias at high multipoles in the cleaned CMB maps presented in this paper.
Similar features are seen even more clearly in the polarization $EE$ and $BB$ full-mission spectra, for which the signal-to-noise ratio is lower. In these cases, the simulations agree well with the data up to $\ell\lesssim200$, after which a negative bias of a few percent is observed in the range $200 \lesssim \ell \lesssim 500$. Then, in the range $500\lesssim\ell\lesssim1500$ the agreement is good, before we see the same negative high-$\ell$ bias as in the temperature case. The same trends are even more prominent in the HMHD and OEHD spectra, which by construction are entirely noise-dominated.
In Fig. \[fig:simspec\_l23\] we focus on the first two multipoles, and compare the observed power to the full simulated distributions in terms of cumulative distribution functions. Overall, we observe acceptable statistical agreement between the data and simulations at these largest scales, with only a few points showing extreme values of 0 or 1; however, even in these cases the observed values lie just at the edge of the simulated histogram. No large outliers are observed. Nevertheless, it is important to note that the effective noise varies greatly between the various analysis pipelines, and it is therefore essential to compare any given data set with its corresponding simulations.
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To summarize, the end-to-end simulations presented and employed in this paper exhibit power biases of several percent with respect to the true observations on intermediate and small scales, while reasonable agreement is observed on large angular scales. These biases originate from corresponding discrepancies at the level of individual frequency bands, as reported in @planck2016-l02 and @planck2016-l03. When employing these simulations for scientific analysis, it is important to verify that the statistic of choice is not sensitive to such percentage-level differences. This will usually be the case for linear or cross-correlation type analyses, but not necessarily for quadratic or auto-correlation type analyses.
### Pixel-space variance analysis
A complementary consistency measure is given by the total variance as measured in pixel space at different pixel resolutions (see, e.g., @Monteserin2008; @Cruz2011; @planck2014-a18). This method normalizes the map with respect to the total variance of the signal plus noise, where the noise variance is estimated through the simulations described above, and the variance of the signal is determined as the value that gives a normalized map variance equal to unity. For the HMHD and OEHD maps, the method simplifies, since the CMB signal is cancelled through the half-difference calculation.
Recognizing the fact that polarization maps are generally noise dominated, we apply the methods described in @planck2016-l08 and Molinari et al. (in preparation) for polarization. These methods include both auto- and cross-estimates for the variance, which is the result of the subtraction between the variance of the $\left<Q^2+U^2\right>$ signal-plus-noise map and the variance of the $\left<Q_N^2+U_N^2\right>$ noise estimated from the MC simulations. For both temperature and polarization, we employ the respective union masks described above. When dealing with HMHD and OEHD maps, we consider the union mask combined with the corresponding unobserved pixel mask.
In Fig. \[fig:simspec\_variance\] we plot the percentage of simulations with a lower variance than the real data, as a function of pixel resolution. The results from this analysis are in good agreement with those found in the power spectrum analysis. In temperature we find a generally good consistency between real data and half difference noise simulations, with few exceptions. We find that only a few simulations have a lower variance for the HMHD map at very large scales and for the OEHD map at intermediate resolutions. At the maximum resolution of $N_{\rm side}=2048$, there is a lack of compatibility with MC simulations for both HMHD and OEHD maps, showing that at high resolution noise in temperature data is poorly described by the simulations. However, given the very high signal-to-noise ratio of the temperature data at all resolutions, a small noise mismatch is irrelevant compared to the CMB cosmic variance.
\[htbp\]
![Consistency between data and simulations as quantified in terms of pixel-space variance for both temperature (left) and polarization (right), and for both full mission maps (top row), HMHD maps (middle row), and for half-mission cross-variance (bottom rows). Coloured lines show results for the four different component-separation pipelines as a function of pixel resolution, $N_{\textrm{side}}$.[]{data-label="fig:simspec_variance"}](figs/DX12_pvalue){width="\columnwidth"}
For the signal-plus-noise data, we observe satisfactory consistency in temperature at high pixel resolutions. At lower resolutions we observe low probabilities, with p values of about 1.0%, which are associated with the well known lack of power on large angular scales. These results are compatible with results reported in the previous release described in [@planck2014-a18]. We have also investigated the higher order moments, skewness and kurtosis in temperature as shown in @planck2016-l07, and find good consistency with Monte Carlo simulations at all resolutions.
For the signal-plus-noise data, we observe satisfactory consistency in temperature at high pixel resolutions. At lower resolutions we observe low probabilities, with $p$ values of about 1.0%, which are associated with the well known lack of power on large angular scales. These results are compatible with results reported in the previous release described in [@planck2014-a18].
In polarization at high resolutions, results are not as robust, due to the noise mismatch. We observe an incompatibility for all the component-separated HMHD and OEHD maps between the MC distribution and real data at intermediate and high resolutions. This suggests that noise in the data (including systematic effects) is not fully characterized by the simulations. At lower resolutions ( $\nside=256$ for HMHD and 128 for OEHD), however, data are compatible with simulations, showing that the noise properties are better represented. In Fig. \[fig:noise\_mismatch\] we show the amplitude of the noise mismatch with respect to the amplitude of the expected CMB variance as a function of pixel resolution. These results give an estimation of the bias due to the noise mismatch in the extraction of the variance from signal plus noise data. The bias is very important at the highest resolution ($N_{\rm side}=2048$), with values of about 40–50% for all the methods. At intermediate resolutions it is of the order of few percent. At large scales the bias is not significant, since half-difference data are compatible with the MC dispersion.
![Amplitude of the noise mismatch in terms of expected signal amplitude, $\textrm{var}_{\mathrm{th}}$, in percentage as a function of the pixel resolution, $N_{\textrm{side}}$ for HMHD polarization data (top plot) and OEHD polarization data (bottom plot). Coloured lines show results for the four different component-separation pipelines. Error bars show the amplitude of the MC dispersion at $\pm$ 1 $\sigma$, showing that where the noise is well characterized the bias is embedded in the uncertainty in the variance extraction and hence it is not significant.[]{data-label="fig:noise_mismatch"}](figs/DX12_P_HMHD_noise_mismatch_compress.pdf "fig:"){width="\columnwidth"} ![Amplitude of the noise mismatch in terms of expected signal amplitude, $\textrm{var}_{\mathrm{th}}$, in percentage as a function of the pixel resolution, $N_{\textrm{side}}$ for HMHD polarization data (top plot) and OEHD polarization data (bottom plot). Coloured lines show results for the four different component-separation pipelines. Error bars show the amplitude of the MC dispersion at $\pm$ 1 $\sigma$, showing that where the noise is well characterized the bias is embedded in the uncertainty in the variance extraction and hence it is not significant.[]{data-label="fig:noise_mismatch"}](figs/DX12_P_OEHD_noise_mismatch_compress.pdf "fig:"){width="\columnwidth"}
In spite of the presence of a noise mismatch, we find that cross- and auto-analyses of the signal-plus-noise maps are in agreement with MC simulations. At intermediate and large scales, auto- and cross-analyses are in good agreement with each other, showing the robustness of the analysis, although with some differences among the component-separation methods due to the presence of residual foregrounds, or systematic effects, or a different impact of the noise mismatch. At high resolution the differences between auto and cross results are mainly due to the noise mismatch, whose impact is more important for the auto analysis. On the other side, the cross-analyses may be biased by a poor description of the correlated noise that we cannot investigate with the above analyses.
In [@planck2016-l08] we consider more detailed analyses of this kind, including also the analysis of the frequency maps, in a way that minimizes the impact of the correlated noise.
### Assessing the impact of simulation noise bias
In order to understand whether these percent-level noise discrepancies are relevant for a given analysis, we strongly recommend considering the following questions while assessing the results.
1. *Which angular scales are relevant for the statistic of choice?* If the statistic is sensitive only to large angular scales ($\ell \lesssim 50$), then the simulations are likely to be adequate. If not, see next question.
2. *Is the statistic of choice sensitive to signal-plus-noise or noise alone?* If the former, then the simulations are likely to be adequate for $\ell \lesssim 1500$ for temperature, and $\ell \lesssim 250$ for polarization; if the latter, then see next question.
3. *Is the statistic of choice sensitive to $\lesssim$5% errors in the noise model?* To quantify this, we recommend applying the statistic of choice to simulations for which the noise contribution is artificially re-scaled either up or down by 5% (see Fig. \[fig:simspec\]), while the signal contribution is unchanged. If the statistic of choice is unable to distinguish between the scaled and the unscaled ensembles, then the statistic is likely robust against the uncertainties in the current simulations. If not, caution is warranted. Typically, linear, cubic, or cross-spectrum type statistics are only marginally sensitive to this type of error in the noise model, whereas quadratic and auto-spectrum type statistics are typically highly sensitive.
Clearly, no general prescription can be given for all analyses, and we therefore stress that caution is warranted when using the end-to-end simulations. That being said, they do provide the most complete description of the uncertainties in the data set currently available, and with an appropriate level of care, they should form the basis of most goodness-of-fit tests with the current data set. For several worked examples of applications of these simulations, see @planck2016-l07.
Foreground template fits
------------------------
Next, we consider residual foreground contamination as measured by correlation between known foreground templates and the cleaned CMB maps. Specifically, for a given cleaned temperature CMB map ${\vec{d}}$, a foreground template ${\vec{t}}$, and the common confidence mask ${\vec{m}}$ (consisting of 0’s and 1’s), we compute the correlation coefficient $$r = \frac{1}{N_{\mathrm{pix}}-1} \sum_{i\in{\vec{m}}} \frac{{\vec{d}}_i -
\left<{\vec{d}}\right>}{\sigma_{{\vec{d}}}} \frac{{\vec{t}}_i - \left<{\vec{t}}\right>}{\sigma_{{\vec{t}}}},$$ where the sum runs over the $N_{\mathrm{pix}}$ pixels not excluded by the mask, $\left<{\vec{d}}\right> = 1/N_{\mathrm{pix}} \sum_i {\vec{d}}_i$, $\sigma_{{\vec{d}}} = \left[1/N_{\mathrm{pix}-1} \sum_i ({\vec{d}}_i-\left<{\vec{d}}\right>)^2\right]^{1/2}$, and similarly for ${\vec{t}}$. All maps are smoothed to a common resolution of $80\arcm$ FWHM, and pixelized at $\nside=128$.
We consider four foreground templates in intensity, namely, the 408MHz @haslam1982 map as processed by @remazeilles2014 for synchrotron emission, the 2018 857-GHz map for thermal dust emission, the @dame2001 map for CO line emission, and the @finkbeiner2003 $H_\alpha$ map for free-free emission. For polarization, we consider the 23-GHz map as a synchrotron tracer. Uncertainties are evaluated from 300 end-to-end simulations. Corresponding results were reported in @planck2014-a11 for the 2015 CMB sky maps.
The results from these calculations are summarized in Table \[tab:tempfit\]. In nearly all cases, we see that the correlation coefficients are lower for the 2018 maps than the corresponding 2015 maps, and most are within the $1\sigma$ confidence limits. The only notable issue is a marginally significant polarization correlation with the synchrotron tracer, ranging in statistical significance between $2.8\sigma$ for to $3.5\sigma$ for . The absolute level of the correlation is low, however, ranging between 3 and 4%.
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Power spectrum comparison
-------------------------
Next, we characterize the cleaned CMB maps in terms of angular power spectra. As above, we employ the [PolSpice]{} estimator for these calculations, and all spectra are evaluated outside the common mask defined in Sect. \[sec:masks\].
Figure \[fig:powspec\_HM\_TT\] shows a comparison of power spectra evaluated from the four cleaned CMB temperature half-mission maps. In the top panel, the solid lines show spectra computed from the half-mission half-sum (HMHS) maps, and thereby contain both signal and noise, while dashed lines show spectra computed from the HMHD, and thereby should contain only instrumental noise and systematic uncertainties. The black solid line shows the best-fit 2018 $\Lambda$CDM model derived from the combination of the low-$\ell$ $TT$, low-$\ell$ $EE$, high-$\ell$ $TT+TE+EE$, and lensing likelihoods [@planck2014-a15]. The bottom panel shows the residuals after subtracting both the best-fit $\Lambda$CDM model (as a signal tracer) and the half-difference spectrum (as a noise tracer) from each of the half-sum spectra.
![Comparison of half-mission temperature power spectra. The top panel shows the half-sum (HMHS; solid lines) and half-difference (HMHD; dashed lines) power spectra, while the bottom panel shows the difference between the half-sum and the best-fit 2018 $\Lambda$CDM and half-difference spectra. The latter residual spectrum is binned with $\Delta\ell=25$.[]{data-label="fig:powspec_HM_TT"}](figs/compsep_TT_hm_comparison_v3_compress.pdf){width="\columnwidth"}
Overall, we observe good agreement among the four pipelines in terms of half-sum spectra up to $\ell \lesssim 1500$. The main notable feature is a small power deficit of about $10\muK^2$ in between $\ell=100$ and 300, corresponding to a relative deficit of 0.2%. At these multipoles, the data are strongly CMB dominated, and algorithmic variations make little difference in terms of overall power. However, at higher multipoles the noise and compact source contributions become relevant, and in that regime the various approaches show slightly different behaviour, with having the largest compact source imprint and the smallest. In this respect it should be noted that while attempts to model and remove the source contributions, , , and all apply point source inpainting as a pre-processing step.
At low multipoles we also see differences among the codes in terms of noise. The lowest noise is achieved by , which also exhibits nearly white noise with a scaling of $\mathcal{O}(\ell^2)$. The highest low-$\ell$ noise – almost an order of magnitude higher than – is seen for for $\ell\lesssim300$. This is not unexpected given the nature of the algorithm. On large angular scales, the frequency weights primarily adjust themselves to suppress foregrounds, while on small scales, they converge to inverse-noise-variance weighting. In this respect, is different from the other three codes in that it explicitly uses estimates of the noise standard deviation to perform inverse-variance noise weighting per pixel. Finally, for we note that the noise decreases around $\ell\approx100$, which corresponds to the multipole at which the three LFI frequencies are excluded at high latitudes (see Appendix \[app:smica\]).
In Fig. \[fig:powspec\_HM\_EE\_BB\] we present a similar comparison for the $EE$ and $BB$ HMHD polarization power spectra. As in Fig. \[fig:powspec\_HM\_TT\], the solid lines includes contributions from both signal and noise. Here we see, at least at the level of visual inspection, that all four codes perform similarly in terms of polarization power spectrum reconstruction, both for HMHS and HMHD spectra. The only marginal outlier is , which exhibits slightly higher $BB$ HMHS and HMHD spectra at multipoles lower than $\ell \lesssim 100$. However, this excess disappears in the difference between the HMHS and HMHD spectrum (bottom panels of Fig. \[fig:powspec\_HM\_EE\_BB\]), suggesting that it is due to a somewhat higher noise level in the map compared to the others, and not a signal bias.
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Finally, in Fig. \[fig:powspec\_HM\_EE\_BB\_lowl\] we show an expansion of the low multipole part of the polarization power spectra without applying any multipole binning. The black solid line in the left panel shows the best-fit 2018 $\Lambda$CDM model for which the posterior mean optical depth of re-ionization is $\tau=0.054\pm0.019$ [@planck2016-l06]. The left and right panels show the $EE$ and $BB$ spectra, respectively. The grey curves indicate corresponding spectra computed from five end-to-end simulations.
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Starting with the $BB$ spectrum, we note that there is an overall significant excess compared to zero. This excesss, however, is reproduced in the simulations, as seen by the non-zero mean of simulations, and therefore reflects the presence of understood residual correlations in the data; see @planck2016-l03 for further discussion. Consequently, we re-emphasize the importance of comparing these data with full end-to-end simulations when subjecting them to cosmological analysis, in order to adequately capture this type of residual noise correlation. A similar noise excess is seen in the $EE$ spectrum for $\ell \gtrsim 8$, with an amplitude similar to the $BB$ spectrum.
The $EE$ spectrum does not not show a clear detection of the reionization peak. As mentioned, the solid black curve shown in Fig. \[fig:powspec\_HM\_EE\_BB\_lowl\] indicates a spectrum for which $\tau=0.054$, and this amplitude is too low to be visually observed in an $\ell$-by-$\ell$ spectrum, in particular in the presence of the noise excess mentioned above. In order to detect this peak, a full likelihood analysis is essential, as presented in @planck2016-l05 and @planck2016-l06.
While the 2018 best-fit value for the optical depth of reionization is $\tau=0.054$, the value used to generate the simulations (which had to be adopted well before the final results were available for computational expense reasons) was $\tau=0.060$. The effect of this difference can be seen in Fig. \[fig:powspec\_HM\_EE\_BB\_lowl\] as an excess of power in the simulations relative to the best-fit model at low multipoles in $EE$. While the difference is within $1\sigma$, it is worth having this issue in mind when studying low-$\ell$ polarization effects with these maps and simulations; see section 3 of @planck2016-l07 for a quantitative analysis of this issue.
A full cosmological likelihood and parameter analysis of the 2018 data is presented in @planck2016-l05, based on cross-spectrum techniques. In this paper, we perform a simple consistency test between the full likelihood analysis and the cleaned CMB maps presented in this paper, by fitting a CMB spectrum (parametrised by an amplitude $A_{\mathrm{CMB}}$ and tilt $n$ relative to the best-fit 2018 $\Lambda$CDM model) and an $\ell^2$ point-source contribution to the difference between the HMHS and HMHD spectra. Explicitly, we adopt the signal model $$D_{\ell} = A_{\mathrm{CMB}} \left(\ell/\ell_{0}\right)^n f_{\ell}^2
D_{\ell}^{\Lambda\mathrm{CDM}} + A_{\mathrm{ps}}
\ell(\ell+1)/(\ell_{\mathrm{ps}}(\ell_{\mathrm{ps}}+1)),$$ where $f_{\ell}$ is the transfer function shown in Fig. \[fig:transferfunctions\], $\ell_0 = 600$ is a pivot multipole for the CMB fit, and $\ell_{\mathrm{ps}}=500$ is a pivot multipole for the point source contribution. The quantity $D_{\ell} {\Lambda\mathrm{CDM}}$ is the best-fit 2018 $\Lambda$CDM power spectrum. Each analysis includes multipoles between $\ell=2$ and 1500 (for which the simulations agree well with the data; see Fig. \[fig:simspec\]), and all spectra are binned with $\Delta\ell=20$. Uncertainties within each bin are defined as the standard deviation of the observed spectrum within the bin. The number of degrees of freedom for the $\chi^2$ is $n_{\mathrm{dof}}=75$. The fit is performed with a simple Metropolis MCMC sampler.
The results from these calculations are summarized in Table \[tab:params\]. Overall, we find good agreement between the cleaned CMB maps and the likelihood analysis, with most $\Lambda$CDM amplitudes consistent with unity within $2\sigma$ and all tilt parameters consistent with zero within $1.5\sigma$. All $\chi^2$s are also reasonable, ranging between 57.8 and 74.0 for 75 degrees of freedom, corresponding to probabilities-to-exceed (PTEs) ranging between 0.07 and 0.49. Finally, as already noted, exhibits the largest point-source contribution, with an amplitude of $A_{\mathrm{ps}}=2.7\pm0.5\muK^2$ at $\ell=500$ in temperature, while and show the smallest contribution, with amplitudes of $A_{\mathrm{ps}}=1.9\muK^2$ at $\ell=500$.
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The real-space N-point correlation functions {#sec:2point_correlation}
--------------------------------------------
A complementary measure of correlations and non-Gaussianity are given by real-space 2- and 3-point correlation functions. These functions are defined as the average product of $N$ observed fields, measured in a fixed relative distance on the sky. In the case of the CMB, the fields correspond to temperature anisotropy $\Delta T$ and two Stokes parameters $Q$ and $U$ describing the linear polarization of the radiation in a given direction (see @planck2016-l07 for more detail).
Because of computational limitations, we restrict our analysis to the pseudo-collapsed and equilateral configurations of the 3-point functions and low resolution CMB maps with a resolution parameter $N_{\rm side}=64$ and smoothed with a $160\arcm$ FWHM Gaussian beam. For both temperature and polarization, we employ the common masks described above, downgraded in the same way as CMB maps. Because we analyse half-difference maps, the common mask is combined with the corresponding unobserved pixel mask. The resulting 2- and 3-point correlation functions for the HMHD and OEHD maps are presented in Fig. \[fig:npt\_commander\] (figures for the remaining component separation maps can be found in the Appendix \[app:npt\_functions\]). We use a simple $\chi^2$ statistic to quantify the agreement between the observed data and the FFP10 noise simulations. Table \[tab:prob\_npt\] lists the significance level in terms of the fraction of simulations with a larger $\chi^2$ value than the observed map. Corresponding analysis for full CMB maps is provided in @planck2016-l07.
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The results indicate consistency between the half-difference maps estimated using the different component-separation methods. Moreover, no statistically significant deviations between data and simulations are found at these angular scales. Note, however, that the confidence regions derived from the noise simulations do vary between methods, indicating that each method results in different effective statistical properties. To avoid biases, it is therefore essential to analyse each map together with the simulations constructed specifically for that map.
Gravitational lensing {#sec:lensing}
---------------------
As an example of science that may be extracted from the cleaned CMB maps presented in this paper, we consider reconstruction of the gravitational lensing potential. For a complete analysis of this topic, we refer the interested reader to @planck2016-l08, from which the following results are reproduced.
Gravitational lensing of CMB photons by large-scale structures induces slight distortions in the statistics of the CMB. In particular, lensing deflections result in a characteristic acoustic peak smoothing signature in the angular power spectrum, and they induce a non-zero four-point CMB correlation function. We use the methodology described in [@planck2016-l08] to reconstruct the lensing power spectrum. With the sensitivity and sky coverage of , this approach constrains the lensing deflection power spectrum to a few percent, with most of the signal coming from temperature observations at high multipoles $\ell \sim 1500$, and these measurements therefore result in a stringent consistency test between the various component-separation methods at small angular scales.
For masking, we employ the union of the intensity and polarization mask recommended in Sect. \[sec:masks\], combined with a Galactic mask allowing $f_{\rm sky} = 0.70$, as well as a mask removing resolved SZ clusters with $\hbox{S/N} > 5$, as given by the 2015 SZ catalogue (@planck2014-a35; for full details, see @planck2016-l08). We consider quadratic lensing estimates built from temperature only ($\hat \phi^{\rm TT}$), as well as the full minimum variance combination ($\hat \phi^{\rm MV}$). The minimum-variance estimator is derived from the full set of quadratic estimators $TT, TE, TB, EE$, and $EB$, which increases the signal-to-noise ratio with respect to $TT$ by roughly 20%. As discussed in Sect. \[sec:cmb\], there is slight power mismatch between data and simulation power on the scales relevant for lensing. To account for this, we add in each case additional power as an isotropic, Gaussian component either to the simulations or to the data.
Figure \[fig:lensing\] shows the our minimum-variance lensing spectrum estimates evaluated from lensing multipoles $8 \leq L \leq 2048 $. Summary amplitude statistics are listed in Table \[tab:lensing\_amplitude\], both on the conservative ($8
\leq L \leq 400$) and high-$L$ ($401 \leq L \leq 2048$) ranges. As we see, the four component-separation methods result in almost identical constraining power. No clear band-power outliers are observed in Fig. \[fig:lensing\], and all summary statistics are consistent with each other within uncertainties. However, all four methods show a lensing power that is slightly tilted with respect to the fiducial model, with slightly less power at high multipoles. More detailed analysis and consistency tests are presented in [@planck2016-l08].
![Lensing reconstruction power spectrum from the four cleaned CMB maps, including lensing multipoles $8 \leq L \leq 2048$ in the minimum variance estimator. For comparison, the black line shows the lensing potential power spectrum adopted for the FFP10 simulation suite.[]{data-label="fig:lensing"}](figs/L04_lensing_compilation_v3_compress.pdf){width="\columnwidth"}
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Analysis of end-to-end simulations
----------------------------------
We finish this CMB-targeted analysis section with a brief discussion of end-to-end simulations, focusing on polarization extraction from the Full Focal Plane (FFP10; [@planck2014-a14]) set. For a corresponding analysis of temperature simulations, see @planck2014-a11.
Unlike the simulations discussed in Sect. \[sec:noise\_consistency\], which only included the CMB and instrumental noise, the simulations considered in this section also includes polarized synchrotron and thermal dust emission. These simulations are processed through each pipeline, allowing each code to estimate spectral parameters (i.e., weights for , and , and spectral indices for ) directly from the simulations.
Figure \[fig:ffp10\_absdiff\] shows the CMB polarization reconstruction error for each of the four CMB analysis pipelines, as evaluated from the end-to-end FFP10 analysis pipeline, defined by $$\Delta P = \sqrt{(Q_{\mathrm{out}}-Q_{\mathrm{in}})^2 +
(U_{\mathrm{out}}-U_{\mathrm{in}})^2},$$ where $Q_{\mathrm{out}}$ and $U_{\mathrm{out}}$ are the estimated Stokes parameters, and $Q_{\mathrm{in}}$ and $U_{\mathrm{in}}$ are the true Stokes parameters. All maps have been smoothed to $80\arcm$ FWHM before computing this quantity, to reduce the impact of instrumental noise.
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In these plots, one may observe generally similar behaviour between and , and between and . Explicitly, and result in slightly lower residuals in the Galactic plane, whereas and appear slightly less sensitive to stripes at high Galactic latitues. As evaluated over the common polarization mask, the standard deviations of the four maps (in alphabetical order) are $0.74\muK$, $0.86\muK$, $0.74\muK$, and $0.75\muK$, respectively.
Polarized foregrounds {#sec:foregrounds}
=====================
We now turn to the scientific characterization of diffuse microwave foregrounds as derived from the 2018 polarization maps; a corresponding discussion of temperature foreground products is given in Appendix \[app:foregrounds\]. Three different algorithms are employed in the following, namely [@eriksen2004; @eriksen2008; @planck2014-a12; @seljebotn2017], [@Remazeilles2011b], and [@cardoso2008].
![(*Top*:) polarization residual maps, ${\vec{d}}_{\nu}-{\vec{s}}_{\nu}$, for each polarized frequency channel. All maps are smoothed to a common resolution of $40\arcm$ FWHM. (*Bottom*:) $\chi^2$ map for the high-resolution polarization analysis, repixelized to resolution $\nside=256$ to reduce noise. []{data-label="fig:comm_pol_residuals"}](figs/res_rc6_030_40arc_n256_Q_v2_compress.pdf "fig:"){width="0.49\columnwidth"} ![(*Top*:) polarization residual maps, ${\vec{d}}_{\nu}-{\vec{s}}_{\nu}$, for each polarized frequency channel. All maps are smoothed to a common resolution of $40\arcm$ FWHM. (*Bottom*:) $\chi^2$ map for the high-resolution polarization analysis, repixelized to resolution $\nside=256$ to reduce noise. []{data-label="fig:comm_pol_residuals"}](figs/res_rc6_030_40arc_n256_U_v2_compress.pdf "fig:"){width="0.49\columnwidth"}\
![(*Top*:) polarization residual maps, ${\vec{d}}_{\nu}-{\vec{s}}_{\nu}$, for each polarized frequency channel. All maps are smoothed to a common resolution of $40\arcm$ FWHM. (*Bottom*:) $\chi^2$ map for the high-resolution polarization analysis, repixelized to resolution $\nside=256$ to reduce noise. []{data-label="fig:comm_pol_residuals"}](figs/res_rc6_044_40arc_n256_Q_v2_compress.pdf "fig:"){width="0.49\columnwidth"} ![(*Top*:) polarization residual maps, ${\vec{d}}_{\nu}-{\vec{s}}_{\nu}$, for each polarized frequency channel. All maps are smoothed to a common resolution of $40\arcm$ FWHM. (*Bottom*:) $\chi^2$ map for the high-resolution polarization analysis, repixelized to resolution $\nside=256$ to reduce noise. []{data-label="fig:comm_pol_residuals"}](figs/res_rc6_044_40arc_n256_U_v2_compress.pdf "fig:"){width="0.49\columnwidth"}\
![(*Top*:) polarization residual maps, ${\vec{d}}_{\nu}-{\vec{s}}_{\nu}$, for each polarized frequency channel. All maps are smoothed to a common resolution of $40\arcm$ FWHM. (*Bottom*:) $\chi^2$ map for the high-resolution polarization analysis, repixelized to resolution $\nside=256$ to reduce noise. []{data-label="fig:comm_pol_residuals"}](figs/res_rc6_070_40arc_n256_Q_v2_compress.pdf "fig:"){width="0.49\columnwidth"} ![(*Top*:) polarization residual maps, ${\vec{d}}_{\nu}-{\vec{s}}_{\nu}$, for each polarized frequency channel. All maps are smoothed to a common resolution of $40\arcm$ FWHM. (*Bottom*:) $\chi^2$ map for the high-resolution polarization analysis, repixelized to resolution $\nside=256$ to reduce noise. []{data-label="fig:comm_pol_residuals"}](figs/res_rc6_070_40arc_n256_U_v2_compress.pdf "fig:"){width="0.49\columnwidth"}\
![(*Top*:) polarization residual maps, ${\vec{d}}_{\nu}-{\vec{s}}_{\nu}$, for each polarized frequency channel. All maps are smoothed to a common resolution of $40\arcm$ FWHM. (*Bottom*:) $\chi^2$ map for the high-resolution polarization analysis, repixelized to resolution $\nside=256$ to reduce noise. []{data-label="fig:comm_pol_residuals"}](figs/res_rc6_100_40arc_n256_Q_v2_compress.pdf "fig:"){width="0.49\columnwidth"} ![(*Top*:) polarization residual maps, ${\vec{d}}_{\nu}-{\vec{s}}_{\nu}$, for each polarized frequency channel. All maps are smoothed to a common resolution of $40\arcm$ FWHM. (*Bottom*:) $\chi^2$ map for the high-resolution polarization analysis, repixelized to resolution $\nside=256$ to reduce noise. []{data-label="fig:comm_pol_residuals"}](figs/res_rc6_100_40arc_n256_U_v2_compress.pdf "fig:"){width="0.49\columnwidth"}\
![(*Top*:) polarization residual maps, ${\vec{d}}_{\nu}-{\vec{s}}_{\nu}$, for each polarized frequency channel. All maps are smoothed to a common resolution of $40\arcm$ FWHM. (*Bottom*:) $\chi^2$ map for the high-resolution polarization analysis, repixelized to resolution $\nside=256$ to reduce noise. []{data-label="fig:comm_pol_residuals"}](figs/res_rc6_143_40arc_n256_Q_v2_compress.pdf "fig:"){width="0.49\columnwidth"} ![(*Top*:) polarization residual maps, ${\vec{d}}_{\nu}-{\vec{s}}_{\nu}$, for each polarized frequency channel. All maps are smoothed to a common resolution of $40\arcm$ FWHM. (*Bottom*:) $\chi^2$ map for the high-resolution polarization analysis, repixelized to resolution $\nside=256$ to reduce noise. []{data-label="fig:comm_pol_residuals"}](figs/res_rc6_143_40arc_n256_U_v2_compress.pdf "fig:"){width="0.49\columnwidth"}\
![(*Top*:) polarization residual maps, ${\vec{d}}_{\nu}-{\vec{s}}_{\nu}$, for each polarized frequency channel. All maps are smoothed to a common resolution of $40\arcm$ FWHM. (*Bottom*:) $\chi^2$ map for the high-resolution polarization analysis, repixelized to resolution $\nside=256$ to reduce noise. []{data-label="fig:comm_pol_residuals"}](figs/res_rc6_217_40arc_n256_Q_v2_compress.pdf "fig:"){width="0.49\columnwidth"} ![(*Top*:) polarization residual maps, ${\vec{d}}_{\nu}-{\vec{s}}_{\nu}$, for each polarized frequency channel. All maps are smoothed to a common resolution of $40\arcm$ FWHM. (*Bottom*:) $\chi^2$ map for the high-resolution polarization analysis, repixelized to resolution $\nside=256$ to reduce noise. []{data-label="fig:comm_pol_residuals"}](figs/res_rc6_217_40arc_n256_U_v2_compress.pdf "fig:"){width="0.49\columnwidth"}\
![(*Top*:) polarization residual maps, ${\vec{d}}_{\nu}-{\vec{s}}_{\nu}$, for each polarized frequency channel. All maps are smoothed to a common resolution of $40\arcm$ FWHM. (*Bottom*:) $\chi^2$ map for the high-resolution polarization analysis, repixelized to resolution $\nside=256$ to reduce noise. []{data-label="fig:comm_pol_residuals"}](figs/res_rc6_353_40arc_n256_Q_v2_compress.pdf "fig:"){width="0.49\columnwidth"} ![(*Top*:) polarization residual maps, ${\vec{d}}_{\nu}-{\vec{s}}_{\nu}$, for each polarized frequency channel. All maps are smoothed to a common resolution of $40\arcm$ FWHM. (*Bottom*:) $\chi^2$ map for the high-resolution polarization analysis, repixelized to resolution $\nside=256$ to reduce noise. []{data-label="fig:comm_pol_residuals"}](figs/res_rc6_353_40arc_n256_U_v2_compress.pdf "fig:"){width="0.49\columnwidth"}\
![(*Top*:) polarization residual maps, ${\vec{d}}_{\nu}-{\vec{s}}_{\nu}$, for each polarized frequency channel. All maps are smoothed to a common resolution of $40\arcm$ FWHM. (*Bottom*:) $\chi^2$ map for the high-resolution polarization analysis, repixelized to resolution $\nside=256$ to reduce noise. []{data-label="fig:comm_pol_residuals"}](figs/chisq_2017_P_n256_v2_compress.pdf "fig:"){width="\columnwidth"}
Internal consistency and goodness-of-fit
----------------------------------------
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![$P$–$P$ scatter plot between the thermal dust polarization amplitude at 353GHz, as estimated with and . Colours indicate the density of points on a logarithmic scale.[]{data-label="fig:gnilc_dust_P"}](figs/gnilc_commander_353_scatter_P_v2.pdf){width="\columnwidth"}
Before considering astrophysical components, it is instructive to consider the internal consistency between the 2018 polarization frequency maps. For this purpose, we employ the model described in Sect. \[sec:commander\], fitting a minimal three-component signal model (CMB, synchrotron, and thermal dust emission) to the seven polarized frequencies between 30 and 353GHz. The synchrotron component is modelled by a single power-law with a free spectral index, $\beta_{\mathrm{s}}$, in the frequency domain, while the thermal dust component is modelled as a modified blackbody with free spectral index, $\beta_{\mathrm{d}}$, and temperature, $T_{\mathrm{d}}$. In the main analyses, the synchrotron spectral index is fixed spatially to $\beta_{\mathrm{s}}=-3.1$, matching the high-latitude temperature result found from the combination of 2015, WMAP, and Haslam data [@planck2014-a12]; as shown in Sect. \[sec:pol\_ind\], the measurements by themselves have little sensitivity to the synchrotron spectral index. For thermal dust, we fix $T_{\mathrm{d}}$ at the result found from the 2018 temperature data in Appendix \[app:foregrounds\]; with a highest frequency of 353GHz, the polarization observations are insensitive to this parameter.
Additionally, we impose a spatial smoothness prior on both synchrotron and thermal dust emission to reduce noise-induced degeneracies between the various components. This takes the form of a Gaussian smoothing kernel with $40\arcm$ FWHM for synchrotron emission and $10\arcm$ FWHM for thermal dust emission. The widths of these priors are chosen to match the resolution at which the data have a significant signal-to-noise ratio; see Appendix \[app:commander\] for further details.
Given this model, the top panels in Fig. \[fig:comm\_pol\_residuals\] show residual maps of the form $\vec d_{\nu} - \vec s_{\nu}$ for each frequency map, all smoothed to a common resolution of $40\arcm$. The colour scales cover $\pm20\muK$ for the LFI channels, and $\pm5\muK$ for the HFI channels. Ideally, each of these maps should be consistent with instrumental noise alone, and for the three LFI channels this appears to be a reasonable approximation. The only clearly visible artefacts in these maps correspond to regions of high foreground amplitudes, which most likely are due to a low level of residual temperature-to-polarization leakage, for instance from bandpass mismatch between individual detectors.
In contrast, significant large-scale residuals may be seen at all four HFI frequencies, with patterns typically aligning with the scanning strategy. Collectively, these features correspond to effective calibration uncertainties that couple the CMB dipole and foregrounds to the reconstructed CMB polarization signal. Although these residuals are significant, their amplitudes are almost an order of magnitude smaller than in the 2015 data. Moreover, the latest end-to-end simulations describe the residuals to a high level of precision [@planck2016-l03].
The bottom panel in Fig. \[fig:comm\_pol\_residuals\] shows the total $\chi^2$ per pixel, as defined by $$\chi^2(p) = \sum_{\nu=1}^{N_{\mathrm{band}}} \left(\frac{d_{\nu}-s_{\nu}(p)}{\sigma_{\nu}(p)}\right)^2,
\label{eq:chisq}$$ downgraded to a $\nside=256$ grid, corresponding to $14\arcm$ pixels. Note that $\sigma_{\nu}(p)$ only accounts for white noise. Thus, the smoothness of the $\chi^2$ map clearly suggests that the 2018 polarization observations are dominated by instrumental white noise on intermediate and small angular scales, not by systematic effects or foreground artefacts.
### Polarization amplitude
Next, we consider the polarization amplitude of synchrotron emission at 30GHz and thermal dust emission at 353GHz, naively defined as $P^{\mathrm{s}} = \sqrt{Q_{\mathrm{s}}^2 + U_{\mathrm{s}}^2}$. As discussed by @plaszczynski2014, this estimator is intrinsically noise-biased; however, since we are only interested in it for comparison and consistency purposes, the noise bias is not critical for this paper. The resulting maps are shown in Figs. \[fig:comm\_pol\_dust\]–\[fig:gnilc\_dust\_P\], as estimated by , , and . For , the synchrotron map is smoothed to $40\arcm$ FWHM and the thermal dust emission map is smoothed to $5\arcm$ FWHM. For , the corresponding smoothing scales are $40\arcm$ and $12\arcm$ FWHM. For the effective angular resolution varies over the sky, depending on the local signal-to-noise ratio. The maps correspond to the amplitudes evaluated at monochromatic reference frequencies, while the and maps correspond to bandpass-integrated maps at 30 and 353GHz, respectively.
Two sets of products are delivered for the 2018 release: (i) the Stokes $I,$ $Q$, and $U$ maps of thermal dust emission at uniform $80'$ resolution, with the associated noise covariance matrix maps ($II$, $IQ$, $IU$, $QQ$, $QU$, and $UU$); and (ii) the Stokes $I,$ $Q$, and $U$ maps of thermal dust emission at variable resolution ($80'$ to $5'$) over the sky, with the associated noise-covariance-matrix maps, along with a beam FWHM map indicating the corresponding variable resolution of the dust over the sky regions. The 2018 dust products are analysed in great detail in [@planck2016-l11B].
The bottom panel of Fig. \[fig:gnilc\_dust\_P\] shows a scatter plot between the and thermal dust amplitudes, both evaluated for a common resolution of $80\arcm$ FWHM. Overall, the agreement is very good, and the Pearson’s correlation coefficient between the two maps is $r=0.999$. Similar good agreement is observed between the and the and maps, except for very high values of $P$, for which applies an inpainting mask during processing to avoid ringing. The main notable difference between the and maps is an overall relative scaling of around 5%, corresponding to the fact that no colour corrections are applied to the map, and it therefore corresponds to the dust signal as observed through the 353-GHz bandpass. This distinction between the two maps is important to bear in mind when subjecting either one to statistical analysis.
Based on these polarization amplitude maps, one can compute the corresponding polarization fraction, defined as $p = P/I$, where $P$ is the polarization amplitude, and $I$ is the corresponding total intensity. This quantity is useful for modelling and characterizing astrophysical emission processes, and iis therefore of great interest to astrophysical theorists. However, it is also highly sensitive to systematic errors in the intensity component, and in particular to the zero level, which is difficult to constrain for the measurements. A careful analysis of the thermal dust polarization fraction derived from the 2018 measurements, including zero level uncertainties, is provided in @planck2016-l11B, and we refer the interested reader to that paper for full details.
![Distribution of spectral indices for polarized synchrotron (top panel) and thermal dust (bottom panel) emission as estimated with without applying any informative Gaussian prior. The synchrotron spectral index shown in this plot is estimated with a $5^{\circ}$ FWHM smoothing scale, and the thermal dust spectral index is estimated with a $3^{\circ}$ FWHM smoothing scale. For the thermal dust case, results are shown both with (green curve) and without (blue curve) applying polarization efficiency corrections at 100–217GHz. The dashed lines in this case indicate Gaussian fits to the central peak.[]{data-label="fig:comm_pol_beta"}](figs/synch_P_beta_priorfree_compress.pdf "fig:"){width="\linewidth"}\
![Distribution of spectral indices for polarized synchrotron (top panel) and thermal dust (bottom panel) emission as estimated with without applying any informative Gaussian prior. The synchrotron spectral index shown in this plot is estimated with a $5^{\circ}$ FWHM smoothing scale, and the thermal dust spectral index is estimated with a $3^{\circ}$ FWHM smoothing scale. For the thermal dust case, results are shown both with (green curve) and without (blue curve) applying polarization efficiency corrections at 100–217GHz. The dashed lines in this case indicate Gaussian fits to the central peak.[]{data-label="fig:comm_pol_beta"}](figs/dust_P_beta_priorfree_v1_compress.pdf "fig:"){width="\linewidth"}
### Synchrotron and thermal dust spectral indices {#sec:pol_ind}
Next, we consider the spectral energy distributions (SEDs) for polarized synchrotron and thermal dust emission. For simplicity, we focus primarily on the effective spectral index for either process, noting that has very limited sensitivity to estimate additional spectral parameters in polarization.
Starting with , we note that the main analysis discussed above is performed with informative (delta function or Gaussian) priors on both $\beta_{\mathrm{s}}$ and $\beta_{\mathbf{d}}$. In order to quantify the intrinsic information content and statistical strength of the data to constrain these parameters at a more basic level, it is useful also to perform *prior-free* runs. The results from such analyses are summarized in Fig. \[fig:comm\_pol\_beta\], for synchrotron emission in the top panel and thermal dust emission in the bottom panel. In either case, the Gaussian prior is removed only on the component in question, not both simultaneously. In all cases, however, a broad uniform prior is imposed in order to exclude completely unphysical values. The synchrotron analysis is performed at a smoothing scale of $5^{\circ}$ FWHM, while the thermal dust analysis is performed at a smoothing scale of $3^{\circ}$ FWHM.
For synchrotron emission, we find a very broad distribution between $\beta_{\mathrm{s}}=-4$ and $-1.5$, with both ends being defined by the uniform prior. There is a weak preference for values between $\beta_{\mathrm{s}}=-3.5$ and $-3.0$, consistent with the value of $\beta_{\mathrm{s}}=-3.1$ found by combining , , and Haslam temperature data in @planck2014-a12, but overall, it is clear that the polarization data by themselves do not significantly constrain the spectral index of synchrotron emission at scales smaller than $5^{\circ}$. For the main analysis, we therefore fix the spectral index for polarized synchrotron emission at the best-fit value derived from the 2015 temperature data, corresponding to $\beta_{\mathrm{s}}=-3.1$. This value is also consistent within the uncertainties with corresponding results derived by @kogut2007, @dunkley2009b, @bennett2012, @fuskeland2014, @vidal2014, and @krachmalnicoff2018.
For thermal dust emission, the situation is more informative, since the HFI data constrain thermal dust emission more strongly than the LFI data constrain synchrotron emission. Focusing for the moment on the blue curve in Fig. \[fig:comm\_pol\_beta\], corresponding to the nominal data set considered in this paper, we observe a clear peak centred around $\beta_{\mathrm{d}} \approx 1.60$, and with a width of $0.10$–$0.15$. The distribution exhibits heavy tails toward both steep and shallow spectral indices, which is typical for noise-dominated data; these pixels are mostly located at high Galactic latitudes, where the dust amplitude is low. Motivated by these results, we adopt a Gaussian prior for the analysis of $\beta_{\mathrm{d}}=1.60\pm0.10$ for the main analysis, acknowledging that the standard deviation quoted above over-estimates the intrinsic scatter in the dust population because of instrumental noise. Note that the uncertainty in this prior refers to the standard deviation of the map, not the error in the mean of the central value.
As mentioned in Sect. \[sec:inputs\], the 2018 HFI polarization measurements are associated with small but non-negligible uncertainties in terms of polarization efficiencies, $\epsilon$. By default, polarization efficiency corrections are not included in the analyses presented in this paper, but instead we assess their impact by comparing results with and without these corrections. The green curve in the bottom panel Fig. \[fig:comm\_pol\_beta\] shows the distribution of $\beta_{\mathrm{d}}$ with application of these corrections at frequencies between 100 and 217GHz. Overall, we see that these polarization efficiencies shift the distribution by $\Delta\beta_{\mathrm{d}} = -0.03$.
The nominal polarization-efficiency corrections described in @planck2016-l03 and @planck2016-l05 do not include any robust estimates for the 353-GHz channel, since the CMB signal that is used to estimate these corrections is faint at this frequency. However, it is reasonable to assume that it is associated with similar uncertainties as the other HFI channels. In Fig. \[fig:comm\_pol\_beta\_353\], we show the $\beta_{\mathrm{d}}$ posterior distributions resulting from changing $\epsilon_{353}$ by 1% in either direction from its nominal value. In this case, we find that a shift of $\epsilon_{353}$ by 1% translates into a change in $\beta_{\mathrm{d}}$ of 0.013.. Combined with the uncertainties arising from the 100- to 217-GHz frequencies, we therefore consider the total systematic uncertainty on $\beta_{\mathrm{d}}$ due to polarization efficiency corrections to be 0.04.
![Effect on the spectral index of polarized thermal dust emission, $\beta_{\mathrm{d}}$, when changing the polarization efficiency correction at 353GHz, $\epsilon_{353}$. A shift of $\epsilon_{353}$ by 1% translates into a change in $\beta_{\mathrm{d}}$ of 0.013.[]{data-label="fig:comm_pol_beta_353"}](figs/dust_P_beta_prior_poleff_v1_compress.pdf){width="\linewidth"}
![Spatial distribution of the spectral index of polarized thermal dust emission, $\beta_{\mathrm{d}}$, as estimated with adopting a smoothing scale of $3^\circ$ FWHM. In the top panel no Gaussian prior is applied. In the bottom panel a Gaussian prior of $\beta_{\mathrm{d}} = 1.60\pm0.10$ is applied. In both cases, the spectral index of synchrotron emission is fixed to $\beta_{\mathrm{s}}=-3.1$.[]{data-label="fig:comm_pol_beta_map"}](figs/dust_beta_P_priorfree_3deg_n256_compress.pdf "fig:"){width="\linewidth"}\
![Spatial distribution of the spectral index of polarized thermal dust emission, $\beta_{\mathrm{d}}$, as estimated with adopting a smoothing scale of $3^\circ$ FWHM. In the top panel no Gaussian prior is applied. In the bottom panel a Gaussian prior of $\beta_{\mathrm{d}} = 1.60\pm0.10$ is applied. In both cases, the spectral index of synchrotron emission is fixed to $\beta_{\mathrm{s}}=-3.1$.[]{data-label="fig:comm_pol_beta_map"}](figs/dust_beta_P_rc6_3deg_n256_compress.pdf "fig:"){width="\linewidth"}
The top panel in Fig. \[fig:comm\_pol\_beta\_map\] shows the spatial distribution of $\beta_{\mathrm{d}}$ from the prior-free analysis without polarization efficiency corrections. In this plot the statistical power of the observations to constrain the spectral index is seen very clearly from position to position, depending on the local dust polarization amplitude. Near the Galactic plane, the data are sufficiently strong to determine the spectral index well per resolution element, while at high latitudes the measurements are fully dominated by instrumental noise. The bottom panel shows the corresponding result when applying the supporting Gaussian prior. From this figure, it is clear that the $\beta_{\mathrm{d}}$ distribution and prior presented above are dominated by measurements in the Galactic plane, where the signal-to-noise ratio is substantially larger than at high Galactic latitudes.
Next, we perform a blind analysis of polarization spectral indices with . This analysis is performed by running with a foreground dimension of $\nfg=2$ (that is, with a two-column foreground emissivity matrix $\tens F$), as defined in Eq. (\[eq:smica:model\]), corresponding to synchrotron and thermal dust emission. Spectral priors are imposed during the multi-frequency fit so that synchrotron emission vanishes at 353GHz and thermal dust emission vanishes at 30GHz.
The results from these calculations are summarized in Fig. \[fig:polsedsmica\] for both $E$-mode and $B$-mode polarization. Parametric best-fits are indicated by dotted lines; these are, however, only the products of post-processing the raw results, and do not correspond to active priors as they do in the Bayesian analysis discussed above. In these particular fits, polarization efficiency corrections are applied to the 100, 143, and 217GHz data, and colour corrections are applied in post-analysis.
![*Top*: Synchrotron and thermal dust SEDs as estimated blindly by . Red and blue curves indicate thermal dust and synchrotron $E$ modes, respectively, and orange and cyan curves indicate corresponding $B$ modes. Dotted lines indicate the best-fit spectra for a power-law fit with $\beta_{\mathrm{s}}=-3.10\pm0.06$ for synchrotron, and a modified blackbody fit with $\beta_{\mathrm{d}}=1.53\pm0.01$ and $T_{\mathrm{d}}=19.6\mathrm{K}$ for thermal dust emission. *Bottom:* Residual spectral energy densities relative to best-fit models, measured in units of the data uncertainty, $\sigma_{\nu}$.[]{data-label="fig:polsedsmica"}](figs/smica_fg_SED_compress.pdf){width="\columnwidth"}
The best-fit spectral parameters derived in this blind manner are $\beta_{\mathrm{s}}=-3.10\pm0.06$ and $\beta_{\mathrm{d}}=1.53\pm0.01$, both corresponding to full-sky averages. Furthermore, these fits provide a statistically sufficient model across the full frequency range, as indicated by the residual spectra shown in the bottom panel of Fig. \[fig:polsedsmica\]. All residuals are within $2\sigma$ of their statistical errors.
The measurements of the polarized thermal dust spectral index are in excellent agreement with the corresponding results presented in @planck2016-l11A, based on both frequency cross-correlation power spectra at high Galactic latitudes and simple colour ratios between the 217- and 353-GHz channels at low Galactic latitudes. At the same time, $\beta_{\mathrm{d}}$ is lower by 0.07 or $3\sigma$ compared to the results presented above. To understand the origin of these differences, it is instructive to take a closer look at the 217/353 colour ratio, which is the fastest, simplest and most transparent estimator available.
The results from this estimator may be summarized as follows. We subtract one of the cleaned CMB maps from the 217- and 353-GHz polarization HM split maps to form two statistically independent foreground-plus-noise maps. We smooth these maps to $3^{\circ}$ FWHM to increase the effective signal-to-noise ratio per pixel. We then compute the cross-polarization amplitude between the two halves of the split, and we finally form the CMB-corrected colour ratio between the 217 and 353GHz maps. Given some estimate of the thermal dust temperature, this ratio may then be easily translated into estimates of the thermal dust spectral index. We adopt a constant temperature of 19.6K in the following.
First, we consider the impact of different CMB estimates produced by each of the four analysis pipelines. With the above procedure, we find median estimates of $\beta_{\mathrm{d}}=1.57$, 1.54, 1.55, and 1.54, when subtracting the , , , and CMB polarization maps, respectively. Different noise-weighting and foreground-modelling assumptions thus account for $\Delta\beta_{\mathrm{d}}\approx0.03$.
Second, the effect of polarization efficiencies has already been addressed above in the context of . We find similar sensitivities to the polarization efficiencies on the 217/353 colour ratio, as the median estimates for each of the four codes when applying these corrections are $\beta_{\mathrm{d}}=1.54$, 1.52, 1.52, and 1.52, corresponding to an effective shift of $\Delta\beta_{\mathrm{d}}\approx0.02$–0.03.
Third and finally, a small effect is due to different bandpass treatments. Specifically, in @planck2016-l11A, bandpass integration effects are taken into account by the so-called colour correction technique, in which a multiplicative correction based on some fiducial spectral parameters is applied to the nominal thermal dust SED at a given reference frequency. The same approach is adopted for the results. In contrast, performs a full integral over the product of the bandpass and the SED for each set of spectral parameters. These two different approaches agree to 0.07% at 143GHz, 0.7% at 217GHz, and 1.3% at 353GHz. In sum, these small differences translate into a net shift of $\Delta\beta_{\mathrm{d}}=0.015$ in terms of the thermal dust spectral index.
Recognizing the significant systematic uncertainties on the thermal dust spectral index from both modelling aspects and polarization efficiencies, we adopt a total systematic uncertainty of 0.05, defined by the above shifts added in quadrature with a statistical uncertainty of 0.02 [@planck2016-l11A]. As a single point estimate, we adopt the average value of the colour-ratio-derived estimates without polarization efficiency corrections, for a total final estimate of $\beta_{\mathrm{d}}=1.55\pm0.05$. This estimate is conservative, and corresponds to marginalizing over all analysis methods and known uncertainties.
### Synchrotron and thermal dust angular power spectra
Finally we consider the angular power spectra of polarized synchrotron and thermal dust emission as estimated by and . We estimate the $EE$ and $BB$ angular cross-spectra outside the common CMB mask for the half-mission split with [XPol]{} (see @tristram2005 and @planck2016-l11A for details). The results from these calculations are summarized in Fig. \[fig:fg\_powspec\]. results are shown in red (for thermal dust emission) and green (for synchrotron emission) and results are shown in orange and light green. For comparison, direct 353-GHz cross-correlation results are shown in purple, derived using the same methodology as in @planck2016-l11A. The solid black lines indicate the best-fit 2018 $\Lambda$CDM CMB spectrum. Dotted coloured lines indicate best-fit power law fits to the spectra, as defined by $$D_{\ell} = q\,\left(\frac{\ell}{80}\right)^\alpha.$$ Overall, we find excellent agreement between the , , and 353-GHz results, demonstrating that the derived component maps are robust with respect to specific algorithmic details for the particular angular ranges and sky coverage considered here.
![$EE$ (top) and $BB$ (bottom) power spectra for synchrotron and thermal dust as computed from the , , and 353-GHz frequency maps; see @planck2016-l11A for algorithmic details. All spectra are evaluated outside the common polarization mask, over 78% of the sky. Dashed lines indicate the best-fit power-law fits for the case, as reported in Table \[tab:fg\_powspec\]. Error bars indicate $3\sigma$ uncertainties. All spectra have been colour corrected to monochromatic reference frequencies of 30GHz for synchrotron and 353GHz for thermal dust emission, respectively.[]{data-label="fig:fg_powspec"}](figs/cls_EE_rc6_v6_compress.pdf "fig:"){width="50.00000%"}\
![$EE$ (top) and $BB$ (bottom) power spectra for synchrotron and thermal dust as computed from the , , and 353-GHz frequency maps; see @planck2016-l11A for algorithmic details. All spectra are evaluated outside the common polarization mask, over 78% of the sky. Dashed lines indicate the best-fit power-law fits for the case, as reported in Table \[tab:fg\_powspec\]. Error bars indicate $3\sigma$ uncertainties. All spectra have been colour corrected to monochromatic reference frequencies of 30GHz for synchrotron and 353GHz for thermal dust emission, respectively.[]{data-label="fig:fg_powspec"}](figs/cls_BB_rc6_v6_compress.pdf "fig:"){width="50.00000%"}
Table \[tab:fg\_powspec\] summarizes the angular power spectra in terms of best-fit power-law models for and , and in terms of the $EE/BB$ ratio, all derived using the same machinery as in @planck2016-l11A. For thermal dust, corresponding results are also given for the direct 353-GHz cross-correlation approach. Power-spectrum amplitudes have been colour corrected to monochromatic reference frequencies of 30 and 353GHz for synchrotron and thermal dust emission, respectively.
-3mm =
Overall, we find excellent agreement in terms of angular power spectra for polarized thermal dust emission between the frequency cross-correlation technique and the and component-separation techniques. The only statistically significant discrepancy is seen for the spatial power-law index parameter, $\alpha$, for which formally a $6\sigma$ difference is observed between and . However, we note that in terms of absolute values the difference is only $\Delta\alpha=0.06$, and no systematic uncertainties are included in these numbers. Finally, it is worth noting that the two analyses are carried out with different angular resolutions, corresponding to $5\arcm$ and $12\arcm$ FWHM respectively. Due to its lower resolution, the analysis is somewhat more sensitive to high-multipole systematics than the and 353GHz analyses.
We also observe excellent agreement between the and maps in terms of polarized synchrotron emission. In addition, we note that the $BB/EE$ ratio measured from the 2018 data is 0.34, which is very similar to the corresponding value of 0.36 estimated from the 2015 data.
Based on the best-fit power spectrum and SED parameters reported above, Fig. \[fig:fg\_cl\_ratio\] summarizes the foreground-to-CMB ratio in terms of the quantity $f(\ell,\nu) = [C_{\ell}^{\mathrm{fg}}(\nu)/C_{\ell}^{\mathrm{CMB}}]^{1/2}$ as a function of both frequency and angular scale. As expected, the overall picture is very similar to that presented from the 2015 data in @planck2014-a12, with one small but notable exception: Because the best-fit value of the optical depth of reionization is lower in the 2018 $\Lambda$CDM model than in the corresponding 2015 model, the relative foregrounds-to-CMB ratio is higher at low $EE$ multipoles, further emphasizing the importance of accurate foreground modelling for large-scale polarization CMB analysis.
![Amplitude ratio between total polarized foregrounds and CMB as a function of both multipole moment and frequency, as defined by $f(\ell,\nu) =
[C_{\ell}^{\mathrm{fg}}(\nu)/C_{\ell}^{\mathrm{CMB}}]^{1/2}$, with parameters derived from 78% of the sky as estimated by . The top and bottom panels show $EE$ and $BB$ spectra, and the black and red contours in the latter corresponds to tensor-to-scalar ratios of $r = 0.0$ and 0.05, respectively.[]{data-label="fig:fg_cl_ratio"}](figs/cl_fg_ratio_EE_v3_compress.pdf "fig:"){width="\columnwidth"}\
![Amplitude ratio between total polarized foregrounds and CMB as a function of both multipole moment and frequency, as defined by $f(\ell,\nu) =
[C_{\ell}^{\mathrm{fg}}(\nu)/C_{\ell}^{\mathrm{CMB}}]^{1/2}$, with parameters derived from 78% of the sky as estimated by . The top and bottom panels show $EE$ and $BB$ spectra, and the black and red contours in the latter corresponds to tensor-to-scalar ratios of $r = 0.0$ and 0.05, respectively.[]{data-label="fig:fg_cl_ratio"}](figs/cl_fg_ratio_BB_v3_compress.pdf "fig:"){width="\columnwidth"}
Conclusions {#sec:conclusions}
===========
In this paper we have presented cleaned CMB temperature and polarization maps derived from the 2018 data set, as well as new polarized synchrotron and thermal dust emission maps. These maps represent a new state-of-the-art characterization of the microwave sky.
The main scientific motivation underlying the work between the 2015 and 2018 data releases has been reduced instrumental systematics. As demonstrated in this and companion papers, the work has been successful, as the updated frequency maps exhibit significantly lower contamination on all angular scales. For polarization, we find that the lower systematics in frequency maps translates directly into lower systematics in CMB and foreground maps. Additionally, new end-to-end CMB-plus-noise simulations have been constructed that more accurately reproduce residual systematics observed in the real data. For full-mission data, these simulations are accurate to $\lesssim3\,\%$ for $\ell\lesssim1500$ in both temperature and polarization. On smaller scales, non-negligible biases are observed, and caution is warranted when subjecting the maps to detailed statistical analysis on scales smaller than $\ell\gtrsim1500$.
It is important to note that the 2018 data release does not represent a globally optimal reduction of the time-ordered data that is ideal for all purposes. In particular, the updated data set does not include single detector maps, and the new frequency maps have complicated bandpass properties [@planck2016-l03]. As a result, accurate reconstruction of astrophysical temperature foreground properties is non-trivial. Thus, while the 2018 release represents a significant step forward in our understanding of the polarized microwave sky compared to the 2015 release, the associated temperature results, for which the astrophysics are richer, do not represent a similar improvement. Indeed, for several intensity applications we anticipate that external users may find the 2015 products more useful than the corresponding 2018 products. One concrete example of this is the astrophysical sky model as presented in @planck2014-a12, which includes intensity estimates of both CO line emission and thermal dust emission. The same considerations apply both to and ; while chronologically formally superseded by the current results, we believe that the 2015 temperature astrophysical foreground models represent more accurate approximations to the true sky than the ones presented in the 2018 data release. To avoid confusion, we therefore do not release the corresponding 2018 foreground temperature products.
Fortunately, these issues are largely unimportant for CMB reconstruction purposes. The analyses presented in this paper and in @planck2016-l06 reach the same conclusion regarding the CMB temperature results, namely that the 2018 CMB temperature data are for all practical purposes statistically consistent with the corresponding 2015 rendition. Of course, this is the direct result of the very high signal-to-noise ratio of the measurements, in that small variations in the processing procedure make very little difference in the final maps compared to the intrinsic sample variance of the true CMB sky.
For large-scale CMB polarization at $\ell\lesssim50$, we find that the 2018 data are compatible with end-to-end simulations. However, it is critical to note that the observations are *not* consistent with uncorrelated white noise at any angular scales. Any statistical analysis of the 2018 polarization data must therefore always be accompanied by a corresponding analysis of the associated end-to-end simulations. In addition, analysis of half-data split sky maps is strongly encouraged in order to probe stability with respect to both noise and residual instrumental systematics.
In addition to improving the large-scale CMB polarization map, the new data processing also results in improved astrophysical polarization results. One concrete example of this is the fact that the 2018 data for the first time allow a pixel-by-pixel estimation of the spectral index of thermal dust emission over the full sky. Corresponding analyses based on previous data sets invariably led to clearly nonphysical results obviously driven by instrumental systematics. With this new data set, we obtain a typical spectral index of polarized thermal dust emission of $\beta_{\mathrm{d}}=1.55\pm0.05$, where the uncertainty accounts both for systematic uncertainties and different analysis techniques. This estimate is largely consistent with comparable results derived from temperature measurements. Also, for polarized synchrotron emission, we are for the first time able to fit the spectral index pixel-by-pixel, and obtain physically meaningful values, even if the signal-to-noise ratio is low; the full-sky averaged synchrotron spectral index for polarized emission is $\beta_{\mathrm{s}}=-3.1\pm0.1$. For thermal dust emission we find a $BB/EE$ angular power spectrum ratio of 0.5, largely independent of sky fraction, while for synchrotron emission we find a lower ratio of 0.34.
In Fig. \[fig:overview\_pol\] we plot the rms of the polarization amplitude as a function of frequency for polarized CMB, synchrotron, and thermal dust emission, evaluated with an angular resolution of $40\arcm$ FWHM. The CMB component is estimated from a simulation drawn from the best-fit 2018 $\Lambda$CDM spectrum, and is dominated by $E$-mode polarization. The synchrotron and thermal dust emission components are based on the sky model, by cross-correlating half-mission sky maps. The dotted lines indicate the sum of the foreground components for three different masks, defined by thresholding the total foreground model evaluated at 70GHz, near the foreground minimum. Three masks are shown, corresponding to 27, 52, and 82% of the sky. The widths of the foreground bands are defined by the two extreme masks. This figure provides a convenient summary of the properties of the polarized sky in the CMB frequencies measured by , and it updates the corresponding polarization panel of figure 51 in @planck2014-a12.
![Polarization amplitude rms as a function of frequency and astrophysical components, evaluated at a smoothing scale of $40\arcm$ FWHM. The green band indicates polarized synchrotron emission, and the red band indicates polarized thermal dust emission. The cyan curve shows the CMB rms for a $\Lambda$CDM model with $\tau=0.05$, and is strongly dominated by $E$-mode polarization. The dashed black lines indicate the sum of foregrounds evaluated over three different masks with $f_{\mathrm{sky}}=0.83$, 0.52, and 0.27. The widths of the synchrotron and thermal dust bands are defined by the largest and smallest sky coverages.[]{data-label="fig:overview_pol"}](figs/overview_pol_v4_compress.pdf){width="\columnwidth"}
Before concluding we briefly summarize some important points regarding limitations and recommended usage of the various component separation products presented in this paper.
- For polarization analysis, the 2018 data products are superior to the 2015 products in all respects, and the new maps entirely supercede the previous release.
- For CMB temperature analysis, we consider the 2015 and 2018 data products as equivalent in terms of overall data quality. Most differences between the two generations of cleaned CMB maps are due to different processing choices, rather than fundamental data quality. For instance, for the 2018 CMB temperature maps are more constrained by data selection issues than the 2015 maps, and as a result the new maps are more contaminated by CO emission. In contrast, for some minor glitches regarding inter-frequency calibration have been resolved in the 2018 maps, and the new maps are therefore somewhat more reliable. For and , only small changes are observed between the two releases. In all cases, the differences are small, typically less than $2\muK$ at high Galactic latitudes with a smoothing scale of $80\arcm$ FWHM.
- For temperature foreground analysis, the 2015 release provides a number of distinct advantages compared to the 2018 release, including no pixelization issues near bright sources in the Galactic plane, more transparent bandpass definitions, and, most importantly, the availability of robust single-bolometer and detector-set maps. For these reasons, we consider the 2015 temperature foreground products from both and to be more reliable than the 2018 products. For the same reason, we anticipate the 2015 temperature data set to continue to play an important role for astrophysical component-separation purposes.
- The noise properties of the observations are complicated both in temperature and polarization, and usage of end-to-end simulations is essential to capture all uncertainties. However, even the best currently available simulations are only accurate to a few percent in power. When employing these simulations for quantitative scientific analysis, it is essential to check that the statistic of choice is not sensitive to this level of uncertainty.
With these caveats in mind, we end our discussion by recalling the original motivation and goal of the mission, namely “…*to measure the fluctuations of the CMB with an accuracy set by fundamental astrophysical limits*” [@planck2005-bluebook]. For temperature, this goal was achieved already with the 2015 release. With the 2018 data release, provides a new state-of-the-art for the field also in terms of polarization.
The Planck Collaboration acknowledges the support of: ESA; CNES, and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MINECO, JA, and RES (Spain); Tekes, AoF, and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); ERC and PRACE (EU). A description of the Planck Collaboration and a list of its members, indicating which technical or scientific activities they have been involved in, can be found at [http://www.cosmos.esa.int/web/planck/planck-collaboration](url). This work has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement numbers 687312, 776282 and 772253.
Commander {#app:commander}
=========
The analysis framework as applied to previous releases is described in detail by @eriksen2004 [@eriksen2008] and @planck2014-a12. This approach implements a standard Bayesian fitting procedure based on Monte Carlo and Gibbs sampling, in which an explicit parametric model including cosmological, astrophysical, and instrumental parameters is fitted to the observations through the posterior distribution.
Due to its very general approach to CMB analysis, it is more appropriate to refer to as a framework rather than as a specific and well-defined algorithm. For instance, the implementation that is employed for the 2017 analysis has been re-written from scratch compared to the 2015 version, and the current version is sometimes referred to as [@seljebotn2017]. The main difference between the old and the new implementations is their different choice of basis functions for the amplitude degrees of freedom, and their different treatment of instrumental beams. While adopted real-space pixels as its fundamental basis set and required uniform angular resolution across frequencies, adopts spherical harmonics as its fundamental basis set and supports different angular resolutions at different frequencies. As a result, the new implementation supports signal reconstruction at the full angular resolution of the observations.
Amplitude sampling algorithm
----------------------------
For full algorithmic specifics regarding the new implementation, see @seljebotn2017; here we review only the main equations. First, we adopt a general signal model on the following form, $$\begin{aligned}
{\vec{s}}_{\nu}(\theta) &= s_{\nu}({\vec{a}}_i, \beta_i, g_{\nu}, {\vec{m}}_{\nu}) \\ & =
g_{\nu} \sum_{i=1}^{N_{\textrm{comp}}} {\tens{F}}_\nu^i(\beta_i){\vec{a}}_{i}\end{aligned}$$ where ${\vec{a}}_i$ is an amplitude vector for component $i$ at a given reference frequency, $\beta_i$ is a general set of spectral parameters for the same component, $g_\nu$ is a multiplicative calibration factor for frequency $\nu$, and ${\vec{m}}_{\nu}$ are monopole and dipole amplitudes. The quantity ${\tens{F}}_\nu^i(\beta_i)$ is a general projection operator that translates from the reference amplitude vector to the basis of the observed data at a given frequency. As such, it accounts for both the choice of basis functions, and for spectral effects such as the frequency dependence of the component in question and unit conversions.
-4mm =
As mentioned above, adopted pixels as its basis set for all diffuse components, requiring identical angular resolution at all frequencies. In this case, the projection operator reduces to the so-called mixing matrix, ${\tens{F}}= {\tens{M}}$, which translates signal amplitudes from a reference frequency to any other observed frequency. In contrast, employs different types of basis functions for different components. For diffuse components, it adopts spherical harmonics, and the projection operator therefore becomes the product of the mixing matrix, which is defined in pixel space, and a spherical harmonics transform, ${\tens{F}}= {\tens{M}}{\tens{Y}}$. For compact objects (radio sources in the current analysis), the map projection is performed through a local real-space [FEBeCoP]{} template per source, ${\tens{B}}_{\mathrm{F}}$, and therefore ${\tens{F}}= {\tens{M}}{\tens{B}}_{\mathrm{F}}$. Finally, fixed template corrections such as monopole, dipole, or zodiacal light corrections, summarized by some overall real-space template matrix per frequency, $T_{\nu}$, are implemented directly as ${\tens{F}}= {\tens{T}}$, and the fitted parameters are thus defined directly as the template amplitude at the respective frequency.
Computationally speaking, by far the most expensive part in is to fit for the linear amplitudes, which corresponds to sampling from the conditional distribution $P({\vec{a}}|{\vec{d}},\ldots)$. As shown by, e.g., @jewell2004 and @wandelt2004, this can be done by solving the so-called Wiener filter equation by conjugate gradients, $$\left({\tens{S}}^{-1} + {\tens{P}}^{\rm T} {\tens{N}}^{-1} {\tens{P}}\right) {\vec{a}}= {\tens{P}}^{\rm T} {\tens{N}}^{-1} {\vec{d}}+ {\tens{P}}^{\rm T} {\tens{N}}^{-1/2} \omega_1 + {\tens{S}}^{-1} \omega_2.
\label{eq:wiener}$$ Here ${\tens{S}}$ is the (prior) covariance matrix of the signal amplitudes, ${\tens{P}}$ is the end-to-end projection operator from amplitude space to data space, ${\tens{N}}$ is the data noise covariance matrix, and $\omega_i$ are Gaussian random vectors with zero mean and unit variance. If the maximum posterior solution is desired rather than a sample drawn from the posterior, one simply sets $\omega_i$ to zero.
The computational expense for solving this equation depends directly on the complexity of the projection operator, ${\tens{P}}$. In most -type analyses, which employ a pixel basis for all components and impose no spatial priors, i.e., ${\tens{S}}=0$, all matrix multiplications are given by diagonal matrices. In contrast, as implemented in , ${\tens{P}}$ involves one spherical harmonic transform per frequency channel, and the computational scaling of the left-hand side increases from $\mathcal{O}(N_{\mathrm{pix}})$ to $\mathcal{O}(N_{\mathrm{pix}}^{3/2})$. Accordingly, the associated CPU time required per sample increases from minutes to tens of hours. This additional cost, however, is very well justified by the new flexibility in terms of beam treatment, which now supports arbitrary resolution at each frequency.
By virtue of being a Gibbs-sampling procedure, requires one sampling step for each parameter under consideration, such as spectral or calibration parameters. However, these parameters are sampled with exactly the same methods in as in , and the details will not be repeated here; see @eriksen2008 and @planck2014-a12.
2018 signal model and priors {#app:commprior}
-----------------------------
As discussed in Sects. \[sec:methods\] and \[sec:inputs\], the maps provided in the 2018 release include only full frequency maps, not individual detector or detector-set maps. This has significant consequences for our ability to reconstruct some important parameters, in particular CO line emission. For this reason, we adopt a simpler signal model in the 2018 analysis than in the 2015 analysis. Explicitly, the basic model considered in the current analysis, as defined in Rayleigh-Jeans temperature units, reads[^11] $$\begin{aligned}
s_{\nu} =\ g_{\nu} \biggl[& {\tens{Y}}{\vec{a}}_{\mathrm{cmb}} \gamma(\nu) \\
&+ {\tens{Y}}{\vec{a}}_{\mathrm{lf}}
\left(\frac{\nu}{\nu_{\mathrm{lf}}}\right)^{\beta_{\mathrm{lf}}(p)} \\
&+ {\tens{Y}}{\vec{a}}_{\mathrm{d}}
\left(\frac{\nu}{\nu_d}\right)^{\beta_{\mathrm{d}}(p)+1}
\left(\frac{e^{h\nu_{\mathrm{d}}/kT_{\mathrm{d}}(p)}-1}{e^{h\nu/kT_{\mathrm{d}}(p)}-1}
\right) \\
&+ {\tens{Y}}{\vec{a}}_{\mathrm{co}} h_{\nu} \\
&+ \sum_{i=1}^{N_{\mathrm{src}}} {\tens{B}}_{\mathrm{F},\nu,i} a_{\mathrm{cs},i}
\left(\frac{\nu}{\nu_{\mathrm{cs}}}\right)^{\alpha_{\mathrm{cs}}(p)}
\\
&+ \sum_{i=1}^{N_{\mathrm{temp}}} {\tens{T}}_{\nu} a_{\mathrm{temp},i}
\biggr],\end{aligned}$$ where the various components correspond to, from top to bottom, CMB, low-frequency/synchrotron emission, thermal dust emission, CO line emission, compact objects, and template corrections. The full model applies only to temperature analysis, since CO line emission, point sources, and template corrections are all omitted from the polarization analysis. For temperature, we refer to the second term as a “low frequency component,” since it includes both synchrotron, free-free, and anomalous microwave emission, while for polarization we refer to it as “synchrotron”, since that is the only component that is significantly detected at low frequencies in polarization.
In the above expression, $\gamma(\nu)$ is the conversion factor between thermodynamic and Rayleigh-Jeans units, $\nu_{\mathrm{lf}} = 30$GHz is the reference frequency for the low-frequency component, $\nu_{\mathrm{d}} = 857$GHz is the thermal dust reference frequency for temperature (353GHz for polarization), $h_{\nu}$ is the CO line ratio between 100 and 217 or 353GHz, respectively, and all other quantities are defined above.
To complete the specification of a model used for Bayesian analysis, we also have to choose priors for the various parameters. Starting with the spectral parameters, we adopt the same types of priors as in previous analyses. Technically speaking, for each parameters these are given as the product of three different priors, each serving a different purpose. First, we impose a uniform prior between two hard limits for numerical reasons; this makes it easier to precompute look-up tables for all unit conversion and bandpass integration quantities. Second, we impose a Jeffreys’ ignorance prior, which effectively normalizes posterior volume effects due to the specific choice of parametrization. Third, and by far most importantly, we adopt Gaussian informative priors with physically motivated means and standard deviations for all spectral parameters. The values of these are listed in Table \[tab:commander\_priors\].
Next, we need to specify spatial priors on the amplitude degrees of freedom. With the new implementation – which models all diffuse components, not just the CMB, in spherical harmonic space – we are now able to impose informative spatial priors on the foregrounds through the signal covariance matrix, ${\tens{S}}$, in Eq. (\[eq:wiener\]). In this paper, we define this matrix in harmonic space in terms of a standard angular power spectrum, ${\cal D}_{\ell}$, per component. In principle, this could be used to enforce physically motivated power spectra for each component, for instance a $\Lambda$CDM spectrum for the CMB, or a power-law spectrum for synchrotron or thermal dust emission. However, in the present analysis, we choose to be minimally constraining, and simply use this new feature to enforce smoothness of the foreground components on small scales. For all components except thermal dust intensity, we implement this by defining a reference prior spectrum given by the shape of a Gaussian smoothing kernel multiplied by an overall amplitude that is larger than the actual sky signal in the high signal-to-noise regime. Thus, the prior takes the form $${\cal D}_{\ell}^{i} = A^i e^{-\ell(\ell+1)\sigma^2},
\label{eq:prior}$$ where $\sigma^2 = \theta_{\mathrm{FWHM}}^2 / (8\ln 2)$, $\theta_{\mathrm{FWHM}}$ is the FWHM of the desired Gaussian smoothing kernel in radians, and $A^i$ is the uniform power spectrum amplitude. This type of prior simply acts as a smooth apodization of the high-$\ell$ spectra, and its main function is to prevent the ringing that would otherwise occur around objects with a sharp cutoff in harmonic space, given by some $\ell_{\mathrm{max}}$. For the special case of thermal dust emission in intensity, we employ a simple cosine apodization between $\ell=5000$ and 6000, in order to retain as much signal as possible. The spatial prior values adopted for the various components are summarized in Table \[tab:commander\_priors\].
Finally, we need to impose priors on the zero levels and dipoles for each map. For HFI zero levels, we adopt the CIB offsets defined in table 6 of @planck2014-a09, while for the LFI we adopt a vanishing monopole at 30GHz. At 44 and 70GHz, we impose no priors on the zero levels, but rather fit them freely, obtaining best-fit values of 17 and 21, respectively. For the HFI channels between 100 and 545GHz, the best-fit zero levels are 12.4, 22.0, 71.0, 431, and 0.346MJy$\textrm{sr}^{-1}$, respectively, while the 857-GHz zero level is fixed. For comparison, the CIB offsets listed in @planck2014-a09 correspond to 12, 21, 68, 451, and 0.35MJy$\textrm{sr}^{-1}$.
We only fit for dipoles in the 70- and 100-GHz channels, a choice determined by inspection of the residual maps resulting from an initial analysis in which no dipoles are fitted. The best-fit -derived amplitudes of the 70- and 100-GHz dipoles are 2.0 and 2.3, respectively.
Sampling compact objects
------------------------
A significant new feature of is its ability to fit compact sources with multi-resolution frequency maps, while at the same time accounting for the full asymmetric beam structure at each frequency. As described above, for a single source this is done through the following parametric model, $$s_{\nu}(p) = {\tens{B}}_{\mathrm{F},\nu} a_{\mathrm{cs}}
\left(\frac{\nu}{\nu_{\mathrm{cs}}}\right)^{\alpha_{\mathrm{cs}}(p)},$$ where ${\tens{B}}_{\mathrm{F},\nu,i}$ is a full [FEBeCoP]{} template evaluated at the pixel closest to the point source in question, ${\vec{a}}$ is the source amplitude in units of mJy, and we assume a simple power-law frequency scaling with a spectral index of $\alpha_{\mathrm{cs}}$ in flux density units (in the current analysis we only consider this component for radio sources, for which a power-law model is a reasonable spectrum). Thus, each source is associated with only two free parameters, the amplitude ${\vec{a}}$ and the spectral index $\alpha$, across all frequencies. This simple two-parameter model, however, is not likely to be adequate for a full fit between 30 and 857GHz for many sources. As a result, when fitting the free parameters, we only include frequencies between 30 and 143GHz in the actual fit; however, the resulting parameters are used to extrapolate to the higher frequencies when fitting other parameters.
Source locations are not identified internally in , but rather defined by external catalogues. Unfortunately, no full-sky, deep, and high-resolution catalogue of radio sources exists for the microwave frequencies, and we therefore construct a hybrid catalogue by combining four different catalogues. First, we include all sources in the AT20G catalogue for declinations below $-15^{\deg}$, for a total of 4499 sources. By virtue of being closest to our frequency range, this catalogue is adopted as an overall reference. Thus, we compute an effective source number density per area of AT20G sources, and adopt this as a threshold density. This threshold is then applied to the GB6 catalogue, including all sources above a flux density defined by requiring that the area number density is the same as for AT20G. This results in 5814 GB6 sources. Next, for sky regions not covered by either AT20G or GB6, we employ the same algorithm to the NVSS catalogue, resulting in 1527 NVSS sources. Finally, we also include all sources found in the PCCS2 catalogue, except for excluding duplicates in the already considered catalogues; this results in 352 unique sources. Thus, at this stage, the full sky has been populated by sources with a nearly uniform number density, for a total of 12192 sources.
It is important to note that the catalogue positions defined above are only used as candidates for source positions. Including a non-existing source will not bias any other parameter, since its relevant parameters are fitted jointly with all other parameters; the only detrimental effect of including too many sources is a slight increase in the overall noise level.
Figure \[fig:comm\_sources\] shows an enlargement of the final compact source map for 30 and 100GHz, generated as described above. The plot shows a $10^\circ \times 10^\circ$ region centred on the South Galactic Pole, for which the scanning strategy provides relatively poor cross-linking. As a result, the asymmetric properties of the 30-GHz beams are clearly visible. Another feature seen in these plots is the large number of overlapping sources in the 30-GHz map. If we included only this single frequency while fitting the spectral properties of the sources, there would be significant degeneracies between such overlapping sources. However, when we include higher frequencies, for which the beams are smaller and neighboring sources overlap less, these degeneracies are effectively broken.
![Enlargement of the compact source map fitted with using real-space spatial [FEBeCoP]{} templates and a power-law spectral model. Shown here is a $10^\circ \times 10^\circ$ region centreed on the South Galactic Pole (SGP), and the top and bottom panels showing the effective point source maps at 30 and 100GHz, respectively. Note the significantly asymmetric beam structures in the 30-GHz map.[]{data-label="fig:comm_sources"}](figs/radio_030_rc6_n1024_zoom_compress.pdf "fig:"){width="1\linewidth"}\
![Enlargement of the compact source map fitted with using real-space spatial [FEBeCoP]{} templates and a power-law spectral model. Shown here is a $10^\circ \times 10^\circ$ region centreed on the South Galactic Pole (SGP), and the top and bottom panels showing the effective point source maps at 30 and 100GHz, respectively. Note the significantly asymmetric beam structures in the 30-GHz map.[]{data-label="fig:comm_sources"}](figs/radio_100_rc6_n2048_zoom_compress.pdf "fig:"){width="1\linewidth"}
Confidence masks {#app:comm_mask}
----------------
For , we establish the following prescription for defining a temperature confidence mask. First, the base temperature mask is defined by smoothing the $\chi^2$ map with a $30\arcm$ FWHM Gaussian beam, suppressing instrumental noise fluctuations, and then thresholding the smoothed map at a value of 50, which corresponds to a roughly $4\sigma$ confidence level at high Galactic latitudes. This mask removes any pixel for which the model obviously breaks down in terms of total $\chi^2$. However, based on frequency residual maps, one does observe residuals corresponding to specific components that are not easily picked up by the total $\chi^2$. To capture these, we augment the base mask with three specifically targeted masks. First, we remove any pixels brighter than $10\,\textrm{mK}$, to eliminate particularly bright radio sources. Second, we exclude by hand the Virgo and Coma clusters and the Crab Nebula. Third, noting that CO emission represents a particularly difficult problem with the current data set, we smooth the 2018 CO emission map shown in Fig. \[fig:comm\_fg\_temp\] with a $30\arcm$ FWHM beam, and exclude any pixels for which the CO amplitude is larger than $50\muK_{\mathrm{RJ}}$ at 100GHz.
The polarization confidence mask is constructed in a similar way, with a few specific modifications. First, a base mask is produced by smoothing and thresholding the $\chi^2$ map shown in Fig. \[fig:comm\_pol\_residuals\]. Second, we remove all pixels for which the polarized thermal dust amplitude smoothed to $3^{\circ}$ FWHM is larger than $20\muK_{\mathrm{RJ}}$ at 353GHz (see Fig. \[fig:comm\_pol\_dust\]); this mask excludes pixels for which large values are observed in the frequency residual maps shown in Fig. \[fig:comm\_pol\_residuals\], but which are not robustly picked up by the $\chi^2$ values. Third, we remove all pixels corresponding to the cosmic ray contaminated ring discussed in Sect. \[sec:fullmission\]. Fourth, we remove particularly bright point sources based on the PCCS2 source catalogue. The resulting masks for both temperature and polarization are shown in Fig. \[fig:dx12\_masks\_commander\].
![ masks in temperature (top) and polarization (bottom).[]{data-label="fig:dx12_masks_commander"}](figs/dx12_v3_commander_mask_int_005a_0512_compress.pdf "fig:"){width="\columnwidth"} ![ masks in temperature (top) and polarization (bottom).[]{data-label="fig:dx12_masks_commander"}](figs/dx12_v3_commander_mask_pol_005a_0512_compress.pdf "fig:"){width="\columnwidth"}
![*Top:* CMB temperature map used for the low-$\ell$ temperature likelihood analysis, smoothed to $60\arcm$ FWHM resolution. The grey region indicates the mask adopted for the likelihood analysis, which retains 86% of the sky. *Bottom:* Difference between the low-$\ell$ likelihood and full-resolution CMB temperature maps, smoothed to $60\arcm$ FWHM resolution.[]{data-label="fig:comm_lowl_vs_fullres"}](figs/comm_likelihood_60arc_compress.pdf "fig:"){width="\columnwidth"} ![*Top:* CMB temperature map used for the low-$\ell$ temperature likelihood analysis, smoothed to $60\arcm$ FWHM resolution. The grey region indicates the mask adopted for the likelihood analysis, which retains 86% of the sky. *Bottom:* Difference between the low-$\ell$ likelihood and full-resolution CMB temperature maps, smoothed to $60\arcm$ FWHM resolution.[]{data-label="fig:comm_lowl_vs_fullres"}](figs/diff_comm_likelihood_fullres_60arc_compress.pdf "fig:"){width="\columnwidth"}
![*Top:* Low-$\ell$ temperature power spectra derived from the low-$\ell$ likelihood map (red) and from the full-resolution map (blue), both evaluated over the low-$\ell$ likelihood mask shown in Fig. \[fig:comm\_lowl\_vs\_fullres\]. *Bottom:* Difference between the two spectra shown in the top panel. []{data-label="fig:comm_cls_lowl_vs_fullres"}](figs/compsep_comm_like_vs_highres_compress.pdf){width="\columnwidth"}
Comparison between low-$\ell$ likelihood and full-resolution maps
-------------------------------------------------------------------
As described in @planck2016-l05, the algorithm is used to generate the low-$\ell$ temperature sky map that feeds the 2018 CMB likelihood, as it was in previous releases. The set-up adopted for that analysis is, however, somewhat different than the one adopted for the main analysis presented in this paper, primarily due to the different angular scales in question. Specifically, since the likelihood map is only used for large angular scales, covering primarily only $\ell\le30$, the full analysis is carried out with , and all input frequency maps are smoothed to a common angular resolution of $40\arcm$ FWHM, similar to the 2015 processing. Finally, the low-$\ell$ analysis internally estimates the CMB power spectrum as one of the parameters in the Bayesian parametric model, and the corresponding Gaussian constrained realization samples [@eriksen2008] provide the necessary inputs for the Blackwell-Rao likelihood estimator employed by the temperature-only likelihood [@chu2005]. For the combined temperature and polarization likelihood, which is map-based rather than power-spectrum-based, a single constrained-realization sample is adopted as the low-$\ell$ likelihood temperature component. We have verified that the choice of the particular sample used has no significant effect on the actual power spectrum outside the analysis mask.
The top panel of Fig. \[fig:comm\_lowl\_vs\_fullres\] shows the low-$\ell$ likelihood temperature map with the corresponding likelihood mask marked in grey. The bottom panel shows the difference map with respect to the full-resolution map, after the latter is smoothed to $40\arcm$ FWHM resolution. Over most of the sky, the absolute difference between the two maps is $\lesssim2\muK$, increasing to $5\muK$ near the Galactic plane. A few bright spots exhibit differences at the 10-$\mu$K level. These differences are dominated by thermal dust emission (see, e.g., Fig. \[fig:comm\_fg\_temp\]), and are in effect due to the different angular resolutions adopted for the two analyses. Since the likelihood analysis is performed at an angular resolution of $40\arcm$ FWHM, the thermal dust spectral index and temperature are also estimated at an angular resolution of $40\arcm$ FWHM. In contrast, the high-resolution analysis estimates the thermal dust spectral index at $10\arcm$ FWHM, and the corresponding temperature at $5\arcm$ resolution. As a consequence, the assumed spectral priors have a relatively larger impact in the high-resolution analysis than in the low-$\ell$ analysis.
Nevertheless, these differences are small in terms of absolute numbers, and have a negligible impact on the derived angular power spectrum. This is explicitly shown in Fig. \[fig:comm\_cls\_lowl\_vs\_fullres\], in which the top panel shows the individual spectra computed from each of the two maps, and the bottom panel shows their difference. Overall, the absolute differences are smaller multipole-by-multipole than $10\muK^2$, corresponding to $\lesssim1\,\%$ in absolute power and $\lesssim0.05\,\sigma$ in terms of cosmic variance. There is also no overall trend in the difference spectrum that might pull systematically on cosmological parameters. The two maps are statistically equivalent in terms of temperature power spectra.
Needlet Internal Linear Combination {#app:nilc}
===================================
The goal of is to estimate the CMB from multi-frequency observations while minimizing the contamination from Galactic and extragalactic foregrounds, and instrumental noise. The method makes a linear combination of the data from the input maps with minimum variance on a frame of spherical wavelets called needlets [@narcowich06localizedtight]. Due to their unique properties, needlets enable localized filtering in both pixel space and harmonic space. Localization in pixel space allows the weights of the linear combination to adapt to local conditions of foreground contamination and noise, whereas localization in harmonic space allows the method to favour foreground rejection on large scales and noise rejection on small scales. Needlets permit the weights to vary smoothly on large scales and rapidly on small scales, which is not possible by cutting the sky into zones prior to processing .
The pipeline [@2012MNRAS.419.1163B; @2013MNRAS.435...18B] is applicable to scalar fields on the sphere, hence we work separately on maps of temperature and the $E$ and $B$ modes of polarization. The decomposition of input polarization maps into $E$ and $B$ is performed on the full sky. At the end of the processing, the CMB $Q$ and $U$ maps are reconstructed from the $E$ and $B$ maps.
Prior to applying , all of the input maps are convolved or deconvolved in harmonic space to a common resolution corresponding to a Gaussian beam of 5 FWHM. Each map is then decomposed into a set of needlet coefficients. For each scale $j$, needlet coefficients of a given map are stored in the form of a single map. The filters $h^{j}_{l}$ used to compute filtered maps are given by $$\begin{aligned}
h^{j}_{l} = \left\{
\begin{array}{rl}
\cos\left[\left(\frac{\ell^{j}_{\mathrm{peak}}-\ell}{\ell^{j}_{\mathrm{peak}}-\ell^{j}_{\mathrm{min}}}\right)
\frac{\pi}{2}\right]& \mathrm{for }\, \ell^{j}_{\mathrm{min}} \le \ell < \ell^{j}_{\mathrm{peak}},\\
\\
1\hspace{0.5in} & \mathrm{for }\, \ell = \ell_{\mathrm{peak}},\\
\\
\cos\left[\left(\frac{\ell-\ell^{j}_{\mathrm{peak}}}{\ell^{j}_{\mathrm{max}}-\ell^{j}_{\mathrm{peak}}}\right)
\frac{\pi}{2}\right]& \mathrm{for }\, \ell^{j}_{\mathrm{peak}} < \ell \le \ell^{j}_{\mathrm{max}}.
\end{array} \right. \end{aligned}$$ For each scale $j$, the filter has compact support between the multipoles $\ell^{j}_{\mathrm{min}}$ and $\ell^{j}_{\mathrm{max}}$ with a peak at $\ell^{j}_{\mathrm{peak}}$ (see Table \[tab:needlet-bands\] and Figure \[fig:needlet-bands\]). The needlet coefficients are computed from these filtered maps on pixels with $\nside$ equal to the smallest power of $2$ larger than $\ell^{j}_{\mathrm{max}}/2$.
-4mm =
![Needlet bands used in the analysis. The solid black line shows the normalization of the needlet bands, i,e., the total filter applied to the original map after needlet decomposition and synthesis of the output map from needlet coefficients.[]{data-label="fig:needlet-bands"}](figs/nilc_windows_v2_compress.pdf){width="\linewidth"}
Due to the deconvolution of sky maps to 5 resolution, noise levels in those maps are boosted. Hence, for each sky map, we consider only those multipoles for which the ratio of the beams of final CMB map and the sky maps are less than $100$. For each needlet scale, only those sky maps are chosen for the reconstruction of CMB whose band limits obtained in this way are less than the band limit of the corresponding needlet scale.
In order to improve the measurement of CMB temperature anisotropy near the Galactic plane, we have used a very small preprocessing mask with a sky fraction of 99.8%. The procedure to generate the preprocessing mask is as follows. First we implement the pipeline on the full-mission data sets. Then we identify the pixels where the CMB is more than $500\,\mu \mathrm{K}_{\mathrm{CMB}}$, and assign a value of 0 to all the pixels that are within $6\arcm$ and a value of 1 to other pixels. We implement this preprocessing mask on the sky maps in the next run of the pipeline. Prior to implementing the pipeline on the sky maps, the mask regions are filled using the Planck Sky Model (PSM), which uses an increasing number of neighbouring pixels to fill regions deeper in the hole. At each iteration (i.e., one row of pixels into the hole), the procedure uses pixels at up to twice the pixel size times the number of iteration.
Estimates of the covariance matrices of needlet coefficients for each scale are computed by smoothing all possible products of needlet coefficients with Gaussian beams. In this way, an estimate of needlet covariances at each point is obtained as a local, weighted average of needlet coefficient products. The FWHMs of the Gaussian windows used for the analysis are chosen to support the computation of statistics; 4225 samples or more samples are averaged. Choosing a smaller FWHM results in excessive error in the covariance estimates, and hence excessive bias. Choosing a larger FWHM results in less localization, and hence some loss of efficiency of the needlet approach.
A patch of angular radius $\theta$ and area $2\pi(1-\cos(\theta))$ contains $N/(4\pi) \times 2\pi\{1-\cos(\theta)\}$ modes. If we wish to have $M$ modes in that patch, we simply solve for the corresponding $\theta$. We chose FWHM$ = 2 \times \theta$ for the Gaussian beam that we use to smooth the covariance matrix. Hence in order to determine the covariance matrix at a particular point, we have given more weight to those pixels that are close to that point, and less weight to those pixels that are far away. However, this strategy is not optimal for the largest scales.
Figure \[fig:needlet-weights\] shows that the $70$, $100$, $143$, and $217$ GHz channels have contributed most to the final reconstruction of the NILC CMB map. However, other channels are also important because these channels are tracers of the foreground signals, and help us to find optimal weights for the final solution.
![Full-sky average of needlet weights for different frequency channels and needlet bands. From top to bottom, the panels show results for temperature, $E$ modes, and $B$ modes.[]{data-label="fig:needlet-weights"}](figs/nilc_weights_i.pdf "fig:"){width="\columnwidth"} ![Full-sky average of needlet weights for different frequency channels and needlet bands. From top to bottom, the panels show results for temperature, $E$ modes, and $B$ modes.[]{data-label="fig:needlet-weights"}](figs/nilc_weights_e.pdf "fig:"){width="\columnwidth"} ![Full-sky average of needlet weights for different frequency channels and needlet bands. From top to bottom, the panels show results for temperature, $E$ modes, and $B$ modes.[]{data-label="fig:needlet-weights"}](figs/nilc_weights_b.pdf "fig:"){width="\columnwidth"}
![Difference of angular power spectra obtained with and without considering the correction to calibration coefficients estimated by .[]{data-label="fig:nilc-calib"}](figs/nilc_tt_calib_delta_bin20_compress.pdf){width="\linewidth"}
Calibration errors are a serious issue for precise measurement of the CMB, as they conspire with the ILC filter to cancel out the CMB. This effect is particularly strong in the high signal-to-noise ratio regime, on large scales in particular. We investigate the impact of this calibration bias by redoing the analysis with slightly modified calibration coefficients, and computing the difference between the CMB spectra estimated in the two cases. We adopt the calibration coefficients determined by in Appendix \[app:smica\]. Figure \[fig:nilc-calib\] shows the impact of calibration on the angular power spectrum of the CMB temperature. Comparison of angular power spectra for two data splits shows that the impact is less than 0.5%.
SEVEM {#app:sevem}
=====
The method produces cleaned CMB maps at different frequencies by subtracting a linear combination of templates constructed internally from the data. In particular, the templates are typically obtained as the subtraction of two close frequency channels, filtered to the same resolution to remove the CMB signal. The cleaning is achieved simply by subtracting a linear combination of the templates $t_j(\vec{x})$ from the data, with coefficients $\alpha_j$ obtained by minimizing the variance outside a given mask: $$T_{\rm c}(\vec{x},\nu)=d(\vec{x},\nu)- \sum_{j=1}^{n_{\rm t}} \alpha_j t_j(\vec{x}).
\label{eq:sevem_basic_formula}$$ Here $n_{\rm }t$ is the number of templates used, while $T_{\rm c}(\vec{x},\nu)$ and $d(\vec{x},\nu)$ correspond to the cleaned and raw maps at frequency $\nu$, respectively. The same expression applies for $T$, $Q$, or $U$. Note that we estimate the linear coefficients $\alpha_j$ independently for $Q$ and $U$ maps, following what was done for the previous release[^12].
The cleaned frequency maps are then combined in harmonic space, taking into account the noise level, resolution, and (optionally) an estimate of the foreground or systematic residuals of each cleaned channel, to produce a final CMB map at the required resolution.
Implementation for temperature
------------------------------
For temperature, we have followed the same procedure as for the 2015 release (see @planck2014-a11 for further details). As before, we clean the 100-, 143-, and 217-GHz frequency channels with four templates, three of them constructed as the difference between two nearby channels ($30-44$, $44-70$, $545-353$), such that the first channel is convolved with the beam of the second one and vice versa, and a fourth template given by the 857-GHz channel, convolved with the 545-GHz beam. The cleaned frequency maps have the same resolution as the corresponding original raw data map. To reduce the contamination from point sources in the templates, we follow the same approach as in the previous release. First, point sources are detected in each frequency map using the Mexican-Hat-Wavelet algorithm [@lopezcaniego2006; @planck2014-a35]. The upper part of Table \[tab:ps\_sevem\] gives the number of point sources detected in intensity and polarization for all the frequency channels over the full-sky, at Galactic latitudes $\left| b\right| > 20\deg $. We then inpaint the holes corresponding to the positions of those point sources in the frequency maps (at their original resolution) involved in the construction of the templates. Note that the size of the hole depends on the amplitude of the detected source and the resolution of the considered channel. The filling is done with a simple diffusive inpainting scheme, which replaces one pixel with the mean value of the neighbouring pixels in an iterative way. To avoid possible inconsistencies when performing the subtraction of two maps to construct a template, the diffusive inpainting is performed for all of the sources detected in both channels. For instance, when constructing the ($30-44$)GHz template, all sources detected at 30 and 44GHz are inpainted in the two frequency maps before subtraction[^13].
-4mm =
In addition, for this release, we also provide a cleaned CMB map for the 70-GHz channel. This map is constructed at its original resolution and $N_{\rm side}=1024$, and has been cleaned with two templates, one constructed as the 30-GHz channel (convolved with the 44-GHz beam) minus the 44-GHz one (convolved with the 30-GHz beam), and a second template obtained as the difference between the 353 and 143 channels, constructed at a resolution equal to that of the 70-GHz channel. This second template has been chosen to trace the emission of the thermal dust, but avoding, as far as possible, the CO contamination (which is mostly present in the 100- and 217-GHz maps). Point source emission in the templates has also been reduced with the inpainting mechanism already described.
-3mm =
The coefficients of the linear combination used for cleaning the frequency maps are given in Table \[table:sevem\_coef\_T\]. They have been calculated by minimizing the variance of the corresponding cleaned map outside a mask that excludes the brightest 1% of the sky and all the point sources detected in intensity. The cleaning procedure introduces a certain level of correlation between the 100-, 143-, and 217-GHz cleaned frequency maps, due to the use of the same templates, but one frequency map is not used to clean the others. The cleaned 70-GHz channel is, however, more correlated, since it is part of one of the templates used to clean the higher frequency channels. Moreover, the 143-GHz map is also used to clean the 70-GHz channel. Therefore, one should bear in mind these correlations when carrying out analyses with the cleaned single frequency maps. A possible way to reduce the correlations introduced by the cleaning process would be, when possible, to use pairs of cleaned frequency maps constructed with different splits, although this would be at the expense of decreasing the signal-to-noise ratio (e.g., to work with the cleaned 143-GHz even-ring and with the 217-GHz odd-ring maps, since the templates are constructed with the corresponding split).
Following the same approach as in the previous release, after the frequency maps are cleaned, they are inpainted, in a first step, at the positions of the point sources identified in the corresponding raw maps. In a second step, the Mexican-Hat-Wavelet algorithm is again run on the cleaned frequency maps, and the newly detected sources (see lower part of Table \[tab:ps\_sevem\]) are further inpainted. The combined area inpainted outside the confidence mask for the 143- and 217-GHz channels (those used to construct the final CMB map) corresponds to around $0.4\%$ of the sky, while it is fully covered by the common confidence mask. Note that the same inpainting strategy is applied to the simulations processed through the pipeline, to make sure that any possible effect introduced by this procedure is statistically taken into account. Finally, the monopole and dipole are removed from the full-sky cleaned maps (note that this is different from the previous release, where monopole and dipole were removed outside the confidence mask). The cleaned intensity maps for the 70-, 100-, 143-, and 217-GHz channels are shown in Fig. \[fig:sevem\_freqmaps\].
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The final CMB map is constructed by combining the cleaned 143- and 217-GHz maps in harmonic space.[^14] In particular, the maps are weighted at each multipole, taking into account the noise level and resolution of the maps, as well as a rough estimation of the expected foreground residuals. This estimation has been updated with respect to the previous release by using the FFP8 simulations. The total weights are shown in Fig. \[fig:weights\_sevem\_I\]. The resolution of the combined map corresponds to a Gaussian beam of 5 FWHM and resolution $\nside = 2048$, with maximum multipole $\ell_{\mathrm{max}} = 4000$. A monopole and a dipole are also removed from the full-sky map.
![Weights used to combine the cleaned single-frequency maps into the final CMB map for temperature, corresponding to 143GHz (blue line) and to 217GHz (green line). The weights do not sum to unity because they include the effect of deconvolving the beams of the frequency maps and convolving with the 5 Gaussian beam of the final map. []{data-label="fig:weights_sevem_I"}](figs/SEVEM_weights_I_compress.pdf){width="\columnwidth"}
Implementation for polarization
-------------------------------
A similar procedure is applied independently to the frequency maps of the Stokes $Q$ and $U$ parameters to obtain cleaned polarization CMB maps, which are aftewards combined in harmonic space to produce the final maps. Given the narrower frequency coverage available for polarization, a different choice of templates needs to be defined in this case. In the previous release, only two cleaned channels (100 and 143GHz) were combined to produce the final polarization map. However, due to the significant improvement of the data in polarization for the current release, we are now able to clean the 217-GHz channel and to include this map in the final combination. This produces a significant improvement in the signal-to-noise ratio of the cleaned CMB polarization maps with respect to the previous version. In addition, in the updated pipeline, the cleaned maps are produced at full resolution ($\nside=2048$ instead of $\nside=1024$ for the 100-, 143-, and 217-GHz channels, as well as the combined map). As for the previous release, a cleaned 70-GHz map is also provided at its native resolution. To reduce point source contamination, inpainting similar to that in the previous release is performed.
The first step of the pipeline is to inpaint the positions of the sources detected in those channels that will be used to construct templates. These point sources are detected using a non-blind approach, among the intensity candidates, using the filtered fusion technique [@argueso2009]. The upper part of Table \[tab:ps\_sevem\] shows the number of sources detected in polarization in each of the frequency channels. The size of the holes to be inpainted takes into account both the amplitude of the source and the beam of the channel. As in the intensity case, when performing the subtraction of two maps to construct a template, the diffuse inpainting is performed for all of the sources detected in both channels. Note that the inpainting is always done at the native resolution of the channel and independently for $Q$ and $U$ maps.
Once the maps have been inpainted, each template is constructed as the subtraction of two frequency channels processed to a common resolution. Given the smaller number of channels in polarization, the maps to be cleaned are also used to construct templates. In this sense, the cleaned maps at different frequencies are, in general, less independent than in the intensity case (the exception is the 70-GHz channel, which is not used as part of the templates for polarization). Six templates (one of them at two different resolutions) are generated to produce cleaned maps at 70, 100, 143, and 217GHz. In particular, to trace the synchrotron emission, the ($30-44$)GHz template is constructed, where the 30-GHz map is smoothed with the 44-GHz beam and vice versa. To trace the thermal dust, templates are produced at ($217-143$), ($217-100$), and ($143-100$) with 1 resolution (this smoothing is included in order to increase the signal-to-noise ratio of the template), and at ($353-217$) and ($353-143$) with 10 resolution. In addition, this last template is also constructed at the resolution of the 70-GHz beam, in order to clean that channel. The produced templates are then subtracted from the (non-inpainted) raw data at their native resolution. Table \[table:sevem\_coef\_P\] shows the list of templates used to clean each map, as well as the corresponding coefficients of the linear combination. These coefficients have been obtained by minimizing the variance of each cleaned map outside a mask excluding the brightest 3% of the sky and all the point sources detected in polarization. Note that to clean the 100- and 143-GHz maps, the same combination of templates as in the previous release is used, although now templates and cleaned maps are produced at $\nside=2048$.
Once the frequency maps are cleaned, inpainting at the position of the point sources detected at each of those channels is carried out. Moreover, once these cleaned inpainted maps are ready, the non-blind point source detection algorithm is run again on them and additional point sources detected (see lower part of Table \[tab:ps\_sevem\]). These newly identified sources are also inpainted. This second iteration of the algorithm was performed for intensity in the previous release but not for polarization; in this version it is done for both cases. The joint area inpainted outside the mask in the three cleaned channels used to produce the combined maps corresponds to around a $0.04\%$ of the sky, and is fully covered by the common confidence mask. As for intensity, exactly the same inpainting procedure is applied to the simulations processed through the pipeline, to account for any possible effects introduced by this step. The cleaned $Q$ and $U$ maps for the 70-, 100-, 143-, and 217-GHz channels are shown in Fig. \[fig:sevem\_freqmaps\]. The maps have been smoothed with a Gaussian beam with 80 FWHM resolution to allow for better visualization.
-3mm =
The last step is to combine the cleaned single-frequency maps in order to produce the final $Q$ and $U$ cleaned CMB maps. This is done by combining in harmonic space the cleaned 100, 143, and 217-GHz maps. The weights take into account the noise of each channel and its resolution. In addition, recognizing the fact that the 217-GHz channel is likely to be somewhat more susceptible to large-scale systematic residuals than the other two channels, we also introduce a relative down-weighting of the 217-GHz channel on the largest scales. This can be seen in Fig. \[fig:weights\_sevem\_P\], where the harmonic weights are given for the 100- (red), 143- (blue), and 217-GHz (green) channels. The same weights are applied for $E$ and $B$. The resolution of the combined map corresponds to a Gaussian beam of FWHM 5 and resolution $\nside = 2048$, with a maximum multipole $\ell_{\mathrm{max}} = 3000$.
![Weights used to combine the cleaned single-frequency maps into the final CMB maps for polarization. The different lines correspond to 100- (red), 143- (blue), and 217-GHz (green) channels. The weights do not sum to unity because they include the effect of deconvolution by the beams of the frequency maps and convolving with the 5 Gaussian beam of the final map. []{data-label="fig:weights_sevem_P"}](figs/SEVEM_weights_E_compress.pdf){width="\columnwidth"}
Masks {#sec:sevem_mask}
-----
In temperature, the confidence mask is generated following a similar procedure to that of the previous release. Specifically, we define the mask by thresholding maps constructed as the difference between two different CMB reconstructions. As in 2015, we construct these differences at $\nside=256$, with resolution given by a Gaussian beam with $\hbox{FWHM} =30\arcm$. In particular, three combinations are considered: the cleaned ($217-143$)GHz and ($143-100$)GHz maps and the difference between two cleaned, combined CMB maps, whose linear coefficients have been obtained by minimizing the variance outside two different masks. From each of these maps, one mask is constructed by removing the brightest pixels (and its direct neighbours) down to a certain threshold. The three masks are multiplied to produce the final confidence mask, which is then smoothed with a Gaussian beam of 1$^\circ$ to avoid sharp edges and upgraded to full resolution. The thresholds that define the masks are chosen by looking at the amplitude of the extrema and the dispersion of the cleaned 100-, 143-, and 217-GHz channels and the combined map after applying the considered mask, trying to find a compromise between reducing the values of these quantities while keeping a reasonable sky fraction. In particular, thresholds removing between 8 and 10% of the sky were found to be adequate for the differences considered. In addition, a small region near the Galactic plane with a relatively high contamination, but that was not captured with these values of the thresholds, was manually masked by applying a circle of 03 radius. This removed around 350 additional pixels, without modification of the thresholds, which would lead to a larger reduction of the area allowed by the mask. The final confidence mask in intensity leaves a suitable sky fraction of 83.8%, and is shown in the top panel of Fig. \[fig:dx12\_masks\_sevem\].
![ masks in temperature (top) and polarization (bottom).[]{data-label="fig:dx12_masks_sevem"}](figs/Figure_SEVEM_int_mask_compress.pdf "fig:"){width="\columnwidth"} ![ masks in temperature (top) and polarization (bottom).[]{data-label="fig:dx12_masks_sevem"}](figs/Figure_SEVEM_pol_mask_compress.pdf "fig:"){width="\columnwidth"}
![ masks in temperature (top) and polarization (bottom) for inpainted point sources.[]{data-label="fig:dx12_masks_inpainting_sevem"}](figs/Figure_SEVEM_combined_inpainting_mask_temp_compress.pdf "fig:"){width="\columnwidth"} ![ masks in temperature (top) and polarization (bottom) for inpainted point sources.[]{data-label="fig:dx12_masks_inpainting_sevem"}](figs/Figure_SEVEM_combined_inpainting_mask_pol_compress.pdf "fig:"){width="\columnwidth"}
In polarization, given the lower signal-to-noise ratio of the reconstructed CMB maps, a different approach from that of intensity needs to be considered to identify the reliable regions of the sky. Several aspects of the approach to construct the polarization confidence mask have been modified with respect to the previous release, and the new method is described here in detail. In particular, we have defined the confidence mask as the product of two individual masks: one specific mask based on the achieved CMB reconstruction, and a second one customized to avoid the regions more contaminated by thermal dust.
For the specific mask, the first step is to downgrade the CMB reconstructed maps ($Q$ and $U$) to a resolution equivalent to a Gaussian beam with $\hbox{FWHM} = 90\arcm$ and $\nside=128$. From these maps, we estimate locally the rms of $P$ (i.e., $\sqrt{Q^2+U^2}$) at each position by caculating the rms of the pixels included in a circle with a given radius centred on the considered pixel. We then estimate the expected rms of $P$ for a map containing only CMB and noise. For the noise, this is obtained by estimating this quantity locally for the odd-even half-difference map, processed through the pipeline, at the resolution being considered, using the same procedure as for the cleaned maps. For the CMB, we simply obtained the rms of $P$, averaging over simulations. Since the CMB and noise are independent, their rms values are added quadratically. The ratio between the rms of the cleaned maps over that expected for a CMB-plus-noise map is then constructed. Pixels with larger ratios are expected to be more contaminated; the specific mask is defined by those pixels above a given threshold. This mask is then smoothed with a Gaussian beam of $\hbox{FWHM}=90\arcm$ to avoid sharp boundaries, and upgraded to $\nside=2048$. We explored several values for the radius of the circle (to locally estimate the rms) and for the amplitude of the threshold, finding that a value of 15 pixels (at $\nside=128$) for the radius and a threshold of 1.5 produced good results.
To construct the dust mask, we use the raw 353-GHz channel, smoothed at a resolution of 90 and $\nside=128$. The rms of $P$ is obtained at each pixel as explained above, and a fixed fraction of pixels with the largest rms values is included in the mask. This mask is again smoothed with a Gaussian beam of 90 and upgraded to $\nside=2048$. To construct this mask, we have chosen a radius for estimating the rms of four pixels and excluded 15% of the sky. Finally, the specific mask and the dust mask are multiplied together, passing 80.3% of the sky. The confidence mask in polarization is shown in the bottom panel of Fig. \[fig:dx12\_masks\_sevem\].
We conclude with some additional comments about the best way to deal with inpainted pixels. The most conservative approach is to explicitly exclude all of the inpainted areas from the analysis. This mplies the inclusion of a large number of holes in the confidence mask, which can be damaging for certain analyses, especially those performed in harmonic space. The diffusive inpainting strategy considered above seems to effectively reduce the emission from detected point sources while, at the same time, not introducing evident artefacts in the cleaned maps (recall that we are filling small holes, corresponding to scales where the background is usually smooth). Therefore, we have only masked those inpainted pixels which are directly excluded by the general algorithm used to construct the confidence mask. For intensity, this leaves a joint inpainted area outside the mask in the two cleaned channels (143 and 217GHz) used to construct the final CMB map of around 0.4% of the sky. For polarization, the corresponding joint area (from the cleaned 100-, 143-, and 217-GHz channels) also covers around 0.04% of the sky. Moreover, for both intensity and polarization, the exact same procedure is applied to the simulations processed through the pipeline, to ensure that any unexpected spurious effects are statistically taken into account. We believe that this is a good way to proceed in order to find a compromise between reducing point-source contamination in the cleaned maps and providing a well-behaved confidence mask for CMB analysis. This is the same approach used in the previous release. Nonetheless, for certain types of analysis, as for example the local study of compact objects, it may be necessary to discard, or at least to be aware of, the inpainted regions. For these cases, we also provide masks of the pixels inpainted in each of the cleaned frequencies, as well as the joint mask for those channels that are used to construct the final CMB maps. The joint masks of inpainted pixels are given in Fig. \[fig:dx12\_masks\_inpainting\_sevem\] for intensity (top) and polarization (bottom). Note that additional inpainting is also performed during the template construction, but those positions are not included in these masks since those pixels are not directly inpainted on the cleaned maps. Finally, we point out that the masks for inpainted point sources given in Fig. \[fig:dx12\_masks\_inpainting\_sevem\] have been included in the confidence common masks (Fig. \[fig:commonmask\]), to reduce possible point source contamination in all the CMB maps. If it is desired to carry out an analysis of the CMB maps without explicitly including point source holes in the mask, the confidence masks should be considered.
Spectral Matching Independent Component Analysis () {#app:smica}
===================================================
The general operation of (Spectral Matching Independent Component Analysis; [@delabrouille2003; @cardoso2008]) and the main changes with respect to the 2015 release are summarized in Sect. \[sec:smica\]. In this appendix, we provide additional implementation details. There are several masking and pre-processing operations whose specifics vary depending on the target map (CMB or foregrounds, temperature or polarization), but the general methodology is the same, following these steps:
1. Preprocessing of the input maps by point source subtraction and masking/inpainting. This step also includes additional masking (Galactic plane, etc.).
2. Estimation of the spectral statistics $\widehat\bC_\ell$ via Eq. (\[eq:smica:scm\]) from the spherical harmonic coefficients computed from the preprocessed maps, possibly with some additional masking to remove particularly bright objects.
3. Fitting of a model to beam-corrected $\widehat\bC_\ell$, from which the harmonic weights $\vec{w}_\ell$ are computed.
4. Computation of the spherical harmonic coefficients from the preprocessed maps and linear combination as per Eq. (\[eq:smica:ilc\]), to synthesize a map with a specified effective Gaussian beam.
5. Determination of a “confidence mask”.
The specifics of the production of each map are given below.
Temperature analysis {#app:smica:temp}
--------------------
The 2018 temperature map is a hybrid of two complementary CMB renderings, namely $X_\textrm{high}$, which includes only HFI observations, and is specialized for high Galactic latitudes, and intermediate and small angular scales, and $X_\textrm{full}$, which includes all channels, and provides us with additional content. The final temperature map is then constructed as a weighted sum of these two maps, following Eq. \[eq:smica:merge\]. The two sky areas to be hybridized are defined by a smooth mask shown at Fig. (\[fig:smica:tmask\]). In polarization, we do not resort to such a hybrid scheme.
#### Recalibration
As in previous releases, a preliminary fit (calibration run) is conducted, with calibration coefficients left unconstrained at 100 and 217GHz. This fit involves only HFI channels, is limited to the first peak ($30\leq\ell\leq 300$), and involves spectral matrices estimated over a clean part of the sky. It yields relative calibration coefficients 1.0004 at 100GHz and 1.0005 at 217GHz. These values are consistent with the results reported in @planck2016-l03 and @planck2016-l05.
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![The transition mask used to combine the $X_\textrm{high}$ and the $X_\textrm{full}$ CMB renderings.[]{data-label="fig:smica:tmask"}](figs/mask_transition_compress.pdf){width="\columnwidth"}
#### Preprocessing
The input maps are preprocessed for point sources as follows. In the maps from 30GHz to 353GHz, we try to fit and subtract the strongest point sources detected at the 5$\sigma$ level in the PCCS2 catalogue (@planck2014-a35). Any point source with an unsatisfactory fit is left “as is” in the map. In a second step, in each of the input maps from 44GHz to 353GHz, we mask all the point sources detected at more than 50$\sigma$ (unless they have already been subtracted in the previous step). The masked areas at all frequencies are then combined to form a common point-source mask. In addition to that point-source mask, we include a small mask, hereafter “the Galactic mask”, blocking the Galactic plane, plus a small number of selected regions (such as the LMC). The resulting “preprocessing mask” is shown in Figure \[fig:smica\_processing\_masks\]. In order to minimize leakage in the subsequent computation of spherical harmonic coefficients, the masked areas under this common mask are filled in by a simple diffusive inpainting procedure.
#### Spectral statistics
The computation of the spherical harmonic coefficients entering in the spectral statistics $\widehat\bC_\ell$ differs between $X_\textrm{high}$ and $X_\textrm{full}$. For $X_\textrm{high}$, we apply an apodized version of the transition mask, while for $X_\textrm{full}$, we use the full sky. In both cases, we use the preprocessed maps with additional masking of bright objects or regions. For $X_\textrm{full}$, which invloves all frequency channels, the point source mask is augmented with all the sources detected at more than 50$\sigma$ at frequencies 30GHz, 545GHz, and 857GHz, and the new holes are again filled in by diffusive inpainting. We also mask part of Galactic region using an apodized version of the Galactic mask. For $X_\textrm{high}$, which invloves only HFI channels, we mask all the point sources detected at 5$\sigma$ at frequencies 100GHz, 143GHz, and 217GHz, even if already subtracted. The resulting holes are then apodized over 30.
#### Spectral fits
For producing the $X_\textrm{high}$ map, processing is conducted, fitting the spectral covariance matrices $\widehat\bC_\ell$ over the multipole range $25\leq\ell\leq 1000$. For this fit, the calibration is kept fixed at the values found in the calibration run. The free parameters are the (binned) CMB spectrum ${\bC}_\ell^{\rm cmb}$, the positive matrices $\tens P_\ell$, and the $6\times \nfg$ foreground emissivity matrix $\tens F$.
For producing the $X_\textrm{full}$ map, a first run is devoted to estimating the foreground emissivity matrix $\tens F$, and a recalibration factor for the 70-GHz channel (this factor is found to be 1.0019). This fit is conducted over the multipole range $2\leq\ell\leq 150$. In a second run, we fit (binned versions of) $\bC_\ell^{\rm cmb}$ and $\tens P _\ell$ over the multipole range $10\leq\ell\leq 1000$, keeping fixed the calibration (vector $\vec a$) and the foreground emissivity matrix $\tens F$.
#### Map synthesis
The fits produce parametric estimates of $\bC_\ell$, from which spectral weights $\vec{w}_\ell$ are readily obtained. They are shown in Figure \[fig:smica\_T\_weights\] for $X_\textrm{high}$ (left panel), $X_\textrm{full}$ (middle panel). Those weights are applied to spherical harmonic coefficients computed from the preprocessed input maps. The spatial transition weights used to hybridize $X_\textrm{high}$ and $X_\textrm{full}$ are shown in Fig. \[fig:smica:tmask\].
#### Confidence mask
The confidence mask combines a point source mask and a Galactic mask determined by a procedure similar to the one used for the 2015 release. It is documented in the Explanatory Supplement.
#### Inpainting
Final inpainting of the CMB maps is no longer performed in the pipeline, but is carried out through a procedure common to all methods, as described in Sect. \[sec:masks\].
#### SZ-free CMB map
A CMB map free of SZ contamination is is produced by a simple adaptation of Eq. (\[eq:smica:ilc\]) as follows. That expression yields weights $\vec{w}_\ell$, which, at each multipole $\ell$, mimimize the output power while enforcing unit gain towards the CMB signal. In other words, it is the minimizer of $\vec{w}_\ell\adj \bC_\ell \vec{w}_\ell$ subject to $\vec{w}_\ell\adj \ba=1$. One can solve the same problem with the additional constraint that the weights should also cancel the SZ signal, that is, enforcing the additional constraint $\vec{w}_\ell\adj \bb=0$ where $\bb$ denotes the SZ emission law. The minimizer of $\vec{w}_\ell\adj \bC_\ell \vec{w}_\ell$ subject to $\vec{w}_\ell\adj \ba=1$ and $\vec{w}_\ell\adj \bb=0$ is easily found in closed form (see @Remazeilles2011a) as $$\label{eq:smica:ilcnosz}
\vec{w}_\ell = \bC_\ell\inv \bG (\bG\adj \bC_\ell\inv \bG)\inv \vec{c}$$ where $\bG = [ \ba\ \bb ]$ and $\vec{c} = [ 1\ 0]\adj$.
![Difference between the CMB map and its SZ-free version. The patch shown is $20 \deg\times 20\deg$ centered on $(l, b) = (46\pdeg3, 53\deg)$.[]{data-label="fig:diffszmap"}](figs/smica_patch_sz_compress.pdf){width="\columnwidth"}
![Angular spectra for the CMB (blue lines), the CMB SZ-free version (green lines), and their difference spectra (red lines), computed on the confidence mask. Half-mission cross-spectra (solid line) and half-mission difference spectra (dashed line) are shown to assess the signal and noise differences between the two CMB maps. []{data-label="fig:diffszspec"}](figs/cl_sz_conf_compress.pdf){width="\columnwidth"}
Figure \[fig:diffszmap\] shows an enlargement of the difference between the CMB maps derived with and without SZ projection. Figure \[fig:diffszspec\] compares the angular power spectra of these two maps.
#### Changes with respect to the 2015 release
Figure \[fig:cmb\_diff\_release\_maps\] shows, for all pipelines, the differences in CMB temperature maps from 2015 to 2018. In the case, the difference could have three origins: changes in the input data, changes in the pipeline, and changes in recalibration procedure. We show here that the difference is mostly due to recalibration by producing a CMB map, referred to as the “2015b map”, obtained from the 2015 data by running the 2015 pipeline with the sole exception that, as for the 2018 release, the 30GHz and 44GHz channels are *not* recalibrated. Figure \[fig:smica\_diff2015\] shows the differences from the 2015 map to this 2015b map (top panel) and from this 2015b map to the 2018 map, while the bottom panel of Fig. \[fig:cmb\_diff\_release\_maps\] shows the difference from the 2015 map to the 2018 map. These three pairwise comparisons make it clear that, in temperature, most of the differences between 2015 and 2018 should be attributed to changes in calibration, rather than to changes in the pipeline.
![CMB difference maps in temperature at 80 resolution. [*Top:*]{} Difference between the 2015 released map and the 2015b map (without recalibration of the 44-GHz channel). [*Bottom:*]{} Difference between the 2015b and the 2018 map. []{data-label="fig:smica_diff2015"}](figs/smica_diff_2015-2015b_nocb_compress.pdf "fig:"){width="\columnwidth"} ![CMB difference maps in temperature at 80 resolution. [*Top:*]{} Difference between the 2015 released map and the 2015b map (without recalibration of the 44-GHz channel). [*Bottom:*]{} Difference between the 2015b and the 2018 map. []{data-label="fig:smica_diff2015"}](figs/smica_diff_2015b-2017_compress.pdf "fig:"){width="\columnwidth"}
Polarization analysis
---------------------
#### CMB reconstruction.
We now turn to the construction of polarization maps, and start with the CMB map. First, a significant modification to the 2018 pipeline is the fact that $E$ and $B$ modes are now processed independently; in contrast, the 2015 analysis fitted and filtered these modes jointly. uses all seven polarized channels. When producing either $E$-mode or $B$-mode CMB maps, the foreground emission is taken to have maximal dimension: $\nfg=7-1=6$.
The input maps are preprocessed as follows. First, in each of the input frequency maps, all point sources detected at the 5$\sigma$ level are masked and the holes are filled by diffusive inpainting. Second, the bright pixels (with amplitude ten times larger than the standard deviation of the map) are similarly masked and inpainted. Finally, a small Galactic mask – obtained by thresholding a combination of the 30-GHz and 353-GHz maps – is applied. The resulting mask, shown in Figure \[fig:smica\_processing\_masks\], covers 97% of the sky. In the 2015 release, the same processing mask was used for polarization and intensity.
As for temperature, we proceed in two steps. A first fit is performed to estimate the foreground emissivity matrix $\tens F$ over the range $5\leq\ell\leq 150$. A second fit is then performed in the range $2\leq\ell\leq 1000$ over parameters $C_\ell^{\rm cmb}$ and $\tens P_\ell$, while $\tens F$ is kept fixed at the value found in the first run. The right panel of Fig. \[fig:smica\_T\_weights\] shows the resulting harmonic weights. These result from spectral statistics $\hat\bC_\ell$ computed from the preprocessed maps without additional masking, unlike in the temperature case.
#### Confidence mask.
A polarization confidence mask has been produced and released, but appears not to be conservative enough. For that reason, we recommend using the common confidence mask to analyse polarized CMB maps.
Polarized foreground reconstruction.
------------------------------------
The results presented in Sect. \[sec:pol\_ind\], regarding the polarized dust and synchrotron emission, are based on a dedicated, blind fit with a foreground emissivity matrix $\tens F$ composed only of $\nfg=2$ columns. The total foreground contribution to a spectral covariance matrix $\bC_\ell$ being $\tens F \tens P_\ell \tens F \adj$, a blind fit can only determine the factors $\tens F$ and $\tens P_\ell$ up to multiplication by an invertible $2\times2$ matrix $\tens T$. Indeed, for any such matrix $\tens T$, one can define $ \tilde{\tens P}_\ell=\tens T \tens P_\ell \tens T \adj$ and $\tilde{\tens F} =\tens F \tens T\inv$ and see that the transfomed pair $(\tilde{\tens F}, \tilde{\tens P}_\ell)$ contributes as much as the original pair $({\tens F}, {\tens P}_\ell)$ to the spectral covariance matrix, since, by construction $ \tens F \tens P_\ell \tens F \adj = \tilde {\tens F} \tilde{\tens P}_\ell \tilde{\tens F} \adj $. Therefore the likelihood is insensitive to the value of $\tens T$. Since a blind fit is (by definition) conducted without constraining either $\tens F$ nor $\tens P_\ell$, the matrix $\tens T$ cannot be determined from the data without imposing extra constraints. This degeneracy could be fixed by constraining ${\tens P}_\ell$ to be diagonal, but this would be equivalent to fitting a (wrong) model of uncorrelated synchrotron and dust emissions. We choose instead to fix the degeneracy as follows. We conduct a blind fit and, in a post processing step, we select (without affecting the quality of the fit) a matrix $T$ given by $$\tens T
= \left[
\begin{array}{cc}
\tens F_{30, 1} & \tens F_{30, 2} \\ \tens F_{353, 1} & \tens F_{353, 2}
\end{array}
\right]\,,$$ so that the first row of $\tilde{\tens F} = \tens F\tens T\inv$ becomes $[ 0 , 1 ]$ and its last row becomes $[1, 0]$. In other words, we fix the indeterminacy in the blind fit of a two-template foreground model by assuming that the entire foreground signal at 30GHz is only synchrotron, and that the entire foreground signal at 353GHz is only thermal dust. We checked that performing a second fit where the synchrotron contribution at 353GHz is not zero but an extrapolated value (and similarly for dust at 30GHz), has no significant effect on fitted values and, unsurprisingly, that it does not affect either of the reconstructed maps.
Maps of polarized dust and synchrotron emission are synthesized from harmonic coefficients computed over the full sky, except for point sources detected at 5$\sigma$, which are masked and inpainted. This is carried out independently for each input map. The $Q$ and $U$ maps are synthesized with an effective Gaussian beam of 3 (FWHM) for synchroton and 12 for dust. The SEDs of dust and synchrotron emission shown in Fig. \[fig:polsedsmica\] are determined from a dedicated fit based on spectral covariance matrices computed from about 70% of the sky.
{#app:gnilc}
The formalism of has been described in detail in [@Remazeilles2011b] and [@planck2016-XLVIII]. The main characteristics can be summarized as follows: (i) a map at given frequency is a weighted linear combination (ILC) of the frequency maps having minimimum variance; (ii) performs localized analysis in both harmonic space and pixel space via needlet (spherical wavelet) decomposition [@Narcowich2006], and as such it adapts component separation to local conditions of contamination both over the sky and over angular scale; and (iii) uses not only spectral information, but also spatial information (angular power spectra) of the non-Galactic components (CIB, CMB, and noise) in order to disentangle the Galactic signal from the CIB, CMB, and noise contamination. Therefore, is a blind, model-independent, data-driven component-separation method, in the sense that there is no prior assumption/parametrization of the Galactic foreground properties.
There are, however, a few differences in the processing steps between intensity and polarization. For intensity, the processing is identical to that of [@planck2016-XLVIII], i.e., the prior information is both spectral and spatial, and consists of the best-fit CMB temperature power spectrum, $C_\ell^{\rm \Lambda CDM}$ [@planck2014-a13], the CIB best-fit auto/cross power spectra across frequency pairs, $C_\ell^{\rm CIB}(\nu_1,\nu_2)$ [@planck2013-pip56], and the noise power spectra across frequencies, $C_\ell^{\rm noise}(\nu)$. For polarization, prior information is only spectral for the CMB, consisting of the CMB SED,[^15] while the noise prior is still spectral and spatial, comprising the noise power spectra at each frequency. In practice, noise power spectra are derived from the half-difference of the first and second halves of each stable pointing period (“rings”) of , in which the sky emission cancels out and leaves an estimate of the full-survey noise.
From those prior power spectra, we simulate Gaussian realizations of the CMB map, $y^{\rm CMB}(p)$, the correlated CIB maps, $y_\nu^{\rm CIB}(p)$, and the noise maps, $y_\nu^{\rm noise}(p)$, where $\nu$ denotes the frequencies and $p$ the pixels. The simulated total “nuisance” map is defined as $$y_\nu(p)
\equiv g_\nu\, y^{\rm CMB}(p) + y_\nu^{\rm CIB}(p) + y_\nu^{\rm noise}(p)$$ for intensity, where $g_\nu$ is the derivative of a blackbody with respect to temperature, and $$y_\nu(p)
\equiv y_\nu^{\rm noise}(p)$$ for polarization, since the CIB is assumed to be unpolarized and we have no spatial information on the CMB polarization.
We perform a needlet decomposition of both the simulated nuisance maps and the frequency maps. We thus define ten needlet windows, $\{h^{(j)}_\ell\}_{1\leq j \leq 10}$, as Gaussian bandpass filters in harmonic space to perform component separation on different ranges of multipoles independently.[^16] The spherical harmonic coefficients of the simulated maps, $y_\nu(p)$, are bandpass-filtered as $h^{(j)}_\ell\,a_{\ell m }(\nu)$. The inverse spherical harmonic transform of the filtered coefficients produces ten needlet maps, $y_\nu^{(j)}(p)$ (one for each needlet scale), for each frequency. Each needlet map, $y_\nu^{(j)}(p)$, contains temperature fluctuations at the specific range of angular scales probed by the associated needlet window, with statistical properties determined by the prior power spectra at these scales.
For each needlet scale $(j)$, we compute the covariance matrix of the nuisance map (noise for polarization; CMB plus CIB plus noise for intensity) in each pixel $p$ for all pairs of frequencies $a$ and $b$ as: $$\left[ {\rm R_n}^{(j)} (p) \right]_{a\,b}
= \sum_{p' \in \mathcal{D}^{(j)}(p)}\, y_a^{(j)}(p')\, y_b^{(j)}(p')\,,$$ where in practice the pixel domain, $\mathcal{D}^{(j)}(p)$, is defined by the convolution in real space of the product of needlet maps, $y_a^{(j)}(p)\, y_b^{(j)}(p)$, with a Gaussian kernel whose the width is a function of the needlet scale considered. Note that the prior covariance matrix of the nuisance map, ${\rm R_n}(p)$, is blind about the particular realization of CMB, CIB, and noise that is found in the observed data.
Similarly, the data ( frequency maps), $d_\nu(p)$, are decomposed onto the same needlet frame, and the frequency-frequency covariance matrix of the data is computed in each pixel for each needlet scale as: $$\left[ {\rm \widehat{R}_d}^{(j)} (p) \right]_{a\,b}
= \sum_{p' \in \mathcal{D}^{(j)}(p)}\, d_a^{(j)}(p')\, d_b^{(j)}(p')\,.$$
As described in [@planck2016-XLVIII], the prior power spectra are thus used to obtain a model of the frequency-frequency covariance matrix, ${\rm R_n}$, of the nuisance contribution (CIB, CMB, and noise) to the total data covariance matrix, $\widehat{\rm R}_{\rm d}$ ($9\times 9$ matrices for intensity, $7\times 7$ for polarization). The signal-to-nuisance ratio, where signal stands for Galactic emission, is obtained via the matrix ${\rm R_n}^{-1/2}\widehat{\rm R}_{\rm d}{\rm R_n}^{-1/2}$, which is estimated locally over the sky and over different ranges of angular scales via needlet decomposition of the maps. The eigenstructure of the matrix ${\rm R_n}^{-1/2}\widehat{\rm R}_{\rm d}{\rm R_n}^{-1/2}$ allows us to discriminate those eigenvalues that are close to unity (therefore corresponding to nuisance) from those that correspond to the contribution of Galactic emission.[^17] This allows us to estimate the local dimension, $m$, of the Galactic signal subspace over the sky and over scales, i.e., the finite number of independent (not physical) components[^18] onto which the correlated Galactic emission can be decomposed. We note ${\rm U_S}$ the matrix collecting the selected subset of $m$ eigenvectors of the matrix ${\rm R_n}^{-1/2}\widehat{\rm R}_{\rm d}{\rm R_n}^{-1/2}$ that form an orthogonal basis of the Galactic signal subspace.
The data[^19] are then projected onto the identified Galactic signal subspace, and an $m$-dimensional ILC is performed on the projected data in order to further minimize any part of the nuisance that did not project orthogonally to the Galactic subspace: $$\label{eq:gnilc_dust}
\widehat{s}^{\,\,\rm dust\, (j)}_\nu (p)
= \sum_{\nu'} {\rm W}_{\nu\nu'}^{(j)}(p) \, d_{\nu'}^{(j)} (p)\,.$$ The matrix of weights can be written in compact form as [@Remazeilles2011b]: $$\label{eq:gnilc_weights}
{\rm W}
= {\rm F}\,\left( {\rm F}^t \, {\rm \widehat{R}_d}^{-1} \, {\rm F}\right)^{-1}\, {\rm F}^t \, {\rm \widehat{R}_d}^{-1} \,,$$ with the estimated foreground mixing matrix, ${\rm F}$, given by $${\rm F} = {\rm R_n}^{1/2}\, {\rm U_S} \,.$$ For polarization, where there is no prior on the CMB power spectra, the ILC is replaced by a constrained ILC [@Remazeilles2011a], for which the vector of weights in frequency is constrained to be orthogonal to the CMB SED. In practice, this is done through a Gram-Schmidt orthogonalization of the set of eigenvectors collected in matrix ${\rm U_S}$ with respect to the CMB SED vector $g_{\nu}$. This constraint ensures that the GNILC weights (Eq. \[eq:gnilc\_weights\]) project out any CMB polarization signal in the reconstructed dust polarization map.
The filters (Eq. \[eq:gnilc\_weights\]) are invariant if $\rm F$ is replaced by $\rm F\,T$ for any invertible matrix $\rm T$. Therefore, the true foreground mixing matrix does not need to be known by ; the only useful information is a set of independent components onto which the correlated Galactic emission can be decomposed.
The estimated needlet maps of dust emission (Eq. \[eq:gnilc\_dust\]) are then synthesised to reconstruct the complete dust maps, as follows. The spherical harmonic coefficients, $\widehat{a}_{\ell m}^{(j)}(\nu)$, of the needlet dust maps, $\widehat{s}^{\,\,\rm dust\, (j)}_\nu (p)$, are again bandpass-filtered by the needlet windows as $h_\ell^{(j)} \widehat{a}_{\ell m}^{(j)}(\nu)$. The filtered coefficients are then inverse-spherical-harmonic transformed into maps, and coadded across needlet scales to form the complete dust map, accounting for all the angular scales.
has many advantages over template subtraction, parametric methods, or smoothing procedures. First, it is a one-shot component-separation method that does not rely on subtraction of any template, such as a CMB template map, coming from another component-separation process. This prevents the propagation of CMB foreground residuals (e.g., dust and CIB residuals in the CMB map) to the reconstructed Galactic map.
The second advantage is related to noise filtering in polarization maps, where performs better than a simple smoothing. Given that is a minimum-variance linear combination of frequency maps, the overall noise level in the maps will always be lower than the noise level in smoothed maps at the same frequency and equal resolution: $$\begin{aligned}
{1\over \sigma^2_{\rm \gnilc}(353\,\hbox{GHz})} = {1\over \sigma^2_{\Planck}(30\,\hbox{GHz})} + ... + {1\over \sigma^2_{\Planck}(353\,\hbox{GHz})},\end{aligned}$$ where $\sigma_{\rm \gnilc}(353\,\hbox{GHz})$ is the noise rms in the 353-GHz map, and $\sigma_{\Planck}(353\,\hbox{GHz})$ is the noise rms in the 353-GHz map. Moreover, a simple smoothing of the 353-GHz $Q$ and $U$ maps will mitigate CMB $E$ and $B$ modes but not cancel them on large scales, and there is no reliable CMB $B$-mode template to be subtracted. Conversely, is an orthogonal projection to the flat CMB SED, and therefore cancels out any CMB $E$- and $B$-mode polarization at all angular scales.
Third, filtering is performed locally over the sky and over scales via wavelet decomposition. This enables optimization of the component-separation process given local variations of contamination over the sky and over scales.
Finally, the method is blind, since it does not rely on any assumption about Galactic foregrounds. Most important, allows for outputting Galactic foreground maps at all frequencies, e.g., at 100–143GHz, without relying on the extrapolation of high-frequency templates with arbitrary emission laws. This is particularly useful in the context of decorrelation effects and searches for primordial $B$ modes [@Tassis2015; @planck2016-L], where we can no longer rely on simple emission laws to extrapolate dust foregrounds to CMB frequencies.
Intensity foregrounds {#app:foregrounds}
=====================
In this appendix, we review the temperature foreground products derived by and from the 2018 frequency maps. As discussed in Sect. \[sec:inputs\] and elsewhere, these results are not intended for scientific analysis, but are included here for reference and completeness purposes.
analysis
---------
We start our discussion with a review of the intensity analysis. For a summary of the methodology and model definitions used in this work, see Sect. \[sec:commander\] and Appendix \[app:commander\]. In short we fit a parametric five-component model to the 2018 data by maximizing the standard Bayesian posterior. The 2018 model includes the following components: (1) CMB; (2) a single power-law foreground model with a free spectral index per pixel to describe the sum of low-frequency foregrounds (synchrotron, free-free, and anomalous microwave emission); (3) a modified blackbody with a free emissivity and temperature to describe thermal dust; (4) a line-emission component at 100, 217, and 353GHz, with fixed line ratios between channels to describe CO emission; and (5) a catalogue of 12192 known point source positions, each source being fitted with a free flux density and spectral index.
We first consider the parameters of the derived astrophysical model in intensity, starting with the point source component, which represents one of the most novel aspects of the 2018 model compared to previous versions.
Starting with the amplitude maps, the most notable difference with earlier results is caused by the explicit inclusion of a radio point source component in the latest model. Each object in this component is associated with an overall flux density and spectral index across all frequencies, while the spatial projection into each frequency component is performed through a full [FEBeCoP]{} calculation, accounting for the asymmetric beam profile in the respective frequency channel. Only frequencies up to and including 143GHz are included when fitting the flux densities and spectral indices, to avoid biases from modelling errors at high frequencies. However, the resulting model is also extrapolated to higher frequencies when fitting other components. Infrared and sub-mm sources are not explicitly modelled in this approach, since they are well described for the frequencies within the diffuse thermal dust component, which has 5 FWHM resolution.
As described in Appendix \[app:commander\], the total catalogue used in this work represents a combination of four separate source catalogs, three of which (AT20G, GB6, and NVSS; [@murphy2010; @gregory1996; @condon1998]) are selected to cover disjoint regions of the sky, and the fourth (PCSS2; [@planck2014-a35]) includes microwave sources that are not detected by any of the former three. In Table \[tab:ptsrc\_summary\], we provide summary statistics for the fits produced in the current analysis, broken down according to reference catalogue. From left to right, columns show: (1) catalogue name; (2) catalogue reference frequency; (3) total number of sources used in our combined catalogue; (4) number of sources statistically detected by in the 2018 data; (5) average flux density recalibration factor relative to the reference catalogue (no colour corrections are applied); and (6) Pearson’s $r$ correlation coefficient evaluated between the reference catalogue and -estimated flux densities.
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Several interesting features may be seen in Table \[tab:ptsrc\_summary\]. Starting with the PCCS2 sources [@planck2014-a35], the correlation between the and PCCS2 flux densities at 30GHz is very high, with a Pearson’s correlation coefficient of 0.99. However, the best-fit relative amplitude between the two catalogues is $a=0.867$. Part of this is due to the fact that the flux densities are intrinsically colour corrected, and therefore correspond to a monochromatic reference frequency of 30GHz, whereas the PCCS2 values correspond to flux densities directly observed in the 30-GHz map without colour correction. Considering that the effective frequency of the 30-GHz channel for a flat-spectrum source with a spectral energy distribution proportional to $\nu^{-2}$ is 28.4GHz, the difference in amplitude is expected to be roughly $(28.4/30)^{2}\approx0.90$. In addition, the analysis takes into account the full asymmetry of the beams, and also exploits all frequencies between 30 and 143GHz in the fit, while the PCCS2 catalogue only considers a symmetric Gaussian beam model, and employs the LFI 30-GHz observations alone.
The fits exhibit a slightly lower correlation coefficient relative to the AT20G source catalogue at 20GHz, with a numerical value of $r=0.74$ and a detection rate of 91%. However, the flux-density calibration is very good, with a relative normalization factor of $a=0.977$. At 4.85GHz, the correlation with the GB6 catalogue flux densities is again very slightly weaker at $r=0.69$, and this time the detection rate is 59%, with a relative normalization of $a=0.56$. Finally, this general trend of weakening correlations becomes even stronger at lower frequencies, with the NVVS catalogue at 1.4GHz only having a correlation coefficient of $r=0.10$ and a relative normalization of $a=0.163$. However, the detection rate remains fairly high, at 72%. NVSS and thus agree on the existence of the set of sources, but disagree significantly on their amplitudes. This is, of course, not unexpected, when extrapolating all the way from 1.4GHz to 30–143GHz. The point source component as evaluated for the 30GHz channel is plotted in the top right panel of Fig. \[fig:comm\_fg\_temp\].
Next, we consider the amplitude parameter maps of the diffuse foreground components, as shown in Fig. \[fig:comm\_fg\_temp\]. Starting with the top left panel, this figure shows the joint low-frequency foreground component, which includes synchrotron, free-free, and anomalous microwave emission as evaluated at 30GHz and smoothed to a resolution of $40\arcm$ FWHM.[^20] A similar low-frequency foreground map was presented in @planck2013-p06, derived from the 2013 data, and the most visually striking difference between these two maps is the absence of small-scale compact objects in the updated map. This is of course due to the fact that these sources are explicitly fitted out in the new model. The resulting source amplitude map at 30GHz is shown in the top right panel.
The bottom left panel of Fig. \[fig:comm\_fg\_temp\] shows the thermal dust amplitude map evaluated at 857GHz and smoothed to $10\arcm$ FWHM. Visually speaking, this map is nearly identical to the corresponding 2015 map, since the thermal dust component is strongly dominated by the 545- and 857-GHz HFI frequency maps, and these have only changed by one or two percent since the last release (see Fig. \[fig:dx11\_vs\_dx12\_hfi\]).
At a strictly visual level, the same holds true for the CO component, shown in the bottom-right panel of Fig. \[fig:comm\_fg\_temp\]. However, in this case the reconstruction quality of the new map is notably worse than in the corresponding 2015 map, as shown in the top panel of Fig. \[fig:comm\_co\]. This figure shows scatter plots between the Dame et al. CO survey map [@dame2001] and the CO 2015 (cyan dots) and 2018 (grey dots) maps. Two effects are notable. First, we note that the slopes are different between the two maps, corresponding simply to the different overall normalization conventions adopted for the two maps. In particular, for the 2015 analysis we employed conversion factors between $\muK_{\mathrm{CMB}}$ and $\mathrm{K}_{\mathrm{RJ}}\, \mathrm{km}\,\mathrm{s}^{-1}$ derived directly from the bandpasses measured on the ground [@planck2013-p03d]. This is significantly more complicated with the single-CO line model employed in the current analysis, and with the 2018 co-added frequency maps. The scale of the current CO amplitude map is therefore instead directly set by regressing against the Dame et al. map, and the resulting scatter plot therefore by definition has a slope of unity.
![$T$–$T$ scatter plots between the @dame2001 $J$=1$\rightarrow$0 map and the 2015 (blue dots) and 2018 (grey dots) CO maps. Note that the 2018 map has been directly calibrated to the Dame et al. map, and is therefore expected to have unity slope by construction, while the 2015 map was calibrated using the bandpasses; this difference explains the overall shift in slopes. The lower level of scatter around the best-fit slope in the 2015 map is due to including single-bolometer and detector-set maps, as opposed to the 2018 map, which exclusively uses co-added frequency maps.[]{data-label="fig:comm_co"}](figs/commander_co_scatter_v1_compress.pdf "fig:"){width="\columnwidth"}\
More important than this choice of normalization, however, is the width and shape of the two scatter plots. Specifically, while the 2015 scatter plot exhibits a very tight overall correlation and no visually notable outliers, the 2018 scatter plot is broader overall and exhibits several outliers toward higher amplitudes in the map. The reasons for this weaker correlation have already been discussed in Sect. \[sec:inputs\] and @planck2016-l03, and can be summarized as being due to the lack of single-bolometer HFI maps and inaccuracies in the CO template corrections used during mapmaking. As described in Appendix \[app:commander\], the CO map is used as a tracer for CO emission in the confidence mask.
Finally, we consider the spectral parameters for various components, shown in the left column of Fig. \[fig:spectral\_params\] for the low-frequency and thermal dust components. These can be compared to similar maps presented in the 2013 and 2015 releases [@planck2013-p06; @planck2014-a12]. Starting with the low-frequency spectral-index map, the two most notable changes with respect to the corresponding 2013 products are different priors on spectral index ($\beta_{\mathrm{lf}}=-2.9\pm0.3$ in 2013 versus $\beta_{\mathrm{lf}}=-3.1\pm0.5$ in 2018), resulting in a darker map at high latitudes, and an overall higher signal-to-noise ratio resulting from the inclusion of four-years of LFI observations in these new maps, as opposed to only 14 months in 2013, resulting in larger areas being data-driven. Otherwise, the two maps are largely consistent.
![ 2018 foreground spectral parameters. Rows show, from top to bottom, the low-frequency spectral index at a $40\arcm$ FWHM smoothing scale, the thermal dust spectral index at $10\arcm$ FWHM, and the thermal dust temperature at $5\arcm$ FWHM, respectively.[]{data-label="fig:spectral_params"}](figs/synch_beta_2018_rc6_v3_compress.pdf "fig:"){width="\columnwidth"}\
![ 2018 foreground spectral parameters. Rows show, from top to bottom, the low-frequency spectral index at a $40\arcm$ FWHM smoothing scale, the thermal dust spectral index at $10\arcm$ FWHM, and the thermal dust temperature at $5\arcm$ FWHM, respectively.[]{data-label="fig:spectral_params"}](figs/dust_beta_2018_rc6_v3_compress.pdf "fig:"){width="\columnwidth"}\
![ 2018 foreground spectral parameters. Rows show, from top to bottom, the low-frequency spectral index at a $40\arcm$ FWHM smoothing scale, the thermal dust spectral index at $10\arcm$ FWHM, and the thermal dust temperature at $5\arcm$ FWHM, respectively.[]{data-label="fig:spectral_params"}](figs/dust_T_2018_rc6_v3_compress.pdf "fig:"){width="\columnwidth"}
Relatively speaking, larger changes are seen for the thermal dust spectral parameters when compared to the 2015 model presented in @planck2014-a12. Starting with the emissivity or spectral index, $\beta_\mathrm{d}$, one can see bright CO-like structures in the 2018 version, for instance near the Fan region at $(l,b)=(140^{\circ},10^{\circ})$; this indicates a stronger degeneracy between CO and thermal dust in the 2018 map than in the 2015 map, and results most likely from the lack of single-bolometer maps in the 2018 analysis. Similarly, one can see a strong dark region extending from the North to the South Ecliptic Pole in the new map. This feature is well-known in mapmaking, and arises from bandpass mismatch between different bolometers used to create a single map. Although the most recent mapmaking process makes a great effort to suppress this effect [@planck2016-l03], the lack of single-bolometer and detector-set maps carries a significant price for subsequent component separation: while it was possible to remove single bolometers for which this effect was particularly pronounced in 2015 (see figure 2 of [@planck2014-a12]), only full frequency maps are available in the 2018 analysis.
![(*Top*): thermal dust intensity map at 353GHz with spatially varying angular resolution. (*Bottom*): $T$–$T$ scatter plot between the thermal dust intensity , both smoothed to a common angular resolution of $80\arcm$ FWHM. An offset of $421\muK$ has been subtracted from the map in both panels (see main text for details). []{data-label="fig:gnilc_dust_T"}](figs/gnilc_T_353_varres_RJ_v3_compress.pdf "fig:"){width="\columnwidth"}\
![(*Top*): thermal dust intensity map at 353GHz with spatially varying angular resolution. (*Bottom*): $T$–$T$ scatter plot between the thermal dust intensity , both smoothed to a common angular resolution of $80\arcm$ FWHM. An offset of $421\muK$ has been subtracted from the map in both panels (see main text for details). []{data-label="fig:gnilc_dust_T"}](figs/gnilc_commander_353_scatter_T_v3.pdf "fig:"){width="\columnwidth"}
At high latitudes, the most notable effect is a brighter overall distribution of small-scale fluctuations, which correspond to small-scale cosmic infrared background (CIB) fluctuations. When interpreting these fluctuations, however, it is important to recall that the two-parameter $\beta$–$T$ modified blackbody model exhibits a strong degeneracy between the spectral index and temperature in the low signal-to-noise regime. The fluctuations seen in the 2018 $\beta$ map were thus also present in the 2015 rendition, but in that case were seen in the temperature map. The main reason for the apparent shift is the choice of thermal dust temperature prior, or, to be more precise, the angular resolution at which it is fitted. In 2015 the thermal dust temperature was fitted at $40\arcm$ FWHM, while in the updated analysis it is fitted at $5\arcm$ FWHM. As a result, the 2015 temperature map had higher effective signal-to-noise per resolution element, and therefore less dependence on the prior and more structure at high latitudes. In contrast, the 2018 temperature map has less signal-to-noise per resolution element, stronger prior dependency, and accordingly also shows less structure at high latitudes, as the temperature is driven to the prior mean, and fluctuations are instead captured in the spectral index map. In general, we caution against over-interpreting the individual parameters of the modified blackbody model in the low signal-to-noise regime, since small changes in the input can lead to relatively large variations in parameter values. In contrast, the resulting SED arising from the parameters is robust.
For completeness, we note that the best-fit CO line ratio between 100 and 217GHz (353GHz) is $h_{2018}=0.58$ ($h_{353}=0.20$), as estimated by from the 2018 data set. For comparison, the corresponding 2013 values for these two parameters were $h_{2018}=0.595$ and $h_{353}=0.295$. However, for the reasons discussed above, we do not attach physical significance to the lower value found in the new data set, but rather recommend continued usage of the previous values when using results for astrophysical analysis and forecasts.
Before concluding our discussion, we emphasize that while we do not consider the 2018 intensity foreground analysis to be as robust as the corresponding 2015 analysis, this has only a very small effect on the corresponding CMB reconstruction after accounting explicitly for CO emission in the confidence mask (see Sec. \[app:comm\_mask\]). As far as CMB reconstruction is concerned, the only important factor is whether the sum of the apparent foregrounds may be modelled within the parameter space of the Bayesian model; whether or not those best-fit values represents the physically true sky is irrelevant. This is of course also precisely why blind CMB reconstruction methods, such as , , and , perform very well. Nevertheless, the fact that the 2018 intensity products appear reasonable, even though inferior, compared to the 2015 products is reassuring.
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Thermal dust intensity maps and their zero levels
-------------------------------------------------
Finally, we compare the thermal dust intensity maps derived with and . Specifically, the top panel of Fig. \[fig:gnilc\_dust\_T\] shows the thermal dust intensity map evaluated at 353GHz, and the bottom panel shows a scatter plot between the and estimates, where the model has been integrated over the 353-GHz channel bandpass. Overall, we observe good agreement between the two estimates.
The behaviour at low intensities is particularly interesting because it is sensitive to how the zero level of each map has been set. By construction, the frequency maps delivered by the HFI DPC and used for component separation have a Galactic zero level consistent with an intensity of the dust foreground at high Galactic latitudes proportional to the column density of the ISM traced by the 21-cm emission of [${{\normalfont\textsc{Hi}}}$]{} at low column densities. In the case of , the processing does not adjust the monopoles contained in the input maps, the largest of which is the CIB monopole. Therefore, the zero levels of the resulting dust maps need to be adjusted prior to Galactic applications. This has been accomplished here, just as in @planck2016-XLVIII, by correlation with the [${{\normalfont\textsc{Hi}}}$]{} map at high latitude, following the methodology set out in @planck2013-p03f and @planck2013-p06b. At 353GHz, 421 is subtracted. In the case of , the zero level at each frequency is solved for explicitly within the component separation processing, with priors set equal to the value of the CIB monopole (see Appendix \[app:commprior\]). The offset found at 353GHz is 431, separate from the thermal dust emission model. Given these zero level adjustments, the agreement at low intensities is satisfactory.
Especially for applications at low intensity, it critical to appreciate that there are significant uncertainties in the zero levels of the thermal dust intensity maps derived from the 2015 and 2018 frequency maps, as discussed in Section 6.1.1 of @planck2014-a12, and of , as discussed in Section 2.2 of @planck2016-l11B, including the possibility of dust associated with ionized gas. These uncertainties need to be evaluated and then propagated in any subsequent analyses using these thermal dust maps. Ideally, the uncertainties can be reduced through improved methods of zero level determination, such as exploitation of correlations with external data sets, including [${{\normalfont\textsc{Hi}}}$]{} and optical extinction [e.g., @planck2016-XLVIII], or via spatial spectral variations [@wehus2014].
Extra CMB plots {#app:splits}
===============
In this Appendix, we present supporting plots relevant for the CMB discussion. These complement and elucidate the analyses and results presented in the main text, and are useful for reference purposes.
First, Figs. \[fig:cmb\_oehd\_maps\] and \[fig:cmb\_hmhd\_maps\] show odd-even and half-mission half-difference maps, and as such, they represent our preferred tracers of noise and instrumental systematics, respectively. The former exhibit very few large-scale correlated features, whereas the latter show clear signatures of both the scanning strategy at high latitudes and Galactic contamination through calibration and leakage effects at low latitudes.
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Next, Fig. \[fig:cmb\_maps\_zoom\] shows a $20^\circ \times 20^\circ$ zoom-in of the four cleaned CMB maps, centered on the North Ecliptic Pole. The polarization pattern expected from a typical $E$-mode signal (’+’-type in Stokes $Q$, and ’$\times$’-type in Stokes $U$) is clearly visible.
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Figures \[fig:cmb\_maps\_zoom\_oehd\] and \[fig:cmb\_maps\_zoom\_hmhd\] show enlargements of the odd-even and half-mission half-difference maps for the same region. In these maps, notable qualitative differences between the four CMB maps are observed, perhaps the most striking of which is the effect of different point source treatments adopted by the four pipelines. For instance, in the half-mission splits one can clearly see bright source residuals in the temperature maps for , , and , but not for . These are due to changes in the amplitude of point sources between both periods of observations, which show up when subtracting the half-mission splits. does not present these residuals because it explicitly inpaints known sources positions in each split, and therefore it reduces significantly this contaminant emission in the half-mission data before constructing the half-difference maps. In the case of the polarization maps, one can also see outlines of the processing mask adopted for inpainting in that case.
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Another type of qualitative difference is seen between on the one side, and the other three codes on the other side. accounts explicitly for spatial variations in instrumental sensitivity at each frequency during Wiener filtering, which corresponds to evaluating an exact inverse-noise-variance weighting pixel-by-pixel in the different channels. This procedure produces somewhat more uniform effective residual maps than the other three codes.
Next, Figs. \[fig:commander\_inpaint\]–\[fig:smica\_inpaint\] show a single Gaussian-constrained realization evaluated for each of the cleaned CMB maps, with the inpainting mask shown in Fig. \[fig:inpaint\] applied. The temperature maps are shown at $5\arcm$ FWHM resolution, and the polarization maps are shown at $80\arcm$ FWHM resolution. These maps are primarily intended for presentation purposes, rather than scientific analysis, since their noise properties are complicated. If similar constrained realizations are required for quantitative analysis, we recommend users to employ a Gibbs sampler, for instance as implemented in , to produce an ensemble of such realizations, which then collectively may be used to propagate uncertainties.
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N-point functions {#app:npt_functions}
=================
Here we present 2-point and 3-point correlation functions for the HMHD and OEHD maps. These complement analyses and figures presented in the main text (Sect. \[sec:2point\_correlation\]). Figures \[fig:npt\_nilc\], \[fig:npt\_sevem\], and \[fig:npt\_smica\] show the correlation functions for half-differences of the , , and maps, respectively.
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[^1]: (<http://www.esa.int/Planck>) is a project of the European Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states and led by Principal Investigators from France and Italy, telescope reflectors provided through a collaboration between ESA and a scientific consortium led and funded by Denmark, and additional contributions from NASA (USA).
[^2]: Note that the numerical value derived for the spectral index of polarized synchrotron emission is not directly comparable to the mean of the low-frequency component spectral index map derived in intensity, since the latter also includes free-free and spinning dust emission.
[^3]: The decision on whether to use a Gaussian kernel or a mild exponential high-$\ell$ cut-off for prior apodization is determined by the effective signal-to-noise ratio of the component in question.
[^4]: I should not be confused with the “Constrained ILC” method [@Remazeilles2011a], which was designed to extract SZ-free CMB temperature anisotropies.
[^5]: <http://healpix.jpl.nasa.gov>
[^6]: behaves differently from the other codes with respect to unobserved pixels. It applies per-pixel inverse noise weighting per frequency channel, and unobserved pixels in a given channel are simply given zero weight in the parametric fits.
[^7]: We remove a common estimate of the CMB signal in order to highlight the foreground and residual monopole and dipole contents of each map. Visually identical results would be obtained by adopting any of the other solutions as a reference instead of .
[^8]: <http://pla.esac.esa.int>
[^9]: The four cleaned frequency maps (from 70 to 217GHz) provided by are also shown in Fig. \[fig:sevem\_freqmaps\].
[^10]: For the particular case of , the evaluation of the transfer function is in principle affected by the pixels inpainted in the cleaned frequency maps. Since those pixels are all excluded in the common confidence mask, we have evaluated this function from full-sky CMB simulations without applying this inpainting. Therefore, this effective transfer function should be a good approximation for the regions passed by the common mask. However, if a transfer function is needed for a region of the sky that contains inpainted pixels, it is recommended to re-evaluate this function for that particular sky coverage, taking into account the inpainting. Although the effect is very small, it can be noticeable, especially for agressive masks that remove only a small fraction of the Galaxy, since in the Galactic regions a relatively large number of sources are inpainted.
[^11]: For simplicity, bandpass integration and unit conversion effects are omitted from this expression. Such effects are handled as in earlier implementations, through construction of fast, splined look-up tables based on direct bandpass convolution for the relevant parameters; see @planck2013-p03d for an overview of the basic equations.
[^12]: In principle, it would be desirable to estimate the linear coefficients taking into account the spinorial character of the $Q$ and $U$ components, since this allows us to keep the physical coherence of the foreground residuals, following, for instance, the method proposed by [@fernandez-cobos2016]. However, in practice, this does not seem to have any significant impact on the results from data and, therefore, for simplicity, we work with independent coefficients for $Q$ and $U$ maps.
[^13]: Note that if a map is used to construct more than one template, several inpainted versions of that map will be constructed in the appropriate way in order to match the pair.
[^14]: In principle one could also include the cleaned 70- and 100-GHz maps in the combined solution. However, given the lower resolution and higher noise level of these channels, the improvement in the signal-to-noise ratio of the combined map is modest. Taking into account also that the addition of these channels could potentially introduce contamination from low-frequency foregrounds or CO emission, we decided to combine only the 143- and 217-GHz cleaned channels in the final map.
[^15]: Given that the amplitude of the CMB $B$-mode power spectrum is unknown, we cannot use spatial information on the CMB as a prior when performing on polarization data.
[^16]: The needlet windows satisfy the relation $\sum_{j=1}^{10} (h^{(j)}_\ell)^2 = 1$ to ensure the conservation of the total power when synthesizing all the needlet maps to reconstruct the complete map.
[^17]: In practice, the distinction between the two sets of eigenvalues is performed via the Akaike Information Criterion, which prevents the method from overfitting the foreground subspace.
[^18]: Those independent components are related to the subset of eigenvectors, or principal components, of the matrix ${\rm R_n}^{-1/2}\widehat{\rm R}_{\rm d}{\rm R_n}^{-1/2}$ for which the associated eigenvalues depart from unity.
[^19]: Needlet coefficients of frequency maps.
[^20]: Although all components are formally estimated without internal smoothing during the analysis, the resulting maps are completely noise dominated on small scales. In practice, each component map therefore needs to be smoothed to the resolution corresponding to the most relevant frequency map for visualization purposes.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Mobile wireless network research focuses on scenarios at the extremes of the network connectivity continuum where the probability of all nodes being connected is either close to unity, assuming connected paths between all nodes (mobile ad hoc networks), or it is close to zero, assuming no multi-hop paths exist at all (delay-tolerant networks). In this paper, we argue that a sizable fraction of networks lies between these extremes and is characterized by the existence of *partial paths*, multi-hop path segments that allow forwarding data closer to the destination even when no end-to-end path is available. A fundamental issue in such networks is dealing with disruptions of end-to-end paths. Under a stochastic model, we compare the performance of the established end-to-end retransmission (ignoring partial paths), against a forwarding mechanism that leverages partial paths to forward data closer to the destination even during disruption periods. Perhaps surprisingly, the alternative mechanism is not necessarily superior. However, under a stochastic monotonicity condition between current future path length, which we demonstrate to hold in typical network models, we manage to prove superiority of the alternative mechanism in stochastic dominance terms. We believe that this study could serve as a foundation to design more efficient data transfer protocols for partially-connected networks, which could potentially help reducing the gap between applications that can be supported over disconnected networks and those requiring full connectivity.'
author:
- |
[Simon Heimlicher, Merkouris Karaliopoulos, Hanoch Levy[^1] and Thrasyvoulos Spyropoulos]{}\
[Computer Engineering and Networks Laboratory]{}\
[ETH Zurich, Switzerland]{}
title: |
On Leveraging Partial Paths\
in Partially-Connected Networks\
[ **Technical Report TR-303**]{}
---
[^1]: Hanoch Levy is on leave absence from the School of Computer Science, Tel Aviv University, Israel
|
{
"pile_set_name": "ArXiv"
}
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[**** ]{}\
Marcel Ausloos ^1,2,$\natural$^, Roy Cerqueti ^3,$\natural$,\*^\
**1** School of Business, University of Leicester, University Road. Leicester, LE1 7RH, United Kingdom. Email: [email protected]\
**2** GRAPES – Group of Researchers for Applications of Physics in Economy and Sociology. Rue de la Belle Jardinière 483, B-4031, Angleur, Belgium. Email: [email protected]\
**3** University of Macerata, Department of Economics and Law. Via Crescimbeni 20, I-62100, Macerata, Italy. Tel.: +39 0733 258 3246; Fax: +39 0733 258 3205. Email: [email protected].\
$\natural$ These authors contributed equally to this work.
\* Corresponding author
Abstract {#abstract .unnumbered}
========
A mere hyperbolic law, like the Zipf’s law power function, is often inadequate to describe rank-size relationships. An alternative theoretical distribution is proposed based on theoretical physics arguments starting from the Yule-Simon distribution. A modeling is proposed leading to a universal form. A theoretical suggestion for the “best (or optimal) distribution”, is provided through an entropy argument. The ranking of areas through the number of cities in various countries and some sport competition ranking serves for the present illustrations.
Introduction
============
Approaches of hierarchical type lie behind the extensive use of models in theoretical physics [@Hansen14], the more so when extending them into new “applications” of statistical physics ideas [@roehner2007driving; @stauffer04], e.g. in complex systems [@kwapien2012physical] and phenomena, like in fluid mechanics [@nonl4; @gadomski2000kinetic] mimicking agent diffusion. In several studies, researchers have detected the validity of power laws, for a number of characteristic quantities of complex systems [@West; @z1; @pnas1; @pnas2; @ausloos2010physa]. Such studies, at the frontier of a wide set of scientific contexts, are sometimes tied to several issues of technical nature or rely only on the exploration of distribution functions. To go deeper is a fact of paramount relevance, along with the exploration of more grounding concepts.
The literature dealing with the rank-size rule is rather wide: basically, papers in this field discuss why such a rule should work (or does not work). Under this perspective, Pareto distribution and power law, whose statement is that there exists a link of hyperbolic type between rank and size, seem to be suitable for this purpose. In particular, the so-called first Zipf’s law [@z1], which is the one associated to a unitary exponent of the power law, has a relevant informative content, since the exponent can be viewed as a proxy of the balance between outflow and inflow of agents.
The theoretical explanation of the Zipf’s law has been the focus of a large number of important contributions [@Simon55; @gabaix; @Gabaix99b; @GabaixIoannides04; @Brakmanetal99; @HillJASA69.74.1017]. However, the reason why Zipf’s law is found to be a valid tool for describing rank-sizes rule is still a puzzle. In this respect, it seems that no theoretical ground is associated to such a statistical property of some sets of data [@Fujitaetal99; @FujitaThisse00].
Generally, Zipf’s law cannot be viewed as a universal law, and several circumstances rely on data whose rank and size relationship is not of hyperbolic nature. Such a statement is true even in the urban geography case, - the one of the original application of the Zipf’s law, for the peculiar case of cities ranking. Remarkable breakdowns has been assessed e.g. in [@CEU31.07.648benguigui; @Peng; @Matlabaetal; @ZDMA; @jsmte; @pnas3]. A further example is given by the number $N_{c,p}$ of cities ($c$) per provinces ($p$) in Italy ($\sum_{c,p} N_{c,p}=$ 8092), see the log-log plot of the data from 2011 in Fig. \[fig:Plotnogood9ZMPpwldexp\]: the (110) provinces are ranked by decreasing order of “importance”, i.e. their number of cities. Fits by (i) a power law, (ii) an exponential and (iii) a Zipf-Mandelbrot (ZM) function [@FAIRTHORNE]
$$\label{ZMeq3}
y(r)=\hat{c}/(\rho+r)^{\omega} \;\equiv \; [c/(\rho+r)]^{\omega},$$
$r$ being the rank.
The fits are, the least to say, quite unsatisfactory in particular in the high rank tail, essentially because data usually often presents an inflection point.
Therefore, no need to elaborate further that more data analysis can bring some information on the matter.
The paper is organized as follows. In Section \[alternative\], an alternative to a hyperbolic rank-size law and its above “improvements” are discussed: the data can be better represented by an (other than Zipf’s law) analytic empirical law, allowing for an inflection point.
Next, we introduce a universal form, allowing for a wider appeal, in Sect. \[universal\], based on a model thereafter presented in Sect. \[modelBeta\]. Such a general law can be turned into a frequency or probability distribution. Thus, the method suggests to consider a criterion of possibly optimal organization through the notion of relative distance to full disorder, i.e., a ranking criterion of entity distributions based on the entropy (Section \[sec:entropy\]). Section \[conclusions\] allows us to conclude and to offer suggestions for further research lines.
An alternative to a hyperbolic rank-size law {#alternative}
=============================================
In the context of best-fit procedures, rank-size theory allows to explore the presence of regularities among data and their specified criterion-based ranking [@Jefferson39]. Such regularities are captured by a best-fit curve. However, as observed in Fig. \[fig:Plotnogood9ZMPpwldexp\], the main problem strangely resides in missing the distribution high rank tail behavior. No doubt, that this partially arises because most fit algorithms take better care of the high values (on the $y$-axis) than the small ones. More drastically, a cause stems in the large rank $r$ tail which is usually supposed to extend to infinity, see Eq. (\[PWLwithcutoff\]), but each system is markedly always of finite size. Therefore, more complicated laws containing a power factor, like the stretched exponential or exponential cut-off laws should be considered inadequate.
We emphasize that we are in presence of data which often exhibits an inflection point.
The presence of an inflection point means that there is a change in the concavity of the curve, even if the slope remains with the same (negative) sign for the whole range. Thus, one could identify two regimes in the ranked data, meaning that the values are clustered in two families at a low and high ranks. In such cases, the finite cardinality $N$ of the dataset leads to a collapse of the upper regime at rank $r_M\equiv N$. Nevertheless, the Yule-Simon distribution [@Pwco3], $$\label{PWLwithcutoff}
y(r)= d \;r^{-\alpha} \; e^{-\lambda r},$$ could be arranged in an appropriate way, according to a Taylor expansion as in [@jsmte]. Eq. (\[PWLwithcutoff\]) can be then rewritten as $$\label{Lavalette3a}
y(r)= \kappa_3\; \frac{(N\;r)^{- \gamma}} { (N-r+1)^{-\xi} },$$ as discussed by Martinez-Mekler et al. [@Mekler] for rank-ordering distributions, - in the arts and sciences; see also more recent work on the subject [@1606.01959v1LavalbeyondZipfMiramontes] with references therein. This Eq. (\[Lavalette3a\]) is a (three-parameter) generalization of $$\label{Lavalette2}
y(r)=K\; \Big(\frac{N\cdot r}{N-r +1}\Big)^{-\beta}\;\equiv\; \kappa_2\; \Big(\frac{r }{N-r +1}\Big)^{-\beta},$$ the (t-parameter) function used when considering the distribution of impact factors in bibliometrics studies [@RRPh49.97.3popescu; @Glottom6.03.83popescu; @JoI1.07.155Mansilla; @JQL18.11.274Voloshynovska; @MAJAQM], i.e., when $ \gamma \equiv \xi$, and recently applied to religious movement adhesion [@MALavPRE]. Notice that there is no fundamental reason why the decaying behavior at low rank should have the same exponent as the collapsing regime at high rank: one should [*a priori*]{} admit $\gamma$ $\neq$ $\xi$.
![Relationship between the number $N_{c,p}$ of IT cities (8092) per provinces (110) on a log-log scale. The ranking criterion is the one associated to the number of cities (high rank when the number of cities is high). The reference year is 2011. Several fits are shown: power law, exponential and Zipf-Mandelbrot function, Eq. (\[ZMeq3\]). The corresponding correlation coefficients are given; different colors and symbols allow to distinguish cases.[]{data-label="fig:Plotnogood9ZMPpwldexp"}](1-Fig1.pdf){height="15.8cm" width="12.2cm"}
![Semi-log plot of the number of cities in IT provinces, $N_{c,p}$; the provinces are ranked by their decreasing “order of importance”, for various years; the 2007, 2008-2009 and 2010-2011; data are displaced by an obvious factor for better readability; the best 3-parameter function, Eq. (\[Lavalette3a\]), fit is shown. Parameter values are obtained by fits through Levenberg-Marquardt algorithms with a 0.01% precision.[]{data-label="fig:PlotliloNcp3f"}](2-Plot22liloNcpLav3_01.pdf){height="15.8cm" width="12.2cm"}
In fact, such an alternative law is easily demonstrated to be an appropriate one for describing size-rank data plots. For example, reconsider the IT $N_{c,p}$ case, shown in Fig. \[fig:Plotnogood9ZMPpwldexp\], redrawn on a semi-log plot in Fig. \[fig:PlotliloNcp3f\]. (All fits, in this communication, are based on the Levenberg-Marquardt algorithm [@Levenberg1944; @Marquardt1963; @Lourakis2011] with a 0.01% imposed precision and after testing various initial conditions for the regression process.) The rank-size relationship appears to follow a flipped noid function around some horizontal mirror or axis. Notice that similar behaviors are observed for different years, although the number of $N_{c,p}$ yearly differs. Incidently, note that, in this recent time, the official data claims a number of 103 provinces in 2007, with an increase by 7 units (BT, CI, FM, MB, OG, OT, VS, in conventional notations) thereafter, leading to 110 provinces. The number of municipalities has also been changing, between 2009 and 2010, whence the rank of a given province is not constant over the studied years.
Other illustrating topics and generalization {#universal2}
---------------------------------------------
In view of taking into account a better fit at low and high rank, one can generalize Eq. (\[Lavalette3a\]) to a five parameter free equation
$$\label{Lavalette5}
y(r)= \kappa_5\; \frac{(N\;(r+\Phi))^{- \gamma}} { (N+1-r+\Psi)^{-\xi} },$$
where the parameter $\Phi$ takes into account Mandelbrot generalization of Zipf’s law at low rank, see Eq. (\[ZMeq3\]), while $\Psi$ allows some flexibility at the highest rank.
In particular, the shape of the curve in Eq. (\[Lavalette5\]) is very sensitive to the variations of $\Phi$ and $\Psi$. As the parameter $\Phi$ increases, the relative level of the sizes at high ranks is also increased. This means that the presence of outliers at high ranks is associated to high values of $\Phi$. If one removes such outliers from the dataset and implements a new fit procedure, one obtains a lower level of the calibrated $\Phi$ and a flattening of the curve at low ranks. In [@AusloosandCerqueti15], the authors have found something similar in a different context; they have denoted the major (upsurging) outlier at rank $1$ by “king” and called the other outliers at ranks $2,3,\dots$ as “viceroys”. The removal of outliers necessarily leads to a more appealing fit , in terms of visualization and $R^2$, when such a procedure is implemented through power laws. In this respect, the introduction of a further parameter – $\Phi$, in this case – serves as adjustment term at high ranks, and represents an improvement of the previous theory. Indeed, the parameter $\Psi$ acts analogously to $\Phi$, but at a low rank. In particular, an increase of $\Psi$ is associated to a flattening of the five parameter curve of Eq. (\[Lavalette5\]) at medium and low ranks. Such a flattening is due to sizes at low ranks which are rather close to those at medium ranks. This phenomenon has been denoted in [@Ausloos13] as “queen” and “harem” effect, - to have in mind the corresponding “king” and “viceroys” effects at low ranks. The queen and harem effect is responsible of the deviations of the power law from the empirical data at a low rank. Thus, the parameter $\Psi$ also constitutes an adjustment term at low ranks and is an effective improvement of the performance of the fitting procedure. Substantially, the specific sense of $\Psi$ should be also read in terms of “generalization” and “in view of best fit”. Usually, one is not sure about the 0 at the origin of axes. Our $\Phi$ corresponds to the $\rho$ of Mandelbrot (see Eq. (\[ZMeq3\])), for which Mandelbrot gives no interpretation: it is only a mathematical trick. Thus, by “symmetry”, we introduce a $\Psi$ at high rank. It allows some flexibility due to possible sharp decays, due to outliers at high ranks. This also allows to move away from strict integers, and open the functions to continuous space as done in Sect. \[universal\].
We have compared the fits conceptualized in Eq. (\[Lavalette3a\]) and Eq. (\[Lavalette5\]) for the specific IT $N_{c,p}$ case (compare Fig. \[fig:PlotliloNcp3f\] and Fig. \[2nd-5\] and Table \[table2nd\]). Even if both these laws are visually appealing and exhibit a high level of goodness of fit, the $R^2$ associated to Eq. (\[Lavalette3a\]) is slightly lower than that of Eq. (\[Lavalette5\]). Thus, we can conclude that the five-parameters law, Eq. (\[Lavalette5\]), performs better than the three-parameters one, Eq. (\[Lavalette3a\]).
![Semi-log plot of the number of cities in IT provinces, $N_{c,p}$; the provinces are ranked according to their decreasing “order of importance”, for various years; the 2007 and 2010-2011 data are displaced by an obvious factor of 10 for better readability; the best 5-parameter function, Eq. (\[Lavalette5\]), fit is shown. Parameter values are obtained by fits through Levenberg-Marquardt algorithms with a 0.01% precision.[]{data-label="2nd-5"}](3new-Plot022liloNcpLav5_01.pdf){height="15.8cm" width="12.2cm"}
[|l|l|l|l|]{}\
&2007 &2008/2009 & 2010/2011\
$N$&103&110&110\
$\kappa_3 N^{-\gamma}$&2.049&18.177&182.265\
$\gamma$&0.301&0.316&0.316\
$\xi$&0.597&0.615&0.614\
$R^2$ &0.99240& 0.99445& 0.99441\
\
&2007 &2008/2009 & 2010/2011\
$N$&110&110&110\
$ \kappa_5 N^{-\gamma} $&3.971&33.709&332.71\
$\gamma$&0.373&0.387&0.386\
$\xi$&0.499&0.527&0.529\
$\Psi$&-7.441&0.608&0.640\
$\Phi$&0.945&0.926&0.906\
$R^2$ &0.99402& 0.99631& 0.99623\
\[table2nd\]
Even though one could display many figures describing the usefulness of the above, let us consider two cases, e.g. in sport matter.
- Consider the ranking of countries at recent Summer Olympic Games: Beijing 2008 and London 2012. The ranking of countries is performed trough the number of “gold medals”, but one can also consider the total number of medals, - thus considering a larger set of countries. A country rank is of course varying according to the chosen criterion. It is also true that due to subsequent analysis of athlete urine and other doping search tests, the attribution of medals may change with time. We downloaded the data available on Aug. 13, 2012, from
$ http://www.bbc.co.uk/sport/olympics/2012/medals/countries$.
Interestingly, the number of gold medals has not changed between Beijing and London, i.e. 302, but due to the “equivalence of athletic scores”, the total number of medals is slightly different : 958 $ \rightarrow$ 962. Moreover, the number of countries having received at least a gold medal is the same (54), but the total number of honored countries decreased from 86 to 85. Obviously, in contrast to the administrative data on IT provinces ranking, there is much “equality between countries” in Olympic Games; therefore a strict rank set contains many empty subsets. It is common to redefine a continuous (discrete) index $i$ in order to rank the countries. Moreover, the rank distributions are much positively skewed (skewness $\sim$ 3) with high kurtosis ($\ge 10$). Therefore, the inflection points occur near $r=r_M/2$ and for a size close to the median value. On Fig. \[fig:Plot15GoldBjGLDNLav4\] and Fig. \[fig:Plot15TotalBjGLDNLav4\], such a ranking for Olympic Games medals is displayed, both for the Gold medal ranking and the overall (“total”) medal ranking. Reasonably imposing $\Phi=0$, the parameters of Eq. (\[Lavalette5\]) lead to remarkable fits, even though the collapsing behavior of the function occurs outside the finite $N$ range. We have tested that a finite $\Phi $ does not lead to much regression coefficient $R^2$ improvement.
- In other sport competitions, the “quality” of teams or/and countries is measured through quantities which are not discrete values. For example, in soccer, more than 200 federations (called “Association Members”, $\sim$ countries) are affiliated to the FIFA
($http://www.fifa.com/worldranking/procedureandschedule/menprocedure/index.html$). The FIFA Country ranking system is based on results over the previous four years since July 2006. It is described and discussed in [@IJMPCFIFAMARCAGNV] to which we refer the reader for more information. Note that a few countries have zero FIFA coefficients. Interestingly the skewness and kurtosis of the FIFA coefficient distributions are rather “well behaved” (close to or $\le$1.0), while the coefficient of dispersion is about 250. From previous studies, it can be observed that the low rank (“best countries”) are well described by a mere power law, including the Mandelbrot correction to the Zipf’s law. However, the high tail behavior is poorly described. We show in Fig. \[fig:Plot61FIFA1213Lav5liloN206\] that the generalized equation is much better indeed. From a sport analysis point of view, one might wonder about some deviation in the ranking between 170 and 190.
![Semi-log plot of the number of Gold medals obtained by countries at Beijing (BJG) and London (LDN) recent Summer Olympic Games, as ranked according to their decreasing “order of importance” index $i$; the best 4-parameter fitting function is displayed, Eq. (\[Lavalette5\]), with $\Phi=0$. []{data-label="fig:Plot15GoldBjGLDNLav4"}](4-ex3-Plot15GoldBjGLDNLav4.pdf){height="15.8cm" width="12.2cm"}
![Semi-log plot of the total number of medals obtained by countries at Beijing (BJG) and London (LDN) recent Summer Olympic Games, as ranked according to their decreasing “order of importance” index $i$; the best 4-parameter fitting function is displayed, Eq. (\[Lavalette5\]), with $\Phi=0$. []{data-label="fig:Plot15TotalBjGLDNLav4"}](5-ex4-Plot15TotalBjGLDNLav4.pdf){height="15.8cm" width="12.2cm"}
![Semi-log plot of the FIFA countries ranked by their decreasing “order of importance” through the FIFA coefficient; the best 5-parameter function, Eq. (\[Lavalette5\]), is shown. []{data-label="fig:Plot61FIFA1213Lav5liloN206"}](6-ex5-Plot61FIFA1213Lav5liloN206.pdf){height="15.8cm" width="12.2cm"}
Universal form {#universal}
---------------
These displays suggest to propose some universal vision as presented next.
It is easily observed in Eq. (\[Lavalette3a\]) that a change of variables $u \; \equiv \; r/(N+1)$, leads to
$$\label{Lavalette3u}
y_1(u)= \hat{\kappa_3}\; \Big[ u^{-\gamma} (1-u)^{\xi} \Big]$$
However, in so doing, $u\in [ 1/(N+1), N/(N+1)]$.
In order to span the full $[0,1]$ interval, it is better to introduce the reduced variable $w$, defined as $w\equiv
(r-1)/(r_M-1)$, where $r_M$ is the maximum number of entities. Moreover, in order to fully generalize the empirical law, in the spirit of ZM, Eq. (\[ZMeq3\]), at low rank, a parameter $\phi$ can be introduced. In the same spirit, we admit a fit parameter $\psi$ allowing for possibly better convergence at $u\simeq1$; we expect, $\mu \sim 1/r_M$.
Thus, we propose the universal form $$\label{Lavalette5u}
y_2(w)= \eta\;\; (\phi+w)^{-\zeta}\;\;\Big[1-w+\psi\Big]^{\chi},$$ for which the two exponents $\chi$ and $\zeta$ are the theoretically meaningful parameters. The amplitude $\eta$ represents a normalizing factor, and can be then estimated. Indeed, by referring to the case $\chi \in (0,+\infty)$ and $\zeta\in (0,1)$ and posing $\tilde{w}=\phi+w$ and $u=1+\phi+\psi$, we can write $$\begin{aligned}
\label{etaLavalette5ri}\nonumber
\eta=\Big[ \int_{\tilde{w}_0}^{\tilde{w}_1} \tilde{w}^{-\zeta}
(u-\tilde{w})^{\chi} \;d\tilde{w} \Big]^{-1} \equiv \\
\frac{1}{(1+\phi+\psi)^{1+\chi-\zeta}}\;\;\frac{1}{[B_{t}(1-\zeta,1+\chi)]_{t_0}^{t_1}}
\;\;,\end{aligned}$$ with $t_0 = \phi/(1+\psi+\phi)$, and $t_1 =(1+
\phi)/(1+\psi+\phi)$, and where $B_t (x, y) $ is the incomplete Euler Beta function [@AbraSteg; @GradRyz; @PearsonBTables], itself easily written, when $t=1$, in terms of the Euler Beta function, $$B(x, y)\equiv B_1(x, y) = \frac{ \Gamma(x) \Gamma(y)}{\Gamma(x+y)};$$ $\Gamma(x)$ being the standard Gamma function.
![Semi-log plot of the number of cities, $N_{c,p}$ and $N_{c,d}$, ranked by decreasing order of “importance” -in the sense of “number of cities”- of provinces (in BE, BG, and IT) or departments (in FR); the best function fit, Eq. (\[Lavalette5u\]), is shown; parameter values are found in Table \[Tablecitprovreg\].[]{data-label="fig:Plot23BEGFRITuNcpdL"}](7-ex6-Plot23BEGFRITuNcpdL.pdf){height="15.8cm" width="12.2cm"}
The function in Eq. (\[Lavalette5u\]) is shown on Fig. \[fig:Plot23BEGFRITuNcpdL\] to describe different cases, with various orders of magnitude, i.e., a semi-log plot of the number of cities in a province, $N_{c,p}$ or in a department, $N_{c,d}$, ranked by decreasing order of “importance”, for various countries (BE, BG, FR, IT). The reference year is 2011. In such cases, $\phi\equiv0$, obviously, thereby much simplifying Eq. (\[etaLavalette5ri\]), whence reducing the fit to a three free parameter search.
For completeness, the main statistical indicators for the number of cities ($N_c $), in the provinces ($N_{c,p}$), regions ($N_{c,r}$) or departments ($N_{c,d}$) in these (European) countries, in 2011 is given in Table \[Tablecitprovreg\]. Notice that the distributions differ: the median ($m$) is sometimes larger (or smaller) than the mean ($\mu$), while the kurtosis and skewness can be positive or negative. Yet the fits with Eq. (\[Lavalette5u\]) seem very fine. The large variety in these characteristics is an [*a posteriori*]{} argument in favor of having examined so many cases.
Modelization {#modelBeta}
------------
The presented argument is of wide application as the reader can appreciate. However, the vocabulary in this modeling section can be adequately taken from the jargon of city evolution for better phrasing and for continuing with the analyzed data.
A preferential attachment process can be defined as a settlement procedure in urn theory, where additional balls are added and distributed continuously to the urns (areas, in this model) composing the system. The rule of such an addition follows an increasing function of the number of the balls already contained in the urns.
In general, such a process contemplates also the creation of new urns. In such a general framework, this model is associated to the Yule-Simon distribution, whose density function $f$ is $$f(a;b) = b\,B(a, b+1),$$ being $a$ and $b$ real nonnegative numbers.
The integral $\int^1_0 x^a\;(1-x)^b\;dx$ represents the probability of selecting $a+b+1$ real numbers such that the first one coincides with $x$, from the second to the $a+1$-th one numbers are less or equal to $x$ and the remaining $b$ numbers belong to $[x,1]$.
In practical words, newly created urn starts out with $k_0$ balls and further balls are added to urns at a rate proportional to the number $k$ that they already have plus a constant $a\ge -k_0$. With these definitions, the fraction $P(k)$ of urns (areas) having $k$ balls (cities) in the limit of long time is given by $$P(k) = \frac{ B(k+a;b)}{B(k_0+a;b-1)}$$ for $k\ge0$ (and zero otherwise). In such a limit, the preferential attachment process generates a “long-tailed” distribution following a hyperbolic (Pareto) distribution, i.e. power law, in its tail.
It is important to note that the hypothesis of continuously increasing urns is purely speculative, even if it is widely adopted in statistical physics. Indeed, such an assumption contrasts with the availability of resources, and the growth of the number of settlements is then bounded. Therefore, as in Verhulst’s modification [@Vlog3] of the Keynesian expansion model of population, a “capacity factor” must be introduced in the original Yule process, thereby leading to the $u$ term in Eq. (\[Lavalette5\]) and its subsequent interpretation.
Entropy connection {#sec:entropy .unnumbered}
==================
One can consider to have access to a sort of “probability” for finding a certain “state” (size occurrence) at a certain rank, through $$\label{pr}\nonumber
p(w) \sim y_2(w)\sim \frac{(\phi+w)^{-\zeta}(1-w+\psi)^{\chi}}{
(1+\phi+\psi)^{\chi-\zeta+1}B(\chi+1,1-\zeta)}\;,$$ the denominator resulting from Eq. (\[etaLavalette5ri\]).
Thereafter, one can obtain something which looks like the Shannon entropy [@shannon] : $ S\equiv -\int p(w)\; ln
(p(w))$. It has to be compared to the maximum disorder number, i.e. $ln (N)$. Whence we define the relative distance to the maximum entropy as $$\label{d}d= \frac {S
}{ln(N)}-1.$$ As a illustration, the only case of the ranking of cities in various countries is discussed. Values are reported in Table \[Tablecitprovreg\]. It is observed that the FR and IT $d$-values are more extreme than those of BG and BE. This corroborates the common knowledge that the former two countries have too many cities, in contrast to the latter two.
Thus, in this particular case, this distance concept based on the universal ranking function with the two exponents $\zeta$ and $\chi$ shows its interest, e.g. within some management or control process. It can be conjectured without much debate that this concept can be applied in many other cases.
It is relevant to note that the entropy argument can be extended in a natural way to the $q$-Tsallis statistics analysis. Such an extension could add further elements to the thermodynamic interpretation of the proposed rank-size analysis. More in details, rank-size law might be associated to $q$-Tsallis distribution through a generalization of the central limit theorem for a class of non independent random variables (see e.g. [@moyanotsallis] and [@naumiscocho2007]). However, the Tsallis approach is well-beyond the aim of the present study, and we leave this issue to future research.
Conclusions
===========
This paper provides a basically three parameter function for the rank-size rule, based on preferential attachment considerations and strict input of finite size sampling. The analysis of the distribution of municipalities in regions or departments has proven the function value after its mapping into “dimensionless variables”. It seems obvious that the approach is very general and not limited to this sort of data. Other aspects suggest to work on theoretical improvements of the rank-size law connections, through ties with thermodynamics features, e.g., entropy and time-dependent evolution equations ideas.
0.2cm [**Acknowledgements**]{} 0.2cm This paper is part of scientific activities in COST Action IS1104, “The EU in the new complex geography of economic systems: models, tools and policy evaluation” and in COST Action TD1210 ’Analyzing the dynamics of information and knowledge landscapes’.
$ $ BE BG FR IT
--------------------- --------- -------- -------- -------- --
$N_c $ 589 264 36683 8092
$N_x$ ($x=p,d$) 11 28 101 110
Min 19 10 1 6
Max 84 22 895 315
Mean ($\mu$) 53.55 9.429 363.2 73.56
Median 64 10 332 60
Std Dev ($\sigma$) 20.32 4.273 198.3 55.34
Skewness -0.311 0.781 0.332 1.729
Kurtosis -1.045 1.480 -0.286 3.683
$\mu/\sigma$ 2.635 2.207 1.832 1.329
$3(\mu-m)/\sigma$ -1.543 -0.401 0.472 0.735
$N$ 11 28 101 110
$\kappa_5$ 7.49 10.28 84.69 203.8
$\gamma$ -0.160 0.157 0.133 0.386
$\xi$ 0.631 0.310 0.653 0.529
$\Psi$ -0.0399 -0.985 -0.999 0.640
$\Phi$ 17.56 -0.820 0.265 0.906
$R^2$ 0.958 0.975 0.990 0.996
$ln(N_x)$ ($x=p,d$) 2.3979 3.3322 4.6151 4.7005
$d$ 0.1587 0.1959 0.3793 0.2451
: Statistical characteristics of the distribution of the Number of cities $N_c$, number of provinces $N_p$ or departments, $N_d$ (in FR), in 2011, in 4 European countries; relevant fit exponents with Eq. (\[Lavalette5\]), and entropic distance $d$.
\[Tablecitprovreg\]
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---
abstract: 'We study how the dominant single and double SM-like Higgs ($h$) production at future $e^+e^-$ colliders is modified in the Georgi-Machacek (GM) model. On imposing theoretical, indirect and direct constraints, significant deviations of $h$-couplings from their SM values are still possible; for instance, the Higgs-gauge coupling coupling can be corrected by a factor $\kappa_{hVV}\in[0.93,1.15]$ in the allowed parameter space. For the Higgs-strahlung $e^+e^-\to hZ$ and vector boson fusion processes $e^+e^-\to h\nu\bar{\nu},~he^+e^-$, the cross section could increase by $32\%$ or decrease by $13\%$. In the case of associated production with a top quark pair $e^+e^-\to ht\bar{t}$, the cross section can be enhanced up to several times when the custodial triplet scalar $H_3^0$ is resonantly produced. In the meanwhile, the double Higgs production $e^+e^-\to hhZ~(hh\nu\bar{\nu})$ can be maximally enhanced by one order of magnitude at the resonant $H_{1,3}^0$ production. We also include exclusion limits expected from future LHC runs at higher energy and luminosity and discuss their further constraints on the relevant model parameters. We find that the GM model can result in likely measurable deviations of Higgs production from the SM at future $e^+e^-$ colliders.'
author:
- 'Bin Li$^{1}$'
- 'Zhi-Long Han$^{2}$'
- 'Yi Liao$^{1,3,4}$'
title: 'Higgs production at future $e^+e^-$ colliders in the Georgi-Machacek model'
---
Introduction {#sec:introduction}
============
The discovery of a 125 GeV scalar at the Large Hadron Collider (LHC) [@Aad:2012tfa; @Chatrchyan:2012xdj] confirmed the Higgs mechanism of electroweak symmetry breaking in the standard model (SM) [@Englert:1964et; @Higgs:1964ia; @Higgs:1964pj; @Guralnik:1964eu; @Higgs:1966ev; @Kibble:1967sv]. Yet the sector that triggers the symmetry breaking remains to be disclosed. With an elementary scalar the SM is also confronted with issues such as gauge hierarchy and flavor problems. To solve or relax some of these issues, there have been various theoretical attempts beyond the SM whose phenomenologies have been extensively studied, such as in the minimal supersymmetric model (MSSM) [@Nilles:1983ge; @Haber:1984rc; @Djouadi:2005gj; @Cao:2012fz], two Higgs doublet models (THDM) [@Ma:2000cc; @Davidson:2009ha; @Haba:2011nb; @Branco:2011iw; @Wang:2016vfj; @Guo:2017ybk], little Higgs models [@ArkaniHamed:2001nc; @ArkaniHamed:2002pa; @ArkaniHamed:2002qx; @Low:2002ws; @Han:2003wu], and composite Higgs models [@Agashe:2004rs; @Gripaios:2009pe; @Marzocca:2012zn], to mention a few with a modified Higgs sector.
To unravel the symmetry breaking sector it is necessary to measure the interactions of the Higgs boson with itself and other related particles. While this is believed to be very challenging at the LHC [@Duhrssen:2004cv], future lepton colliders provide an avenue to study these interactions at a reasonably good precision due to cleaner environment [@Lafaye:2017kgf]. There have been several proposals for next-generation $e^+e^-$ colliders that are under active studies, including the Circular Electron-Positron Collider (CEPC) [@CEPC-SPPCStudyGroup:2015csa; @CEPC-SPPCStudyGroup:2015esa], the Future Circular Collider (FCC-ee) [@Gomez-Ceballos:2013zzn], the Compact Linear Collider (CLIC) [@Aicheler:2012bya; @Abramowicz:2016zbo], and the International Linear Collider (ILC) [@Baer:2013cma; @Asner:2013psa; @Fujii:2015jha]. These colliders are planned to operate at a center of mass (CM) energy ranging from about 250 GeV to 3 TeV, thereby making accessible most of dominant production processes of the Higgs boson.
At an $e^+e^-$ collider of high enough energy one could study simultaneously the interactions of the Higgs boson $h$ with itself and with gauge bosons $W^\pm,~Z$ or even fermions such as the top quark $t$. This would be very helpful for us to build an overall picture on the symmetry breaking sector and gain a hint on possible physics that goes beyond the SM. The $hZZ$ coupling can be measured via the Higgs-strahlung process $e^+e^-\to hZ$ and the $ZZ$ fusion process $e^+e^-\to (ZZ\to h)e^+e^-$, while the $hWW$ coupling can be probed via the $WW$ fusion process $e^+e^-\to (WW\to h)\nu\bar{\nu}$. The top Yukawa coupling can possibly be extracted from the associated production process $e^+e^-\to ht\bar{t}$. And finally, the Higgs pair production processes $e^+e^-\to hh\nu\bar{\nu},~hhZ$ provide an access to the trilinear coupling of the Higgs boson. These processes are generally modified by new interactions or new heavy particles, and precise measurements on them could help us identify the imprints of physics beyond the SM [@Yue:2003yk; @Yue:2005av; @Liu:2006rc; @Arhrib:2008jp; @LopezVal:2009qy; @Asakawa:2010xj; @Heng:2013wia; @Yang:2014tia; @Yang:2014gca; @Liu:2014uua; @Han:2015orc; @Antusch:2015gjw; @Kanemura:2016tan; @Khosa:2016jly; @DeCurtis:2017gzi; @Guo:2017ugk; @Gu:2017ckc].
In this work we will study the above mentioned dominant single and double Higgs production at future $e^+e^-$ colliders in the Georgi-Machacek (GM) model [@Georgi:1985nv; @Chanowitz:1985ug]. The model is interesting because it introduces weak isospin-triplet scalars in a manner that preserves the custodial $SU(2)_V$ symmetry. While this symmetry guarantees that the $\rho$ parameter is naturally unity at the tree level, the arrangement of the triplet scalars allows them to develop a vacuum expectation value (VEV) as large as a few tens of GeV. After spontaneous symmetry breaking there remain ten physical scalars that can be approximately classified into irreducible representations of $SU(2)_V$, one quintuplet, one triplet and two singlets. These multiplets couple to gauge bosons and fermions differently. In particular, the couplings of the SM-like Higgs boson $h$ can be significantly modified, and processes involving $h$ receive additional contributions from new scalars as intermediate states. It is a distinct feature of the GM model compared to a scalar sector with only singlet and doublet scalars that the $h$ couplings to fermions and gauge bosons may be enhanced, or always enhanced in the case of the quartic couplings to a gauge boson pair.
The GM model has been extended by embedding it in more elaborate theoretical scenarios such as little Higgs [@Chang:2003un; @Chang:2003zn] and supersymmetric models [@Cort:2013foa; @Vega:2017gkk], by generalizing it to larger $SU(2)$ multiplets [@Logan:2015xpa] or including dark matter [@Campbell:2016zbp; @Pilkington:2017qam]. The phenomenology of exotic scalars has previously been studied, including searches for exotic scalars and the application of a variety of constraints on the model parameter space [@Haber:1999zh; @Godfrey:2010qb; @Logan:2010en; @Chang:2012gn; @Chiang:2012cn; @Kanemura:2013mc; @Englert:2013zpa; @Englert:2013wga; @Chiang:2013rua; @Efrati:2014uta; @Chiang:2014hia; @Godunov:2015lea; @Chiang:2015rva; @Chiang:2015amq; @Degrande:2015xnm; @Arroyo-Urena:2016gjt; @Blasi:2017xmc; @Zhang:2017och; @Chiang:2017vvo; @Degrande:2017naf; @Logan:2017jpr; @Krauss:2017xpj; @Sun:2017mue]; in particular, previous works on $e^+e^-$ colliders [@Chiang:2015rva; @Zhang:2017och] mainly concentrated on the custodial quintuplet particles. When these exotic scalars are heavy, it is difficult to produce them directly even at LHC, but we will show that they could be probed indirectly at $e^+e^-$ colliders via modifications to the SM-like Higgs production processes. If the new scalars are light enough, they could contribute as resonances in those processes and thus affect them more significantly. In either case, high energy $e^+e^-$ colliders could provide a viable way to test the GM model.
This paper is organized as follows. We recall in the next section the basic features in the Higgs sector of the Georgi-Machacek model. We discuss in Section \[sec:constraints\] various constraints on the model parameter space coming from current and future LHC runs as well as from low energy precision measurements and theoretical considerations. Then we investigate various SM-like Higgs production processes in Section \[sec:phenomenology\] at the $500~{{\rm GeV}}$ and $1~{{\rm TeV}}$ ILC. We reserve for our future work a comparative study of electron colliders operating at various energies and luminosities which requires a detailed simulation of the relevant processes. We finally summarize our main results in Section \[sec:conclusion\]. Some lengthy coefficients and Higgs trilinear couplings are delegated to Appendixes \[app:htt\], \[app:trilinear\], and \[app:hhvv\].
The Georgi-Machacek Model {#sec:GM model}
=========================
The model contains the usual Higgs doublet scalar $\phi$ with hypercharge $Y=1/2$ and introduces a new complex triplet scalar $\chi$ with $Y=1$ and a new real triplet scalar $\xi$ with $Y=0$. To make the custodial symmetry manifest, it is convenient to recast them in a matrix form: $$\begin{aligned}
\Phi=
\begin{pmatrix}
\phi^{0*}&\phi^+\\
-\phi^- &\phi^0
\end{pmatrix},~
\Delta=
\begin{pmatrix}
\chi^{0*}&\xi^+ &\chi^{++}\\
-\chi^- &\xi^0 &\chi^+\\
\chi^{--}&-\xi^-&\chi^0
\end{pmatrix},\end{aligned}$$ using the phase convention, $\chi^{--}=(\chi^{++})^*,~\chi^-=(\chi^+)^*,~\xi^-=(\xi^+)^*,~\phi^-=(\phi^+)^*$, and $\xi^0=(\xi^0)^*$. The matrices $\Phi$ and $\Delta$ transform under $SU(2)_L\times SU(2)_R$ as $\Phi\to U_L\Phi U_R^\dagger$ and $\Delta\to U_L\Delta U_R^\dagger$ with $U_{L,R}=\exp(i\theta_{L,R}^a T^a)$, where, for $\Phi$, $T^a=\tau^a/2$ with $\tau^a$ being the Pauli matrices, and for $\Delta$, $T^a=t^a$ are $$\begin{aligned}
t^1=\frac{1}{\sqrt{2}}
\begin{pmatrix}
0&~~1&~~0\\
1&~~0&~~1\\
0&~~1&~~0
\end{pmatrix},~
t^2=\frac{1}{\sqrt{2}}
\begin{pmatrix}
0&~~-i&~~0\\
i&~~0&~~-i\\
0&~~i&~~0
\end{pmatrix},~
t^3=
\begin{pmatrix}
1&~~0&~~0\\
0&~~0&~~0\\
0&~~0&~~-1
\end{pmatrix}.\end{aligned}$$
The most general scalar potential invariant under $SU(2)_L\times SU(2)_R\times U(1)_Y$ is given by [@Hartling:2014zca] $$\begin{aligned}
\nonumber
V_H=&
\frac{1}{2}\mu_2^2{{\rm tr}}(\Phi^\dagger \Phi)+\frac{1}{2}\mu_3^2{{\rm tr}}(\Delta^\dagger \Delta)+\lambda_1[{{\rm tr}}(\Phi^\dagger\Phi)]^2+\lambda_2{{\rm tr}}(\Phi^\dagger\Phi){{\rm tr}}(\Delta^\dagger \Delta)
\\
\nonumber
&+\lambda_3{{\rm tr}}(\Delta^\dagger \Delta \Delta^\dagger \Delta)+\lambda_4[{{\rm tr}}(\Delta^\dagger \Delta)]^2-\lambda_5{{\rm tr}}(\Phi^\dagger \tau^a \Phi \tau^b){{\rm tr}}(\Delta^\dagger t^a \Delta t^b)
\\
&-M_1{{\rm tr}}(\Phi^\dagger\tau^a\Phi\tau^b)(U\Delta U^\dagger)_{ab}-M_2{{\rm tr}}(\Delta^\dagger t^a \Delta t^b)(U \Delta U^\dagger)_{ab},\end{aligned}$$ where all free parameters are real and the matrix $U$ is [@Aoki:2007ah] $$U=
\begin{pmatrix}
-\frac{1}{\sqrt{2}}&~~0&~~\frac{1}{\sqrt{2}}\\
-\frac{i}{\sqrt{2}}&~~0&~~-\frac{i}{\sqrt{2}}\\
0&~~1&~~0
\end{pmatrix}.$$ The spontaneous symmetry breaking is triggered by the VEVs $\langle\Phi\rangle=1_2v_\phi/\sqrt{2}$ and $\langle\Delta\rangle=1_3v_\Delta$. As usual, the weak gauge bosons obtain masses from the kinetic terms of the scalars $$\label{equ:lkin}
\mathcal{L}=\frac{1}{2}{{\rm tr}}[(D^\mu \Phi)^\dagger D_\mu \Phi]+\frac{1}{2}{{\rm tr}}[(D^\mu \Delta)^\dagger D_\mu \Delta],$$ where $$\begin{aligned}
\nonumber
D_\mu\Phi &= \partial_\mu \Phi + ig_2 A_\mu^a \frac{\tau^a}{2}\Phi
-ig_1B_\mu\Phi \frac{\tau^3}{2},
\\
D_\mu\Delta &= \partial_\mu \Delta + ig_2 A_\mu^a t^a\Delta
-ig_1B_\mu\Delta t^3,\end{aligned}$$ with $g_{2,1}$ being the gauge couplings of $SU(2)_L\times U(1)_Y$. Their squared masses are $m_W^2=g_2^2v^2/4$ and $m_Z^2=(g_1^2+g_2^2)v^2/4$ where $$\begin{aligned}
v^2=v_\phi^2+8v_\Delta^2,\end{aligned}$$ which should be identified with $1/(\sqrt{2}G_F)$ where $G_F$ is the Fermi constant. The parameter $\rho=1$ is thus established at the tree level. Since the custodial symmetry is explicitly broken by hypercharge and Yukawa couplings, divergent radiative corrections to $\rho$ will generally appear at one loop within the framework of the GM model [@Gunion:1990dt].
Restricting our discussions to the tree level, the custodial symmetry is respected by the scalar spectra so that the scalars can be classified into irreducible representations of $SU(2)_V$. Excluding the would-be Nambu-Goldstone bosons to be eaten up by the weak gauge bosons, they are decomposed into a quintuplet, a triplet and two singlets. Denoting the real and imaginary parts of the neutral components of the original fields after extracting out the VEVs, $$\begin{aligned}
\phi^0=\frac{1}{\sqrt{2}}(v_\phi+\phi^r+i\phi^i),~
\chi^0=v_\Delta+\frac{1}{\sqrt{2}}(\chi^r+i\chi^i),~
\xi^0=v_\Delta+\xi^r,\end{aligned}$$ the quintuplet states are [@Hartling:2014zca] $$\begin{aligned}
H_5^{++}&=\chi^{++},
\nonumber
\\
H_5^+&=\frac{1}{\sqrt{2}}(\chi^+-\xi^+),
\nonumber
\\
H_5^0&=\sqrt{\frac{2}{3}}\xi^r-\sqrt{\frac{1}{3}}\chi^r,\end{aligned}$$ plus $H_5^{--,-}=(H_5^{++,+})^*$. The triplet states are $$\begin{aligned}
H_3^+&=-s_H\phi^++c_H\frac{1}{\sqrt{2}}(\chi^++\xi^+),
\nonumber
\\
H_3^0&=-s_H\phi^i+c_H\chi^i,\end{aligned}$$ plus $H_3^-=(H_3^+)^*$, where the doublet-triplet mixing angle $\theta_H$ is defined by $$\begin{aligned}
\label{eqn:chsh}
c_H\equiv\cos \theta_H=\frac{v_\phi}{v},~
s_H\equiv\sin \theta_H=\frac{2\sqrt{2}v_\Delta}{v}.\end{aligned}$$ We denote the quintuplet and triplet masses as $m_5$ and $m_3$ respectively. At the Lagrangian level, the quintuplet scalars couple to the electroweak gauge bosons but not to fermions (i.e., with $H_5VV$ but without $H_5f\bar{f}$ couplings), while the opposite is true for the triplet scalars (with $H_3f\bar{f}$ but without $H_3VV$ couplings).
The two custodial singlets mix by an angle $\alpha$ into the mass eigenstates $$\begin{aligned}
h&=c_\alpha \phi^r-s_\alpha \left(\sqrt{\frac{1}{3}}\xi^r+\sqrt{\frac{2}{3}}\chi^r\right),
\nonumber
\\
H_1^0&=s_\alpha \phi^r + c_\alpha \left(\sqrt{\frac{1}{3}}\xi^r+\sqrt{\frac{2}{3}}\chi^r\right),\end{aligned}$$ with $c_\alpha=\cos\alpha$ and $s_\alpha=\sin\alpha$. We assume that the lighter state $h$ is the observed 125 GeV scalar [@Aad:2012tfa; @Chatrchyan:2012xdj; @Aad:2015zhl]. The angle is determined by [@Hartling:2014zca] $$\begin{aligned}
\label{eq:mixing_angle_singlet}
\sin 2\alpha =\frac{2M_{12}^2}{m_{1}^2-m_h^2},~
\cos 2\alpha =\frac{M_{22}^2-M_{11}^2}{m_{1}^2-m_h^2},\end{aligned}$$ where $m_{h,1}$ are the masses of $h$ and $H_1^0$ respectively, and in terms of the scalar couplings and VEVs, $$\begin{aligned}
\nonumber
M_{11}^2=&8\lambda_1v_\phi^2,
\\
\nonumber
M_{12}^2=&\frac{\sqrt{3}}{2}v_\phi\left[-M_1+4(2\lambda_2-\lambda_5)v_\Delta\right],\\
M_{22}^2=&\frac{M_1v_\phi^2}{4v_\Delta}-6M_2v_\Delta+8(\lambda_3+3\lambda_4)v_\Delta^2.\end{aligned}$$
Constraints on Parameter Space {#sec:constraints}
==============================
![Distribution of survived points in the $\alpha-v_\Delta$ plane under theoretical and indirect constraints generated by GMCALC. The black, red, yellow, and green points are further excluded by direct searches for $H_1^0$, $H_5$, and $H_3$ and by Higgs signal strength analysis respectively, while the circular blue points pass all current constraints. The green, magenta, orange, and black curves are exclusion bounds expected from projection results of 14 TeV LHC with $300~{{\rm fb}}^{-1}$, HL-LHC with $3000~{{\rm fb}}^{-1}$, and ILC at 250 and 500 GeV.[]{data-label="fig:constraints"}](fig00.pdf){width="0.45\linewidth"}
There are generally many free parameters in the scalar potential with an extended scalar sector. In this section, we will perform a combined analysis based on theoretical considerations and indirect and direct constraints to obtain the allowed regions for the parameters that are most relevant for our later Higgs production processes. The theoretical constrains are mainly derived from the requirement of perturbativity and vacuum stability [@Aoki:2007ah; @Hartling:2014zca], while the indirect ones cover the oblique parameters ($S,~T,~U$), the $Z$-pole observables ($R_b$), and the $B$-meson observables [@Hartling:2014aga; @Hartling:2014zca]. Among the $B$-meson observables ($B_s^0-\bar{B}_s^0$ mixing, $B_s^0\to \mu^+\mu^-$, and $b\to s\gamma$), the decay $b\to s\gamma$ currently sets the strongest bound. All of these constraints have been implemented in the calculator GMCALC [@Hartling:2014xma] for the GM model, which will be applied as our starting point. On top of this we will impose up-to-date direct experimental constraints which cover the searches for a heavy neutral Higgs boson ($H_1^0$), custodial triplet bosons ($H_3$) and quintuplet bosons ($H_5$), and the signal strength analysis of the SM-like Higgs boson ($h$). For the signal strength analysis, we will also include the constraints expected from future runs of LHC and the proposed ILC. In Fig. \[fig:constraints\] we show how survived points in the $\alpha-v_\Delta$ plane evolve with the inclusion of various constraints. With theoretical and indirect constraints alone, a $v_\Delta$ as large as $60~{{\rm GeV}}$ is still allowed. When the constraints from direct searches for $H_1^0$ (black points), $H_5$ (red), $H_3$ (yellow) and from the Higgs signal strength (green) are included, more and more points are excluded. At this stage, we have $v_\Delta\lesssim40~{{\rm GeV}}$, and $\alpha<0$ is preferred. Also shown in the figure are the future prospects of constraints derived from Higgs signal strength measurements at 14 TeV LHC with $300~{{\rm fb}}^{-1}$ (green curve), High-Luminosity LHC (HL-LHC) with $3000~{{\rm fb}}^{-1}$ (magenta) [@ATLAS:HL-LHC], ILC with $1150~{{\rm fb}}^{-1}$ at $250~{{\rm GeV}}$ (orange) and $1600~{{\rm fb}}^{-1}$ at $500~{{\rm GeV}}$ (black) [@Asner:2013psa]. It is clear that a wide parameter space will be within the reach of future ILC operations. In the following subsections we will describe these direct experimental constraints in more detail.
Singlet Searches
----------------
![Distribution of survived points shown in the $v_\Delta-m_1$ plane (left) and $\alpha-m_1$ plane (right). The red, blue, and green points are excluded by the $H_1^0\to WW,
~ZZ,~hh$ searches respectively.[]{data-label="fig:H1"}](figH1.pdf "fig:"){width="0.45\linewidth"} ![Distribution of survived points shown in the $v_\Delta-m_1$ plane (left) and $\alpha-m_1$ plane (right). The red, blue, and green points are excluded by the $H_1^0\to WW,
~ZZ,~hh$ searches respectively.[]{data-label="fig:H1"}](figH1-2.pdf "fig:"){width="0.465\linewidth"}
In the GM model, the heavy neutral Higgs boson $H_1^0$ can decay to a pair of vector bosons when it is above the threshold. Searches for heavy resonances decaying to a $WW$ [@ATLAS:2016kjy] and $ZZ$ [@ATLAS:2016npe] pair are performed by the ATLAS Collaboration using the data collected at $\sqrt{s}=13~{{\rm TeV}}$ with an integrated luminosity of $13.2~{{\rm fb}}^{-1}$. For the gluon fusion (ggF) and vector boson fusion (VBF) production of the heavy Higgs, the corresponding cross sections are calculated as $$\begin{aligned}
\label{equ:ggf}
\nonumber
\sigma_{\text{ggF}}^{\text{GM}}=&\sigma_{\text{The}}(gg\to H_1^0)\times\kappa_{H_1^0f\bar{f}}^2\times \text{BR}_{\text{GM}}(H_1^0\to VV),\\
\sigma_{\text{VBF}}^{\text{GM}}=&\sigma_{\text{The}}(qq\to H_1^0)\times\kappa_{H_1^0VV}^2\times \text{BR}_{\text{GM}}(H_1^0\to VV),\end{aligned}$$ where $$\begin{aligned}
\label{equ:heavy higgs}
\nonumber
\kappa_{H_1^0f\bar{f}}&\equiv \frac{g_{H_1^0f\bar{f}}}{g_{hf\bar{f}}^\text{SM}}
=\frac{s_\alpha}{c_H},
\\
\kappa_{H_1^0VV}&\equiv \frac{g_{H_1^0VV}}{g_{hVV}^\text{SM}}
=s_\alpha c_H+\frac{2\sqrt{6}}{3}c_\alpha s_H,\end{aligned}$$ with $g$s denoting the couplings in the SM and the GM model. The theoretical cross sections of a SM-like heavy Higgs $\sigma_{\text{The}}(gg\to H_1^0)$ and $\sigma_{\text{The}}(qq\to H_1^0)$ have been tabulated in Ref. [@Heinemeyer:2013tqa], while the branching ratio $\text{BR}_{\text{GM}}(H_1^0\to VV)$ is obtained using GMCALC.
When $m_1>2m_h$, $H_1^0$ can also decay into a Higgs pair $hh$, which may greatly enhance the Higgs pair production at the LHC. The cross section for resonant production of a Higgs boson pair is given by [@Chang:2017niy] $$\sigma(pp\to H_1^0 \to hh) = \sigma_{\text{The}}(gg\to H_1^0)\times\kappa_{H_1^0f\bar{f}}^2\times \text{BR}_{\text{GM}}(H_1^0\to hh),$$ where $\text{BR}_{\text{GM}}(H_1^0\to hh)$ is also calculated by GMCALC. Recently, a search for resonant Higgs boson pair production ($H_1^0\to hh$) has been performed by the CMS Collaboration [@Sirunyan:2017guj] in the $b\bar{b}\ell\ell \nu\nu$ final state.
In Fig. \[fig:H1\] we show in the $v_\Delta-m_1$ and $\alpha-m_1$ planes the survived points upon applying the constraints from the direct searches $H_1^0\to WW$, $H_1^0\to ZZ$, and $H_1^0\to hh$. While the $H_1^0\to WW,~ZZ$ searches remove only a small portion of points, a considerable portion is excluded by the $H_1^0\to hh$ search. Due to large variations of BR($H_1^0\to hh$) in our scan [@Chang:2017niy], no clear dependence of the exclusion bounds on $v_\Delta$ and $\alpha$ is visible.
Triplet Searches
----------------
![Distribution of survived points shown in the $v_\Delta-m_3$ plane (left) and $\alpha-m_3$ plane (right). The green, blue, red, and purple points are excluded by the $H_3^0\to hZ$, $H_3^0\to t\bar{t}$, $H^\pm\to tb$, and $t\to H_3^+ b$ searches respectively.[]{data-label="fig:H3"}](figH3.pdf "fig:"){width="0.45\linewidth"} ![Distribution of survived points shown in the $v_\Delta-m_3$ plane (left) and $\alpha-m_3$ plane (right). The green, blue, red, and purple points are excluded by the $H_3^0\to hZ$, $H_3^0\to t\bar{t}$, $H^\pm\to tb$, and $t\to H_3^+ b$ searches respectively.[]{data-label="fig:H3"}](figH3-2.pdf "fig:"){width="0.465\linewidth"}
The signature of a neutral triplet Higgs boson $H_3^0$ has been considered in Ref. [@Chiang:2015kka]. Without direct couplings to gauge bosons, the promising signature is $gg\to H_3^0\to Zh$ for $m_Z+m_h<m_3<2m_t$, or $gg\to H_3^0 \to t\bar{t}$ for $m_3>2m_t$ [@Chiang:2015kka]. The charged triplet Higgs boson $H_3^+$ can decay into $\tau^+\nu$ for $m_3<m_t$ or into $t\bar{b}$ for $m_3>m_t+m_b$ [@Chiang:2012cn]. We therefore consider the following direct searches:
- Search for a CP-odd Higgs boson $H_3^0$ decaying to $hZ$ [@Aad:2015wra; @TheATLAScollaboration:2016loc].
- Search for a heavy Higgs boson $H_3^0$ decaying to a top quark pair [@Aaboud:2017hnm].
- Search for charged Higgs bosons decaying via $t\to H_3^+(\to \tau^+\nu)b$ or $H_3^\pm\to tb$ [@Aad:2014kga; @Khachatryan:2015qxa].
Our results are shown in Fig. \[fig:H3\] in the $v_\Delta-m_3$ and $\alpha-m_3$ planes. Among the constraints from those searches, $H_3^0\to hZ$ sets the most stringent one. In particular, in the mass region $200\lesssim m_3\lesssim 500~{{\rm GeV}}$, where BR($H_3^0\to hZ$) is dominant or relatively large, $v_\Delta$ can be pushed down as low as $10~{{\rm GeV}}$ under certain circumstances.
Quintuplet Searches
-------------------
Being independent of the singlet mixing angle $\alpha$, the constraints on the quintuplet scalars are only sensitive to the VEV $v_\Delta$ and their mass $m_5$. In Ref. [@Chiang:2015kka] the constraint on $v_\Delta$ has been obtained as a function of the exotic Higgs boson mass via various channels for the additional neutral scalars in the GM model. In this paper, we adopt the constraints from the decay channels $H_5^0\to\gamma\gamma$ and $H_5^0\to ZZ$ through the VBF mechanism.
In Refs. [@Aad:2015nfa; @CMS:2016szz] a search was performed for heavy charged scalars decaying to $W^\pm$ and $Z$ bosons at $\sqrt{s}=13~{{\rm TeV}}$ LHC. The upper limits on the cross section for the production of charged Higgs bosons times branching fractions to $W^\pm Z$ are transformed to the exclusion limits on $v_\Delta$ versus $m_5$ in the GM model.
![Constraints from searches for quintuplet Higgs bosons on the $v_\Delta-m_5$ plane. Points above exclusion curves are eliminated.[]{data-label="fig:h5"}](figH5.pdf){width="0.45\linewidth"}
The experimental constraints on the $H_5^{++}$ mass were studied in the GM model in Ref. [@Chiang:2014bia] by recasting the ATLAS measurement of the cross section for the like-sign diboson process $pp\to W^\pm W^\pm jj$. The $W^+W^+W^-W^-$ vertex is effectively modified by mediations of the doubly-charged Higgs bosons $H^{\pm\pm}$. That the relevant $W^\pm W^\pm H^{\mp\mp}$ vertex is proportional to $v_\Delta$ can be used to exclude parameter space on the plane of $v_\Delta$ and $m_5$. In this work we also take into account the latest search for like-sign $W$ boson pairs by the CMS [@Sirunyan:2017ret].
Additional subsidiary constraints are as follows:
- An absolute lower bound on the doubly-charged Higgs mass from the ATLAS like-sign dimuon data was obtained in Ref. [@Kanemura:2014ipa], which gives $m_5\gtrsim 76$ GeV.
- An upper bound on $s_H$ for $m_5\sim 76-100~{{\rm GeV}}$ can be obtained using the results of a decay-model-independent search for new scalars produced in association with a $Z$ boson from the OPAL detector at the LEP collider [@Abbiendi:2002qp].
- A limit $\tan \theta_H < 10/3$ is imposed to avoid a nonperturbative top quark Yukawa coupling [@Barger:1989fj].
In Fig. \[fig:h5\], we summarize the constraints from searches for quintuplet Higgs bosons on the $v_\Delta-m_5$ plane. In the low mass region $76~{{\rm GeV}}<m_5<110~{{\rm GeV}}$, the most stringent constraint comes from the neutral Higgs boson decay $H_5^0\to \gamma\gamma$ through the VBF production. In the mass interval $110-200~{{\rm GeV}}$, the like-sign diboson process $pp\to W^\pm W^\pm jj$ via the doubly-charged Higgs boson $H_5^{\pm\pm}$ gives the best bound. Above $200~{{\rm GeV}}$, the most severe constraint is set by the latest search for like-sign $W$ boson pairs, which could exclude $v_\Delta\gtrsim 15~{{\rm GeV}}$ when $m_5\sim 200~{{\rm GeV}}$.
Higgs Signal Strengths
----------------------
![Scatter plot of the Higgs signal strength $\chi^2$ fit with a $2\sigma$ range shown in the $\alpha-v_\Delta$ plane.[]{data-label="fig:chi"}](fig02.pdf){width="0.45\linewidth"}
The signal strengths of the SM-like Higgs boson production and decay in various channels can provide significant constraints on its couplings to the SM particles in the GM model [@Belanger:2013xza]. The signal strength for a specific production and decay channel $i\to h\to f$ is defined as $$\begin{aligned}
\mu_{i}^f
=\frac{\sigma_i\times \text{BR}_f}{\sigma^{\text{SM}}_i\times \text{BR}^{\text{SM}}_f},
\end{aligned}$$ where $\sigma_i~(\sigma^{\text{SM}}_i)$ is the reference value (SM prediction) of the Higgs production cross section for $i\to h$, and $\text{BR}_f~(\text{BR}^{\text{SM}}_f)$ the branching ratio for the decay $h\to f$. We include the production channels via the gluon fusion ($ggF$), the vector boson fusion (VBF), the associated production with a vector boson ($Vh$) and with a pair of top quarks ($tth$), and the decay channels $h\to\gamma\gamma,~ZZ,~W^\pm W^\mp,~\tau^\pm\tau^\mp,~b\bar{b}$. With the experimental values of $\mu_X^{\text{exp}}$ and standard deviation $\Delta \mu_X^{\text{exp}}$ [@Khachatryan:2016vau], we build a $\chi^2$ value for each allowed point as $$\chi^2=\sum_X\left(\frac{\mu_X^{\text{exp}}-\mu_X}{\Delta \mu_X^{\text{exp}}}\right)^2,$$ where the sum extends over all channels mentioned above.
From Eqs. (\[eqn:hvv\], \[eqn:khtt\]), we are aware that the SM-like Higgs couplings involved in the signal strengths depend only on the triplet VEV ($v_\Delta$) and the singlet mixing angle ($\alpha$). Therefore the constraints on $v_\Delta$ and $\alpha$ can be directly extracted from data without specifying other parameters. In Fig. \[fig:chi\] we show the scatter plot of $\chi^2$ values on the $\alpha-v_\Delta$ plane within a $2\sigma$ range. It is clear that the measurement of the Higgs signal strengths is most sensitive to the region with large $v_\Delta$ and large $|\alpha|$, where large deviations of Higgs couplings from the SM take place. Hence the large $\chi^2$ region, e.g., $v_\Delta\sim50$ GeV and $\alpha\sim-40^\circ$ would be excluded by future operations of LHC [@Chiang:2015amq].
Future Experimental Constraints
-------------------------------
For completeness we also presented in Fig. \[fig:constraints\] the constraints of the Higgs signal strength $\chi^2$ fit on the $\alpha-v_\Delta$ plane based on the projection results from 14 TeV LHC with an integrated luminosity of $300~{{\rm fb}}^{-1}$ (LHC@300) and $3000~{{\rm fb}}^{-1}$ (HL-LHC@3000) [@ATLAS:HL-LHC] and from ILC with an integrated luminosity of $1150~{{\rm fb}}^{-1}$ at 250 GeV (ILC250) and $1600~{{\rm fb}}^{-1}$ at 500 GeV (ILC500) [@Asner:2013psa]. The LHC (HL-LHC) result is performed on the ATLAS detector with only statistical and experimental systematic uncertainties taken into account. The expected precision is given as the relative uncertainty in the signal strength with the central values all assumed to be unity. In principle, this assumption applies only to the SM, but we employ these anticipated results as a reference so that we could be clear to what extent they can impose a constraint on the parameter space. It is clear from Fig. \[fig:constraints\] that the constrains from LHC@300, HL-LHC@300, ILC250, and ILC500 gradually become more and more stringent. Basically speaking, the deviations in Higgs couplings are determined by $\alpha$ and $v_\Delta$. Our fitting results show that LHC@300 (HL-LHC@3000) could approximately exclude $v_\Delta\gtrsim 30~(20)~{{\rm GeV}}$ while ILC250 and ILC500 could further push it down to about $v_\Delta\gtrsim 10~{{\rm GeV}}$.
Let us consider the scale factors $\kappa_{hVV}$ and $\kappa_{hf\bar{f}}$ as an example under the assumption that no obvious deviations in Higgs couplings from SM values will be observed. At LHC@300, $\kappa_{hVV}$ and $\kappa_{hf\bar{f}}$ could be constrained within the ranges $[0.92,1.08]$ and $[0.93,1.13]$ respectively, while at LHC@3000 they could be further narrowed down to $\kappa_{hVV}\in[0.97,1.04]$ and $\kappa_{hf\bar{f}}\in[0.93,1.06]$. At $e^+e^-$ colliders, e.g., ILC, precision measurements of the Higgs productions and decays can rigorously constrain the parameter space. At ILC250 where $e^+e^-\to hZ$ dominates, measurements of the Higgs signal strength in this channel could give $\kappa_{hVV}\in[0.998,1.02]$ and $\kappa_{hf\bar{f}}\in[0.995,1.007]$. And at ILC500 measurements via the $e^+e^-\to hZ$ and $e^+e^-\to h\nu_e\bar{\nu}_e$ channels would yield $\kappa_{hVV}\in[0.998,1.01]$ and $\kappa_{hf\bar{f}}\in[0.998,1.004]$. On the other hand, if the on-going LHC observes a certain hint of Higgs coupling deviation , for instance, $\kappa_{hVV}>1$ for the most optimistic case, the future operation of ILC will be hopeful to confirm it. In this way, the GM model would be strongly favored by the deviation $\kappa_{hVV}>1$.
Higgs Production at $e^+e^-$ Colliders {#sec:phenomenology}
======================================
In this section we will study the dominant single and double SM-like Higgs production at future $e^+e^-$ colliders in the GM model. For comparison we first reproduce in Fig. \[fig:cs\_ee\_hX\] the dominant Higgs production cross sections in the SM. The Higgs-strahlung (HS) process ($e^+e^-\to hZ$) is dominant for a CM energy $\sqrt{s}<500~{{\rm GeV}}$. At higher energy, the Higgs production is dominated by the $WW$ fusion process ($e^+e^-\to h\nu_e\bar{\nu}_e$), while the $ZZ$ fusion process ($e^+e^-\to he^+e^-$) also becomes significant. The subdominant processes such as $e^+e^-\to ht\bar{t}$, $e^+e^-\to hhZ$ and $e^+e^-\to hh \nu_e\bar{\nu}_e$ provide access to the top Yukawa coupling and the Higgs trilinear self-coupling.
![Cross section as a function of CM energy for dominant Higgs production processes in the SM at $e^+e^-$ colliders with $m_h=125$ GeV.[]{data-label="fig:cs_ee_hX"}](fig021.pdf){width="0.45\linewidth"}
It is clear that due to the $s$-channel topology (see Figs. \[fig:feyn-single\], \[fig:feyn-htt\], \[fig:feyn-hhz\]), the $hZ$, $ht\bar{t}$, and $hhZ$ production cross sections become maximal near the thresholds and decrease as collision energy goes up. On the contrary, the VBF ($h\nu_e\bar{\nu}_e$, $he^+e^-$, and $hh \nu_e\bar{\nu}_e$) cross sections increase as $\ln\sqrt{s}$ due to their $t$-channel topology (see Figs. \[fig:feyn-single\], \[fig:feyn-hhvv\]). These dominant processes can be divided into three types according to the couplings involved, namely the Higgs couplings to gauge bosons (Fig. \[fig:feyn-single\]) and to the top quark (Fig. \[fig:feyn-htt\]), and the trilinear Higgs self-couplings (Figs. \[fig:feyn-hhz\], \[fig:feyn-hhvv\]), which we now investigate in detail for the GM model.
Production via Higgs-strahlung and vector boson fusion
------------------------------------------------------
![Feynman diagrams for Higgs production via HS (a) and VBF (b, c) at $e^+e^-$ colliders.[]{data-label="fig:feyn-single"}](feyn-single.pdf "fig:"){width="0.9\linewidth"}\
In Fig. \[fig:feyn-single\] we depict the Feynman diagrams for single Higgs production at $e^+e^-$ colliders that involves only Higgs couplings to weak gauge bosons. The amplitudes for both HS and VBF processes are modified in the GM model by the same ratio of the Higgs-gauge couplings from the SM, $$\label{eqn:hvv}
\kappa_{hVV}=\frac{g_{hVV}}{g_{hVV}^{\text{SM}}}
= c_\alpha c_H-\frac{2\sqrt{6}}{3}s_\alpha s_H,$$ where $c_H$ and $s_H$ are defined in Eq. (\[eqn:chsh\]) in terms of $v_\Delta$. We can thus extract $\kappa_{hVV}^2$ by measuring these cross sections and set constraints on the parameters $v_\Delta$ and $\alpha$,
![Predicted value of $\kappa_{hVV}$ in the $v_\Delta-\alpha$ plane after imposing all the constraints in Sec. \[sec:constraints\].[]{data-label="fig:khVV"}](fig01.pdf "fig:"){width="0.45\linewidth"}\
The predicted value of $\kappa_{hVV}$ in the $v_\Delta-\alpha$ plane is shown in Fig. \[fig:khVV\]. It is obvious that an $\mathcal{O}(10\%)$ deviation of $\kappa_{hVV}$ from unity is still viable. Although the allowed $\kappa_{hVV}$ is in a range of about 0.93-1.15, most of the survived points tend to have $\kappa_{hVV}>1$. The LHC@300 will mostly exclude $\kappa_{hVV}\gtrsim1.1$, while HL-LHC@3000 could probe down to $\kappa_{hVV}\gtrsim1.05$. The scale factor $\kappa_{hVV}$ is expected to be measured at future $e^+e^-$ colliders with high precision. For example, the measurements for the $hVV$ couplings may reach an accuracy of 1% at CEPC (250 GeV, 5 ab$^{-1}$) and ILC (500 GeV, 500 fb$^{-1}$) [@CEPC-SPPCStudyGroup:2015esa; @Asner:2013psa]. Hence, the GM model could be probed indirectly if a large enough deviation of $\kappa_{hVV}$ from unity is measured. Especially, a measured $\kappa_{hVV}>1$ would be strong evidence in favor of the GM model. If $\kappa_{hVV}$ turns out to be consistent with unity, a precise measurement of it would put a stringent bound on the GM model parameter space.
The cross section for the HS process is [@Barger:1993wt] $$\sigma(e^+e^-\to Zh)=\frac{G_F^2m_Z^4}{96\pi s}(V_e^2+A_e^2)
\frac{\sqrt{\beta}(\beta+12 r_Z)}{(1-r_Z)^2}\kappa_{hVV}^2,$$ where the $Z$ couplings to the fermion $f$ of electric charge $Q_f$ are $V_f=2I_3^f-4Q_fs_W^2,~A_f=2I_3^f$, with $I_3^f=\pm1/2$ being the third weak isospin of the left-handed fermion $f$, and $s_W^2=\sin^2\theta_W$ with $\theta_W$ being the Weinberg angle. And $\beta=(1-r_Z-r_h)^2-4r_Zr_h$ is the usual two-body phase space function with $r_{h,Z,W}\equiv m_{h,Z,W}^2/s$, etc.
![Distribution of relative corrections $\delta\sigma/\sigma_{\text{SM}}$ in the $v_\Delta-\alpha$ plane for HS and VBF processes.[]{data-label="fig:hZ-hvv"}](fighzhvv.pdf){width="0.45\linewidth"}
The total cross section for the $WW~(ZZ)$ fusion process is [@Djouadi:1996uj] $$\sigma_{VV}=\frac{G_F^3m_V^4}{64\sqrt{2}\pi^3}\int_{r_h}^{1}dx\int_{x}^{1}
dy\frac{(V_V^2+A_V^2)^2f(x,y)+4V_V^2A_V^2g(x,y)}{[1+(y-x)/r_V]^2}\kappa_{hVV}^2,
\label{eq:sigmaVV}$$ where $V$ denotes either $W$ or $Z$, $V_W=A_W=\sqrt{2}~(V_Z=V_e,~A_Z=A_e)$ for the $WW~(ZZ)$ fusion respectively, and $$\begin{aligned}
f(x,y)&=\left(\frac{2x}{y^3}-\frac{1+2x}{y^2}+\frac{2+x}{2y}-\frac{1}{2}\right)
\left[\frac{z}{1+z}-\ln(1+z)\right]+\frac{x}{y^3}\frac{z^2(1-y)}{1+z},\\
g(x,y)&=\left(-\frac{x}{y^2}+\frac{2+x}{2y}-\frac{1}{2}\right)\left[\frac{z}{1+z}-\ln(1+z)\right],\end{aligned}$$ with $z=y(x-r_h)/(xr_V)$.
The HS and VBF processes are within the reach of all future $e^+e^-$ colliders mentioned in Sec. \[sec:introduction\]. Their cross sections are corrected by the same factor of $\kappa_{hVV}^2$, so that their relative corrections compared to the SM $\delta\sigma/\sigma_{\text{SM}}=\kappa_{hVV}^2-1$ are the same and independent of the collision energy. From Fig. \[fig:hZ-hvv\], we see that the cross sections could maximally increase by 32% or decrease by 13% with the current constraints. Even if no clear deviation would be found at LHC@300 or HL-LHC@3000, an increase of up to 10% in the HS and VBF cross section would still be possible at the ILC.
Associated Production with Top Quarks {#subsec:htt}
-------------------------------------
![Feynman diagrams for $ht\bar t$ production at $e^+e^-$ colliders.[]{data-label="fig:feyn-htt"}](feyn-htt.pdf "fig:"){width="0.9\linewidth"}\
The associated Higgs production with a top quark pair is an important process to measure the top quark Yukawa coupling at a linear collider [@Gaemers:1978jr; @Djouadi:1992gp]. Compared to the SM case, the existing interactions are modified and in addition there is a new contribution in the GM model that is mediated by the CP-odd heavy Higgs $H_3^0$, see Fig. \[fig:feyn-htt\]. The scale factor to the SM $h\bar f f$ coupling is, $\kappa_{hf\bar{f}}=c_\alpha/c_H$, and those involving the custodial triplet $H_3$ are $$\begin{aligned}
\nonumber
&&-\kappa_{H_3^0 d\bar{d}}=\kappa_{H_3^0 u\bar{u}} = \tan\theta_H,\\
&&\kappa_{H_3^0hZ}=\kappa_{H_3^+hW^-}=c_\alpha s_H+\frac{2\sqrt{6}}{3}s_\alpha c_H,
\label{eqn:khtt}\end{aligned}$$ which appear in the Feynman rules as follows: $$\begin{aligned}
\label{eq:definition}
\nonumber
H_3^0\bar ff&:&\kappa_{H_3^0 f\bar{f}}\frac{m_f}{v}\gamma_5,
\\
\nonumber
H_3^0hZ^\mu&:&\kappa_{H_3^0hZ}\frac{e}{2s_Wc_W}(p_h-p_3)_\mu,
\\
H_3^- hW^{+\mu}&:&-i\kappa_{H_3^+hW^-}\frac{e}{2s_W}(p_h-p_3)_\mu,\end{aligned}$$ where $p_h~(p_3)$ is the incoming momentum of $h~(H_3^{0,+})$. The scanned results for these $\kappa$s are shown in Fig. \[fig:khtt\] in the $v_\Delta-\alpha$ plane. For the current constraints, we see that a deviation from unity as large as $\mathcal{O}(\pm10\%)$ is still allowed for $\kappa_{hf\bar{f}}$. The scale factor $\kappa_{H_3^0 f\bar{f}}$ does not depend on $\alpha$, and its magnitude can maximally reach about 0.54 and vanishes in the limit of $v_\Delta\to 0$, while $\kappa_{H_3^0hZ}$ lies in the interval of $-0.2$ to $0.6$. The future operation of LHC@300 will be capable of excluding points with $\kappa_{hf\bar{f}}\lesssim 0.975$ and $|\kappa_{H_3^0 f\bar{f}}|\gtrsim 0.35$, while ILC500 has the ability to constrain $\kappa_{hf\bar{f}}\approx 1$, $|\kappa_{H_3^0 f\bar{f}}|\lesssim 0.1$, and $\kappa_{H_3^0hZ}\in[0,0.2]$.
![Predicted values of $\kappa_{hf\bar{f}}$, $|\kappa_{H_3^0f\bar{f}}|$, and $\kappa_{H_3^0hZ}$ in the $v_\Delta-\alpha$ plane after imposing all the constraints in Sec. \[sec:constraints\].[]{data-label="fig:khtt"}](fig03.pdf "fig:"){width="0.32\linewidth"} ![Predicted values of $\kappa_{hf\bar{f}}$, $|\kappa_{H_3^0f\bar{f}}|$, and $\kappa_{H_3^0hZ}$ in the $v_\Delta-\alpha$ plane after imposing all the constraints in Sec. \[sec:constraints\].[]{data-label="fig:khtt"}](fig07-H3ff.pdf "fig:"){width="0.31\linewidth"} ![Predicted values of $\kappa_{hf\bar{f}}$, $|\kappa_{H_3^0f\bar{f}}|$, and $\kappa_{H_3^0hZ}$ in the $v_\Delta-\alpha$ plane after imposing all the constraints in Sec. \[sec:constraints\].[]{data-label="fig:khtt"}](fig07-H3hZ.pdf "fig:"){width="0.32\linewidth"}
The cross section in the GM model can be adapted from those in the SM and MSSM [@Djouadi:1992gp] $$\begin{aligned}
\label{equ:htt}
\nonumber
\frac{d\sigma(e^+e^-\to ht\bar{t})}{dx_h}=&
N_c\frac{\sigma_0}{(4\pi)^2}\Bigg\{\Big[Q_e^2Q_t^2+
\frac{2Q_eQ_tV_eV_t}{16c_W^2s_W^2(1-r_Z)}
+\frac{(V_e^2+A_e^2)(V_t^2+A_t^2)}{256c_W^4s_W^4(1-r_Z)^2}\Big]G_1
\\
+&\frac{V_e^2+A_e^2}{256c_W^4s_W^4(1-r_Z)^2}
\Big[A_t^2\sum_{i=2}^{6}G_i+V_t^2(G_4+G_6)\Big]+
\frac{Q_eQ_tV_eV_t}{1-r_Z}G_6+G_7\Bigg\}.\end{aligned}$$ Here $\sigma_0=4\pi\alpha_\text{QED}^2/(3s)$, $\alpha_\text{QED}$ is the fine structure constant, $N_c=3$, $x_h=2E_h/\sqrt{s}$ with $E_h$ the Higgs boson energy. The explicit expressions for the coefficients $G_i (i=1,\cdots 7)$ are given in Appendix \[app:htt\].
![Cross section for $e^+e^-\to ht\bar{t}$ as a function of the $H_3^0$ mass $m_3$ at $\sqrt{s}=500~{{\rm GeV}}$ (left) and 1000 GeV (right). The horizontal line indicates the SM result.[]{data-label="fig:htt"}](fightt500.pdf "fig:"){width="0.45\linewidth"} ![Cross section for $e^+e^-\to ht\bar{t}$ as a function of the $H_3^0$ mass $m_3$ at $\sqrt{s}=500~{{\rm GeV}}$ (left) and 1000 GeV (right). The horizontal line indicates the SM result.[]{data-label="fig:htt"}](fightt1000.pdf "fig:"){width="0.44\linewidth"}
In Fig. \[fig:htt\] we present the cross section for $e^+e^-\to ht\bar{t}$ as a function of the $H_3^0$ mass $m_3$ at $\sqrt{s}=500,~1000~{{\rm GeV}}$. In the parameter region $2m_t<m_3<\sqrt{s}-m_h$, the subprocess $e^+e^-\to hH_3^0,~H_3^0\to t\bar{t}$ is resonant, where the cross section can be strongly enhanced by several times. When the mass of $H_3^0$ is far above the resonant region, the cross section reduces gradually to the SM value. Interestingly, in the resonant region with enhanced cross section, we actually have $\kappa_{hf\bar{f}}\gtrsim1$ for most of the survived points, while above this region the cross section decreases with the decrease of $\kappa_{hf\bar{f}}$. Combined with Fig. \[fig:khtt\], most of points with $\kappa_{hf\bar{f}}<1$ will be excluded by LHC@300 and HL-LHC@3000. Hence, large decrease of the $ht\bar{t}$ production might not be possible. On the other hand, for $H_3^0$ resonantly produced, most of allowed points actually have $\kappa_{hf\bar{f}}\approx1$. Therefore, even if no large deviation in the Higgs coupling is observed at ILC250 or ILC500, significant enhancement of the $ht\bar{t}$ production may still be viable. At CLIC the expected accuracy for cross section is about $8.4\%$ with an integrated luminosity of 1.5 ab$^{-1}$ at $\sqrt{s}=1.4~{{\rm TeV}}$ [@Abramowicz:2016zbo], and at ILC the accuracy could reach $28\%~(6.3\%)$ at $500~(1000)~{{\rm GeV}}$ [@Asner:2013psa]. There is thus a good chance to test the $ht\bar{t}$ production at these high energy machines.
Double Higgs Production
-----------------------
![Same as Fig. \[fig:khtt\], but for $\kappa_{hhh}$, $\kappa_{hhVV}$, $\kappa_{H_1^0hh}$ and $\kappa_{H_1^0VV}$ []{data-label="fig:kH10"}](fig04.pdf "fig:"){width="0.45\linewidth"} ![Same as Fig. \[fig:khtt\], but for $\kappa_{hhh}$, $\kappa_{hhVV}$, $\kappa_{H_1^0hh}$ and $\kappa_{H_1^0VV}$ []{data-label="fig:kH10"}](fig06.pdf "fig:"){width="0.45\linewidth"}\
![Same as Fig. \[fig:khtt\], but for $\kappa_{hhh}$, $\kappa_{hhVV}$, $\kappa_{H_1^0hh}$ and $\kappa_{H_1^0VV}$ []{data-label="fig:kH10"}](fig05.pdf "fig:"){width="0.45\linewidth"} ![Same as Fig. \[fig:khtt\], but for $\kappa_{hhh}$, $\kappa_{hhVV}$, $\kappa_{H_1^0hh}$ and $\kappa_{H_1^0VV}$ []{data-label="fig:kH10"}](fig07.pdf "fig:"){width="0.45\linewidth"}
To study the Higgs self-interactions in the symmetry breaking sector, it is indispensable to measure the Higgs pair production at future $e^+e^-$ colliders. In this subsection we consider possible production mechanisms in the GM model. These processes include the double Higgs-strahlung process $e^+e^-\to hhZ$ in Fig. \[fig:feyn-hhz\] and the Higgs pair production via the $W$ boson fusion $e^+e^-\to hh\nu_e\bar{\nu}_e$ in Fig. \[fig:feyn-hhvv\] while ignoring the much smaller $Z$ boson fusion [@Abramowicz:2016zbo; @Baer:2013cma]. These processes involve the scale factors $$\begin{aligned}
\kappa_{hhVV}&=&1+\frac{5}{3}s_\alpha^2,\end{aligned}$$ $\kappa_{H_1^0VV}$ shown in Eq. (\[equ:heavy higgs\]), and $\kappa_{hhh}=g_{hhh}/g_{hhh}^{\text{SM}}$, $\kappa_{H_1^0hh}= g_{H_1^0hh}/g_{hhh}^{\text{SM}}$, where $g_{hhh}^{\text{SM}}=3m_h^2/v$, and $g_{hhh},~g_{H_1^0hh}$ are given in Appendix \[app:trilinear\]. In Fig. \[fig:kH10\], we show the scanned results for these scale factors in the allowed parameter space. Compared to their SM counterparts, the $hhVV$ coupling is always enhanced due to $\kappa_{hhVV}\geq 1$, whereas the $hhh$ coupling can even change its sign. For those points with $\kappa_{hhh}<0$, LHC@300 will exclude most of them while HL-LHC@3000 could eliminate them altogether. The precision expected to be reachable at the ILC for the measurement of $g_{hhh}$ is of $\mathcal{O}(10\%)$ [@Fujii:2015jha], so the effect of the GM model should be observable. For the new couplings involving $H_1^0$, the magnitude of $\kappa_{H_1^0hh}$ can be so large that the processes are significantly enhanced but perturbation theory may not apply there, while the scale factor $\kappa_{H_1^0VV}$ ranges from $-0.2$ to $0.6$. Considering the limit expected to be available at LHC@300 (HL-LHC@3000), $\kappa_{H_1^0 hh}\lesssim 10$ ($\kappa_{H_1^0 hh}\lesssim 5$) should be satisfied.
### Double Higgs-Strahlung
The differential cross section for the process $e^+e^-\to hhZ$ can be written as [@Djouadi:1996ah; @Osland:1998hv; @Djouadi:1999gv] $$\begin{aligned}
\label{equ:sigma-hhz}
\frac{d\sigma(e^+e^-\to hhZ)}{dx_1dx_2}=&
\frac{G_F^3m_Z^6}{384\sqrt{2}\pi^3s}(V_e^2+A_e^2)\frac{\mathcal{A}}{(1-r_Z)^2}.\end{aligned}$$ In the above equation, $x_{1,2}=2E_{1,2}/\sqrt{s}$ with $E_{1,2}$ being the energies of the Higgs bosons, and the shortcuts $x_3=2-x_1-x_2$, $y_i=1-x_i~(i=1,2,3)$, $r_{1,3}=m_{1,3}^2/s$ are used. The function $\mathcal{A}$ can be expressed in the form [@Osland:1998hv] $$\begin{aligned}
\label{eqn:calA}
\mathcal{A}=r_Z\left\{\frac{1}{2}|a|^2f_a+|b(y_1)|^2f_b+2\text{Re}[ab^*(y_1)]g_{ab}
+\text{Re}[b(y_1)b^*(y_2)]g_{bb}\right\}+\{x_1,y_1\leftrightarrow x_2,y_2\},\end{aligned}$$ where $$\begin{aligned}
\label{equ:ab}
\nonumber
a=&\frac{1}{2}\bigg(\frac{\kappa_{hVV}\kappa_{hhh}}{y_3+r_Z-\tilde{r}_h}
+\frac{\kappa_{H_1^0VV}\kappa_{H_1^0hh}}{y_3+r_Z-\tilde{r}_1}\bigg)\frac{3m_h^2}{m_Z^2}
+\bigg(\frac{\kappa_{hVV}^2}{y_1+r_h-\tilde{r}_Z}+
\frac{\kappa_{hVV}^2}{y_2+r_h-\tilde{r}_Z}\bigg)+\frac{\kappa_{hhVV}}{2r_Z},\\
b(y)=&\frac{1}{2r_Z}\bigg(\frac{\kappa_{hVV}^2}{y+r_h-\tilde{r}_Z}+
\frac{\kappa_{H_3^0hZ}^2}{y+r_h-\tilde{r}_3}\bigg).\end{aligned}$$ Here $\tilde{r}_i~(i=h,Z,H_1,H_3)$ includes the total decay width of the particle $i$ in its mass squared, i.e., $\tilde{r}_i=(m_i^2-im_i\Gamma_i)/s$ when a resonance is crossed over. The coefficients $f$ and $g$ in Eq. (\[eqn:calA\]) are $$\begin{aligned}
\nonumber
f_a=&x_3^2+8r_Z,
\\
\nonumber
f_b=&(x_1^2-4r_h)[(y_1-r_Z)^2-4r_Zr_h],
\\
\nonumber
g_{ab}=&r_Z[2(r_Z-4r_h)+x_1^2+x_2(x_2+x_3)]-y_1(2y_2-x_1x_3),
\\
\nonumber
g_{bb}=&(y_3-x_1x_2-x_3r_Z-4r_hr_Z)(2y_3-x_1x_2-4r_h+4r_Z)
\\
&+r_Z^2(4r_h+6-x_1x_2)+2r_Z(r_Z^2+y_3-4r_h).\end{aligned}$$
![Feynman diagrams for $hhZ$ production at $e^+e^-$ colliders.[]{data-label="fig:feyn-hhz"}](feyn-hhz.pdf "fig:"){width="0.9\linewidth"}\
![Cross section for $e^+e^-\to hhZ$ as a function of the $H_1^0$ mass $m_1$ (left panel) and $H_3^0$ mass $m_3$ (right) at $\sqrt{s}=500~{{\rm GeV}}$ (upper) and 1 TeV (lower). The horizontal line indicates the SM value.[]{data-label="fig:hhz"}](fighhz500-1.pdf "fig:"){width="0.45\linewidth"} ![Cross section for $e^+e^-\to hhZ$ as a function of the $H_1^0$ mass $m_1$ (left panel) and $H_3^0$ mass $m_3$ (right) at $\sqrt{s}=500~{{\rm GeV}}$ (upper) and 1 TeV (lower). The horizontal line indicates the SM value.[]{data-label="fig:hhz"}](fighhz500-3.pdf "fig:"){width="0.45\linewidth"}\
![Cross section for $e^+e^-\to hhZ$ as a function of the $H_1^0$ mass $m_1$ (left panel) and $H_3^0$ mass $m_3$ (right) at $\sqrt{s}=500~{{\rm GeV}}$ (upper) and 1 TeV (lower). The horizontal line indicates the SM value.[]{data-label="fig:hhz"}](fighhz1000-1.pdf "fig:"){width="0.45\linewidth"} ![Cross section for $e^+e^-\to hhZ$ as a function of the $H_1^0$ mass $m_1$ (left panel) and $H_3^0$ mass $m_3$ (right) at $\sqrt{s}=500~{{\rm GeV}}$ (upper) and 1 TeV (lower). The horizontal line indicates the SM value.[]{data-label="fig:hhz"}](fighhz1000-3.pdf "fig:"){width="0.45\linewidth"}\
The double Higgs boson production is sensitive to the triple Higgs coupling $g_{hhh}$, which cannot be probed by the single Higgs boson production. In addition, the heavy CP-even Higgs boson $H_1^0$ contributes, thereby enabling sensitivity to the $H_1^0hh$ coupling as well. The total cross section is shown in Fig. \[fig:hhz\] at the energy $\sqrt{s}=500~{{\rm GeV}}$ and 1 TeV. Compared to the SM case, the on-shell production of heavy scalars $H_1^0$ and $H_3^0$ followed by decays $H_1^0\to hh$ and $H_3^0\to hZ$ plays a dominant role in the resonant region of the parameter space, where the cross section can increase by more than one order of magnitude. Since in the resonance region, most of the currently allowed points have $\kappa_{hhh}\approx 1$, the future measurements at LHC@300 and HL-LHC@3000 would be hard to exclude such points. In the non-resonance region, the cross section at $\sqrt{s}=500~{{\rm GeV}}$ is positively correlated with $\kappa_{hhh}$, leading to enhanced cross section with $\kappa_{hhh}>1$ or suppressed cross section with $\kappa_{hhh}<1$. The cross section at $\sqrt{s}=1~{{\rm TeV}}$ can be either enhanced or suppressed even with $\kappa_{hhh}\approx 1$.
### Double Higgs from Vector Boson Fusion
![Feynman diagrams for double Higgs production from $WW$ fusion at $e^+e^-$ colliders.[]{data-label="fig:feyn-hhvv"}](feyn-hhvv.pdf "fig:"){width="0.9\linewidth"}\
Besides the resonant $WW$ fusion for single $h$ production, there exists non-resonant $WW$ fusion production of a pair of $h$, $e^+e^-\to hh\nu_e \bar{\nu}_e$, shown in Fig. \[fig:feyn-hhvv\]. The GM model modifies existing interactions in the SM and introduces new contributions due to $H_1^0$ and $H_3^\pm$ exchanges. The cross section in the effective $W$ approximation can be written as [@Osland:1998hv; @Djouadi:1999gv] $$\begin{aligned}
\sigma(e^+e^-\to hh\nu_e\bar{\nu}_e)=\int_{x_\text{min}}^{1}dx\frac{dL}{dx}\hat{\sigma}_{WW}(x),\end{aligned}$$ where $x_\text{min}=4r_h(=4m_h^2/s)$, the differential luminosity function is [@Osland:1998hv] $$\begin{aligned}
\frac{dL(x)}{dx}=\frac{G_F^2m_W^4}{8\pi^4}
\frac{1}{x}[-(1+x)\ln x-2(1-x)],\end{aligned}$$ and the cross section for the subprocess is [@Osland:1998hv] $$\begin{aligned}
\nonumber
\hat{\sigma}_{WW}(x)=&
\frac{G_F^2\hat{s}}{64\pi}\bigg[4\beta_h\Big(
\frac{3r_h\kappa_{hVV}\kappa_{hhh}}{1-r_h}
+\frac{3r_h\kappa_{H_1^0VV}\kappa_{H_1^0hh}}{1-r_1}
+\kappa_{hhVV}\Big)^2
\\
\nonumber
&+2\Big(\frac{3r_h\kappa_{hVV}\kappa_{hhh}}{1-r_h}
+\frac{3r_h\kappa_{H_1^0VV}\kappa_{H_1^0hh}}{1-r_1}
+\kappa_{hhVV}\Big)(\kappa_{hVV}^2F_1+\kappa_{H_3^\pm hW}^2F_2)
\\
&+\beta_h^{-1}(\kappa_{hVV}^4F_3+\kappa_{H_3^\pm hW}^4F_4
+4\kappa_{hVV}^2\kappa_{H_3^\pm hW}^2F_5)\bigg],
\label{eqn:subWW}\end{aligned}$$ where the functions $F_i~(i=1,\dots,5)$ are reproduced in Appendix \[app:hhvv\] and $\beta_h=(1-4r_h)^{1/2}$. Note that here $r_i~(i=h,~Z,~H_1,~H_3)$ are defined with respect to $\hat s=xs$, e.g., $r_1=m_1^2/\hat s$.
The total cross section for $e^+e^-\to hh\nu_e\bar{\nu}_e$ is shown in Fig. \[fig:hhvv\] at $\sqrt{s}=500~{{\rm GeV}}$ and 1 TeV respectively. It varies in a wide range as $\kappa_{hhh}$ and $\kappa_{H_1^0hh}$ do, indicating its sensitivity to the values of trilinear couplings. Similarly to the case of the double Higgs-strahlung process, the cross section is strongly enhanced in the parameter range where the fusion subprocess, $WW\to H_1^0\to hh$, is resonant also with $\kappa_{hhh}\approx1$. Out of the resonant region the cross section decreases with increasing $\kappa_{hhh}$ due to destructive interference between the scalar and gauge parts of the amplitude. From Fig. \[fig:kH10\], one sees that $\kappa_{hhh}<1$ would not be favored by HL-HLC@3000. In this case, we expect a smaller cross section in the non-resonance region.
![Cross section for $e^+e^-\to hh\nu_e\bar{\nu}_e$ as a function of the $H_1^0$ mass $m_1$ at $\sqrt{s}=500~{{\rm GeV}}$ (left panel) and 1 TeV (right). The horizontal line indicates the SM value.[]{data-label="fig:hhvv"}](fighhvv500.pdf "fig:"){width="0.45\linewidth"} ![Cross section for $e^+e^-\to hh\nu_e\bar{\nu}_e$ as a function of the $H_1^0$ mass $m_1$ at $\sqrt{s}=500~{{\rm GeV}}$ (left panel) and 1 TeV (right). The horizontal line indicates the SM value.[]{data-label="fig:hhvv"}](fighhvv1000.pdf "fig:"){width="0.45\linewidth"}
Conclusions {#sec:conclusion}
===========
In this paper, we have investigated the dominant single and double SM-like Higgs ($h$) production at future $e^+e^-$ colliders in the Georgi-Machacek (GM) model. We comprehensively considered various constraints that are currently available, including theoretical, indirect, and direct experimental constraints on the model parameters. In particular, we updated the constraints from searches for heavy singlet ($H_1^0$), triplet ($H_3$), and quintuplet ($H_5$) Higgs scalars and from SM-like Higgs signal strengths with the latest LHC results. The GM model parameter space now becomes more tightly constrained, for instance, only $v_\Delta\lesssim40~{{\rm GeV}}$ with $-30^\circ\lesssim\alpha\lesssim0^\circ$ is allowed now.
Our analysis of the Higgs production has been done in the allowed parameter regions established as above. Besides modifications to the SM couplings, the SM-like Higgs production receives new contributions from new scalars such as $H_1^0,~H_3^0$, and can deviate significantly from the prediction in the SM. Our numerical results have the following key features:
- For single Higgs production such as $e^+e^-\to hZ,~h\nu_e\bar{\nu}_e,~he^+e^-$, the production mechanisms are the same as in SM with the only modification occurring in the Higgs-gauge boson coupling $hVV$. Their cross sections are modified by a factor ranging from 0.86 to 1.32.
- For associated Higgs production with a top pair $e^+e^-\to ht\bar{t}$, there is a new contribution via $e^+e^-\to hH_3^0$ with $H_3^0\to t\bar{t}$. When $2m_t<m_3<\sqrt{s}-m_h$ is fulfilled, $H_3^0$ becomes resonantly produced, thus enhancing the cross section by up to three times.
- For the double Higgs-strahlung process $e^+e^-\to hhZ$, both $H_1^0$ and $H_3^0$ contribute. When resonantly produced, they can lead to an order of magnitude enhancement in the cross section. In the non-resonant region, large deviations are still possible due to a large modification to the trilinear Higgs coupling.
- For the double Higgs production via vector boson fusion $e^+e^-\to hh\nu_e\bar{\nu}_e$, both $H_1^0$ and $H_3^\pm$ enter but only $H_1^0$ can be on-shell produced. In the resonant region more than an order of magnitude enhancement is viable, and in the non-resonant region the cross section can maximally increase or decrease by an order of magnitude.
Anticipating that LHC will be operated at slightly higher energy and with larger integrated luminosity, we have also estimated its future impact on Higgs couplings and production at electron colliders. We found that LHC@300 and HL-LHC@3000 have the ability to exclude some of the parameter space that is currently allowed. When the new scalars in the GM model are light enough to be resonantly produced at electron colliders, large enhancement in the SM-like Higgs production is possible though its couplings are close to the SM values. In the non-resonance region, the double Higgs production channels are expected to deviate significantly from the SM under the strict future bounds from LHC. If the on-going LHC experiments could observe some deviations of the Higgs couplings, future electron colliders would be capable of confirming it; if not, precise measurements of the Higgs properties at future electron colliders could probe most of the currently allowed parameter space of the GM model. Compared to other popular models such as MSSM and THDM, the GM model can enhance or suppress Higgs couplings to vector bosons and fermions, and drastically change the trilinear Higgs coupling from the SM prediction. The modification of these Higgs couplings leads to distinguished phenomenology for Higgs production processes at $e^+e^-$ colliders. Furthermore, light scalars such as $H_1^0$ and $H_3^0$ can be on-shell produced in certain processes, which provides a good chance to hunt new particles. We thus expect that the model could be tested with high precision and be discriminated from some other new physics scenarios.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work was supported in part by the Grants No. NSFC-11575089 and No. NSFC-11025525, by The National Key Research and Development Program of China under Grant No. 2017YFA0402200, and by the CAS Center for Excellence in Particle Physics (CCEPP). We thank the anonymous referee for suggesting the inclusion of projected limits that are expected from future LHC operations.
Coefficients for $\sigma(e^+e^-\to ht\bar{t})$ {#app:htt}
==============================================
The seven coefficients $G_i~(i=1,\cdots,7)$ appearing in Eq. (\[equ:htt\]) can be obtained by slight adaption from the MSSM case [@Djouadi:1992gp]. The Higgs radiation from the top quark yields the first two coefficients: $$\begin{aligned}
\nonumber
G_1=&\frac{2g_t^2}{s^2(\beta_t^2-x_h^2)x_h}
\bigg\{-4\beta_t(4m_t^2-m_h^2)(2m_t^2+s)x_h+(\beta_t^2-x_h^2)
\Big[16m_t^4+2m_h^4\\
\nonumber
&-2m_h^2sx_h+s^2x_h^2-4m_t^2(3m_h^2-2s-2sx_h)\Big]
\ln\Big(\frac{x_h+\beta_t}{x_h-\beta_t}\Big)\bigg\},\\
\nonumber
G_2=&\frac{-2g_t^2}{s^2(\beta_t^2-x_h^2)x_h}\bigg\{\beta_t x_h\Big[-96m_t^4+24m_t^2m_h^2-s(m_h^2-s-sx_h)(\beta_t^2-x_h^2)\Big]\\
&+2(\beta_t^2-x_h^2)\Big[24m_t^4+2(m_h^4-m_h^2sx_h)-m_t^2(14m_h^2-12sx_h-sx_h^2)\Big]
\ln\Big(\frac{x_h+\beta_t}{x_h-\beta_t}\Big)\bigg\}.\end{aligned}$$ The other five coefficients all involve the Higgs radiation from the $Z$ boson including its interference with the radiation from the top quark: $$\begin{aligned}
\nonumber
G_3=&\frac{-2\beta_t g_Z^2m_t^2}{m_Z^2(m_h^2-m_Z^2+s-sx_h)^2}
\{4m_h^4+12m_Z^4+2m_Z^2sx_h^2+s^2(x_h-1)x_h^2
\\
\nonumber
&-m_h^2[8m_Z^2+s(x_h^2+4x_h-4)]\},
\\
\nonumber
G_4=&\frac{\beta_t g_Z^2m_Z^2}{6(m_h^2-m_Z^2+s-sx_h)^2}[48m_t^2+12m_h^2+s(24-\beta_t^2-24x_h+3x_h^2)],
\\
\nonumber
G_5=&\frac{4g_{t}g_Zm_t}{m_Zs(m_h^2-m_Z^2+s-sx_h)}\bigg\{\beta_t s\Big[6m_Z^2-x_h(m_h^2+s-sx_h)\Big]
\\
\nonumber
&+2\Big[m_h^2(m_h^2-3m_Z^2+s-sx_h)+m_t^2(12m_Z^2-4m_h^2+sx_h^2)\Big]
\ln\Big(\frac{x_h+\beta_t}{x_h-\beta_t}\Big)\bigg\},
\\
\nonumber
G_6=&\frac{-8g_{t}g_Zm_tm_Z}{s(m_h^2-m_Z^2+s-sx_h)}\bigg[\beta_t s+(4m_t^2-m_h^2+2s-sx_h)\ln\Big(\frac{x_h+\beta_t}{x_h-\beta_t}\Big)\bigg],
\\
\nonumber
G_7=&\frac{-g_{H_3^0t\bar{t}}\kappa_{hH_3^0Z}}{m_h^2-m_{H_3}^2+s-sx_h}
\bigg\{2\beta_t(4m_h^2-sx_h)\Big[\frac{(m_h^2+s-sx_h)g_{H_3^0t\bar{t}}
\kappa_{H_3^0hZ}}{m_h^2-m_{H_3}^2+s-sx_h}-\frac{2m_tg_{Z}}{m_Z}\Big]
\\
&+\frac{2 g_{t}}{s}[2sx_h\beta_t(sx_h-s-m_h^2)-4(m_h^2(sx_h-s-m_h^2)+m_t^2(4m_h^2-sx_h^2))]
\ln\Big(\frac{x_h+\beta_t}{x_h-\beta_t}\Big)\bigg\}.\end{aligned}$$ The relevant $h$ couplings are defined in terms of their $\kappa$ factors: $$\begin{aligned}
g_Z= \frac{m_Z}{v}\kappa_{hVV},~g_{t}= -\frac{m_t}{v}\kappa_{ht\bar{t}},
~g_{H_3^0t\bar{t}}=\frac{m_t}{v}\kappa_{H_3^0 t\bar{t}}.\end{aligned}$$ The kinematical variable $\beta_t$ is $$\begin{aligned}
\beta_t=\bigg\{\frac{[x_h^2-(x_h^\text{min})^2](x_h^\text{max}-x_h)}
{x_h^\text{max}-x_h+4r_t}\bigg\}^{1/2},\end{aligned}$$ where $x_h^\text{min}=2r_h^{1/2}$ and $x_h^\text{max}=1-4r_t+r_h$.
Trilinear Higgs Couplings {#app:trilinear}
=========================
The trilinear Higgs couplings used in our calculation include $$\begin{aligned}
\label{eq:trilinear}
\nonumber
g_{hhh}=&~24\lambda _1 c_{\alpha }^3 v_{\phi }+\frac{3\sqrt{3}}{2} c_{\alpha }^2 s_{\alpha } \left[M_1+4\left(\lambda _5-2 \lambda _2\right) v_{\Delta}\right]+6 \left(2\lambda _2-\lambda _5\right) c_{\alpha } s_{\alpha }^2 v_{\phi }\\\nonumber
&+4 \sqrt{3} s_{\alpha }^3\left[M_2-2 \left(\lambda _3+3 \lambda _4\right) v_{\Delta}\right],\\\nonumber
g_{H_1^0hh}=&~4 \left(6\lambda _1-2 \lambda _2+\lambda _5\right) c_{\alpha }^2 s_{\alpha }
v_{\phi } +2 \left(2 \lambda _2-\lambda _5\right) s_{\alpha }^3v_{\phi }\\
&+ \sqrt{3} c_{\alpha } s_{\alpha }^2 \left[M_1-4 M_2+4 \left(2 \lambda _3-2 \lambda_2+6 \lambda _4+\lambda _5\right) v_{\Delta}\right] \\ \nonumber
&-\frac{\sqrt{3}}{2} c_{\alpha }^3 \left[M_1+4\left(\lambda _5-2 \lambda _2\right) v_{\Delta }\right].\end{aligned}$$ In the decoupling limit, $s_\alpha,v_\Delta,M_1,M_2\to 0$, and $\lambda_1\to m_h^2/(8v^2)$, we have $g_{hhh}\to 24 v\lambda_1\to 3m_h^2/v$ and $g_{H_1^0hh}\to -\sqrt{3}M_1/2\to 0$.
Coefficients for $\sigma(e^+e^-\to hh\nu_e\bar{\nu}_e)$ {#app:hhvv}
=======================================================
The coefficients used in the calculation of $e^+e^-\to hh\nu_e\bar{\nu}_e$ are [@Osland:1998hv] $$\begin{aligned}
\nonumber
F_1=&~8[2r_W+(r_h-r_W)^2]l_W-4\beta_h(1+2r_h-2r_W),\\\nonumber
F_2=&~8(r_h-r_3)^2l_3-4\beta_h(1+2r_h-2r_w),\\\nonumber
F_3=&~8\beta_h[2r_W+(r_h-r_W)^2][2r_W+1-3(r_h-r_W)^2]\frac{l_W}{a_W}\\\nonumber
&+16[2r_W+(r_h-r_W)^2]^2y_W+16\beta_h^2(1+a_W)^2,\\\nonumber
F_4=&~8\beta_h(r_h-r_3)^2[1-3(r_h-r_3)^2]\frac{l_3}{a_3}\\\nonumber
&+16(r_h-r_3)^4y_3+16\beta_h^2(1+a_3)^2,\\
F_5=&~\frac{\beta_h}{4}(z_Wl_W+z_3l_3)+8\beta_h^2(1+a_W)(1+a_3),\end{aligned}$$ where $$\begin{aligned}
\nonumber
l_{W,3}=&~\ln\frac{1-2r_h+2r_{W,3}-\beta_h}{1-2r_h+2r_{W,3}+\beta_h},\\\nonumber
y_{W,3}=&~\frac{2\beta_h^2}{(1-2r_h+2r_{W,3})^2-\beta_h^2},\\\nonumber
a_{W,3}=&~-\frac{1}{2}+r_h-r_{W,3},\\\nonumber
z_W=&~\frac{(1+2a_W)^2}{a_3-a_W}[8r_W+(1+2a_W)^2]+\frac{(1-2a_W)^2}{a_3+a_W}[8r_W+(1+2a_W)^2],\\
z_3=&~-\frac{(1+2a_3)^2}{a_3-a_W}[8r_W+(1+2a_3)^2]+\frac{(1+2a_3)^2}{a_3+a_W}[8r_W+(1-2a_3)^2].\end{aligned}$$
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Let $\phi : S^1\times D^2\to S^1$ be the natural projection. An oriented knot $K\hookrightarrow V = S^1\times D^2$ is called an almost closed braid if the restriction of $\phi$ to K has exactly two (non-degenerate) critical points (and K is a closed braid if the restriction of $\phi$ has no critical points at all).
We introduce new isotopy invariants for closed braids and almost closed braids in the solid torus V. These invariants refine finite type invariants. They are still calculable with polynomial complexity with respect to the number of crossings of K. Let the solid torus V be standardly embedded in the 3-sphere and let A be the axis of the complementary solid torus $S^3\setminus V$. We give examples which show that our invariants can detect non-invertibility of 2-component links $K\cup A\hookrightarrow S^3$. Notice that all quantum link invariants fail to do so and that it is not known wether there are finite type invariants which can detect non-invertibility of 2-component links.[^1]
author:
- Thomas Fiedler
title: Isotopy invariants for closed braids and almost closed braids via loops in stratified spaces
---
[*to Séverine*]{}
Introduction
============
This paper is a new and much shortened version of the preprint [@F03].As well known, the isotopy problem for closed braids in the solid torus reduces to the conjugacy problem in braid groups (see [@M78]). The latter problem is solved, but in general only with exponential complexity with respect to the braid lenght (see [@B-B04] and references therein). It is therefore interesting to construct invariants which distinguish conjugacy classes of braids and which are calculable with polynomial complexity. Finite type invariants for knots in the solid torus are an example of such invariants (see [@V90], [@V01], [@BN], [@G96], [@F01] and references therein).
In this paper we construct another class of calculable invariants.We give now a brief outline of our approach: let $\hat \beta\subset V$ be a closed braid (i.e. the restriction of $\phi$ to $\hat \beta$ has no critical points) and such that $\hat \beta$ is a knot. We fix a projection $pr : V \to S^1\times I$. Let $M(\hat \beta)$ be the infinite dimensional) space of all closed braids which are isotopic to $\hat \beta$ in V. $M(\hat \beta)$ has a natural stratification. The strata $\sum^{(1)}$ of codimension 1 are just the braid diagrams which have in the projection $pr$ either exactly one ordinary triple point or exactly one ordinary autotangency.We call the corresponding strata $\sum^{(1)}(tri)$ and respectively $\sum^{(1)}(tan)$. First, we associate to a closed braid in a canonical way a loop in $M(\hat \beta)$, called the [*canonical loop* ]{} and denoted $rot(\hat \beta)$.Next, we associate to the canonical loop an oriented singular link in a thickened torus. This link is called the [*trace graph* ]{} and it is denoted by $TL(\hat \beta)$. All singularities of $TL(\hat \beta)$ are ordinary triple points. These triple points correspond exactly to the intersections of $rot(\hat \beta)$ with $\sum^{(1]}(tri)$. There is a natural coorientation on $\sum^{(1)}(tri)$ and, hence, each triple point in $TL(\hat \beta)$ has a sign. To each triple point corresponds a knot diagram which has just ordinary crossings and exactly one triple crossing. We use the position of the ordinary crossings with respect to the triple crossing in the [*Gauss diagram* ]{} in order to construct [*weights* ]{} for the triple points. We associate to the loop $rot(\hat \beta)$ then [*weighted intersection numbers* ]{} with $\sum^{(1)}(tri)$. These intersection numbers turn out to be finite type invariants. We call them [*one-cocycle invariants* ]{}.As all finite type invariants the one-cocycle invariants have a natural degree. However, they are rather special finite type invariants, as shows the following proposition.
Let $\hat \beta$ be a closed n-braid which is a knot and let c be the word lenght of $\beta \in B_n$ (with respect to the standard generators of $B_n$). Then all one-cocycle invariants of degree d vanish for $\hat \beta$ if $d \geq c + n^2 - n - 1$.
One has to compare this proposition with the following well known fact: the trefoil has non-trivial finite type invariants of arbitrary high degree. Consequently, the one-cocycle invariants define a natural filtered subspace in the filtered space of all finite type invariants for those knots in the solid torus, which are closed braids. There is a very simple procedure, coming from singularity theory, in order to construct all one-cocycle invariants for closed braids, without solving a big system of equations. They verify automatically the marked 4T-relations (compare [@G96] and also [@F01]).
However , the one-cocycle invariants can be refined considerably. When we deform $\hat \beta$ in V by a generic isotopy then $rot(\hat \beta)$ in $M(\hat \beta)$ deforms by a generic homotopy. The following lemma is our key observation.
The loop $rot(\hat \beta)$ is never tangential to $\sum^{(1)}(tan)$.
It follows from this lemma that the connected components of the natural resolution of $TL(\hat \beta)$ (i.e. the abstract union of circles where the branches in the triple points are separated) are isotopy invariants of $\hat \beta \hookrightarrow V$. We apply now our theory of one-cocycle invariants but only to those triple points in $TL(\hat \beta)$ where three [*given* ]{} components of $TL(\hat \beta)$ intersect. The resulting invariants are called [*character invariants* ]{}. They are no longer of finite type but they are still calculable with polynomial complexity with respect to the braid lenght.
The set of finite type invariants (in particular , the set of one-cocycle invariants) can be seen as a trivial local system on $M(\hat \beta)$. In contrast to this, the set of character invariants is in general a non-trivial local system on $M(\hat \beta)$.
Alexander Stoimenow has written a computer program in order to calculate character invariants. It turns out that already character invariants of linear complexity can sometimes detect non-invertibility of closed braids (i.e. the closed braid together with the axis of the complementary solid torus in $S^3$ is a non-invertible link in $S^3$). This implies in particular that character invariants are in general not of finite type.
We observe that the most simple character invariants are still well defined for almost closed braids.
.
The basic notions of our one parameter approach to knot theory, namely the space of non-singular knots, its discriminant, the stratification of the discriminant, the coorientation of strata of low codimension, the canonical loop, the trace graph, the equivalence relation for trace graphs, are worked out in all details in our joint work with Vitaliy Kurlin [@F-K06]. Therefore we concentrate in this paper only on the construction of the new invariants.
[*Acknowledgements* ]{}
I am grateful to Stepan Orevkov and Vitaliy Kurlin for many interesting discussions. I am especially grateful to Alexander Stoimenow for writing his computer program and for calculating interesting examples.
Basic notions of one parameter knot theory
==========================================
In this section we recall briefly the basic notions of our theory. All details with complete proofs can be found in [@F-K06] (even in a much more general setting).
The space of closed braids and its discriminant
-------------------------------------------------
We work in the smooth category and all orientable manifolds are actually oriented. We fix once for all a coordinate system in $\mathbb{R}^3$ : $(\phi , \rho , z )$. Here, $(\phi ,\rho ) \in S^1\times \mathbb{R}^+$ are polar coordinates of the plane $\mathbb{R}^2 = \{ z = 0 \}$. A closed n-braid $\hat \beta$ is a knot in the solid torus $V = \mathbb{R}^3 \setminus z-axes$, such that $\phi : \hat \beta \to S^1$ is non-singular and $[\hat \beta] = n \in H_1(V)$. Let $M(\hat \beta)$ be the infinit dimensional space of all closed braids (with respect to $\phi$) which are isotopic to $\hat \beta$ in V. Let $M_n$ be the union of all spaces $M(\hat \beta)$. A well known theorem of Artin (see e.g. [@M78]) says that two closed braids in the solid torus are isotopic as links in the solid torus if and only if they are isotopic as closed braids. Therefore it is enough to consider only isotopies through closed braids. Let $pr : \mathbb{R}^3 \setminus z-axes \to \mathbb{R}^2 \setminus 0$ be the canonical projection $(\phi ,\rho , z) \to (\phi ,\rho )$.Each closed braid is then represented by a knot diagram with respect to $pr$. A generic closed braid $\hat \beta$ has only ordinary double points as singularities of $pr(\hat \beta)$. Let $\sum$ be the discriminant in $M(\hat \beta)$ which consists of all non-generic diagrams of closed braids isotopic to $\hat \beta$.
The discriminant $\sum$ has a natural stratification: $\sum = \sum^{(1)} \cup \sum^{(2)} \cup ...$, where $\sum^{(i)}$ are the union of all strata of codimension i in $M(\hat \beta)$.
(Reidemeisters theorem for closed braids)
$\sum^{(1)} = \sum(tri) \cup \sum(tan)$,
where $\sum(tri)$ is the union of all strata which correspond to diagrams with exactly one ordinary triple point (besides ordinary double points) and $\sum(tan)$ is the union of all strata which correspond to diagrams with exactly one ordinary autotangency.
In the sequel we need also the description of $\sum^{(2)}$.
$\sum^{(2)} = \sum_1 \cup \sum_2 \cup \sum_3 \cup \sum_4$,
where $\sum_1$ is the union of all strata which correspond to diagrams with exactly one ordinary quadruple point, $\sum_2$ is the union of all strata which correspond to diagrams with exactly one ordinary autotangency through which passes another branch transversally, $\sum_3$ corresponds to the union of all strata of diagrams with an autotangency in an ordinary flex, $\sum_4$ is the union of all transverse intersections of strata from $\sum^{(1)}$.
The canonical loop
------------------
We identify $\mathbb{R}^3 \setminus z-axes$ with the standard solid torus $V = S^1 \times D^2 \hookrightarrow \mathbb{R}^3 \setminus z-axes$. We identify the core of V with the unit circle in $\mathbb{R}^2$.
Let $rot(V)$ denote the $S^1$-parameter family of diffeomorphismes of V which is defined in the following way: we rotate the solid torus monotoneously and with constant speed around its core by the angle t , $t \in [0 ,2\pi]$, i.e. all discs $( \phi = const ) \times D^2$ stay invariant and are rotated simoultaneously around their centre.
Let $\hat \beta$ be a closed braid.
The [*canonical loop* ]{} $rot(\hat \beta) \in M(\hat \beta)$ is the oriented loop induced by $rot(V)$.
Notice that the whole loop $rot(\hat \beta)$ is completely determined by an arbitrary point in it.
The following lemma is an immediat corollary of the definition of the canonical loop .
Let $\hat \beta_s , s\in [0 ,1 ]$ , be an isotopy of closed braids in the solid torus. Then $rot(\hat \beta_s ) , s \in [0 ,1 ]$, is a homotopy of loops in $M(\hat \beta )$.
Evidently, the canonical loop can be defined for an arbitrary link in V in exactly the same way. However, in the case of closed braids we can give an alternative combinatorial definition, which makes concret calculations much easier.
Let $\Delta \in B_n$ be Garside’s element, i.e. $\Delta^2$ is a generator of the centre of $B_n$ (see [@B74]). Geometrically, $\Delta^2$ is the full twist of the n strings.
Let $\gamma \in B_n$ be a braid with closure isotopic to $\hat \beta$. Then the [*combinatorial canonical loop* ]{} $rot(\gamma)$ is defined by the following sequence of braids:
$\gamma \to \Delta\Delta^{-1}\gamma \to \Delta^{-1}\gamma\Delta \to \dots \to \Delta^{-1}\Delta\gamma' \to \gamma'
\to \Delta\Delta^{-1}\gamma' \to \Delta^{-1}\gamma'\Delta \to \dots \to \Delta^{-1}\Delta\gamma \to
\gamma$Here, the first arrow consists only of Reidemeister II moves, the second arrow is a cyclic permutation of the braid word (which corresponds to an isotopy of the braid diagram in the solid torus) and the following arrows consist of “pushing $\Delta$ monotoneously from the right to the left through the braid $\gamma$”. We obtain a braid $\tilde \gamma$ and we start again.
We give below a precise definition in the case $n = 3$. The general case is a straightforward generalization which is left to the reader. $\Delta = \sigma_1\sigma_2\sigma_1$ for $n = 3$. We have just to consider the following four cases:$\sigma_1\Delta = \sigma_1(\sigma_1\sigma_2\sigma_1) \to \sigma_1(\sigma_2\sigma_1\sigma_2) = \Delta\sigma_2$$\sigma_2\Delta = (\sigma_2\sigma_1\sigma_2)\sigma_1 \to (\sigma_1\sigma_2\sigma_1)\sigma_1 = \Delta\sigma_1$
$\sigma_1^{-1}\Delta = \sigma_1^{-1}(\sigma_1\sigma_2\sigma_1) \to \sigma_2\sigma_1 \to (\sigma_1\sigma_1^{-1})\sigma_2\sigma_1 \to \sigma_1(\sigma_2\sigma_1\sigma_2^{-1}) = \Delta\sigma_2^{-1}$
$\sigma_2^{-1}\Delta = \sigma_2^{-1}(\sigma_1\sigma_2\sigma_1) \to (\sigma_1\sigma_2\sigma_1^{-1})\sigma_1 \to \sigma_1\sigma_2 \to \sigma_1\sigma_2\sigma_1\sigma_1^{-1} = \Delta\sigma_1^{-1}$.
Notice, that the sequence is canonical in the case of a generator and almost canonical in the case of an inverse generator. Indeed, we could replace the above sequence $\sigma_1^{-1}\Delta \to \Delta\sigma_2^{-1}$ by
$\sigma_1^{-1}(\sigma_1\sigma_2\sigma_1) \to \sigma_2\sigma_1 \to \sigma_2\sigma_1\sigma_2\sigma_2^{-1} \to (\sigma_1\sigma_2\sigma_1)\sigma_2^{-1}$.
But it turns out that the corresponding canonical loops in $M(\hat \beta)$ differ just by a homotopy which passes once transversally through a stratum of $\sum_2^{(2)}$.
Let c be the word lenght of $\gamma$. Then we use exactly $2c(n-2)$ braid relations (or Reidemeister III moves) in the combinatorial canonical loop. This means that the corresponding loop in $M(\hat \beta)$ cuts $\sum^{(1)}(tri)$ transversally in exactly $2c(n-2)$ points.
One easily sees that the combinatorial canonical loop $rot(\gamma)$ from Definition 1 is homotopic in $M(\hat \beta)$ without touching $\sum^{(1)}(tan)$ (i.e. we never make in the one parameter family a Reidemeister II move forwards and just after that the same move backwards) to the geometrical canonical loop $rot(\hat \beta)$ from Definition 2.
The trace graph
---------------
The trace graph $TL(\hat \beta)$ is our main combinatorial object. It is an oriented singular link in a thickened torus. All its singularities are ordinary triple points.
Let $\hat \beta_t , t \in S^1$, be the (oriented) family of closed braids corresponding to the canonical loop $rot(\hat \beta)$. We assume that the loop $rot(\hat \beta)$ is a generic loop. Let $\{ p_1^{(t)}, p_2^{(t}, \dots , p_m^{(t)} \}$ be the set of double points of $pr(\hat \beta_t) \subset S_\phi^1 \times \mathbb{R}_\rho$. The union of all these crossings for all $t \in S^1$ forms a link $TL(\hat \beta) \subset (S_\phi^1 \times \mathbb{R}_\rho^+) \times S_t^1$ (i.e. we forget the coordinate $z(p_i^{(t)})$). $TL(\hat \beta)$ is non-singular besides ordinary triple points which correspond exactly to the triple points in the family $pr(\hat \beta_t)$. A generic point of $TL(\hat \beta)$ corresponds just to an ordinary crossing $p_i^{(t)}$ of some closed braid $\hat \beta_t$. Let $t: TL(\hat \beta) \to S_t^1$ be the natural projection. We orient the set of all generic points in $TL(\hat \beta)$ (which is a disjoint union of embedded arcs) in such a way that the local mapping degree of t at $p_i^{(t)}$ is $+1$ if and only if $p_i^{(t)}$ is a positive crossing (i.e. it corresponds to a generator of $B_n$ , or equivalently , its [*writhe* ]{} $w(p_i^{(t)}) = +1$).
The arcs of generic points come together in the triple points and in points corresponding to an ordinary autotangency in some $pr(\hat \beta)$. But one easily sees that the above defined orientations fit together to define an orientation on the natural resolution $TL\tilde (\hat \beta)$ of $TL(\hat \beta)$ (compare also [@Fi01]). $TL\tilde(\hat \beta)$ is a union of oriented circles, called [*trace circles* ]{}.We can attach [*stickers* ]{} $i \in \{ 1, 2, \dots ,n-1 \}$ to the edges of $TL(\hat \beta)$ in the following way: each edge of $TL(\hat \beta)$ corresponds to a letter in a braid word. Indeed, each generic point in an edge corresponds to an ordinary crossing of a braid projection and , hence , to some $\sigma_i$ or some $\sigma_i^{-1}$ . We attache to this edge the number i . The information about the exponent $+1$ or $-1$ is contained in the orientation of the edge.
We identify $H_1(V)$ with $\mathbb{Z}$ by sending the core of V to the generator $+1$. If $\hat \beta$ is a knot then we can attache to each trace circle a [*homological marking* ]{} $a \in H_1(V)$ in the following way: let p be a crossing corresponding to a generic point in the trace circle. We smooth p with respect to the orientation of the closed braid. The result is an oriented 2-component link. The component of this link which contains the undercross which goes to the overcross at p is called $p^+$. We associate now to p the homology class $a = [p^+] \in H_1(V)$ (compare also [@F93]). One easily sees that $a \in \{ 1 ,2 , \dots , n-1\}$ and that the class a does not depend on the choice of the generic point in the trace circle. Indeed, the two crossings involved in a Reidemeister II move have the same homological marking and a Reidemeister III move does not change the homological marking of any of the three involved crossings.
Evidently, crossings with different homological markings belong to different trace circles. Surprisingly, the inverse is also true. The following lemma is essentially due to Stepan Orevkov.
Let $TL(\hat \beta)$ be the trace graph of the closure of a braid $\beta \in B_n$, such that $\hat \beta$ is a knot. Then $TL\tilde (\hat \beta)$ splitts into exactly n-1 trace circles. They have pairwise different homological markings.
Consequently, the trace circles are characterised by their homological markings. Notice, that the set of homological markings is independent of the word lenght of the braid.
A higher order Reidemeister theorem for trace graphs of closed braids
----------------------------------------------------------------------
A [*trihedron* ]{} is a 1-dimensional subcomplex of $TL(\hat \beta)$ which is contractible in the thickened torus and which has the form as shown in Figure 1.
A [*tetrahedron* ]{} is a 1-dimensional subcomplex of $TL(\hat \beta)$ which is contractible in the thickened torus and which has the form as shown in Figure 2.
A [*trihedron move* ]{} is shown in Figure 3.
A [*tetrahedron move* ]{} is shown in Figure 4.
The rest of $TL(\hat \beta) \hookrightarrow S^1 \times S^1 \times \mathbb{R}^+$ is unchanged under the moves. The stickers on the edges change in the canonical way.
Notice, that a trihedron move corresponds to a generic homotopy of the canonical loop which passes once through an ordinary tangency with a stratum of $\sum^{(1)}(tri)$. A tetrahedron move corresponds to a generic homotopy of the canonical loop which passes transversally once through a stratum of $\sum_1^{(2)}$, i.e. corresponding to an ordinary quadruple point.
The [*equivalence relation for trace graphs $TL(\hat \beta)$* ]{} is generated by the following three operations:
\(1) isotopy in the thickened torus
\(2) trihedron moves
\(3) tetrahedron moves
The following important Reidemeister type theorem is a particular case of Theorem 1.10. in [@F-K06].
Two closed braids (which are knots) are isotopic in the solid torus if and only if their trace graphs in the thickened torus are equivalent.
Notice, that not all representatives of an equivalence class of a trace graph correspond to the canonical loop (which is very rigid) of some closed braid. However, one easily sees that all representatives correspond to loops in the space $M(\hat \beta)$.
A generic homotopy of a loop in $M(\hat \beta)$ could of course be tangent at some point to $\sum^{(1)}(tan)$. One easily sees that this would imply a Morse modification of the trace graph and, hence, change the components of its natural resolution. The corner stone of the theory developed in this paper is the fact, that this does not happen for the (very rigid) homotopies of $rot(\hat \beta)$ which are induced by generic isotopies of $\hat \beta$ in V. Consequently, the trace circles are isotopy invariants for closed braids (compare Lemma 3.4. in [@F-K06]).
A homotopy of loops $\gamma_s , s \in [0 , 1]$ , in $M(\hat \beta)$ is called a [*tan-transverse homotopy* ]{} if no loop $\gamma_s$ is tangential to $\sum^{(1)}(tan)$. Let S be any loop in $M(\hat\beta)$. Its tan-transverse homotopy class is denoted by $[S]_{t-t}$.
There can be loops $\gamma_s$ in a tan-transverse homotopy which are tangential to $\sum^{(1)}(tri)$. For the trace graphs associated to the loops $\gamma_s$ this corresponds to a trihedron move.
In order to define our invariants we need more precise information about isotopies of trace graphs.
A [*time section* ]{} in the thickened torus $(S^1_\phi \times \mathbb{R}^+_\rho) \times S^1_t$ is an annulus of the form $(S^1_\phi \times \mathbb{R}^+_\rho) \times \{ t = const \}$.
The intersection of $TL(\hat \beta)$ with a generic time section corresponds to the crossings of the closed braid $\hat \beta_t$. Using the orientation and the stickers on $TL(\hat \beta)$ we can read off a cyclic braid word for $\hat \beta$ in each generic time section. The tangent points of $TL(\hat \beta)$ with time sections correspond exactly to the Reidemeister II moves in the one parameter family of diagrams $\hat \beta_t , t \in S^1$. A triple point in $TL(\hat \beta)$ slides over such a tangent point if and only if the canonical loop passes in a homotopy transversally through a stratum of $\sum_2^{(2)}$. We illustrate this in Figure 5.
When the canonical loop passes transversally through a stratum of $\sum_3^{(2)}$ then the trace graph changes as shown in Figure 6.
Finally, when the canonical loop passes transversally through a stratum of $\sum_4^{(2)}$ then the t-values of triple points or tangencies with time sections are interchanged. We show an example in Figure 7.
One-cocycle invariants
======================
In this section we introduce our one-cocycle invariants. They are a special class of finite type invariants for knots in the solid torus.
Gauss diagrams for closed braids with a triple crossing
--------------------------------------------------------
Let $f : S^1 \to \hat \beta$ be a generic orientation preserving diffeomorphisme. Let p be any crossing of $\hat \beta$. We connect $f^{(-1)}(p) \in S^1$ by an oriented chord, which goes from the undercross to the overcross and we decorate it by the writhe $w(p)$. Moreover, we attache to the chord the homological marking $[p^+]$. The result is called a [*Gauss diagram* ]{} for $\hat \beta$ (compare e.g. [@PV] and [@F01]).
One easily sees that $\hat \beta$ up to isotopy is determined by its Gauss diagram and the number $n = [\hat \beta] \in H_1(V)$. (Notice, that the trace graph of a closed braid is always transverse to the $\phi$-sections.)
A [*Gauss sum of degree d* ]{} is an expression assigned to a diagram of a closed braid which is of the following form:
$\sum$ function( writhes of the crossings)
where the sum is taken over all possible choices of d (unordered) different crossings in the knot diagram such that the chords without the writhes arising from these crossings build a given subdiagram with given homological markings. The marked subdiagrams (without the writhes) are called [*configurations*]{}. If the function is the product of the writhes, then we will denote the sum shortly by the configuration itself. We need to define Gauss diagrams for knots with an ordinary triple point too. The triple point corresponds to a triangle in the Gauss diagram of the knot. Notice, that the preimage of a triple point has a natural ordering coming from the orientation of the $\mathbb{R}^+$-factor. One easily sees that this order is completely determined by the arrows in the triangle.
We provide each stratum of $\sum^{(1)}(tri)$ with a coorientation which depends only on the non-oriented underlying curves $pr(\hat \beta)$ in $S^1 \times \mathbb{R}^+$. Consequently, for the definition of the coorientation we can replace the arrows in the Gauss diagram simply by chords.
The [*coorientation* ]{} of the strata in $\sum^{(1)}(tri)$ is given in Figure 8.
Notice that the second line in Figure 8 does not occure for closed braids, but it does occure for almost closed braids. The two coorientations are choosen in such a way that they fit together in $\sum^{(2)}_2$.
There are exactly two types of triple points without markings. We show them in Figure 9.
We attache now the homological markings to the three chords. Let $a,b \in \{1 , 2 , \dots , n-1\}$ be fixed. Then the markings are as shown in Figure 10. We encode the types of the marked triple points by $(a ,b)^-$ and $(a ,b)^+$. The union of the corresponding strata of $\sum^{(1)}(tri)$ are encoded in the same way.
One-cocycle invariants of degree one
------------------------------------
We will construct in a canonical way one-cocycles on the space $M_n$ (the space of all closed n-braids which are knots). We obtain numerical invariants when we evaluate these cocycles on the homology class represented by the canonical loop.
Let $n, d \in \mathbb{N}^*$ be fixed. Let $(a,b)^\pm$ be a fixed [*type of marked triple point*]{} as shown in Figure 10. Here $a, b, a+b \in \mathbb{Z}/n\mathbb{Z}$.
A [*configuration I of degree d*]{} is an abstract Gauss diagram without writhes which contains exactly $d-1$ arrows marked in $\mathbb{Z}/n\mathbb{Z}$ besides the sub-diagram $(a,b)^\pm$.
Let $\{ I_i \}$ be the finite set of all configurations of degree $d$ with respect to $(a,b)^\pm$. Let $\Gamma_{(a,b)^\pm} = \sum_{i}
{\epsilon_i I_i}$ be a linear combination with each $\epsilon_i \in
\{ 0, +1, -1 \}$. (The type of the triple point is [*always*]{} fixed in any cochain $\Gamma$).
$\Gamma_{(a,b)^\pm}$ gives rise to a [*1-cochain of degree $d$*]{} by assigning to each oriented generic loop $S \subset M_n$ an integer $\Gamma_{(a,b)^\pm}(S)$ in the following way: $$\Gamma_{(a,b)^\pm}(S) =
\sum_{s_i \in S \cap \Sigma^{(1)}(tri) \textrm{of type $(a,b)^\pm$}}
{w(s_i)(\sum_{i}{\epsilon_i (\sum_{D_i}{\prod_{j}{w(p_j)}))}}}$$ where $D_i$ is the set of unordered $(d-1)$-tuples $(p_1, \dots, p_{d-1})$ of arrows which enter in $I_i$ in the Gauss diagram of $s_i$ .
If $\Gamma_{(a,b)^\pm}(S)$ is invariant under each generic deformation of $S$ through a stratum of $\Sigma^{(2)}$, then $\Gamma_{(a,b)\pm}$ is a 1-cocycle.
[*Proof:*]{} In this case, $\Gamma_{(a,b)^\pm}(S)$ is invariant under homotopy of $S$. Indeed, tangent points of $S$ with $\Sigma^{(1)}(tri)$ correspond just to trihedron moves. The two triple points give the same contribution to $\Gamma_{(a, b)^\pm}(S)$ but with different signs. A tangency with $\Sigma^{(1)}(tan)$ does not change the contribution of the triple points at all. This implies the invariance under homology of $S$ because the contributions to $\Gamma_{(a,b)\pm}(S)$ of different diagrams with triple points are not related to each other. $\Box$
A [*cohomology class*]{} in $H^1(M_n; \mathbb{Z})$ is [*of degree $d$*]{} if it can be represented by some 1-cocycle $\Gamma_{(a,b)\pm}$ of degree at most $d$.
The above definition induces a filtration on a part of $H^1(M_n;
\mathbb{Z})$.
Let $M$ be the (disconnected) space of all embeddings $f: S^1
\hookrightarrow \mathbb{R}^3$. Vassiliev [@V90] has introduced a filtration on a part of $H^1(M; \mathbb{Z})$ using the discriminant $\Sigma_{sing}$. It is not difficult to see that the space of all (unparametrized) differentiable maps of the circle into the solid torus is contractible. Indeed, there is an obvious canonical homotopy of each (perhaps singular) knot to a multiple of the core of the solid torus.
The core of the solid torus is invariant under $rot_{S^1}(V)
\times rot_{D^2}(V)$. Thus, the above space is star-like. Therefore, Alexander duality could be applied and Vassilievs approach could be generalized for knots in the solid torus too. It would be interesting to compare his filtration with our filtration.
In the next sections, we will construct 1-cocycles $\Gamma_{(a,b)\pm}$ in an explicit way.
Let $\beta \in B_n$ be such that its closure $\hat \beta \hookrightarrow
V$ is a knot.
The space of finite type invariants of degree 1 is of dimension $[n/2]$ (here $[.]$ is the integer part). It is generated by the Gauss diagram invariants $W_a(\hat \beta) = \sum_{}{w(p)}$, where $a \in
\{ 1, 2, \dots, [n/2] \}$. The sum is over all crossings with fixed homological marking a.
[*Proof:*]{} It follows from Goryunov’s [@G96] generalization of finite type invariants for knots in the solid torus that the invariants of degree 1 correspond just to marked chord diagrams with only one chord. Obviously, all these invariants can be expressed as Gauss diagram invariants:
$W_a(\hat \beta) = \sum_{}{w(p)}$, $a \in \{ 1, \dots, n-1 \}$
(see also [@F93], and Section 2.2 in [@F01].)
Let us define $V_a(\hat \beta) := W_a(\hat \beta) - W_{n-a}(\hat \beta)$ for all $a \in \{ 1, \dots, n-1 \}$. We observe that $V_a(\hat \beta)$ is invariant under switching crossings of $\hat \beta$. Indeed, if the marking of the crossing $p$ was $[p] = a$, then the switched crossing $p^{-1}$ has marking $[p^{-1}] = n-a$, but $w(p)=
-w(p^{-1})$. But every braid $\beta \in B_n$ is homotopic to $\gamma = \prod_{i=1}
^{n-1}{\sigma_i}$. A direct calculation for $\gamma$ shows that $V_a(\hat \gamma) \equiv 0$. It is easily shown by examples that $W_a, a \in \{1, \dots, [n/2] \}$ (seen as invariants in $\mathbb{Q}$) are linearly independent. $\Box$
Let $a, b \in \mathbb{Z}/n\mathbb{Z}$ be fixed. Consider the union of all cooriented strata of $\Sigma^{(1)}$ which correspond to triple points of type either $(a, b)^-$ or $(a, b)^+$. The closure in $M_n$ of each of these sets form integer cycles of codimension 1 in $M_n$.
Otherwise stated, $\Gamma_{(a,b)^+}$ and $\Gamma_{(a,b)^-}$ both define integer 1-cocycles of degree 1. $\Gamma_{(a,b)^\pm}(S)$ is in this case by definition just the algebraic intersection number of $S$ with the corresponding union of strata of $\Sigma^{(1)}(tri)$.
[*Proof:*]{} According to Section 2, we have to prove that the cooriented strata fit together in $\Sigma^{(2)}_1$ and $\Sigma^{(2)}_2$. The first is evident , because at a stratum of $\Sigma^{(2)}_1$ just four strata of $\Sigma^{(1)}(tri)$ intersect pairwise transversally. For the second, we have to distinguish 24 cases. Three of them are illustrated in Figure 11. The whole picture in a normal disc of $\Sigma^{(2)}_2$ is then obtained from Figure 12.
All other cases are obtained from these three by inverting the orientation of the vertical branch, by taking the mirror image (i.e. switching all crossings), and by choosing one of two possible closings of the 3-tangle (in order to obtain an oriented knot). In all cases, one easily sees that the two adjacent triple points are always of the same marked type and that the coorientations fit together. $\Box$
$\Gamma_{(a,b)^+}$ defines a non-trivial 1-cohomology class of degree 1 if and only if $a \not= b$ and $a + b \leq n-1$.
$\Gamma_{(a,b)^-}$ defines a non-trivial 1-cohomology class of degree 1 if and only if $a \not= b$ and $a + b \geq n+1$.
The following identities hold:
(\*) $\Gamma_{(a,b)^+} + \Gamma_{(b,a)^+} \equiv 0$
(\*) $\Gamma_{(a,b)^-} + \Gamma_{(b,a)^-} \equiv 0$
[*Proof:*]{} For closed braids, the markings are all in $\{ 1, \dots, n-1 \}$. Therefore, if $a+b>n-1$ in $\Gamma_{(a,b)^+}$ or $a+b<n+1$ in $\Gamma_{(a,b)^-}$, then there is no such triple point at all and the 1-cocycle is trivial. $\Gamma_{(a,a)^\pm} \equiv 0$ is a special case of the identities.
Examples show that all the remaining 1-cocycles are non-trivial (we will see lots of examples later).
In order to prove the identities, we use the following Gauss diagram sums (see also section 1.6 in [@F01]): $$I^+_{(a,b)} = \sum_{} w(p)w(q), \qquad I^-_{(a,b)} = \sum_{} w(p)w(q)$$ Here, the first sum is over all couples of crossings which form a subconfiguration as shown in Figure 13. The second sum is over all couples of crossings which form a subconfiguration as shown in Figure 14. (Here, a and b are the homological markings.) These sums applied to diagrams of $\hat \beta$ are not invariants. Let $S \subset M(\hat \beta)$ be a generic loop. Then $I^\pm_{(a,b)}$ is constant except when $S$ crosses $\Sigma^{(1)}(tri)$ in strata of type $(a,b)^\pm$ or $(b,a)^\pm$. At each such intersection in positive (resp., negative) direction, $I^\pm_{(a,b)}$ changes exactly by $-1$ (resp., $+1$). Indeed, the configurations of the three crossings which come together in a triple point are shown in Figure 15. In each of the four cases, exactly one pair $p, q$ of crossings contributes to one of the sums $I$.
After drawing all possible triple points, it is easily seen that $p, q$ must verify: $w(p)w(q) = -1$ for the first two cases and $w(p)w(q)=+1$ for the last two cases. Notice that the type of the triple point is completely determined by the sub-configuration shown in Figure 13 and 14. Thus, the sums $I$ are constant by passing all types of triple points except those shown in Figure 15. The generic loop $S$ intersects $\Sigma$ only in strata that correspond to triple points or to autotangencies. An autotangency adds to the Gauss diagram always one of the sub-diagrams shown in Figure 16.
The two arrows do not enter together in the configurations shown in Figure 13 and 14. If one of them contributes to such a configuration, then the other contributes to the same configuration but with an opposite sign.
Therefore, for any $\hat \beta_1$, $\hat \beta_2 \in M(\hat \beta)
\setminus \Sigma$, the difference $I^\pm_{(a,b)}(\hat \beta_1) -
I^\pm_{(a,b)}(\hat \beta_2)$ is just the algebraic intersection number of an oriented arc from $\hat \beta_1$ to $\hat \beta_2$ with the union of the cycles of codimension one $(a,b)^+ \cup (b,a)^+$ (resp., $(a,b)^- \cup (b,a)^-$). Hence, for each loop $S$, these numbers are 0. The identities (\*) follow and the proof of the proposition is complete.$\Box$
Obviously, for closed 2-braids, there are no 1-cocycles of any degree $d$ at all (because there are never triple points). The previous proposition implies that there are no non-trivial 1-cocycles of degree one for closed 3-braids, but that in general there are $2((n-3)+(n-5)+(n-7)+
\dots)$ such cocycles.
We will see later that if we apply a 1-cocycle of degree one to the canonical class $[rot(\hat \beta)]$ then we obtain a finite type invariant of degree one for $\hat \beta$. Proposition 3 says that there are exactly $[n/2]$ such invariants which are independent. But the number of non-trivial 1-cocycles of degree one is quadratic in $n$. Therefore, there are lots of relations between them when they are restricted to the canonical class. The following question seems to be interesting: Are the relations (\*) from Proposition 3 the only relations in general between the 1-cocycles of degree one?
Let $\hat \beta$ be the closure of the 4-braid $\beta = \sigma_1
\sigma_2^{-1}\sigma_3^{-1}$. We consider $$\Gamma_{(1,2)^-} and \Gamma_{(2,1)^-}$$ $$\Gamma_{(2,3)^+} and \Gamma_{(3,2)^+}$$ A calculation by hand gives: $$\Gamma_{(1,2)^-}(rot(\hat \beta)) =
\Gamma_{(2,3)^+}(rot(\hat \beta)) = -1$$ and $$\Gamma_{(2,1)^-}(rot(\hat \beta)) =
\Gamma_{(3,2)^+}(rot(\hat \beta)) = +1$$ Therefore, all four 1-cocycles of degree 1 are non-trivial.
One-cocycles of degree two
--------------------------
Let $\beta \in B_n$ such that $\hat \beta \hookrightarrow V$ is a knot and let $a \in \mathbb{Z}/n\mathbb{Z}$ be fixed. Let $I$ be a configuration of degree 2, i.e. besides the marked triple point there is exactly one marked arrow.
An [*adjacent configuration of $I$*]{} is any configuration which is obtained by sliding exactly one of the end points of the arrow over exactly one of the vertices of the triangle and which preserves the markings. An example is shown in Figure 17. A [*chain of adjacent configurations*]{} is a sequence of configurations such that any two consecutive configurations are adjacent.
A configuration $I$ of degree 2 is called [*braid impossible*]{} if it never occures as a subconfiguration of a Gauss diagram of a closed $n$-braid which is a knot. Otherwise it is called [*braid possible*]{}
Any configuration $I$ which contains one of the sub-configurations in Figure 18 is braid impossible.
[*Proof:*]{} This follows immediately from the proof of Proposition 4.2 in [@Fi03].$\Box$
There are other configurations $I$ which are braid impossible. For example, it follows from Proposition 4.3 in [@Fi03] that $I$ is braid impossible for $n=3$ if it contains one of the sub-configurations drawn in Figure 19.
A configuration $I$ is called [*rigid*]{} if [*all*]{} adjacent configurations are braid impossible. Otherwise it is called [*flexible*]{}.
Let $a \in \{ 1, 2, \dots ,[(n-1)/2] \}$ be fixed.The configurations shown in Figure 20 are all braid possible and rigid.
[*Proof:*]{} It is easy to show that the above configurations are braid possible by just taking examples. Let us show that e.g. the first of them is rigid. Indeed, all four adjacent configurations contain the third sub-configuration of Figure 18. Thus, by Lemma 6, they are not braid possible. The proof in the other cases is analogous.$\Box$
Let $I$ be a braid possible and rigid configuration. Let $I'$ be any configuration obtained from $I$ by reversing some arrows and replacing for each of these arrows its marking $x$ by $n-x$. We say that $I'$ is [*derived from*]{} $I$. If the triangle of $I'$ corresponds still to a Reidemeister III move, then $I'$ is also braid possible and rigid. This is evident from the fact that the set of closed braids is invariant under arbitrary changings of crossings. Therefore, the above construction allows to get lots of braid possible rigid configurations starting from the three cases in Figure 20 (examples are shown in Figure 21).
Let $\Gamma(S)$ be a 1-cochain of degree 2. Obviously, $\Gamma(S)$ is invariant under deformation of $S$ through all strata in $\Sigma^{(2)}$ besides $\Sigma^{(2)}_1$ and $\Sigma^{(2)}_2$ (see the proof of Proposition 3). Let us consider quadruple points. We have to guarantee that $\Gamma(S_m)=0$ if $S_m$ is the boundary of a small normal disc of a stratum in $\Sigma^{(2)}_1$. Each triple point occurs exactly twice in $S_m$, say at $s_1, s_2 \in S_m$., and with different signs $w(s_1)=-w(s_2)$.
The Gauss diagrams of the closed braids are the same at $s_1$ and $s_2$ besides exactly three crossings $p_1, p_2, p_3$ which are now in another position with respect to the triple point (and to each other). We illustrate this in Figure 22. Let $I$ be a configuration which contains the triple point and only one of the three crossings, say $p_1$. One easily sees that the triple point together with $p'_1$ is then an adjacent configuration. For example, $p_1$ and $p'_1$ are in the same position with respect to the crossing between the branches 2 and 3 (see Figure 22). [*Thus, if $n>3$, then for each configuration $\epsilon_iI_i$ in $\Gamma$ (see Definition 12), $\Gamma$ has to contain all adjacent configurations of $I_i$ and they have to enter all with the same coefficient $\epsilon_i$.*]{} We call this condition on $\Gamma$ the [*quadruple-condition*]{}. If $\Gamma$ verifies the [*quadruple-condition*]{}, then $\Gamma(S_m)=0$. In particular, this implies that the triple point together with $p_1$ or $p_2$ or $p_3$ is never a rigid configuration! It remains to study a loop $S_m$ which is the boundary of a normal disc for a stratum in $\Sigma^{(2)}_2$ (see Figure 12). The two triple points in $S_m$ are of the same type and have different signs as already explained in the proof of Lemma 5. But their Gauss diagrams differ by exactly one crossing as shown in Figure 23. But now $p$ and $p'$ are different crossings. The homological markings $[p]=[p']$ coincide but $w(p)=-w(p')$.
Therefore, the configurations on the left-hand side and on the right-hand side of Figure 23 should enter in $\Gamma$ with opposite coefficients $\epsilon_i$. Notice that the marking $[p]$ of $p$ coincides with at least one of the markings of the triple point.
A configuration $I$ is called [*t-invariant*]{} if the marking of the arrow is different from all three markings of the triple point.
Let $\Gamma' \subset \Gamma$ be the t-invariant part of $\Gamma$. Then, evidently, $\Gamma'(S_m)=0$.
Let us now consider configurations which are not t-invariant. If the two configurations in Figure 23 could be related by a chain of adjacent configurations then our method would break down. Indeed, in order to guarantee invariance under passing a quadruple point, they should enter in $\Gamma$ with the same coefficient $\epsilon$. But in order to guarantee invariance under passing an autotangency with a transverse branch they should enter in $\Gamma$ with opposite coefficients. The following surprising lemma implies that this does not occur for some types of triple points.
Let the type of the triple point in Figure 23 be one of the types shown in Figure 20 (or one obtained from them as explained in Remark 8). Then at least one of the two configurations in Figure 23 is rigid.
[*Proof:*]{} We have to distinguish two cases for the markings.
[*Case 1*]{} All three markings of the triple point are different. This is the case in $II$ and $III$ of Figure 20 if $3a \not= n$.
[*Case 2*]{} There are exactly two markings which are equal, hence we are in $I$ of Figure 20 or in $II$ or $III$ with $3a=n$. (Remember that markings of braid possible configurations are always non-zero).
The crossings $p$ and $p'$ form together a sub-configuration as shown in Figure 18. The crossings $p$ and $p'$ interchange the place in the triangle (corresponding to the triple point) and one easily sees that both arrows $p$ and $p'$ always move in the same direction on the circle. Therefore, in Case 1, we have the couples of configurations as shown in Figure 24. We see that the configurations on the left-hand side are always rigid and those on the right-hand side are always flexible. In Case 2, we have the couples of configurations shown in Figure 25.
The cases $I$ and $III$ with $3a=n$ are equivalent to the cases shown in Figure 25. We see that both configurations are always rigid.$\Box$
[*Thus, for each configuration $\epsilon_iI_i$ in $\Gamma$ which is shown on the right-hand side of Figure 24 or 25, $\Gamma$ has to contain also the corresponding rigid configuration on the left-hand side of Figure 24 or 25. This configuration has to enter in $\Gamma$ with the coefficient $-\epsilon_i$.*]{} We call this condition on $\Gamma$ the [*t-condition*]{}.
Let $\Gamma$ be a 1-cochain of degree 2 that satisfies the quadruple-condition and the t-condition. Then, $\Gamma$ is a 1-cocycle of degree 2.
[*Proof:*]{} We have proven that $\Gamma(S)$ is invariant under all homotopies of $S$. The rest of the proof follows from Lemma 8. $\Box$
The case of closed 3-braids is very special because there do not appear any quadruple points in isotopies. Therefore we do not need the quadruple-condition. It follows easily from the proof of Lemma 8 that the 1-cochain in Figure 26 defines a 1-cocycle of degree 2. (For closed 3-braids, the homological markings of the triple point are determined by the arrows.) An easy calculation yields: $$\Gamma(rot(\hat {\sigma_1\sigma_2^{-1}}))=+2$$ Therefore, $\Gamma$ defines a non-trivial 1-cohomology class of degree 2.
It follows from Theorem 4 that the $\Gamma$ shown in Figure 27 defines a 1-cocycle of degree 2 for closed 4-braids. A calculation yields: $$\Gamma(rot(\hat {\sigma_1\sigma_2^{-1}\sigma_3^{-1}}))=-1$$ Therefore, $\Gamma$ is a non-trivial class of degree 2.
[*Observation:*]{} Let $a \in \{ 1, \dots ,[n/2]-1 \}$ be fixed and let $I_a^{(1)}$ be the configuration shown in Figure 28. This configuration is derived from the configuration $I$ in Figure 20 (see Remark 8). Therefore, the configuration $I_a^{(1)}$ is a rigid braid-possible configuration. But moreover, $I_a^{(1)}$ is a t-invariant configuration and hence, $I_a^{(1)}$ [*already defines a 1-cocycle of degree 2*]{}. In the next section, we will see how to generalize $I_a^{(1)}$ in order to obtain non-trivial 1-cocycles of arbitrary degree in a very simple way.
One-cocycles of arbitrary degree
--------------------------------
For the degree $d>2$, other strata in $\Sigma^{(2)}$ also impose conditions on $\Gamma$.
$\Gamma$ verifies the [*tan-condition*]{} if no configuration $I$ in $\Gamma$ contains any sub-configuration as shown in Figure 16.
If $\Gamma$ verifies the [*tan-condition*]{} then $\Gamma(S)$ is invariant under deformation of $S$ through points in $\Sigma^{(1)}(tri) \cap \Sigma^{(1)}(tan)$. The next definition is a straightforward generalization of Definition 14.
An [*adjacent configuration*]{} of a braid possible configuration $I$ is any configuration that is obtained in the following way: one chooses an arrow among the $(d-1)$ that are not part of the triangle, and one slides an endpoint of this arrow over exactly one endpoint of another arrow (the latter may belong to the triangle). All markings of arrows are preserved. The resulting configuration has to be also braid possible.
$\Gamma$ verifies the [*tri-condition*]{} if for each (braid possible) configuration $\epsilon I$ in $\Gamma$, all adjacent configurations of $I$ are also contained in $\Gamma$, with the same coefficient $\epsilon$.
If $\Gamma$ verifies the tri-condition, then $\Gamma(S)$ is invariant under deformation of $S$ through points in $\Sigma^{(1)}(tri)
\cap \Sigma^{(1)}(tri)$. The quadruple-condition for degree 2 is generalized for arbitrary degree in the obvious way. Notice that if $\Gamma$ satisfies the tri-condition then it satisfies automatically the quadruple-condition.
Obviously, the strata of $\Sigma^{(2)}_3$ do not impose any condition on $\Gamma$. The t-condition for degree 2 has to be generalized in the following way.
The configurations on the same line in Figure 29 are called [*associated configurations*]{} if besides the shown sub-configuration, the rest of the configurations are identical. Notice that in the small arcs of the circle there are no other endpoints of arrows.
(Of course we extend all definitions to the configurations derived from those of Figure 29 by the Remark 8). If a configuration $I$ is different from all configurations in Figure 29 (and of their derived configurations), then the associated configuration will be the empty configuration.
$\Gamma$ satisfies the [*t-condition*]{} if each of its configurations $\epsilon I$ occurs together with $-\epsilon I'$ where $I'$ is the associated configuration.
Let $\Gamma$ be a 1-cochain of degree $d$ that satisfies the t, tri, tan-conditions. Then $\Gamma$ is a 1-cocycle of degree $d$.
[*Proof:*]{} It is completely similar to the proof of Theorem 4.$\Box$
Let us take a 1-cocycle of degree 1, and consider the sum of all diagrams that are obtained from this 1-cocycle by adding one single arrow with a new marking, in any possible position with respect to the triangle. This sum is a 1-cocycle of degree 2. But this 1-cocycle is not interesting because it is just the product of the 1-cocycle of degree 1 with an invariant of degree 1 (see Proposition2). In order to get a 1-cocycle $\Gamma$ which do not decompose into products of 1-cocycles of lower degrees we need that [*not all*]{} possible positions of an arrow with a fixed marking enter into $\Gamma$ with the same coefficient.
[*Observation:*]{} Let $\beta \in B_n$ and let $a \in \{ 1, \dots, n-1 \}$ be fixed. Then the sub-configurations shown in Figure 30 are [*locally rigid*]{}, i.e. none of the two arrows can slide over the other one in the small pictured arcs, because the resulting configuration would not be braid possible. Moreover, they verify the tan-condition. Using this observation, it is easy to construct non-decomposing 1-cocycles of arbitrary degrees. We give an example in the following proposition.
The configuration in Figure 31 defines a 1-cocycle of odd degree $d$ if $3a = n$.
[*Proof:*]{} Obviously, the whole configuration is rigid. Moreover, no arrow with marking $n-a$ can become an arrow of the triangle (by passing $\Sigma^{(2)}_2$), because it is always separated from the triangle by an arrow with marking $a$. Evidently, no arrow with marking $a$ can become an arrow of the triangle (by passing $\Sigma^{(2)}_2$). No arrow can slide over the triangle, because $3a = n$. $\Box$ We generalize now the 1-cocycle $I^{(1)}_a$ from the end of the previous section.
Let $d \in \mathbb{N}^*$ be odd and let $a \in \{ 1, 2, \dots, [n/2]-1 \}$ be fixed. The configuration $I_a^{(d)}$ is defined in Figure 32 (in the actual picture, we took $d=5$). The arrows with markings $a$ and $n-a$ are alternating in the figure.
$I^{(d)}_a$ defines a 1-cocycle of degree $d+1$.
[*Proof:*]{} It is completely analogous to the proof of Proposition 4. $\Box$
Let $d \in \mathbb{N}^*$ be even, let $n$ be divisible by 3 and let $a=n/3$. The 1-cochain $\Gamma_{n/3}^{(d)}$ is defined in Figure 33 (in the actual picture, we took $d=4$).
$\Gamma^{(d)}_{n/3}$ defines a 1-cocycle of degree $d+1$.
[*Proof:*]{} The first configuration in $\Gamma^{(d)}_{n/3}$ is still rigid. But exactly one of the arrows with marking $n/3$ can be interchanged now with exactly one of the arrows in the triangle (by passing $\Sigma^{(2)}_2$). The result is the second configuration in the figure. This configuration is no longer rigid. The remaining configurations in Figure 33 are just all the adjacent configurations. We need that $3a=n$ in order to guarantee that in the last configuration the arrow $n/3$ cannot slide further over some of the remaining two vertices of the triangle.$\Box$
For example, the braid possible sub-configuration $II$ in Figure 20 with $a = n/3$ is not contained in $\Gamma^{(d)}_{n/3}$. Such considerations imply easily that in fact $\Gamma^{(d)}_{n/3}$ is not a product of 1-cocycles of lower degrees.
If $n$ is not divisible by 3 then we replace $\hat \beta$ by a $3k$-cable, $k \in \mathbb{N}^*$, and take $a=kn$.
The mirror image $\Gamma^{(d)}_{n/3}!$ of $\Gamma^{(d)}_{n/3}$ (obtained by reversing all arrows, including those of the triangles, and replacing all markings by their opposites) is of course also a 1-cocycle of degree $d+1$.
Let $\beta = \sigma_1^{-1}\sigma_2^3 \in B_3$. Then, $\Gamma_1^{(2)}!(rot(\hat \beta))=-1$. This shows that the above 1-cocycles are not always trivial.
Let $K \hookrightarrow V$ be a closed braid which is a knot and let $\Gamma$ be a 1-dimensional cohomology class of degree $d$. Let $[rot(K)]$ be its canonical class. Then, $\Gamma([rot(K)])$ is a $\mathbb{Z}$-valued finite type invariant of degree at most $d$ for $K \hookrightarrow V$.
[*Proof:*]{} We have shown that $\Gamma([rot(K)])$ depends only on the isotopy type of $K \hookrightarrow V$. The cocycle invariant $\Gamma([rot(K)])$ is calculated as some sum $\sum_{s_i}$ over triple points $s_i$ in $rot(K)$. Therefore, it suffices to prove that this sum $\sum_{}$ for each triple point $s_i$ is of finite type (even if it is not invariant). If $\Gamma$ is of degree $d$ then $\sum_{s_i}$ depends only on the triple point and of configurations of $d-1$ other crossings. This means that in order to calculate a summand in $\sum_{s_i}$ we can switch all other crossings besides the triple point and the fixed $d-1$ crossings. The result will not change. This implies immediately that each $\sum_{s_i}$ is of degree $d$ (see [@PV] and also [@F01]).$\Box$
We call $\Gamma([rot(K)])$ a [*1-cocycle invariant of degree $d$.*]{}
Let $K \hookrightarrow V$ be a closed 4-braid. Using Theorem 6, Proposition 2, Proposition 3 and the examples of the next section, one easily calculates that $$\Gamma_{(2,1)^-}([rot(K)]) \equiv \Gamma_{(3,2)^+}([rot(K)])
\equiv W_1(K)-W_2(K)$$
The 1-cocycle invariants have the following nice property, which can be used to estimate from below the length of conjugacy classes of braids.
[**Proposition 1**]{} [*Let the knot $K = \hat \beta \hookrightarrow V$ be a closed $n$-braid and let $c(K)$ be its minimal crossing number, i.e. its minimal word length in $B_n$. Then all 1-cocycle invariants of degree $d$ vanish for*]{} $$d \geq c(K) + n^2 - n - 1.$$
[*Proof:*]{} Assume that the word length of $\beta$ is equal to $c(K)$. We can represent $[rot(K)]$ by the following isotopy which uses shorter braids. $$\beta \to \Delta\Delta^{-1}\beta \to \Delta^{-1}\beta\Delta
\to \Delta^{-1}\Delta\beta' \to \beta' \to \Delta\Delta^{-1}\beta'
\to \Delta^{-1}\beta'\Delta \to \Delta^{-1}\Delta\beta \to \beta$$
Here, $\beta'$ is the result of rotating $\beta$ by $\pi$ i.e. each $\sigma_i^{\pm 1}$ is replaced by $\sigma_{n-i}^{\pm 1}$. Obviously, $c(\Delta) = \frac{n(n-1)}{2}$. Thus, each Gauss diagram which appears in the isotopy has no more than $c(K) + n^2 - n$ arrows. Indeed, we create a couple of crossings by pushing $\Delta$ through $\beta$ only after having eliminated a couple of crossings before (see the previous section). Therefore, for each diagram with a triple point there are at most $c(K)+n^2-n-3$ other arows, and hence, each summand in a 1-cocycle of degree $d$ is already zero if $d \geq
c(K) + n^2 - n - 1$.$\Box$
The invariants are not functorial under cabling
-----------------------------------------------
Bar-Natan, Thang Le, Dylan Thurston [@B-T-T] and independently S. Willerton [@Wi] have given a formula for the Kontsevich integral of the cable of a knot in $\mathbb{R}^3$. This formula has the following corollary:
(Bar-Natan, Thang Le, D. Thurston - Willerton) Let $K, K'\hookrightarrow \mathbb{R}^3$ be knots which have the same Vassiliev invariants up to a fixed degree $d$. Then, all (the same) cables of $K$ and $K'$ have the same Vasiliev invariants up to degree $d$.
In other words, it is useless to cable knots in purpose to distinguish them by Vassiliev invariants.
Our 1-cocycle invariants of degree $d$ form a subset of all finite type invariants of degree $d$ for knots in the solid torus. For example, as already mentioned, there are no non-trivial 1-cocycle invariants at all for closed 2-braids and there are no non-trivial 1-cocycle invariants of degree 1 for closed 3-braids. However, it turns out that cabling is a usefull operation for the subset of 1-cocycle invariants of degree $d$.
The closed 2-braids $\hat \sigma_1$, $\hat \sigma_1^{-1}$ can not be distinguished by any 1-cocycle invariants. However, $Cab_2(\hat \sigma_1)$ and $Cab_2(\hat \sigma_1^{-1})$ can be distinguished by a 1-cocycle invariant of degree 1.
[*Proof:*]{} $Cab_2(\hat \sigma_1)$ and $Cab_2(\hat \sigma_1^{-1})$ can be represented respectively by the 4-braids $\beta = \sigma_3\sigma_2\sigma_1\sigma_2^2$ and $\beta'= \sigma_3^{-1}\sigma_2\sigma_1^{-1}\sigma_2^{-2}$. A calculation by hand shows:
$$\Gamma_{(2,1)^-}([rot(\hat \beta)]) =
\Gamma_{(3,2)^+}([rot(\hat \beta)]) =+1$$
and
$$\Gamma_{(2,1)^-}([rot(\hat \beta')]) =
\Gamma_{(3,2)^+}([rot(\hat \beta')]) =-3$$
$\Box$
Obviously, closed 2-braids are classified by the unique invariant $W_1$ of degree 1. The braid $\hat \sigma_1$ is obtained from $\hat \sigma_1^{-1}$ by multiplying $\sigma_1^{-1}$ with $\sigma_1^2$. >From this, one easily concludes that for all $k \in \mathbb{Z}$,
$$\Gamma_{(2,1)^-}([rot(Cab_2(\sigma_1^{2k+1}))]) \equiv
\Gamma_{(3,2)^+}([rot(Cab_2(\sigma_1^{2k+1}))]) \equiv 1+4k$$
Therefore, the unique non-trivial 1-cocycle of degree 1 for the 2-cable of 2-braids classifies also closed 2-braids.
Do all finite type invariants of closed braids and of local knots arise as linear combinations of 1-cocycle invariants of appropriate cables?
Homological estimates for the number of braid relations in one-parameter families of closed braids
--------------------------------------------------------------------------------------------------
Our 1-cocycles can be used in order to obtain information about one-parameter families of closed braids which are knots. Let $K$ be a closed n-braid which is a knot. Let $S$ be a generic loop in $M(K)$.
The \*-[*length*]{} $b([S])$ of $[S] \in H_1(M(K); \mathbb{Z})$ is the minimal number of triple points in $S$ among all unions of generic loops $S$ in $M(K)$ which represent $[S]$.
Let $a, b \in \{ 0, 1, \dots, n \}$ with $a<b$. Then $$b([S]) \geq 2 \sum_{(a,b)^+}{\vert \Gamma_{(a,b)^+}([S])\vert}
+2\sum_{(a,b)^-}{\vert \Gamma_{(a,b)^-}([S])\vert}$$
[*Proof:*]{} Each $\Gamma_{(a,b)^+}$, $\Gamma_{(a,b)^-}$ is a (in general non-trivial) 1-cocycle of degree 1 (see Proposition 3). Each triple point in $S$ contributes by $\pm 1$ to the value of such a cocycle. The inequality follows now from the relations (\*) in Proposition 3.$\Box$
Let $\beta = \sigma_1\sigma_2^{-1}\sigma_3^{-1} \in B_4$. For all $m \in \mathbb{Z}$, we have $b(m[rot(\hat \beta)])
\geq 4 \vert m \vert$. This follows immediately from Example 2.
Theorem 7 does not contain any information in the case of closed 3-braids (because all 1-cocycles of degree 1 are trivial). Therefore, we use the 1-cocycle $\Gamma$ of degree 2 from Example 2. Let $\beta = \sigma_1\sigma_2^{-1} \in B_3$. One has $\Gamma(rot(\hat \beta)) = +2$. It is easily seen that this implies $\Gamma !(rot(\hat \beta)) = +2$ too, where $\Gamma !$ is the mirror image of $\Gamma$ (see Remark 10). Thus, for all $m \in \mathbb{Z} \setminus 0$, $m(rot(\hat \beta))$ intersects both types of strata in $\Sigma^{(1)}(tri)$ (compare Figure 9). But the closure of the union of all strata of the same type in $\Sigma^{(1)}(tri)$ are trivial cycles of codimension 1 in $M(\hat \beta)$ and hence, $m(rot(\hat \beta))$ intersects each of these two strata in at least two points. It follows that $b(m[rot(\hat \beta)]) \geq 4$. The canonical loop shows that this inequality is sharp for $m = \pm 1$.
Character invariants
====================
In this section we introduce new easily calculable isotopy invariants for closed braids.
Remember, that a generic isotopy $\hat \beta_s , s \in [0 ,1]$, induces a tan-transverse homotopy $rot(\hat \beta_s) , s \in [0 ,1]$, of the canonical loops. The trace circles for different parameter s are in a natural one-to-one correspondence. Consequently, we can give [*names* ]{} $x_i$ to the circles of $TL\tilde(\hat \beta_0)$ and extend these names in a unique way on the whole family of trace circles.
Let $\{x_1 ,x_2 ,\dots \}$ be the set of named trace circles. Obviously, for each circle $x_i$ there is a well defined homological marking $h_i \in H_1(V)$. Let $[x_i] \in H_1(T^2)$ be the homology class represented by the circle $x_i$ (with its natural orientation induced from the orientation of the trace graph).
Character invariants of degree one
----------------------------------
In this section, we use the named cycles, i.e. the trace circles, in order to refine the 1-cocycles of degree 1 which were defined in Section 3.2.
Let K be a closed n-braid which is a knot. Let $S \subset M(K)$ be a generic loop and let $X = \{ x_1, \dots, x_m \}$ be the corresponding set of cycles of named crossings. Let $x_{i_1}, x_{i_2}, x_{i_3} \in X$ be fixed. We do not assume that they are necessarily different. Let $h_{i_1},
h_{i_2}, h_{i_3}$ be the corresponding homological markings.
A [*character of degree 1 of $S$*]{}, denoted by $$C_{(h_{i_1}, h_{i_2})^\pm(x_{i_1}, x_{i_2}, x_{i_3})}(S)$$ or sometimes shortly $C(S)$, is the algebraic intersection number of $S$ with the strata $(h_{i_1}, h_{i_2})\pm$ in $\Sigma^{(1)}(tri)$ and such that the crossings of the triple point belong to the named cycles as shown in Figure 34. We call the unordered set $\{ x_{i_1}, x_{i_2}, x_{i_3} \}$ the [*support*]{} of the character $C$.
Evidently, in order to obtain a non-trivial intersection number we need that $h_{i_3} = h_{i_1} + h_{i_2} - n$ in $(h_{i_1}, h_{i_2})+$ and that $h_{i_3} = h_{i_1} + h_{i_2}$ in $(h_{i_1}, h_{i_2})-$ (see Figure 10). It follows also that $x_{i_1} = x_{i_2} = x_{i_3}$ implies that necessarily $h_{i_1} = h_{i_2} = h_{i_3} =0$ or $h_{i_1} = h_{i_2} = h_{i_3} = n$.
Notice that for characters of degree 1 the relations (\*) from Proposition 3 are no longer valid. For example, $C_{(h_{i_1}, h_{i_2})^+(x_{i_1}, x_{i_2}, x_{i_3})}(S)$ can be non-trivial for $h_{i_1} = h_{i_2}$ and even for $x_{i_1} = x_{i_2}$.
Let $S, S' \subset M(K)$ be generic loops and let $\{ x_1, \dots, x_m \}$, $\{ x'_1, \dots, x'_{m'} \}$ be the corresponding sets of named cycles. If $S$ and $S'$ are tan-transverse homotopic, then $m = m'$ and there is a bijection $\sigma: \{ x_1, \dots, x_m \} \to \{ x'_1, \dots, x'_{m'} \}$ which preserves the homological markings $h_i$ as well as the homology classes $[x_i]$ and such that $$C_{(h_{i}, h_{j})^\pm(x_{i_1}, x_{i_2}, x_{i_3})}(S) =
C_{(h_i, h_j)^\pm(\sigma (x_{i_1}), \sigma (x_{i_2}), \sigma (x_{i_3}))}(S')$$ for all triples $(x_{i_1}, x_{i_2}, x_{i_3})$.
[*Proof:*]{} This is an immediate consequence of Lemma 5 and the fact that the trace circles are isotopy invariants.$\Box$
In other words, characters $C(S)$ of degree one are invariants of $[S]_{t-t}$.
Character invariants of arbitrary degree for loops of closed braids
-------------------------------------------------------------------
We refine the results of the sections 3.3 and 3.4. in a straightforward way.Let $S \subset M(K)$ be an oriented generic loop and let $X = \{ x_1, \dots, x_m \}$ be the corresponding set of named cycles.
Let $I$ be a configuration of degree $d$ (see Definition 11) and let $(x_{i_1}, \dots, x_{i_{d+2}})$ be a fixed $(d+2)$-tuple of elements in $X$ (not necessarily distinct). A [*named configuration*]{} $I_{(x_{i_1}, \dots, x_{i_{d+2}}), \phi }$ is the configuration $I$ together with a given bijection $\phi$ of $(x_{i_1}, \dots, x_{i_{d+2}})$ with the $d+2$ arrows in $I$ and such that $x_{i_1}, x_{i_2}, x_{i_3}$ are the arrows of the triangle exactly as in the previous section. Of course, different bijections give in general different named configurations.
A [*character chain*]{} $\Gamma$ of degree $d$ is a linear combination $\Gamma = \sum_i{\epsilon_i I_i}, \epsilon_i \in \{ +1, -1, \}$ of named configurations $I_i$ of degree $d$ which are all defined with the same $d+2$-tuple $(x_{i_1}, \dots, x_{i_{d+2}})$ where $x_{i_1}, x_{i_2}, x_{i_3}$ are the arrows of the triangle as shown in Figure 34. We refine the definitions and conditions of Sections 3.3. and 3.4. in the obvious way. [*Adjacent named configurations*]{} are obtained by sliding the arrows and preserving the names.
In [*associated named configurations*]{}, the two arrrows which interchange have the same name. We show an example in Figure 35. The [*named quadruple and tri-conditions*]{} are now: $\Gamma$ contains all named adjacent configurations and they enter $\Gamma$ with the same sign.
The [*t-condition*]{} is now: $\Gamma$ contains all named associated configurations and they enter $\Gamma$ with different signs.
The [*tan-condition*]{} is now: no named configuration $I_i$ in $\Gamma$ contains any sub-configuration as shown in Figure 36. Notice that sub-configurations as shown in Figure 36 are allowed if the arrows have different names $x_i \not= x_j$, even if they have the same homological markings $h_i = h_j$.
Let $S, S' \subset M(K)$ be generic loops and let $\{ x_1, \dots, x_m \}$, $\{ x'_1, \dots, x'_m \}$ be the corresponding sets of named cycles. If $S$ and $S'$ are tan-transverse homotopic, then there is a bijection $$\sigma: \{ x_1, \dots, x_m \} \to \{ x'_1, \dots, x'_m \}$$ which preserves the homological markings $h_i$ as well as the homology classes $[x_i]$ and such that $$C_{(x_{i_1}, \dots, x_{i_{d+2}})}(S) =
C_{(\sigma (x_{i_1}), \dots, \sigma(x_{i_{d+2}}))}(S')$$ for all character chains of degree $d$ which satisfy the named quadruple, tri, t, tan-conditions.
[*Proof:*]{} This is completely analogous to the proofs of Theorems 5 and 8.$\Box$
The character chains $C(S)$ of the above theorem are called [*characters of degree $d$*]{}. They are invariants of $[S]_{t-t}$.
Here, all names $x_i$ are different.
The named configuration shown in Figure 37 defines a character of degree 5 for closed $n$-braids if $h_1 = h_2 = h_4 \leq [\frac {n-1}{2}]$. Indeed, the configuration is braid possible and rigid (see Lemmas 6 and 7) and neither $x_2$ nor $x_4$ can become an arrow of the triangle by passing $\Sigma^{(2)}_2$.
Let $h_i ,h_j$ and a type of triple point, e.g. $(h_i ,h_j)^+$, be fixed. It follows immediately from the definitions that
(\*\*) $\Gamma_{(h_i ,h_j)^+} = \sum_{(x_k ,x_l ,x_m)} C_{(h_i ,h_j)+}(x_k ,x_l ,x_m )$
where $h(x_k) = h_i$ and $h(x_l) = h_j$.
Hence, character invariants define splittings of one-cocycle invariants. However, the set of character invariants on the right hand side of (\*\*) is not an ordered set. Therefore we have to consider the set of character invariants as a local system on $M(\hat \beta)$.
It follows from Lemma 3 that the names $x_i$ are determined by their homological markings $h_i$, and that there are exactly n-1 trace circles.
However, this is in general no longer true in the case of multiples of the canonical loop.
Let $l \in \mathbb{N}$ be fixed and let $l rot(\hat \beta)$ be the loop which is defined by going $l$ times along the canonical loop. Let $TL(l\hat\beta)$ denote the corresponding trace graph (the t-coordinate in the thickened torus covers now $l$ times the t-circle).
We will show in a simple example that the local system of character invariants is in general non trivial if $l > 1$.
Let $\beta = \sigma_2\sigma_1^{-1}\sigma_2\sigma_1^{-1} \in B_3$. We will write shortly $\beta = 2\bar12\bar1$.
The combinatorial canonical loop for $l = 2$ is given by the following sequence (where we write the names of the crossings just below the crossings).
$2\bar 12\bar 1 \hspace{0.3cm} \to \hspace{0.3cm}
\bar 1\bar 2\bar 12\bar 12(\bar 11)21 \hspace{0.3cm} \to \hspace{0.3cm}
\bar 1\bar 2\bar 12\bar 1221 \hspace{0.3cm} \to \hspace{0.3cm}
\bar 1\bar 2\bar 12\bar 121(\bar 121) \hspace{0.4cm} *_1\to$\
$abcd \hspace{1.3cm}
zyxabcdxyz \hspace{1.4cm}
zyxabcyz \hspace{1.5cm}
zyxabcu_1u_1yz$\
$\bar 1\bar 2\bar 12\bar 1(212)1\bar 2 \hspace{0.3cm} *_2\to
\bar 1\bar 2\bar 12(\bar 11)211\bar 2 \to
\bar 1\bar 2\bar 12211\bar 2 \to
\bar 1\bar 2\bar 121(\bar 121)1\bar 2 \hspace{0.3cm} *_3\to$ $zyxabcu_1zyu_1 \hspace{1.2cm}
zyxabzu_1cyu_1 \hspace{0.5cm}
zyxau_1cyu_1 \hspace{0.5cm}
zyxau_2u_2u_1cyu_1$\
$\bar 1\bar 2\bar 1(212)1\bar 21\bar 2 \hspace{0.5cm} *_4\to
(\bar 1\bar 2\bar 1121)1\bar 21\bar 2 \hspace{0.3cm} \to \hspace{0.3cm}
1\bar 21\bar 2 \hspace{0.2cm} \to \hspace{0.2cm}
\bar 1\bar 2\bar 11\bar 21(\bar 212)1 \hspace{0.2cm} *_5\to$ $zyxau_2cu_1u_2yu_1 \hspace{0.9cm}
zyxcu_2au_1u_2yu_1 \hspace{0.6cm}
u_1u_2yu_1 \hspace{0.5cm}
z_1y_1x_1u_1u_2yu_1x_1y_1z_1$\
$\bar 1\bar 2\bar 11\bar 2112(\bar 11) \hspace{0.3cm} \to \hspace{0.3cm}
\bar 1\bar 2\bar 11\bar 2112 \hspace{0.3cm} \to \hspace{0.3cm}
\bar 1\bar 2\bar 11\bar 21(121)\bar 1 \hspace{0.3cm}*_6\to \hspace{0.3cm}
\bar 1\bar 2\bar 11(\bar 212)12\bar 1$ $z_1y_1x_1u_1u_2yy_1x_1u_1z_1 \hspace{0.2cm}
z_1y_1x_1u_1u_2yy_1x_1 \hspace{0.2cm}
z_1y_1x_1u_1u_2yy_1x_1v_1v_1 \hspace{0.3cm}
z_1y_1x_1u_1u_2yv_1x_1y_1v_1$\
$*_7\to
\bar 1\bar 2\bar 1112(\bar 11)2\bar 1 \to
\bar 1\bar 2\bar 11122\bar 1 \to
\bar 1\bar 2\bar 11(121)\bar 12\bar 1 \hspace{0.3cm} *_8\to
(\bar 1\bar 2\bar 1121)2\bar 12\bar 1$ $z_1y_1x_1u_1v_1yu_2x_1y_1v_1 \hspace{0.4cm}
z_1y_1x_1u_1v_1yy_1v_1 \hspace{0.3cm}
z_1y_1x_1u_1v_1yv_2v_2y_1v_1 \hspace{0.3cm}
z_1y_1x_1u_1v_2yv_1v_2y_1v_1$\
$\to 2\bar 12\bar 1$\
$v_1v_2y_1v_1$
We have the identifications: $d=x$, $b=z$, $x=c$, $y=u_2$, $z=a$, $u_1=z_1$, $u_2=x_1$, $x_1=u_1$, $y_1=v_2$, $z_1=y$. This gives us:
$2\bar 12\bar 1 \to 2\bar 12\bar 1$\
$aacc \hspace{0.6cm} v_1v_2v_2v_1$
together with the names $u_1=z_1=x_1=u_2=y$.
The second rotation gives us:
$2\bar 12\bar 1 \to 2\bar 12\bar 1 \to 2\bar 12\bar 1$\
$aacc \hspace{0.4cm} v_1v_2v_2v_1 \hspace{0.4cm}
v_3v_4v_4v_3$
together with the identification $v_1=v_2$, $y_3=v_4$, $z_2=v_2$ and with the names $u_1=z_1=x_1=u_2=y$, $u_3=z_3=x_3=u_4=y_2$. The monodromy (i.e. how the set of crossings is maped to itself after the rotation) implies now: $a=v_3=v_4=c$.
Therefore, we have exactly four named cycles: $a, v_1, u_1, u_3$ for
$2\bar 12\bar 1 \to 2\bar 12\bar 1$ with $l = 2$.
Consequently, we have
$2\bar12\bar1 \to 2\bar12\bar1 \to 2\bar12\bar1$ with the names $aaaa \to v_1v_1v_1v_1 \to aaaa$.
Hence, $rot(\hat \beta)$ acts by interchanging $a$ and $v_1$ (as well as $u_1$ and $u_3$). Consequently, the local system is non trivial in this example.
The examples
------------
The following examples are calculated by Alexander Stoimenow using his program in c++. His program is available by request (see [@S]).
Let $\beta = \bar12\bar1^32^3 \in B_3$. ( $\hat \beta$ represents the knot $8_9$.)
We want to show that the link $\hat \beta \cup$ (core of complementary solid torus) is not invertible in $S^3$. This is equivalent to show that $\beta$ is not conjugate to $\beta_{inverse} = 2^3\bar1^32\bar1$ (compare [@F01]).
Because $\beta$ is a 3-braid , the homological markings are in $\{1 ,2\}$.
Character invariants of degree one for $l = 1$ and $l = 2$ do not distinguish $\hat \beta$ from $\hat \beta_{inverse}$. However, for $l = 3$ we obtain three different named cycles $x_1 ,x_2 ,x_3$ of homological marking 1 and three different named cycles $y_1 ,y_2 ,y_3$ of marking 2.
We consider the set of nine character invariants of degree one which are of the following form (see Figure 38) , where $i ,j \in \{1 ,2 ,3\}$.
For $\hat \beta$ we obtain the set $\{-1 ,-1 ,-1 ,-1 ,-1 ,-1 ,2 ,2 ,2 \}$ and for $\hat \beta_{inverse}$ we obtain the set $\{1 ,1 ,1 ,1 ,1 ,1 ,-2 ,-2 ,-2\}$. Evidently, even without knowing the local system, there is no bijection of the trace circles for $\hat \beta$ and those for $\hat \beta_{inverse}$ which identifies the above sets. Consequently, $\hat \beta$ and $\hat \beta_{inverse}$ are not isotopic.
The knot $9_5$ can be represented as a 8-braid with 33 crossings. Character invariants of degree one for $l = 2$ show that the braid is not invertible in the same way as in the previous example.
The knot $8_6$ can be represented as a 5-braid with 14 crossings. Character invariants of degree one for $l = 2 ,4 ,6$ show that it is not invertible as a 5-braid. (It does not work for for $l = 1 ,3 ,5$.)
The knot $8_{17}$ is not invertible as a 3-braid, which is shown with $l = 4$. (It does not work with $l = 1 ,2 ,3$.)
Let $b \in P_5$ be Bigelow’s braid ( see [@Bi] ). It has trivial Burau representation. Let $s = \sigma_1\sigma_2\sigma_3\sigma_4$. The braids s and bs have the same Burau representation. This is still true for their 2-cables, i.e. we replace each strand by two parallel strands. Character invariants of degree one for $l = 2$ show that the (once positively half-twisted) 2-cables of the above braids are not conjugate, and, consequently, the braids s and bs are not conjugate either.
A refinement of character invariants of degree one
--------------------------------------------------
The number of triple points in a trace graph can only change by trihedron moves, as follows from Theorem 3 .
A [*generalized trihedron* ]{} is a trihedron which might have other triple points on the edges.
Figure 39 shows a tetrahedron move which transformes a trihedron into a generalized trihedron. The generalized trihedron has still exactly two vertices.
Evidently, the number of generalized trihedrons does not change under tetrahedron moves. Let $E$ be the set of all triple points in the trace graph $TL(\hat \beta)$ which are not vertices of generalized trihedrons. The following lemma is an immediate consequence of Theorem 3.
The set $E$, and, hence $card(E)$, is an isotopy invariant of closed braids $\hat \beta$.
Moreover, for each element of $E$ we have the additional structure defined before: type , sign, markings, names.
It follows from Theorem 3 and the geometric interpretation of generalized trihedrons in [@F-K06] that the two vertices of a generalized trihedron have always different signs. Consequently, character invariants of degree one count just the algebraic number of elements in $E$ which have a given type and given names. But already the geometric number of such elements in $E$ is an invariant as shows Lemma 9.
Let $C^{+(-)}_{(h_i ,h_j)^{+(-)}}(x_k ,x_l ,x_m)$ be the number of all positive (respectively negative) triple points in $E$ of given type $(h_i ,h_j)^{+(-)}$ and with given names $x_k ,x_l ,x_m $. We call these the [*positive (respectively negative) character invariants*]{}.
The following proposition is now an immediate consequence of Lemma 9 and Definition 30.
The positive and the negative character invariants are isotopy invariants of closed braids.
Let us consider $\beta = \sigma_2\sigma_1^{-1} \in B_3$. Its trace graph $TL(\hat \beta)$ is shown in Figure 40. One easily sees that it does not contain any generalized trihedrons. Consequently, all four triple points are in $E$. There are exactly two names $x_1$ and $x_2$. They correspond to the homological markings $h_1 = 1$ and $h_2 = 2$.
One easily calculates that two of the triple points are of type $(1 ,1)^-$ and they have different signs. The other two are of type $(2 ,2)+$ and they have different signs too. Consequently, all character invariants of degree one are zero.
However, we have $C^+_{(1 ,1)^-}(x_1 ,x_1 ,x_2) = C^-_{(1 ,1)^-}(x_1 ,x_1 ,x_2) = 1$, and\
$C^+_{(2 ,2)^+}(x_2 ,x_2 ,x_1) = C^-_{(2,2)^+}(x_2 ,x_2 ,x_1) = 1$.Consequently, the positive and negative character invariants contain in this example more information than the character invariants of degree one (for $l = 1$).
There is not yet a computer program available in order to calculate these invariants in more sophisticated examples.
Homotopical estimates for the number of braid relations in one parameter families of closed braids
--------------------------------------------------------------------------------------------------
In section 3.6. we have used the 1-cocycles in order to estimate the $*$-length $b([S])$ for classes $[S] \in H_1(M(K); \mathbb{Z})$.
Let $S \subset M(K)$ be a generic loop.
The $*$-[*length*]{} $b([S]_{t-t})$ of the tan-transvers homotopy class is the minimal number of triple points in $S$ amongst all generic loops in $M(K)$ which represent $[S]_{t-t}$.
Let $\{ x_1, \dots, x_m \}$ be the set of essential named cycles of $S$. Let $C_{(x_{i_1}, x_{i_2}, x_{i_3})}(S)$, $\{ i_1, i_2, i_3 \} \subset \{ 1, \dots, m\}$ be the set of all characters of degree one for $S$. Then
$$b([S]_{t-t}) \geq \sum_{\{ i_1,i_2,i_3 \} \subset \{ 1, \dots, m \}}
{\vert C_{(x_{i_1}, x_{i_2}, x_{i_3})}(S) \vert}$$
[*Proof:*]{} Different triples $(x_{i_1}, x_{i_2}, x_{i_3})$ correspond to different strata of $\Sigma^{(1)}(tri)$. Locally, each intersection index of $S$ with such a stratum is equal to $\pm 1$. The result follows.$\Box$
One easily calculates that for $S=2rot(\hat {\sigma_2\sigma_1^{-1}})$ there are exactly eight non-trivial characters of degree one. Each of them is equal to $\pm 1$. This can be generalized straightforwardly for arbitrary $l$ with $\vert l \vert \geq 2$. We obtain the following proposition:
For all integers $l$ such that $\vert l \vert \geq 2$, we have: $$b([l.rot(\hat {\sigma_2\sigma_1^{-1}})]_{t-t}) = 4\vert l \vert$$
[*Proof:*]{} This follows immediately from Theorem 10 together with a direct calculation which shows that
$$b([l.rot(\hat {\sigma_2\sigma_1^{-1}})]_{t-t}) \leq 4\vert l \vert$$
$\Box$
Characters of higher degree can be used, of course, in the same way as 1-cocycles of higher degree were used to estimate $b([S])$ (see section 3.6.).
Character invariants of degree one for almost closed braids
===========================================================
Let $K \hookrightarrow V = \mathbb{R}^3 \setminus z-axes$ be an oriented knot such that the restriction of $\phi$ to $K$ has exactly two critical points. In this case $K$ is called an [*almost closed braid*]{}. Necessarily, one of the critical points is a local maximum and the other is a local minimum. In analogy to the case of closed braids we consider almost closed braids up to isotopy through almost closed braids.
We consider the (geometric) canonical loop $rot(K)$ and the corresponding trace graph $TL(K)$. $TL(K)$ is an oriented link with triple points and exactly four boundary points (corresponding to the oriented tangencies at the critical points of $\phi$).
There are more types of moves for $TL(K)$ as in the case of closed braids (see [@F-K06]). But it turns out that only two of these additional moves are relevant for the construction of the invariants.
A [*band move*]{} is shown in Figure 41.
A [*unknot move* ]{} is shown in Figure 42.
A band move corresponds to a branch which passes transversally through an ordinary cusp and an unknot move corresponds to the case that the maximum and the minimum of $\phi$ have the same critical value. Notice, that there are no [*extrem pair moves* ]{}, i.e. two maxima or two minima of $\phi$ with the same critical value (for all this compare [@F-K06]).An extreme pair move induces a Morse modification of index 1 of the trace graph. Such modifications are difficult to control. We restrict ourselfs to almost closed braids in order to avoid them.
The unknot component $x_i$ of $TL(K)$ which was created by an unknot move, has always $[x_i] = 0$ in $H_1(T^2)$. Notice, that in a band move there is always involved one component $x_i$ which has boundary. Let $X = \{x_1 ,x_2 ,\dots \}$ be the set of all closed trace circles of $TL\tilde(K)$ which represent non-trivial homology classes in $H_1(T^2)$.
The above observations together with Theorem 8 and Theorem 1.10 from [@F-K06] imply immediately the following theorem.
Each character invariant of degree one with names $x_l ,x_k ,x_m \in X$ is an isotopy invariant for almost closed braids.
Evidently, we can apply the above theorem to $lrot(K)$ with arbitrary $l \in \mathbb{N}$ exactly as in the case of closed braids.
[99]{} Bigelow S. : The Burau representation is not faithful for n = 5 , Geom. Topol. 3 (1999) , 397-404.
Bar-Natan D. , On the Vassiliev knot invariants, Topology 34 (1995), 423-472.
Bar-Natan D. , Thang T. Q. Le , Thurston D. : Two applications of elementary knot theory to Lie algebras and Vassiliev invariants , Geom. Topol. 7 (2003) , 1-31.
Birman J . : Braids , Links and Mapping class groups , Annals of Mathematics Studies 82 , Princeton University Press (1974).
Birman J. , Brendle T. : Braids : A survey , mathGT/0409205 (2004).
Fiedler T. : A small state sum for knots , Topology 32 (1993) , 281-294.
Fiedler T. : Gauss diagram invariants for Knots and Links , Mathematics and Its Applications 532 ,Kluwer Academic Publishers(2001).
Fiedler T. : Isotopy invariants for smooth tori in 4-manifolds , Topology 40 (2001) , 1415-1435.
Fiedler T. : Gauss diagram invariants for knots which are not closed braids , Math. Proc. Camb. Phil. Soc. 135(2003) , 335-348. Fiedler T. : One parameter knot theory , preprint 262, Lab. Math. Emile Picard ,UPS ,(2003).
Fiedler T. , Kurlin V. : A one-parameter approach to knot theory , math.GT/0606381.
Goryunov V. : Finite order invariants of framed knots in a solid torus and in Arnold’s $J^+$-theory of plane curves , “Geometry and Physics” , Lect. Notes in Pure and Appl. Math. (1996), 549-556.
Morton H. : Infinitely many fibered knots having the same Alexander polynomial , Topology 17 (1978) ,101-104.
Polyak M. ,Viro O. : Gauss diagram formulas for Vassiliev invariants , Internat. Math. Res. Notes 11 (1994), 445-453.
Stoimenow A. : www.kurims.kyoto-u.ac.jp/ stoimeno/
Vassiliev V. : Cohomology of knot spaces , Adv. in Sov. Math. , Theory of Singularities and its Appl. , A.M.S. Providence, R.I. (1990) , 23-69.
Vassiliev V. : Combinatorial formulas of cohomology of knot spaces , Moscow Math. Journal 1 (2001) , 91-123.
Willerton S. : The Kontsevich integral and algebraic structures on the space of diagrams , “Knots in Hellas 98” , Series on Knots and Everything 24 , World Scientific (2000), 530-546.
Laboratoire Emile Picard
Université Paul Sabatier
118 ,route de Narbonne
31062 Toulouse Cedex 09, France
[email protected]
[^1]: 2000 [*Mathematics Subject Classification*]{}: 57M25.[*Key words and phrases*]{}: Closed braids, Vassiliev invariants, character invariants, non-invertibility of links
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Many important physical phenomena involve subtle signals that are difficult to observe with the unaided eye, yet visualizing them can be very informative. Current motion magnification techniques can reveal these small temporal variations in video, but require precise prior knowledge about the target signal, and cannot deal with interference motions at a similar frequency. We present DeepMag an end-to-end deep neural video-processing framework based on gradient ascent that enables automated magnification of subtle color and motion signals from a specific source, even in the presence of large motions of various velocities. While the approach is generalizable, the advantages of DeepMag are highlighted via the task of video-based physiological visualization. Through systematic quantitative and qualitative evaluation of the approach on videos with different levels of head motion, we compare the magnification of pulse and respiration to existing state-of-the-art methods. Our method produces magnified videos with substantially fewer artifacts and blurring whilst magnifying the physiological changes by a similar degree.'
author:
- Weixuan Chen
- Daniel McDuff
bibliography:
- 'sample-bibliography.bib'
date: June 2018
title: |
DeepMag: Source Specific Motion Magnification\
Using Gradient Ascent
---
{width="\textwidth"}
Introduction
============
Revealing subtle signals in our everyday world is important for helping us understand the processes that cause them. Magnifying small temporal variations in video has applications in both basic science (e.g., visualizing physical processes in the world), engineering (e.g., identifying the motion of large structures) and education (e.g, teaching scientific principals). To provide an illustration, physiological phenomena are often invisible to the unaided eye, yet understanding these processes can help us detect and treat negative health conditions. Pulse and respiration magnification specifically, are good exemplar tasks for video magnification as physiological phenomena cause both subtle color and motion variations. Furthermore, larger rigid and non-rigid motions of the body often mask the subtle variations, which makes the magnification of physiological signals non-trivial.
Several methods have been proposed to reveal subtle temporal variations in video. *Lagrangian* methods for video magnification [@liu2005motion] rely on accurate tracking of the motion of particles (e.g., via optical flow) over time. These approaches are computationally expensive and will not work effectively for color changes. *Eulerian* video magnification methods do not rely on motion estimation, but rather magnify the variation of pixel values over time [@wu2012eulerian]. This simple and clever approach allows for subtle signals to be magnified that might otherwise be missed by optical flow. Subsequent iterations of such approaches have improved the method with phase-based representations [@wadhwa2013phase], matting [@elgharib2015video], second-order manipulation [@Zhang2017], and learning-based representations [@oh2018learning]. However, all these approaches use frequency properties to separate the target signal from noise, so they require precise prior knowledge about the signal frequency. Furthermore, if the signal of interest is at a similar frequency to another signal (for example if head motions are at a similar frequency as the pulse signal) an Eulerian approach will magnify both and cause numerous artifacts (see Fig. \[fig:overview\]).
To address these problems, we present a generalized approach for magnifying color and motion variations in videos that feature other periodic or random motions. Our method leverages a convolutional neural network (CNN) as a video motion discriminator to separate a specific source signal even if it overlaps with other motion sources in the frequency domain. Then the separated signal can be magnified in video by performing gradient ascent [@erhan2009visualizing] in the input space of the CNN, with the other motion sources untouched. To adapt the gradient ascent method to the video magnification task, several methodological innovations are introduced including adding L1 normalization and sign correction. The whole algorithm proves to work effectively even in the presence of interference motions with large magnitudes and velocities. Fig. \[fig:overview\] shows a comparison between the proposed method and previous approaches.
While our method can generally be applied to any type of color or motion magnification task, magnifying physiological changes on the human body without impacting other aspects of the visual appearance is an especially interesting use case with numerous applications in and of itself. In medicine and affective computing the photoplethysmogram (PPG) and respiration signals are used as unobtrusive measures of cardiopulmonary performance. Visualizing these signals could help in the understanding vascular disease, heart conditions (e.g., arterial fibrillation) [@chan2016diagnostic] and stress responses. For example, jugular venous pressure (JVP) is analyzed by studying subtle motions of the neck. This is challenging for clinicians and video-magnification could offer a practical aid. Another application is in the design of avatars [@suwajanakorn2017synthesizing]. Synthetic embodied agents may fall into the “uncanny valley" [@mori1970uncanny] or be easily detected as “spoofs" if they do not exhibit accurate physiological responses, including respiration, pulse rates and blood flow that can be recovered using video analysis [@poh2010non]. Our method presents the opportunity to not only magnify signals but also synthesize them at different frequencies within a video.
The main contributions of this paper are to: (1) present our novel end-to-end framework for video magnification based on a deep convolutional neural network and gradient ascent, (2) demonstrate recovery of the pulse and respiration waves and magnification of these signals in the presence of large rigid head motions, (3) systematically quantitatively and qualitatively compare our approach with state-of-the-art motion magnification approaches under different rigid motion conditions.
Related Work
============
Video Motion Magnification
--------------------------
Lagrangian video magnification approaches involve estimation of motion trajectories that are then amplified [@liu2005motion; @wang2006cartoon]. However, these approaches require a number of complex steps including, performing a robust registration, frame intensity normalization, tracking and clustering of feature point trajectories, segmentation and magnification. Another approach, using temporal sampling kernels can aid visualization of time-varying effects within videos [@fuchs2010real]. However, this method involves video downsampling and relies on high framerate input videos.
The neat Eulerian video magnification (EVM) approach proposed by Wu et al. [@wu2012eulerian] combines spatial decomposition with temporal filtering to reveal time varying signals without estimating motion trajectories. However, it uses linear magnification that only allows for relatively small magnifications at high spatial frequencies and cannot handle spatially variant magnification. To counter the limitation, Wadhwa et al. [@wadhwa2013phase] proposed a non-linear phase-based approach, magnifying phase variations of a complex steerable pyramid over time. Replacing the complex steerable pyramid [@wadhwa2013phase] with a Riesz pyramid [@wadhwa2014riesz] produces faster results. In general, the linear EVM technique is better at magnifying small color changes, while the phase-based pipeline is better at magnifying subtle motions [@Wu2012web]. Both the EVM and the phase-EVM techniques rely on hand-crafted motion representations. To optimize the representation construction process, a learning-based method [@oh2018learning] was proposed, which uses convolutional neural networks as both frame encoders and decoders. With the learned motion representation, fewer ringing artifacts and better noise characteristics have been achieved.
One common problem with all the methods above is that they are limited to stationary objects, whereas many realistic applications would involve small motions of interest in the presence of large ones. After motion magnification, these large motions would result in large artifacts such as haloes or ripples, and overwhelm any small temporal variation. A couple of improvements have been proposed including a clever layer-based approach called DVMAG [@elgharib2015video]. By using matting, it can amplify only a specific region of interest (ROI) while maintaining the quality of nearby regions of the image. However, the approach relies on 2D warping (either affine or translation-only) to discount large motions, so it is only good at diminishing the impact of motions parallel to the camera plane and cannot deal with more complex 3D motions such as the human head rotation. The other method addressing large motion interferences is video acceleration magnification (VAM) [@Zhang2017]. It assumes large motions to be linear on the temporal scale so that magnifying the motion acceleration via a second-order derivative filter will only affect small non-linear motions. However, the method will fail if the large motions have any non-linear components, and ideal linear motions are rare in real life, especially on living organisms.
Another problem with all the previous motion magnification methods is that they use frequency properties to separate target signals from noise, so they typically require the frequency of interest to be known a priori for the best results and, as such, have at least three parameters (the frequency bounds and a magnification factor) that need to be tuned. If there are motion signals from different sources that are at similar frequencies (e.g., someone is breathing and turning their head), it is previously not possible to isolate the different signals.
Gradient Ascent for Feature Visualization
-----------------------------------------
Opposite to gradient descent, gradient ascent is a first-order iterative optimization algorithm that takes steps proportional to the positive of the gradient (or approximate gradient) of a function. Since neural networks are generally differentiable with respect to their inputs, it is possible to perform gradient ascent in the input space by freezing the network weights and iteratively tweaking the inputs towards the maximization of an internal neuron firing or the final output behavior. Early works found that this technique can be used to visualize network features (showing what a network is looking for by generating examples) [@erhan2009visualizing; @simonyan2013deep] and to produce saliency maps (showing what part of an example is responsible for the network activating a particular way) [@simonyan2013deep].
A recent famous application of gradient ascent in feature visualization is Google DeepDream [@mordvintsev2015deepdream]. It maximizes the L2 norm of activations of a particular layer in a CNN to enhance patterns in images and create a dream-like hallucinogenic appearance. It should be noted that applying gradient ascent independently to each pixel of the inputs commonly produces images with nonsensical high-frequency noise, which can be improved by including a regularizer that prefers inputs that have natural image statistics. Also, following the same idea of DeepDream, not only a network layer but also a single neuron, a channel, or an output class can be set as the objective of gradient ascent. For a comprehensive discussion of various regularizers and different optimization objectives used in feature visualization tasks see [@olahfeature].
None of the previous works have applied gradient ascent to motion magnification or any task related to motions in video. In contrast to DeepDream and similar visualization tools, our method maximizes the output activation of a CNN in motion representations computed from frames instead of in raw images.
Video-Based Physiological Measurement
-------------------------------------
Over the past decade video-based physiological measurement using RGB cameras has developed significantly [@mcduff2015survey]. For instance, physiological parameters such as heart rate (HR) and breathing rate (BR) have been accurately extracted from facial videos in which subtle color changes of the skin caused by blood circulation can be amplified and analyzed (a.k.a., imaging plethysmography) [@verkruysse2008remote; @poh2010non; @poh2011advancements; @de2013robust; @Tarassenko2014; @Wang2016b]. Similar metrics have also been extracted by analyzing subtle face motions associated with the blood ejection into the vessels (a.k.a., imaging ballistocardiography) [@Balakrishnan2013] as well as more prominent chest volume changes during breathing [@Tan2010; @Janssen2016].
Early work on imaging plethysmography identified that spatial averaging of skin pixel values from an imager could be used to recover the blood volume pulse [@takano2007heart]. The strongest pulse signal was observed in the green channel [@verkruysse2008remote], but a combination of color channels provides improved results [@poh2010non; @mcduff2014improvements]. Combining these insights with face tracking and signal decomposition enables a fully automated recovery of the pulse wave and heart rate [@poh2010non].
In the presence of dynamic lighting and motion, advancements were needed to successfully recover the pulse signal. Leveraging models grounded in the optical properties of the skin has improved performance. The CHROM [@de2013robust] method uses a linear combination of the chrominance signals. It makes the assumption of a standardized skin color profile to white-balance the video frames. The Pulse Blood Vector (PBV) method [@de2014improved] relies on characteristic blood volume changes in different regions of the frequency spectrum to weight the color channels. Adapting the facial ROI can improve the performance of iPPG measurements as blood perfusion varies in intensity across the body [@Tulyakov2016]
Few approaches have made use of supervised learning for video-based physiological measurement. Formulating the problem is not trivial and performance has been modest [@osman2015supervised; @monkaresi2014machine]. Recent advances in deep neural video analysis offer opportunities for recovering accurate physiological measurements. Recently, Chen and McDuff [@chen2018deepphys] presented a supervised method using a convolutional attention network that provided state-of-the-art measurement performance and generalized across people. Our video magnification algorithm is based on a novel framework that allows recovery of pulse and respiratory waves using such a convolutional architecture.
Methods
=======
{width="\linewidth"}
Video Magnification Using Gradient Ascent
-----------------------------------------
Fig. \[fig:architechture\] shows the workflow of the proposed video magnification algorithm using gradient ascent. Similar to previous video magnification algorithms, it reads a series of video frames $C(t),~t=1,2,\cdots,T$, magnifies a specific subtle motion in them, and outputs frames of the same dimension $\widetilde{C}(t),~t=1,2,\cdots,T$.
The first step of our algorithm is computing the input motion representation $X_1(t)$ from the original video frames $C(t),~t=1,2,\cdots,T$. $X_1(t)$ represents any change happening between two consecutive frames $C(t)$ and $C(t+1)$. Common motion representations include frame difference and optical flow. Different motion representations can emphasize different aspects of motions. For example, the physio-logy-based motion representation called normalized frame difference [@chen2018deepphys] was proposed to capture skin absorption changes robustly under varying rigid motions. On the other hand, optical flow based on the brightness constancy constraint is good at representing object displacements, but largely ignores the light absorption changes of objects. As a general framework for video magnification, our algorithm supports any type of motion representation.
In realistic videos the motion representations are comprised of multiple motions from different sources. For example, unconstrained facial video recordings commonly contain not only respiration movements and pulse-induced skin color changes but also head rotations and facial expressions. As we are only interested in magnifying one of these motions at a time, a video magnification algorithm should have the ability to separate the target motion from the others in the motion representation. Previous methods have typically used frequency-domain characteristics of the target motion in separation, so they rely on precise prior knowledge about the motion frequency (e.g. the exact heart rate). Furthermore, if any other motion overlaps with the target motion in frequency, it will still be magnified and cause artifacts. To improve the specificity of magnification and reduce the dependence on prior knowledge, we propose to use a deep convolutional neural network (CNN) to model the relationship between the motion representation and the motion of interest. As shown in Fig. \[fig:architechture\], the CNN has the input motion representation $X_1(t)$ as its input, and the first-order derivative $y(t)$ of the target motion signal $p(t)$ as its output. For many motion types, there are available datasets with paired videos and ground truth motion signals (e.g., facial videos with pulse and respiration signals measured from medical devices). Therefore, the weights $\theta$ of the CNN can be determined by training it on one of these datasets. It has been shown in [@chen2018deepphys] that CNNs trained in this way have good generalization ability over different objects (human subjects), different backgrounds, and different lighting conditions.
As the CNN has established the relationship between the input motion representation $X_1(t)$ and the target motion signal $p(t)$, magnification of $p(t)$ in $X_1(t)$ can be achieved by amplifying the L2 norm of its first-order derivative $y(t)$ and then propagating the changes back to $X_1(t)$ using gradient ascent. The process can be expressed as $$\label{eq:I}
X_{n+1} = X_{n}+\gamma\nabla\|y(X_{n}|\theta)\|_2,~n=1,2,\cdots,N-1$$ in which $N$ is the total number of iterations and $\gamma$ is the step size. $\theta$ is the weights of the CNN, which are frozen during gradient ascent. $\nabla\|y(X_{n}|\theta)\|_2$ is the gradient of $\|y(t)\|_2$ with respect to $X_n(t)$, which is the direction to which $X_n(t)$ can be modified to specifically magnify the target motion rather than the other motions. Note that both $X_{n}$ and $y$ correspond to time point $t$ in (\[eq:I\]), but $t$ is omitted for conciseness.
The vanilla gradient ascent in (\[eq:I\]) is appropriate for magnifying a single motion representation $X_1(t)$ at time $t$. However, for video magnification, a series of motion representations $X_1(t) ,~t=1,2,\cdots,T$ need to be processed and magnified to the same level. Since the magnitude of the gradient is sensitive to the surface shape of the objective function (i.e. a point on a steep surface will have high magnitude whereas a point on the fairly flat surface will have low magnitude), it is not guaranteed that the accumulated gradient will be proportional to the original motion amplitude. Therefore, we apply L1 normalization to the gradient $$\label{eq:II}
X_{n+1} = X_{n}+\gamma\frac{\nabla\|y(X_{n}|\theta)\|_2}{\|\nabla\|y(X_{n}|\theta)\|_2\|_1}$$ so that only the gradient direction is kept and the gradient magnitude is controlled by the step size $\gamma$.
Another problem with (\[eq:I\]) is that motions in opposite directions contribute equivalently to the L2 norm of $y(t)$. As a result, the target motion might be amplified in terms of the absolute amplitude but 180-degrees out of phase. To address the problem, we correct the signs of the gradient to always match the signs of the input motion representation $$\label{eq:III}
X_{n+1} = X_{n}+\gamma\frac{\nabla\|y(X_{n}|\theta)\|_2\odot sgn(X_{n}\odot\nabla\|y(X_{n}|\theta)\|_2)}{\|\nabla\|y(X_{n}|\theta)\|_2\|_1}$$ in which $sgn(\cdot)$ is the sign function and $\odot$ is element-wise multiplication.
Summing up the changes of $X_n(t)$ in all the iterations, we get the final expression of the magnified motion representation: $$\label{eq:IV}
X_N = X_1+\sum_{n=1}^{N-1}{\gamma\frac{\nabla\|y(X_{n}|\theta)\|_2\odot sgn(X_{n}\odot\nabla\|y(X_{n}|\theta)\|_2)}{\|\nabla\|y(X_{n}|\theta)\|_2\|_1}}$$ There are only two hyper-parameters $\gamma$ and $N$, which can be tuned to change the magnification factor. Finally, the magnified motion representation can be combined with previous frames to iteratively generate the output video. The complete algorithm is summarized in Algorithm \[alg1\].
$C(t),~t=1,2,\cdots,T$ is a series of video frames, $\mathcal{M}$ is a motion representation estimator, $\theta$ is the pre-trained CNN weights for predicting a target motion signal $y$, $\gamma$ is the step size, and $N$ is the number of iterations Compute motion representation: $X_1(t)\gets \mathcal{M}(C(t),C(t+1))$ Compute gradient: $G_n(t)\gets \nabla\|y(X_{n}(t)|\theta,t)\|_2$ L1 normalization: $G_n(t)\gets G_n(t)/\|G_n(t)\|_1$ Sign correction: $G_n(t)\gets G_n(t)\odot sgn(G_n(t)\odot X_n(t))$ Gradient ascent: $X_{n+1}(t)\gets X_n(t)+\gamma G_n(t)$ $\widetilde{C}(1)=C(1)$ Reconstruct magnified frame $\widetilde{C}(t+1)\gets\mathcal{M}^{-1}(\widetilde{C}(t),X_N(t))$
![We used two exemplar tasks to illustrate the benefits of DeepMag. a) Color (Blood flow) magnification. b) Motion (respiration) magnification. These two tasks require different input motion representations and CNN architectures due to the nature of the motion signals.[]{data-label="fig:cnn"}](images/cnn.pdf){width="0.5\linewidth"}
Example I: Color Magnification
------------------------------
One example of applying our proposed algorithm is in the magnification of subtle skin color changes associated with the cardiac cycle. As blood flows through the skin it changes the light reflected from it. A good motion representation for these color changes is normalized frame difference [@chen2018deepphys], which is summarized below.
For modeling lighting, imagers and physiology, previous works used the Lambert-Beer law (LBL) [@lam2015robust; @Xu2014a] or Shafer’s dichromatic reflection model (DRM) [@Wang2016b]. We build our motion representation on top of the DRM as it provides a better framework for separating specular reflection and diffuse reflection. Assume the light source has a constant spectral composition but varying intensity. We can define the RGB values of the $k$-th skin pixel in an image sequence by a time-varying function: $$\label{eq:1}
\pmb{C}_k(t)=I(t) \cdot (\pmb{v}_s(t)+\pmb{v}_d(t))+\pmb{v}_n(t)$$ where $\pmb{C}_k(t)$ denotes a vector of the RGB values; $I(t)$ is the luminance intensity level, which changes with the light source as well as the distance between the light source, skin tissue and camera; $I(t)$ is modulated by two components in the DRM: specular reflection $\pmb{v}_s(t)$, mirror-like light reflection from the skin surface, and diffuse reflection $\pmb{v}_d(t)$, the absorption and scattering of light in skin-tissues; $\pmb{v}_n(t)$ denotes the quantization noise of the camera sensor. $I(t)$, $\pmb{v}_s(t)$ and $\pmb{v}_d(t)$ can all be decomposed into a stationary and a time-dependent part through a linear transformation [@Wang2016b]: $$\label{eq:2}
\pmb{v}_d(t) = \pmb{u}_d \cdot d_0 + \pmb{u}_p \cdot p(t)$$ where $\pmb{u}_d$ denotes the unit color vector of the skin-tissue; $d_0$ denotes the stationary reflection strength; $\pmb{u}_p$ denotes the relative pulsatile strengths caused by hemoglobin and melanin absorption; $p(t)$ denotes the BVP. $$\label{eq:3}
\pmb{v}_s(t) = \pmb{u}_s \cdot (s_0+s(t))$$ where $\pmb{u}_s$ denotes the unit color vector of the light source spectrum; $s_0$ and $s(t)$ denote the stationary and varying parts of specular reflections. $$\label{eq:4}
I(t) = I_0 \cdot (1+i(t))$$ where $I_0$ is the stationary part of the luminance intensity, and $I_0\cdot i(t)$ is the intensity variation observed by the camera. The stationary components from the specular and diffuse reflections can be combined into a single component representing the stationary skin reflection: $$\label{eq:5}
\pmb{u}_c \cdot c_0 = \pmb{u}_s \cdot s_0 + \pmb{u}_d \cdot d_0$$ where $\pmb{u}_c$ denotes the unit color vector of the skin reflection and $c_0$ denotes the reflection strength. Substituting (\[eq:2\]), (\[eq:3\]), (\[eq:4\]) and (\[eq:5\]) into (\[eq:1\]), produces: $$\label{eq:6}
\pmb{C}_k(t)=I_0\cdot (1+i(t)) \cdot
(\pmb{u}_c \cdot c_0+\pmb{u}_s \cdot s(t)+\pmb{u}_p \cdot p(t))+\pmb{v}_n(t)$$ As the time-varying components are much smaller (i.e., orders of magnitude) than the stationary components in (\[eq:6\]), we can neglect any product between varying terms and approximate $\pmb{c}_k(t)$ as: $$\label{eq:7}
\pmb{C}_k(t)\approx \pmb{u}_c \cdot I_0 \cdot c_0 \cdot (1+i(t)) +
\pmb{u}_s \cdot I_0 \cdot s(t)+\pmb{u}_p \cdot I_0 \cdot p(t)+\pmb{v}_n(t)$$ The first step in computing our motion representation is spatial averaging of pixels, which has been widely used for reducing the camera quantization error $\pmb{v}_n(t)$ in (\[eq:7\]). We implemented this by downsampling every frame to $L$ pixels by $L$ pixels using bicubic interpolation. Emperical evidence shows that bicubic interpolation preserves the color information more accurately than linear interpolation [@mcduff2018super]. Selecting $L$ is a trade-off between suppressing camera noise and retaining spatial resolution ([@wang2015exploiting] found that $L = 36$ was a good choice for face videos.) The downsampled pixel values will still obey the DRM model only without the camera quantization error: $$\label{eq:8}
\pmb{C}_l(t)\approx \pmb{u}_c \cdot I_0 \cdot c_0 + \pmb{u}_c \cdot I_0 \cdot c_0 \cdot i(t) +
\pmb{u}_s \cdot I_0 \cdot s(t)+\pmb{u}_p \cdot I_0 \cdot p(t)$$ where $l=1,\cdots,L^2$ is the new pixel index in every frame.
Then we need to reduce the dependency of $\pmb{C}_l(t)$ on the stationary skin reflection color $\pmb{u}_c \cdot I_0 \cdot c_0$, resulting from the light source and subject’s skin tone. In (\[eq:8\]), $\pmb{u}_c \cdot I_0 \cdot c_0$ appears twice. It is difficult to eliminate the second term as it interacts with the unknown $i(t)$. However, the first time-invariant term, which is usually dominant, can be removed by taking the first order derivative of both sides of (\[eq:8\]) with respect to time: $$\label{eq:9}
\pmb{C}'_l(t)\approx \pmb{u}_c \cdot I_0 \cdot c_0 \cdot i'(t) +
\pmb{u}_s \cdot I_0 \cdot s'(t)+\pmb{u}_p \cdot I_0 \cdot p'(t)$$
One problem with this frame difference representation is that the stationary luminance intensity level $I_0$ is spatially heterogeneous due to different distances to the light source and uneven skin contours. The spatial distribution of $I_0$ has nothing to do with physiology, but is different in every video recording setup. Thus, $\pmb{C}'_l(t)$ was normalized by dividing it by the temporal mean of $\pmb{C}_l(t)$ to remove $I_0$: $$\label{eq:10}
\frac{\pmb{C}'_l(t)}{\overline{\pmb{C}_l(t)}}\approx \pmb{1} \cdot i'(t) + diag^{-1}(\pmb{u}_c)\pmb{u}_s \cdot \frac{s'(t)}{c_0} +
diag^{-1}(\pmb{u}_c)\pmb{u}_p \cdot \frac{p'(t)}{c_0}$$ where $\pmb{1}=[1~1~1]^T$. In (\[eq:10\]), $\overline{\pmb{C}_l(t)}$ needs to be computed pixel-by-pixel over a short time window to minimize occlusion problems and prevent the propagation of errors. We found it was feasible to compute it over two consecutive frames so that (\[eq:10\]) can be expressed discretely as: $$\label{eq:11}
\pmb{X}_1(l,t) = \frac{\pmb{C}'_l(t)}{\overline{\pmb{C}_l(t)}}\sim\frac{\pmb{C}_l(t+1)-\pmb{C}_l(t)}{\pmb{C}_l(t+1)+\pmb{C}_l(t)}$$ which is the normalized frame difference we used as motion representation.
The CNN we used for extracting pulse signals from the motion representation is shown in Fig. \[fig:cnn\] (a). The pooling layers are 2x2 average pooling, and the convolution layers have a stride of one. All the layers use ReLU as the activation function. Note that bounded activation function such as tanh and sigmoid are not suitable for this task, as they will limit the extent to which the motion representation can be magnified in the gradient ascent.
After gradient ascent, the input motion representation $\pmb{X}_1(l,t)$ was magnified as $\pmb{X}_N(l,t)$, from which we could reconstruct the magnified video. The first step of reconstruction is to denoise the output motion representation by filtering the accumulated gradient: $$\widetilde{\pmb{X}_N}(l,t) = \pmb{X}_1(l,t) + \mathcal{F}(\pmb{X}_N(l,t) - \pmb{X}_1(l,t))$$ in which $\mathcal{F}$ is a zero-phase band-pass filter. Note that unlike previous motion magnification methods the function of the filter here is not to select the target motion but to remove low and high frequency noise, so the filter bands do not need to precisely match the motion frequency in the video and can be chosen conservatively. Specifically, a 6th-order Butterworth filter with cut-off frequencies of 0.7 and 2.5 Hz was used to generally cover the normal heart rate range (42 to 150 beats per minute). Then we applied the inverse operation of (\[eq:11\]) to reconstruct the downsampled version of the frames $\widetilde{\pmb{C}_l(t)}$: $$\widetilde{\pmb{C}_l}(t+1) = \frac{1+\widetilde{\pmb{X}_N}(l,t)}{1-\widetilde{\pmb{X}_N}(l,t)} \cdot\widetilde{\pmb{C}_l}(t), ~\widetilde{\pmb{C}_l}(1)=\pmb{C}_l(1)$$ Finally, $\pmb{C}_l(t)$ was upsampled back to the original video resolution: $$\widetilde{\pmb{C}_k}(t) = \pmb{C}_k(t) - \mathcal{U}(\pmb{C}_l(t)) + \mathcal{U}(\widetilde{\pmb{C}_l}(t))$$ in which $\mathcal{U}$ is an image upsampling operator.
Example II: Motion Magnification
--------------------------------
Our second example is amplifying subtle motions on the human body induced by respiration. We used phase variations in a complex steerable pyramid to represent the local motions in a video. The complex steerable pyramid [@Simoncelli1992; @Portilla2000] is a filter bank that breaks each frame of the video $C(t)$ into complex-valued sub-bands corresponding to different scales and orientations. The basis functions of this transformation are scaled and oriented Gabor-like wavelets with both cosine- and sine-phase components. Each pair of cosine- and sine-like filters can be used to separate the amplitude of local wavelets from their phase. Specifically, each scale $r$ and orientation $\theta$ is a complex image that can be expressed in terms of amplitude $A$ and phase $\phi$ as: $$A(r,\theta,t)e^{i\phi(r,\theta,t)}$$ We take the first-order temporal derivative of the local phases $\phi$ computed in this equation as our input motion representation: $$\label{eq:12}
X_1(r,\theta,t)=\phi(r,\theta,t+1)-\phi(r,\theta,t)$$ For small motions, these phase variations are approximately proportional to displacements of image structures along the corresponding orientation and scale [@gautama2002phase]. To lower computational cost, we computed a pyramid with octave bandwidth and four orientations ($\theta=0^\circ,45^\circ,90^\circ,135^\circ$). Using half-octave or quarter-octave bandwidth and more orientations would enable our algorithm to amplify more motion details, but would require significantly greater computational recourses. In theory, $X_1(r,\theta,t)$ contains $r=1,2,\cdots,R$ scales of representations in different spatial resolutions, and extracting the target respiration motion from them would need $R$ different CNNs to fit different input dimensions. However, we found that $X_1(r,\theta,t)$ and the amplified $X_N(r,\theta,t)$ on different scales were approximately proportional to $0.5^r$, so it is possible to only process one scale $r=r_0$ and interpolate the other scales with it.
The CNN we used for extracting respiration signals from the motion representation is shown in Fig. \[fig:cnn\] (b). The neural network is deeper than the one used for pulse magnification, because the input motion representation for respiration has a higher dimension. The pooling layers and convolution layers are of the same type as in Fig. \[fig:cnn\] (a). As we met the dying ReLU problem (ReLU neurons were stuck in the negative side and always output 0) in our experiments, the activation functions of all the layers were replaced with scaled exponential linear units (SELU) [@klambauer2017self].
After gradient ascent, the input motion representation $X_1(r_0,\theta,t)$ was magnified as $X_N(r_0,\theta,t)$, from which we could reconstruct the magnified video. Unlike in Example I, the phase variations were reconstructed by reversing (\[eq:12\]) before denoising: $$\widetilde{\phi}(r_0,\theta,t+1) = X_N(r_0,\theta,t)+\widetilde{\phi}(r_0,\theta,t),~
\widetilde{\phi}(r_0,\theta,1)=\phi(r_0,\theta,1)$$ Then the reconstructed phase was denoised by band-pass filtering and $2\pi$ phase clipping: $$\widetilde{\phi}(r_0,\theta,t) = \phi(r_0,\theta,t)
+\mathcal{F}(\widetilde{\phi}(r_0,\theta,t))\cdot\frac{sgn(2\pi-|\phi(r_0,\theta,t)|)+1}{2}$$ The filter $\mathcal{F}$ is a 6th-order zero-phase Butterworth filter with cut-off frequencies of 0.16 and 0.5 Hz for generally covering the normal breathing rate range (10 to 30 beats per minute). The magnified phase of the other scales can be interpolated by exponentially scaling the filtered term: $$\label{eq:13}
\widetilde{\phi}(r_0,\theta,t) = \phi(r_0,\theta,t)
+\mathcal{F}(\widetilde{\phi}(r_0,\theta,t))\cdot\frac{sgn(2\pi-|\phi(r_0,\theta,t)|)+1}{2}\cdot(\frac{1}{2})^{r-r_0}$$ Finally, the magnified video frame $\widetilde{C}(t)$ can be reconstructed from all the scales of the complex steerable pyramid with their phase updated as (\[eq:13\]).
![Exemplary frames from the four tasks of our video dataset. Note the different backgrounds and head rotation speeds.[]{data-label="fig:frame_example"}](images/frame_example.pdf){width="0.75\linewidth"}
Data
====
We used the dataset collected by Estepp et al. [@estepp2014recovering] for testing our approach. Videos were recorded with a Basler Scout scA640-120gc GigE-standard, color camera, capturing 8-bit, 658x492 pixel images, 120 fps. The camera was equipped with 16 mm fixed focal length lens. Twenty-five participants (17 males) were recruited to participate for the study. Nine individuals were wearing glasses, eight had facial hair, and four were wearing makeup on their face and/or neck. The participants exhibited the following estimated Fitzpatrick Sun-Reactivity Skin Types [@fitzpatrick1988validity]: I-1, II-13, III-10, IV-2, V-0. Gold-standard physiological signals were measured using a BioSemi ActiveTwo research-grade biopotential acquisition unit.
We used videos of participants during a set of four, five-minutes tasks for our analysis. Two of the tasks (A and D) were performed in front of a patterned background and two (B and C) were performed in front of a black background. The four tasks were designed to capture different levels of head rotation about the vertical axis (yaw). Examples of frames from the tasks can be seen in Figs. \[fig:frame\_example\]. **Task A:** Participants stayed still allowing for small natural motions. **Task B:** Participants performed a 120-degree sweep centered about the camera at a speed of 10 degrees/sec. **Task C:** Similar to Task B but with a speed of 30 degrees/sec. **Task D:** Participants were asked to reorient their head position once per second to a randomly chosen targets positioned in 20-degree increments over a 120-degree arc. Thus simulating random head motion.
Evaluation
==========
We compare the color magnification results to Eulerian video magnification [@wu2012eulerian] and video acceleration magnification [@Zhang2017], and compare the motion magnification results to phase-based Eulerian video magnification [@wadhwa2013phase] and video acceleration magnification (EVM and phase-based EVM perform poorly for motion magnification and color magnification respectively). In each case we perform qualitative evaluations similar to that presented in prior work. In addition, we perform a quantitative evaluation by assessing the image quality of the resulting videos. Prior work has generally not considered quantitative evaluations.
For obtaining our own results, the CNN model was either trained and tested on different time periods of the same videos (participant-dependent) or trained and tested on videos of different human participants (participant-independent), both using a 20% holdout rate for testing. The qualitative and quantitative results we show in the following sections are always from video excerpts in the test set. To achieve a fair comparison, all the compared methods used the same filter bands: \[0.7 Hz, 2.5 Hz\] for pulse color magnification, and \[0.16 Hz, 0.5 Hz\] for respiration motion magnification. Since VAM uses difference of Gaussian (DoG) filters defined by a single pass-band frequency, we adopted the center frequencies of the physiology frequency bands ($\sqrt{0.7\times 2.5}=1.3~Hz$ for pulse, and $\sqrt{0.16\times 0.5}=0.28~Hz$ for respiration) as its filtering parameters. In the color magnification baselines, video frames were decomposed into multiple scales using a Gaussian pyramid with the intensity changes in the fourth level amplified (following the source code released by [@wu2012eulerian]). All the motion magnification baselines used complex steerable pyramids with octave bandwidth and four orientations. The magnification factors of all the methods were tuned to be visually the same on task A without head motion interferences.
![Scan line comparisons of color magnification methods for a Task D video: a) original video, b) Eulerian video magnification, c) video acceleration magnification, d) Our method. The yellow line shows the source of the scan line in the frames. The section of video shown was 15 seconds in duration. Our method produces clearer magnification of the color change due to blood flow and significantly fewer artifacts.[]{data-label="fig:scanline_hr"}](images/scanline_hr.pdf){width="0.72\linewidth"}
![Original and magnified traces of a pixel (the yellow dot) in three color channels of a Task B video (a) red channel (b) green channel (c) blue channel. Magnified traces using different step sizes $\gamma$ are shown in different colors. The notches in the traces correspond to when the participant rotated her head to the far left/right and the pixel was no longer on the skin. Our method amplified the subtle color changes of the pixel only when it was on the skin, and kept the relative magnitudes of the pulse in three color channels with the green channel one being the strongest.[]{data-label="fig:factor_hr"}](images/factor_hr.pdf){width="0.65\linewidth"}
Color Magnification
-------------------
We apply our method to the task of magnifying the photoplethysmogram. In this task the target variable for training the CNN was the gold standard contact PPG signal. The input motion representation was 36 pixels $\times$ 36 pixels $\times$ 3 color channels. In terms of the hyper-parameters of gradient ascent, the number of iterations $N$ was chosen to be 20, and the step size $\gamma$ was chosen to be $6\times 10^{-5}$. We found these choices provided a moderate magnification level, equivalent to the magnification using EVM. Different choices of these hyper-parameters will be discussed in the following sections.
Fig. \[fig:scanline\_hr\] shows a qualitative comparison between our method and the baseline methods. The human participant in the video reoriented his head once per second to a random direction. In the horizontal scan line of the input video, only the head rotation is visible and the subtle color changes of the skin corresponding to pulse cannot be seen with the unaided eye. In the results of the baseline methods, strong motion artifacts are introduced. This is because the complex head motion is not distinguishable from the pulse signal in the frequency domain, so it is amplified along with the pulse. Since the pulse-induced color changes are several orders of magnitude weaker than the head motion, they are completely buried by the motion artifacts in the amplified video. The VAM scan line (Fig. \[fig:scanline\_hr\] (c)) shows slightly fewer artifacts than the EVM scan line (Fig. \[fig:scanline\_hr\] (b)) as the head rotation was occasionally semi-linear. On the other hand, our algorithm uses a deep neural network to separate the pulse signal from the head motion, and uses gradient ascent to specifically amplify it. Consequently, its scan line (Fig. \[fig:scanline\_hr\] (d)) preserves the morphology of the head rotation while revealing the periodic color changes clearly on the skin.
To show the magnification effects on different colors and different object surfaces, we drew the original and magnified traces of a pixel in three color channels of a video in Fig. \[fig:factor\_hr\]. The human participant in the video rotated her head left and right, so the selected pixel was on her forehead in half of time and was on the black background in the other half of time (corresponding to the notches in the traces). First, the pulse-induced color changes were only magnified when the pixel was on the skin surface, which proved the good spatial specificity of our algorithm. Second, the magnified pulse signal has much higher amplitude in the green channel than in the other channels. This is consistent with previous findings that the amplitude of the human pulse is approximately 0.33:0.77:0.53 in RGB channels under a halogen lamp [@de2013robust], and verifies that our algorithm faithfully kept the original physiological property in magnification. Third, we changed the chosen step size $\gamma$ to its multiples ($0.5\gamma$, $2\gamma$ and $4\gamma$) with the number of iterations $N$ unaltered, and visualized the resulting pixel traces also in Fig. \[fig:factor\_hr\]. There is a clear trend that longer step sizes lead to higher amplitudes of the magnified pulse.
[rcccc|cccc|cccc|cccc]{}\
Task & A & B & C & D & A & B & C & D & A & B & C & D & A & B & C & D\
EVM [@wu2012eulerian] & 36.5 & 35.1 & 24.8 & 20.3 & .975 & .957 & .853 & .779 & - & - & - & - & - & - & - & -\
Phase-EVM [@wadhwa2013phase] & - & - & - & - & - & - & - & - & 31.1 & 25.9 & 24.6 & 23.5 & .907 & .775 & .726 & .780\
VAM [@Zhang2017] & 36.6 & 36.4 & 26.7 & 22.5 & .976 & .969 & .892 & .809 & 30.6 & 26.8 & 24.6 & 23.2 & .900 & .800 & .720 & .770\
DeepMag - P. Dep. & 38.2 & **42.8** & **42.8** & **38.5** & **.981** & **.987** & **.987** & **.981** & 33.3 & **41.5** & 41.4 & **34.1** & **.940** & **.980** & **.979** & **.952**\
DeepMag - P. Ind. & **38.3** & 42.7 & 42.6 & **38.5** & **.981** & **.987** & **.987** & **.981** & **33.4** & **41.5** & **41.4** & 34.0 & **.940** & .979 & **.979** & .951\
To perform a quantitative evaluation of video quality we used two metrics: peak signal-to-noise ratio (PSNR) and structural similarity (SSIM). In both cases we calculated the metrics on every frame of the tested videos, and took their averages across all participants within each task. The reference frame in each case was the corresponding frame from the original, unmagnified video. Table \[tab:video\_quality\] shows a comparison of the video quality metrics for the baselines and our method. Although the magnified blood flow or respiration will naturally cause the metrics to be lower, we found that artifacts in the generated videos had a much more significant impact on their values than the magnified physiology. Thus, lower PSNR and SSIM values indicate more artifacts and lower quality. According to the table, our methods achieve both higher PSNR and SSIM than the baseline methods, which verify the ability of our methods to magnify subtle color changes with motion artifact suppressed. On task A containing limited head motions, the metrics of the baseline methods are very close to those of our method. However, as the head rotation becomes faster and random on more difficult tasks, the video quality of the baseline outputs dramatically decreases. This is because their algorithms amplify any motion lying in the filter band and does so indiscriminately. The magnification thus leads to significant artifact when large head motions are present. On the other hand using our method, the video quality is maintained at almost the same level on different tasks. Both PSNR and SSIM are only slightly lower on Task A and Task D, because the patterned background is more vulnerable to artifacts than the black one. The difference between the participant-dependent results and the participant-independent results is also very small, suggesting that our algorithm has good generalization ability and can be successfully applied to new videos containing different human participants without additional tuning.
![Scan line comparisons of motion magnification methods for a Task B video: a) original video, b) phase-based Eulerian video magnification, c) video acceleration magnification, d) Our method. The yellow line shows the source of the scan line in the frames. The section of video shown was 15 seconds in duration. Our method produces comparable magnification of the respiration motion and significantly fewer artifacts and blurring.[]{data-label="fig:scanline_br"}](images/scanline_br.pdf){width="0.72\linewidth"}
![Original and magnified traces of a pixel (the red dot) in the phase representation $\phi(r_0,\theta,t)$ of a Task C video along four orientations (a) $\theta=0^{\circ}$ (b) $\theta=45^{\circ}$ (c) $\theta=90^{\circ}$ (d) $\theta=135^{\circ}$. Magnified traces using different step sizes $\gamma$ are shown in different colors. The pixel exhibits a respiration movement mainly in the vertical direction, so its magnified phase traces have the highest amplitude along the $\theta=90^{\circ}$ orientation.[]{data-label="fig:factor_br"}](images/factor_br.pdf){width="0.65\linewidth"}
Motion Magnification
--------------------
We apply our method to the task of magnifying respiration motions. In this task the target variable for training the CNN was the gold standard respiration signal measured via the chest strap. Given the subtle nature of the motions we found that a higher dimension input motion representation was needed than for the PPG magnification. As shown in Fig. \[fig:cnn\], the motion representation was in 123 pixels $\times$ 123 pixels $\times$ 4 orientations. The gradient ascent hyper-parameters $N$ and $\gamma$ were chosen to be 20 and $3.6\times 10^{-3}$ to produce moderate magnification effects.
Fig. \[fig:scanline\_br\] shows a qualitative comparison between our method and the baseline methods. The human participant in the video rotated his head at a speed of 10 degrees/sec. A vertical scanline on his shoulder was drawn along with time to show the respiration movement. In the input video, the respiration movement is very subtle. Both our method and the baseline methods greatly increased its magnitude (Fig. \[fig:scanline\_br\] (b) (c) (d)). However, the baseline methods cannot clearly distinguish the phase variations caused by respiration and by head rotation, so it also amplified the head rotation and blurred the participant’s face. Our method is based on a better motion discriminator learned via the CNN so that the head motions are not amplified.
To show the intermediate phase variations and different magnification effects along different orientations, we drew the original and magnified traces of a pixel in the phase representation $\phi(r_0,\theta,t)$ (Fig. \[fig:factor\_br\]). Since the selected pixel is on the shoulder of the human participant, the respiration movement is mainly in the vertical direction. As a result, the amplified phase variations corresponding to breathing have the highest amplitude along $\theta=90^{\circ}$ (Fig. \[fig:factor\_br\] (c)) and the lowest amplitude along $\theta=0^{\circ}$ (Fig. \[fig:factor\_br\] (a)). We also changed the chosen step size $\gamma$ to its multiples ($0.5\gamma$, $2\gamma$ and $4\gamma$) with the number of iterations $N$ unaltered, and visualized the resulting phase traces in Fig. \[fig:factor\_br\]. The figure suggests that the magnification level always increases along with the step size.
The same quantitative metrics as those for color magnification were computed and shown in Table \[tab:video\_quality\]. They also generally follow the same pattern as in the color magnification analysis: The video quality of the baseline methods is impacted by the level of head motions, while our method is considerably more robust. There is no significant difference between our participant-dependent results and participant-independent results.
------------- ------------- ---------- ----------- ----------- ------------- ---------- ----------- -----------
Step size $0.5\gamma$ $\gamma$ $2\gamma$ $4\gamma$ $0.5\gamma$ $\gamma$ $2\gamma$ $4\gamma$
Pulse 43.2 42.6 41.6 39.9 0.987 0.987 0.986 0.986
Respiration 42.0 41.4 40.6 39.6 0.982 0.979 0.974 0.965
------------- ------------- ---------- ----------- ----------- ------------- ---------- ----------- -----------
: Video quality measured via Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity (SSIM) for Task C videos magnified to different levels.[]{data-label="tab:mag_factor"}
![Learning curves: (a) The change of the CNN loss with different numbers of iterations $N$ and different step sizes $\gamma$. (b) The change of the CNN loss with different products of $N$ and $\gamma$.[]{data-label="fig:learningcurve"}](images/learningcurve.pdf){width="0.8\linewidth"}
Magnification Factors
---------------------
The magnification factor of our algorithm is controlled by two hyper-parameters, the number of iterations $N$ and the step size $\gamma$. In Fig. \[fig:factor\_hr\] and Fig. \[fig:factor\_br\], we chose the same $N$ and tuned $\gamma$ to be different multiples. The resulting magnification levels were always higher when $\gamma$ was longer. However, there is a trade-off in the selection of $\gamma$, as a higher magnification factor also introduces more artifacts. Table \[tab:mag\_factor\] shows the average video quality metrics PSNR and SSIM for our output videos on an exemplary task (Task C) with different choices of $\gamma$. For both the pulse and respiration magnification tasks, the video quality decreases to different extents with the increase of $\gamma$. Given that artifacts considerably reduce the PSNR and SSIM metrics (as shown in Table \[tab:video\_quality\]), the fact that the values do not change dramatically with $\gamma$ shows that few artifacts are introduced with increasing magnification.
To quantitatively analyze the effects of $N$ and $\gamma$ on the magnification factor, we drew exemplary learning curves for one of our videos in Fig. \[fig:learningcurve\] (a) with different choices of parameters. The curves show the changes of our CNN loss, the L2 norm of the differential motion signal, which is a good estimate of the target motion magnitude. According to the learning curves, both $N$ and $\gamma$ positively correlate with the motion magnitude, and the relationship between $N$ and the motion magnitude is semi-linear. However, a longer step size with fewer iterations is not equivalent to a shorter step with more iterations. In Fig. \[fig:learningcurve\] (b), we show how the loss changes along with the product of $N$ and $\gamma$, which suggests that relatively small step sizes and more iterations can increase the magnification factor more efficiently.
![(a) Time series and histograms of the L1 norms of the input motion representation $X_{1}$ for a 30-second video. (b) Time series and histograms of the L1 norms of the motion gradient $\nabla\|y(X_{1}|\theta)\|_2$ for the same video.[]{data-label="fig:comp_l1"}](images/comp_l1.pdf){width="0.8\linewidth"}
![Pixel-wise correlation coefficients between the input and magnified motion representations in the respiration magnification task, with the sign correction mechanism (b) and without the sign correction mechanism (c).[]{data-label="fig:flip"}](images/flip.pdf){width="0.8\linewidth"}
Gradient Ascent Mechanisms
--------------------------
Compared with traditional gradient ascent, we added two new mechanisms to adapt the approach to the task of video magnification: L1 normalization and sign correction. Here we show experimental results to support the necessity of these mechanisms.
The goal of applying L1 normalization is to make sure every frame in a video is magnified to the same level. To achieve this goal, the gradient $\nabla\|y(X_{n}|\theta)\|_2$ in (\[eq:I\]) needs to be approximately proportional to the motion representation $X_{n}$. However, it was not the case without L1 normalization. Fig. \[fig:comp\_l1\] shows the time series and histograms of the L1 norms of $X_{1}$ and $\nabla\|y(X_{1}|\theta)\|_2$ for a 30-second video. It is obvious that the distribution of the motion representation is Gaussian while the distribution of the gradient is highly skewed. To correct the distribution of the gradient to match the motion representation, it needs to be L1 normalized.
In Fig. \[fig:flip\], we show the pixel-wise correlation coefficients between the input and the magnified motion representations in the respiration magnification task, with and without the sign correction mechanism. When there is no sign correction, the correlation coefficients have both positive and negative values (Fig. \[fig:flip\] (b)). As introduced in Section 3.1, the negative values appear because the target motion could be amplified with its direction reversed. In the example in Fig. \[fig:flip\] (b), most of the negative values happen on the background, which are negligible as the background has nearly no motion to amplify, but some of them are on the human body, which will cause the output video to be blurry on magnification. After sign correction is applied, all the correlation coefficients become positive (Fig. \[fig:flip\] (c)).
Conclusions
===========
Revealing subtle signals in our everyday world is important for helping us understand the processes that cause them. We present a novel single deep neural framework for video magnification that is robust to large rigid motions. Our method leverages a CNN architecture that enables magnification of a specific source signal even if it overlaps with other motion sources in the frequency domain. We present several methodological innovations in order to achieve our results, including adding L1 normalization and sign correction to the gradient ascent method.
Pulse and respiration magnification are good exemplar tasks for video magnification as these physiological phenomena cause both subtle color and motion variations that are invisible to the unaided eye. Our qualitative evaluation illustrates how the PPG color changes and respiration motions can be clearly magnified. Comparisons with baseline methods show that our proposed architecture dramatically reduces artifacts when there are other rotational head motions present in the videos.
In a systematic quantitative evaluation our method improves the PSNR and SSIM metrics across tasks with different levels of rigid motion. By magnifying a specific source signal we are able to maintain the quality of the magnified videos to a greater extent.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This paper introduces a formal metalanguage called the lambda-q calculus for the specification of quantum programming languages. This metalanguage is an extension of the lambda calculus, which provides a formal setting for the specification of classical programming languages. As an intermediary step, we introduce a formal metalanguage called the lambda-p calculus for the specification of programming languages that allow true random number generation. We demonstrate how selected randomized algorithms can be programmed directly in the lambda-p calculus. We also demonstrate how satisfiability can be efficiently solved in the lambda-q calculus.'
author:
- |
Philip Maymin[^1]\
Harvard University
date: 'December 31, 1996 '
title: Extending the Lambda Calculus to Express Randomized and Quantumized Algorithms
---
Introduction
============
This paper presents three formal language calculi, in increasing order of generality. The first one, the $\lambda $-calculus, is an old calculus for expressing functions. It is the basis of the semantics for many functional programming languages, including Scheme [@R4RS]. The second one, the $%
\lambda ^{p}$-calculus, is a new calculus introduced here for expressing randomized functions. Randomized functions, instead of having a unique output for each input, return a distribution of results from which we sample once. The third one, the $\lambda ^{q}$-calculus, is a new calculus introduced here for expressing quantumized functions. Quantumized functions also return a distribution of results, called a *superposition*, from which we sample once, but $\lambda ^{q}$-terms have signs, and identical terms with opposite signs are removed before sampling from the result. Thus, superpositions can appear to shrink in size whereas distributions cannot. The $\lambda ^{p}$-calculus is an extension of the $\lambda $-calculus. The $%
\lambda ^{q}$-calculus is an extension of the $\lambda ^{p}$-calculus. The $%
\lambda ^{q}$-calculus is the most general but it is best presented in reference to the intermediary $\lambda ^{p}$-calculus.
Although much research has been done on the hardware of quantum computation (c.f. [@deutsch; @85], [@deutsch; @89], [@simon]), none has focused on formalizing the software. Quantum Turing machines [@deutsch; @85] have been introduced but there has been no quantum analogue to Church’s $\lambda $-calculus. The $\lambda $-calculus has served as the basis for most programming languages since it was introduced by Alonzo Church [@church] in 1936. It and other calculi make the implicit assumption that a term may be innocuously observed at any point. Such an assumption is hard to separate from a system of rewriting rules because to rewrite a term, you must have read it. However, as has been pointed out by Deutsch [@deutsch; @92], any physical system is a computer. We may prepare it in some state, let it evolve according to its dynamics, and observe it periodically. Here, the notion of observation is crucial. One of the goals of these calculi is to make observation explicit in the formalism itself.
The intension of the $\lambda ^{p}$- and the $\lambda ^{q}$-calculi is to formalize computation on the level of *potentia* discussed by Heisenberg [@heisenberg]. Heisenberg’s quantum reality is a two-world model. One world is the world of *potentia*, events that haven’t happened but could. The other world is the world of actual events that have occured and been observed. A goal of these calculi is to allow easy expression of algorithms that exist and operate in the world of *potentia* yet are, at conclusion, observed.
To this end, collections (distributions and superpositions) should be thought of with the following intuition. A collection is a bunch of terms that do not communicate with each other. When the collection is observed, at most one term in each collection will be the result of the observation. In the $\lambda ^{q}$-calculus, the terms in a collection have signs, but still do not communicate with each other. The observation process somehow removes oppositely-signed terms. The key point is that in neither calculus can one write a term that can determine if it is part of a collection, how big the collection is, or even if its argument is part of a collection. Collections can be thought of as specifications of parallel terms whose execution does not depend on the execution of other terms in the same collection.
The Lambda Calculus
===================
This section is a review of the $\lambda $-calculus and a reference for later calculi.
The $\lambda $-calculus is a calculus of functions. Any computable single-argument function can be expressed in the $\lambda $-calculus. Any computable multiple-argument function can be expressed in terms of computable single-argument functions. The $\lambda $-calculus is useful for encoding functions of arbitrary arity that return at most one output for each input. In particular, the $\lambda $-calculus can be used to express any (computable) *algorithm*. The definition of algorithm is usually taken to be Turing-computable.
Syntax
------
The following grammar specifies the syntax of the $\lambda $-calculus. $$\begin{tabular}{|ll|}
\hline
$
\begin{array}{ll}
x & \in \text{\emph{Variable}} \\
M & \in \text{\emph{LambdaTerm}} \\
w & \in \text{\emph{Wff}}
\end{array}
$ & $
\begin{array}{l}
\text{Variables} \\
\text{Terms of the }\lambda \text{-calculus} \\
\text{Well-formed formulas of the }\lambda \text{-calculus}
\end{array}
$ \\
& \\
$
\begin{array}{lll}
M & ::= & x \\
& \,\,\,| & M_{1}M_{2} \\
& \,\,\,| & \lambda x.M \\
& & \\
w & ::= & M_{1}=M_{2}
\end{array}
$ & $
\begin{array}{l}
\text{variable} \\
\text{application} \\
\text{abstraction} \\
\\
\text{well-formed formula}
\end{array}
$ \\ \hline
\end{tabular}
\label{lambda syntax}$$
To be strict, the subscripts above should be removed (e.g., the rule for well-formed formulas should read $w::=M=M$) because $M_{1}$ and $M_{2}$ are not defined. However, we will maintain this incorrect notation to emphasize that the terms need not be identical.
With this abuse of notation, we can easily read the preceding definition as: a $\lambda $-term is a variable, or an application of two terms, or the abstraction of a term by a variable. A well-formed formula of the $\lambda $-calculus is a $\lambda $-term followed by the equality sign followed by a second $\lambda $-term.
We also adopt some syntactic conventions. Most importantly, parentheses group subexpressions. Application is taken to be left associative so that the term $MNP$ is correctly parenthesized as $\left( MN\right) P$ and not as $M\left( NP\right) .$ The scope of an abstraction extends as far to the right as possible, for example up to a closing parenthesis, so that the term $\lambda x.xx$ is correctly parenthesized as $\left( \lambda x.xx\right) $ and not as $\left( \lambda x.x\right) x.$
Substitution
------------
We will want to substitute arbitrary $\lambda $-terms for variables. We define the substitution operator, notated$~M\left[ N/x\right] $ and read “$%
M $ with all free occurences of $x$ replaced by $N$.” The definition of the free and bound variables of a term are standard. The set of free variables of a term $M$ is written $FV\left( M\right) $. There are six rules of substitution, which we write for reference. $$\begin{array}{ll}
1.\;x\left[ N/x\right] \equiv N & \\
2.\;y\left[ N/x\right] \equiv y & \text{for variables }y\not{\equiv}x \\
3.\;\left( PQ\right) \left[ N/x\right] \equiv \left( P\left[ N/x\right]
\right) \left( Q\left[ N/x\right] \right) & \\
4.\;\left( \lambda x.P\right) \left[ N/x\right] \equiv \lambda x.P & \\
5.\;\left( \lambda y.P\right) \left[ N/x\right] \equiv \lambda y.\left(
P\left[ N/x\right] \right) & \text{if }y\not{\equiv}x\text{ and }y\notin
FV\left( N\right) \\
6.\;\left( \lambda y.P\right) \left[ N/x\right] \equiv \lambda z.\left(
P\left[ z/y\right] \left[ N/x\right] \right) & \text{if }y\not{\equiv}x\text{
and }y\in FV\left( N\right) \text{ } \\
& \text{and }z\notin FV(P)\bigcup FV\left( N\right)
\end{array}
\label{substitution}$$
This definition will be extended in both subsequent calculi.
Reduction {#notions of reduction}
---------
The concept of *reduction* seeks to formalize rewriting rules. Given a relation $R$ between terms, we may define the one-step reduction relation, notated$~\rightarrow _{R},$ that is the contextual closure of $R.$ We may also define the reflexive, transitive closure of the one-step reduction relation, which we call $R$-reduction and notate$~\twoheadrightarrow _{R},$ and the symmetric closure of $R$-reduction, called $R$-interconvertibility and notated$~=_{R}.$
The essential notion of reduction for the $\lambda $-calculus is called $%
\beta $-reduction. It is based on the $\beta $-relation, which is the formalization of function invocation. $$\beta \triangleq \left\{ \left( \left( \lambda x.M\right) N,M\left[
N/x\right] \right) \,\,\,|\,\,\,M,N\in LambdaTerm,\,x\in Variable\right\}
\label{beta}$$
There is also the $\alpha $-relation that holds of terms that are identical up to a consistent renaming of variables. $$\alpha \triangleq \left\{ \left( \lambda x.M,\lambda y.M\left[ y/x\right]
\right) \,\,\,|\,\,\,M\in LambdaTerm,\,y\notin FV\left( M\right) \right\}$$ We will use this only sparingly.
Evaluation Semantics
--------------------
By imposing an evaluation order on the reduction system, we are providing meaning to the $\lambda $-terms. The evaluation order of a reduction system is sometimes called an operational semantics or an evaluation semantics for the calculus. The evaluation relation is typically denoted $\rightsquigarrow
.$
We use call-by-value evaluation semantics. A *value* is the result produced by the evaluation semantics. Call-by-value semantics means that the body of an abstraction is not reduced but arguments are evaluated before being passed into abstractions.
There are two rules for the call-by-value evaluation semantics of the $%
\lambda $-calculus. $$\begin{aligned}
&&\frac {}{v\rightsquigarrow v}\text{(Refl)\qquad \qquad (for }v\text{ a
value)} \\
&&\frac{M\rightsquigarrow \lambda x.P\quad N\rightsquigarrow N^{\prime
}\quad P\left[ N^{\prime }/x\right] \rightsquigarrow v}{MN\rightsquigarrow v}%
\text{(Eval)}\end{aligned}$$
Reference Terms
---------------
The following $\lambda $-terms are standard and are provided as reference for later examples.
Numbers are represented as Church numerals. $$\begin{aligned}
\underline{0} &\equiv &\lambda x.\lambda y.y \\
\underline{n} &\equiv &\lambda x.\lambda y.x^{n}y\end{aligned}$$
where the notation $x^{n}y$ means $n$ right-associative applications of $x$ onto $y.$ It is abbreviatory for the term $
\begin{array}{l}
\underbrace{x(x(\cdots (x}y))) \\
\,n\text{ times}
\end{array}
.$ When necessary, we can extend Church numerals to represent both positive and negative numbers. For the remainder of the terms, we will not provide definitions. The predecessor of Church numerals is written $\underline{\text{%
P}}.$ The successor is written $\underline{\text{S}}.$
The conditional is written $\underline{\text{IF}}.$ If its first argument is truth, written $\underline{\text{T}},$ then it returns its second argument. If its first argument is falsity, written $\underline{\text{F}},$ then it returns its third argument. A typical predicate is $\underline{\text{0?}}$ which returns $\underline{\text{T}}$ if its argument is the Church numeral $%
\underline{\text{0}}$ and $\underline{\text{F}}$ if it is some other Church numeral.
The fixed-point combinator is written $\underline{\text{Y}}.$ The primitive recursive function-building term is written $\underline{\text{PRIM-REC}}$ and it works as follows. If the value of a function $f$ at input $n$ can be expressed in terms of $n-1$ and $f\left( n-1\right) ,$ then that function $f$ is primitive recursive, and it can be generated by providing $\underline{%
\text{PRIM-REC}}$ with the function that takes the inputs $n-1$ and $f\left(
n-1\right) $ to produce $f\left( n\right) $ and with the value of $f$ at input $0.$ For example, the predecessor function for Church numerals can be represented as $\underline{\text{P}}\equiv \underline{\text{PRIM-REC}}%
\,\left( \lambda x.\lambda y.x\right) \,\underline{\text{0}}.$
The Lambda-P Calculus
=====================
The $\lambda ^{p}$-calculus is an extension of the $\lambda $-calculus that permits the expression of *randomized* algorithms. In contrast with a computable algorithm which returns at most one output for each input, a randomized algorithm returns a *distribution* of answers from which we sample. There are several advantages to randomized algorithms.
1. Randomized algorithms can provide truly random number generators instead of relying on pseudo-random number generators that work only because the underlying pattern is difficult to determine.
2. Because they can appear to generate random numbers arbitrarily, randomized algorithms can model random processes.
3. Given a problem of finding a suitable solution from a set of possibilities, a randomized algorithm can exhibit the effect of choosing random elements and testing them. Such algorithms can sometimes have an *expected* running time which is considerably shorter than the running time of the computable algorithm that tries every possibility until it finds a solution.
Syntax {#section:lambda-p syntax}
------
The following grammar describes the $\lambda
^{p}$-calculus. $$\begin{tabular}{|ll|}
\hline
$
\begin{array}{ll}
x & \in \text{\emph{Variable}} \\
M & \in \text{\emph{LambdaPTerm}} \\
w & \in \text{\emph{WffP}}
\end{array}
$ & $
\begin{array}{l}
\text{Variables} \\
\text{Terms of the }\lambda ^{p}\text{-calculus} \\
\text{Well-formed formulas of the }\lambda ^{p}\text{-calculus}
\end{array}
$ \\
& \\
$
\begin{array}{lll}
M & ::= & x \\
& \,\,\,| & M_{1}M_{2} \\
& \,\,\,| & \lambda x.M \\
& \,\,\,| & M_{1},M_{2} \\
& & \\
w & ::= & M_{1}=M_{2}
\end{array}
$ & $
\begin{array}{l}
\text{variable} \\
\text{application} \\
\text{abstraction} \\
\text{collection} \\
\\
\text{well-formed formula}
\end{array}
$ \\ \hline
\end{tabular}
\newline
\label{lambda-p syntax}$$
Note that this grammar differs from the $\lambda $-calculus only in the addition of the fourth rule for terms. Therefore, all $\lambda $-terms can be viewed as $\lambda ^{p}$-terms.
A $\lambda ^{p}$-term is a variable, or an application of two terms, or the abstraction of a term by a variable, or a collection of two terms. It follows that a term may be a collection of a term and another collection, so that a term may actually have many nested collections.
We adhere to the same parenthesization and precedence rules as the $\lambda $-calculus with the following addition: collection is of lowest precedence and the comma is right associative. This means that the expression $\lambda
x.x,z,y$ is correctly parenthesized as $\left( \lambda x.x\right) ,(z,y)$.
We introduce abbreviatory notation for collections. Let us write $\left[
M_{i}^{i\in S}\right] $ for the collection of terms $M_{i}$ for all $i$ in the finite, ordered set $S$ of natural numbers. We will write $a..b$ for the ordered set $\left( a,a+1,\ldots ,b\right) .$ In particular, $\left[
M_{i}^{i\in 1..n}\right] $ represents $M_{1},M_{2},\ldots ,M_{n}$ and $%
\left[ M_{i}^{i\in n..1}\right] $ represents $M_{n},M_{n-1},\ldots ,M_{1}$. More generally, let us allow multiple iterators in arbitrary contexts. Then, for instance, $$\left[ \lambda x.M_{i}^{i\in 1..n}\right] \equiv \lambda x.M_{1},\lambda
x.M_{2},\ldots ,\lambda x.M_{n}$$ and $$\left[ M_{i}^{i\in 1..m}N_{j}^{j\in 1..n}\right] \equiv
\begin{array}{c}
M_{1}N_{1},M_{1}N_{2},\ldots ,M_{1}N_{n}, \\
M_{2}N_{1},M_{2}N_{2},\ldots ,M_{2}N_{n}, \\
\vdots \\
M_{m}N_{1},M_{m}N_{2},\ldots ,M_{m}N_{n}
\end{array}
.$$
Note that $\left[ \lambda x.M_{i}^{i\in 1..n}\right] $ and $\lambda x.\left[
M_{i}^{i\in 1..n}\right] $ are not the same term. The former is a collection of abstractions while the latter is an abstraction with a collection in its body. Finally, we allow this notation to hold of non-collection terms as well by identifying $\left[ M_{i}^{i\in 1..1}\right] $ with $M_{1}$ even if $%
M_{1}$ is not a collection. To avoid confusion, it is important to understand that although this “collection” notation can be used for non-collections, we do not extend the definition of the word *collection.* A *collection* is still the syntactic structure defined in grammar (\[lambda-p syntax\]).
With these additions, every term can be written in this bracket form. In particular, we can write a collection as $\left[ \left[ M_{i}^{i\in
S_{i}}\right] _{j}^{j\in S}\right] ,$ or a collection of collections. Unfortunately, collections can be written in a variety of ways with this notation. The term $M,N,P$ can be written as $\left[ M_{i}^{i\in
1..3}\right] $ if $M_{1}\equiv M$ and $M_{2}\equiv N$ and $M_{3}\equiv P;$ as $\left[ M_{i}^{i\in 1..2}\right] $ if $M_{1}\equiv M$ and $M_{2}\equiv
N,P;$ or as $\left[ M_{i}^{i\in 1..1}\right] $ if $M_{1}\equiv M,N,P.$ However, it cannot be written as $\left[ M_{i}^{i\in 1..4}\right] $ for any identification of the $M_{i}.$ This observation inspires the following definition.
\[dfn: cardinality\]The *cardinality* of a term $M,$ notated$%
~\left| M\right| ,$ is that number $k$ for which $\left[ M_{i}^{i\in
1..k}\right] \equiv M$ for some identification of the $M_{i}$ but $\left[
M_{i}^{i\in 1..\left( k+1\right) }\right] \not{\equiv}M$ for any identification of the $M_{i}$.
Note that the cardinality of a term is always strictly positive.
Alternative Syntax {#alternative syntax}
------------------
We present an alternative syntax for the $\lambda
^{p}$-calculus that is provably equivalent to the one given above under certain assumptions. We will call the temporary calculus whose syntax we define below the $\lambda ^{p^{\prime }}$-calculus to distinguish it from the one we will ultimately adopt.
The following grammar describes the syntax of the $\lambda ^{p^{\prime }}$-calculus. $$\begin{tabular}{|ll|}
\hline
$
\begin{array}{ll}
x & \in \text{\emph{Variable}} \\
M & \in \text{\emph{LambdaTerm}} \\
C & \in \text{\emph{LambdaP}}^{\prime }\text{\emph{Term}} \\
w & \in \text{\emph{WffP}}^{\prime }
\end{array}
$ & $
\begin{array}{l}
\text{Variables} \\
\text{Terms of the }\lambda \text{-calculus} \\
\text{Terms of the }\lambda ^{p^{\prime }}\text{-calculus} \\
\text{Well-formed formulas of the }\lambda ^{p^{\prime }}\text{-calculus}
\end{array}
$ \\
& \\
$
\begin{array}{lll}
M & ::= & x \\
& \,\,\,| & M_{1}M_{2} \\
& \,\,\,| & \lambda x.M \\
& & \\
C & ::= & M \\
& \,\,\,| & C_{1},C_{2} \\
& \,\,\,| & C_{1}C_{2} \\
& & \\
w & ::= & C_{1}=C_{2}
\end{array}
$ & $
\begin{array}{l}
\text{variable} \\
\text{application} \\
\text{abstraction} \\
\\
\text{term} \\
\text{construction} \\
\text{collection application} \\
\\
\text{well-formed formula}
\end{array}
$ \\ \hline
\end{tabular}
\newline
\label{lambda-p' syntax}$$
The syntax of the $\lambda ^{p}$-calculus, grammar (\[lambda-p’ syntax\]), allows the same terms and well-formed formulas as the syntax of the $\lambda
^{p^{\prime }}$-calculus, grammar (\[lambda-p syntax\]), if we identify abstractions of collections with the appropriate collection of abstractions, and applications of collections with collections of applications.
If we have, in the $\lambda ^{p}$-calculus, that $$\left[ \lambda x.M_{i}^{i\in 1..n}\right] \equiv \lambda x.\left[
M_{i}^{i\in 1..n}\right] \label{abstraction identity}$$ and $$\left[ M_{i}^{i\in 1..m}N_{j}^{j\in 1..n}\right] \equiv \left[ M_{i}^{i\in
1..m}\right] \left[ N_{j}^{j\in 1..n}\right] \noindent
\label{application identity}$$ then $w$ is a well-formed formula in the $\lambda ^{p}$-calculus if and only if it is a well-formed formula in the $\lambda ^{p^{\prime }}$-calculus.
It is sufficient to show that an arbitrary $\lambda ^{p^{\prime }}$-term is a $\lambda ^{p}$-term and vice versa.
First we show that an arbitrary $\lambda ^{p^{\prime }}$-term is a $\lambda
^{p}$-term by structural induction. From (\[lambda-p’ syntax\]), a $%
\lambda ^{p^{\prime }}$-term $C$ is either a $\lambda $-term, a construction, or a collection application. If $C$ is a $\lambda $-term, then it is a $\lambda ^{p}$-term. If it is a construction $C_{1},C_{2}$, then $C$ is a $\lambda ^{p}$-collection term $C_{1},C_{2}$ because $C_{1}$ and $C_{2}$ are $\lambda ^{p}$-terms by the induction hypothesis. Finally, if $C$ is a collection application $C_{1}C_{2}$, then, since $C_{1}$ and $C_{2}$ are $%
\lambda ^{p}$-terms by the induction hypothesis and $C_{1}C_{2}$ is a $%
\lambda ^{p}$-application term, $C$ is a $\lambda ^{p}$-term.
Now we show that an arbitrary $\lambda ^{p}$-term is a $\lambda ^{p^{\prime
}}$-term by structural induction. If $M$ is a $\lambda ^{p}$-term, it is either a variable, an application, a collection, or an abstraction. If $M$ is a variable, then it is a $\lambda $-term and therefore a $\lambda
^{p^{\prime }}$-term. If $M$ is an application $PQ,$ then by the induction hypothesis $P$ and $Q$ are $\lambda ^{p^{\prime }}$-terms, so that $PQ$ is a $\lambda ^{p^{\prime }}$-collection application and $M$ is a $\lambda
^{p^{\prime }}$-term. If $M$ is a collection, then by the induction hypothesis it is a collection of $\lambda ^{p}$-terms that are $\lambda
^{p^{\prime }}$-terms, so that $M$ is also a $\lambda ^{p^{\prime }}$-construction.
If $M$ is an abstraction $\lambda x.N,$ then by the induction hypothesis, $N$ is a $\lambda ^{p^{\prime }}$-term. Therefore, $N$ is either a $\lambda $-term, a construction, or a collection application. If $N$ is a $\lambda $-term, then $M$ is a $\lambda ^{p^{\prime }}$-term. If $N$ is a $\lambda
^{p^{\prime }}$-construction, then it is also a $\lambda ^{p}$-collection term by the first part of this proof. Therefore, $M$ is an abstraction over a collection, and by assumption (\[abstraction identity\]) is identical to a collection over abstractions. By the induction hypothesis, each of the abstractions in the collection is a $\lambda ^{p^{\prime }}$-term, so the collection itself is a $\lambda ^{p^{\prime }}$-construction. Therefore, $M$ is a $\lambda ^{p^{\prime }}$-term. Finally, if $N$ is a collection application, then by assumption (\[application identity\]) it is identical to a collection of applications. Therefore, $N$ is a $\lambda ^{p}$-collection term. By the same reasoning as in the previous case, it follows that $M$ is a $\lambda ^{p^{\prime }}$-term.
This completes the proof. The $\lambda ^{p}$-calculus seems more expressive than the $\lambda
^{p^{\prime }}$-calculus because it allows terms to be collections of other terms. We have seen that with the two assumptions (\[abstraction identity\]) and (\[application identity\]), the two calculi are equally expressive. Without these assumptions, some abstractions can be expressed in the $%
\lambda ^{p}$-calculus that cannot be expressed in the $\lambda ^{p^{\prime
}}$-calculus. Are these assumptions justifiable?
The first assumption (\[abstraction identity\]) states that the abstraction of a collection is syntactically identical to the collection of the abstractions. For example, the term $\lambda x.\left( x,xx\right) $ is claimed to be identical to the term $\lambda x.x,\lambda x.xx.$ Given our intuitive understanding of what these terms represent, it is indisputable that these two terms are equal in a semantic sense. Applying each term to arbitrary inputs ought to yield statistically indistinguishable results. However, the question is: should we identify these terms on a syntactic level? Certainly the two $\lambda $-terms $\lambda x.x$ and $\lambda
x.\left( \lambda y.y\right) x$ are semantically equivalent, but we do not identify them on a syntactic level.
The other assumption (\[application identity\]) states that the application of two collections is the collection of all possible applications of terms in the two collections. For example, the term $\left(
M,N\right) (P,Q)$ is claimed to be identical to the term $MP,MQ,NP,NQ.$ Again, these two would, given our intuition, be equal in the statistical sense, but should they represent the same syntactic structure? Even if the two terms represent the same thing in the real world, that is, if they share the same denotation, it does not follow that they should be syntactically identical. $\beta $-reduction preserves denotation, but we do not say that a term and what it reduces to are syntactically identical.
On the other hand, we do want to identify some terms that are written differently. For example, the order of terms in a collection ought not distinguish terms. The terms $M,N$ and $N,M$ should be identified, given our intuitive understanding of what these terms mean. Identifying unordered terms is not uncommon and is done in other calculi [@abadi-cardelli]. How do we decide whether to identify certain pairs of terms or to define a notion of reduction for them?
We want to identify terms when the differences do not affect computation and result from the limiting nature of writing. Identifying collections with different orders is a workaround for the sequentiality and specificity of the comma operator. Together with the grammar, such an identification clarifies the terms of discourse. However, this reasoning does not apply to the assumptions (\[abstraction identity\]) and (\[application identity\]) because these assumptions are trying to identify terms that bear only a semantic relationship to each other.
Much as we refrain from identifying a term with its $\beta $-reduced form, we do not want to identify an application (abstraction) of collections with a collection of applications (abstractions). We may choose instead to capture this relationship in the form of a relation and associated reductions. Such is the approach we will adopt here.
Of the two grammars, we choose the $\lambda ^{p}$-calculus because it is the more general one. We will define the $\gamma $-relation to hold of an application of collections and a collection of applications. However, we will neither identify nor provide a relation for the analogous abstraction relationship of the pair of terms identified in assumption (\[abstraction identity\]), because such a step would be redudant. By the observation function we will define in §\[lambda-p observation\], observing an abstraction of collections is tantamount to observing a collection of abstractions, so no new power or expressibility would be gained.
Syntactic Identities
--------------------
We define substitution of terms in the $\lambda ^{p}$-calculus as an extension of substitution of terms in the $\lambda $-calculus. In addition to the six rules of the $\lambda $-calculus, we introduce one for collections. $$\left( P,Q\right) \left[ N/x\right] \equiv \left( P\left[ N/x\right]
,Q\left[ N/x\right] \right) \label{substitution-p}$$
We identify terms that are collections but with a possibly different ordering. We also identify nested collections with the top-level collection. The motivation for this is the conception that a collection is an unordered set of terms. Therefore we will not draw a distinction between a set of terms and a set of a set of terms.
We adopt the following axiomatic judgement rules. $$\begin{aligned}
&&\dfrac {}{M,N\equiv N,M}\text{(ClnOrd)} \\
&&\dfrac {}{\left( M,N\right) ,P\equiv M,(N,P)}\text{(ClnNest)}\end{aligned}$$
With these axioms, ordering and nesting become innocuous. As an example here is the proof that $A,(B,C),D\equiv A,C,B,D.$ For clarity, we parenthesize fully and underline the affected term in each step. $$\begin{array}{llll}
\underline{A,((B,C),D)} & \equiv & ((\underline{B,C}),D),A & \text{(ClnOrd)}
\\
& \equiv & (\underline{(C,B),D}),A & \text{(ClnOrd)} \\
& \equiv & \underline{(C,(B,D)),A} & \text{(ClnNest)} \\
& \equiv & A,(C,(B,D)) & \text{(ClnOrd)}
\end{array}$$
We now show that ordering and parenthesization are irrelevant in general.
\[theorem:order/paren invariance\]If the $n$ ordered sets $S_{i},$ $%
1\leq i\leq n,$ are distinct and $\Pi $ is a permutation of the ordered set $%
1..n$, then $\left[ \left[ M_{i}^{i\in S_{i}}\right] _{j}^{j\in 1..n}\right]
\equiv \left[ M_{i}^{i\in S_{1}S_{2}\cdots S_{n}}\right] $ and $\left[
M_{i}^{i\in 1..n}\right] \equiv \left[ M_{i}^{i\in \Pi }\right] ,$ where juxtaposition of ordered sets denotes extension (e.g., $\left( 1..3\right)
\left( 5..7\right) =\left( 1,2,3,5,6,7\right) $).
We prove this theorem by induction on $n.$
There are two base cases. When $n=1,$ the claim holds trivially. When $n=2,$ the claim follows from the $\left( \text{ClnOrd}\right) $ axiom.
For the inductive case, we consider $M\equiv \left[ M_{i}^{i\in 1..\left(
n+1\right) }\right] \equiv M_{1},\left[ M_{i}^{i\in 2..\left( n+1\right)
}\right] $ and assume the claim holds for all collections of $n$ or fewer terms, and that $n\geq 2.$
To show parenthesization invariance, we write $M\equiv P,Q$ where $P\equiv
\left[ P_{i}^{i\in 1..n}\right] $ and $Q$ are collections. It will be sufficient to show that $M\equiv \left[ P_{i}^{i\in 1..(n-1)}\right] ,\left(
P_{n},Q\right) $. By the induction hypothesis, we may parenthesize $P$ arbitrarily. We choose to parenthesize $P$ left-associatively as $$\left( \left( \left( P_{1},P_{2}\right) ,P_{3}\,\cdots \,P_{n-2}\right)
,P_{n-1}\right) ,P_{n}.$$ Then, by the $\left( \text{ClnNest}\right) $ axiom, $M\equiv \left( \left(
\left( P_{1},P_{2}\right) ,P_{3}\,\cdots \,P_{n-2}\right) ,P_{n-1}\right)
,\left( P_{n},Q\right) ,$which is identical to $\left[ P_{i}^{i\in
1..(n-1)}\right] ,\left( P_{n},Q\right) $ by the reordering allowed by the induction hypothesis. This completes the proof of parenthesization invariance.
To show reordering invariance, note that the permutation $\Pi $ of $%
1..\left( n+1\right) $ either has $1$ as its first element or it does not. If it does, then by the induction hypothesis, $\left[ M_{i}^{i\in 2..\left(
n+1\right) }\right] $ can be reordered in an arbitrary manner, so that $%
M\equiv \left[ M_{i}^{i\in \Pi }\right] .$ If it does not, then the first element of $\Pi $ is an integer $k$ between $2$ and $n-1.$ By the induction hypothesis, we can reorder $\left[ M_{i}^{i\in 2..\left( n+1\right) }\right]
$ as $M_{k},\left[ M_{i}^{i\in 2..\left( n+1\right) -\{k]}\right] ,$ where we write $2..\left( n+1\right) -k$ for the ordered set $\left( 2,3,\ldots
,k-1,k+1,\ldots ,n+1\right) .$ Then, underlying the affected term, we get $$\begin{array}{lll}
M & \equiv \underline{M_{1},M_{k},\left[ M_{i}^{i\in 2..\left( n+1\right)
-\{k]}\right] } & \text{by reordering} \\
& \equiv \left( \underline{M_{1},M_{k}}\right) ,\left[ M_{i}^{i\in 2..\left(
n+1\right) -\{k]}\right] & \text{by the }\left( \text{ClnNest}\right) \text{
axiom} \\
& \equiv \underline{\left( M_{k},M_{1}\right) ,\left[ M_{i}^{i\in 2..\left(
n+1\right) -\{k]}\right] } & \text{by the }\left( \text{ClnOrd}\right) \text{
axiom} \\
& \equiv M_{k},M_{1},\left[ M_{i}^{i\in 2..\left( n+1\right) -\{k]}\right] &
\text{by the }\left( \text{ClnNest}\right) \text{ axiom}
\end{array}$$
Then, by the induction hypothesis, $M_{1},\left[ M_{i}^{i\in
2..\left( n+1\right) -\{k]}\right] $ can be reordered arbitrarily so that $M$ can be reordered to fit the permutation $\Pi $. This completes the proof of reordering invariance and of the theorem. Aside, it no longer matters that we took the comma to be right associative since, with these rules, any arbitrary parenthesization of a collection does not change the syntactic structure.
Because of this theorem, we can alter the abbreviatory notation and allow arbitrary unordered sets in the exponent. This allows us to write, for instance, $\left[ M_{i}^{i\in 1..n-\{j\}}\right] \equiv M_{1},M_{2},\ldots
,M_{j-1},M_{j+1},\ldots ,M_{n}$ where $a..b$ is henceforth taken to be the unordered set $\left\{ a,a+1,\ldots ,b\right\} $ and the subtraction in the exponent represents set difference.
This also subtly alters the definition of *cardinality* (\[dfn: cardinality\]). Whereas before the cardinality of a term like $\left(
x,y\right) ,z$ was 2, because of this theorem, it is now 3. Because every $%
\lambda ^{p}$-term is finite, the cardinality is well-defined.
Reductions
----------
The relation of collection application is called the $\gamma $-relation. It holds of a term that is an application at least one of whose operator or operand is a collection, and the term that is the collection of all possible pairs of applications.
$$\gamma ^{p}\triangleq \left\{
\begin{array}{l}
\left( \left[ M_{i}^{i\in 1..m}\right] \left[ N_{j}^{j\in 1..n}\right]
,\left[ M_{i}^{i\in 1..m}N_{j}^{j\in 1..n}\right] \right) \\
\text{such that }M_{i},N_{j}\in LambdaPTerm,\,m>1\text{ or }n>1
\end{array}
\right\} \label{gamma-p}$$
The $\gamma $-relation is our solution to the concerns of §\[alternative syntax\] regarding claim (\[application identity\]). We will omit the superscript except to disambiguate from the $\gamma $-relation of the $\lambda ^{q}$-calculus.
We generate the reduction relations as described in §\[notions of reduction\] to get the relations of $\gamma $ -reduction in one step $%
\rightarrow _{\gamma },$ $\gamma $-reduction $\twoheadrightarrow _{\gamma },$ and $\gamma $-interconvertibility $=_{\gamma }.$
The $\gamma $-relation is Church-Rosser.
\[thm: gamma-p CR\]For $\lambda ^{p}$-terms $M,R,S,$ if $%
M\twoheadrightarrow _{\gamma }R$ and $M\twoheadrightarrow _{\gamma }S$ then there exists a $\lambda ^{p}$-term $T$ such that $R\twoheadrightarrow
_{\gamma }T$ and $S\twoheadrightarrow _{\gamma }T.$
This is shown in the standard way by proving the associated strip lemma.
As a result of this theorem, $\gamma $-normal forms, when they exist, are unique. We now show that all terms have $\gamma $-normal forms.
\[thm: gamma-p NF exists\]For every $\lambda ^{p}$-term $M$ there exists another $\lambda ^{p}$-term $N$ such that $M\twoheadrightarrow _{\gamma }N$ and $N$ has no $\gamma $-redexes.
The proof is by structural induction on $M$.
If $M\equiv x$ is a variable, there are no $\gamma $-redexes, so $N\equiv M$.
If $M\equiv \lambda x.P$ is an abstraction, then the only $\gamma $-redexes, if any, are in $P.$ By the induction hypothesis, there exists a term $%
P^{\prime }$ such that $P\twoheadrightarrow _{\gamma }P^{\prime }$ and $%
P^{\prime }$ has no $\gamma $-redexes. Then $M\twoheadrightarrow _{\gamma
}\lambda x.P^{\prime }\equiv N$ and $N$ has no $\gamma $-redexes either.
If $M\equiv PQ$ is an application, then there exist terms $P^{\prime
},Q^{\prime }$ such that $P\twoheadrightarrow _{\gamma }P^{\prime }$ and $%
Q\twoheadrightarrow _{\gamma }Q^{\prime }$ and neither $P^{\prime }$ nor $%
Q^{\prime }$ have $\gamma $-redexes. Let $\left[ P_{i}^{i\in 1..\left|
P^{\prime }\right| }\right] \equiv P^{\prime }$ and $\left[ Q_{i}^{i\in
1..\left| Q^{\prime }\right| }\right] \equiv Q^{\prime }.$ Then $%
M\twoheadrightarrow _{\gamma }P^{\prime }Q^{\prime }\rightarrow _{\gamma
}\left[ P_{i}^{i\in 1..\left| P^{\prime }\right| }Q_{j}^{j\in 1..\left|
Q^{\prime }\right| }\right] \equiv N$ where none of the $P_{i}$ or $Q_{j}$ are collections, by the definition of cardinality. Also, since neither $%
P^{\prime }$ nor $Q^{\prime }$ had $\gamma $-redexes, none of the $P_{i}$ or $Q_{j}$ have $\gamma $-redexes either. Therefore, $N$ does not have any $%
\gamma $-redexes.
If $M\equiv \left[ M_{i}^{i\in 1..\left| M\right| }\right] $ is a collection, then for each $M_{i}$ there exists a term $N_{i}$ such that $%
M_{i}\twoheadrightarrow _{\gamma }N_{i}$ and $N_{i}$ has no $\gamma $-redexes. Then $M\twoheadrightarrow _{\gamma }\left[ N_{i}^{i\in 1..\left|
M\right| }\right] \equiv N$ and since none of the $N_{i}$ have $\gamma $-redexes, neither does $N.$
This exhausts the cases and completes the proof. Therefore, all $\lambda ^{p}$-terms have normal forms and they are unique , so we may write $\gamma \left( M\right) $ for the $\gamma $-normal form of a $\lambda ^{p}$-term $M.$
We extend the $\beta $-relation to apply to collections. $$\beta ^{p}\triangleq \left\{
\begin{array}{l}
\left( \left( \lambda x.M\right) \left[ N_{i}^{i\in S}\right] ,\left[
M\left[ N_{i}^{i\in S}/x\right] \right] \right) \\
\text{such that }M\text{ and }\left[ N_{i}^{i\in S}\right] \in
LambdaPTerm,\,x\in Variable
\end{array}
\right\} \label{beta-p}$$ where $\left[ M\left[ N_{i}^{i\in S}/x\right] \right] $ is the collection of terms $M$ with $N_{i}$ substituted for free occurrences of $x$ in $M,$ for $%
i\in S.$
One-step $\beta $-reduction in the $\lambda ^{p}$-calculus $\rightarrow
_{\beta ^{p}}$ is different from that of the $\lambda $-calculus because the grammar is extended. (Note that we omit the superscript on $\beta $-reduction when there is no ambiguity about which calculus is under consideration.) Therefore, we need to prove that the $\beta $-relation is still Church-Rosser in the $\lambda ^{p}$-calculus. This is easy to do but not helpful because the appropriate notion of reduction in the $\lambda ^{p}$-calculus is not $\beta $-reduction, but $\beta $-reduction with $\gamma $-reduction to normal form after each $\beta $-reduction step.
We define the relation $\beta \gamma $ which is just like the $\beta $-relation except that the resultant term is in $\gamma $-normal form. $$\beta \gamma \triangleq \left\{ \left( M,\gamma \left( N\right) \right)
\,\,\,|\,\,\,\left( M,N\right) \in \beta ^{p}\right\} \label{beta-gamma-p}$$
The $\beta \gamma $-relation is Church-Rosser.
\[thm: beta-gamma-p CR\]For $\lambda ^{p}$-terms $M,R,S,$ if $%
M\twoheadrightarrow _{\beta }R$ and $M\twoheadrightarrow _{\beta }S$ then there exists a $\lambda ^{p}$-term $T$ such that $R\twoheadrightarrow
_{\beta }T$ and $S\twoheadrightarrow _{\beta }T.$
Again the proof follows the standard framework.
Evaluation Semantics
--------------------
We extend the call-by-value evaluation semantics of the $\lambda $-calculus.
There are three rules for the call-by-value evaluation semantics of the $%
\lambda ^{p}$-calculus. We modify the definition of a value $v$ to enforce that $v$ has no $\gamma $-redexes.
$$\begin{aligned}
&&\dfrac {}{v\rightsquigarrow v}\text{(Refl)\qquad \qquad (for }v\text{ a
value)} \\
&&\dfrac{\gamma \left( M\right) \rightsquigarrow \lambda x.P\quad \gamma
\left( N\right) \rightsquigarrow N^{\prime }\quad \gamma \left( P\left[
N^{\prime }/x\right] \right) \rightsquigarrow v}{MN\rightsquigarrow v}\text{%
(Eval)} \\
&&\dfrac{\gamma \left( M\right) \rightsquigarrow v_{1}\quad \gamma \left(
N\right) \rightsquigarrow v_{2}}{\left( M,N\right) \rightsquigarrow \left(
v_{1},v_{2}\right) }\text{(Coll)}\end{aligned}$$
Observation {#lambda-p observation}
-----------
We define an observation function $\Theta $ from $\lambda ^{p}$-terms to $\lambda $-terms. We employ the random number generator $RAND$, which samples one number from a given set of numbers. $$\begin{aligned}
\Theta \left( x\right) &=&x \\
\Theta \left( \lambda x.M\right) &=&\lambda x.\Theta \left( M\right) \\
\Theta \left( M_{1}M_{2}\right) &=&\Theta \left( M_{1}\right) \Theta \left(
M_{2}\right) \\
\Theta \left( M\equiv \left[ M_{i}^{i\in 1..\left| M\right| }\right] \right)
&=&M_{RAND(1..\left| M\right| )}\end{aligned}$$
The function $\Theta $ is total because every $\lambda ^{p}$-term is mapped to a $\lambda $-term. Note that for an arbitrary term $T$ we may write $\Theta \left( T\right) =T_{RAND(S)}$ for some possibly singleton set of natural numbers $S$ and some collection of terms $\left[ T_{i}^{i\in
S}\right] .$
\[defn: statistically indistinguishable\]We say that $\Theta \left(
M\right) =M_{RAND\left( S\right) }$ is *statistically indistinguishable* from $\Theta \left( N\right) =N_{RAND\left( S^{\prime }\right) }$, written $%
\Theta \left( M\right) \stackrel{d}{=}\Theta \left( N\right) ,$ if there exist positive integers $m$ and $n$ and a total, surjective mapping $\varphi
$ between $S$ and $S^{\prime }$ such that, for each $k\in S,$ $M_{k}\equiv
N_{\varphi \left( k\right) }$ and $$\frac{\left| \left[ M_{i}^{i\in \left\{ j\,\,\,|\,\,\,M_{j}\equiv
M_{k}\right\} }\right] \right| }{\left| S\right| }=\frac{\left| \left[
N_{i}^{i\in \left\{ \varphi \left( j\right) \,\,\,|\,\,\,M_{j}\equiv
M_{k}\right\} }\right] \right| }{\left| S^{\prime }\right| }$$ that is, if the proportion of terms in $M$ identical to $M_{k}$ is the same as the proportion of terms in $N$ identical to $N_{\varphi \left( k\right)
}, $ for all $k\in S.$ In particular, if the mapping $\varphi $ is an isomorphism, $\Theta \left( M\right) $ is said to be *statistically identical* to $\Theta \left( N\right) $, written $\Theta \left( M\right)
\stackrel{D}{=}\Theta \left( N\right) .$
Because the mapping $\varphi $ is total and surjective, statistical indistinguishability is a symmetric property. Note that statististical identity is a restatement of the theorem of parenthesization and ordering invariance (\[theorem:order/paren invariance\]). We now show that observing a $\lambda ^{p}$-term is statistically indistinguishable from observing its $\gamma $-normal form.
\[thm: gamma-p preserves stats. ind.\]If $M\twoheadrightarrow _{\gamma
}N $ then $\Theta \left( M\right) \stackrel{d}{=}\Theta \left( N\right) .$
The proof is by structural induction on $M$. The induction hypothesis is stronger than required and states that if $M\twoheadrightarrow _{\gamma }N$ then $\Theta \left( M\right) $ and $\Theta \left( N\right) $ are statistically identical.
If $M$ is a variable then $M\equiv N$ and $\Theta \left( M\right) =\Theta
\left( N\right) .$ In particular, $\Theta \left( M\right) \stackrel{D}{=}%
\Theta \left( N\right) .$
If $M\equiv \lambda x.P$ is an abstraction then $N\equiv \lambda x.P^{\prime
}$ where $P\twoheadrightarrow _{\gamma }P^{\prime }$. By the induction hypothesis, $\Theta \left( P\right) \stackrel{D}{=}\Theta \left( P^{\prime
}\right) ,$ and by definition $\Theta \left( M\right) =\lambda x.\Theta
\left( P\right) \stackrel{D}{=}\lambda x.\Theta \left( P^{\prime }\right)
=\Theta \left( \lambda x.P^{\prime }\right) =\Theta \left( N\right) $.
If $M\equiv PQ$ is an application, then either $N$ is an application or a collection. If $N$ is an application, then $N\equiv P^{\prime }Q^{\prime }$ and, since $P\twoheadrightarrow _{\gamma }P^{\prime }$ and $%
Q\twoheadrightarrow _{\gamma }Q^{\prime },$ we have by the induction hypothesis that $\Theta \left( P\right) \stackrel{D}{=}\Theta \left(
P^{\prime }\right) $ with isomorphism $\varphi _{P}$ and $\Theta \left(
Q\right) \stackrel{D}{=}\Theta \left( Q^{\prime }\right) $ with isomorphism $%
\varphi _{Q}$. We write $$\begin{aligned}
\Theta \left( P\right) &=&P_{RAND\left( S_{P}\right) } \\
\Theta \left( P^{\prime }\right) &=&P_{RAND\left( S_{P^{\prime }}\right)
}^{\prime } \\
\Theta \left( Q\right) &=&Q_{RAND\left( S_{Q}\right) } \\
\Theta \left( Q^{\prime }\right) &=&Q_{RAND\left( S_{Q^{\prime }}\right)
}^{\prime } \\
\Theta \left( M\right) &=&M_{RAND\left( S_{M}\right) } \\
\Theta \left( N\right) &=&N_{RAND\left( S_{N}\right) }\end{aligned}$$
and note that it is sufficient to exhibit an isomorphism between $%
\Theta \left( M\right) =\Theta \left( PQ\right) $ and $\Theta \left(
N\right) =\Theta \left( P^{\prime }Q^{\prime }\right) .$ Without loss of generality, let $S_{p}=S_{p^{\prime }}=1..p$ and $S_{q}=S_{q^{\prime }}=1..q$ so we can write, for $i\in 1..\left( pq\right) ,$$$\begin{aligned}
M_{i} &\equiv &P_{1+\left( i-1\right) \backslash q}Q_{1+\left( i-1\right)
\func{mod}q} \\
N_{i} &\equiv &P_{1+\left( i-1\right) \backslash q}^{\prime }Q_{1+\left(
i-1\right) \func{mod}q}^{\prime } \label{Ni}\end{aligned}$$ where $x\backslash y=\left\lfloor \frac{x}{y}\right\rfloor .$ Remember that by theorem (\[theorem:order/paren invariance\]), order and parenthesization does not matter. Given isomorphisms $\varphi _{P}$ and $%
\varphi _{Q},$ we need to find an isomorphism, $\varphi ,$ between the $%
M_{i} $ and $N_{i}.$ We rewrite $$\begin{aligned}
M_{i} &\equiv &P_{\varphi _{P}\left( 1+\left( i-1\right) \backslash q\right)
}^{\prime }Q_{1+\left( i-1\right) \func{mod}q} \\
&\equiv &P_{\varphi _{P}\left( 1+\left( i-1\right) \backslash q\right)
}^{\prime }Q_{\varphi _{Q}\left( 1+\left( i-1\right) \func{mod}q\right)
}^{\prime } \\
&\equiv &N_{1+\left( \varphi _{P}\left( 1+\left( i-1\right) \backslash
q\right) -1\right) q+\left( \varphi _{Q}\left( 1+\left( i-1\right) \func{mod}%
q\right) -1\right) }\end{aligned}$$
where the last identity follows from rewriting identity (\[Ni\]) as $$N_{1+\left( j-1\right) q+\left( k-1\right) }\equiv P_{j}Q_{k}$$
where $j\in 1..p,\,k\in 1..q.$ Thus, the isomorphism $\varphi
\left( i\right) =1+\left( \varphi _{P}\left( 1+\left( i-1\right) \backslash
q\right) -1\right) q+\left( \varphi _{Q}\left( 1+\left( i-1\right) \func{mod}%
q\right) -1\right) $ satisfies the definition of statistical identity (\[defn: statistically indistinguishable\]) so $\Theta \left( M\right)
\stackrel{D}{=}\Theta \left( N\right) .$
Finally, if $M\equiv \left[ M_{i}^{i\in 1..\left| M\right| }\right] $ is a collection, then $N\equiv \left[ N_{i}^{i\in 1..\left| M\right| }\right] $ must be a collection, too, where each $M_{i}\twoheadrightarrow _{\gamma
}N_{i}.$ By the induction hypothesis, $\Theta \left( M_{i}\right) \stackrel{D%
}{=}\Theta \left( N_{i}\right) $ for each $i\in 1..\left| M\right| .$ In particular, for each $M_{i}$ there is an $N_{j}$ such that $M_{i}\equiv
N_{j}.$ The isomorphism follows by identifying each $i$ with the appropriate $j.$
This exhausts the cases and completes the proof.
Observational Semantics
-----------------------
We provide another type of semantics for the $\lambda ^{p}$-calculus called its *observational semantics.* A formalism’s observational semantics expresses the computation as a whole: preparing the input, waiting for the evaluation, and observing the result. The observational semantics relation between $\lambda ^{p}$-terms and $\lambda $-terms is denoted$~\multimap $. It is given by a single rule for the $\lambda ^{p}$-calculus. $$\frac{M\rightsquigarrow v\quad \Theta \left( v\right) =N}{M\multimap N}\text{%
(ObsP)} \label{obs-p}$$
Examples
--------
A useful term of the $\lambda ^{p}$-calculus is a random number generator. We would like to define a term that takes as input a Church numeral and computes a collection of numerals from to . This can be represented by the following primitive recursive $\lambda ^{p}$-term. $$\underline{\text{R}}\equiv \underline{\text{PRIM-REC}}\,\left( \lambda
k.\lambda p.\left( k,p\right) \right) \,\underline{\text{0}}$$
Then for instance $\underline{\text{R}}\,\underline{\text{3}}%
=\left( 3,2,1,0\right) .$
The following term represents a random walk. Imagine a man that at each moment can either walk forward one step or backwards one step. If he starts at the point $0$, after $n$ steps, what is the distribution of his position? $$\underline{\text{W}}\equiv \underline{\text{PRIM-REC}}\,\left( \lambda
k.\lambda p.\left( \underline{\text{P}}p,\underline{\text{S}}p\right)
\right) \,\underline{\text{0}}$$
We assume we have extended Church numerals to negative numbers as well. This can be easily done by encoding it is a pair. We will show some of the highlights of the evaluation of $\underline{\text{W}}\,\underline{\text{3%
}}.$ Note that $\underline{\text{W}}\,\underline{\text{1}}=\left( \underline{%
-1},\underline{1}\right) .$$$\begin{array}{lll}
\underline{\text{W}}\,\underline{\text{3}} & = & \underline{\text{P}}\left(
\underline{\text{W}}\,\underline{\text{2}}\right) ,\underline{\text{S}}%
\left( \underline{\text{W}}\,\underline{\text{2}}\right) \\
& = & \underline{\text{P}}\left( \underline{\text{P}}\left( \underline{\text{%
W}}\,\underline{\text{1}}\right) ,\underline{\text{S}}\left( \underline{%
\text{W}}\,\underline{\text{1}}\right) \right) ,\underline{\text{S}}\left(
\underline{\text{P}}\left( \underline{\text{W}}\,\underline{\text{1}}\right)
,\underline{\text{S}}\left( \underline{\text{W}}\,\underline{\text{1}}%
\right) \right) \\
& = & \underline{\text{P}}\left( \underline{\text{P}}\left( \underline{\text{%
W}}\,\underline{\text{1}}\right) ,\underline{\text{S}}\left( \underline{%
\text{W}}\,\underline{\text{1}}\right) \right) ,\underline{\text{S}}\left(
\underline{\text{P}}\left( \underline{\text{W}}\,\underline{\text{1}}\right)
,\underline{\text{S}}\left( \underline{\text{W}}\,\underline{\text{1}}%
\right) \right) \\
& = & \underline{\text{P}}\left( \underline{\text{P}}\left( \underline{-1},%
\underline{1}\right) ,\underline{\text{S}}\left( \underline{-1},\underline{1}%
\right) \right) ,\underline{\text{S}}\left( \underline{\text{P}}\left(
\underline{-1},\underline{1}\right) ,\underline{\text{S}}\left( \underline{-1%
},\underline{1}\right) \right) \\
& = & \underline{\text{P}}\left( \left( \underline{-2},\underline{0}\right)
,\left( \underline{0},\underline{2}\right) \right) ,\underline{\text{S}}%
\left( \left( \underline{-2},\underline{0}\right) ,\left( \underline{0},%
\underline{2}\right) \right) \\
& = & \left( \left( \underline{-3},\underline{-1}\right) ,\left( \underline{%
-1},\underline{1}\right) \right) ,\left( \left( \underline{-1},\underline{1}%
\right) ,\left( \underline{1},\underline{3}\right) \right) \\
& \equiv & \left( \underline{-3},\underline{-1},\underline{-1},\underline{1},%
\underline{-1},\underline{1},\underline{1},\underline{3}\right)
\end{array}$$
Observing $\underline{\text{W}}\,\underline{\text{3}}$ yields $%
\underline{-1}$ with probability $\frac{3}{8},$ $\underline{1}$ with probability $\frac{3}{8},$ $\underline{-3}$ with probability $\frac{1}{8}$, and $\underline{3}$ with probability $\frac{1}{8}.$
The Lambda-Q Calculus
=====================
The $\lambda ^{q}$-calculus is an extension of the $\lambda ^{p}$-calculus that allows easy expression of *quantumized* algorithms. A quantumized algorithm differs from a randomized algorithm in allowing negative probabilities and in the way we sample from the resulting distribution.
Variables and abstractions in the $\lambda ^{q}$-calculus have *phase*. The phase is nothing more than a plus or minus sign, but since the result of a quantumized algorithm is a distribution of terms with phase, we call such a distribution by the special name *superposition*. The major difference between a superposition and a distribution is the observation procedure. Before randomly picking an element, a superposition is transformed into a distribution by the following two-step process. First, all terms in the superposition that are identical except with opposite phase are cancelled. They are both simply removed from the superposition. Second, the phases are stripped to produce a distribution. Then, an element is chosen from the distribution randomly, as in the $\lambda ^{p}$-calculus.
The words *phase* and *superposition* come from quantum physics. An electron is in a superposition if it can be in multiple possible states. Although the phases of the quantum states may be any angle from $0{{}^{\circ
}}$ to $360{{}^{\circ }}$, we only consider binary phases. Because we use solely binary phases, we will use the words *sign* and *phase* interchangeably in the sequel.
A major disadvantage of the $\lambda ^{p}$-calculus is that it is impossible to compress a collection. Every reduction step at best keeps the collection the same size. Quantumized algorithms expressed in the $\lambda ^{q}$-calculus, on the other hand, can do this as easily as randomized algorithms can generate random numbers. That is, $\lambda ^{q}$-terms can contain subterms with opposite signs which will be removed during the observation process.
Syntax {#syntax-1}
------
The following grammar describes the $\lambda ^{q}$-calculus. $$\begin{tabular}{|ll|}
\hline
$
\begin{array}{ll}
S & \in \text{\emph{Sign}} \\
x & \in \text{\emph{Variable}} \\
M & \in \text{\emph{LambdaQTerm}} \\
w & \in \text{\emph{WffQ}}
\end{array}
$ & $
\begin{array}{l}
\text{Sign, or phase} \\
\text{Variables} \\
\text{Terms of the }\lambda ^{q}\text{-calculus} \\
\text{Well-formed formulas of the }\lambda ^{q}\text{-calculus}
\end{array}
$ \\
& \\
$
\begin{array}{lll}
S & ::= & + \\
& \,\,\,| & - \\
& & \\
M & ::= & Sx \\
& \,\,\,| & M_{1}M_{2} \\
& \,\,\,| & S\lambda x.M \\
& \,\,\,| & M_{1},M_{2} \\
& & \\
w & ::= & M_{1}=M_{2}
\end{array}
$ & $
\begin{array}{l}
\text{positive} \\
\text{negative} \\
\\
\text{signed variable} \\
\text{application} \\
\text{signed abstraction} \\
\text{collection} \\
\\
\text{well-formed formula}
\end{array}
$ \\ \hline
\end{tabular}
\newline
\label{lambda-q syntax}$$
Terms of the $\lambda ^{q}$-calculus differ from terms of the $\lambda ^{p}$-calculus only in that variables and abstractions are *signed*, that is, they are preceded by either a plus (+) or a minus (-) sign. Just as $%
\lambda $-terms could be read as $\lambda ^{p}$-terms, we would like $%
\lambda ^{p}$-terms to be readable as $\lambda ^{q}$-terms. However, $%
\lambda ^{p}$-terms are unsigned and cannot be recognized by this grammar.
Therefore, as is traditionally done with integers, we will omit the positive sign. An unsigned term in the $\lambda ^{q}$-calculus is abbreviatory for the same term with a positive sign. With this convention, $\lambda ^{p}$-terms can be seen as $\lambda ^{q}$-terms all of whose signs are positive. Also, so as not to confuse a negative sign with subtraction, we will write it with a logical negation sign ($\lnot $). With these two conventions, the $\lambda ^{q}$-term $+\lambda x.+x-\!x$ is written simply $\lambda x.x\lnot
x.$
Instead of these conventions, we could just as well have rewritten the grammar of signs so that the positive sign was spelled with the empty string (traditionally denoted by the Greek letter $\epsilon $) and the negative signs was spelled with the logical negation sign. We would have gotten the alternative grammar below. $$\begin{array}{lll}
S^{\prime } & ::= & \epsilon \\
& \,\,\,| & \lnot
\end{array}
\label{alternative grammar for signs}$$
However, this would have suggested an asymmetry between positive and negative signs and allowed the interpretation that negatively signed terms are a “type” of positively signed terms. On the contrary, we want to emphasize that there are two distinct kinds of terms, positive and negative, and neither is better than the other. There is no good reason why $\lambda
^{p}$-terms should be translated into positively signed $\lambda ^{q}$-terms and not negatively signed ones. This arbitrariness is captured better as a convention than a definition.
Finally, we adhere to the same parenthesization and precedence rules as the $%
\lambda ^{p}$-calculus. In particular, we continue the use of the abbreviatory notation $\left[ M_{i}^{i\in S}\right] $ for collections of terms, although we will not recast the parenthesization and ordering invariance theorem (\[theorem:order/paren invariance\]) for terms of the $%
\lambda ^{q}$-calculus. The modifications to the proof are mild.
Syntactic Identities
--------------------
We want to give a name to the relationship between two terms that differ only in sign.
\[defn: opposite\]A $\lambda ^{q}$-term $M$ is the *opposite* of a $%
\lambda ^{q}$-term $N$, written $M\equiv \overline{N},$ if either $$M\equiv S_{1}x\text{ and }N\equiv S_{2}x$$
where $S_{1}$ and $S_{2}$ are different signs, or $$M\equiv S_{1}\lambda x.M^{\prime }\text{ and }N\equiv S_{2}\lambda
x.N^{\prime }$$
where $S_{1}$ and $S_{2}$ are different signs, and $%
M^{\prime }\equiv N^{\prime }.$
Note that not all terms have opposites but if $M\equiv \overline{N}
$ then it follows that $N\equiv \overline{M}.$
We define substitution of terms in the $\lambda ^{q}$-calculus as a modification of substitution of terms in the $\lambda ^{p}$-calculus. We rewrite the seven rules of the $\lambda ^{p}$-calculus to take account of the signs of the terms. First, we introduce the function notated by sign concatenation, defined by the following four rules: $$\begin{aligned}
++ &\mapsto &+ \\
+- &\mapsto &- \\
-+ &\mapsto &- \\
-- &\mapsto &+\end{aligned}$$
Notating this in our alternative syntax for signs (\[alternative grammar for signs\]), these rules can be summarized by the single rewrite rule $$\lnot \lnot \mapsto \epsilon$$
because the concatenation of a sign $S$ with $\epsilon $ is just $%
S $ again. Now we can use this function in the following substitution rules. $$\begin{array}{ll}
1.\;\left( Sx\right) \left[ N/x\right] \equiv SN & \text{for variables }y\not%
{\equiv}x \\
2.\;\left( Sy\right) \left[ N/x\right] \equiv Sy & \\
3.\;\left( PQ\right) \left[ N/x\right] \equiv \left( P\left[ N/x\right]
\right) \left( Q\left[ N/x\right] \right) & \\
4.\;\left( S\lambda x.P\right) \left[ N/x\right] \equiv S\lambda x.P & \\
5.\;\left( S\lambda y.P\right) \left[ N/x\right] \equiv S\lambda y.\left(
P\left[ N/x\right] \right) & \text{if }y\not{\equiv}x\text{ and }y\notin
FV\left( N\right) \\
6.\;\left( S\lambda y.P\right) \left[ N/x\right] \equiv S\lambda z.\left(
P\left[ z/y\right] \left[ N/x\right] \right) & \text{if }y\not{\equiv}x\text{
and }y\in FV\left( N\right) \\
& \text{and }z\notin FV(P)\bigcup FV\left( N\right) \\
7.\;\left( P,Q\right) \left[ N/x\right] \equiv \left( P\left[ N/x\right]
,Q\left[ N/x\right] \right) &
\end{array}
\label{substitution-q}$$
Where did we use the sign concatenation function in the above substitution rules? It is hidden in rule (1). Consider $\left( \lnot x\right) \left[
\lnot \lambda y.y/x\right] \equiv \lnot \lnot \lambda y.y.$ This is not a $%
\lambda ^{q}$-term by grammar (\[lambda-q syntax\]). Applying the sign concatenation function yields $\lambda y.y,$ which is a $\lambda ^{q}$-term. However, we could not have rewritten rule (1) to be explicit about the sign of $N$ because $N$ may be an application or a collection and therefore not have a sign.
Reduction {#reduction}
---------
The $\gamma $-relation of the $\lambda ^{q}$-calculus is of the same form as that of the $\lambda ^{p}$-calculus.
$$\gamma ^{q}\triangleq \left\{
\begin{array}{l}
\left( \left[ M_{i}^{i\in 1..m}\right] \left[ N_{j}^{j\in 1..n}\right]
,\left[ M_{i}^{i\in 1..m}N_{j}^{j\in 1..n}\right] \right) \\
\text{such that }M_{i},N_{j}\in LambdaQTerm,\,m>1\text{ or }n>1
\end{array}
\right\} \label{gamma-q}$$
We omit the superscript when it is clear from context if the terms under consideration are $\lambda ^{p}$-terms or $\lambda ^{q}$-terms. We still write $\gamma \left( M\right) $ for the $\gamma $-normal form of $M.$ Theorems (\[thm: gamma-p CR\]) and (\[thm: gamma-p NF exists\]) are easily extendible to terms of the $\lambda ^{q}$-calculus so $\gamma \left(
M\right) $ is well-defined.
We extend the $\beta $-relation to deal properly with signs.
$$\beta ^{q}\triangleq \left\{
\begin{array}{l}
\left( \left( S\lambda x.M\right) N,SM\left[ N/x\right] \right) \\
\text{such that }S\in \text{\emph{Sign}},S\lambda x.M\text{ and }N\in
LambdaQTerm
\end{array}
\right\}$$
We refer to $\beta $-reduction for the $\lambda $-calculus, the $%
\lambda ^{p}$-calculus, and the $\lambda ^{q}$-calculus all with the same notation when there is no risk of ambiguity.
Evaluation Semantics
--------------------
We modify the call-by-value evaluation semantics of the $\lambda ^{p}$-calculus.
There are three rules for the call-by-value evaluation semantics of the $%
\lambda ^{q}$-calculus. $$\begin{aligned}
&&\frac {}{v\rightsquigarrow v}\text{(Refl)\qquad \qquad (for }v\text{ a
value)} \\
&&\frac{\gamma \left( M\right) \rightsquigarrow S\lambda x.P\quad \gamma
\left( N\right) \rightsquigarrow N^{\prime }\quad \gamma \left( SP\left[
N^{\prime }/x\right] \right) \rightsquigarrow v}{MN\rightsquigarrow v}\text{%
(Eval)} \\
&&\frac{\gamma \left( M\right) \rightsquigarrow v_{1}\quad \gamma \left(
N\right) \rightsquigarrow v_{2}}{\left( M,N\right) \rightsquigarrow \left(
v_{1},v_{2}\right) }\text{(Coll)}\end{aligned}$$
Observation {#lambda-q observation}
-----------
We define an observation function $\Xi $ from $%
\lambda ^{q}$-terms to $\lambda $-terms as the composition of a function $%
\Delta $ from $\lambda ^{q}$-terms to $\lambda ^{p}$-terms with the observation function $\Theta $ from $\lambda ^{p}$-terms to $\lambda $-terms defined in (\[lambda-p observation\]). Thus, $\Xi =\Theta \circ \Delta $ where we define $\Delta $ as follows. $$\begin{aligned}
\Delta \left( Sx\right) &=&x \\
\Delta \left( S\lambda x.M\right) &=&\lambda x.\Delta \left( M\right) \\
\Delta \left( M_{1}M_{2}\right) &=&\Delta \left( M_{1}\right) \Delta \left(
M_{2}\right) \\
\Delta \left( M\equiv \left[ M_{i}^{i\in 1..\left| M\right| }\right] \right)
&=&\left[ \Delta \left( M_{i}^{i\in \left\{ i\,\,\,|\,\,\,M_{i}\not{\equiv}%
\overline{M_{j}}\,\,\text{for }j\in 1..\left| M\right| \right\} }\right)
\right] \label{Collection case for delta}\end{aligned}$$
The key is in the case (\[Collection case for delta\]) where the argument to $\Delta $ is a collection. In this case, the function $\Delta $ does two things. First, it removes those pairs of terms in the collection that are opposite. Then, it recursively applies itself to each of the remaining terms.
Note that unlike the observation function $\Theta $ of the $\lambda ^{p}$-calculus, the observation function $\Xi $ of the $\lambda ^{q}$-calculus is not total. For some $\lambda ^{q}$-term $M$, $\Xi \left( M\right) $ does not yield a $\lambda $-term. An example of such a term is $M\equiv x,\lnot x$ because $\Delta \left( M\right) $ is the collection $M$ with all pairs of opposites removed. However, the empty collection is not a $\lambda ^{p}$-term. Therefore, some $\lambda ^{q}$-terms cannot be observed. The non-totality of the observation function $\Xi $ does not limit the $\lambda
^{q}$-calculus because careful programming can always insert a unique term into a collection prior to observation to ensure observability. There is thus no need to add distinguished tokens to the $\lambda ^{q}$-calculus such as or .
Because $\Xi =\Theta \circ \Delta ,$ the definition of statistical indistinguishability (\[defn: statistically indistinguishable\]) applies to $\Xi \left( M\right) $ and $\Xi \left( N\right) $ as well, if both $%
\Delta \left( M\right) $ and $\Delta \left( N\right) $ exist. Although observing a $\lambda ^{p}$-term is statistically indistinguishable from observing its $\gamma $-normal form, observing a $\lambda ^{q}$-term is, in general, statistically distinguishable from observing its $\gamma $-normal form.
Observational Semantics
-----------------------
The observational semantics for the $\lambda ^{q}$-calculus is similar to that of the $\lambda ^{p}$-calculus (\[obs-p\]). It is given by a single rule. $$\frac{M\rightsquigarrow v\quad \Xi \left( v\right) =N}{M\multimap N}\text{%
(ObsQ)} \label{obs-q}$$
Examples
--------
We provide one example. We show how satisfiability may be solved in the $%
\lambda ^{q}$-calculus. We assume possible solutions are encoded some way in the $\lambda ^{q}$-calculus and there is a term $\underline{\text{CHECK}_{f}}
$ that checks if the fixed Boolean formula $f$ is satisfied by a particular truth assignment, given as the argument. The output from this is a collection of $\underline{\text{T}}$ (truth) and $\underline{\text{F}}$ (falsity) terms. We now present a term that will effectively remove all of the $\underline{\text{F}}$ terms. It is an instance of a more general method. $$\underline{\text{REMOVE-F}}\equiv \lambda x.\,\underline{\text{IF}}%
\,x\,x\,\left( x,\lnot x\right)$$
We give an example evaluation. $$\begin{array}{lll}
\underline{\text{REMOVE-F}}\,\left( \underline{\text{F}},\underline{\text{T}}%
,\underline{\text{F}}\right) & \equiv & \left( \lambda x.\,\underline{\text{%
IF}}\,x\,x\,\left( x,\lnot x\right) \right) \left( \underline{\text{F}},%
\underline{\text{T}},\underline{\text{F}}\right) \\
& \twoheadrightarrow _{\gamma } & \left(
\begin{array}{l}
\left( \lambda x.\,\underline{\text{IF}}\,x\,x\,\left( x,\lnot x\right)
\right) \underline{\text{F}}, \\
\left( \lambda x.\,\underline{\text{IF}}\,x\,x\,\left( x,\lnot x\right)
\right) \underline{\text{T}}, \\
\left( \lambda x.\,\underline{\text{IF}}\,x\,x\,\left( x,\lnot x\right)
\right) \underline{\text{F}}
\end{array}
\right) \\
& \twoheadrightarrow _{\beta } & \left( \left( \underline{\text{F}},\lnot
\underline{\text{F}}\right) ,\underline{\text{T}},\left( \underline{\text{F}}%
,\lnot \underline{\text{F}}\right) \right) \\
& \equiv & \left( \underline{\text{F}},\lnot \underline{\text{F}},\underline{%
\text{T}},\underline{\text{F}},\lnot \underline{\text{F}}\right)
\end{array}$$
Observing the final term will always yield $\underline{\text{T}}.$ Note that the drawback to this method is that if $f$ is unsatisfiable then the term will be unobservable. Therefore, when we insert a distinguished term into the collection to make it observable, we risk observing that term instead of $\underline{\text{T}}.$ At worst, however, we would have a fifty-fifty chance of error.
Specifically, consider what happens when the argument to $\underline{\text{%
REMOVE-F}}$ is a collection of $\underline{\text{F}}^{\prime }$s$.$ Then $%
\underline{\text{REMOVE-F}}\,\underline{\text{F}}=\left( \underline{\text{F}}%
,\lnot \underline{\text{F}}\right) .$We insert $\underline{\text{I}}\equiv
\lambda x.x$ which, if we observe, we take to mean that either $f$ is unsatisfiable or we have bad luck. Thus, we observe the term $\left(
\underline{\text{I}},\underline{\text{F}},\lnot \underline{\text{F}}\right)
. $ This will always yield $\underline{\text{I}}.$ However, we cannot conclude that $f$ is unsatisfiable because, in the worst case, the term may have been $\left( \underline{\text{I}},\underline{\text{REMOVE-F}}\,%
\underline{\text{T}}\right) =\left( \underline{\text{I}},\underline{\text{T}}%
\right) $ and we may have observed $\underline{\text{I}}$ even though $f$ was satisfiable. We may recalculate until we are certain to an arbitrary significance that $f$ is not satisfiable.
Therefore, applying $\underline{\text{REMOVE-F}}$ to the results of $%
\underline{\text{CHECK}_{f}}$ and then observing the result will yield $%
\underline{\text{T}}$ only if $f$ is satisfiable.
Conclusion
==========
We have seen two new formalisms. The $\lambda ^{p}$-calculus allows expression of randomized algorithms. The $\lambda ^{q}$-calculus allows expression of quantumized algorithms. In these calculi, observation is made explicit, and terms are presumed to exist in some Heisenberg world of *potentia*.
This work represents a new direction of research. Just as the $\lambda $-calculus found many uses, the $\lambda ^{p}$-calculus and the $\lambda ^{q}$-calculus may help discussion of quantum computation in the following ways.
1. Quantum programming languages can be specified in terms of the $%
\lambda ^{q}$-calculus and compared against each other.
2. Algorithms can be explored in the $\lambda ^{q}$-calculus on a higher level than quantum Turing machines, which, like classical Turing machines, are difficult to program.
3. An exploration of the relationship between the $\lambda ^{q}$-calculus and quantum Turing machines, quantum computational networks, or other proposed quantum hardware, may provide insights into both fields.
We have seen some algorithms for the $\lambda ^{p}$-calculus and the $%
\lambda ^{q}$-calculus. It should not be difficult to see that the $\lambda
^{p}$-calculus can simulate a probabilistic Turing machine and that the $%
\lambda ^{q}$-calculus can simulate a quantum Turing machine. It should also follow that a probabilistic Turing machine can simulate the $\lambda ^{p}$-calculus, with the exponential slowdown that comes from computing in the world of reality rather than the world of *potentia.* However, it is not obvious that a quantum Turing machine can simulate the $\lambda ^{q}$ -calculus. An answer to this question, whether positive or negative, will be interesting. If quantum computers can simulate the $\lambda ^{q}$-calculus efficiently, then the $\lambda ^{q}$-calculus can be used as a programming language directly. As a byproduct, satisfiability will be efficiently solvable. If quantum computers cannot simulate the $\lambda ^{q}$-calculus efficiently, knowing what the barrier is may allow the formulation of another type of computer that can simulate it.
[9]{} Abadi, Martín and Luca Cardelli. *A Theory of Objects*. Monographs in Computer Science, David Gries and Fred B. Schneider editors. Springer-Verlag, New York: 1996. Chapter 6.
Barendregt, Hendrik Pieter. *The lambda calculus: its syntax and semantics.* North-Holland, New York: 1981.
Church, Alonzo. *An unsolvable problem of elementary number theory.* American Journal of Mathematics 58, 1936, pp.345-363.
*Revised*$^{4}$* Report on the Algorithmic Language Scheme.* William Clinger and Jonathan Rees editors. November 2, 1991.
Deutsch, David. *Quantum theory, the Church-Turing principle and the universal quantum computer*. Proc. R. Soc. Lond. A400, 1985, pp.97-117.
Deutsch, David. *Quantum computational networks*. Proc. R. Soc. Lond. A425, 1989, pp.73-90.
Deutsch, David. *Quantum computation*. Physics World, June, 1992, pp.57-61.
Heisenberg, Werner. *Physics and philosophy.* Harper & Brothers, New York: 1958.
Simon, Daniel. *On the power of quantum computation*. Proceedings of the 35th Annual Symposium on the Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, CA: 1994.
[^1]: The author’s email address is `[email protected]`.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the propagation of three-dimensional (3D) bipolar ultrashort electromagnetic pulses in an inhomogeneous array of semiconductor carbon nanotubes. The heterogeneity is represented by a planar region with an increased concentration of conduction electrons. The evolution of the electromagnetic field and electron concentration in the sample are governed by the Maxwell’s equations and continuity equation. In particular, non-uniformity of the electromagnetic field along the axis of the nanotubes is taken into account. We demonstrate that, depending on values of parameters of the electromagnetic pulse approaching the region with the higher electron concentration, the pulse is reflected from the region or passes it. Specifically, our simulations demonstrate that, after interacting with the higher-concentration area, the pulse can propagate steadily, without significant spreading. The possibility of such ultrashort electromagnetic pulses propagating in arrays of carbon nanotubes over distances significantly exceeding characteristic dimensions of the pulses makes it possible to consider them as 3D solitons.'
author:
- 'Eduard G. Fedorov'
- 'Alexander V. Zhukov'
- Roland Bouffanais
- 'Alexander P. Timashkov'
- 'Boris A. Malomed'
- 'Herv[é]{} Leblond'
- Dumitru Mihalache
- 'Nikolay N. Rosanov'
- 'Mikhail B. Belonenko'
title: 'Propagation of three-dimensional bipolar ultrashort electromagnetic pulses in an inhomogeneous array of carbon nanotubes'
---
Introduction
============
Nowadays, carbon nanotubes (CNTs)—quasi-one-dimensional macromolecules of carbon—are considered as promising objects with a potential for applications to the development of the elemental base of modern electronics, including nanocircuits usable in neurocomputers [@1]. Strong interest in these materials, starting from the moment of their discovery [@2; @3], is due to their unique physical properties (e.g., see Refs. [@4; @5; @6; @7; @8; @9]), which, in addition to the above-mentioned potential for the use in electronics, pave the way to a wide range of possibilities for the creation of ultra-strong composite materials, fuel cells, chemical sensors, and optical devices (such as displays, LEDs, and transparent conductive surfaces), etc. From the viewpoint of optoelectronic applications, specific features of the electronic structure of CNTs are of unique interest. The nonparabolicity of the dispersion law of conduction electrons (i.e., the energy dependence on the quasimomentum) in nanotubes makes it possible to observe a number of unique electromagnetic phenomena, including nonlinear diffraction and self-focusing of laser beams [@10; @11], as well as the propagation of solitary electromagnetic waves [@12] at field strengths starting from $\sim 10^{3}-10^{4}$ V/cm. In this connection, it is relevant to mention that possibilities offered by modern laser technologies for the generation of powerful electromagnetic radiation with specified properties—including ultrashort laser pulses with the duration on the order of several half-cycles of field oscillations [@13; @14]—have stimulated studies of the propagation of electromagnetic waves in various novel media [@15; @16; @17; @18; @19; @20; @21; @22; @23; @23a; @23b], CNTs being one of them.
The possibility of propagation of solitary electromagnetic waves in arrays of CNTs has been first theoretically established in a one-dimensional (1D) model based on an assumption of uniformity of the field along the nanotube axis [@12]. Actually, this approximation is only valid in a very narrow range of values of the underlying parameters. Subsequent studies aimed at investigating increasingly more realistic models describing the evolution of the electromagnetic field in nanotube arrays, taking into regard various physical factors affecting the dynamics of the electromagnetic wave. Adding the complexity to the model’s framework proceeded in several directions: i) increasing the dimensionality of the setting, ii) taking into account non-uniformity of the electromagnetic wave field, and iii) including various inhomogeneities of the medium. The investigation of these factors was carried out both independently, to clarify the role of each factor, and, subsequently, considering combined effects of multiple factors.
Naturally, the first factor considered to make the model more realistic was the dimensionality of the model. The propagation of electromagnetic waves in arrays of CNTs was studied in the framework of 1D [@12; @24], 2D [25,26,27,28]{}, and 3D [@29; @30] models. It was established that bipolar ultrashort electromagnetic pulses propagate, in a stable fashion, in the form of breather-like" light bullets over distances far exceeding the characteristic size of the pulses along the direction of their motion. The topicality of studying the propagation of electromagnetic waves in arrays of nanotubes in the inhomogeneous model is due, in particular, to the need to take transverse diffraction into account. Moreover, as in reality, a solitary electromagnetic wave in a real physical system is bounded in all directions, which implies non-uniformity of the field in any direction. Thus, the necessity of constructing a 3D model for the evolution of the electromagnetic field is obvious. In particular, non-uniformity of the field of the electromagnetic pulse along the axis of the nanotubes is an inherent ingredient of the 3D model. The construction of a model involving the latter feature was first reported in Refs. [@28; @29; @30]. In particular, the propagation of ultrashort pulses in homogeneous arrays of CNTs has been investigated in 2D [@28] and 3D [@29] models. One of the main results of these studies is the prediction of redistribution of conduction electrons, leading to specific variations of the density of conduction electrons. Among possible practical applications of this phenomenon in micro- and nano-electronics, one can envisage the manufacturing of highly accurate chemical sensors [@29] based on specifically designed arrays of semiconductor CNTs. As shown in Ref. [30]{}, such dynamical inhomogeneities in the electron subsystem of nanotubes can also underlie a complex medium-mediated mechanism of interaction of colliding electromagnetic pulses in the array of nanotubes.
Among factors that significantly affect the evolution of electromagnetic waves in arrays of semiconductor CNTs, a noteworthy one is the presence of various impurities and inhomogeneities. In Ref. [@31], the propagation of a bipolar electromagnetic pulse in the 2D geometry, and in the presence of a multilevel impurity uniformly distributed over the sample, was investigated. It was shown that doping the medium with an impurity of this type leads to a modification of the characteristics of the propagating pulse as compared to the case of propagation in a pure" sample.
Specific sample-doping format deserve particular consideration—in particular, localized introduction of impurities, limited to a certain part of the sample’s volume, which implies the creation of an inhomogeneity. Such defects of the medium, which are not initially associated with the action of the electromagnetic field, can be, for example, implemented as metallic inclusions or layers containing an increased concentration of conduction electrons relative to the concentration in the homogeneous part of the sample. This kind of heterogeneity can emerge either as a consequence of technological failures, at the stage of sample manufacturing, or as a result of purposeful formation of the inhomogeneity with intended properties. The features of the interaction of 2D unipolar light bullets" with metallic inclusions in nanotube arrays have been theoretically studied in Refs. [@32; @33]. The interaction of a bipolar ultrashort electromagnetic pulse with a layer of an increased electron concentration in the 2D model was studied both under the assumption of the uniformity of the field along the nanotube axis [@34], and also taking into account non-uniformity of the field [@35]. As a result, the selective nature of the interaction of the pulse with the static inhomogeneity of the medium has been established: a decrease of the pulse’s duration and increase of its amplitude facilitates its passage through a layer of an increased electron concentration, while pulses with a longer duration and smaller amplitude can be reflected by this layer.
Thus, the propagation of electromagnetic pulses in inhomogeneous media—taking into account the non-uniformity of the pulses’ field—are of significant interest. Indeed, there is a need to search for physical effects that can be used for the development of new components of the elemental base of optoelectronics, developing schemes for optical information processing and nondestructive testing systems, etc. Some peculiarities of light-matter interaction mentioned above may be employed for the development of ultra-fast optical transistors, switches, logic elements, transmission and signal delay lines, soliton memory elements [@35a], using ultra-short pulses as bits of data. Different outcomes of data processing operations may be associated with different regimes of the pulse propagation in the medium (e.g., different outcomes of the collision of a pulse with a layer of increased conductivity, embedded in the bulk of a semiconductor structure). It is therefore topical to implement such concepts using some of the most promising available materials, such as, in particular, graphene-based materials. In this connection, it is relevant to address effects of the static inhomogeneity in the electron subsystem of the array of semiconductor CNTs on the propagation of an ultrashort bipolar electromagnetic pulses in the most realistic 3D geometry. In this work, we consistently take into account factors that affect the dynamics of the ultrashort laser pulses, generalizing previously considered particular cases in the framework of an integrated model.
The system’s configuration and key assumptions
==============================================
We consider the propagation of a solitary electromagnetic wave in a volumetric array of a single-walled semiconductor carbon nanotubes (CNTs) embedded in a homogeneous dielectric medium. The nanotubes considered here are of the zigzag" type $(m,0)$, where integer $m$ (not a multiple of three for semiconductor nanotubes) determines their radius, $R=mb\sqrt{3}/2\pi $, with $b=1.42\times 10^{-8}\mathrm{~cm}$ being the distance between neighboring carbon atoms [@4; @5; @6; @7]. The CNTs are arranged in such a way that their axes are parallel to the common $x$-axis, and distances between adjacent nanotubes are much larger than their diameter. The latter assumption allows one to neglect the interaction between CNTs [@36]. Moreover, it allows us to consider the system as an electrically quasi-1D one, in which electron tunneling between neighboring nanotubes may be neglected, and electrical conductivity is possible only along the axis of the nanotubes. We define the configuration of the system in such a way that the pulse propagates through the CNT array in the direction perpendicular to their axes (for the definiteness’ sake, along the $z$-axis), while the electric component of the wave field, $\mathbf{E}=\{E,0,0\}$, is collinear with the $x$-axis (see Fig. \[fig1\]).
![The schematic plot of the setup and the associated coordinate system.[]{data-label="fig1"}](Figure-1.eps){width="60.00000%"}
For a wide range of values of the system’s parameters, the characteristic distance at which an appreciable change in the field of a bipolar electromagnetic pulse occurs, is significantly greater than both the distance between neighboring nanotubes and the length of the conduction-electron’s path along the axis of the nanotubes. On the other hand, with the nanotube radius $R\approx 5.5\times 10^{-8}\mathrm{~cm}$ and $m=7$, the characteristic distance between the nanotubes, sufficient to exclude the overlap of the electron wave functions of adjacent ones, i.e., substantially exceeding $R$, may still be negligibly small in comparison with the wavelength $\lambda $ of the electromagnetic radiation. In particular, this assumption is definitely valid for the infrared radiation, with $\lambda >1$ $\mathrm{\mu }$m. Under this condition, the nanotube array, which is a discrete structure at the microscopic level, may nevertheless be considered as a quasi-uniform medium for the propagation of electromagnetic waves. In this case, for length scales comparable to the pulse’s dimensions, the array of CNTs may be considered as a uniform continuous medium. In other words, the electromagnetic field in the system—specifically, ultra-short pulses carried by the infrared wavelength—are not affected by the discrete structure in the medium, there being no scattering (or even recurrent scattering) of electromagnetic radiation on inhomogeneities or irregularities of the CNT volumetric array. Of course, the scattering will appear in the framework of a microscopic theory (see, e.g., Ref. [@Lagendijk]), which should be a subject for a separate work.
Another assumption that we adopt here concerns the time duration of the electromagnetic pulse, $T_{S}$, the relaxation time of the conduction current along the nanotube axis, $t_{\mathrm{rel}}$, and also the time of the observation of the light propagation in the system, $t$ —we assume $T_{S}\ll t<t_{\mathrm{rel}}$. If this condition is met, it is possible to neglect field decay, thus enabling the collisionless approximation in describing the lossless evolution of the field [@12].
Governing equations
===================
Given the orientation of the coordinate system relative to the nanotube axis, as defined in Fig. \[fig1\], the electron energy spectrum for CNTs takes the form [@29; @30] $$\epsilon (p_{x},s)=\gamma _{0}\sqrt{1+4\cos \left( p_{x}\frac{d_{x}}{\hbar }\right) \cos \left( \pi \frac{s}{m}\right) +4\cos ^{2}\left( \pi \frac{s}{m}\right) }, \label{1}$$where the electron quasimomentum is $\mathbf{p}=\left\{ p_{x},s\right\} $, $s=1,2,\dots ,m$ being an integer characterizing the momentum quantization along the perimeter of the nanotube, with $m$ being the number of hexagonal carbon cycles which form the circumference of the CNT, $\gamma _{0}$ is the overlap integral, and $d_{x}=3b/2$.
Equation for the vector potential
---------------------------------
The electromagnetic field in the CNT array is governed by Maxwell’s equations [@37; @38], from which, taking into account the Lorentz gauge condition, we obtain the wave equation for the spatiotemporal evolution of the vector field potential: $$\frac{\varepsilon }{c^{2}}\frac{\partial ^{2}\mathbf{A}}{\partial t^{2}}-
\frac{\partial ^{2}\mathbf{A}}{\partial x^{2}}-\frac{\partial ^{2}\mathbf{A}
}{\partial y^{2}}-\frac{\partial ^{2}\mathbf{A}}{\partial z^{2}}=\frac{4\pi
}{c}\mathbf{j}, \label{2}$$with $\mathbf{A}=\left\{ A,0,0\right\} $, where $\mathbf{j}=\left\{
j,0,0\right\} $ is the current density, $c$ the speed of light in vacuum, and $\varepsilon $ the average relative dielectric constant of the medium (see, e.g., Refs. [@25; @27; @28; @29; @30]).
We emphasize that the system under consideration has a nonzero electric conductivity only along the $x$–axis, while in the $\left( y,z\right) $ plane the current is absent due to the negligible coupling between neighboring nanotubes. Thus, since the second and third components of the conduction current $\mathbf{j}$ are zero, Eq. admits the existence of zero solutions for the second and third components of the vector potential. We use this fact to define the vector potential as being collinear to axes of the nanotubes.
The conduction current density $j$ along the nanotube axis is determined by applying the approach used in Refs. [@39; @40], which yields$$j=2e\sum_{s=1}^{m}\int\limits_{-\pi \hbar /d}^{\pi \hbar
/d}v_{x}f(p_{x},s)dp_{x}, \label{3}$$where $e<0$ is the electron charge, $v_{x}$ is the electron velocity and $f(p_{x},s)$ is the electron distribution function with respect to quasimomenta $p_{x}$ and numbers $s$ characterizing the quantization of the electron’s momentum along the perimeter of a nanotube. The integration over the quasimomentum in Eq. (\[3\]) is carried out within the first Brillouin zone.
Using the expression for the electron energy to determine their velocity as $v_{x}=\partial \epsilon (p_{x},s)/\partial p_{x}$, and taking into account the electron distribution $f(p_{x},s)$ according to the Fermi–Dirac statistics, we derive from Eq. , an expression for the current density (for more details see Ref. [@30]): $$j=-en\frac{d_{x}}{\hbar }\gamma _{0}\sum_{r=1}^{\infty }G_{r}\sin \left[ r\frac{d_{x}}{\hbar }\left( A\frac{e}{c}+e\int\limits_{0}^{t}\frac{\partial
\phi }{\partial x}dt^{\prime }\right) \right] , \label{4}$$where $n=n(x,y,z,t)$ is the local value of the concentration of conduction electrons, $\phi $ is the scalar potential (self-consistent equations for the quantities $n$ and $\phi $ are derived in Sec. III.B and Sec. III.C, respectively), and coefficients $G_{r}$ are given by $$G_{r}=-r\frac{\displaystyle\sum\limits_{s=1}^{m}{\displaystyle\frac{\delta
_{r,s}}{\gamma _{0}}}\displaystyle\int_{-\pi }^{+\pi }\cos (r\kappa )\left\{
1+\exp \left[ \frac{\theta _{0,s}}{2}+\sum_{q=1}^{\infty }\theta _{q,s}\cos
\left( q\kappa \right) \right] \right\} ^{-1}\text{d}\kappa }{\displaystyle\sum_{s=1}^{m}\int_{-\pi }^{+\pi }\left\{ 1+\exp \left[ \frac{\theta _{0,s}}{2}+\sum_{q=1}^{\infty }\theta _{q,s}\cos \left( q\kappa \right) \right]
\right\} ^{-1}\text{d}\kappa }. \label{5}$$Here $\theta _{r,s}=\delta _{r,s}(k_{B}T)^{-1}$, while $T$ is the temperature, $k_{B}$ the Boltzmann constant, and $\delta _{r,s}$ are coefficients of the Fourier decomposition [@41] of spectrum : $$\delta _{r,s}=\frac{d_{x}}{\pi \hbar }\int_{-\pi \hbar /d_{x}}^{-\pi \hbar
/d_{x}}\epsilon (p_{x},s)\cos \left( r\frac{d_{x}}{\hbar }p_{x}\right) \text{
d}p_{x}. \label{6}$$
The evolution of the vector potential of the field in the system is determined by the projection of Eq. onto the nanotube axis, which, taking into account expression and after introducing dimensionless variables, takes the following form: $$\frac{\partial ^{2}\Psi }{\partial \tau ^{2}}-\left( \frac{\partial ^{2}\Psi
}{\partial \xi ^{2}}+\frac{\partial ^{2}\Psi }{\partial \upsilon ^{2}}+\frac{
\partial ^{2}\Psi }{\partial \zeta ^{2}}\right) +\eta \sum_{r=1}^{\infty
}G_{r}\sin \left[ r\left( \Psi +\int\limits_{0}^{\tau }\frac{\partial \Phi }{
\partial \xi }d\tau ^{\prime }\right) \right] =0, \label{7}$$where $\eta =n/n_{\mathrm{bias}}=\eta (\xi ,\upsilon ,\zeta ,\tau )$ is the reduced (dimensionless) density of conduction electron, $n_{\mathrm{bias}}$ is the concentration of conduction electrons in the homogeneous part of the sample in the absence of electromagnetic fields, $\Psi =Aed_{x}/\left(
c\hbar \right) $ is the projection of the scaled vector potential onto the $x $-axis, $\Phi =\phi \sqrt{\varepsilon }ed_{x}/(c\hbar )$ is the dimensionless scalar potential, $\tau =\omega _{0}t/\sqrt{\varepsilon }$ is the scaled time, $\xi =x\omega _{0}/c$, $\upsilon =y\omega _{0}/c$ and $\zeta =z\omega _{0}/c$ are the scaled coordinates, and $$\omega _{0}\equiv 2\frac{|e|d_{x}}{\hbar }\sqrt{\pi \gamma _{0}n_{\mathrm{\
bias}}}. \label{8}$$Thus, Eq. describes the evolution of the vector potential of the self-consistent electromagnetic field in the CNT array: the field is fully determined by the density of the conduction current \[see Eq. \], and the conduction current is, in turn, affected by the field \[see Eq. \].
Equation for the electron density
---------------------------------
In the general case, the electromagnetic field in the system under consideration is non-uniform in space. Indeed, the field of an ultrashort electromagnetic pulse propagating in an array of nanotubes is localized at each moment in a small (moving) region of space. The non-uniformity is invisible on the scale of the nanotube radius $\sim 5\times 10^{-8}\mathrm{\
~cm}$, or even for the distance between neighboring nanotubes $\sim
10^{-7}-10^{-6}\mathrm{~cm}$. However, it is significant at the wavelength scales of the infrared radiation, $\lambda $ $\sim 1\mathrm{\ \mu }$m.
The spatial non-uniformity of the field along the nanotube axis determines the dependence of the current density on coordinate $x$, as it follows from expression for the current density. Since the total charge in the sample is conserved, and the change in its bulk density $\rho =en$ obeys the continuity equation $\nabla \mathbf{j}+\partial \rho /\partial t=0$ [37,38]{}, the non-uniformity of the current density causes a temporal change in electron density, as per $$\frac{\partial n}{\partial t}=-\frac{1}{e}\frac{\partial j}{\partial x}.
\label{9}$$We stress that, as the system considered here is supposed to be electrically quasi-one-dimensional, i.e., the conductivity is only effective along the nanotubes axis, given the negligible overlap of the electron wave functions of neighboring nanotubes, the field non-uniformity along directions orthogonal to the nanotube axes does not affect the distribution of the electron concentration in the sample.
Substituting Eq. for the projection of the current density onto the axis of nanotubes into Eq. , and passing to the dimensionless notation (the same as in Ref. [@30]), we obtain an equation governing the evolution of the electron concentration under the action of the electromagnetic pulse: $$\frac{\partial \eta }{\partial \tau }=\alpha \sum_{r=1}^{\infty }G_{r}\frac{
\partial }{\partial \xi }\left\{ \eta \sin \left[ r\left( \Psi
+\int\limits_{0}^{\tau }\frac{\partial \Phi }{\partial \xi }d\tau ^{\prime
}\right) \right] \right\} , \label{10}$$with $\alpha \equiv d_{x}\gamma _{0}\sqrt{\varepsilon }/c\hbar $, the other quantities being defined in Eq. .
Equation for the scalar potential field
---------------------------------------
The system as a whole being electro-neutral, the redistribution of the electron concentration in the sample, due to the action of the non-uniform field along the axis of the nanotubes, is equivalent to appearance of regions of high and low electron concentration relative to the initial equilibrium distribution, $n_{0}=n(\xi ,\upsilon ,\zeta ,\tau _{0})$, taken (at the the initial time $\tau _{0}$) prior to the entrance of the an electromagnetic pulse into the sample. Thus, the local concentration of electrons, $n(\xi ,\upsilon ,\zeta ,\tau )$, may be represented as the sum of the initial equilibrium value $n_{0}$ and the concentration of the additional" charge, $\delta n(\xi ,\upsilon ,\zeta ,\tau
)=n-n_{0}$, with density $\delta \rho =e\delta n=e(n-n_{0})$. Note that $\delta \rho \neq 0$ implies a local imbalance between the negative charge of free electrons and the positive charge of holes. The local imbalanced charge perturbs the distribution of the field according to the driven wave equation for the scalar potential, which follows from Maxwell’s equations [@37; @38]: $$\frac{\varepsilon }{c^{2}}\frac{\partial ^{2}\phi }{\partial t^{2}}-\left(
\frac{\partial ^{2}\phi }{\partial x^{2}}+\frac{\partial ^{2}\phi }{\partial
y^{2}}+\frac{\partial ^{2}\phi }{\partial z^{2}}\right) =\frac{4\pi }{
\varepsilon }\delta \rho . \label{11}$$Using the same dimensionless notations as above, Eq. can be written as $$\frac{\partial ^{2}\Phi }{\partial \tau ^{2}}-\left( \frac{\partial ^{2}\Phi
}{\partial \xi ^{2}}+\frac{\partial ^{2}\Phi }{\partial \upsilon ^{2}}+\frac{
\partial ^{2}\Phi }{\partial \zeta ^{2}}\right) =\beta (\eta -\eta _{0}),
\label{12}$$where $\beta =1/\alpha =c\hbar /\left( d_{x}\gamma _{0}\sqrt{\varepsilon }\right) $ \[see Eq. \], and $\eta _{0}=n_{0}/n_{\mathrm{bias}}=\eta
(\xi ,\upsilon ,\zeta ,\tau _{0})$ is the dimensionless local value of the concentration of conduction electrons at the initial instant of time in the absence of the field.
Thus, the evolution of the field in the CNT array, taking into account the redistribution of the conduction-electron density, is governed by the system of equations , , and , which provide a self-consistent model for the evolution of the electromagnetic field and electronic subsystem in the array.
The localization of the electromagnetic pulse
---------------------------------------------
Upon obtaining the numerical solution to equations , , and it is possible to calculate the electric field, as $\mathbf{E}=-c^{-1}\partial \mathbf{A}/\partial t-\nabla \phi $ (see, e.g., Refs. [37, 38]{}). Taking into account that the vector potential has a nonzero component only along the nanotube axis (see the description of the system configuration above), one can write expressions for the components of the electric field as follows:
$$E_{x}=E_{0}\left( \frac{\partial \Psi }{\partial \tau }+\frac{\partial \Phi
}{\partial \xi }\right) ,\quad E_{y}=E_{0}\frac{\partial \Phi }{\partial
\upsilon },\quad E_{z}=E_{0}\frac{\partial \Phi }{\partial \zeta },
\label{13}$$
where $E_{0}\equiv -\hbar \omega _{0}/ed_{x}\sqrt{\varepsilon }$. Thus, the electric field in the CNT array is not, generally, collinear with the nanotube axis. However, the $\ y$ and $z$ components of the electric field, orthogonal to the nanotube axis, do not affect the dynamics of electrons, due to the absence of conductivity of the system in these directions. Thus, only component $E_{x}$, which affects the dynamics of the electronic subsystem, is relevant to the description of the solitary electromagnetic wave. The energy density of this component is$$I=E_{x}^{2}=I_{0}\left( \frac{\partial \Psi }{\partial \tau }+\frac{\partial
\Phi }{\partial \xi }\right) ^{2}, \label{14}$$where $I_{0}=E_{0}^{2}$, and we have made use of the first expression in Eq. (\[13\]). The position of a local maxima of this quantity identifies the instantaneous location of the ultrashort electromagnetic pulse.
Non-uniformity of the electron concentration
============================================
We assume that the CNT array contains a localized inhomogeneity, in the form of a region with an increased concentration of conduction electrons, exceeding the bulk concentration $n_{\mathrm{bias}}$ in the homogeneous sample. As mentioned above, such a local defect can be created by introducing donor impurities at the stage of the fabrication of the sample, subject to the condition of the electro-neutrality of the entire system. Accordingly, the initial electron density (in the absence of an electromagnetic pulse) is $n_{0}=n(\xi ,\upsilon ,\zeta ,\tau _{0})$. We stress that, in the absence of the electromagnetic pulse, each segment of the sample is locally electro-neutral, even if the electron concentration is inhomogeneous. Namely, in the region of increased electron density, the hole concentration is higher too, compensating the charge of the free electrons.
We assume that the region of the increased electron concentration is a narrow layer parallel to the nanotube axis, and its thickness $\delta z_{
\mathrm{imp}}$ is much smaller than the spatial dimension of the electromagnetic pulse in the direction of its propagation along the $z$-axis. In addition, we also consider this narrow layer as being indefinitely extended in the $x$ and $y$, as shown in Fig. \[fig2\].
![The layer of increased concentration of conduction electrons.[]{data-label="fig2"}](Figure-2.eps){width="50.00000%"}
The respective dimensionless electron concentration in the sample in the absence of the electromagnetic field is approximated by the Gaussian profile (see Ref. [@35]): $$\eta (\xi ,\upsilon ,\zeta ,\tau _{0})=\eta (\zeta )=1+\left( \eta _{\mathrm{\ imp}}^{\mathrm{max}}-1\right) \exp \left\{ -\left( \frac{\zeta }{\delta
\zeta _{\mathrm{imp}}}\right) ^{2}\right\} , \label{15}$$where $\eta _{\mathrm{imp}}^{\mathrm{max}}=n_{\mathrm{imp}}^{\mathrm{max}}/n_{\mathrm{bias}}$, $n_{\mathrm{imp}}^{\mathrm{max}}$ being the maximum electron concentration in the region of inhomogeneity, and $\delta \zeta _{
\mathrm{imp}}$ is a dimensionless parameter determined by the characteristic half-thickness of the region of the increased electron concentration, $\delta \zeta _{\mathrm{imp}}=\omega _{0}\delta z_{\mathrm{imp}}/c$. The concentration of conduction electrons is assumed to be constant in the $\left( x,y\right) $ plane.
The initial electron density in the system determines the corresponding scalar potential. Taking into account the fact that $\eta \equiv \eta _{0}$ holds initially, the right-hand side of Eq. vanishes. As a result, Eq. produces a constant solution. Since the scalar potential is always determined up to an arbitrary constant [@37; @38], its initial value may be fixed to be zero: $$\Phi (\xi ,\upsilon ,\zeta ,\tau _{0})=0, \label{16}$$which we assume to be the initial distribution of the scalar potential in the system.
The initial form of the electromagnetic pulse
=============================================
We now assume that the electromagnetic pulse propagates in the array of CNTs, with the $\xi $-component of the dimensionless vector potential at the initial instant of time $\tau =\tau _{0}$ defined as follows: $$\Psi (\xi ,\upsilon ,\zeta ,\tau _{0})=\Psi _{B}(\zeta ,\tau _{0})\exp \left[
-\frac{(\xi -\xi _{0})^{2}+(\upsilon -\upsilon _{0})^{2}}{w_{0}^{2}}\right] ,
\label{17}$$where $\Psi _{B}(\zeta ,\tau _{0})$ is the $\xi $-component at $\xi =\xi
_{0} $ and $\upsilon =\upsilon _{0}$, with $\xi _{0}$ and $\upsilon _{0}$ being dimensionless pulse’s coordinates, along the $\xi $- and $\upsilon $-axes, respectively, at the initial instant of time, and the initial transverse half-width $w_{0}$ of the pulse.
Profile $\Psi _{B}(\zeta ,\tau _{0})$ is chosen as a breather of the sine-Gordon equation, i.e., a non-topological oscillating soliton [@42]: $$\Psi _{B}(\zeta ,\tau _{0})=4\arctan \left\{ \left( \frac{1}{\Omega ^{2}}
-1\right) ^{1/2}\frac{\sin \chi }{\cosh \mu }\right\} , \label{18}$$where
$$\begin{aligned}
\chi & \equiv \sigma \Omega \frac{\tau _{0}-(\zeta -\zeta _{0})U}{\sqrt{
1-U^{2}}}, \label{19} \\
\mu & \equiv \sigma \left[ \tau _{0}U-(\zeta -\zeta _{0})\right] \sqrt{\frac{
1-\Omega ^{2}}{1-U^{2}}}, \label{20}\end{aligned}$$
$U=u/v_{0}$ is the ratio of the initial propagation velocity $u$ of the electromagnetic pulse (breather) along the $\zeta $-axis within the sine-Gordon approximation, and the linear speed of light in the medium, $v_{0}=c/\sqrt{\varepsilon }$. Further, $\zeta _{0}$ is the dimensionless coordinate of the breather along the $\zeta $-axis at moment $\tau =\tau
_{0} $, $\Omega <1$ is a free parameter, which determines the breather’s oscillation frequency (scaled by frequency $\omega _{B}=\omega _{0}\Omega $ in physical units), and $\sigma =\sqrt{G_{1}}$ \[coefficients $G_{j}$ are calculated as per Eq. \].
The basic argument in favor of the choice of the initial condition in the form of Eqs. and is that the equation for the vector potential may be considered as a non-uniform generalization of the sine-Gordon equation. As the sine-Gordon equation gives rise to breather solutions in the form of Eq. , it is reasonable to assume the possibility of the propagation of solitary waves in a form close to breathers. This assumption is amply justified by results reported in Refs. [@25; @26; @27; @28; @29; @30]. The second factor in Eq. corresponds to the Gaussian distribution of the field in the plane $(\xi ,\zeta )$ perpendicular to the propagation direction of the electromagnetic pulse. The choice of the Gaussian distribution for this field is justified by a wide range of applicability of the Gaussian waveforms [@43; @44; @45; @46; @47].
The component of the electric field along the nanotube axis, taking into account the expression at the initial instant of time, has the form of $$\begin{aligned}
E_{x} &=&4E_{0}\frac{\sigma \sqrt{1-\Omega ^{2}}}{\sqrt{1-U}^{2}}\left\{
\frac{\cos \chi \cosh \mu -U\left( \Omega ^{-2}-1\right) ^{1/2}\sin \chi
\sinh \mu }{\cosh ^{2}\mu +\left( \Omega ^{-2}-1\right) \sin ^{2}\chi }
\right\} \notag \\
&\times &\exp \left[ -\frac{(\xi -\xi _{0})^{2}+(\upsilon -\upsilon
_{0})^{2} }{w_{0}^{2}}\right] . \label{21}\end{aligned}$$Equations – describe a short wave packet, consisting of a carrier wave and an envelope. The carrier, which accounts for the internal oscillations [@25; @29] of the pulse, is determined by the oscillating behavior of the function $\sin \chi $, while the envelope accounts for the exponential behavior of the function $\cosh \mu $. In the case of a few-cycle pulse, variation scales of both the envelope and carrier of the pulse have the same order of magnitude, hence its profile varies periodically with the frequency of the carrier. The pulse given by Eq. is categorized as a bipolar" one, as the sign of this field component changes periodically.
We emphasize that the initial parameters of the electromagnetic pulse at $\tau =\tau _{0}$ are given under the assumption that the pulse is still located at a sufficient distance from the inhomogeneity layer, where the scaled concentration of conduction electrons is different from $1$. From the experimental viewpoint, the effective optical frequency, $\omega
_{\mathrm{opt}}$, and the characteristic duration of the pulse, $T_{S}$, are relevant parameters characterizing the shape of the electromagnetic pulse [@35].
The optical frequency $\omega _{\mathrm{opt}}$ is determined as follows. As said above, the internal vibrations of breather are represented by the function $\sin \chi $. We represent the argument (see Eq. ) of this function in the form $$\chi =\omega _{c}t_{0}+k_{\mathrm{wave}}(z-z_{0}), \label{22}$$where $k_{\mathrm{wave}}$ is the wave vector. In dimensionless form and using Eq. , we have: $$\chi =\frac{\omega _{0}}{\sqrt{\varepsilon }}\frac{\sigma \Omega }{\sqrt{
1-U^{2}}}t_{0}-U\frac{\sigma \Omega }{\sqrt{1-U^{2}}}(z-z_{0}). \label{23}$$By comparing Eq. and Eq. , we obtain the following expression for the carrier-wave’s frequency: $$\omega _{c}=\frac{\omega _{0}}{\sqrt{\varepsilon }}\frac{\sigma \Omega }{
\sqrt{1-U^{2}}}. \label{24}$$
It must, however, be noticed that frequency $\omega _{c}$ coincides with the frequency at which the optical spectrum reaches its maximum in the slowly-varying-envelope approximation only. For few- and single-cycle pulses, the latter frequency is larger than the former, the ratio between them increasing as the number of cycles decreases, up to $\approx 1.66$. Further, due to the strong nonlinear behavior, the spectrum is not conserved in the course of the propagation, and the frequency at which it reaches its maximum oscillates. According to Eqs. –, i.e., in the framework of the sine-Gordon approximation, the amplitude of the oscillations can reach $\pm 0.14\omega _{c}$.
Duration $T_{S}$ of the electromagnetic pulse (wave packet) with vector potential is determined by the factor $\cosh \mu $. The usual FWHM definition of $T_{S}$ is the time during which the instantaneous amplitude of the running" envelope, measured at a fixed point, exceeds half of its peak value. In the few-cycle regime, the definition of $T_{S}$ should be implemented numerically, using, e.g., the standard deviation. Further, even this standard definition can give rise to ambiguities, as there is some discrepancy between the durations of fields $\Psi $ and $E_{x}$, the ratio of which may become $1.3$ in the single-cycle regime.
Therefore, it is more convenient do define the pulse duration in terms of the slowly-varying-envelope approximation. According to the definition of $\mu $, the role of the characteristic normalized duration of the pulse may be played by$$\tau _{S}=\frac{1}{\sigma U}\sqrt{\frac{1-U^{2}}{1-\Omega ^{2}}}.
\label{tauS}$$The fact that $\tau _{S}\sim 1/U$ at $U\rightarrow 0$ corresponds to the slowly-varying-envelope approximation limit, the expression for the pulse in this limit being $$E_{xB}=E_{0}\frac{\partial \Psi _{B}}{\partial \tau }=4E_{0}\frac{\sigma
\sqrt{1-\Omega ^{2}}}{\sqrt{1-U^{2}}}\left\{ \frac{\cos \chi \cosh \mu }{
\cosh ^{2}\mu +\left( \Omega ^{-2}-1\right) \sin ^{2}\chi }\right\} ,
\label{ESVEA}$$at $\xi =\xi _{0}$, $\zeta =\zeta _{0}$. It is close to the usual sech-shaped pulse, which would be obtained by neglecting the term $\sin
^{2}\chi $ in the denominator: $$E_{xB\mathrm{sech}}=4E_{0}\frac{\sigma \sqrt{1-\Omega ^{2}}}{\sqrt{1-U^{2}}}
\frac{\cos \chi }{\cosh \mu }, \label{Esech}$$but does not coincide with it. Furthermore, an explicit result for the envelope does not follow straightforwardly from Eq. . Therefore, we opt to define the pulse’s width in terms of the sech approximation . In this case, the ratio between the FWHM duration, $T_{S}$, and the half width at the maximum value of the $\mathrm{sech}$ function, which is precisely $\tau_{S}$ in normalized form, is well known to yield the $2\ln (2+\sqrt{3})$ factor.
Based on these considerations, we obtain the following duration of the electromagnetic pulse: $$T_{S}=2\ln (2+\sqrt{3})\frac{\sqrt{\varepsilon }}{\omega _{0}\sigma U}\sqrt{
\frac{1-U^{2}}{1-\Omega ^{2}}}. \label{25}$$To summarize, the pulse’s shape of is fully characterized by the dimensionless speed $U$ and frequency of internal oscillations $\Omega $, which determine the carrier frequency $\omega _{c}$ and characteristic duration $T_{S}$.
The number of cycles in the pulse can then be defined as $N_{p}=T_{S}/T_{c}$, where $T_{c}=2\pi /\omega _{c}$ is the period of the carrier wave. Then, according to Eqs. and , $$N_{p}=\frac{\ln \left( 2+\sqrt{3}\right) \Omega }{\pi \sqrt{1-\Omega ^{2}}U}.
\label{nbp}$$It is thus seen that the parameter $U$ essentially defines the few- or sub-cycle character of the pulse, from the slowly-varying-envelope approximation at $U\rightarrow 0$ to the deeply sub-cycle configuration at $U\rightarrow 1$. Due to the above-mentioned discrepancy between $\omega _{c}$ and the actual maximum of the optical spectrum, $N_{p}$ defined as per Eq. (\[nbp\]) underestimates the number of optical cycles, by approximately $8\% $ in the limit of the single-cycle regime ($N_{p}=1$), which is obtained for $U=0.242$. At $U\rightarrow 1$, $N_{p}$ approaches $0.24$, while the value computed from the maximum of the optical spectrum is $0.46$. Obviously, the concept of the number of cycles is ambiguous in the sub-cycle regime.
In the sub-cycle regime, with $T_{S}<T_{c}$, the oscillations of $\sin \chi $ cannot be considered as a carrier wave anymore. Then, the central frequency of the pulse is mainly determined by the inverse $1/T_{S}$ of its duration, as can be checked by numerically computing the optical spectrum, which amounts to the computation of the Fourier transform of $E_{x}$. Hence, we define the optical frequency as $\omega _{\mathrm{opt}}=\omega _{c}$ if $N_{p}>1$, i.e., if $U<0.242$, and $\omega _{\mathrm{opt}}=2\pi /T_{S}$ otherwise. The corresponding wavelength in vacuum is $\lambda _{\mathrm{opt}}=cT_{c}$ for $U<0.242$ and $\lambda _{\mathrm{opt}}=cT_{S}$ for $U>0.242$. It can easily be checked that the maximum value of field $E_{x}$ is $$E_{\mathrm{max}}=4E_{0}\frac{\sigma \sqrt{1-\Omega ^{2}}}{\sqrt{1-U^{2}}},
\label{eq:Em}$$which increases with $U$, and diverges at $U\rightarrow 1$, hence, $U$ may be also considered as a measure of the pulse’s peak intensity, $I_{p}$. Within the slowly-varying-envelope approximation, $$I_{p}=\frac{c\sqrt{\epsilon }}{8\pi }E_{\mathrm{max}}^{2}=\frac{2c\sqrt{
\epsilon }E_{0}^{2}}{\pi }\frac{\sigma ^{2}(1-\Omega ^{2})}{1-U^{2}}.
\label{eq:Ip}$$Although the actual intensity depends on the wave’s velocity and may differ from this expression, we will use Eq. in the few- and subcycle regimes.
Transmission and reflection coefficients
========================================
As a result of the interaction with the layer of increased electron concentration, the initial electromagnetic pulse is generally split into reflected and transmitted ones. The ratio of the energies of the transmitted and reflected wave packets depends on various parameters, including the initial characteristics of the incident pulse, and also parameters of the scattering layer. To quantify this, we calculate the transmission and reflection coefficients, $K_{\mathrm{pass}}$ and $K_{\mathrm{refl}}$, as per [@35] $$K_{\mathrm{pass}}=\frac{\int_{0}^{+\infty }d\zeta \int_{-\infty }^{+\infty
}d\upsilon \int_{-\infty }^{+\infty }d\xi I(\xi ,\upsilon ,\zeta ,\tau
_{\infty })}{\int_{-\infty }^{+\infty }d\zeta \int_{-\infty }^{+\infty
}d\upsilon \int_{-\infty }^{+\infty }d\xi I(\xi ,\upsilon ,\zeta ,\tau
_{\infty })}, \label{26}$$
$$K_{\mathrm{refl}}=\frac{\int_{-\infty }^{0}d\zeta \int_{-\infty }^{+\infty
}d\upsilon \int_{-\infty }^{+\infty }d\xi I(\xi ,\upsilon ,\zeta ,\tau
_{\infty })}{\int_{-\infty }^{+\infty }d\zeta \int_{-\infty }^{+\infty
}d\upsilon \int_{-\infty }^{+\infty }d\xi I(\xi ,\upsilon ,\zeta ,\tau
_{\infty })}. \label{27}$$
In Eqs. and , $\tau _{\infty }$ corresponds to any time after the establishment of the stable propagation of the electromagnetic pulse, after its interaction with the layer, when the pulse is already at a sufficiently large distance away from it, so that the field energy density in this layer is negligible in comparison to the maximum energy density of the field, i.e., condition $I(\xi ,\upsilon ,0,\tau _{\infty })\ll I_{\mathrm{max}}\equiv \text{max}\left\{ I(\xi ,\upsilon ,\zeta ,\tau _{\infty
})\right\} $ holds. Coefficient $K_{\mathrm{pass}}$, defined as per Eq. , may be interpreted as the ratio of the energy of the wave packet passing the layer of increased electron concentration to the total field energy in the volume of the sample. Similarly, $K_{\mathrm{refl}}$, defined by Eq. , is the share of the energy of the reflected wave packet in the sum of the energies of the transmitted and reflected packets.
The system considered in this paper is conservative since the collisionless approximation is assumed. Therefore, the energy conservation law imposes constraint $$K_{\mathrm{refl}}+K_{\mathrm{pass}}=1. \label{28}$$Note that, when condition $I(\xi ,\upsilon ,0,\tau _{\infty })\ll I_{\mathrm{\ max}}$ is satisfied, and also by virtue of the energy conservation law, coefficients $K_{\mathrm{pass}}$ and $K_{\mathrm{refl}}$ are time-independent.
If the energy of the wave packet propagating in the original direction significantly exceeds the energy of the wave packet reflected from the inhomogeneity layer, i.e., $K_{\mathrm{pass}}\gg K_{\mathrm{refl}}$, we assume that the pulse has passed through this layer. In the opposite case, when the energy of the reflected wave packet significantly prevails over the energy of the transmitted one, i.e., $K_{\mathrm{pass}}\ll K_{\mathrm{refl}}$, we categorize the pulse as reflected. For certain values of the system’s parameters—in particular, at some threshold" value of the initial peak intensity, $I_{p_{\mathrm{thr}}}$, of the incident pulse—it may split in two wave packets with approximately equal energies, which propagate in opposite directions after the interaction with the inhomogeneity layer, thereby corresponding to $K_{\mathrm{pass}}\approx K_{
\mathrm{refl}}$.
Numerical analysis and discussion of the results
================================================
System parameters
-----------------
For modeling, we used the following realistic values of the parameters of CNTs of the zig-zag type $(m,0)$: $m=7$, $\gamma _{0}=2.7$ eV, $b=1.42\times
10^{-8}$ cm, $d_{x}\approx 2.13\times 10^{-8}$ cm, $n_{\mathrm{bias}}=10^{18} $ cm$^{-3}$. We assume that the CNT array is embedded in a dielectric matrix, and the resulting effective dielectric constant of the system is $\varepsilon =4$, the calculation of coefficients $G_{r}$ given by Eq. in the expression for the conduction current density is performed for temperature $T=293$ K (see, for example, Refs. [@30; @35]).
We note that the use of the collisionless approximation (which makes it possible to neglect dissipative effects, as mentioned above) is justified when the evolution time interval is smaller than the relaxation time $t_{\mathrm{\ rel}}$. In the course of the respective time, $t_{\mathrm{rel}}\simeq 10^{-11}$ s, the electromagnetic pulse passes distance $z\leq ct_{\mathrm{rel }}/\sqrt{\varepsilon }\simeq 0.15$ cm.
When simulating the interaction of the pulse with a layer of the increased electron concentration, we vary values of the parameters within a wide range. In particular, parameter $\eta _{\mathrm{imp}}^{\mathrm{max}}=n_{
\mathrm{imp}}^{\mathrm{max}}/n_{\mathrm{bias}}$, corresponding to the maximum concentration of electrons in the inhomogeneity layer, ranges from $2 $ to $100$. Next, the layer’s dimensionless thickness, $\delta \zeta _{\mathrm{imp}}$, is varied between $0.05$ to $0.5$. Parameter $U$ of the electromagnetic pulse approaching the layer is chosen in the interval of $U\in (0.5;0.999)$. We note that, at velocities $U<0.5$, in the course of time $\sim t_{\mathrm{rel}}$, the pulse passes a negligible distance, which is much smaller than its own spatial width along the $\zeta $-axis. Velocities corresponding to $U>0.999$ are not considered here because of limitations imposed by the numerical scheme.
Frequency $\Omega $ of the internal vibrations of the breather-like electromagnetic pulse is varied in the interval of $\Omega \in (0.1;0.9)$. As it decreases, the width of the pulse along the $\zeta $-axis decreases too, although for $\Omega \leq 0.5$ the corresponding change of the pulse’s profile is insignificant. At $\Omega >0.7$, the width of the pulse becomes comparable with dimensions of the numerical grid, which was chosen in accordance with the characteristic size of real samples of CNT arrays. Lastly, transverse width $w_{0}$ of the pulse is varied between $0.5$ and $2.0$, which leads to no qualitative difference in the character of the interaction of the electromagnetic pulse with the region of increased electron concentration.
Scenarios for the interaction of the electromagnetic pulse with a layer of increased electron concentration
-----------------------------------------------------------------------------------------------------------
Equations , , and do not admit analytical solutions, therefore we carried out numerical simulations to study the propagation of the electromagnetic pulse in the CNT array. To solve this system of equations with initial conditions , , and , we used an explicit finite-difference three-layer scheme of the cross" type described in Refs. [@48; @49; @50], which was adapted by us for the 3D model, using the approach developed in Ref. [35]{}. Here, we do not describe it in detail, as the numerical scheme and the computational algorithm are similar to those presented in a detailed form in Ref. [@35] for the 2D geometry. As a result of the computations, wehave found fields $\Psi (\xi ,\upsilon ,\zeta ,\tau )$, $\eta (\xi ,\upsilon
,\zeta ,\tau )$, and $\Phi (\xi ,\upsilon ,\zeta ,\tau )$, and also calculated the distribution of the field energy density at each instant of time, using Eq. .
The simulations reveal that, depending on values of certain system’s parameters, different scenarios of the interaction of the ultrashort pulse with the layer of increased electron concentration are possible. The pulse may either pass the layer or bounce back from it. Control parameters, whose values determine the result of the interaction, are characteristics of the electromagnetic pulse (in particular, the speed at which it is approaching the inhomogeneity layer) and of the layer itself (its thickness and the concentration of conduction electrons in it). Passing the layer is facilitated both by the increase in the peak intensity of the incident pulse, and by the decrease in the thickness of the layer and concentration of electrons in it. In fact, similar scenarios of the interaction of the pulse with the layer are produced by varying all control parameters.
In what follows, we present a number of key results for the propagation of the ultrashort pulse in the inhomogeneous CNT array for different values of parameters of the pulse and inhomogeneity layer. Figures \[fig3\] and [fig4]{} display the results for the ultrashort pulse interacting with the layer at different values of the pulse’s parameter $U$, while other parameters remain constant: the dimensionless frequency of internal oscillations is $\Omega =0.5$, transverse pulse’s width $w_{0}=1.75$, and characteristics of the inhomogeneity layer $\eta _{\mathrm{imp}}^{ \mathrm{max}}=30$, $\delta \zeta _{\mathrm{imp}}=0.1$.
Figure \[fig3\] illustrates the passage of the electromagnetic pulse through the layer of increased electron concentration. The temporal narrowness of the incident pulse and its peak intensity are determined by $U=0.99$. The duration of such a pulse is $T_{S}\simeq 12.8\times 10^{-15}$ s (see Eq. ), which corresponds to the optical frequency $\omega _{\mathrm{opt}}\simeq 4.9\times 10^{14}$ s$^{-1}$, and wavelength $\lambda _{\mathrm{opt}}\simeq 2.8$ $\mu $m, at the lower limit of the mid-infrared band. The maximum amplitude of the electric field of the pulse is $E_{\mathrm{max}}\simeq 8.6\times 10^{4}$ statvolt/cm $\simeq 2.6\times 10^{9}$ V/m, which follows from Eq. . This electric-field amplitude corresponds to a peak intensity $I_{p}\simeq 1.76\times 10^{19}$ erg$\cdot $cm$^{2} \cdot $s$^{-1}\simeq 1.76$ TW/cm$^{2}$.
Figure \[fig3\] shows the distribution of the energy density of the electric field $I(\xi ,\upsilon _{0},\zeta ,\tau )$ (see Eq. ) in the cross-section $\left( \xi ,\zeta \right) $ (at $\upsilon =\upsilon _{0}$), at different values of the dimensionless time, $\tau =\omega _{0}t/\sqrt{
\varepsilon }$. The energy density of the electric field is represented by ratio $I/I_{0}$, using a suitable colormap, with blue and yellow areas corresponding, respectively, to minimum and maximum values. For values of the system’s parameters selected above, the unit along the $\xi $- and $\zeta $-axes corresponds to distance $\simeq 4.2\times 10^{-4}$ cm. Note that we show the distribution of $I/I_{0}$ only in the cross-section $\left(
\xi ,\zeta \right) $ (for $\upsilon =\upsilon _{0}$), as the pattern of the distribution of the energy density in the cross-section $\left( \upsilon
,\zeta \right) $ is quite similar.
![Distribution of the energy density of the electric field $I(\protect\xi ,\protect\upsilon _{0},\protect\zeta ,\protect\tau )$ in the array of nanotubes at various moments of dimensionless time $\protect\tau =\protect\omega _{0}t/\protect\sqrt{\protect\varepsilon }$, in the course of the passage of the laser pulse through the layer of high electron concentration, located at $\protect\zeta =0$: (a) $\protect\tau =0$, (b) $\protect\tau =3.0$ , (c) $\protect\tau =6.0$, (d) $\protect\tau =9.0$. Dimensional coordinates $\protect\xi =x\protect\omega _{0}/c$ and $\protect\zeta =z\protect\omega _{0}/c$ are plotted along the horizontal and vertical axes. Values of $I/I_{0}$ are mapped with the help of the color scale, yellow and blue areas corresponding, respectively, to the maximum and minimum values of the energy density. []{data-label="fig3"}](Figure-3.eps){width="100.00000%"}
It can be seen from Fig. \[fig3\] that the electromagnetic pulse after interacting with the inhomogeneity layer passes it and continues to stably propagate in the medium in the original direction. Note that in this case, only a negligible fraction of the initial electromagnetic pulse is reflected, in the form of a wave packet with a small amplitude propagating in the opposite direction. The transmission and reflection coefficients in this case are $K_{\mathrm{pass}}\approx 0.8464$ and $K_{\mathrm{refl}}\approx 0.1536$, respectively, satisfying relation $K_{\mathrm{pass}}\gg
K_{ \mathrm{refl}}$, which allows us to speak mainly about the passage of the layer of high electron concentration by the pulse.
Figure \[fig4\] shows the reverse situation, namely the reflection of the electromagnetic pulse, with $U=0.80$, from the layer of increased electron concentration. These values of the parameters of the electromagnetic pulse correspond to duration $T_{S}\simeq 6.7\times 10^{-14}$ s, optical frequency $\omega _{\mathrm{opt}}\simeq 9.3\times 10^{13}$ s$^{-1}$, and wavelength $\lambda _{\mathrm{opt}}\simeq 20~\mu $m, in the mid-infrared range, and the maximum of the electric field $E_{\mathrm{max}}\simeq 2.0\times 10^{4}$ statvolt/cm$\simeq 6.1\times 10^{8}$ V/m, which corresponds to peak intensity $I_{p}\simeq 9.75\times 10^{17}$ erg$\cdot $cm$^{2}\cdot $s$^{-1}\simeq 97.5$ GW/cm$^{2}$. Similarly to Fig. \[fig3\], this figure shows the distribution of the energy density of the electric field $I(\xi
,\upsilon _{0},\zeta ,\tau )$ in cross-section $\left( \xi ,\zeta \right) $ of the CNT array at various values of dimensionless time $\tau $.
![The distribution of the energy density of the electric field, $I(
\protect\xi ,\protect\upsilon _{0},\protect\zeta ,\protect\tau )$ in the array of nanotubes at different values of dimensionless time $\protect\tau $ , when the laser pulse is reflected from the layer of increased electron concentration placed at $\protect\zeta =0$: (a) $\protect\tau =0$, (b) $\protect\tau =3.0$, (c) $\protect\tau =6.0$, (d) $\protect\tau =9.0$. The notation is the same as in Fig. \[fig3\].[]{data-label="fig4"}](Figure-4.eps){width="100.00000%"}
The electromagnetic pulse, having an insufficiently high initial peak intensity, does not pass this region, bouncing back and propagating in the reverse direction. The respective transmission and reflection coefficients are $K_{\mathrm{pass}}\approx 0.075$ and $K_{\mathrm{refl}}\approx 0.9925$. Relation $K_{\mathrm{pass}}\ll K_{\mathrm{refl}}$ in this case may be interpreted as satisfying the criterion for the reflection of the laser pulse from the layer of an increased electron concentration.
As shown by the numerical analysis, the possibility of the passage of the electromagnetic pulse through the layer of the high electron concentration depends not only on the peak intensity of the incident pulse, but also on parameters of the layer, such as its thickness, $\delta \zeta _{\mathrm{imp}} $, and the maximum reduced concentration of electrons in it, $\eta _{\mathrm{\ imp}}^{\mathrm{max}}$.
Figure \[fig5\] shows the dependence of the reflection and transmission coefficients of the electromagnetic pulse on parameter $U$ that determines its initial peak intensity, longitudinal width and duration. As $U$ increases, the transmission coefficient $K_{\mathrm{pass}}$ (the solid curve) increases too and, accordingly, the reflection coefficient $K_{\mathrm{refl}}$ (the dotted curve) decreases. The value $U=U_{\mathrm{thr}}$ of $U$ is such that the two curves intersect, thereby implying the equality between the transmission and reflection coefficients: $$K_{\mathrm{pass}}(U_{\mathrm{thr}})=K_{\mathrm{refl}}(U_{\mathrm{thr}}).$$The latter equation fully defines the threshold value $U_{\mathrm{thr}}$. For values of parameters used in Figs. \[fig3\] and \[fig4\], namely $\eta _{\mathrm{imp}}^{\mathrm{max}}=30$ and $\delta \zeta _{\mathrm{imp}}=0.1 $, the threshold is $U_{\mathrm{thr}}\simeq 0.96$, which corresponds to the peak intensity $I_{p_{\mathrm{thr}}}\simeq 4.6\times 10^{18}$ erg$\cdot $cm$^{2}\cdot $s$^{-1}=460$ GW/cm$^{2}$. At $U\simeq U_{\mathrm{thr}}$, the electromagnetic pulse can be divided in two approximately identical wave packets, one of which continues to move in the original direction, while the other bounces back from the layer of increased electron concentration. Calculations show that the transmission coefficient $K_{\mathrm{pass}}$ increases, and the reflection coefficient $K_{\mathrm{refl}}$ decreases with the increase of $U$, while other parameters are kept constant. Further, $K_{\mathrm{pass}}$ decreases, and $K_{\mathrm{refl}}$ increases with the increase of any parameter of the inhomogeneity layer, *viz*., $\eta _{\mathrm{imp}}^{ \mathrm{max}}$ \[see Fig. 5(a)\] and $\delta \zeta _{\mathrm{imp}}$ \[see Fig. 5(b)\], at a fixed value of $U$.
![Transmission and reflection coefficients, $K_{\mathrm{pass}}$ and $K_{\mathrm{refl}}$ (solid and dashed curves, respectively) vs. parameter $U$ , which characterizes the narrowness and intensity strength of the incident electromagnetic pulse when it interacts with the layer of increased electron concentration: (a) for increasing concentration, namely, for $\protect\delta
\protect\zeta _{\mathrm{imp}}=0.1$: [*1*]{} (red) – $\protect\eta _{\mathrm{imp}}^{
\mathrm{max}}=20$, [*2*]{} (green) – $\protect\eta _{\mathrm{imp}}^{\mathrm{max}}=30$ , [*3*]{} (blue) – $\protect\eta _{\mathrm{imp}}^{\mathrm{max}}=40$; and (b) for increasing values of the layer’s thickness, namely, for $\protect\eta _{
\mathrm{imp}}^{\mathrm{max}}=20$: [*1*]{} (red) – $\protect\delta \protect\zeta _{
\mathrm{imp}}=0.10$, [*2*]{} (green) – $\protect\delta \protect\zeta _{\mathrm{imp}
}=0.15$, [*3*]{} (blue) – $\protect\delta \protect\zeta _{\mathrm{imp}}=0.20$.[]{data-label="fig5"}](Figure-5.eps){width="100.00000%"}
Thus, the result of the interaction of the laser pulse with the layer of the increased electron concentration depends on various parameters, including the initial pulse’s parameter $U$, which controls peak intensity $I_{p}$, optical frequency $\omega _{\mathrm{opt}}$, duration $T_{S}$, the maximum reduced electron concentration in the inhomogeneity layer, $\eta _{\mathrm{\
imp}}^{\mathrm{max}}$, and the characteristic thickness of this layer, $\delta \zeta _{\mathrm{imp}}$.
Qualitatively, these dependencies have a simple physical explanation. As can be seen from Eq. , the current density induced by the pulse is proportional to the conduction electrons concentration. Therefore, with the increase of the carrier concentration of conduction electrons. Therefore, with the increase of the carrier concentration in the impurity band, the current that creates the pulse also increases, i.e., the impurity region becomes more conductive. Even at the level of linearized equations, using the Fresnel formulas for the reflection and transmission coefficients, it is clear that a more conducting medium reflects the electromagnetic wave more efficiently. The obtained dependencies on the speed of the pulses may be qualitatively understood too. As the peak intensity increases, the velocity also increases, reducing the time of the interaction with the impurity band, hence the time available for generating the pulse running in the reverse direction is reduced. Indeed, as can be seen from Eq. , the higher the intensity, the higher the pulse velocity is. From Eq. , one can notice that a higher pulse velocity leads to a reduction in the longitudinal width of the pulse, which consequently yields a shorter pulse duration \[Eq. \]. In summary, a decrease in the pulse duration and longitudinal width results in a shorter time interval during which the pulse interacts and passes through the region with an increased electron concentration.
Another explanation for the particular pulse dynamics observed in the presence of the layer of inhomogeneity is that a higher peak intensity of the pulse makes it easier to overcome the effective potential barrier induced by the impurity band. In Refs. [@50a; @50b] an energy-based analysis of the interaction between the pulse and a layer of increased electron concentration in a semiconductor is reported. The energy of the soliton, necessary to overcome the effective potential barrier created by the impurity region, was defined through the effective Lagrangian of the soliton in a vicinity of an impurity, in the framework of a model based on an inhomogeneous (perturbed) wave equation of the sine-Gordon type, similar to the governing equation used in the present work. In Refs. [@50a; @50b] it was established that larger values of the pulse’s velocity—or, equivalently, of the peak intensity—and smaller values of the pulse duration favor the passage of a pulse through the region of increased conductivity.
In connection to these results, it is relevant to compare them with known results obtained for the 1D sine-Gordon equation with impurities [51,52,53,54]{}. Conclusions formulated in those works indicate that, colliding with an inhomogeneity, the breather is either attenuated, due to the emission of linear waves, or splits into a kink-antikink pair (note that an essential part of those above results was obtained analytically). The difference demonstrated by our results is related to two central factors: (i) the three-dimensionality of our problem, which modifies the dispersion law, and (ii) the nonlinearity of our medium, represented by multiple sine terms. Because of the nonintegrability of the present model, our quasi-soliton suffers radiation losses, which, however, become significant for times much larger than those we are considering here, namely, at times when the relaxation effects in the electronic subsystem become significant. The effects of the three-dimensionality, in particular, imply the necessity to redefine the topological charge, which plays a major role in the dynamics of the 1D models. In the 1D case, the topological charge acts as a selection rule", which prohibits certain decay mechanisms. In the present case, these rules do not apply, because of the three-dimensionality. While the breather-like solutions suffer some radiation losses, as mentioned above, it is not seen in the reflection/transmission coefficients, as the integration is carried out throughout the entire spatial domain, taking the contribution from the radiation field into account.
Thus, based on the results produced by our numerical analysis, it can be asserted that the electromagnetic pulses with relatively low peak intensities cannot pass the layer of increased electron concentration, while the pulses with peak intensities significantly exceeding a certain threshold value overcome the repelling layer. As the peak intensity of the pulse increases, the possibility of its passage through the layer increases too. In other words, an increase in the optical frequency and a decrease in the duration of the electromagnetic pulse contribute to the ability of pulse to pass the layer.
The threshold value of parameter $U_{\mathrm{thr}}$ of the electromagnetic pulse, in turn, depends on a number of factors, including parameters $\eta
_{ \mathrm{imp}}^{\mathrm{max}}$ and $\delta \zeta _{\mathrm{imp}}$ of the layer of increased electron concentration. The possibility of the passage of the inhomogeneity layer by the pulse increases with a decrease in these parameters. These findings refine and generalize conclusions obtained in previous works devoted to the study of the interaction of extremely short pulses with layers of an increased electron concentration in CNT arrays, which were based on 2D models [@34; @35]. The system considered in the present work acts as a filter" for extremely short electromagnetic pulses, selectively transmitting narrow ones (with higher optical frequencies), and reflecting pulses of longer durations, with lower frequencies. This effect may be used as a basis for the operation of optical logic elements and laser field control devices, as well as in the technology of nondestructive quality control of electronic elements based on CNTs.
Conclusions
===========
Key results of this work are summarized as follows:
- It has been established that, as a result of the scattering of the electromagnetic pulse on the layer of increased electron concentration in the array of CNTs (carbon nanotubes), both the passage of the pulse through the layer and reflection from it take place.
- The result of the interaction of the electromagnetic pulse with the layer of increased electron concentration depends on values of the system’s parameters, including the speed (determined by the optical frequency and duration) of the pulse, and also on characteristics of the inhomogeneity layer (its thickness and the excess of the conduction electron concentration with respect to the bulk array).
- The increase in the peak intensity (or increase in the optical frequency, or decrease in the duration) of the electromagnetic pulse, as well as the decrease in the thickness of the inhomogeneity layer and concentration of conduction electrons in it, facilitate the passage of the pulse through this layer.
- After interacting with the layer of high electron concentration, the electromagnetic pulse retains its characteristics, remaining an oscillating bipolar wave packet that can propagate steadily over distances that are noticeably larger than its dimensions along the direction of motion.
A. V. Zhukov and R. Bouffanais are financially supported by the SUTD-MIT International Design Centre (IDC). N. N. Rosanov acknowledges the support from the Russian Foundation for Basic Research, Grant 16-02-00762, and from the Foundation for the Support of Leading Universities of the Russian Federation (Grant 074-U01). M. B. Belonenko acknowledges support from the Russian Foundation for Fundamental Research. E. G. Fedorov is grateful to Prof. Tom Shemesh for his generous support. B. A. Malomed appreciates hospitality of the School of Electrical and Electronic Engineering at the Nanyang Technological University (Singapore).
All the authors contributed equally to this work.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: |
We propose an approach for collective enhancement of precision for remote optical lattice clocks and a way of generation of the Einstein-Podolsky-Rosen (EPR) state of remote clocks. In the first scenario a distributed spin squeezed state (SSS) of $M$ clocks is generated by a collective optical quantum nondemolition measurement on clocks with parallel Bloch vectors. Surprisingly, optical losses which usually present the main limitation to SSS can be overcome by an optimal network design which provides close to Heisenberg scaling of the time precision with the number of clocks $M$. We provide an optimal network solution for distant clocks as well as for clocks positioned in close proximity of each other. In the second scenario, we employ collective dissipation to drive two clocks with oppositely oriented Bloch vectors into a steady state entanglement. The corresponding EPR state provides secret time sharing beyond the projection noise limit between the two quantum synchronized clocks protected from eavesdropping. An important application of the EPR entangled clock pair is remote sensing of, for example, gravitational effects and other disturbances to which clock synchronization is sensitive.\
PACS [32.80.Wr, 37.30.+i, 42.50.Ct, 42.62.Fi]{}
author:
- 'Eugene S. Polzik$^1$ and Jun Ye$^2$'
title: 'Entanglement and spin-squeezing in a network of optical lattice clocks'
---
**Introduction.** Optical atomic clocks provide some of the most precise and accurate physical measurements to date [@Bloom2014; @Nicholson2015; @Katori2015; @Chou2010; @Madej2012]. The precision of optical lattice clocks is presently limited by the available frequency stability of the best lasers [@Hinkley2013; @Bishof2013; @Kessler2012], but quantum noise of uncorrelated atoms lures not far below. With enhanced laser stability and improved measurement protocol to reduce the laser noise [@Borregaard2013; @Rosenband2013; @Kohlhaas2015], the next frontier of precision can be advanced by generating entangled states of the clock atoms. For small number of atoms $N$, a maximally entangled clock operating with GHZ states can reach the Heisenberg limit of stability much faster than the best classical schemes [@Kessler2014; @Komar2014]. Spin-squeezed states (SSS) [@wineland1992] are particularly suitable for improving the precision of optical lattice clocks that operate on large $N$ and currently hold the record for clock precision [@Nicholson2015]. Distant clocks connected into a spin squeezed network can provide a higher collective precision for all users.
Spin squeezing (SS) and entanglement of atomic ensembles have so far been experimentally demonstrated for single ensembles of spins associated with atomic states separated by radio- or microwave frequencies. This was achieved by optical quantum nondemolition (QND) measurement [@hald1999; @kuzmich2000; @julsgaard2001; @appel2009; @schleier2010; @Leroux2010; @takano2009; @thompson2014], by mapping squeezed light onto an atomic ensemble [@appel2008], by atomic interactions in a Bose-Einstein condensate [@esteve2008], and by engineered dissipation [@krauter2011]. Improvement to clock precision beyond the quantum projection noise (QPN) was demonstrated for microwave clocks [@appel2009; @Leroux2010].
Networks of remote clocks offer new possibilities for secret time sharing, remote sensing and interferometry that can take advantage of the unprecedented clock precision. A recent proposal outlined probabilistic generation of GHZ-type of entanglement by single photon communication between distant clocks each containing a small number of $n_q$ qubits as discrete quantum variables [@Kessler2014; @Komar2014]. However, for optical lattice clocks containing macroscopic numbers of atoms $N$ we encounter continuous variables such as spin squeezed and EPR entangled states that can be generated deterministically. QND probing on cyclic transitions has been identified as the condition for Heisenberg scaling with $N$ in Ref. [@Saffman2009] and demonstrated for microwave clocks in Ref. [@appel2009; @thompson2014]. However, Heisenberg scaling with the number of clocks $M$ in a chain is problematic as SSS have low tolerance for losses, in particular to losses of the optical channel for optical QND. Here we demonstrate that an optimal design of the network of cavity-enhanced optical clocks allows to keep Heisenberg scaling with $M$ even in the presence of substantial channel losses. Optical lattice clocks with their long coherence times are ideal for generation of such states. In the second part of the Letter we present a scenario where an Einstein-Podolsky-Rosen (EPR) entangled state of two clocks is generated by engineered dissipation. Such clocks feature the ”synchronized time” protected from any eavesdropper and available only for the participants working together. Both proposals are aimed at optical clocks with a macroscopic number of atoms. As a specific example, we show their feasibility for Sr clocks with realistic experimental parameters.
**A network of clocks in a collective squeezed state.**
The ultimate limit of precision for a clock made of $N$ independent atoms is defined by the the angular uncertainty of a coherent spin state (CSS) of the ensemble (quasi-)spin [@itano1993]. CSS is a product state $|\Psi\rangle = \Pi_{i=1}^{N} \frac{1}{\sqrt{2}}
(|1\rangle_i+|2\rangle_i)$ of uncorrelated atoms oriented in the same direction, $J_x=N/2$. The other two projections of $\hat{J}$ have minimal equal variances allowed by the Heisenberg uncertainty relation: $Var(J_z)=Var(J_y) = J_x/2 = N/4$. Introducing quantum correlations between atoms allows to reduce $Var(J_z)$ to below the QPN limit. For the resulting spin squeezed state (SSS), under the condition that $$\begin{aligned}
\xi = {}&
\frac{Var(J_z)}{J^2}N =Var(X_A)\frac{N}{J}<1
\label{eq:wineland},\end{aligned}$$ the atoms become entangled [@sorensenduan2001], and the corresponding signal-to-noise ratio for spectroscopy is improved by the inverse of $\xi$ [@wineland1992], which is the spin squeezing parameter for metrology. The canonical operators, $X_A=J_z/\sqrt{J},P_A=J_y/\sqrt{J}$ obey the commutation relation $[X_A,P_A]=i$ where $J$ is the length of the mean pseudo-spin vector. The quantum noise limited clock precision defined as the minimal detectable angle of spin rotation for the clock sequence is $\sqrt{Var(J_z)}/J=\sqrt{\xi/N}$.
As shown theoretically [@Saffman2009] and experimentally [@appel2009] for Cs clocks, QND probing of the clock levels using cyclic transitions under a high optical depth leads to Heisenberg scaling of precision with $N$. This approach led to the recent demonstration of SSS for Rb ground state [@thompson2014].
The QND interaction $H \propto X_A X_L$ [@RMP] leads to the input-output relation for photonic canonical variables $X_L$ and $P_L$ $$\label{eq:QND}
P^{out}_L=P^{in}_L+\kappa X_A.$$ with $\kappa = \sqrt{d\eta e^{-\eta}}$ the interaction constant, $d$ the resonant single pass optical depth, $\eta=n_{dr}(\gamma/\Delta)^2\sigma/A$ a parameter describing spontaneous emission caused by the probe, $\sigma$ the resonant dipole cross section, $A$ the beam cross section, $\gamma,\Delta$ the natural linewidth and detuning of the optical transition, $n_{dr}$ the photon number in the QND probe. For QND on a cyclic transition the degree of spin squeezing is $$\xi=\frac{1}{e^{-\eta}\, (1+\kappa^2)}.
\label{eq:etaeqn}$$ This results in $\xi_{min}=\sqrt{e}/{\kappa_{opt}^2}= 2e/d\propto N^{-1}$ achieved for $\eta=1/2$ and the Heisenberg scaling of the clock precision $\sqrt{\xi/N}$ with the atom number. Such scaling for a microwave clock probed on two cyclic transitions has been demonstrated in [@appel2009; @anne].
The unique energy level structure of alkaline earth atoms provides an ideal configuration to implement a QND protocol based on a cyclic transition now in an optical clock. As a specific example, we consider an optical lattice clock operating on the $^1$S$_0$ ($|1\rangle$) - $^3$P$_0$ ($|2\rangle$) transition with $N$ atoms placed in an optical resonator. The collective QND readout is performed on the cyclic $^1$S$_0$ ($|1\rangle$) - $^3$P$_1$ ($|3\rangle$) narrow transition ($\sim$7.4 kHz) in Sr using a far detuned probe (Fig. \[Fig:level\](b)). The clock sequence and the details of the QND measurement are outlined in the Supplementary Material.
![**Clock operation and probe**. The atomic level structure for the optical lattice clock. (a) Traditional destructive readout of the clock state populations in $|1\rangle$ and $|2\rangle$. (b) QND probe of the clock transition $|1\rangle - |2\rangle$ using a far detuned probe on $|1\rangle - |3\rangle$. The wavelengths are given for Sr.[]{data-label="Fig:level"}](figure1.pdf){width="1\columnwidth"}
Towards our aim of demonstrating Heisenberg scaling with the number of distant clocks in a clock chain we consider first a single lattice clock placed in an optical resonator (Fig. \[Fig:cavity\](a)) [@Chen2014]. In the following we assume that the detuning of the atom and cavity resonances is much greater than the vacuum Rabi frequency $\Omega$, which in turn is much greater than the atomic ($\gamma$) and the cavity ($\Gamma$) linewidths.
We consider a standing wave cavity with input/output mirror power transmission coefficients $T_1,T_2$, single pass intracavity losses $\mathcal{L}$ and the detuned probe single pass absorption $d_{\Delta}$. The cavity power transmission coefficient on resonance for small $\mathcal{L},d_{\Delta}$ is $4T_1T_2/(T_1+T_2+ 2\mathcal{L}+2d_{\Delta})^2=4T_1T_2/(T_1+T_2+ 2\mathcal{L})^2[1-2Fd_{\Delta}/\pi]$, where $F=2\pi /(T_1+T_2+ 2\mathcal{L})$ is the cavity finesse. Thus, $d_{\Delta}$, as well as the corresponding phase shift, is enhanced by a factor $2F/\pi$. Depending on details of the experimental realization, the optimal measurement is achieved either in reflection from a single-ended over-coupled cavity with $T_1 \gg T_2, \mathcal{L}, d_{\Delta}$ or with a symmetric cavity in transmission. The atomic absorption, $d_{\Delta}=N/{2n}$ at the optimal $\eta=1/2$, can be reduced by using larger $\Delta$ and photon number $n$.
Eq. \[eq:QND\] is modified in the presence of the cavity. With $n$ photons detected during the interaction time the observed probe phase shift consists of two terms: $$\label{eq:phaseshift}
\varphi = n^{-1/2}+ \sqrt{d e^{-\eta_n}} (\gamma/\Delta)\sqrt{\sigma/A}2F/{\pi}X_A.$$ The first term is the shot noise of detected light, and the second term represents the cavity enhanced phase shift. To derive the cavity-based input-output equation we multiply both sides of Eq.\[eq:phaseshift\] with $\sqrt{n}$, $$\begin{aligned}
\label{eq:cavityenhanced}
P^{out}_L = {}& P^{in}_L+\kappa_{cav}X_A \nonumber\\
= {}& 1 + \sqrt{d n e^{-\eta_n}} (\gamma/\Delta)\sqrt{\sigma/A}2F/{\pi}X_A.\end{aligned}$$ Here, $\kappa_{cav}=\sqrt{d n e^{-\eta}} (\gamma/\Delta)\sqrt{\sigma/A}2F/{\pi}=\sqrt{d \eta_n e^{-\eta_n} }2F/{\pi}$ is the cavity enhanced atom-light interaction constant and $\eta_n$ corresponds to the detected photon number $n$. The relation between the spontaneous emission rate in the cavity and in free space is $\eta_{cav}=\eta_n F/\pi$ for small $\mathcal{L}$. Eq. \[eq:etaeqn\] is then modified with substitutions $\eta \rightarrow \eta_{cav}=1/2$ and $\kappa_{opt} \rightarrow \kappa_{cav}=\sqrt{4dF\eta_{cav}e^{-\eta_{cav}}/\pi}=\sqrt{2dFe^{-1/2}/\pi}$ with the optimal value $\eta=1/2$. For the case of large optical depth and/or finesse, $4d \eta_n F^2/{\pi}^2\gg 1$, we arrive at the squeezed spin variance $\xi_{min}=\sqrt{ e}/{\kappa_{cav}^2}= e\pi/(2dF)\propto (FN)^{-1}$ achieved for $\eta_{cav}=1/2$, valid for $dF\gg 1$. The clock precision is then $\sqrt{\xi_{min}/N}=\sqrt{2\pi eA/(\sigma F)}N^{-1}$. Note that our treatment is limited to $ \sigma F/A<2\pi e$, otherwise the Holstein-Primakoff approximation breaks down when the size of the antisqueezed component becomes comparable with $N$.
![**Entangled clock sequence** (a) Cavity QED used for QND probe of the clock states. (b) The eigenfrequency spectrum for $10^3$ Sr atoms distributed in an optical lattice inside a 5 cm cavity with $F=10^5$. The single-atom effective vacuum Rabi frequency is 16 kHz. (c) A cascaded cavity system to entangle multiple independent spin-squeezed clocks.[]{data-label="Fig:cavity"}](figure3.pdf){width="1\columnwidth"}
In a realistic design for cavity QED with Sr atoms [@Bishof2014], we envision $F=10^5$, length of 5 cm, and $N$ = 1000. The atoms in state $|1\rangle$ are collectively coupled to a single mode of this cavity through state $|3\rangle$. The bare cavity mode is dressed by the presence of the $|1\rangle$-atoms and the resonance spectrum is determined by the atom-cavity detuning ($\Delta$) and the collective vacuum Rabi splitting ($\Omega$) that depends on the number of atoms in $|1\rangle$ (Fig. \[Fig:cavity\](b)). The estimated $\Omega=500$ kHz, which is to be compared with $\Gamma=29$ kHz and $\gamma=7$ kHz. The collective cooperativity factor $\Omega^2/(\Gamma\gamma)=d\,F$ = 1200, leading to an estimated 20 dB of metrologically useful spin squeezing. For Sr with a cyclic optical transition, the actual value of $\Delta$ does not play a fundamental role, but $\Delta\gg\Omega$ can be useful if large values of $n$ are desired.
It follows from the above discussion that in the case of a lossless optical channel connecting $M$ identical clocks, a collective QND probe of the whole network leads to the precision that is a factor of $M$ better than each clock, as opposed to uncorrelated clocks for which the precision improves by a factor of $\sqrt{M}$. Figure \[Fig:cavity\](c) shows an optical probe field passing through a chain of successive optical cavities followed by a single quantum measurement performed at the output. Such interaction generates a collective squeezed state of the entire system of $M$ clocks. A channel with finite losses can be accounted for by the substitution $\kappa =\sqrt{4dF\eta e^{-\eta}/\pi} \rightarrow \kappa_i=\sqrt{4dF_i\eta_i e^{-\eta_i }e^{-r_i}/\pi}$, with the probe induced decoherence for the $i$-th clock $\eta_i$, where $e^{-r_i}$ describes the optical channel transmission from the $i$-th clock to the detector (subscript $cav$ omitted for brevity). The noise of the measurement is still the shot noise of the detected probe whereas the signals due to the spin projection from all clocks add up, so that $S/N$ for the chain is $ \sum\limits_{i=1}^M \sqrt{N}\kappa_i e^{-\eta_i/2}=\sum\limits_{i=1}^M \sqrt{4dNF_i\eta_i e^{-2\eta_i}e^{-r_i}/\pi }$. Maximal collective spin squeezing for the chain can be found by optimizing this expression, given the clock parameters and the channel transmission properties. Consider, for example, $M$ clocks connected with a channel with equal transmission $e^{-r}$ between each pair of clocks ($r_i=(M-i)r$ and total channel transmission is $t=e^{-(M-1)r}$). With the optimal value $\eta_i=ne^{r_i}(\gamma/\Delta)^2\sigma/AF_i/\pi=1/2$, the collective S/N becomes $\sqrt{4dNF_M /\pi e}\sum\limits_{i=1}^M e^{-(M-i)r}= \sqrt{4dNF_M /\pi e}(e^{-Mr}-1)/(e^{-r}-1)$ where the fixed value of $\eta_i$ dictates that the cavity finesse is maximal for the last clock in the chain, $F_i=F_M e^{(i-M)r}$.
Assuming a sufficiently dense chain of distant clocks ($r$$\ll$1 but $Mr$$\gg$1), we reach the precision for the chain $(S/N)^{-1}$ = $(4dNF_M /\pi e)^{-1/2}M^{-1}$$\mid$$\ln{t}$$\mid$$ \propto (NM)^{-1}$$\mid$$\ln{t}$$\mid$. The expression in parenthesis is limited to $\ll$$N$ because the size of the antisqueezed quadrature must be $\ll$$N$. Within this limit we obtain Heisenberg scaling of the precision of the chain with both $N$ and $M$ for any given channel transmission $t$. For example, four clocks probed by QND measurement through a channel with $t=e^{-(M-1)r}=0.5$ ($3$ dB total losses) provide precision improvement of $3.1$, and eight clocks in the same channel give the improvement by $6$. If the ultimate performance of each clock dictates an upper limit on $N$ due to, $e.g.$, atomic interactions [@Nicholson2012], a chain of entangled clocks may provide an optimal solution. Distant clocks in a collective SSS may offer an opportunity for testing sensitive relativistic effects [@brukner].
**EPR entangled clocks.** SSS discussed above allows for determination of one of the two quantum projections of the Bloch vector to better than $\sqrt{J/2}$, which is sufficient for improved clock precision. However, for a pair of suitably designed clocks a more intriguing state is possible where *both* projections are defined better than this limit *with respect to each other*. Clock comparison can thus run significantly better than the conventional synchronous mode [@Bize2000; @Takamoto2011; @Nicholson2012]. Such state of two Bloch vectors (spins) is a special case of the EPR state with the entanglement condition $Var(J_{y1}+J_{y2})+ Var(J_{z1}+J_{z2})<2J$ [@RMP]. It can be realized when the mean spins of the two ensembles are oriented in opposite directions, $J=J_{x1}=-J_{x2}$, as demonstrated for collective magnetic spins [@julsgaard]. For optical clocks the requirement of oppositely oriented mean spins means that the two clocks should be initialized in two opposite clock states (Fig. \[Fig:EPRlevels\]).
![**Transitions driving two clocks into an EPR-entangled pair.** Clock 1 and clock 2 are driven with four phase locked classical fields (solid arrows). Forward scattered quantum fields (dashed arrows) generate entanglement corresponding to the two clock Bloch vectors being exactly antiparallel despite their individual quantum noise.[]{data-label="Fig:EPRlevels"}](figure4.pdf){width="1\columnwidth"}
When the Bloch vector describes a pair of states separated by an optical transition, the conventional QND method of generating an EPR state is not applicable because it would require a direct measurement of the oscillations at an optical frequency. However, as demonstrated for magnetic spin oscillators [@wasilewski2009], the EPR state can be generated by a common dissipation process provided by forward scattering of indistinguishable photons that does not involve any measurement. The interaction Hamiltonian between two atomic ensembles and light that generates an EPR state of the atomic operators $b^{\dagger}_1$ and $b^{\dagger}_2$ is $H \propto \mu_1 a^{\dagger}_{+} b^{\dagger}_1+ \nu_1 a^{\dagger}_{-} b_1 + \mu_2 a^{\dagger}_{-} b^{\dagger}_2 + \nu_2 a^{\dagger}_{+} b_2 + h.c.$ The first (last) two terms describe the creation of photon fields $a_+,a_-$ and corresponding creation/annihilation of the collective atomic excitation $b_1$ ($b_2$). For an optical clock these operators correspond to the collective excitation generated in the lower (upper) state of clock 1 (2). Entanglement is generated under the following conditions [@wasilewski2009]: photons scattered from the two clocks into mode $a_+$ are indistinguishable (same for mode $a_ -$) and $\mu_1=\mu_2, \nu_1=\nu_2$.
The challenge of realization of such interaction for an optical clock transition (or for any collective excitation scheme realized on an optical transition) is that due to the selection rules the above conditions are not feasible with a standard Raman transition (four-wave mixing). It turns out, however, that these conditions can be fulfilled using a six-wave mixing process shown for a specific example of Sr optical clock in (Fig. \[Fig:EPRlevels\]). The use of two-photon driving fields (blue, red, black and green solid arrows) allows to fulfill the condition of indistinguishability for photons $a_+$ ($a_-$) emitted by the two ensembles by choosing the two-photon detunings $\delta_P,\delta_D$ to be the same in both clocks and by phase locking of the lasers (solid arrows). The condition $\mu_1=\mu_2, \nu_1=\nu_2$ for scattering amplitudes in the Hamiltonian can be met by tuning the one-photon detunings $\delta_{S1},\delta_{S2},\delta_{P1},\delta_{P2}$. Similar to SS, the degree of entanglement scales with the optical depth thus benefiting from cavity enhancement as well.
An ideal entangled state of this kind corresponds to the two clock Bloch vectors being exactly antiparallel (Fig. \[Fig:EPRlevels\]). This is to be contrasted with the case of SSS where the Bloch vector direction is defined better than QPN only in the plane in which the squeezing axis lies.
The EPR state can be used for secret time sharing analogous to the quantum key distribution. The clock sequence resembles the standard clock sequence (see Fig. \[Fig:sequence\] in Supplementary Material) with an important inset in step (d). At this step the two clock owners randomly choose either to apply or not to apply the $\pi/2$ rotation around $x$ axis. They then publicly exchange the choice with respect to the $\pi/2$ pulses, but not the results of the clock interrogation. The procedure is repeated several times. In close analogy to the quantum key distribution we can use the measurements in which we have made the same choice of rotations for the relative time measurements with high precision. Each of the clock owners acting separately will achieve a much worse precision, compared to QPN, since one half of the EPR state is a noisy thermal state. If an eavesdropping attempt is made, the combined two-clock precision will be compromised as well.
Another attractive feature of the EPR entangled clocks is the improved capability to check any clock disagreement quickly, enabling an efficient approach for characterization of systematic effects of an unknown clock (2) using a well-calibrated clock (1).
Perhaps the most important application of an EPR pair of clocks is for remote sensing. The EPR correlation can be used to map out electromagnetic field from site 1 to site 2 remotely. Namely, one can slave clock 2 to clock 1 by matching the conditions of clock 1 to that of 2. In fact, this might be the best tool to explore gravitation potential-induced decoherence such as described in Ref. [@pikovski], and it can also serve a potentially important role for a future long-baseline atom inteferometer for gravitational wave detection [@Hogan2015].
We thank A. Sørensen, M. Bishof, X. Zhang, S. Bromley, and M. Barrett for discussions. We acknowledge funding from ERC grant INTERFACE, NIST, DARPA, and NSF-JILA PFC. We dedicate this work to H. J. Kimble, as the initial ideas were formed at a celebration of his scientific achievement in April 2014.
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**SUPPLEMENTARY MATERIAL**
**Quantum nondemolition measurement in an optical clock.** We consider an optical lattice clock operating on the $^1$S$_0$ ($|1\rangle$) - $^3$P$_0$ ($|2\rangle$) transition with $N$ atoms placed in an optical resonator. In a normal clock operation, the population of the two clock states is measured destructively by scattering photons with the strong $^1$S$_0$ - $^1$P$_1$ transition (Fig. \[Fig:level\](a)), permitting a QPN-limited probe of the clock transition. A phase sensitive probe based on this strong transition was implemented to enable a less destructive measurement of the state population [@Lodewyck2009]. Here we consider a collective readout of the cyclic $^1$S$_0$ ($|1\rangle$) - $^3$P$_1$ ($|3\rangle$) narrow transition ($\sim$7.4 kHz) in Sr using a far detuned probe (Fig. \[Fig:level\](b)).
An ensemble of $N$ clock atoms can be described by the collective pseudo-spin vector $\hat{J}$ of spin-$1/2$ particles. $J_z$ is defined by the population difference $\Delta N$, such that: $J_z=\frac 1 2 (N_1-N_2)=\Delta N/2$. Atoms are prepared in a superposition of the two clock states by a $\pi /2$ rotation of $\hat{J}$ around the $y$-axis of the Bloch sphere (Fig. \[Fig:sequence\](b)). The population of $|1\rangle$ is measured by a probe detuned by $\Delta$ from the cyclic $|1\rangle - |3\rangle$ transition. Note that this probe does not cause redistribution of the populations between the clock states, hence $J_z$ is conserved and is a true QND variable [@appel2009; @RMP; @Chen2014]. After a $\pi$-pulse is applied to swap the clock states, the population of $|1\rangle$ is measured again. Under this operation the effect of the imprecision of the $\pi/2$ pulse and fluctuations of $N$ are suppressed. This QND probe will introduce a Stark shift of the clock transition consisting of a mean value and a random shift due to the shot noise of the probe (quantum back action of the measurement). The former can be canceled by choosing the detuning $-\Delta$ for the second measurement of the population. We note that the precision of the $\pi$ rotation should be better than $N^{-1/2}$. This sequence results in creation of an SSS shown as an ellipse in Fig. \[Fig:sequence\](c). Squeezing of $J_z$ is then converted into squeezing of the coherence between the clock states through a $\pi/2$ rotation around the $x$-axis (Fig. \[Fig:sequence\](d)). The atomic spin is then let to precess, as in a standard Ramsey sequence (Fig. \[Fig:sequence\](e)). After a certain precession time, a $\pi/2$ rotation around $y$ is applied (Fig. \[Fig:sequence\](f)), where the population measurement noise is reduced by squeezing.
![**Entangled clock sequence**. The sequence of operations of the clock including generation of a spin squeezed entangled state. Details in the text. []{data-label="Fig:sequence"}](figure2.pdf){width="1\columnwidth"}
**Heisenberg scaling for optical clock precision with the number of atoms.** A coherent spin state (CSS) is a product state of individual uncorrelated atoms oriented in the same direction, for example, $J_x=N/2$, with $|\Psi\rangle = \Pi_{i=1}^{N} \frac{1}{\sqrt{2}}
(|1\rangle_i+|2\rangle_i)$. The other two projections of $\hat{J}$ have minimal equal variances allowed by the Heisenberg uncertainty relation: $Var(J_z)=Var(J_y) = J_x/2 = N/4$. These fluctuations, referred to as QPN, and shown as a circle in Fig. \[Fig:sequence\](b), pose a fundamental limit to the precision of the clock operating on $N$ uncorrelated atoms [@itano1993]. Introducing quantum correlations between atoms allows to reduce $Var(J_z)$ to below the QPN limit. For this SSS, under the condition that $$\begin{aligned}
Var(J_z) {}& < \frac{J^2}{N} \nonumber\\
\Leftrightarrow \xi = {}&
\frac{Var(J_z)}{J^2}N =Var(X_A)\frac{N}{J}<1
\label{eq:SMwineland},\end{aligned}$$ the atoms become entangled [@sorensenduan2001], and the corresponding signal-to-noise ratio for spectroscopy is improved by the inverse of $\xi$ [@wineland1992], which is the spin squeezing parameter for metrology. Here we introduce the canonical operators, $X_A=J_z/\sqrt{J},P_A=J_y/\sqrt{J}$, obeying the commutation relation $[X_A,P_A]=i$ where $J$ is the length of the mean pseudo-spin vector, which is the atomic coherence. The quantum noise limited clock precision defined as the minimal detectable angle of spin rotation for the clock sequence is $\sqrt{Var(J_z)}/J=\sqrt{\xi/N}$.
Eq. \[eq:SMwineland\] shows that $\xi$ is determined by the variance of the squeezed component of the spin and by the mean spin $J$. The input-output relation for light, with similarly defined operators of $X_L$ and $P_L$, assuming the QND interaction $H \propto X_A X_L$ [@RMP] is $P^{out}_L=P^{in}_L+\kappa X_A$. To within a factor of unity the interaction constant $\kappa = \sqrt{d\eta e^{-\eta}}$, and $d$ is the resonant single pass optical depth. Furthermore, $\eta=n_{dr}(\gamma/\Delta)^2\sigma/A$ is a parameter describing spontaneous emission caused by the probe, which leads to the reduction of coherence as $J=e^{-\eta}N/2$. $\sigma$ is the resonant dipole cross section, $A$ is the beam cross section, $\gamma,\Delta$ the natural linewidth and detuning of the optical transition, $n_{dr}$ is the photon number in the QND probe. Taking the reduction of $J$ into account, we get the SSS with $\xi=\frac{1}{e^{-\eta}\, (1+\kappa^2)}$. The minimal value $\xi_{min}=\sqrt{e}/{\kappa_{opt}^2}= 2e/d\propto N^{-1}$ is achieved for $\eta=1/2$, and the optimal $\kappa_{opt}=\sqrt{d/2}$ valid for $d>>1$. The precision of the clock $\sqrt{\xi/N}$ then follows the Heisenberg scaling $1/N$.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We prove new fast learning rates for the one-vs-all multiclass plug-in classifiers trained either from exponentially strongly mixing data or from data generated by a converging drifting distribution. These are two typical scenarios where training data are not iid. The learning rates are obtained under a multiclass version of Tsybakov’s margin assumption, a type of low-noise assumption, and do not depend on the number of classes. Our results are general and include a previous result for binary-class plug-in classifiers with iid data as a special case. In contrast to previous works for least squares SVMs under the binary-class setting, our results retain the optimal learning rate in the iid case.'
author:
- 'Vu Dinh$^,$[^1]'
- 'Lam Si Tung Ho$^{,\star}$'
- Nguyen Viet Cuong
- |
\
Duy Nguyen
- 'Binh T. Nguyen'
bibliography:
- 'tamc15.bib'
title: 'Learning From Non-iid Data: Fast Rates for the One-vs-All Multiclass Plug-in Classifiers'
---
Introduction
============
Fast learning of plug-in classifiers from low-noise data has recently gained much attention [@audibert2007fast; @kohler2007ontherate; @monnier2012classification; @minsker2012plug]. The first fast/super-fast learning rates[^2] for the plug-in classifiers were proven by Audibert and Tsybakov [@audibert2007fast] under the Tsybakov’s margin assumption [@tsybakov2004optimal], which is a type of low-noise condition. Their plug-in classifiers employ the local polynomial estimator to estimate the conditional probability of a label $Y$ given an observation $X$ and use it in the plug-in rule. Subsequently, Kohler and Krzyzak [@kohler2007ontherate] proved the fast learning rate for plug-in classifiers with a relaxed condition on the density of $X$ and investigated the use of kernel, partitioning, and nearest neighbor estimators instead of the local polynomial estimator. Monnier [@monnier2012classification] suggested to use local multi-resolution projections to estimate the conditional probability of $Y$ and proved the super-fast rates of the corresponding plug-in classifier under the same margin assumption. Fast rates for plug-in classifiers were also achieved in the active learning setting [@minsker2012plug].
Nevertheless, these previous analyses of plug-in classifiers typically focus on the binary-class setting with iid (independent and identically distributed) data assumption. This is a limitation of the current theory for plug-in classifiers since (1) many classification problems are multiclass in nature and (2) data may also violate the iid data assumption in practice. In this paper, we contribute to the theoretical understandings of plug-in classifiers by proving novel fast learning rates of a multiclass plug-in classifier trained from non-iid data. In particular, we prove that the multiclass plug-in classifier constructed using the *one-vs-all method* can achieve fast learning rates, or even super-fast rates, with the following two types of non-iid training data: data generated from an *exponentially strongly mixing sequence* and data generated from a *converging drifting distribution*. To the best of our knowledge, this is the first result that proves fast learning rates for multiclass classifiers with non-iid data. Moreover, these learning rates do not depend on the number of classes.
Our results assume a multiclass version of Tsybakov’s margin assumption. In the multiclass setting, this assumption states that the events in which the most probable label of an example is ambiguous with the second most probable label have small probabilities. This margin assumption was previously considered in the analyses of multiclass empirical risk minimization (ERM) classifiers with iid data [@Zhang04statistical] and in the context of active learning with cost-sensitive multiclass classifiers [@agarwal2013selective]. Our results are natural generalizations for both the binary-class and the iid data settings. As special cases of our results, we can obtain fast learning rates for the one-vs-all multiclass plug-in classifiers in the iid data setting and the fast learning rates for the binary-class plug-in classifiers in the non-iid data setting. Our results can also be used to obtain the previous fast learning rates [@audibert2007fast] for the binary-class plug-in classifiers in the iid data setting.
In terms of theory, the extension from binary class to multiclass problem is usually not trivial and depends greatly on the choice of the multiclass classifiers. In this paper, our results show that this extension can be achieved with plug-in classifiers and the one-vs-all method. The one-vs-all method is a practical way to construct a multiclass classifier using binary-class classification [@rifkin2004defense]. This method trains a model for each class by converting multiclass data into binary-class data and then combines them into a multiclass classifier.
Our paper considers two types of non-iid data. Exponentially strongly mixing data is a typical case of identically but not independently distributed data. Fast learning from exponentially strongly mixing data has been previously analyzed for least squares support vector machines (LS-SVMs) [@steinwart2009fast; @Hang2014184] and ERM classifiers [@Hang2014184]. On the other hand, data generated from a drifting distribution (or drifting concept) is an example of independently but not identically distributed data. Some concept drifting scenarios and learning bounds were previously investigated in [@bartlett1992learning; @long1999complexity; @barve1996complexity; @mohri2012new]. In this paper, we consider the scenario where the parameters of the distributions generating the training data converge uniformly to those of the test distribution with some polynomial rate.
We note that even though LS-SVMs can be applied to solve a classification problem with binary data, the previous results for LS-SVMs cannot retain the optimal rate in the iid case [@steinwart2009fast; @Hang2014184]. In contrast, our results in this paper still retain the optimal learning rate for the Hölder class in the iid case. Besides, the results for drifting concepts can also achieve this optimal rate. Other works that are also related to our paper include the analyses of fast learning rates for binary SVMs and multiclass SVMs with iid data [@steinwart2007fast; @Shen07general] and for the Gibbs estimator with $\phi$-mixing data [@pierre2014prediction].
Preliminaries
=============
Settings
--------
Let $\{ (X_i,Y_i) \}_{i=1}^n$ be the labeled training data where ${ X_i \in {\mathbb}{R}^d }$ and $Y_i \in \{1, 2, \allowbreak \ldots, m \}$ for all $i$. In the data, $X_i$ is an observation and $Y_i$ is the label of $X_i$. The binary-class case corresponds to $m = 2$, while the multiclass case corresponds to $m > 2$. For now we do not specify how $\{ (X_i,Y_i) \}_{i=1}^n$ are generated, but we assume that test data are drawn iid from an unknown distribution ${\mathbf}{P}$ on ${\mathbb}{R}^d \times \{1,2,\ldots, m\}$. In Section \[sec:strongly-mixing\] and \[sec:concept-drift\], we will respectively consider two cases where the training data $\{ (X_i,Y_i) \}_{i=1}^n$ are generated from an exponentially strongly mixing sequence with stationary distribution ${\mathbf}{P}$ and where $\{ (X_i,Y_i) \}_{i=1}^n$ are generated from a drifting distribution with the limit distribution ${\mathbf}{P}$. The case where training data are generated iid from ${\mathbf}{P}$ is a special case of these settings.
Given the training data, our aim is to find a classification rule $f : {\mathbb}{R}^d \rightarrow \{1, 2, \ldots, m \}$ whose risk is as small as possible. The risk of a classifier $f$ is defined as $R(f) \triangleq {\mathbf}{P}(Y \ne f(X))$. One minimizer of the above risk is the Bayes classifier $f^*(X) \triangleq \arg\max_j \eta_j (X)$, where ${ \eta_j (X) \triangleq {\mathbf}{P}(Y=j|X) }$ for all $j \in \{ 1, 2, \ldots, m \}$. For any classifier ${\widehat}{f}_n$ trained from the training data, it is common to characterize its accuracy via the excess risk ${\mathcal}{E}({\widehat}{f}_n) \triangleq {\mathbf}{E}R({\widehat}{f}_n) - R(f^*)$, where the expectation is with respect to the randomness of the training data. A small excess risk for ${\widehat}{f}_n$ is thus desirable as the classifier will perform close to the optimal classifier $f^*$ on average.
For any classifier $f$, we write $\eta_f (X)$ as an abbreviation for $\eta_{f(X)}(X)$, which is the value of the function $\eta_{f(X)}$ at $X$. Let ${\mathbf}{1}_{\{ \cdot \}}$ be the indicator function. The following proposition gives a property of the excess risk in the multiclass setting. This proposition will be used to prove the theorems in the subsequent sections.
\[prop:excess-risk\] For any classifier ${\widehat}f_n$, we have ${\mathcal}{E}({\widehat}{f}_n) = {\mathbf}{E}\left[\eta_{f^*}(X)-\eta_{{\widehat}f_n}(X) \right]$, where the expectation is with respect to the randomness of both the training data and the testing example $X$.
$R({\widehat}f_n)-R(f^*)$ $$\begin{aligned}
&=& {\mathbf}{P}( Y \ne {\widehat}f_n (X) ) - {\mathbf}{P}(Y \ne f^*(X)) \,\,\,=\,\,\, {\mathbf}{P}(Y = f^*(X)) - {\mathbf}{P}(Y = {\widehat}f_n(X)) \\
&=& {\mathbf}{E}_{X,Y} \left [ {\mathbf}{1}_{\{Y = f^*(X)\}} - {\mathbf}{1}_{\{Y = {\widehat}f_n(X)\}} \right] = {\mathbf}{E}_X \left [ {\mathbf}{E}_Y \left [ {\mathbf}{1}_{\{Y = f^*(X)\}} - {\mathbf}{1}_{\{Y = {\widehat}f_n(X)\}} \big | X \right ]\right ] \\
&=& {\mathbf}{E}_X \left[ \sum_{j=1}^m {\eta_j (X) \left( {\mathbf}{1}_{\{ f^*(X) = j \}} - {\mathbf}{1}_{\{ {\widehat}f_n(X) = j \}} \right) }\right]
\,\,=\,\, {\mathbf}{E}_X \left[ \eta_{f^*}(X) - \eta_{{\widehat}f_n}(X) \right].\end{aligned}$$ Thus, ${\mathcal}{E}({\widehat}{f}_n) = {\mathbf}{E}\left[\eta_{f^*}(X)-\eta_{{\widehat}f_n}(X) \right]$.
Following the settings for the binary-class case [@audibert2007fast], we assume the following Hölder assumption: all the functions $\eta_j$’s are in the Hölder class $\Sigma(\beta, L, {\mathbb}{R}^d)$. We also assume that the marginal distribution ${\mathbf}{P}_X$ of $X$ satisfies the strong density assumption. The definition of Hölder classes and the strong density assumption are briefly introduced below by using the notations in [@audibert2007fast].
For $\beta > 0$ and $L > 0$, the Hölder class $\Sigma(\beta, L, {\mathbb}{R}^d)$ is the set of all functions $g : {\mathbb}{R}^d \rightarrow {\mathbb}{R}$ that are $\lfloor \beta \rfloor$ times continuously differentiable, and for any $x, x' \in {\mathbb}{R}^d$, we have $|g(x') - g_x(x')| \leq L ||x-x'||^\beta$, where $||\cdot||$ is the Euclidean norm and $g_x$ is the $\lfloor \beta \rfloor^{th}$-degree Taylor polynomial of $g$ at $x$. The definition of $g_x$ can be found in Section 2 of [@audibert2007fast].
Fix $c_0, r_0 > 0$ and $0 < \mu_{\text{min}} < \mu_{\text{max}} < \infty$, and fix a compact set ${\mathcal}{C} \subset {\mathbb}{R}^d$. The marginal ${\mathbf}{P}_X$ satisfies the strong density assumption if it is supported on a compact $(c_0,r_0)$-regular set $A \subseteq {\mathcal}{C}$ and its density $\mu$ (w.r.t. the Lebesgue measure) satisfies: $\mu_{\text{min}} \leq \mu(x) \leq \mu_{\text{max}}$ for $x \in A$ and $\mu(x) = 0$ otherwise. In this definition, a set $A$ is $(c_0,r_0)$-regular if $\bm{\lambda} [A \cap B(x,r)] \geq c_0 \bm{\lambda}[B(x,r)]$ for all $0 < r \leq r_0$ and $x \in A$, where $\bm{\lambda}$ is the Lebesgue measure and $B(x,r)$ is the Euclidean ball in ${\mathbb}{R}^d$ with center $x$ and radius $r$.
Margin Assumption for Multiclass Setting {#sec:margin-assumption}
----------------------------------------
As in the binary-class case, fast learning rates for the multiclass plug-in classifier can be obtained under an assumption similar to Tsybakov’s margin assumption [@tsybakov2004optimal]. In particular, we assume that the conditional probabilities $\eta_j$’s satisfy the following margin assumption, which is an extension of Tsybakov’s margin assumption to the multiclass setting. This is a form of low noise assumption and was also considered in the context of active learning to analyze the learning rate of cost-sensitive multiclass classifiers [@agarwal2013selective].
There exist constants $C_0 > 0$ and $\alpha \geq 0$ such that for all $t > 0$, $${\mathbf}{P}_X( \eta_{(1)}(X) - \eta_{(2)}(X) \leq t) \leq C_0 t^\alpha$$ where $\eta_{(1)}(X)$ and $\eta_{(2)}(X)$ are the largest and second largest conditional probabilities among all the $\eta_j(X)$’s.
The One-vs-All Multiclass Plug-in Classifier {#sec:1-vs-all}
============================================
We now introduce the one-vs-all multiclass plug-in classifier which we will analyze in this paper. Let ${\widehat}\eta_n (X) = ({\widehat}\eta_{n,1} (X), {\widehat}\eta_{n,2} (X), \ldots, {\widehat}\eta_{n,m} (X))$ be an $m$-dimensional function where ${\widehat}\eta_{n,j}$ is a nonparametric estimator of $\eta_j$ from the training data. The corresponding multiclass plug-in classifier ${\widehat}f_n$ predicts the label of an observation $X$ by $${\widehat}f_n (X) = \arg\max_j {{\widehat}\eta_{n,j} (X)}.$$ In this paper, we consider plug-in classifiers where ${\widehat}\eta_{n,j}$’s are estimated using the one-vs-all method and the local polynomial regression function as follows. For each class $j \in \{ 1, 2, \ldots, m \}$, we first convert the training data $\{ (X_i,Y_i) \}_{i=1}^n$ to binary class by considering all $(X_i,Y_i)$’s such that $Y_i \ne j$ as negative (label 0) and those such that $Y_i = j$ as positive (label 1). Then we construct a local polynomial regression function ${\widehat}\eta^{{\,\text{LP}}}_{n,j} (x)$ of order $\lfloor \beta \rfloor$ with some appropriate bandwidth $h > 0$ and kernel $K$ from the new binary-class training data (see Section 2 of [@audibert2007fast] for the definition of local polynomial regression functions). The estimator ${\widehat}\eta_{n,j}$ can now be defined as $${\widehat}\eta_{n,j}(x) \triangleq
\begin{cases}
0 & \mbox{if } {\widehat}\eta^{{\,\text{LP}}}_{n,j}(x) \leq 0 \\
{\widehat}\eta^{{\,\text{LP}}}_{n,j}(x) & \mbox{if } 0< {\widehat}\eta^{{\,\text{LP}}}_{n,j}(x) < 1 \\
1 & \mbox{if } {\widehat}\eta^{{\,\text{LP}}}_{n,j}(x) \geq 1
\end{cases}.$$ In order to prove the fast rates for the multiclass plug-in classifier, the bandwidth $h$ and the kernel $K$ of the local polynomial regression function have to be chosen carefully. Specifically, $K$ has to satisfy the following assumptions, which are similar to those in [@audibert2007fast]: $$\exists c > 0 \text{ such that for all } x \in {\mathbb}{R}^d, \text{ we have } K(x) \geq c {\mathbf}{1}_{ \{ ||x|| \leq c \} },$$ $$\int_{{\mathbb}{R}^d} K(u) du = 1, \,\,\, \sup_{u \in {\mathbb}{R}^d}(1 + ||u||^{2 \beta}) K(u) < \infty, \,\,\, \text{and } \int_{{\mathbb}{R}^d}(1 + ||u||^{4 \beta}) K^2(u) du < \infty.$$ Note that Gaussian kernels satisfy these conditions. The conditions for the bandwidth $h$ will be given in Section \[sec:strongly-mixing\] and \[sec:concept-drift\].
Fast Learning For Exponentially Strongly Mixing Data {#sec:strongly-mixing}
====================================================
In this section, we consider the case where training data are generated from an exponentially strongly mixing sequence [@steinwart2009fast; @modha1996minimum]. Let $Z_i = (X_i, Y_i)$ for all $i$. Assume that $\{ Z_i \}_{i=1}^\infty$ is a stationary sequence of random variables on ${\mathbb}{R}^d \times \{ 1, 2, \ldots, m \}$ with stationary distribution ${\mathbf}{P}$. That is, ${\mathbf}{P}$ is the marginal distribution of any random variable in the sequence. For all $k \geq 1$, we define the $\boldsymbol \alpha$-mixing coefficients [@steinwart2009fast]: $$\boldsymbol \alpha(k) \triangleq \sup_{A_1 \in \sigma_1^t, A_2 \in \sigma_{t+k}^\infty, t \geq 1}{|{\mathbf}{P}(A_1 \cap A_2) - {\mathbf}{P}(A_1){\mathbf}{P}(A_2)|}$$ where $\sigma_a^b$ is the $\sigma$-algebra generated by $\{ Z_i \}_{i=a}^b$. The sequence $\{ Z_i \}_{i=1}^\infty$ is exponentially strongly mixing if there exist positive constants $C_1$, $C_2$ and $C_3$ such that for every $k \geq 1$, we have $$\label{eqn:mixing}
\boldsymbol \alpha(k) \leq C_1 \exp(-C_2 k^{C_3}).$$ We now state some key lemmas for proving the convergence rate of the multiclass plug-in classifier in this setting. Let $n_e \triangleq \left \lfloor \frac{n}{\lceil \{8n/C_2\}^{1/(C_3 + 1)} \rceil} \right \rfloor$ be the effective sample size. The following lemma is a direct consequence of Bernstein inequality for an exponentially strongly mixing sequence [@modha1996minimum].
\[lem\_Bernstein\] Let $\{ Z_i \}_{i=1}^\infty$ be an exponentially strongly mixing sequence and $\phi$ be a real-valued Borel measurable function. Denote $W_i = \phi(Z_i)$ for all $i \geq 1$. Assume that $|W_1| \leq C$ almost surely and ${\mathbf}E[W_1] = 0$. Then for all $n \geq 1$ and $\epsilon >0$, we have $${\mathbf}P^{\otimes n} \left( \left |\frac{1}{n} \sum_{i=1}^n W_i \right | \geq \epsilon \right) \leq 2(1+4e^{-2} C_1) \exp \left( - \frac{\epsilon^2 n_e}{2{\mathbf}E|W_1|^2 + 2 \epsilon C / 3} \right),$$ where ${\mathbf}P^{\otimes n}$ is the joint distribution of $\{Z_i\}_{i=1}^n$ and $C_1$ is the constant in Eq. .
The next lemma is about the convergence rate of the local polynomial regression functions using the one-vs-all method. The proof for this lemma is given in Section \[sec:lem\_mixing\_proof\].
\[lem\_mixing\] Let $\beta$, $r_0$, and $c$ be the constants in the Hölder assumption, the strong density assumption, and the assumption for the kernel K respectively. Then there exist constants $C_4, C_5, C_6 > 0$ such that for all $\delta > 0$, all bandwidth $h$ satisfying $C_6 h^\beta < \delta$ and $0<h\leq r_0/c$, all $j \in \{1,2, \ldots, m\}$ and $n \geq 1$, we have $${\mathbf}P^{\otimes n}(|{\widehat}\eta_{n,j}(x) - \eta_j(x)| \geq \delta) \leq C_4 \, \exp(- C_5 n_e h^d\delta^2)$$ for almost surely $x$ with respect to ${\mathbf}{P}_X$, where $d$ is the dimension of the observations (inputs).
Given the above convergence rate of the local polynomial regression functions, Lemma \[lem:rate\] below gives the convergence rate of the excess risk of the one-vs-all multiclass plug-in classifier. The proof for this lemma is given in Section \[lem:rate\_proof\].
\[lem:rate\] Let $\alpha$ be the constant in the margin assumption. Assume that there exist $C_4, C_5 > 0$ such that ${\mathbf}P^{\otimes n}(|{\widehat}\eta_{n,j}(x) - \eta_j(x)| \geq \delta) \leq C_4 \, \exp(-C_5 a_n \delta^2)$ for almost surely $x$ with respect to ${\mathbf}P_X$, and for all $j \in \{1,2, \ldots, m\}$, $\delta > 0$. Then there exists $C_7 > 0$ such that for all $n \geq 1$, $${\mathcal}{E}({\widehat}{f}_n) = {\mathbf}{E}R({\widehat}f_n) - R(f^*) \leq C_7 a_n^{-(1+\alpha)/2}.$$
Using Lemma \[lem\_mixing\] and \[lem:rate\], we can obtain the following theorem about the convergence rate of the one-vs-all multiclass plug-in classifier when training data are exponentially strongly mixing. This theorem is a direct consequence of Lemma \[lem\_mixing\] and \[lem:rate\] with $h=n_e^{-1/(2 \beta +d)}$ and $a_n=n_e^{2\beta/(2 \beta +d)}$.
\[thm\_fastrate\] Let $\alpha$ and $\beta$ be the constants in the margin assumption and the Hölder assumption respectively, and let $d$ be the dimension of the observations. Let ${\widehat}f_n$ be the one-vs-all multiclass plug-in classifier with bandwidth $h=n_e^{-1/(2 \beta +d)}$ that is trained from an exponentially strongly mixing sequence. Then there exists some constant $C_8 >0$ such that for all $n$ large enough that satisfies $0< n_e^{-1/(2 \beta +d)}\leq r_0/c$, we have $${\mathcal}{E}({\widehat}{f}_n) = {\mathbf}{E} R({\widehat}f_n) - R(f^*) \le C_8 n_e^{-\beta(1+\alpha)/(2 \beta+d)}.$$
The convergence rate in Theorem \[thm\_fastrate\] is expressed in terms of the effective sample size $n_e$ rather than the sample size $n$ since learning with dependent data typically requires more data to achieve the same level of accuracy as learning with independent data (see e.g., [@steinwart2009fast; @cuong2013generalization; @ane2008analysis]). However, Theorem \[thm\_fastrate\] still implies the fast rate for the one-vs-all multiclass plug-in classifier in terms of the sample size $n$. Indeed, the rate in the theorem can be rewritten as $O(n^{-\frac{\beta(1+\alpha)}{2\beta + d} \cdot \frac{C_3}{C_3+1}})$, so the fast learning rate is achieved when $2(\alpha-1/C_3)\beta > (1+1/C_3)d$ and the super-fast learning rate is achieved when $(\alpha - 1 - 2/C_3)\beta>d(1+1/C_3)$.
Fast Learning From a Drifting Concept {#sec:concept-drift}
=====================================
In this section, we consider the case where training data are generated from a drifting concept that converges to the test distribution ${\mathbf}{P}$. Unlike the setting in Section \[sec:strongly-mixing\] where the training data form a stationary sequence of random variables, the setting in this section may include training data that are not stationary. Formally, we assume the training data $\{ Z_i \}_{i=1}^n = \{ (X_i,Y_i) \}_{i=1}^n$ are generated as follows. The observations $X_i$ are generated iid from the marginal distribution ${\mathbf}P_X$ satisfying the strong density assumption. For each $i \ge 1$, the label $Y_i$ of $X_i$ is generated from a categorical distribution on $\{1,2,\ldots,m\}$ with parameters $\eta^i (X_i) \triangleq (\eta^i_1 (X_i), \eta^i_2 (X_i), \ldots, \eta^i_m (X_i))$. That is, the probability of $Y_i = j$ conditioned on $X_i$ is $\eta^i_j (X_i)$, for all $j \in \{1,2,\ldots,m\}$.
Note that from our setting, the training data are independent but not identically distributed. To prove the convergence rate of the multiclass plug-in classifier, we assume that $\|\eta^n_j - \eta_j \|_\infty \triangleq \sup_{x \in {\mathbb}{R}^d} |\eta^n_j(x) - \eta_j(x)| = O(n^{-(\beta+d)/(2\beta + d)})$ for all $j$, i.e., $\eta^n_j$ converges uniformly to the label distribution $\eta_j$ of test data with rate $O(n^{-(\beta+d)/(2\beta + d)})$. We now state some useful lemmas for proving our result. The following lemma is a Bernstein inequality for the type of data considered in this section [@yurinskiui1976exponential].
\[lem\_Bernstein\_ind\] Let $\{ W_i \}_{i=1}^n$ be an independent sequence of random variables. For all $i \ge 1$ and $l > 2$, assume ${\mathbf}E W_i = 0$, ${\mathbf}E |W_i|^2 = b_i$, and ${\mathbf}E|W_i|^l \leq b_i H^{l-2} l! /2$ for some constant $H > 0$. Let $B_n \triangleq \sum_{i=1}^n{b_i}$. Then for all $n \ge 1$ and $\epsilon > 0$, we have $${\mathbf}P^{\otimes n} \left( \left| \sum_{i=1}^n{W_i} \right| \ge \epsilon \right) \leq 2 \exp \left( - \frac{\epsilon^2}{2(B_n + H \epsilon)} \right),$$ where ${\mathbf}P^{\otimes n}$ is the joint distribution of $\{ W_i \}_{i=1}^n$.
The next lemma states the convergence rate of the local polynomial regression functions in this setting. The proof for this lemma is given in Section \[sec:lem:approX\_jnd\_proof\]. Note that the constants in this section may be different from those in Section \[sec:strongly-mixing\].
\[lem:approX\_jnd\] Let $\beta$, $r_0$, and $c$ be the constants in the Hölder assumption, the strong density assumption, and the assumption for the kernel K respectively. Let ${\widehat}\eta_{n,j}$ be the estimator of $\eta_j$ estimated using the local polynomial regression function with ${ h = n^{- 1/(2\beta+d)} }$. If ${ \|\eta^n_j - \eta_j\|_\infty = O(n^{-(\beta+d)/(2\beta + d)}) }$ for all $j$, then there exist constants $C_4, C_5, C_6 > 0$ such that for all $\delta > 0$, all $n$ satisfying $C_6 n^{- \beta/(2\beta+d)} < \delta < 1$ and $0< n^{- 1/(2\beta+d)} \leq r_0/c$, and all $j \in \{1,2, \ldots, m\}$, we have $${\mathbf}P^{\otimes n}(|{\widehat}\eta_{n,j}(x) - \eta_j(x)| \geq \delta) \leq C_4 \, \exp(- C_5 n^{2\beta/(2\beta+d)} \delta^2)$$ for almost surely $x$ with respect to ${\mathbf}{P}_X$, where $d$ is the dimension of the observations.
Note that Lemma \[lem:rate\] still holds in this setting. Thus, we can obtain Theorem \[thm\_fastrate\_drift\] below about the convergence rate of the one-vs-all multiclass plug-in classifier when training data are generated from a drifting concept converging uniformly to the test distribution. This theorem is a direct consequence of Lemma \[lem:rate\] and \[lem:approX\_jnd\] with $a_n=n^{2\beta/(2 \beta +d)}$. We note that the convergence rate in Theorem \[thm\_fastrate\_drift\] is fast when $\alpha \beta > d/2$ and is super-fast when $(\alpha-1)\beta>d$.
\[thm\_fastrate\_drift\] Let $\alpha$ and $\beta$ be the constants in the margin assumption and the Hölder assumption respectively, and let $d$ be the dimension of the observations. Let ${\widehat}f_n$ be the one-vs-all multiclass plug-in classifier with bandwidth $h=n^{-1/(2 \beta +d)}$ that is trained from data generated from a drifting concept converging uniformly to the test distribution. Then there exists some constant $C_8 >0$ such that for all $n$ large enough that satisfies $0< n^{-1/(2 \beta +d)}\leq r_0/c$, we have $${\mathcal}{E}({\widehat}{f}_n) = {\mathbf}{E} R({\widehat}f_n) - R(f^*) \le C_8 n^{-\beta(1+\alpha)/(2 \beta+d)}.$$
Remarks
=======
When training data are exponentially strongly mixing, the convergence rate in Theorem \[thm\_fastrate\] is expressed in terms of the effective sample size $n_e$ rather than the sample size $n$. This is due to the fact that learning with dependent data typically requires more data to achieve the same level of accuracy as learning with independent data (see e.g., [@steinwart2009fast; @cuong2013generalization; @ane2008analysis]). However, Theorem \[thm\_fastrate\] still implies the fast rate for the one-vs-all multiclass plug-in classifier in terms of the sample size $n$. Indeed, the rate in the theorem can be rewritten as $O(n^{-\frac{\beta(1+\alpha)}{2\beta + d} \cdot \frac{C_3}{C_3+1}})$, so the fast learning rate is achieved when $2(\alpha-1/C_3)\beta > (1+1/C_3)d$ and the super-fast learning rate is achieved when $(\alpha - 1 - 2/C_3)\beta>d(1+1/C_3)$.
The rates in Theorem \[thm\_fastrate\] and \[thm\_fastrate\_drift\] do not depend on the number of classes $m$. They are both generalizations of the previous result for binary-class plug-in classifiers with iid data [@audibert2007fast]. More specifically, $C_3 = +\infty$ in the case of iid data, thus we have $n_e = n$ and the data distribution also satisfies the condition in Theorem \[thm\_fastrate\_drift\]. Hence, we can obtain the same result as in [@audibert2007fast].
Another important remark is that our results for the one-vs-all multiclass plug-in classifiers retain the optimal rate $O(n^{-\beta(1+\alpha)/(2\beta + d)})$ for the Hölder class in the iid case [@audibert2007fast] while the previous results in [@steinwart2009fast; @Hang2014184] for LS-SVMs with smooth kernels do not (see Example 4.3 in [@Hang2014184]). Besides, from Theorem \[thm\_fastrate\_drift\], the one-vs-all multiclass plug-in classifiers trained from a drifting concept can also achieve this optimal rate. We note that for LS-SVMs with Gaussian kernels, Hang and Steinwart [@Hang2014184] proved that they can achieve the essentially optimal rate in the iid scenario (see Example 4.4 in [@Hang2014184]). That is, their learning rate is $n^{\zeta}$ times of the optimal rate for any $\zeta > 0$. Although this rate is very close to the optimal rate, it is still slower than $\log n$ times of the optimal rate.[^3]
Technical Proofs
================
Proof of Lemma \[lem\_mixing\] {#sec:lem_mixing_proof}
------------------------------
Fix $j \in \{ 1, \ldots, m \}$. Let $Y'_i \triangleq {\mathbf}{1}_{\{ Y_i = j \}}$ be the binary class of $X_i$ constructed from the class $Y_i$ using the one-vs-all method in Section \[sec:1-vs-all\]. By definition of $\eta_j$, note that $\mathbf{P}[Y'_i = 1 | X_i] = \eta_j(X_i)$. Let $\mu$ be the density of ${\mathbf}P_X$. We consider the matrix ${\mathbf}B \triangleq (B_{s_1,s_2})_{|s_1|,|s_2| \leq \lfloor \beta \rfloor}$ with the elements ${ B_{s_1,s_2} \triangleq \int_{{\mathbb}R^d}{u^{s_1+s_2}K(u) \mu(x + h u)du} }$, and the matrix ${\widehat}{{\mathbf}B} \triangleq ({\widehat}B_{s_1,s_2})_{|s_1|,|s_2| \leq \lfloor \beta \rfloor}$ with the elements ${\widehat}B_{s_1,s_2} \triangleq \linebreak \frac{1}{n h^d} \sum_{i=1}^n (\frac{X_i - x}{h})^{s_1+s_2} K(\frac{X_i - x}{h})$, where $s_1, s_2$ are multi-indices in ${\mathbb}N^d$ (see Section 2 of [@audibert2007fast] for details on multi-index). Let $\lambda_{{\mathbf}B}$ be the smallest eigenvalue of ${\mathbf}B$. Then, there exists a constant $c_1$ such that ${ \lambda_{{\mathbf}B} \geq c_1 > 0 }$ (see Eq. (6.2) in [@audibert2007fast]).
Fix $s_1$ and $s_2$. For any $i = 1, 2, \ldots, n$, we define $$T_i \triangleq \frac{1}{h^d} \left( \frac{X_i - x}{h} \right)^{s_1+s_2} K \left( \frac{X_i - x}{h} \right) - \int_{{\mathbb}R^d}{u^{s_1+s_2}K(u) \mu(x + h u) du}.$$ It is easy to see that ${\mathbf}E[T_1] = 0$, $|T_1| \leq c_2 h^{-d}$, and ${\mathbf}E|T_1|^2 \leq c_3 h ^{-d}$ for some $c_2, c_3 > 0$. By applying Lemma \[lem\_Bernstein\], for any $\epsilon > 0$, we have $$\begin{aligned}
{\mathbf}P^{\otimes n}(|{\widehat}B_{s_1,s_2} - B_{s_1,s_2}| \geq \epsilon) &= {\mathbf}P^{\otimes n} \left( \left| \frac{1}{n} \sum_{i=1}^n T_i \right| \geq \epsilon \right) \\
&\leq 2(1+4e^{-2}C_1) \exp \left( - \frac{\epsilon^2 n_e h^d}{2 c_3 + 2 \epsilon c_2 / 3} \right).\end{aligned}$$ Let $\lambda_{{\widehat}{{\mathbf}B}}$ be the smallest eigenvalue of ${\widehat}{{\mathbf}B}$. From Eq. (6.1) in [@audibert2007fast], we have $$\lambda_{{\widehat}{{\mathbf}B}} \geq \lambda_{{\mathbf}B} - \sum_{|s_1|,|s_2| \leq \lfloor \beta \rfloor} |{\widehat}B_{s_1,s_2} - B_{s_1,s_2}|.$$ Let $M$ be the number of columns of ${\widehat}B$. Then, there exists $c_4 > 0$ such that $$\begin{aligned}
\label{eqn:bound1}
{\mathbf}P^{\otimes n}(\lambda_{{\widehat}{{\mathbf}B}} \leq c_1/2) \leq 2(1+4e^{-2} C_1) M^2 \exp(-c_4 n_e h^d).\end{aligned}$$ Let $\eta^x_j$ be the $\lfloor \beta \rfloor^{th}$-degree Taylor polynomial of $\eta_j$ at $x$. Consider the vector ${ {\mathbf}a \triangleq (a_s)_{|s|\leq \lfloor \beta \rfloor} \in \mathbb{R}^M }$ where $a_s \triangleq \frac{1}{n h^d} \sum_{i=1}^n {[Y'_i - \eta^x_j(X_i)] ( \frac{X_i - x}{h} )^s K ( \frac{X_i - x}{h} )}$. Applying Eq. (6.5) in [@audibert2007fast] for ${ \lambda_{{\widehat}{{\mathbf}B}} \geq c_1/2 }$, we have $$\label{eqn:bound2}
|{\widehat}\eta_{n,j}(x) - \eta_j(x)| \leq |{\widehat}\eta^{{\,\text{LP}}}_{n,j}(x) - \eta_j(x)| \leq \lambda_{{\widehat}{{\mathbf}B}}^{-1} M \max_s|a_s| \leq (2 M / c_1) \max_s|a_s|.$$ We also define: $\displaystyle T^{(s,1)}_i \triangleq \frac{1}{h^d} [Y'_i - \eta_j(X_i)] ( \frac{X_i - x}{h} )^s K ( \frac{X_i - x}{h} )$, and $${\hskip 2cm} T^{(s,2)}_i \triangleq \frac{1}{h^d} [\eta_j(X_i) - \eta^x_j(X_i)] ( \frac{X_i - x}{h} )^s K ( \frac{X_i - x}{h} ).$$ Note that ${\mathbf}{E}[T_1^{(s,1)}]=0$, $|T_1^{(s,1)}| \le c_5 h^{-d}$ and ${\mathbf}{E}|T_1^{(s,1)}|^2 \le c_6 h^{-d}$ for some $c_5, c_6 > 0$. Similarly, $|T_1^{(s,2)}-{\mathbf}{E}T_1^{(s,2)}| \le c_7 h^{\beta-d}+ c_8 h^{\beta} \le c_9 h^{\beta - d}$ and ${\mathbf}{E} |T_1^{(s,2)}-{\mathbf}{E}T_1^{(s,2)}|^2 \le c_{10}h^{2\beta-d}$, for some $c_7$, $c_8$, $c_9$, $c_{10} > 0$. Thus, by applying Lemma \[lem\_Bernstein\] again, for any $\epsilon_1, \epsilon_2 > 0$, we have $${\mathbf}P^{\otimes n} \left( \left|\frac{1}{n}\sum_{i=1}^n { T_i^{(s,1)}}\right| \geq \epsilon_1 \right) \le 2(1+4e^{-2}C_1) \exp{\left( - \frac{\epsilon_1^2 n_e h^d}{2 c_6+ 2 c_5 \epsilon_1/3} \right)}, \text{ and}$$ $$\displaystyle {\mathbf}P^{\otimes n} ( | \frac{1}{n}\sum_{i=1}^n{ (T_i^{(s,2)} {\hskip -1mm} - {\mathbf}{E} T_i^{(s,2)}) } | \geq \epsilon_2 ) \le 2(1+4e^{-2}C_1) \exp{\left(\frac{-\epsilon_2^2 n_e h^d}{2 c_{10} h^{2\beta} + 2 c_{9} h^{\beta}\epsilon_2/3}\right)}.$$ Moreover, $|{\mathbf}{E}T_1^{(s,2)}| \le c_{8} h^{\beta}$. By choosing $h^\beta \leq c_1 \delta/(6 M c_{8})$, there exists $c_{11} > 0$ such that $\displaystyle {\mathbf}P^{\otimes n} \left ( |a_s| \geq \frac{c_1 \delta}{2M} \right )$ $$\begin{aligned}
&\leq& {\mathbf}P^{\otimes n} \left ( \left | \frac{1}{n} \sum_{i=1}^n{T_i^{(s,1)}} \right | \geq \frac{c_1 \delta}{6M} \right )
+ {\mathbf}P^{\otimes n} \left ( \left | \frac{1}{n} \sum_{i=1}^n{(T_i^{(s,2)}-{\mathbf}{E}T_i^{(s,2)})} \right | \geq \frac{c_1 \delta}{6M} \right ) \nonumber \\
&\leq& 4 (1+4e^{-2}C_1) \exp(-c_{11}n_e h^d \delta^2).
\label{eqn:bound3}\end{aligned}$$ Let $C_6 = 6 M c_8 / c_1$. By , , and , there exist $C_4, C_5 > 0$ such that $$\begin{aligned}
& & {\mathbf}P^{\otimes n}(|{\widehat}\eta_{n,j}(x) - \eta_j(x)| \geq \delta) \\
&\leq& {\mathbf}P^{\otimes n}(\lambda_{{\widehat}{{\mathbf}B}} \leq c_1/2) + {\mathbf}P^{\otimes n}(|{\widehat}\eta_{n,j}(x) - \eta_j(x)| \geq \delta, \lambda_{{\widehat}{{\mathbf}B}} > c_1/2) \\
&\leq& C_4 \exp(- C_5 n_e h^d \delta^2).\end{aligned}$$ Note that the constants $C_4, C_5, C_6$ can be modified so that they are the same for all $\delta$, $h$, $j$, and $n$. Thus, Lemma \[lem\_mixing\] holds.
Proof of Lemma \[lem:rate\] {#lem:rate_proof}
---------------------------
Since $\eta_{f^*}(x) -\eta_{{\widehat}f_n}(x) \ge 0$ for all $x \in {\mathbb}R^d$, we denote, for any $\delta>0$, $$A_0 \triangleq \{x \in {\mathbb}R^d : \eta_{f^*}(x) - \eta_{{\widehat}f_n}(x) \le \delta\}, \text{ and }$$ $$A_i \triangleq \{x \in {\mathbb}R^d : 2^{i-1}\delta<\eta_{f^*}(x) - \eta_{{\widehat}f_n}(x) \le 2^i \delta\}, \text{ for } i\geq 1.$$ By Proposition \[prop:excess-risk\], ${\mathbf}{E}R({\widehat}f_n) - R(f^*) = {\mathbf}{E} [(\eta_{f^*}(X)-\eta_{{\widehat}f_n}(X)) \, {\mathbf}{1}_{\{{\widehat}f_n(X)\ne f^*(X)\}}]$ $$\begin{aligned}
&=& \sum_{i=0}^{\infty}{{\mathbf}{E}\left[(\eta_{f^*}(X)-\eta_{{\widehat}f_n}(X)) \, {\mathbf}{1}_{\{{\widehat}f_n(X)\ne f^*(X)\}} \, {\mathbf}{1}_{ \{ X \in A_i \} }\right]} \\
&\le& \delta {\mathbf}{P} \left( 0<\eta_{f^*}(X)-\eta_{{\widehat}f_n}(X) \le \delta \right) \\
& & + \sum_{i=1}^{\infty}{{\mathbf}{E}\left[(\eta_{f^*}(X)-\eta_{{\widehat}f_n}(X)) \, {\mathbf}{1}_{\{{\widehat}f_n(X)\ne f^*(X)\}} \, {\mathbf}{1}_{ \{ X \in A_i \} }\right]}.\end{aligned}$$ Let ${{\widehat}\eta}_{n, {\widehat}f_n}(x)$ denote ${{\widehat}\eta}_{n, {\widehat}f_n(x)}(x)$. For any $x$, since ${{\widehat}\eta}_{n, {\widehat}f_n}(x)$ is the largest among ${{\widehat}\eta}_{n, j}(x)$’s, we have ${ \eta_{f^*}(x)-\eta_{{\widehat}f_n}(x) \le |\eta_{f^*}(x) - {{\widehat}\eta}_{n,f^*}(x)| + |{{\widehat}\eta}_{n, {\widehat}f_n}(x) -\eta_{{\widehat}f_n}(x)| }$. For any $i \ge 1$, we have $$\begin{aligned}
&&{\mathbf}{E}\left[(\eta_{f^*}(X)-\eta_{{\widehat}f_n}(X)) \, {\mathbf}{1}_{\{{\widehat}f_n(X)\ne f^*(X)\}} \, {\mathbf}{1}_{ \{ X \in A_i \}}\right]\\
&\le& 2^i \delta \, {\mathbf}{E}\left[ {\mathbf}{1}_{\{ |\eta_{f^*}(X) - {{\widehat}\eta}_{n,f^*}(X)| + |{{\widehat}\eta}_{n, {\widehat}f_n}(X) -\eta_{{\widehat}f_n}(X)| \ge 2^{i-1}\delta\}}~{\mathbf}{1}_{\{0<\eta_{f^*}(X)-\eta_{{\widehat}f_n}(X)<2^i \delta\}}\right]\\
&\le& 2^i \delta \, {\mathbf}{E}_X [{\mathbf}{P}^{\otimes n}( |\eta_{f^*}(X) - {{\widehat}\eta}_{n,f^*}(X)| + |{{\widehat}\eta}_{n, {\widehat}f_n}(X) -\eta_{{\widehat}f_n}(X)| \ge 2^{i-1}\delta) \cdot \\
& & {\hskip 1.2cm} {\mathbf}{1}_{\{0<\eta_{f^*}(X)-\eta_{{\widehat}f_n}(X)<2^i \delta\}} ]\\
&\le& c_1 2^i \delta \exp \left(- c_2 a_n(2^{i-2}\delta)^2\right) ~ {\mathbf}{P}_X (0<\eta_{f^*}(X)-\eta_{{\widehat}f_n}(X)<2^i \delta),\end{aligned}$$ for some $c_1, c_2 > 0$. We have ${\mathbf}{P}_X (0<\eta_{f^*}(X)-\eta_{{\widehat}f_n}(X)<\delta) \le \linebreak {\mathbf}{P}_X [\eta_{f^*}(X)-\eta_{(2)}(X)<\delta]$, and by the margin assumption, for all $t>0$, we get ${\mathbf}{P}_X [\eta_{f^*}(X)-\eta_{(2)}(X)<t] \le C_0 t^{\alpha}$. Therefore, $$\begin{aligned}
& & {\mathbf}{E}\left[(\eta_{f^*}(X)-\eta_{{\widehat}f_n}(X)) \, {\mathbf}{1}_{\{{\widehat}f_n(X)\ne f^*(X)\}} \, {\mathbf}{1}_{ \{ X \in A_i \}}\right] \\
&\le& c_1 C_0 2^{i(\alpha+1)} \delta^{\alpha+1} \exp \left(-c_2 a_n(2^{i-2}\delta)^2\right).\end{aligned}$$ By choosing $\delta=a_n^{-1/2}$, there exists $C_7 > 0$ that does not depend on $n$ and $$\begin{aligned}
{\mathbf}{E}R({\widehat}f_n) - R(f^*) &\le& C_0 a_n^{-(\alpha+1)/2} + ~2 c_1 C_0 a_n^{-(\alpha+1)/2} \sum_{i\ge 1} 2^{i(\alpha+1)/2}\exp(-c_2 2^{2i-4}) \\
&\le& C_7 a_n^{-(\alpha+1)/2}.\end{aligned}$$
Proof of Lemma \[lem:approX\_jnd\] {#sec:lem:approX_jnd_proof}
----------------------------------
The proof for this lemma is essentially similar to the proof for Lemma \[lem\_mixing\] in Section \[sec:lem\_mixing\_proof\], except that we use the Bernstein inequality for iid random variables to bound ${\mathbf}P^{\otimes n}(|{\widehat}B_{s_1,s_2} - B_{s_1,s_2}| \geq \epsilon)$ and thus obtain ${\mathbf}P^{\otimes n}(\lambda_{{\widehat}{{\mathbf}B}} \leq c_1/2) \leq 2 M^2 \exp(-c_4 n h^d)$ as an analogy of Eq. in Section \[sec:lem\_mixing\_proof\]. Besides, Eq. can be obtained in the same way as in Section \[sec:lem\_mixing\_proof\]. To obtain the bound similar to Eq. , we define $$\begin{aligned}
T^{(s,1)}_i &\triangleq& \frac{1}{h^d} [Y'_i - \eta^i_j(X_i)] (\frac{X_i - x}{h})^s K(\frac{X_i - x}{h}) \\
T^{(s,2)}_i &\triangleq& \frac{1}{h^d} [\eta^i_j(X_i) - \eta_j(X_i)] (\frac{X_i - x}{h})^s K(\frac{X_i - x}{h}) \\
T^{(s,3)}_i &\triangleq& \frac{1}{h^d} [\eta_j(X_i) - \eta^x_j(X_i)] (\frac{X_i - x}{h})^s K(\frac{X_i - x}{h}).\end{aligned}$$ Note that ${\mathbf}{E}[T_i^{(s,1)}] = 0$, $|T_i^{(s,1)}| \le c_5 h^{-d}$, and ${\mathbf}{E}|T_i^{(s,1)}|^2 \le c_6 h^{-d}$ for some $c_5, c_6 > 0$. Thus, ${\mathbf}{E}|T_i^{(s,1)}|^l \le (c_5 h^{-d})^{l-2} {\mathbf}{E}|T_i^{(s,1)}|^2 \le H_1^{l-2} {\mathbf}{E}|T_i^{(s,1)}|^2 l!/2$, where $H_1 \triangleq c_5 h^{-d}$ and $l > 2$. Similarly, $|T_i^{(s,2)}-{\mathbf}{E}T_i^{(s,2)}| \le c_7 h^{- d}$ and $\text{Var}[T_i^{(s,2)}] \le c_8 h^{2-d}$ for some $c_7, c_8 > 0$. Thus, ${\mathbf}{E}|T_i^{(s,2)}-{\mathbf}{E}T_i^{(s,2)}|^l \le H_2^{l-2} \text{Var}[T_i^{(s,2)}] l!/2$, for $H_2 \triangleq c_7 h^{- d}$ and $l > 2$. Furthermore, $|T_i^{(s,3)}-{\mathbf}{E}T_i^{(s,3)}| \le c_9 h^{\beta-d}$ and $\text{Var}[T_i^{(s,3)}] \le c_{10}h^{2\beta-d}$ for some $c_9, c_{10} > 0$. Hence, ${\mathbf}{E}|T_i^{(s,3)}-{\mathbf}{E}T_i^{(s,3)}|^l \le H_3^{l-2} \text{Var}[T_i^{(s,3)}] l!/2$ for $H_3 \triangleq c_9 h^{\beta-d}$ and $l > 2$. Thus, from Lemma \[lem\_Bernstein\_ind\], $${\mathbf}P^{\otimes n} (\frac{1}{n}\sum_{i=1}^n{| T_i^{(s,1)}|} \geq \epsilon_1 ) \le 2 \exp (-\frac{n h^d \epsilon_1^2}{2(c_6+ c_5 \epsilon_1)})$$ $${\mathbf}P^{\otimes n} (\frac{1}{n}\sum_{i=1}^n|{ T_i^{(s,2)} - {\mathbf}{E} T_i^{(s,2)}|} \geq \epsilon_2 ) \le 2 \exp ( -\frac{n h^d \epsilon_2^2}{ 2(c_8 h^2 + c_7 \epsilon_2)})$$ $${\mathbf}P^{\otimes n} (\frac{1}{n}\sum_{i=1}^n|{ T_i^{(s,3)} - {\mathbf}{E} T_i^{(s,3)}|} \geq \epsilon_3 ) \le 2 \exp (-\frac{n h^d \epsilon_3^2}{2 (c_{10} h^{2\beta} + c_9 h^{\beta}\epsilon_3 )}),$$ for all $\epsilon_1, \epsilon_2, \epsilon_3 > 0$. Moreover, ${\mathbf}{E}|T_i^{(s,3)}| \le c_{11} h^{\beta}$ for some $c_{11} > 0$, and $\frac{1}{n}\sum_{i=1}^n{{\mathbf}{E}|T_i^{(s,2)}|} \le O(h^{-d} \frac{1}{n}\sum_{i=1}^n{\|\eta^i_j - \eta\|_\infty}) \leq O(h^{-d} \frac{1}{n} \sum_{i=1}^n i^{-(\beta + d)/(2\beta + d)})
\leq O(h^{-d} \frac{1}{n} (1 + \int_{u=1}^n u^{-(\beta + d)/(2\beta + d)} du))
\leq O(h^{-d} n^{-(\beta + d)/(2\beta + d)}) \leq c_{12} h^{\beta}$ for some $c_{12} > 0$ since $h = n^{- 1/(2\beta+d)}$. Thus, we can obtain the new Eq. as ${\mathbf}P^{\otimes n} \left ( |a_s| \geq \frac{c_1 \delta}{2M} \right ) \le 6 \exp(- c_{13} n h^d \delta^2)$ for some $C_6 > 0$ and $c_{13} > 0$. And from the new Eq. , , and , we can obtain Lemma \[lem:approX\_jnd\].
[^1]: These authors contributed equally to this work.
[^2]: Fast learning rate means the trained classifier converges with rate faster than $n^{-1/2}$, while super-fast learning rate means the trained classifier converges with rate faster than $n^{-1}$.
[^3]: The optimal rates in Example 4.3 and 4.4 of [@Hang2014184] may not necessarily be the same as our optimal rate since Hang and Steinwart considered Sobolev space and Besov space instead of Hölder space.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In the present study we have demonstrated epitaxial stabilization of the metastable magnetically-hard $\varepsilon$-Fe$_2$O$_3$ phase on top of a thin MgO(111) buffer layer grown onto the GaN (0001) surface. The primary purpose to introduce a 4nm-thick buffer layer of MgO in between Fe$_2$O$_3$ and GaN was to stop thermal migration of Ga into the iron oxide layer. Though such migration and successive formation of the orthorhombic GaFeO$_3$ was supposed earlier to be a potential trigger of the nucleation of the isostructural $\varepsilon$-Fe$_2$O$_3$, the present work demonstrates that the growth of single crystalline uniform films of epsilon ferrite by pulsed laser deposition is possible even on the MgO capped GaN. The structural properties of the 60nm thick Fe$_2$O$_3$ layer on MgO / GaN were probed by electron and x-ray diffraction, both suggesting that the growth of $\varepsilon$-Fe$_2$O$_3$ is preceded by formation of a thin layer of $\gamma$-Fe$_2$O$_3$. The presence of the magnetically hard epsilon ferrite was independently confirmed by temperature dependent magnetometry measurements. The depth-resolved x-ray and polarized neutron reflectometry reveal that the 10nm iron oxide layer at the interface has a lower density and a higher magnetization than the main volume of the $\varepsilon$-Fe$_2$O$_3$ film. The density and magnetic moment depth profiles derived from fitting the reflectometry data are in a good agreement with the presence of the magnetically degraded $\gamma$-Fe$_2$O$_3$ transition layer between MgO and $\varepsilon$-Fe$_2$O$_3$. The natural occurrence of the interface between magnetoelectric $\varepsilon$- and spin caloritronic $\gamma$- iron oxide phases can enable further opportunities to design novel all-oxide-on-semiconductor devices.'
author:
- Victor Ukleev
- Mikhail Volkov
- Alexander Korovin
- Thomas Saerbeck
- Nikolai Sokolov
- Sergey Suturin
title: 'Stabilization of [$\varepsilon$-Fe$_2$O$_3$]{} epitaxial layer on MgO(111)/GaN via an intermediate $\gamma$-phase'
---
The magnetic-on-semiconductor heterostructures attract a lot of interest nowadays due to the vast opportunities they provide for designing novel functional spintronic devices for magnetic memory applications and bio-inspired computing [@prinz1990hybrid; @ohno1999electrical; @wolf2001spintronics; @yuasa2004giant; @kent2015new; @grollier2016spintronic; @dieny2017perpendicular]. Placing a multiferroic layer with controllable magnetization/polarization in contact with a semiconductor adds the functionality of controlling optical, electronic and magnetic properties of the heterostructure by applied voltage [@scott2007data; @gajek2007tunnel; @ortega2015multifunctional; @hu2017understanding]. One of the rare examples of material with spontaneous room-temperature magnetization and electric polarization is the metastable iron(III) oxide polymorph [$\varepsilon$-Fe$_2$O$_3$]{} [@gich2014multiferroic; @ohkoshi2015nanometer; @katayama2017chemical; @xu2018origin]. Quite recently, the crystalline layers of [$\varepsilon$-Fe$_2$O$_3$]{} have been successfully synthesized on a number of oxide substrates [@gich2010epitaxial; @gich2014multiferroic; @thai2016stabilization; @hamasaki2017crystal; @corbellini2017epitaxially; @viet2018specific] and GaN(0001) [@suturin2018tunable]. The structural and magnetic properties of the iron oxide films drastically depend on the composition of the neighboring buffer layer, the chosen substrate and the growth temperature. The feasibility to synthesize as much as four different iron oxide phases: [$\varepsilon$-Fe$_2$O$_3$]{}, Fe$_3$O$_4$, $\alpha$-Fe$_2$O$_3$ and [$\gamma$-Fe$_2$O$_3$]{} on GaN(0001) by fine adjustment of growth parameters has been recently demonstrated [@suturin2018tunable]. It has been shown that stabilization of the [$\varepsilon$-Fe$_2$O$_3$]{} phase requires elevated growth temperature that leads to formation of a few nanometer-thick Ga-rich magnetically soft transition layer at the interface between the iron oxide film and the GaN substrate [@ukleev2018unveiling]. Later on, a very similar Ga/Fe substitution phenomena have been observed in yttrium iron garnet (YIG) films grown at above 700$^\circ$C onto a gadolinium gallium garnet (GGG) [@suturin2018role]. Although $Pna2_1$ Ga-substituted epsilon-ferrite GaFeO$_3$ is isostructural to [$\varepsilon$-Fe$_2$O$_3$]{} [@abrahams1965crystal] and promotes further growth of the desired phase, its magnetic ordering temperature and coercivity field are somewhat lower than those of [$\varepsilon$-Fe$_2$O$_3$]{} [@katayama2017chemical]. This can potentially reduce the magnetoelectric and magnetooptical performance of the functional devices based on the [$\varepsilon$-Fe$_2$O$_3$]{}/ GaN heterostructures.
In the present study, we have successfully introduced an epitaxial MgO buffer between the [$\varepsilon$-Fe$_2$O$_3$]{} and GaN layers to eliminate Ga migration into the iron oxide film. The resulting structural and magnetic properties of the fabricated heterostructure were probed by complementary x-ray diffraction (XRD), x-ray reflectometry (XRR), vibrating sample magnetometry (VSM), and polarized neutron reflectometry (PNR). An outcome of the epitaxial stabilization of [$\varepsilon$-Fe$_2$O$_3$]{} on the MgO buffer is a technological advantage that provides further opportunities to integrate the promising epsilon ferrite into epitaxial Fe [@goryunov1995magnetic; @klaua2001growth; @yuasa2004giant; @raanaei2008structural; @moubah2016discrete], Fe$_3$O$_4$ [@anderson1997surface; @margulies1997origin; @gao1997growth; @kim1997selective; @voogt1999no], $\alpha$-Fe$_2$O$_3$ [@gao1997growth; @gao1997synthesis; @kim1997selective] and [$\gamma$-Fe$_2$O$_3$]{} [@gao1997growth; @voogt1999no] heterostructures and superlattices grown on MgO substrates.
![(Color online) Atomic force microscopy images of the surface morphology at consecutive growth stages (from bottom to top): GaN, MgO/GaN and [$\varepsilon$-Fe$_2$O$_3$]{}/MgO/GaN.[]{data-label="afm"}](AFM.png){width="6.5cm"}
The substrates used in this work were commercial sapphire Al$_2$O$_3$ (0001) wafers with a 3$\mu$m-thick Ga terminated GaN (0001) layer grown on top by means of metalorganic vapour-phase epitaxy (MOVPE). The GaN surface showed a step-and-terrace surface morphology (Fig. \[afm\]) as confirmed by atomic force microscopy (AFM). The oxide layers were grown by pulsed laser deposition (PLD) from MgO and Fe$_2$O$_3$ targets ablated using a KrF laser. The crystallinity and epitaxial relations of the grown layers were controlled by in-situ high energy electron diffraction (RHEED) reciprocal space 3D mapping. With this technique [@suturin2016] one obtains a 3D reciprocal space map from a sequence of conventional RHEED images taken during the azimuthal rotation of the sample. Thus obtained sequence of the closely spaced spherical cuts through the reciprocal space can be then compiled into a uniform 3D map and shown in the easy interpreted form of planar cuts and projections. The side cuts and plan views of the reciprocal space maps obtained at each growth stage are shown in the same scale in Fig. \[rheed\]. The expected positions of the reciprocal lattice nodes are indicated with circles on the the left halves of the maps.
The 4nm thick MgO layer was deposited onto GaN in 0.02 mbar of oxygen at the substrate temperature of 800$^\circ$C. As confirmed by atomic force microscopy (Fig. \[afm\]), the MgO coverage on GaN is smooth and sufficiently uniform to serve as a diffusion barrier. The epitaxial relations extracted from RHEED are as follows: GaN(0001) $||$ MgO(111); GaN\[1-10\] $||$ MgO$\pm$\[11-2\] (Fig. \[rheed\]). The two possible MgO orientations arise due to the symmetry reduction occuring at the interface: from GaN(0001) C$_6$ to MgO(111) C$_3$. Reflections on the RHEED map of MgO are streaky corresponding to the semi-flat surface.
![(Color online) In-situ reflection high-energy electron diffraction maps obtained at consecutive growth stages: MgO/GaN, [$\gamma$-Fe$_2$O$_3$]{}/MgO/GaN and [$\varepsilon$-Fe$_2$O$_3$]{}/[$\gamma$-Fe$_2$O$_3$]{}/MgO/GaN. Shown in the same scale are the side cuts (top) and plan view projections (bottom) of the reciprocal space. The modeled reflection positions are shown with circles.[]{data-label="rheed"}](RHEED.png){width="6cm"}
{width="14cm"}
A 60nm thick iron oxide layer was grown onto the surface of MgO(111) in 0.2 mbar of oxygen at the substrate temperature of 800$^\circ$C following the approach described in our previous report [@suturin2018tunable]. It was discovered that unlike when grown directly on GaN, the iron oxide layer on MgO nucleates in gamma rather than in epsilon phase. Upon deposition of 3-5nm of iron oxide, the RHEED reciprocal space maps start showing a distinct $2\times2$ pattern of streaks characteristic for the spinel [$\gamma$-Fe$_2$O$_3$]{} lattice (Fig. \[rheed\]) oriented with the \[111\] axis perpendicular to the surface and the \[11-2\] axis parallel to MgO \[11-2\] and GaN\[1-10\]. The diffraction map remains streaky corresponding to the still flat surface.
The preference of the [$\gamma$-Fe$_2$O$_3$]{} over [$\varepsilon$-Fe$_2$O$_3$]{} is naturally related to the cubic symmetry of both lattices. The phase choice mechanisms for the Fe$_2$O$_3$ / MgO(111) system might be similar to those of the Fe$_2$O$_3$ / MgO (001) system where [$\gamma$-Fe$_2$O$_3$]{} is known to be the dominant phase [@gao1997growth; @huang2013epitaxial; @sun2014effect]. It is noteworthy that a thin $\gamma$-like transition layer was also observed during the nucleation of $\alpha$- and [$\varepsilon$-Fe$_2$O$_3$]{} directly on GaN [@suturin2018tunable]. Though the diffraction patterns of that layer bore resemblance to FeO, the spacing between the adjacent (111) layers of oxygen was very similar to [$\gamma$-Fe$_2$O$_3$]{}.
When the total thickness of the iron oxide reaches about 10nm, the $2\times2$ streak pattern gets gradually replaced by the $6\times1$ streak pattern which is an unmistakable fingerprint of the [$\varepsilon$-Fe$_2$O$_3$]{} phase. This pattern persists until the growth is stopped at 60nm of the iron oxide total thickness (Fig. \[rheed\]). The pattern is dotty rather than streaky in agreement with the few nm surface roughness measured by AFM (Fig. \[afm\]). The [$\varepsilon$-Fe$_2$O$_3$]{} lattice is oriented with the polar \[001\] axis perpendicular to the surface and the easy magnetization \[100\] axis parallel to the one of the three equivalent GaN \[1-10\] in-plane directions resulting in three crystallographic domains at $120 ^\circ$ to each other. It is essential that the growth temperature at this stage is no less than 800$^\circ$C otherwise nucleation of [$\varepsilon$-Fe$_2$O$_3$]{} phase does not occur.
To accurately study the crystal structure of the film volume we have applied X-ray diffraction in addition to the surface sensitive RHEED. The XRD measurements were carried out at the BL-3A beamline, KEK Photon Factory (Tsukuba, Japan). The 3D reciprocal space maps were compiled from a series of diffraction patterns taken with a Pilatus 100K two-dimensional detector during a multi-angle rotation performed on a standard 4-circle Euler diffractometer. The linear and planar cuts through the 3D maps obtained across the reciprocal space specular region are shown in Fig. \[xrd\]. The series of [$\varepsilon$-Fe$_2$O$_3$]{} 002$\cdot$N and [$\gamma$-Fe$_2$O$_3$]{} 111$\cdot$N reflections are easily identifiable in addition to the reflections of the underlying Al$_2$O$_3$ and GaN. We do not observe distinctly the reflections of MgO as they considerably overlap with those of [$\gamma$-Fe$_2$O$_3$]{}. Moreover the MgO layer is 15 times thinner than Fe$_2$O$_3$ and has about 1.5 times lower scattering length density for x-rays.
The derived out-of-plane lattice constant of epsilon ferrite $c=9.43$[Å]{} is in agreement with our earlier studies of [$\varepsilon$-Fe$_2$O$_3$]{} / GaN [@suturin2018tunable]. The (111) interplane distance in [$\gamma$-Fe$_2$O$_3$]{} is in agreement with the bulk lattice constant of [$\gamma$-Fe$_2$O$_3$]{} $a=8.33$Å. The in-plane lattice arrangement becomes clear from the analysis of the reciprocal space region containing the off-specular [$\varepsilon$-Fe$_2$O$_3$]{} 20N reflections. The [$\varepsilon$-Fe$_2$O$_3$]{} lattice shows a 1% in-plane expansion towards $a=5.14$Å and $b=8.86$Å. The [$\gamma$-Fe$_2$O$_3$]{} lattice shows a 1.5% lattice expansion towards the equivalent cubic lattice constant of $a=8.47$Å. The in-plane expansion is not surprising taking into account the fact that the in-plane periodicity in GaN is about 8.5% larger than in Fe$_2$O$_3$ [@suturin2018tunable]. The observed in-plane and out-of-plane reflection widths may be used to judge on the strain relaxation and minimal crystallographic domain size in the grown films. The strain relaxation if present would involve a distribution of lattice parameters in the system and would cause reflection broadening that is proportional to the magnitude of the wave vector $Q_z$. Even if such a broadening is present in our system, it is below the experimental resolution as all the observed reflections are of the same shape and width. Such effect can be attributed to the finite size of the coherent crystallographic domains within the crystal lattice and is typical for the nanostructured samples. Measuring the in-plane and out-of-plane reflection widths (see the insets in Fig. \[xrd\]) one can conclude that the minimal coherent domains of [$\varepsilon$-Fe$_2$O$_3$]{} are shaped as (width$\times$height) 14nm $\times$ 35nm columns (in agreement with Ref. [@ukleev2018unveiling]) while those of [$\gamma$-Fe$_2$O$_3$]{} look like 33nm$\times$10nm disks. The reduced coherent thickness of [$\varepsilon$-Fe$_2$O$_3$]{} film suggests that a transition layer with a mixed lattice structure exist at the [$\gamma$-Fe$_2$O$_3$]{}/[$\varepsilon$-Fe$_2$O$_3$]{} interface. The lateral coherence between the the adjacent nucleation sites is substantially reduced because the surface cell of the iron oxides is larger than that of MgO. Compared to [$\gamma$-Fe$_2$O$_3$]{} the coherent domain of [$\varepsilon$-Fe$_2$O$_3$]{} is smaller because of the larger surface cell and the lack of the C$_3$ symmetry. Thus the antiphase boundaries are formed more frequently in [$\varepsilon$-Fe$_2$O$_3$]{}.
The magnetometry measurements were carried out using a Quantum Design PPMS vibrating-sample magnetometer (VSM). The magnetic field was applied in the sample plane along the \[100\] easy magnetization axis of one of the three [$\varepsilon$-Fe$_2$O$_3$]{} domains. Fig. \[squid\] shows the hysteresis loops measured in the temperature range of 5-400K and corrected for the linear diamagnetic contribution of the substrate. The observed values of saturation magnetization were about 130emu/cm$^3$ at $T\,=\,5$K and 100emu/cm$^3$ at $T\,=\,400$K which is consistent with what was reported for [$\varepsilon$-Fe$_2$O$_3$]{} nanoparticles [@jin2004giant] and [$\varepsilon$-Fe$_2$O$_3$]{} thin film grown on SrTiO$_3$ (STO) [@gich2014multiferroic], YSZ [@corbellini2017epitaxially; @knivzek2018spin] and GaN[@ukleev2018unveiling], and predicted from ab-initio calculations [@xu2018origin].
![(Color online) In-plane hysteresis $M(B)$ curves of 70nm-thick [$\varepsilon$-Fe$_2$O$_3$]{}/MgO film measured at 5-400K. Shown are curves (a) as measured and (b) decomposed to the hard and soft components. To express the magnetization in emu/cm$^3$ the curves in (a) are normalized to the expected film thickness of 70nm. The hard and soft component curves in (b) are normalized to the thicknesses of 60nm corresponding to the thickness of [$\varepsilon$-Fe$_2$O$_3$]{} layer and 70nm corresponding to the total thickness of the sample.[]{data-label="squid"}](SQUID1.png "fig:"){width="8.5cm"} ![(Color online) In-plane hysteresis $M(B)$ curves of 70nm-thick [$\varepsilon$-Fe$_2$O$_3$]{}/MgO film measured at 5-400K. Shown are curves (a) as measured and (b) decomposed to the hard and soft components. To express the magnetization in emu/cm$^3$ the curves in (a) are normalized to the expected film thickness of 70nm. The hard and soft component curves in (b) are normalized to the thicknesses of 60nm corresponding to the thickness of [$\varepsilon$-Fe$_2$O$_3$]{} layer and 70nm corresponding to the total thickness of the sample.[]{data-label="squid"}](SQUID2.png "fig:"){width="8.5cm"}
{width="16.5cm"}
The wasp-waist magnetization loops shown in Fig. \[squid\]a are typical for [$\varepsilon$-Fe$_2$O$_3$]{} films and nanoparticles and can be qualitatively decomposed to hard and soft component loops (Fig. \[squid\]b) by subtracting $2M_{soft}/\pi \cdot$ arctan($B$/$B_{soft}$) function with temperature-independent $M_{soft}\,=\,71$ emu/cm$^3$ and $B_{soft}$=62mT. These parameters were unambiguously derived from manual optimization aimed at making the remaining hard component smooth and monotonous in the vicinity of zero magnetic field.
The value of $M_{soft}\,=\,71$ emu/cm$^3$ observed for the soft magnetic component is in general agreement with the presence of [$\gamma$-Fe$_2$O$_3$]{} sublayer buried below the main layer of [$\varepsilon$-Fe$_2$O$_3$]{} as observed by XRD, RHEED and PNR. The magnetization plotted in Fig. \[squid\]b is normalized to the total film thickness of 70 nm. Taking into account the reported values of $M_s$=300-400 emu/cm$^3$ for [$\gamma$-Fe$_2$O$_3$]{} / MgO, the soft loop can be attributed to a layer of [$\gamma$-Fe$_2$O$_3$]{} having thickness of 12-14 nm. This is comparable though slightly higher than the thickness estimated from RHEED and PNR (see the details below).
The hard component hysteresis loops show a large saturation field of 1.2-1.8T characteristic of [$\varepsilon$-Fe$_2$O$_3$]{}. The coercive field gradually increases as the sample is cooled down - from 0.27T at 400K to 0.66T at 5K. The loop shape is typical for the system with three uniaxial domains at 120 deg to each other. At saturation the magnetization is collinear to the field in all three domains $M_s^{sum}=3\cdot M_s$. From saturation to zero field the magnetization gradually decreases to $2/3\cdot M_s^{sum}$ as the the magnetization in the two non collinear domains returns to the equilibrium state at 120deg to the field. From this state the magnetization reversal is gradually completed towards the negative saturation. Notably, the magnetic phase transition to an incommensurate state that is often observed in [$\varepsilon$-Fe$_2$O$_3$]{} nanoparticles, as dramatic shrinkage of the loop at $T\approx100-150$K [@gich2006high; @tseng2009nonzero; @garcia2017unveiling; @ohkoshi2017large], has not been observed in [$\varepsilon$-Fe$_2$O$_3$]{} films - neither on GaN nor on the other substrates. The absence of a sharp phase transition in films can be caused by the variation of the magnetic properties across the film depth. Thus, a temperature-dependent investigation of the depth resolved magnetic structure of [$\varepsilon$-Fe$_2$O$_3$]{} films by neutron or resonant x-ray diffraction is highly desired to address this issue.
The XRR measurement was performed on the Panalytical X’Pert PRO x-ray diffractometer at room temperature using Cu $K_\alpha$ (1.5406Å) radiation to determine the electron scattering length density (SLD) profile $\rho_e$ of the film as a function of distance from the GaN surface $z$. The specular reflectance was measured in the range of incident angles between 0.5 to 3.5degrees covering the $Q_z$ range from 0.075 to 0.5Å$^{-1}$.
The neutron reflectometry experiments were performed at the D17 setup [@saerbeck2018recent; @illdata2018] (ILL, Grenoble, France) in polarized time-of-flight mode. Sample temperature and magnetic field were controlled by an Oxford Instruments 7T vertical field cryomagnet equipped with single-crystalline sapphire windows. Neutrons with wavelengths of $4-16$Å were used to ensure the constant polarization of $P_0\,>\,99\%$. Three different incident angles (0.8, 1.5 and 3.7degrees) were chosen to access the $Q_z$ range from 0.017 to 0.17Å$^{-1}$. Intensity of the reflected beam was collected by two-dimensional $^3$He position-sensitive detector. The data was integrated using a method taking into account the sample curvature or beam divergence [@saerbeck2018recent; @cubitt2015improved]. Non-spin-flip reflectivities $R^+$ and $R^-$, where +(-) denotes the incident neutron spin alignment parallel (antiparallel) to the direction of applied magnetic field, were acquired without polarization analysis. The detailed description of the reflectometry techniques can be found elsewhere [@ankner1999polarized; @zabel2008polarized].
Figure \[xrrpnr\]a shows x-ray reflectivity (room temperature) and neutron reflectivity ($T=5$K) curves plotted as a function of momentum transfer $Q_z$. The neutron reflectivity curves were measured at the characteristic characteristic points of the $M(B)$ loop marked as ($1-4$) in Fig.\[squid\]. The PNR curves shown in Fig. \[xrrpnr\]a are measured in applied magnetic fields of $B=0.025$T (state 1 in remanence) and $B=2$T (state 3 in saturation). The XRR and PNR curves were simultaneously fitted using GenX software [@bjorck2007genx]. The simplest model, for which the fitting routine converges, corresponds to a stack consisting of the GaN substrate, the MgO buffer, the transition iron oxide layer with an unspecified density and the main [$\varepsilon$-Fe$_2$O$_3$]{} layer. The depth-profiles of the x-ray ($\rho_e$) and nuclear neutron ($\rho_n$) scattering length densities (SLDs) extracted from the refined model are shown in Fig. \[xrrpnr\]b. The profiles reflect the chemical composition and density of the layers as well as the structural roughness of the interfaces. The root mean square (RMS) roughness of all the interfaces is below 15Å. Notably, we observe the transition layer at the iron oxide/MgO interface with thickness of $105\pm10$Å and reduced x-ray and neutron nuclear SLDs compared to the main [$\varepsilon$-Fe$_2$O$_3$]{} volume of the film. This looks natural as [$\gamma$-Fe$_2$O$_3$]{} having the same chemical formula as [$\varepsilon$-Fe$_2$O$_3$]{} is by 3.4% less dense due to the presence of iron vacancies in the inverted spinel structure. The comparably low SLD of the MgO layer gives a few nm wide reduction of $\rho_e$ and $\rho_n$ located on the SLD profile at $z=0$.
The magnetization profile of the heterostructure is encoded in the dependence of the spin-asymmetry ratio ($R^+$-$R^-$)/($R^+$+$R^-$) on $Q_z$. Fitting it against the model gives the depth profile of the magnetic contribution to the neutron SLD $\rho_m$ (Å$^{-2}$) which can be converted to magnetization $M$ (emu/cm$^3$) using the following formula: $M=\,3505\cdot 10^5 \cdot \rho_m$ [@zhu2005modern]. The measured and fitted spin-asymmetry ratios are shown in Fig. \[xrrpnr\]c for the two magnetic states 2 and 3 on the lower branch of the hysteresis loop (see Fig. \[squid\]): with partially switched magnetization ($B=+0.5$T) and in full saturation ($B=+2$T). The fitted model suggests that the iron oxide film is divided into two magnetically different sub-systems: the main [$\varepsilon$-Fe$_2$O$_3$]{} layer with a saturation magnetization of $M_{s1} \approx 56$emu/cm$^3$ and an interfacial layer with $M_{s2}\approx70$emu/cm$^3$ (Fig. \[xrrpnr\]d). Using the PNR data obtained at 5K we are able to track the magnetization behavior of individual sublayers as the system is magnetized from the negative remanence (state 1) to full saturation (state 3) and back to the positive remanence (state 4). As shown in (Fig. \[xrrpnr\]d) the magnetization of the softer interface layer is switched between $B=0.025$T (state 1) and $B=0.5$T (state 2) and reaches saturation of 70emu/cm$^3$ at $B=2$T. The magnetization of the much harder [$\varepsilon$-Fe$_2$O$_3$]{} layer switches somewhere between $B=0.5$T (state 2) and $B=2$T (state 3). As the magnetically hard component of the hysteresis loop is not completely closed in the maximum applied positive of 2T (Fig. \[squid\]b), the PNR curves measured at $B=2$T (state 3) and $B=0.025$T (state 4) belong to the minor branch of the hysteresis. Magnetization of 56emu/cm$^3$ is found at $B=2$T, which is slightly smaller that the saturation moment. Going back to positive remanence of the minor loop (state 4), the magnetization of both interface and bulk layers start slowly decreasing (faster for the interface layer).
Sequential switching of interface $\gamma$- and main $\varepsilon$- layers in principle reflects a step-like shape of the hysteresis loops observed by VSM magnetometry (Fig. \[squid\]). It must be noted that the maximum magnetization for [$\varepsilon$-Fe$_2$O$_3$]{} layer derived from PNR is about twice lower than the highest reported values for [$\varepsilon$-Fe$_2$O$_3$]{} but in good agreement with the maximum magnetization observed in the decomposed VSM loop shown in Fig. \[squid\]b. The maximum magnetization of the [$\gamma$-Fe$_2$O$_3$]{} layer derived from PNR is about 5 times lower than the expected 300-400emu/cm$^3$ reported for [$\gamma$-Fe$_2$O$_3$]{}/MgO layers [@gao1997growth; @huang2013epitaxial; @sun2014effect], and cannot completely explain the soft-magnetic component observed by VSM. Magnetic degradation of the transition [$\gamma$-Fe$_2$O$_3$]{} layer can be possibly explained by the size effect [@orna2010origin], epitaxial strain [@yang2010strain; @bertinshaw2014element; @gibert2015interfacial] or large number of the antiphase boundaries [@rigato2007strain; @ramos2009artificial] between the nano-columns in the plane of the layer and at the interface with main [$\varepsilon$-Fe$_2$O$_3$]{} film.
The much higher magnetization of the soft magnetic component observed in VSM suggests that another soft magnetic phase is likely present in the sample that cannot be distinguished in the PNR experiment. Similar effect was also observed in [$\varepsilon$-Fe$_2$O$_3$]{} grown directly on GaN [@ukleev2018unveiling]. The most plausible candidates are homogeneously distributed minor fractions of polycrystalline [$\gamma$-Fe$_2$O$_3$]{} and Fe$_3$O$_4$ [@jin2005formation; @lopez2016growth; @corbellini2017effect] not pronounced in XRD data. Again, one must also take into account the columnar structure of the [$\varepsilon$-Fe$_2$O$_3$]{} films containing considerable concentration of the antiphase boundaries. As was pointed out in Ref. [@sofin2011anomalous] the antiphase boundaries in iron oxides may account for the soft magnetic behavior. The magnetic moments located in minor phase fractions of small volume, or at the antiphase boundaries in the sample plane that cannot be resolved with PNR, which is a laterally averaging technique, because the disordered moments at boundaries and minor phase fractions are highly diluted, but integrated into the magnetization measured by VSM. We suggest that the deposition of small ($\mu$m-scale) iron particulates ejected from the PLD target is the most plausible scenario, that have been also observed for other PLD films [@haindl2016situ; @zhai2018weak; @grant2018particulate].
In conclusion, we have demonstrated the possibility to epitaxially grow single crystal [$\varepsilon$-Fe$_2$O$_3$]{} thin film on MgO(111) surface by pulsed laser deposition. In contrast to the previously investigated non-buffered [$\varepsilon$-Fe$_2$O$_3$]{}/GaN(0001) system, where the interfacial GaFeO$_3$ magnetically degraded layer was reported to form due to Ga diffusion[@ukleev2018unveiling] from GaN, the [$\varepsilon$-Fe$_2$O$_3$]{} / MgO / GaN system has advantage of exploiting the diffusion blocking MgO barrier. Though formation of the orthorhombic GaFeO$_3$ was supposed earlier to be a potential trigger of the nucleation of the isostructural [$\varepsilon$-Fe$_2$O$_3$]{}, the present work demonstrates that the growth of single crystalline uniform films of epsilon ferrite by pulsed laser deposition is possible even without the aid of Ga. Still the aid of Ga seems important as on GaN the [$\varepsilon$-Fe$_2$O$_3$]{} layer could be nucleated with a transition layer of few angstrom thickness while on MgO the growth of [$\varepsilon$-Fe$_2$O$_3$]{} film is preceded by nucleation of a 10nm thick layer of another iron oxide phase. A complimentary combination of electron and x-ray diffraction, x-ray reflectometry and polarized neutron reflectometry techniques allowed unambiguous identification of this phase as $P4_132$ ($P4_332$) cubic [$\gamma$-Fe$_2$O$_3$]{}. This phase is known to show magnetoelectric functionality [@cheng2016enhancements] and spin Seebeck effect [@jimenez2017spin] and can enable further opportunities to design the novel all-oxide heterostructure magnetoelectric and spin caloritronic devices.
We are grateful to Institut Laue-Langevin for provided neutron and x-ray reflectometry beamtime (proposal No.: 5-54-244 [@illdata2018]). Synchrotron x-ray diffraction experiment was performed at KEK Photon Factory as a part of the proposal No. 2018G688. We thank Dr. Tian Shang for the assistance with magnetization measurements. The part of the study related to PNR and XRR was partially supported by SNF Sinergia CRSII5-171003 NanoSkyrmionics. The part of the study related to growth technology and diffraction studies was supported by Russian Foundation for Basic Research grant 18-02-00789. The open access fee was covered by FILL2030, a European Union project within the European Commission’s Horizon 2020 Research and Innovation programme under grant agreement No. 731096.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'By reaching through shrouding blastwaves, efficiently discovering off-axis events, and probing the central engine at work, gravitational wave (GW) observations will soon revolutionize the study of gamma-ray bursts. Already, analyses of GW data targeting gamma-ray bursts have helped constrain the central engines of selected events. Advanced GW detectors with significantly improved sensitivities are under construction. After outlining the GW emission mechanisms from gamma-ray burst progenitors (binary coalescences, stellar core collapses, magnetars, and others) that may be detectable with advanced detectors, we review how GWs will improve our understanding of gamma-ray burst central engines, their astrophysical formation channels, and the prospects and methods for different search strategies. We place special emphasis on multimessenger searches. To achieve the most scientific benefit, GW, electromagnetic, and neutrino observations should be combined to provide greater discriminating power and science reach.'
address:
- '$^{1}$Department of Physics & Columbia Astrophysics Laboratory, Columbia University, New York, NY 10027, USA'
- '$^{2}$Center for Gravitation and Cosmology, University of Wisconsin-Milwaukee, Milwaukee, WI 53211, USA'
author:
- 'I. Bartos$^{1}$, P. Brady$^{2}$, S. Márka$^{1}$'
title: 'How Gravitational-wave Observations Can Shape the Gamma-ray Burst Paradigm'
---
Introduction
============
Gamma-ray bursts (GRBs) are the brightest electromagnetic explosions in the Universe (e.g., [@Meszaros2012]). For the short time of their activity, they outshine all other sources in the sky in gamma rays. These energetic explosions originate from cataclysmic cosmic events, whose “inner engine" that drives the observed emission is confined to volumes mere tens of kilometers across, as indicated by the duration and variability of gamma-ray emission (e.g., [@1999PhR...314..575P]). This inner engine is hidden from direct electromagnetic observations, even though the variability of the emission is directly related to the engine’s activity. Gravitational waves (GWs), on the other hand, carry information directly from the inner engine to the observer.
With the near completion of advanced GW detectors [@2010CQGra..27h4006H; @2011CQGra..28k4002A; @2006CQGra..23S.207W; @2011IJMPD..20.1755K], we will soon be able to directly probe the dynamical processes leading to the creation of GRBs, which are less (or not) accessible via other messengers. The Advanced LIGO [@2010CQGra..27h4006H] and Virgo [@AdV] detectors are planned to begin observation in 2015, albeit initially below design sensitivity, gradually reaching their design sensitivity around the end of the decade [@LVCcommissioning]. Upon reaching their design sensitivity, the Advanced LIGO and Virgo detectors are expected to be about 10 times more sensitive (at their most sensitive frequency band around $\sim150\,$Hz) than the initial LIGO/Virgo detectors. Further, advanced detectors will have a wider sensitive frequency band, reaching down to $\sim10\,$Hz. This will improve their sensitivity to wide-band sources such as compact binaries. The global GW detector network will further increase its reach in both distance and source direction reconstruction by the addition of further detectors [@2012arXiv1210.6362N], such as the Japanese KAGRA (formerly LCGT; [@2011IJMPD..20.1755K]) detector that is under construction, as well as the planned third LIGO observatory in India [@LVCcommissioning].
There are a number of mechanisms conjectured to result in a GRB. Potential mechanisms include the formation of a central object surrounded by an accretion disk from the merger of binary neutron stars [@1984PAZh...10..422B; @1986ApJ...308L..43P; @1989Natur.340..126E] or a neutron star and a low-mass black hole [@2007PhR...442..166N], the collapse of the core of a massive star [@Wo:93; @2006ARAA..44..507W; @hjorth:11], the global reconfiguration of the magnetic fields in magnetized neutron stars [@1992ApJ...392L...9D; @1992AcA....42..145P; @2007PhR...442..166N], the accretion-induced collapse of white dwarfs following accretion from a non-degenerate companion [@2008MNRAS.385.1455M; @2009arXiv0908.1127M] or following the merger of a white dwarf binary [@2004ApJ...601L.167M; @2009arXiv0908.1127M], or even cosmic strings [@1987ApJ...316L..49B]. Identifying the mechanisms behind the creation of individual GRBs will be greatly aided by GW detections.
This review intends to survey the prospects of GW measurements in understanding the central engines of GRBs. While our theoretical view of GW emission mechanisms connected to GRBs is rapidly evolving, the main directions, the opportunities and limitations of observational GW astrophysics are becoming clearer. With this review we aim to provide guidance to (i) astronomers interested in utilizing the capabilities of GW measurements to complement electromagnetic or neutrino observations in understanding GRBs, and (ii) GW scientists interested in the open questions in GRB astrophysics that can be addressed through GW measurements.
Compact binary mergers, the likely progenitors of most short GRBs (e.g., [@2007PhR...442..166N; @2011NewAR..55....1B], although see [@2008MNRAS.385.1455M] for alternative models), are one of the primary targets of GW searches (e.g., [@2009PhRvD..79l2001A]). Advanced GW detectors will be able to, among others, measure the properties of the binary [@1994PhRvD..49.2658C], directly probe the dynamics of the central engine, and correlate the binary’s masses and orientation with prompt and delayed electromagnetic counterparts. These observations will constrain the central engine mechanism, provide standard sirens for measuring cosmological parameters [@1986Natur.323..310S; @1993ApJ...411L...5C; @1996PhRvD..53.2878F], and constrain the nature of nuclear matter (see Section \[section:cbcphysics\]).
GW emission from isolated central engines, such as stellar core collapses, accretion induced collapses or magnetars, is typically at higher frequencies and at lower amplitudes than for compact binaries. GW emission depends on the development of varying quadrupole moment, which is highly source and model dependent. The detection of GWs from isolated objects would help us understand the internal processes in massive stellar collapses, including the development of differentially rotating protoneutron stars, the nature of nuclear matter, or the creation of massive accretion disks (Sections \[section:collapsarphysics\], \[section:magnetarphysics\] & \[section:millisecondmagnetars\]).
For many topics touched upon here, we can refer the interested reader to existing, excellent reviews; on GRB blastwaves [@Meszaros2012; @2007PhR...442..166N; @Waxman00; @Meszaros2002; @Piran2005]; on short and long GRB observations ; on models of central engines involving core-collapse events [@Ott2008], neutron star mergers [@2007NJPh....9...17L; @Lattimer:2006], and isolated neutron stars [@2009astro2010S.229O; @corsi09]; on GW emission from core-collapse events [@Ott2008; @2003LRR.....6....2N] and binary mergers [@Cutler:2001; @2002grg..conf...72C; @2009CQGra..26k4004F; @2010CQGra..27k4002D; @lrr-2011-6]; on short GRB rates [@2007PhR...442..166N], long GRB rates [@Kalogera:2004tn; @Kalogera:2004nt], and compact object merger rates [@2010CQGra..27q3001A]. Some investigations of isotropic prompt and delayed electromagnetic counterparts have recently been summarized in [@2012ApJ...746...48M] (for short GRBs).
This review is organized as follows. Section \[section:emissionprocesses\] outlines GW emission processes from GRB progenitors, focusing on the emission scenarios that may produce signals which could be detectable with advanced GW detectors. Section \[section:astrophysics\] presents some of the intriguing astrophysical questions that could be answered through the detection of GWs from GRB progenitors, in some cases in coincidence with other messengers. In Section \[section:observationalstrategies\] we review GW observation strategies from single GW to multimessenger searches, also outlining prospects for the advanced detector era. Finally, Section \[section:conclusion\] we briefly summarizes the presented results and the near future of GRB astrophysics with GWs.
Gravitational-wave emission processes in GRB progenitors {#section:emissionprocesses}
========================================================
In this section, we outline the main GRB progenitor models with an emphasis on their expected GW signature. It should be clear that, depending on the progenitor model, the GW emission from GRB progenitors varies widely with respect to predicted signal strength and characteristic frequency.
Compact binary coalescence
--------------------------
The majority of short-hard GRBs are thought to be powered by the merger of NS-NS or NS-BH binaries [@1984SvAL...10..177B; @1986ApJ...308L..43P; @1989Natur.340..126E; @2007NJPh....9...17L; @2010CQGra..27k4002D; @2012arXiv1204.4919Z; @2008arXiv0809.1602K; @2012arXiv1210.8152G]. Compact binaries are also strong GW sources: the GWs produced in the minutes before the merger should be detectable, by second generation instruments (i.e., Advanced LIGO and Virgo), reliably out to $\sim 450\,$Mpc for NS-NS binaries, and further for NS-BH systems [@2010CQGra..27q3001A; @2010ApJ...716..615O].
Theoretical studies provide detailed information about the binary evolution and the gravitational waveform expected from these systems. Moreover, numerical simulations, which include the physics of the NS matter, continue to improve (see Refs. [@Lattimer:2006; @2009CQGra..26k4004F; @2010CQGra..27k4002D; @lrr-2011-6] for reviews). Here we present an overview of compact binary evolution, which is summarised in Fig. \[fig:CBCflowchart\]. At the highest level, there can be four phases: the inspiral phase during which the orbit shrinks due to energy and angular momentum loss via GWs; the merger phase during which dynamical effects and microphysics of the NS become important; the accretion phase when material stripped from the NS(s) accretes onto the merged objects; and the ringdown phase when the merged objects settle down to an unperturbed (Kerr) BH or a NS. The details vary by NS-NS and NS-BH merger and by mass (and other parameters) of the merging objects. The accretion stage is likely essential to the formation of short GRBs from binary coalescences, although the formation and evolution of protomagnetars following NS-NS merger has also been proposed as a plausible central engine [@2008MNRAS.385.1455M]. Figure \[fig:CBCspectrum\] presents the main features of GW emission in a schematic spectrum of the GW effective amplitude, separately for NS-NS and BH-NS mergers. Below we discuss the different stages of the evolution in more detail.
### Inspiral phase
— Early in the evolution of a compact binary system, the two objects are separated by a relatively large distance compared to their radii. The binary elements spiral towards each other by losing angular momentum via the emission of GWs. During the early stages of the inspiral, the two compact objects can be approximated as point masses for the purposes of GW emission [@lrr-2011-6]. At the late stages of the inspiral, the internal structure of the objects become increasingly important. For example, the tidal deformation of NSs in a binary system can (slightly) affect the orbital period (and therefore the gravitational waveform) in the late inspiral phase [@2008PhRvD..77b1502F; @2009PhRvD..79l4033R; @2009PhRvD..80f4037K; @2010PhRvL.105z1101B; @2011PhRvD..84b4017B]. Further, general relativistic spin-spin or spin-orbit coupling can cause the binary’s orbital plane to precess, affecting the binary’s evolution and GW emission [@1994PhRvD..49.6274A; @2004PhRvD..70d2001V; @2006PhRvD..74l2001L].
Nevertheless, the dominant features of the GW signal from the inspiral phase are captured by neglecting the spins and internal structure of the binary elements. As the objects spiral together, their orbital frequency increases producing a GW signal that sweeps upward in frequency. About $\sim15$ minutes before merger, the GW from the inspiral of a NS-NS binary begins to sweep upward from $\sim10\,$Hz through the band of Earth-based GW interferometers. The effective amplitude $h_{\mathrm{eff}} \equiv f | \tilde{h}(f) |$ of the GW signal from a binary system decreases as $h_{\mathrm{eff}} \propto f^{-1/6}$ [@2010PhRvL.104n1101K], up to a mass-dependent cutoff frequency $f_{\mathrm{cut}}\sim1-3\,$kHz [@1998PhRvD..57.4535F; @2007PhRvL..99l1102O; @2010PhRvL.104n1101K]. The frequency ranges $\lesssim1\,$kHz and $1-3\,$kHz are traditionally considered the inspiral and early merger phases, respectively. For $f\lesssim f_{\mathrm{cut}}$ the merger retains a binary-like structure and consequently emits relatively strong GWs [@2010PhRvL.104n1101K].
Advanced detectors will be able to detect a NS-NS inspiral up to $D_h \sim 450\,$Mpc, while NS-BH inspirals will be detectable up to $D_h \sim950\,$Mpc [@2010CQGra..27q3001A] (the distances are given for untriggered searches, with optimal source orientation and direction; for further details see Section \[section:allsky\]). The effective survey volume determined by averaging over sky location and inclination of the sources is $\sim 4\pi (D_h / 2.26)^3/3$ [@2010CQGra..27q3001A]. Using the current best-guess rates of mergers, this gives tens of NS-NS and a few NS-BH binaries detected with advanced detectors each year [@2010CQGra..27q3001A]. Additional advanced detectors, such as KAGRA [@2011IJMPD..20.1755K] or LIGO India [@2012CQGra..29l4012W], can significantly increase this range [@2012arXiv1210.6362N]. Third generation detectors are expected to reach an order of magnitude farther than advanced detectors, i.e. to several Gpc, and hence will be able to observe tens of thousands of events a year (e.g., [@2011GReGr..43..409A]).
### Merger phase {#subsubsection:mergerphase}
— Depending on the binary system, the merger can progress in multiple distinct directions with qualitatively different GW and gamma-ray emission. The formation of a massive accretion disk is probably crucial to the generation of a GRB from binary mergers[^1] [@2010CQGra..27k4002D]. Accretion disks can be created via the tidal disruption of a NS at some point during the merger [@2004MNRAS.351.1121R; @2006PhRvD..73f4027S; @2009PhRvD..79d4030S; @2009PhRvD..79d4024E; @2010PhRvD..82d4049K; @2010PhRvD..81f4026F]. Alternatively, for NS-NS mergers, material with centrifugal support can be left outside the newly formed BH. Whether an accretion disk is formed or not depends on the binary properties (mass, spin, etc.), as well as the NS equation of state (EOS). A disk mass of $\sim0.01\,$M$_\odot$, where M$_\odot$ is the mass of the sun, is probably sufficient to supply the energy for the creation of a short GRB [@2010PhRvL.104n1101K; @2012arXiv1210.8152G]. In a recent comparison between numerical simulations and observation, Giacomazzo et al. [@2012arXiv1210.8152G] found that the observed emission of short GRBs implies torus masses $\lesssim0.01\,$M$_\odot$, which favors “high-mass" NS-NS mergers ($M_{total}\gtrsim3\,$M$_\odot$). Further, they find that BH-NS mergers, while cannot be excluded, would require a very rapidly spinning BH (with spin $\gtrsim0.9$).
Here we list the possible outcomes of the merger and outline the scenarios that can lead to them.
1. [**BH with no accretion disk**]{} — for BH-NS binaries, if the radius of the innermost stable circular orbit (ISCO) is greater than the tidal disruption radius, the NS plunges into the BH before it could be tidally disrupted, resulting in a BH with no accretion disk. This will be the case for binaries with relatively high BH:NS mass ratio ($\gtrsim5:1$) [@2009PhRvD..79d4030S; @2009PhRvD..79d4024E; @2010CQGra..27k4002D; @2010PhRvL.105k1101C]. This ratio strongly depends on the BH spin (due to the change in the location of the ISCO) and the NS EOS [@2010PhRvD..82d4049K; @2011PhRvD..84f4018K; @2011ApJ...727...95P; @2012arXiv1207.6304F].
For NS-NS binaries with equal masses, if the binary mass exceeds a threshold $M_{\mathrm{thr}}$, the NSs will promptly collapse to a BH upon merger [@2008PhRvD..78h4033B; @2011PhRvD..83l4008H], leaving essentially no accretion disk behind [@2006PhRvD..73f4027S; @2010CQGra..27k4105R; @2011PhRvD..83l4008H]. $M_{\mathrm{thr}}$ depends on the NS EOS. Kiuchi et al. [@2009PhRvD..80f4037K] found that a BH is promptly formed if total mass of the binary system is $\gtrsim 3\,$M$_\odot$.
In this scenario no accretion-powered GRB will be produced, although there are possible channels through which even such a system can emit electromagnetic radiation [@2011ApJ...742...90M; @2012ApJ...755...80P].
2. [**BH with accretion disk**]{} — For BH-NS binaries with relatively low mass ratio ($\lesssim4:1$), the NS will be tidally disrupted before falling into the BH, which leads to the formation of a massive accretion disk [@2008PhRvD..77h4015S; @2009PhRvD..79d4030S; @2009PhRvD..79d4024E; @2010CQGra..27k4002D]. For spinning BHs, the mass ratio below which disruption occurs is even higher [@2010PhRvD..82d4049K; @2010PhRvL.105k1101C; @2010CQGra..27k4106D; @2011PhRvD..84f4018K], and the mass of the formed disk can greatly depend on BH spin and spin-alignment [@2010PhRvD..82d4049K; @2011PhRvD..83b4005F; @2011PhRvD..84f4018K; @2012arXiv1207.6304F]. Prior to tidal disruption, the emitted GW frequency reaches $f_{\mathrm{tidal}}\sim2-4\,$kHz[^2] [@2011PhRvD..84f4018K; @2012PhRvD..85d4061L].
For NS-NS binaries with unequal NS masses and sufficiently large total mass ($\gtrsim 3\,$M$_\odot$ [@2009PhRvD..80f4037K; @2010CQGra..27k4105R; @2011PhRvL.107e1102S]), the less massive NS will be tidally disrupted, followed by the more massive NS’s prompt collapse into a BH, leaving a potentially massive accretion disk behind [@2008PhRvD..78h4033B].
In this scenario, after the merger of a NS-NS binary, if the forming disk features azimuthal variations, GWs may be emitted by the material orbiting the central object around a peak amplitude $f_{\mathrm{peak}}\sim5-6\,$kHz [@2010PhRvL.104n1101K]. For BH-NS binaries, $f_{\mathrm{peak}}$ can be significantly lower, $\sim$inversely proportional to the total mass of the system [@2009CQGra..26a5009H]. For instance a BH-NS binary with $5\,$M$_\odot$ BH mass has $f_{\mathrm{peak}}\approx 1-2\,$kHz, depending on, e.g., the BH spin [@2010PhRvD..82d4049K; @2011PhRvD..84f4018K].
This scenario is a good candidate for the creation of accretion-powered GRBs.
3. [**Hypermassive NS formation**]{} — NS-NS binaries with total mass $\lesssim3$M$_\odot$ (this mass threshold depends on the NS EOS [@2009PhRvD..80f4037K; @2011PhRvD..83l4008H; @2011PhRvL.107e1102S]) will not promptly collapse into a BH, but will first form a so-called *hypermassive* NS [^3] [@2005PhRvD..71h4021S; @2006PhRvD..73f4027S; @2008PhRvD..78h4033B], supported by differential rotation and thermal pressure [@2010CQGra..27k4002D; @2010PhRvL.104n1101K; @2011PhRvL.107e1102S]. The hypermassive NS eventually collapses into a BH with a delay of $1\,$ms-1s due to (i) losing angular momentum via GWs or magnetic processes [@2011PhRvD..83d4014G] (magnetorotational instability [@2006PhRvL..96c1101D] or magnetic winding [@2011ApJ...732L...6R]), (ii) its rotation becoming more uniform due to magnetic braking and viscosity [@2000ApJ...544..397S; @2004PhRvD..69j4030D], in which case differential rotational no longer supports the NS, and/or (iii) cooling due to, e.g., neutrino emission, so thermal pressure provides less support against the gravitational pull [@2011PhRvL.107e1102S; @2011PhRvL.107u1101S].
A rapidly rotating hypermassive NS will assume a (non-axisymmetric) ellipsoidal shape [@2008PhRvD..78h4033B], which is energetically favorable over a spheroid [@2010CQGra..27k4002D]. Such an ellipsoidal hypermassive NS will emit a strong GW signal at twice its (quasiperiodic) rotational frequency $f_{\mathrm{qpd}}\sim2-4\,$kHz [@2007PhRvL..99l1102O; @2009PhRvD..80f4037K; @2011PhRvD..83l4008H; @2011PhRvL.107e1102S].
Hypermassive NSs may leave a massive accretion disk behind after collapsing into a BH [@2006PhRvD..73f4027S; @2011PhRvD..83l4008H; @2012arXiv1204.6240R]. It seems that the outcome depends on the binary mass. As we saw above, for high-mass NSs, the merger promptly forms a BH. At the highest masses for which a hypermassive NS is formed, a very short-lived NS is formed, with likely suppressed accretion disk formation [@2011PhRvD..83l4008H]. For lower-mass binaries, the formed hypermassive NS is longer-lived, resulting in stronger GW emission. These lower-mass binaries also result in the formation of a more massive accretion disk.
In this scenario, in which a hypermassive NS is formed, a significant amount of GW energy can be emitted from the hypermassive NS at around its quasiperiodic rotation frequency. Quasiperiodic GW emission from a hypermassive NS would be detectable with advanced detectors from $\sim20\,$Mpc [@2005PhRvL..94t1101S; @2011PhRvL.107e1102S], especially because it would be accompanied by an inspiral phase with significantly higher signal-to-noise ratio (SNR). This scenario is a good candidate for the creation of GRBs, with either BH-torus or protomagnetar central engines.
4. [**Formation of stable NS**]{} — If the total mass $M_{binary}$ of a NS-NS binary is below the mass limit $M_{NS,max}$ of non-rotating NSs, or if the merged NS’s mass is $<M_{NS,max}$ due to, e.g., tidal disruption, a stable, long-lived NS can form from the merger. While typical observed NS-NS binary masses [@1999ApJ...512..288T] are likely above $M_{NS,max}\gtrsim2\,$M$_\odot$ [@2010Natur.467.1081D], it is plausible that some mergers end up forming a stable NS.
Accretion disks themselves may contribute to the GW emission of binaries [@2003MNRAS.341..832Z; @2005MNRAS.356.1371Z; @2007PhRvD..75d4016N; @2011PhRvL.106y1102K]. BH-torus systems can be unstable to non-axisymmetric perturbations that may give rise to non-axisymmetric torus structure, resulting in GW emission at a few-hundred Hz. See also Section \[section:accretiondiskinstabilities\] for GW emission from accretion disks.
### Ringdown phase
— When a BH is formed, or when substantial matter with non-axisymmetric structure (i.e., NS or fragmented accretion disk) falls into the BH, the event horizon of the BH is initially perturbed. Subsequently, it quickly approaches the non-perturbed (Kerr) solution via the emission of GWs. This process is called ringdown, with characteristic frequencies of $f_{\mathrm{peak}}\approx 6.5-7\,$kHz (e.g., [@2009PhRvD..80f4037K; @2011PhRvD..83l4008H]) [^4], given the typical mass of the binary ($\sim2-3\,$M$_\odot$).
At and above $f_{\mathrm{peak}}$, the GW spectrum is qualitatively independent of the properties of the binary (although quantitatively the characteristic frequency decreases with increasing binary mass). It decays exponentially due to the $m=2$ quasinormal mode oscillation of the final black hole [@1998PhRvD..57.4535F; @2010CQGra..27k4002D]. For solar-mass binaries, GW radiation from the BH ringdown is undetectable with advanced GW interferometers due to its high frequency.
We note that BH ringdown can be suppressed or even choked in the presence of intense accretion onto the BH, as is likely the case for unequal-mass NS-NS binaries [@2010CQGra..27k4105R].
Core Collapse {#section:collapsars}
-------------
Massive stars develop an inert iron core supported by non-thermal degeneracy pressure in their center as the final stage of nuclear fusion. The growing iron core becomes gravitationally unstable upon reaching a mass around the Chandrasekhar mass ($\sim1.44\,$M$_\odot$) due to a softening of the EOS as the degenerate fermions become relativistic [@2007PhR...442...38J], collapsing into a NS or a BH. Such a collapsing stellar system may result in a supernova explosion (a core-collapse supernova), and/or a GRB. The central engine driving GRB emission may be an accreting BH (e.g., [@Wo:93]) or a rapidly spinning, strongly magnetized protoneutron star (a *millisecond proto-magnetar*; e.g., [@2011MNRAS.413.2031M] and references therein).
The core collapse of a massive star may emit GWs through various mechanisms [@2003LRR.....6....2N; @Ott2008]. Below we focus on the mechanisms producing GWs that may be sufficiently strong to be detected at distances relevant to GRBs, i.e. that may be detectable from $\gg10\,$Mpc with 2$^{nd}$ or $3^{rd}$ generation GW interferometers. A schematic diagram of these emission processes are shown in Fig. \[fig:collapsarGW\].
### Rotational instabilities in protoneutron stars {#section:protoneutronstar}
— For massive stars with initial stellar masses $10\,$M$_\odot\lesssim M \lesssim25\,$M$_\odot$, the collapsing core is expected to form a so-called protoneutron star [@2003ApJ...591..288H]. The resulting rotating protoneutron star can be unstable to non-axisymmetric deformations, potentially giving rise to copious GW emission [@1998LRR.....1....8S; @2000ApJ...542..453S; @2002ApJ...565..430F; @2003CQGra..20R.105A; @2007PhRvD..75d4023B; @2007CQGra..24S.171M; @2010CQGra..27k4104C]. The onset of rotational instabilities depends on the rotational rate of the star, which can be conveniently parameterized by $\beta \equiv T_{\mathrm{rot}}/|W|$, i.e. the ratio of the star’s rotational kinetic ($T_{\mathrm{rot}}$) and gravitational potential ($W$) energy [@1994PhRvL..72.1314H]. The resulting non-axisymmetric structure may be a bar-like $m=2$ mode, giving rise to a characteristic GW emission. Higher $m$ modes may also arise, albeit they have longer growth time [@1983PhRvL..51...11F]. While there are still various uncertainties in the evolution and role of rotational instabilities (e.g., non-linear mode-coupling effects can severely limit the deformation [@2007PhRvD..75d4023B], or the role of viscosity and neutrino cooling), the emerging picture is that rotational instabilities are likely viable emitters of GWs and could play an important role in the future detection and understanding of GRB progenitors through their GW signature.
The energy potentially available for GW emission is abundant. The rotational energy of a typical NS with 1$\,$kHz rotational frequency is $\sim10^{-2}$M$_\odot\,$c$^2$ (e.g., [@2001ApJ...550..426L]). Even a fraction of this energy, if radiated away in GWs, could be detectable at large distances ($\gg10\,$Mpc) with advanced detectors. The protoneutron star may also accrete supernova fallback material. Such an accretion further increases the angular momentum and energy available that may be radiated away via GWs [@2002MNRAS.333..943W; @2008arXiv0809.1602K; @2012arXiv1207.3805P].
The amplitude of a GW signal emitted by a rotating bar scales as $h\sim M R^2 f^2 /d$, where $M$, $R$, $f$ and $d$ are the mass, radius, GW frequency (i.e. twice the rotational frequency), and distance of the NS, respectively [@2008arXiv0809.1602K]. The energy radiated away in GWs, in the Newtonian quadrupole approximation, can be estimated as (see, e.g., [@2008arXiv0809.1602K])
$$E_{\mathrm{GW}} \approx 10^{-2} M_\odot c^2 \left(\frac{\epsilon}{0.2}\right)^2 \left(\frac{f}{2\,\mbox{kHz}}\right)^6\left(\frac{M}{1.4\,M_{\odot}}\right)\left(\frac{R}{12\,\mbox{km}}\right)^2\left(\frac{\tau}{0.1\,\mbox{s}}\right),
\label{equation:barmodeEGW}$$
where $\epsilon$ is the ellipticity of the bar and $\tau$ is the duration of the presence of the instability. Protoneutron stars are subject to different rotational instabilities, the two main categories being dynamical and secular instabilities. A star is dynamically unstable if it is unstable to non-axisymmetric perturbations even in the absence of dissipation (i.e. if a slightly non-axisymmetric shape that conserves angular momentum is energetically favorable). A star is secularly unstable if it is unstable to non-axisymmetric perturbations only if dissipative effects are relevant, i.e. if the change towards a non-axisymmetric shape requires the radiation or redistribution of angular momentum (see [@CPA:CPA3160200203; @Lebovitz19981407] for interesting historical overviews). Below we discuss these instabilities further in detail.
1. [**Dynamical instabilities**]{} – Rapidly rotating stars will be subject to dynamical instabilities driven by hydrodynamical and gravitational effects [@1998LRR.....1....8S; @2005ApJ...618L..37W; @2007PhRvD..75d4023B]. Dynamical instability is the “simplest" form of NS instabilities, since its development is quick (on the time scale of the rotational period of the NS), and does not require dissipation. A uniformly rotating, classical fluid body becomes dynamically unstable at a rotation rate $\beta \gtrsim0.27 \equiv \beta_{\mathrm{dyn}}$ [^5] [@Chandrasekhar:207046]. Stability conditions are essentially the same for relativistic stars, for which $\beta_{\mathrm{dyn}}\sim0.24$ [@2007PhRvD..75d4023B; @2008arXiv0809.1602K]. Furthermore, differentially rotating stars are subject to non-axisymmetric instabilities even at much slower rotation with $\beta\lesssim0.09$ . Such low-$\beta$ instabilities in differentially rotating protoneutron stars are probably analogous to the Papaloizou-Pringle instability ([@1984MNRAS.208..721P]; also see Section \[section:accretiondiskinstabilities\]): the protoneutron star core is surrounded by a fluid rotating at the frequency of a non-axisymmetric mode of the core, hence exciting this mode [@2005ApJ...618L..37W]. As numerical simulations so far have been too short to capture the long-term behavior of some dynamical instabilities, they are not conclusive in terms of the total energy emitted via GWs from dynamical instabilities. Nevertheless, GW emission seems to be fast relative to the cooling time of the protoneutron star, or compared to energy loss due to viscosity. Consequently, if competing mechanisms that radiate away angular momentum (e.g., magnetic fields) are weak, GWs can carry away a significant fraction of the protoneutron star’s rotational energy, producing a signal that may be detectable from $$D\gtrsim 60\,\mbox{Mpc}\left(\frac{E_{\mathrm{GW}}}{10^{-2}\,\mbox{M}_\odot\mbox{c}^2}\right)^{1/2}\left(\frac{f}{1\,\mbox{kHz}}\right)^{-1}$$ for narrow-band, circularly polarized GW signals from optimal source direction[^6]. The frequency scaling of the distance only applies to $f\gtrsim300\,$Hz.
2. [**Secular instabilities**]{} – Stars with lower rotation rates (i.e. for which the faster-developing dynamical instabilities are not prevalent) can be subject to secular (i.e. dissipation-driven) non-axisymmetric instabilities [@1995ApJ...442..259L; @1998PhRvL..80.4843L; @2009ApJ...702.1171C]. Dissipation can occur via gravitational radiation [@1998PhRvL..80.4843L] or fluid viscosity [@1998LRR.....1....8S; @2008arXiv0809.1602K]. GW emission drives frame-dragging (so-called Chandrasekhar-Friedman-Schutz) instabilities of modes that are retrograde with respect to the star but prograde with respect to the observer [@1970PhRvL..24..611C; @1978ApJ...222..281F]. Among GW-driven instabilities, fundamental $f$-mode bar instabilities have the shortest growth time: $0.1\,\mbox{s} \lesssim \tau_{\mathrm{GW}}\lesssim7\times10^4\,$s for $0.27\gtrsim\beta\gtrsim0.15$ [@1995ApJ...442..259L]. In the uniformly rotating approximation of a relativistic star, the protoneutron star becomes unstable to GW-driven $m=2$ $f$-mode instabilities for $\beta\gtrsim0.06-0.09\equiv \beta_{\mathrm{sec}}$ depending on the EOS and stellar mass [@1999ApJ...510..854M] (compare with the Newtonian limit of $\beta\gtrsim0.14$ [@1970PhRvL..24..611C; @1995ApJ...442..259L]). At lower rotation rates, stars become unstable to higher-multipole $f$ modes, albeit higher modes have longer growth times [@1983PhRvL..51...11F; @2012arXiv1209.5308P]. A recent work of Passamonti et al. [@2012arXiv1209.5308P] indicates that the most unstable $f$ modes developing in the aftermath of a supernova explosion may in fact be $l=m=3$ and 4 modes.
As angular momentum is radiated away through GWs (and potentially other channels), the protoneutron star’s rotation frequency and therefore the emitted GW frequency decreases, sweeping towards the most sensitive band of LIGO-like detectors. Some analytical and numerical results indicate that such a GW signal may be detectable at large distances, up to $\sim100\,$Mpc with advanced interferometers [@1995ApJ...442..259L; @2004PhRvD..70h4022S; @2004ApJ...617..490O; @2010PhRvD..81h4055Z]. Some other recent simulations are also promising, even though they only cover the first few milliseconds after core collapse and therefore cannot capture the long-term evolution of the bar mode [@2007PhRvL..98z1101O; @2007CQGra..24..139O]. Nevertheless, some recent, realistic simulations (e.g., [@2012arXiv1209.5308P; @2012arXiv1206.6604C; @2012PhRvD..85b4030Z; @2012arXiv1203.3590L]) predict much smaller detectable range.
Rotating protoneutron stars are also unstable to GW-driven *r-mode* oscillations (the restoring force being the Coriolis force) at any (i.e. arbitrarily low) rotation rate [@1998ApJ...502..708A; @2002CQGra..19.1247O; @2008arXiv0809.1602K; @2011PhRvL.107j1101H]. $R$ modes are important only if their growth time is less than the damping time of viscous forces. Further, $r$-mode instability is expected to be saturated at low amplitude due to dissipative effects [@2011GReGr..43..409A], and is suppressed in the presence of magnetic fields [@2000ApJ...531L.139R]. Nevertheless, under favorable conditions, GW signals from protoneutron star $r$ modes may be detectable for several years after core collapse [@2009PhRvD..79j4003B]. Given such a long duration, the emitted GW signal may be integrated for a measurement of $\sim1\,$yr that would give a detectable signal to distances of $\sim30\,$Mpc [@2009PhRvD..79j4003B] (or to even farther if the protoneutron star is a *strange quark star* [@2011GReGr..43..409A; @2005PhRvL..95u1101O]). Further calculations (see [@2008arXiv0809.1602K] and references therein) suggest that the saturation (i.e. maximum) amplitude of $r$-mode instabilities is limited by its non-linear coupling to other inertial modes.
3. [**Magnetic distortion**]{} – Toroidal magnetic fields ($B\gtrsim10^{12}\,$G) can distort a rotating NS into a prolate shape [@2001MNRAS.327..639I; @2002PhRvD..66h4025C; @2003PhRvD..67l4026I; @2004ApJ...600..296I; @2012arXiv1207.4035F]. Such configuration is unstable to the growth of the angle between the NS’s angular momentum and the magnetic axis. This angle increases until the angular momentum and magnetic axis are orthogonal. The resulting configuration is a rotating, non-axisymmetric body that is an efficient emitter of GWs [@2002PhRvD..66h4025C; @2009MNRAS.398.1869D]. Fast rotating magnetars can also lose spin energy via magnetic dipole radiation and/or magnetized, relativistic winds. Nevertheless, observations indicate that such energy losses are not typical ([@2009MNRAS.398.1869D] and references therein), suggesting that GW emission may be relevant in the early phase of newly born magnetars. Given that magnetic distortions indeed result in efficient GW emission, Dall’Osso et al. [@2009MNRAS.398.1869D] find that such a GW signal can be detected out to the Virgo cluster with advanced detectors.
Besides their initial rotation, protoneutron stars can accrete material from the infalling matter after core collapse [@1989ApJ...346..847C]. It is possible that NSs radiate away this angular momentum that they gain from accretion via GWs, e.g., via dynamical and/or secular instabilities discussed above. Such GW signal may be detectable to $\gtrsim10\,$Mpc with advanced detectors [@2012arXiv1207.3805P].
### Accretion disk non-axisymmetric instabilities {#section:accretiondiskinstabilities}
— Upon the core collapse of a massive star ($\gtrsim 30\,$M$_{\odot}$ [@2006RPPh...69.2259M]), a plausible scenario is the formation of a central BH surrounded by an accretion disk [@1993ApJ...405..273W; @2011ApJ...737....6S; @2012arXiv1206.5927S]. Such a BH-torus system can be the source of copious GW emission if the disk assumes a finite quadrupole structure due to non-axisymmetric instabilities [@2003LRR.....6....2N]. The emergence of such a non-axisymmetric structure on the time scales comparable to the lifetime of the accretion disk requires a stellar progenitor with sufficiently high angular momentum. A high rotation rate is also required for the creation of a GRB (e.g., [@2005NatPh...1..147W]). Below we outline some of the possible scenarios through which non-axisymmetric instabilities in accretion disks may result in strong GW emission, and address whether existing or future GW detectors can probe that signal.
As an example to the sensitivity of a GW search to accretion-disk non-axisymmetric instabilities, we consider a 3M$_\odot$ BH with a $0.01\,$M$_\odot$ clump in its accretion disk. Such a system emits GWs similar to an isolated low-mass binary coalescence, and may be seen to a horizon distance (for definition see Section \[section:allsky\]) [@2010ApJ...716..615O] $$D_h\simeq 55\,\mbox{Mpc}\,(\mathcal{M}_c/0.1\,\mbox{M}_\odot)^{5/6}
\label{equation:horizonscaling}$$ for an advanced detector with optimal source position and orientation (this distance is reduced by a factor 2.26 for average sky location and orientation), where $\mathcal{M}_c=(m_1m_2)^{3/5}/(m_1+m_2)^{1/5}$ is the so-called *chirp mass* ($\mathcal{M}_c=0.1\,$M$_\odot$ for the example above).
1. [**Disk fragmentation via gravitational instability**]{} – Accretion disks with sufficiently large angular momenta are gravitationally unstable [@1994ApJ...420..247W; @2007ApJ...658.1173P; @2004PhRvD..69j4016D]. A gravitationally unstable disk will fragment if the disk cooling time is sufficiently short ($\lesssim$ orbital period) [@2001ApJ...553..174G]. Relevant accretion disks cool rapidly, e.g., via neutrinos or strong winds [@2007NJPh....9...17L], favoring fragmentation, although the emergence of magnetorotational instability (MRI) in the disk can heat the disk through ohmic dissipation [@2000ApJ...530..464F]. The resulting fragmented disk will emit a strong, chirp-like GW signal [@2003ApJ...589..861K]. Viscosity and GW emission drive the angular momentum loss of the disk. Consequently, low disk viscosity favors stronger GW emission, which may be detectable from $\sim100\,$Mpc with Advanced LIGO/Virgo [@2007ApJ...658.1173P]. The GW frequency at its highest SNR is probably in the $10^2-10^3\,$Hz range, depending on the strength of viscous forces. The duration of GW emission may be similar to that of gamma-ray emission. Furthermore, the tidal disruption of fragments in the accretion disk may be behind some of the X-ray flares observed in GRB afterglows [@2007ApJ...658.1173P]. Accretion disk fragmentation has been observed in approximate numerical simulations [@2004PhRvD..69j4016D; @2011PhRvD..84b4022G]. It is not clear, however, whether such a fragmentation would occur in more realistic cases as well, unless the pressure support of the star is drastically reduced, e.g., via neutrino losses [@2011PhRvD..84b4022G].
2. [**Non-axisymmetric structure via Papaloizou-Pringle instability**]{} – Differentially rotating accretion disks can be subject to global non-axisymmetric instabilities. One such instability, discovered by Papaloizou & Pringle [@1984MNRAS.208..721P], develops in accretion disks in which azimuthal pressure gradients (due to high internal temperature) give rise to differential rotation [@1984MNRAS.208..721P; @1986MNRAS.221..339G; @1996MNRAS.281..119C; @1997ASPC..121...90B], i.e. the angular velocity $\Omega(R)$ of the disk depends on the radius $R$. In such disks, large-scale spiral pressure waves can emerge with fixed pattern speed $\Omega_p$. If there is a radius $R_c$, the so-called corotation radius, at which the pattern speed is equal to the disk rotational rate (i.e. $\Omega(R_c)=\Omega_p$), the wave within $R_c$ will propagate slower than the rotation of the disk, in fact decreasing the mechanical energy of the disk, resulting in a so-called *negative energy wave* (e.g., [@1997ASPC..121...90B]). Such negative energy wave can interact, at the corotation radius, with the *positive energy wave* that develops at $R>R_c$. The negative energy wave can increase its amplitude by *losing* energy, which in turn can increase the amplitude of the positive energy wave, thus feeding the instability. See, e.g., [@1997ASPC..121...90B] for an expressive description of the phenomenon.
The Papaloizou-Pringle instability gives rise to an ($m=1$) non-axisymmetric structure on a dynamical time scale (i.e. over a time period comparable to the rotation period). Such non-axisymmetric structure can persist for much longer than the dynamical time scale, resulting in strong GW emission. The Papaloizou-Pringle instability and the resulting non-axisymmetric structure have been observed in 3D relativistic simulations of BH-torus systems [@2011PhRvD..83d3007K; @2011PhRvL.106y1102K]. These simulations indicate that BH-accretion disk systems subject to the Papaloizou-Pringle instability emit GWs in the $10^2-10^3\,$Hz frequency range that may be detectable up to $\sim100\,$Mpc with Advanced LIGO/Virgo [@2011PhRvL.106y1102K]. Strong magnetic fields present in the accretion disk can enhance the instability for thick disks (and may suppress it for thin disks) [@2011MNRAS.410.1617F]. Nevertheless, we note that the Papaloizou-Pringle instability was found in simulations for initially axisymmetric tori [@2011PhRvL.106y1102K]. Such complete axial symmetry may not develop in compact binary mergers. Numerical simulations of compact binary mergers have not yet shown signs of the development of the Papaloizou-Pringle instability (e.g., [@2011ApJ...732L...6R]).
3. [**Suspended accretion**]{} – In order for accretion around a central, rotating BH to continue for the time scales of long GRBs, it has been suggested that accretion may be “suspended" (i.e. slowed down or temporarily halted; e.g., [@2001ApJ...552L..31V; @2001ApJ...552L..31V; @2003ApJ...584..937V]). Such suspended accretion would be achieved through magnetic fields, which transfer some of the rotational energy of the BH to the disk. A fraction of this rotational energy could then be radiated away through GWs [@2001PhRvL..87i1101V; @2004PhRvD..69d4007V]. In order to emit GWs, the accretion disk needs to lose its axial symmetry, e.g., through fragmentation. GW radiation from suspended accretion has been suggested to carry away as much as $E_{\mathrm{GW}}\sim10^{-2}\,$M$_\odot$c$^2$ in the sensitive frequency band of LIGO/Virgo-like interferometers over a duration comparable to the duration of long GRBs ($\sim30\,$s) [@2001ApJ...552L..31V; @2003ApJ...584..937V]. Suspended accretion, nevertheless, requires highly ordered accretion, which may be difficult to achieve due to the development of MRI heating or disk turbulence (e.g., [@2005ApJ...622.1008K]). Simulations to date (e.g., [@2011MNRAS.413.2031M]) have not provided support for suspended accretion, even in the presence of $\sim10^{15}$G magnetic fields.
### Fragmentation of collapsing core {#section:fragmentationofcollapsingcore}
— In very rapidly rotating stars, infalling matter may fragment even before the formation of a BH-torus system. It has been suggested that, in analogy with observed phenomena in star formation, the collapsing core may fragment and form two or more compact objects [@1995MNRAS.273L..12B; @2002ApJ...579L..63D; @2003ApJ...589..861K]. Such fragmentation was observed in relativistic numerical simulations of approximate pre-supernova cores [@2006PhRvL..96p1101Z; @2007PhRvD..76b4019Z; @2006astro.ph..8028R]. Such core fragmentation would give rise to strong, characteristic GW emission, similarly to the case of binary mergers. Nevertheless, the rotation rate necessary for such a fragmentation seems to be difficult to achieve with current stellar models [@2003LRR.....6....2N].
Magnetars {#subsection:magnetars}
---------
Highly magnetized neutron stars (magnetars) are thought to be the engine behind soft gamma repeaters (SGRs) and anomalous X-ray pulsars (AXPs) [@1992ApJ...392L...9D; @1995MNRAS.275..255T; @1996ApJ...473..322T; @1999ApJ...510L.115K]. Some magnetars occasionally produce so-called *giant flares* that resemble short GRBs ( and references therein). Giant flares are much less energetic than typical short GRBs that are at cosmological distances, hence they are only detectable from the Milky Way or nearby galaxies, up to $\lesssim40\,$Mpc with current instruments . The lack of excess short-GRB population from the direction of these galaxies indicates that magnetars may only be responsible for a small sub-population of short GRBs [@2005Natur.434.1107P].
Giant flares are much more common (with much weaker gamma emission) than other types of GRBs, as indicated by the rate of observed galactic and nearby giant flares. Due to their high rate, they may be detectable with GW observatories even though they are much weaker sources of GWs than other, extragalactic GRBs.
Giant flares (as well as SGR activity or abrupt changes in the NS spin period; [@1996Natur.382..518C]) are thought to be the result of so-called *starquakes*: the tectonic activity (cracking) of the NS crust [@1996Natur.382..518C], which is followed by the reconfiguration of the NS’s magnetic fields [@2005Natur.434.1107P]. Starquakes induce seismic vibrations in the NS, causing the observed quasi-periodic oscillations (QPOs) in the X-ray tails of giant flares .
Alternatively, it has been suggested [@2006MNRAS.368L..35L; @2011MNRAS.410.1036V] that QPOs cannot be driven by oscillations in the NS crust due to the quick ($\sim1\,$s) dissipation of the vibration via Alfvén waves into the neutron star interior. Nevertheless, the similarities between the statistical properties of SGR events and earthquakes may provide further evidence for the crustal origin of SGR events [@2005Natur.434.1107P].
NS seismic vibrations result in the emission of GWs [@1998MNRAS.299.1059A; @2001MNRAS.320..307K; @2001MNRAS.327..639I; @2007PhRvD..76f2003A; @2011PhRvD..83j4014C], although this emission may be weak [@2005MPLA...20.2799H]. In the optimal scenario for GW emission, magnetic reconfiguration can liberate $10^{48}-10^{49}\,$erg of crustal elastic energy, which is the upper limit for the energy radiated away via GWs [@2011PhRvD..83j4014C; @2001MNRAS.327..639I]. This energy can be even higher if the NS is of strange quark matter [@2011PhRvD..83j4014C].
Quasi-periodic oscillations (QPOs) with various frequencies from $\sim10-10^3\,$Hz and durations up to $\sim100\,$s have been observed in every giant flare X-ray afterglow so far . Some of these QPOs may be connected to the seismic modes of the NSs . Of special interest are QPOs around $\sim100\,$Hz, which fall into the most sensitive frequency band of LIGO/Virgo. Possible GW emission of $\sim10^{44}\,$erg at this frequency would be detectable out to $\sim10\,$kpc with Advanced LIGO/Virgo (given 10$\times$ sensitivity improvement compared to initial detectors) [@2008PhRvL.101u1102A]. It is possible that GWs are emitted for a significantly longer time (days to months) than the observed electromagnetic QPO [@2011PhRvD..83h1302K], making longer-term GW searches necessary.
A fraction of the energy from magnetic reconfiguration may excite NS fundamental quadrupolar fluid modes (*f modes*; e.g., [@2011ApJ...735L..20L; @2011ApJ...736L...6C], although see [@2012arXiv1206.6604C]). If NS $f$ modes are excited, they are damped by GWs on a very short time scale ($\sim200\,$ms; [@2011PhRvD..83j4014C] and references therein), which is shorter than most other potential damping mechanisms. Therefore, most of the energy in the $f$ modes may be radiated away via GWs [@2011PhRvD..83j4014C]. NS $f$ modes oscillate at $1-2\,$kHz (for stiff EOS [^7]), not too far from the most sensitive band of Advanced LIGO/Virgo [@2001MNRAS.320..307K]. Given the most extreme case in which $10^{49}\,$erg being transferred into $f$ modes that oscillate at $\sim1\,$kHz, the resulting GW could be detected up to $\lesssim2\,$Mpc with Advanced LIGO/Virgo (given 10$\times$ sensitivity improvement compared to initial detectors) [@2008PhRvL.101u1102A]. “Stacking" the GW signals from multiple events can extend this distance even farther [@2009ApJ...701L..68A; @2011ApJ...734L..35A]. If the excitation energy is comparable to the observed giant flare energy ($\sim10^{46}\,$erg) and if not all energy is radiated away through GWs, only galactic sources would be detectable with advanced GW detectors. Furthermore, recent magnetohydrodynamic simulations of magnetized NSs indicate that only a small fraction of the released energy is converted to $f$ modes, which would make the detection of $f$-mode GWs less likely [@2012PhRvD..85b4030Z; @2012arXiv1203.3590L].
Millisecond protomagnetars {#section:millisecondmagnetars}
--------------------------
While the commonly favored model for the central engine behind GRBs is a rapidly accreting BH [@Wo:93; @1999ApJ...526..152F; @2001ApJ...557..949N; @2010ApJ...713..800L], there are observational indications [@2006Natur.444.1053G; @2006Natur.444.1044G] that a fraction of short GRBs may originate from highly magnetized, rapidly spinning protoneutron stars, so called *millisecond protomagnetars* [@2007RMxAC..27...80T; @2008MNRAS.385.1455M; @2009arXiv0908.1127M; @2009MNRAS.396.2038B; @2012MNRAS.419.1537B]. Further, outflows driven by such protomagnetars could also lead to the creation of long GRBs [@2004ApJ...611..380T; @2007ApJ...659..561M; @2011MNRAS.413.2031M]. The emergence of a millisecond protomagnetar from various cosmic events is therefore a plausible step in the creation of some GRBs.
The possibility that some GRBs are driven by rapidly spinning protomagnetars opens the door to alternative GRB progenitors. While traditionally considered progenitors, i.e. (i) the core collapse of massive stars [@2004ApJ...611..380T; @2007ApJ...659..561M; @2009MNRAS.396.2038B; @2011MNRAS.413.2031M] and (ii) NS-NS mergers, can themselves lead to the creation of millisecond protomagnetars, other cosmic events, such as (iii) the accretion-induced collapse of white dwarfs [@1991ApJ...367L..19N; @2006ApJ...644.1063D; @2008MNRAS.385.1455M; @2009arXiv0908.1127M], or the merger of two white dwarfs [@2001MNRAS.320L..45K; @2008MNRAS.385.1455M], are also plausible progenitors.
Millisecond protomagnetars may emit a sizable fraction of their rotational energy via GWs [@2002PhRvD..66h4025C; @2005ApJ...634L.165S; @2009MNRAS.398.1869D; @2011PhRvD..84b3002K], although they may also lose angular momentum via magnetic processes (e.g., [@2006PhRvL..96c1101D; @2006PhRvD..74j4026S]). Fast rotation can result in the emergence of rotational instabilities, especially if the protoneutron star is differentially rotating (see also Section \[section:protoneutronstar\]). If a millisecond magnetar indeed drives a GRB, it needs to maintain fast rotation for durations comparable to the duration of the GRB, potentially allowing sufficient time for rotational instabilities to emerge and result in GW emission. If a significant fraction of the rotational energy ($\sim10^{-2}$M$_\odot\,$c$^2$; e.g., [@2001ApJ...550..426L]) is converted into gravitational radiation, the resulting GW signal could be detected to a distance of $\sim 100\,$Mpc with advanced detectors (see also Section \[section:protoneutronstar\]).
The rotation rate of millisecond magnetars, and therefore the emitted GW strength and frequency, likely depends on the cosmic event they originate from. Accretion-induced collapse of white dwarfs that accrete mass and angular momentum from a non-degenerate companion may produce the most rapidly rotating protomagnetars. Due to continued accretion, such white dwarfs will be rapidly rotating upon collapse, resulting in a rapidly rotating protomagnetar, and a strong GW signal. On the other hand, the remnant of a binary white dwarf merger will likely collapse to a magnetar with a significant delay ($\sim10^6\,$yr), allowing the remnant to lose most of its angular momentum via stellar winds [@2007MNRAS.380..933Y; @2012arXiv1207.0512S; @2012ApJ...748...35S]. Binary white dwarf mergers are therefore not likely to be significant sources of GWs.
GRB astrophysics with gravitational waves {#section:astrophysics}
=========================================
This section addresses the question: What can the detection of GW signals add to our understanding of the physics of GRBs and their progenitors? Detectable electromagnetic radiation, our main source of information on GRBs, is observable only from relatively large distances ($\gtrsim10^{12}-10^{13}\,$cm) from the central engine. On the other hand, GWs are created right at the central engine, and can convey information about it without being distorted or absorbed by matter on their way to the observer. Here we outline some of the questions of interest that could be addressed through observing the GW signature of GRBs.
Compact binary coalescence {#section:cbcphysics}
--------------------------
1. [**Progenitors of short GRBs**]{} – As compact binary mergers will be detectable from hundreds of Mpc with the Advanced LIGO-Virgo network [@2010CQGra..27q3001A], within these distances it will be possible to confirm or rule out compact binaries as GRB progenitors. While identification based on electromagnetic signals is not always straightforward [@2012arXiv1204.4919Z], the detection of GWs will be likely essential to unambiguously reveal the nature of the progenitor [@Cutler:2001]. By connecting each measured source with the presence or absence of detectable GW emission, and by exploring the connections between observed electromagnetic counterparts and their binary progenitors, the central engine will be probed in unprecedented ways. For example, observing GRBs in coincidence with those binaries for which numerical simulations predict the formation of accretion disks would indicate a connection between accretion disks and GRBs. Already, analyses of GW data from initial detectors targeting gamma-ray bursts have helped rule out binary mergers as the central engines of selected events [@070201; @051103].
2. [**Population prospects**]{} – The populations of short GRBs [@2007ApJ...664.1000B], as well as compact binaries [@2010CQGra..27q3001A], are highly uncertain. The observation of GWs from compact binary coalescences could be an effective way of determining the source population. Due to the very large field-of-view of GW detectors and the very weak beaming of GW emission, they can be used efficiently to locate practically all sources within their well-determined horizon distance of hundreds of megaparsecs. As source distances can also be determined using GWs (see next subsection), the binary coalescence population could also be mapped as a function of redshift, although only with the sensitivity of third-generation GW detectors [@2009arXiv0906.4151S].
3. [**Cosmological parameters**]{} – The detection of a binary coalescence GW signal from an observed short GRB could be used to determine the distance and redshift of the source [@1986Natur.323..310S; @1993PhRvD..48.4738M; @2010ApJ...725..496N]. In fact, the determination can be done independently of the cosmological distance ladder. Distances of short GRBs can be reconstructed to a $\sim10-30\%$ precision for $\lesssim500\,$Mpc for NS-NS, and for $\lesssim1.5\,$Gpc for NS-BH binaries using the Advanced LIGO-Virgo detector network [@2010ApJ...725..496N]. Reconstructed source distances can then be used to accurately reconstruct the luminosity of short GRBs, as well as to test Hubble relation [@1986Natur.323..310S], with one year of observation with advanced detectors potentially allowing for a $\sim2\%$ precision [@2006PhRvD..74f3006D]. Further, in combination with measurements of the cosmic microwave background, reconstructed binary merger distances could be used to constrain the dark energy equation of state [@2006PhRvD..74f3006D].
4. [**Jet angular structure**]{} – The angular structure of relativistic jets from short GRBs is poorly constrained [@2011ApJ...732L...6R]. The rate of binary mergers detected through GWs, together with the rate of short GRBs for which a binary progenitor can be confirmed with GWs, could be used to determine the opening angle of short GRBs.
Further, the polarization of the inspiral GW signals in principle could be used to characterize the viewing angle of observed GRBs [@1538-4357-585-2-L89]. The GW polarization from a binary inspiral depends on the viewing angle compared to the rotational axis of the binary. Towards the rotational axis, the GW signal is circularly polarized, while the polarization becomes elliptical for off-axis observers, eventually becoming linear for observers in the equatorial plane. For sufficiently strong GW signals from the inspiral phase, reconstructing the eccentricity of the GW polarization could provide information on the opening angle. With a large number of measurements, the angular structure of the jet could be mapped as well [@1538-4357-585-2-L89]. These correlations can be compared to similar constraints that might follow from the detailed multiband light curve of the blastwave [@2011ApJ...733L..37V]. Nevertheless, due to the relatively weak change in GW polarization with angle, the GW polarization may only be identified for large ($\gg1^\circ$) angular differences, and/or for very high SNR ($\gtrsim100$), making the utility of GW polarization limited.
5. [**Neutron Star Equation of State**]{} – Nuclear forces have a profound influence on the structure of NSs (e.g., [@1998nucl.th...4027A; @2001ApJ...550..426L]). They determine, among other things, the relation between NS mass and radius, or the NS mass limit (for the rotating and non-rotating cases). The EOS of matter is poorly constrained at NS densities [@2008PhRvD..77b1502F]. The observation of NSs with masses up to $\sim2\,$M$_\odot$ [@2005ApJ...634.1242N] imply a stiff EOS (a NS with soft EOS would not be able to support this much mass).
The evolution of NS-NS/BH-NS mergers strongly depends on the NSs’ nuclear EOS. Therefore, the GW signal from binary coalescences can be used to determine/constrain the EOS. For instance, if a NS is tidally disrupted during a binary merger, the orbital frequency at which tidal disruption occurs can be used to determine the radius of the NS. This, together with the NS mass reconstructed from the inspiral gravitational waveform [@1994PhRvD..49.2658C], can be used to constrain the nuclear EOS [@2000PhRvL..84.3519V; @2007PhR...442..109L; @2010PhRvD..82d4049K; @2011PhRvD..84f4018K; @2012PhRvD..85d4061L; @2012PhRvL.108i1101M].
The tidal deformation of NSs in a binary system can affect the gravitational waveform even prior to the merger phase. The GW energy spectrum during the last few orbits of a NS-NS binary prior to merger can be used to determine the compactness ratio (mass/radius) of the NSs [@2002PhRvL..89w1102F; @2012PhRvL.108i1101M]. For these last few orbits of the binary, the GW frequency is at a high but still reasonably sensitive frequency for LIGO-type detectors, making the analysis of these last few orbits feasible.
The tidal deformation of a NS in the inspiral phase can be described by one parameter (the so-called *Love number* [@2008PhRvD..77b1502F]), which is effectively the ratio of the star’s induced quadrupole moment to the quadrupole moment of the perturbing tidal gravitational field of the binary companion. Flanagan & Hinderer [@2008PhRvD..77b1502F] showed that the nuclear EOS of NSs can be constrained even through the early inspiral phase due to the effect of tidal deformation on the waveform. Read et al. [@2009PhRvD..79l4033R] simulated the inspiral phase of NS-NS binaries, showing that the NS EOS can be constrained (the NS radius can be determined to within $\sim1$km precision, which, together with the NS mass, would rule out a part of the EOS parameter space) with advanced GW detectors for a source at a distance of $100\,$Mpc. BH-NS mergers can similarly be used to gain information on the NS EOS [@2011PhRvD..84j4017P; @2012PhRvD..85d4061L].
For some NS-NS mergers, collapse to a BH is delayed and a hypermassive NS is formed (see Section \[subsubsection:mergerphase\]). During the merger, non-axisymmetric oscillation modes of the hypermassive NS are excited, resulting in GW emission [@1994PhRvD..50.6247Z; @1999PhRvD..60j4021A; @2002PhRvD..65j3005O; @2002PThPh.107..265S; @2005PhRvD..71h4021S; @2011MNRAS.418..427S; @2012PhRvL.108a1101B; @2012PhRvD..86f3001B]. The waveform of the resulting gravitational radiation depends on the nuclear EOS. Consequently, the observed waveform can be used to significantly constrain the EOS [@2002PhRvD..65j3005O; @2002PThPh.107..265S; @2005PhRvD..71h4021S; @2011MNRAS.418..427S; @2012PhRvL.108a1101B; @2012PhRvD..86f3001B]. The GW signature of hypermassive-NS oscillations could be detected with SNR$=2$ out to $20-45\,$Mpc (depending on the EOS), which may be sufficient to constrain the EOS (at such distances, the source can be confirmed with high accuracy via GWs from the inspiral phase) [@2012PhRvL.108a1101B].
Further, it is possible that the inner part of a NS becomes strange quark matter at high densities. A star made of strange quark matter can be self-bound, a marked difference from hadronic NSs that are gravitationally bound [@Lattimer:2006]. The GW emission of a strange quark star is substantially different from that of hadronic NSs [@2007PhR...442..109L], hence GWs could be used to determine whether NS interiors may contain strange quark matter [@2002MNRAS.337.1224A; @2004MNRAS.349.1469O; @2005PhRvD..71f4012L; @2011PhRvL.107u1101S].
6. [**Magnetic fields in protoneutron stars**]{} – Magnetic fields are suspected to play a significant role in the origin of short GRBs [@2008PhRvD..78b4012L; @2010CQGra..27k4002D; @2011ApJ...732L...6R; @2012ApJ...755...80P], and are known to vary between individual pulsars (e.g., from their spindown rate [@1977ApJ...215..302F]). In the inspiral phase, internal magnetic fields in NSs (for realistic field strengths) have negligible effect on the dynamics of the binary coalescence [@2000ApJ...537..327I; @2008PhRvD..78b4012L; @2008PhRvD..78h4033B; @2009MNRAS.399L.164G; @2010PhRvL.105k1101C]. While magnetic tension in the NS can reduce tidal deformation, this effect will be too weak to be detectable with planned GW detectors for realistic pre-merger magnetic strengths ($\lesssim10^{14}\,$G) [@2009MNRAS.399L.164G]. For BH-NS binaries, magnetic fields seem not to alter the dynamics and therefore the gravitational waveform [@2010PhRvL.105k1101C].
Magnetic fields can significantly affect the post-merger behavior of NS-NS binaries, if a hypermassive NS is formed after the merger. Upon merger, magnetic fields will be amplified via Kelvin-Helmholtz instabilities in the shear layer between the merging NSs [@2006Sci...312..719P; @2008PhRvD..78h4033B], or later via differential rotation [@2006PhRvL..96c1101D; @2008PhRvD..78b4012L]. Even if magnetic fields are small prior to merger, the amplified fields can be strong enough to affect the evolution of the merger remnant and therefore GW emission. Magnetic fields in the hypermassive NS compete with GWs in dissipating angular momentum from the hypermassive NS. The decrease of angular momentum eventually results in the “delayed collapse" ($\sim10\,$ms$-1\,$s, depending on the mass and the EOS) of the hypermassive NS into a BH (e.g., [@2008PhRvD..78b4012L; @2012PhRvL.108a1101B]).
By probing the magnetic properties of a large number of NSs in binary mergers, one could determine, e.g., the highest achievable magnetic field frozen in a stationary NS ([@Lattimer:2006]; e.g., via the time difference between binary merger and GRB).
7. [**Accretion disks**]{} – The large-amplitude, quasiperiodic GW emission from instabilities in the accretion disk, probably together with the detection of the earlier inspiral phase from the binary, can be utilized to reconstruct properties of the accretion disk, even if no electromagnetic signal is observed from the source. Beyond reconstructing the mass and lifetime of the accretion disk, the development of the instabilities is sensitive to the properties of the accretion disk (see Section \[section:accretiondiskinstabilities\]), i.e. the development of the instabilities by itself can already be informative.
Core collapse {#section:collapsarphysics}
-------------
1. [**Progenitors of long GRBs**]{} – It is believed that the origin of at least some long GRBs are the core collapses of massive stars . This could be confirmed if the GWs in coincidence with a long GRB progenitor were detected. Nevertheless, if no GW emission is detected in coincidence with long GRBs, this does not rule out the core-collapse model, although may constrain core-collapse dynamics. If GWs are detected in coincidence with a long GRB, the internal evolution of a core collapse, which may largely depend on the progenitor, could also be examined. For example, the formation of a protomagnetar or a BH-torus system, or the fragmentation of the massive stellar core may be differentiated via GWs.
2. [**Neutron star equation of state**]{} – The mass, radius and rotation rate of NSs have a profound effect on their potential GW emission through rotational instabilities. Detecting the GW signature of these instabilities can be used to constrain the mass-radius relation, rotation, and therefore the EOS, of NSs (e.g., [@2011PhRvD..83f4031G]).
Further, rotational instabilities in NSs result in qualitatively different GW emission for conventional and strange quark NSs [@2011GReGr..43..409A; @2005PhRvL..95u1101O]. Whether a NS is composed of strange quark matter could be inferred from the detected GW signature of the NS’s rotational instability.
3. [**Neutron star internal physics**]{} – The evolution of the gravitational waveform from protoneutron star rotational instabilities strongly depends on the nuclear EOS at NS densities, as well as the physical parameters of the protoneutron star. Differential rotation inside a protoneutron star , temperature (e.g., [@2009PhRvD..79j4003B]), viscosity and neutrino cooling [@2009PhRvD..79j4003B] may all leave their mark on the evolution of rotational instabilities and the resulting GW emission. Further, if a hypermassive (or supramassive; see Section \[subsubsection:mergerphase\] for their definition) NS collapses into a BH due to either losing angular momentum or accretion, the rotational frequency prior to collapse provides information on the NS EOS [@2007PhR...442..109L]. This rotational frequency may be inferred from the GW signal from the rotating NS [@2011PhRvD..83f4031G; @2012arXiv1207.3805P].
4. [**Accretion physics**]{} – Because GW emission from massive stellar collapses can be produced by the accretion onto the central object following the collapse, the GW signal therefore can provide insight into accretion physics: (i) the mechanisms that transport angular momentum; (ii) instabilities; and (iii) interaction with the black hole.
For BH-torus systems the strength of viscosity (and possibly the strength of other processes through which the torus loses angular momentum) can be determined, as these are competing effects for angular momentum loss: The total loss of angular momentum can be inferred from the rate of change of the GW frequency, while the loss through GWs can be inferred from the GW signal amplitude.
Accretion of matter by the protoneutron star results in extended GW emission due to the intake of angular momentum [@1998ApJ...501L..89B; @2007PhRvD..76h2001A]. As the time scale and nature of accretion is likely different for accretions from supernova fallback material or material from a companion star, the extended GW signal from rotational instabilities will carry important information on the accretion mechanism.
5. [**Magnetic fields in neutron stars**]{} – Sufficiently strong magnetic fields inside a protoneutron star can be competing with GWs in radiating angular momentum away. As NS spindown may be reflected in the GW frequency, measuring the GW amplitude and NS spindown may provide information on the strength and nature of magnetic fields present in the NS.
Magnetars {#section:magnetarphysics}
---------
The detection of GWs in coincidence with a giant flare from a magnetar could provide information on the processes that lead to flaring in magnetars (i.e. tectonic activity [@1996Natur.382..518C] or global reconfiguration [@2001MNRAS.327..639I] vs origin from the magnetosphere [@2006MNRAS.368L..35L]). Non-detection, nevertheless, does not necessarily rule out these models. Upon detection, the frequency of GWs from NSs excited by, e.g., starquakes, could be used to infer the NS mass, radius and EOS [@2011MNRAS.414.3014C; @2012MNRAS.423..811C; @2012PhRvL.108t1101S], or potentially even the strength of magnetic fields inside the NS [@2011PhRvD..83h1302K].
Millisecond protomagnetars {#section:protomagnetarphysics}
--------------------------
The detection of millisecond magnetars’ GW signature in coincidence with a GRB would confirm their presence and role as the central engine behind some GRBs. Further, the GW signature of millisecond magnetars may indicate their origin (e.g., typical rotation rates may differ for different mechanisms; see Section \[section:millisecondmagnetars\]). Advanced GW detectors will also be able to determine, out to $\sim450\,$Mpc, whether the progenitor of a short GRB was a binary merger of a NS with another NS or a BH [@2010CQGra..27q3001A]. The confirmation of a compact binary merger would rule out the other scenarios in which millisecond magnetars can form, e.g., white dwarf accretion-induced collapses, or white dwarf binary mergers.
Observational strategies and prospects {#section:observationalstrategies}
======================================
The success of detection of GWs from GRB progenitors depends on the strength of the emitted GWs, the sensitivity of GW observatories, as well as the observation strategies used to separate GW signals from the background. The search strategy, e.g., the use of multiple messengers, can also add to the information one can extract from the source.
Various transient GW search strategies exist, aiming to find the GW signature of GRBs. These strategies differ mainly in their prior assumptions on the gravitational waveform. In the most general case, one can look for so-called GW bursts with minimal assumptions on the waveform, namely defining a maximum signal duration and bandwidth [@Klimenko:2005; @Xpipeline1367-2630-12-5-053034; @RollinsThesis]. These generic burst searches can be modified to include additional information about the source. For instance if one considers an accretion-type emission, circular (on-axis) or linear (off-axis) polarizations can be required from GW signals, increasing detection sensitivity for signals satisfying these conditions. Other model-dependent assumptions include, e.g., GW signals rapidly evolving characteristic frequency [@2010CQGra..27s4017C], such as in the case of compact binary mergers.
For compact binary coalescences, the gravitational waveform for a given mass configuration can be accurately calculated for the initial inspiral and final ringdown phases, and to some extent the merger phase [@2011PhRvD..83l2005A]. For such systems with properties in a highly limited parameter space, it is beneficial to use a signal waveform template bank and perform matched filtering based analysis (e.g., [@2010PhRvD..82j2001A]).
Below we review the GW search strategies targeting the progenitors of GRBs. We indicate the prospects of such searches through presenting projected sensitivity estimates based on observational results with initial detectors. Advanced detectors will be $\sim10\times$ more sensitive than initial detectors, the exclusion distances therefore will be $\sim10\times$ greater, corresponding to a $\sim$1000-fold improvement in achievable source rate constraints. Further, advanced detectors will be sensitive in a wider frequency band. Additionally, the advanced detector network will eventually be larger, with the inclusion of KAGRA and probably LIGO India. These improvements will further increase the achievable sensitivity of the advanced detector network [@2012arXiv1210.6362N], and will further improve signal parameter reconstruction (e.g., source direction; [@2011PhRvD..83j2001K]) [@2012arXiv1210.6362N].
Gravitational wave search strategies
------------------------------------
Search strategies for GW transients can be divided into four main categories based on their use of other messengers.
1. **Untriggered** searches, in which no information is used from other messengers. The analyses use GW data alone to find astrophysical sources (e.g., [@2010PhRvD..81j2001A; @2010PhRvD..82j2001A]).
2. **Externally-triggered** searches, in which one specifically looks for GW signals from sources confirmed via other messengers. For instance one can search for GW signals from the progenitors of detected GRBs, using their location, time, and other parameters (e.g., [@2005PhRvD..72d2002A; @S2S3S4; @2008ApJ...681.1419A; @2010ApJ...715.1438A; @2010ApJ...715.1453A; @2011ApJ...734L..35A]).
3. **Electromagnetic follow-up** searches, in which GW signal candidates are used to trigger electromagnetic follow-up searches with other telescopes (e.g., [@2003ApJ...591.1152S; @2007AAS...211.9903P; @2008CQGra..25r4034K; @2010AAS...21540606S]).
4. **Multimessenger** searches, in which one uses (typically sub-threshold) signal candidates from multiple observatories of different messengers in a joint search (e.g., [@2009IJMPD..18.1655V; @Baret20111; @PhysRevLett.107.251101; @2012PhRvD..85j3004B]).
In the following we briefly review some of the GW search strategies and past GW searches for GRBs, organized along the above categories.
### Untriggered searches {#section:allsky}
— Untriggered transient searches provide a generic way to identify plausible GW sources without relying on the detection of other messengers. While GRBs are highly beamed (e.g., [@2001ApJ...562L..55F; @2005Natur.437..845F; @2006ApJ...650..261S; @2006ApJ...653..468B; @2006ApJ...653..462G; @Liang2007; @2012ApJ...756..189F]), the beaming of GW emission from GRB progenitors is weak (e.g., [@1538-4357-585-2-L89]). For realistic GRB beaming angles, there will likely be more observations of GWs from off-axis GRB progenitors than GW observations triggered by GRBs [@2012arXiv1206.0703C].
To help quantify the source population detectable with untriggered (and other) searches, it is useful to characterize the sensitivity of a GW observatory to a GW source with the so-called *horizon distance* $D_h$. The horizon distance is defined as the distance at which a GW source from optimal orientation (such that emission towards the Earth is the strongest) and optimal location (i.e. the most sensitive direction of the GW detector) is observed with a given SNR (typically chosen to be $\rho=8$ for untriggered searches; see also Section \[subsubsection:extrig\]) by a single GW detector. Compared to this optimal orientation and location, the orientation and direction-averaged distance at which the source produces the $\rho=8$ is $\sim D_h/2.26$ (both direction and orientation-averaging contribute a factor $\sim1.5$).
For a NS-NS binary with 1.35 M$_\odot$ mass for each NS, the projected horizon distance for untriggered searches with advanced detectors (upon non-detection) is [@2010ApJ...716..615O] $$D_h^{untrig.}\simeq 450\,\mbox{Mpc}\,(\mathcal{M}_c/1.2\,\mbox{M}_\odot)^{5/6},
\label{eq:Dh}$$ where $\mathcal{M}_c=(m_1m_2)^{3/5}/(m_1+m_2)^{1/5}$ is the so-called chirp mass (i.e. the horizon distance only depends on this specific combination of the masses of the two compact objects). For higher-mass binaries, the horizon distance is somewhat lower than the prediction of Eq. \[eq:Dh\] (see [@2012arXiv1206.0703C]).
### Externally triggered searches {#subsubsection:extrig}
— External triggers in GW searches provide additional information and increased search sensitivity for GW analyses [@1993ApJ...417L..17K; @FMR99]. For example an external GRB trigger reduces the temporal and spatial extent in which one has to search for a GW signal [@2009RPPh...72g6901A]. A detected external trigger can also be used to set constraints on the GW signal. For example a short GRB implies that the GW signal is probably a binary merger.
Chen & Holz [@2012arXiv1206.0703C] estimates that for GRB beaming factor $f_b\leq7.5$ (corresponding to opening angle $\theta=30^\circ$), the number of triggered and untriggered detections of GWs from GRB progenitors would be comparable. Further, even for greater beaming factors (i.e. smaller opening angles), externally triggered searches will enable the detection of some additional, particularly interesting, sources. The extra information in external triggers can be taken into account in the horizon distance by effectively lowering the SNR threshold (i.e. by keeping the false-alarm rate constant). The results of Chen & Holz imply that the sensitivity of externally triggered searches is greater than that of untriggered searches by a factor $\sim1.3$ (for inspiral searches; this factor is similar for other types of searches as well), i.e. $$D_h^{ext.trig.}\approx 1.3D_h^{untrig.}.$$ The same sensitivity increase was found by Dietz et al. [@2012arXiv1210.3095D]. While this increased horizon distance means that many more sources fall within observable reach, the actual number of detected sources will be decreased by the beaming of the electromagnetic (or other) signal, as well as the sky coverage of the available electromagnetic telescopes. On the other hand, all triggered sources will be detected face-on, while untriggered sources can take any orientation, which adds to the sensitivity of triggered searches compared to untriggered searches (for binary mergers, the GW amplitude is $\sim1.5$ times greater face-on than averaged over all directions). Chen & Holz [@2012arXiv1206.0703C] find that the externally-triggered observation rate is a sizable addition, although less than, the untriggered GW observation rate, while Kelley et al. [@2012arXiv1209.3027K] find that GRB-triggered searches present no appreciable addition to the total rate [@2012arXiv1209.3027K]. While both of these studies approximate GW background as purely Gaussian, real GW background features a non-Gaussian tail that may increase the importance of externally triggered and follow-up searches.
Previous GRB-triggered GW searches include searches for both unmodeled GW bursts and compact binary mergers. For example the short GRBs 070201 and 051103 had electromagnetic positions overlapping nearby galaxies (Andromeda and M81, respectively). No binary merger counterpart was found in either case, ruling out a binary progenitor in those galaxies, favoring an SGR model (a binary progenitor from a more distant galaxy is not ruled out, albeit it is unlikely) [@0004-637X-681-2-1419; @2012ApJ...755....2A]. Nevertheless, it is worth noting that, given their expected rate of occurrence, a binary merger within the distance of Andromeda or M81 ($<4\,$Mpc) would be extremely unlikely. For GRB-triggered GW searches with initial LIGO/Virgo, see also, e.g., [@2012ApJ...760...12A].
Several GW searches aimed to identify GWs in coincidence with SGR flares [@2007PhRvD..76f2003A; @2008PhRvL.101u1102A; @2009ApJ...701L..68A; @2011ApJ...734L..35A]. Most recently Abadie *et al.* [@2011ApJ...734L..35A] used 1279 flares from six magnetars as triggers, aiming to identify GW signals from neutron-star $f$-mode ringdowns or other GW-producing mechanisms. The search set constraints on the energy of GW emission from these flares comparable to some giant flares’ electromagnetic energies, and an order of magnitude below previously existing limits. For a nearby magnetar (SGR 0501+4516) located at $\sim1-2\,$kpc that emitted a large number of flares during the observation period, the obtained upper limit on $f$-mode GW emission was $\sim 10^{47}$ erg. The upper limit for GW emission at $\sim10^2$ Hz, i.e. within most sensitive frequency band of LIGO/Virgo, was $\sim3\times10^{44}$ erg, a promising limit compared to some theoretical upper limits on GW emission from SGRs [@2011PhRvD..83j4014C]. With Advanced LIGO/Virgo, one can expect, beyond more stringent limits on $f$-mode emission, constraints on the maximum emissible *total* energy from other NS oscillation modes (we note that these upper limit are constraining only if a substantial fraction of total energy goes into GWs, which is debated; e.g., [@2012arXiv1206.6604C]).
### Electromagnetic follow-up
— With the rise of a global GW detector network, it became possible to reconstruct the direction of a GW signal candidate, albeit with a relatively large uncertainty. Such reconstruction enables the use of GW signal candidates in triggering electromagnetic follow-up observations [@2003ApJ...591.1152S; @2007AAS...211.9903P; @2008CQGra..25r4034K]. There are various scientific advantages of such a follow-up search:
1. As electromagnetic observations cannot continuously cover the whole sky with high sensitivity, GWs can be used to guide these telescopes and point towards the right direction at the right time.
2. Often the most interesting and strongest emission from an electromagnetic transient occurs in a short period after its onset. Since GW emission is, in many cases, expected to precede the onset of electromagnetic emission, telescopes have the chance to commence follow-up observation in a very early emission stage, or even catch the onset of the electromagnetic event.
3. GW signal candidates will often have too low significance to be unambiguously identified as extraterrestrial signals. The observation of an electromagnetic follow-up event can enhance the significance of the joint observation, making detection more probable.
4. Similarly to other multimessenger searches, information from the different messengers can enhance the information (and therefore science) that one can extract from the source.
Electromagnetic follow-up searches of GW event candidates is one of the most promising and one of the fastest-evolving areas of GW astrophysics (see for recent observational development with the LIGO-Virgo GW detectors). See Section \[section:EMcounterpart\] below for a more detailed description of the potentially detectable electromagnetic counterparts and search strategy.
The sensitivity of electromagnetic follow-up searches will differ from the sensitivity of externally triggered searches due to (i) the limited reach of electromagnetic transient observations and (ii) the finite sky area that is scanned by observatories following up the GW event candidate. Taking (i) into account in defining the horizon distance, one gets $$D_h^{follow\mbox{-}up}\approx \mbox{min}\left\{D_h^{ext.trig.},D_{\mathrm{EM}}\right\},$$ where $D_{\mathrm{EM}}$ is the reach of the electromagnetic follow-up observatory. Note that $D_h^{ext.trig.}$ refers to on-axis sources, while follow-up observations are the most interesting for off-axis sources (due to the greater source rate). For off-axis directions, GW emission is typically weaker by a factor $\sim1.5$. The rate of follow-up observations further depends on sky coverage of the follow-up observatories, as well as the beaming of the electromagnetic signal. For a survey of the electromagnetic detectability of GW event candidates, see [@2012arXiv1210.6362N].
### Multimessenger analyses
— Multimessenger analyses generalize the idea behind externally triggered searches by combining information from sub-threshold signal candidates of various messengers into one, more powerful analysis. While in many cases each type of sub-threshold messenger would individually have too low significance to be identified as an astrophysical signal of interest, the combination of individual significances can greatly enhance the analysis’ potential of identifying astrophysical events (see Section \[chapter:GWHENsearch\] for a multimessenger search).
The *Astrophysical Multimessenger Observatory Network* (AMON) initiative [@AMON] plans to combine sub-threshold triggers of multiple messengers in a low-latency coherent multi-messenger analysis. AMON is planned to combine multiple messengers to identify candidates that would be sub-threshold events for individual observatories. The extracted information would be used, besides claiming detection itself, to initiate electromagnetic follow-up observations, which can add to the number of coincidentally observable messengers.
The sensitivity of multimessenger searches depends on the false alarm statistics of the different detectors included in the search, as well as the types of messengers. Transient messengers with amplitude signatures, such as GWs or high-statistics photon signals, will have a relatively well defined horizon distance, beyond which the detection of such a messenger is highly unlikely. Other, discrete messengers, such as high-energy neutrinos, will have no such horizon distance, since there is a finite probability of detecting at least one neutrino even from a very distant source with expected signal strength $\ll1$ neutrino. These distant sources can significantly contribute to the total number of detectable multimessenger sources, especially in the case in which their presence can be confirmed with other messengers (see, e.g., [@2012PhRvD..85j3004B]). For such discrete messengers, the GW-multimessenger horizon distance is comparable to that of externally triggered searches: $$D_h^{multimessenger}\sim D_h^{ext.trig.}.$$ The rate of detected multimessenger sources will be limited by beaming, and the probabilistic nature of observing at least 1 discrete messenger (e.g., high-energy neutrino [@PhysRevLett.107.251101]).
Electromagnetic counterparts {#section:EMcounterpart}
----------------------------
This section outlines the observable electromagnetic emission of GRB progenitors, focusing on counterparts other than gamma rays, and their utility in GW searches.
Due to the highly beamed emission of most GRBs[^8], the majority of GRBs are off-axis, i.e. their prompt emission cannot be observed from the earth. Nevertheless, weaker, off-axis electromagnetic emission from GRB progenitors could be detected during GW follow-up observations, if the electromagnetic emission is present beyond a few minutes after the prompt emission (i.e. the delay between detecting a GW signal candidate and pointing a telescope towards the expected source location).
Most compact object mergers observed by LIGO/Virgo will not be accompanied by observable GRBs due to the small gamma-ray beaming angle. For this reason, the last few years have seen extensive investigation into rapid follow-up search strategies for more isotropic, but more difficult to identify, electromagnetic counterparts . Follow-up searches aim to minimize the time between the identification of a GW event candidate and the start time of the observation with electromagnetic telescopes in the required directions. The time delay of sending a request for follow-up search to electromagnetic telescopes after a GW event was $\sim$tens of minutes for initial searches (mostly due to the manual confirmation of event selection). This delay will likely be reduced to $\sim$minutes for searches with advanced detectors [@2012SPIE.8448E..0QS].
The uncertainty in direction reconstruction of GW event candidates with the planned advanced detector network will likely be in the $\sim$tens of square degrees range (e.g., [@2011PhRvD..83j2001K; @2012arXiv1210.6362N]). With such an uncertainty, telescopes with large fields of view will need to be utilized, focusing on electromagnetic emission that is ongoing for hours-days so the telescopes can scan through the interesting sky area. Existing galaxy catalogs can also be used to largely decrease the sky area that needs to be mapped (see Section \[subsection:hostgalaxies\]). We note here that, besides GW observations, the observation of the prompt gamma-ray emission itself has a position resolution of a few degrees, i.e. comparable to the resolution of GW observations. Afterglow observations are therefore also important for the direction reconstruction of observed GRBs.
Beyond direction reconstruction, GRB afterglows also promise to carry important information connected to GW radiation. Corsi & Mészáros [@2009ApJ...702.1171C] pointed out that the observed X-ray plateau in the afterglow of some GRBs may be connected to the formation of a highly magnetized millisecond pulsar that can lose angular momentum through GWs during a 10$^3-10^4\,$s period (see also [@2001ApJ...552L..35Z]).
Several promising emission models have been identified which predict longer-lived, weakly (or not at all) beamed emission that can be detected on a distance scale comparable to the sensitivity range of advanced GW detectors. In the following list of these models, we focus on optical counterparts of compact binary coalescences. For a survey of the detectability of some of these emission models, see, e.g., [@2012arXiv1210.6362N].
1. [**Electromagnetic remnants from compact binary mergers**]{} — NS-NS and NS-BH mergers can eject energetic (sub-relativistic or relativistic) outflows. These energetic outflows can interact with matter in the interstellar medium, creating a long-lasting radio signal [@2011Natur.478...82N; @2012arXiv1204.6242P]. Radio remnants from dynamically ejected sub-relativistic material can appear months to years after the binary merger, and can last for several years [@2012arXiv1204.6242P]. A deep radio survey can identify these radio remnants out to $\sim300\,$Mpc [@2011Natur.478...82N; @2012arXiv1204.6242P]. Mildly/ultra-relativistic outflows may create even brighter radio emission, on the time scale of weeks [@2012arXiv1204.6242P]. The detectability of the radio transient strongly depends on circum-merger density, and therefore the signal may be lower for binaries that left their galaxy prior to merger.
An interesting case is shocks generated within the NSs upon the merger of a NS-NS binary [@2011PhRvL.107e1102S; @2012PhRvD..86f4032P], which can plausibly drive ultra-relativistic outflows. These shock waves, besides heating the NSs, result in shock breakout from the NSs’ surface, driving a nearly omnidirectional, ultra-relativistic outflow [@2012arXiv1209.5747K]. The outflow decelerates in the ambient medium, resulting in a bright X-ray flare seconds after the merger that could be detectable with, e.g., Swift XRT [@2005SSRv..120..165B]. The flare gradually changes to optical and radio as the outflow decelerates, potentially detectable with, e.g., Pan-STARRS [@2002SPIE.4836..154K] and EVLA [@2011ApJ...739L...1P].
2. [**Macronovae/kilonovae**]{} — Matter ejected from compact binary encounters may produce electromagnetic optical or near-infrared transients that are suitable for follow-up searches of GW signal candidates [@1998ApJ...507L..59L; @2005astro.ph.10256K; @2005ApJ...634.1202R; @2010MNRAS.406.2650M; @2011ApJ...734L..36S; @2011ApJ...736L..21R; @2011ApJ...738L..32G; @2012ApJ...746...48M; @2012arXiv1204.6240R; @2012arXiv1204.6242P]. Matter ejected by binary mergers (tidal tails or accretion disk outflows; e.g., [@2005astro.ph.10256K]), initially at nuclear densities, expands and undergoes r-process nucleosynthesis, producing heavier, radioactive elements. The decay of such elements produces an isotropic emission that lasts for days, named “macronova" [@2005astro.ph.10256K] or “kilonova" [@2010MNRAS.406.2650M]. This is an appealing model, because it suggests that virtually all LIGO detected NS-NS or BH-NS mergers would have detectable optical counterparts. However, finding these counterparts in the large GW directional uncertainty region would likely require a combination of powerful survey telescopes and follow-up spectroscopy, as used by Palomar Transient Factory, and Pan-Starrs, or the future LSST [@2012IAUS..285..358M; @2012arXiv1209.3027K; @2012arXiv1204.6242P; @2012ApJ...746...48M].
3. [**X-ray/optical afterglow**]{} — The relativistic outflow from a GRB central engine interacts with matter in the circumburst medium, emitting a long-lasting electromagnetic afterglow in a wide range of frequencies (e.g., [@2011ApJ...733L..37V; @2012ApJ...746...48M]). At X-ray and optical frequencies (lasting from minutes to a day), afterglow emission is the strongest shortly after the prompt GRB in the direction of the relativistic outflow (i.e. on-axis), and rapidly decays afterwards [@2011ApJ...733L..37V]. On-axis X-ray emission is detectable (e.g., with the Swift XRT telescope) over a period of $\sim1\,$day after the prompt burst for source distances out to the horizon distance of GW telescopes [@2012ApJ...759...22K]. Off-axis X-ray and optical emission, far from the axis of rotation, peaks at a significantly lower flux, making the detection of off-axis X-ray less useful in multimessenger searches [@2011ApJ...733L..37V]. For radio frequencies around $\sim10\,$GHz, on-axis emission for long GRBs peaks at 3-6 days in the source rest frame after the prompt burst [@2012ApJ...746..156C], while off-axis emission peaks on a similar time scale.
4. [**Magnetic interaction between BH & NS**]{} — For NS-BH binaries, once the BH enters the magnetosphere of the NS, it can interact with the NS magnetic field [@2011ApJ...742...90M]. For strong enough NS magnetic fields ($\sim10^{12}\,$G), such an interaction results in the copious emission of electromagnetic radiation, via a mechanism analogous to the Blandford-Znajek mechanism [@1977MNRAS.179..433B]. The electromagnetic radiation may resemble an extremely short ($\mathcal{O}(\mbox{ms})$) GRB , and could be detected from distances of $\gtrsim60\,$Mpc using the Swift or Fermi satellites [@2011ApJ...742...90M]. Although this mechanism produces a short electromagnetic emission, it is expected to be less beamed than the “standard" gamma-ray emission.
5. [**Magnetic interaction between NS & NS**]{} — The magnetic interaction of NSs in NS-NS binaries can drive an electromagnetic (X-rays or gamma-rays) signal within the last few seconds prior to merger through electric dissipation [@2012ApJ...755...80P]. As the non-magnetic NS moves through the magnetosphere of the other, highly magnetized NS ($\sim10^{12}-10^{14}\,$G), the induced electromagnetic force across the non-magnetic NS sets up a circuit connecting the two stars. Some of this electric potential dissipates either in the surface layer of the magnetic NS or in the space between the two stars, depending on the resistivity of the space between the two stars. A part of the dissipated energy is carried away through electromagnetic emission, which can reach up to $\sim10^{44}\,$erg$\,$s$^{-1}$ in the last $\lesssim1\,$s prior to merger [@2011PhRvD..83l4035L; @2012ApJ...757L...3L].
6. [**Neutron star flares induced by tidal crust cracking**]{} — NSs in inspiraling binary systems are subject to tidal deformation. When this deformation exceeds a critical level, the crust of the NS cracks, possibly resulting in a NS flare similar to observed flares [@1992ApJ...398..234K; @2010PhRvD..82b4016P; @2010ApJ...723.1711T; @2012ApJ...749L..36P] (see Section \[subsection:magnetars\] on magnetar giant flares). Such flare would represent an isotropic electromagnetic counterpart for binary mergers. Observed short-GRB precursors may be created by crust cracking [@2010ApJ...723.1711T].
Neutrino counterpart
--------------------
Many GW sources, and in particular GRBs, are expected to be copious emitters of neutrinos [@0004-637X-657-1-383; @0004-637X-690-2-1681; @1997PhRvL..78.2292W; @2001PhRvL..87q1102M; @ET; @P1000062]. Astrophysical neutrino emission is expected within two main sub-groups based on two distinct emission mechanisms, with two distinct energy ranges. MeV energy neutrinos (with energies $\epsilon_\nu \lesssim 100\,$MeV) are produced in the extremely hot, dense central regions of core-collapse supernovae and probably GRBs. High-energy neutrinos (with energies $\epsilon_\nu \gtrsim 100\,$GeV) are expected to be emitted by shock accelerated particles in relativistic outflows driven by the central engine of the GRB (e.g., [@1997PhRvL..78.2292W]). So far only MeV energy astrophysical neutrinos have been confirmed, and in one instance, for supernova 1987A [@PhysRevLett.58.1490; @PhysRevLett.58.1494].
One of the advantages of joint GW-neutrino searches [@PhysRevLett.107.251101; @2012PhRvD..85j3004B] is that GW and neutrino detectors (MeV and high energy) continuously observe the whole sky (one high-energy neutrino detector observing half the sky), recording signal candidates without the need to “point” the detector in a particular direction. Such full sky coverage is of special importance for multimessenger searches as the sky coverage for each messenger needs to overlap for a joint search. Further, while electromagnetic follow-up searches require low-latency response from electromagnetic telescopes, GW-neutrino observations can be performed with high latency without loss of information. Nevertheless, fast analysis of joint GW-neutrino observations can also enable the electromagnetic follow-up of joint event candidates.
### MeV neutrinos
— In core-collapse supernovae, most of the released gravitational binding energy is emitted in a burst of $\sim$MeV neutrinos (e.g., ). Both MeV neutrino [@PhysRevLett.104.251101; @2011arXiv1108.0171I] and GW emissions [@Ott2008; @0004-637X-707-2-1173] are expected to commence near core bounce within a few milliseconds ($\lesssim 10$ ms). This temporal correlation is orders of magnitude tighter than correlation with electromagnetic signals [@Baret20111; @P1000062], and can greatly enhance the sensitivity of a search for joint GW-MeV neutrino sources. MeV neutrino emission peaks within a fraction of a second after bounce, while emission continues for up to tens of seconds as the protoneutron star cools and contracts .
The joint detection of GWs and MeV neutrinos could provide constraints on the core-collapse supernova mechanism as well as information on the properties of matter at high energies and densities [@P1000062]. For example the neutrino spectrum from core-collapse supernovae depend both on the nuclear EOS as well as the spin of the core [@0004-637X-685-2-1069]. One can break this degeneracy using the additional information available in the GW channel, inferring information on both the EOS and the spin of the core. A potential secondary collapse due to hadron-quark phase transition in the protoneutron star during its post-bounce evolution [@PhysRevLett.102.081101] would also result in characteristic neutrino and GW emissions [@PhysRevLett.102.081101; @MNR:MNR14056].
Long and short GRB central engines are also probably strong emitters of MeV neutrinos [@0004-637X-657-1-383; @0004-637X-690-2-1681; @2011ApJ...737....6S; @2012arXiv1206.5927S; @2012arXiv1206.5927S]. As the progenitors of long GRBs are likely the progenitors of core-collapse supernovae as well , emission from long GRBs may commence similarly to the case of core-collapse supernovae. In the core-collapse scenario, neutrino emission from the core ceases upon black hole formation, while a newly formed accretion disk will become a significant source, as a large fraction of the accretion energy can be carried away by neutrinos [@0004-637X-657-1-383].
Compact binary mergers – the likely progenitor of short GRBs – are expected to emit MeV neutrinos [@0004-637X-657-1-383; @0004-637X-690-2-1681] from the accretion disk around the central black hole. In the case of NS-NS binary mergers, a hypermassive NS may form prior to BH formation, and further produce neutrino [@0004-637X-690-2-1681] as well as GWs [@2008PhRvD..78h4033B] emission. MeV neutrino emission from compact binary mergers is expected to commence at the beginning with the merger phase, in coincidence with the GW burst due to the merger. This GW burst signal is preceded by the much longer inspiral phase, and followed by the ringdown phase. Such temporal coincidence can be utilized in a joint search. Nevertheless, for typical distances of binary mergers, MeV neutrinos would be difficult to detect [@2003MNRAS.342..673R].
Several large-scale MeV neutrino detectors are currently in operation. These include Super-Kamiokande (Japan) [@2003NIMPA.501..418T], KamLAND (Japan) [@KamLAND], LVD (Italy) [@Aglietta:272638], Borexino (Italy) [@Alimonti2002205] and Baksan (Russia) [@2009arXiv0910.0738N]. Further, the IceCube high-energy neutrino detector [@IceCubeAhrens2004507] is also capable of detecting bursts of MeV neutrinos [@2011arXiv1108.0171I], albeit without the ability to reconstruct the source direction. Recognizing the importance of multimessenger observations, Super-Kamiokande, LVD, Borexino and IceCube are members of the Supernova Early Warning System (SNEWS) [@2004NJPh....6..114A]. These observatories send real-time triggers of detected supernova candidate events, which are distributed to the astronomer community to allow for low-latency follow-up electromagnetic (or other) searches.
The most sensitive current MeV neutrino detectors (Super-Kamiokande and IceCube) are able to detect the MeV neutrino signal of a supernova from up to $\sim$100 kpc [@2009PhRvD..80h7301H]. The expected event number within this distance, however, is relatively small outside of the Milky Way. Planned megaton detectors, such as LBNE [@2010JPhCS.203a2109M] or Hyper-Kamiokande [@2003nipb.conf..307N], could detect supernovae from up to $\lesssim$10 Mpc [@2005PhRvL..95q1101A] with a supernova rate of $\sim1$ per year. Multimessenger searches of MeV neutrinos and GWs could further increase these detectable supernova rates [@2010CQGra..27h4019L], and would provide increased confidence in a detected signal. This can be especially important when no electromagnetic counterpart is detected.
### High-energy neutrinos {#chapter:GWHENsearch}
— Non-thermal, high-energy ($\gg$GeV) neutrinos are thought to be created within relativistic outflows driven by the central engine [@1997PhRvL..78.2292W; @1999ApJ...521L.117E; @2001PhRvL..87q1102M; @2002bjgr.conf...30M; @2003PhRvL..90x1103R; @2003PhRvD..68h3001R; @2005PhRvL..95f1103A; @2005MPLA...20.2351R; @2006ApJ...651L...5M]. Emission from early GRB afterglows is also plausible (e.g., [@2007PhRvD..76l3001M]). The emission mechanism is likely similar for both long and short GRBs, the former expected to be the stronger emitter. Core-collapse supernovae with rapidly rotating cores [@2005MPLA...20.2351R; @HEN2005PhysRevLett.93.181101], magnetars [@2003ApJ...595..346Z; @2005ApJ...633.1013I] and millisecond protomagnetars [@2009PhRvD..79j3001M] are also thought to emit high-energy neutrinos. While recent limits obtained with the IceCube detector constrain some emission models [@2012Natur.484..351A], the standard fireball picture for neutrino emission remains viable [@PhysRevLett.108.231101; @2012ApJ...752...29H].
As their production mechanism is connected to that of gamma photons, high-energy neutrinos are probably beamed similarly to that of gamma ray emission. This decreases the rate of detectable joint sources similarly to the case of electromagnetic emission.
For a joint detection, an especially interesting class of sources are GRBs with little or no detectable electromagnetic counterpart, which are therefore only detectable via GWs and neutrinos. Such sources include so-called *choked* GRBs. In the core-collapse supernova scenario, gamma-ray emission is observable only if the relativistic outflow from the central engine, that is responsible for the production of gamma-rays, breaks out of the star [@2008PhRvD..77f3007H; @2012PhRvD..85j3004B]. The outflow, in order to advance, needs to be driven by an (active) central engine. If the breakout time of the outflow is longer than the duration of the active central engine, the outflow stalls, producing no observable gamma-ray emission [@2001PhRvL..87q1102M]. High-energy neutrinos are produced within the outflow and can escape through the stellar envelope, producing similar neutrino emission as for “successful” GRBs. Other interesting sources include low-luminosity (LL) GRBs, that can frequently go unobserved if their luminosity falls below the threshold of gamma-ray telescopes. LL GRBs, while weaker neutrino sources than their high-luminosity (HL) counterparts, are thought to be much more frequent, making their overall neutrino flux comparable or even surpass the flux from conventional HL GRBs [@2006ApJ...651L...5M; @2007APh....27..386G; @2007PhRvD..76h3009W]. Besides being interesting for sources not observed through their electromagnetic emission, GW-high-energy neutrino observations can provide information on the internal structure of the progenitor, as well as the properties and dynamics of the relativistic outflow (e.g., [@2003PhRvD..68h3001R]).
A potentially interesting subclass of GRBs for joint GW-high energy neutrino observations is low-luminosity GRBs [@2006Natur.442.1014S; @Liang2007; @2006ApJ...651L...5M; @2011ApJ...739L..55B] that may be a distinct population from high-luminosity GRBs [@Liang2007; @2007ApJ...659.1420T; @2011ApJ...739L..55B]. It was suggested that more massive stars may form a black hole after collapse, and produce a high-luminosity, high-Lorentz factor outflow, while less massive stars may form a neutron star after collapse, and drive an outflow with low luminosity and Lorentz factor [@2006Natur.442.1018M; @2007ApJ...659.1420T]. More recently, Bromberg et al. [@2011ApJ...739L..55B] found that estimated jet breakout times for low-luminosity GRBs are much longer than the observed durations, in marked difference with high-luminosity GRBs, suggesting that low-luminosity GRBs may be “choked", i.e. their relativistic outflow stalls before breaking out of the stellar envelope (see also [@2012arXiv1206.0700P; @2012ApJ...749..110B]). Low-luminosity GRBs are thought to be an order of magnitude more common than their high-luminosity counterparts, with lower electromagnetic luminosity and smaller beaming angle [@2006Natur.442.1014S; @Liang2007], making them a promising source type for multimessenger detections.
Joint searches for GWs and high-energy neutrinos are well-suited for multimessenger analyses, since both messengers have typically sub-threshold significances. For this purpose Baret et al. [@2012PhRvD..85j3004B] developed a multimessenger analysis method that combines the significance and other information from the two messengers.
Host galaxies {#subsection:hostgalaxies}
-------------
The location of potential or confirmed host galaxies of GRBs can be utilized in GW searches to enhance sensitivity as well as in event selection or localization. Long GRBs are found to occur in star-forming regions of distant galaxies, in accordance with their association to the deaths of massive stars . Consequently, the rate of long GRBs is correlated to the blue luminosity of galaxies . Long GRBs are actually found in the very brightest regions of their host galaxies, which are significantly fainter and more irregular than typical host galaxies of core-collapse supernovae, suggesting that they are associated with extremely massive stars, and galaxies with limited chemical evolution [@2006Natur.441..463F].
The distribution of short GRBs seem to follow that of old ($\sim1\,$Gyr) stellar populations within normal star forming as well as elliptical galaxies [@2007PhR...442..166N; @2009ApJ...690..231B; @2011NewAR..55....1B], in stark contrast with the distribution of long-GRB host galaxies [@2007PhR...442..166N]. It is highly unlikely that short and long GRBs are drawn from the same underlying population [@2009ApJ...690..231B]. Nevertheless, the rate of short GRBs in a given galaxy is correlated with the optical light of their host, and to less extent with the blue luminosity of the galaxy [@2010ApJ...708....9F], indicating that short-GRB progenitors have a wide age distribution of several Gyr. As neutron stars may experience an initial kick at birth [@1998AA...332..173P; @1998ApJ...496..333F; @2002ApJ...579L..63D], binary mergers can take place far away from the star-forming region where they originate from. For kick velocities of $\mathcal{O}(100\,\mbox{km}\,\mbox{s}^{-1})$ and inspiral times of several Gyr, binaries could travel large distances before their merger [@2007PhR...442..166N; @2010ApJ...725L..91K], traveling far outside their host galaxies. Typical predicted distances are $\sim10-100\,$kpc [@2007ApJ...664.1000B], the predicted distance distribution being well-matched by observed short-GRB distributions [@2010ApJ...722.1946B].
Magnetars [@1995MNRAS.275..255T; @2007PhR...442..166N] form another possible progenitor type of short GRBs, making up less than $\sim$one-third of their population [@2009ApJ...690..231B]. They are weaker and softer gamma-ray emitters than other short GRBs, making them identifiable mostly at smaller distances, within the Milky Way and the Large Magellanic Cloud. Magnetars are thought to be created in supernova explosions [@2011AdSpR..47.1326H]. As they can receive an initial kick velocity during the supernova explosion similarly to radio pulsars, only a fraction of them may be near its respective supernova remnant, in accordance with observations [@2011AdSpR..47.1326H]. Magnetars are observed to be mostly young ($\sim$10$^4$ yr) objects [@2011AdSpR..47.1326H], therefore they did not travel far from the star forming region of their respective supernovae. Consequently, extragalactic magnetar flares can be expected to occur within the star-forming regions of galaxies.
The expected distribution of GRBs can be used to enhance GW searches through identifying preferential GW source directions (e.g., [@2012PhRvD..85j3004B; @2012arXiv1210.6362N]). A priori information on source distribution has been used in multiple GW-GRB searches .
The galaxy distribution plays a significant role in electromagnetic follow-up searches. Since most GW signal candidates have a reconstructed direction with an uncertainty of a few degrees (which may be scattered over a larger area on the sky), the sky area of the possible GW source direction is greater than the field-of-view of most astronomical instruments used for follow-up observations. To decrease the surveyed area on the sky, the directions in which electromagnetic follow-up is performed is down-selected using the directions of galaxies overlapping with GW sky area of the possible GW source direction [@2008CQGra..25r4034K].
In multimessenger searches, the distribution of galaxies can be used to increase the significance of astrophysical signals as well as to reject background events whose directions do not overlap with the direction of a galaxy (e.g., [@2012PhRvD..85j3004B]). The significance of event candidates can be enhanced by weighing events with the a priory probability distribution of multimessenger sources (e.g., based on the luminosity and distance of galaxies).
Galaxy sky locations are especially important for GW sources that are detectable only within $\sim100\,$Mpc. On this distance scale, existing galaxy catalogs are more complete (e.g., [@0264-9381-28-8-085016]), and the number of galaxies is small within the sky area defined by the directional uncertainty of GW observations ($\sim$ few degrees; [@2012arXiv1210.6362N]). The use of galaxy direction on this scale could therefore significantly add to the sensitivity of these searches. Galaxy catalogs can still aid searches for sources at farther distances, although the sensitivity gain significantly reduces with distance [@2012arXiv1210.6362N].
Discussion & Outlook {#section:conclusion}
====================
In this review we presented an overview of the major directions in which gravitational-wave (GW) astronomy has the potential of advancing our understanding of GRBs and their progenitors. We discussed the different promising directions that can lead to GW emission from GRB progenitors that is detectable with second-generation GW observatories. We also discussed some of the major scientific questions that could be addressed upon the detection of GWs from these sources. Second generation GW observatories, commencing their operation in the next few years (e.g., [@LVCcommissioning]), will be sensitive enough to reach distances for which the detection of GW signals from GRB progenitors becomes possible.
The GW signal expected from compact binary coalescences, the likely progenitors of short GRBs, is fairly well understood as a function of the (often unknown) source properties. The detection of GWs can provide estimates or constraints on these source properties. We presented a detailed overview of the questions that can be addressed about short-GRB progenitors upon the detection of their GW signature.
The core-collapse of massive stars, the likely progenitor of long GRBs, has less constrained theoretical GW emission models. This is partly due to the lack of our detailed understanding of the progenitor (e.g., how cores maintain their high rotation rate prior to collapse), and partly due to the complex dynamics of the collapsing core and the resulting compact object (protoneutron star or black hole) and accretion disk, whose understanding requires further theoretical and numerical studies. Nevertheless, the core-collapse dynamics may lead to GW signals that are detectable on distance scales comparable to the lower end of the long-GRB distance scale ($\gtrsim100\,$Mpc). For example rotational instabilities in differentially rotating protoneutron-stars, as well as global instabilities in tori around black holes, both lead to rotating, non-axisymmetric systems with potentially strong GW emission.
Starquakes in highly-magnetized neutron stars, likely responsible for producing a fraction of short GRBs, result in seismic oscillations in the neutron star, which in turn may produce GWs detectable with second generation GW observatories. The detection (or non-detection) of such GWs coincident with the flaring activity of neutron stars can be informative, e.g., on the properties of matter at nuclear densities, as well as on the properties of astrophysical neutron stars.
The authors gratefully acknowledge the excellent comments, help, and suggestions of Alessandra Corsi, Jolien Creighton, Riccardo DeSalvo, Raymond Frey, Kunihito Ioka, Jonah Kanner, Kostas Kokkotas, Koutarou Kyutoku, Ilya Mandel, Brian Metzger, Kohta Murase, Ehud Nakar, Tsvi Piran, Maurice Van Putten, Luciano Rezzolla, Stephan Rosswog, Yuichiro Sekiguchi, Peter Shawhan and Masaru Shibata. The authors especially thank Richard O’Shaughnessy for his detailed comments throughout the preparation of the manuscript. IB & SM are grateful for the generous support of Columbia University in the City of New York and the National Science Foundation under cooperative agreement PHY-0847182. ROS & PB are supported by NSF award PHY-0970074 and the UWM Research Growth Initiative. ROS & PB appreciate the hospitality of the UCSB KITP, supported by the NSF with award NSF PHY11-25915.
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[^1]: Millisecond magnetars represent plausible alternative central engine candidates for some GRBs [@2007RMxAC..27...80T; @2008MNRAS.385.1455M; @2009MNRAS.396.2038B; @2012MNRAS.419.1537B]. See also, e.g., [@ADM:ME2002] and references therein, for further alternative models.
[^2]: Mass shedding commences much earlier, at significantly GW frequencies than tidal disruption. Note that Ref. [@2000PhRvL..84.3519V] identified mass shedding as the cutoff frequency, which underestimates the tidal disruption frequency [@2012PhRvD..85d4061L].
[^3]: Neutron stars are called hypermassive if they exceed the mass limit of rigidly rotating NSs [@2000ApJ...528L..29B]. NSs that are above the mass limit of non-rotating NSs, but whose mass could be supported by rigid rotation, are called *supramassive*; e.g., [@2006PhRvL..96c1101D]. For a binary with low-mass ($\sim1\,$M$_\odot$) NSs, it is possible that the resulting post-merger NS mass can be supported even without differential rotation (i.e. it is not hypermassive). In this case the NS can be long lived (see, e.g., [@2010ApJ...724L.199O]).
[^4]: Compare to the orbiting frequency $f_{\mathrm{ISCO}}=4.4(M/M_\odot)^{-1}\,$kHz of a particle around a Schwarzschild BH of mass $M$ at the innermost stable circular orbit (ISCO). The orbital frequency greatly depends on the BH spin (e.g., [@2012PhRvD..85f2002R] for the specific dependence).
[^5]: Note that a uniformly rotating, constant density star cannot reach such high $\beta$. The maximum rotation of a compact star is limited by mass-shedding to $\beta\sim0.1$ [@2003CQGra..20R.105A].
[^6]: Given $10\times$ sensitivity improvement for advanced detectors; based on the results of [@2008PhRvL.101u1102A] that used only one GW detector – multiple detectors could see even farther.
[^7]: The density of NSs with stiff EOS changes relatively slowly with pressure (as opposed to soft EOS).
[^8]: Beaming of short GRBs in general is not well constrained [@2007PhR...442..166N]. Nevertheless, beaming has been confirmed for a few short GRBs [@2006ApJ...653..468B; @2012ApJ...756..189F], indicating a highly beamed jet with half opening angles $\lesssim 10^\circ$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Total radiative thermal neutron-capture $\gamma$-ray cross sections for the $^{182,183,184,186}$W isotopes were measured using guided neutron beams from the Budapest Research Reactor to induce prompt and delayed $\gamma$ rays from elemental and isotopically-enriched tungsten targets. These cross sections were determined from the sum of measured $\gamma$-ray cross sections feeding the ground state from low-lying levels below a cutoff energy, E$_{\rm crit}$, where the level scheme is completely known, and continuum $\gamma$ rays from levels above E$_{\rm crit}$, calculated using the Monte Carlo statistical-decay code DICEBOX. The new cross sections determined in this work for the tungsten nuclides are: $\sigma_{0}(^{182}{\rm W}) = 20.5(14)$ b and $\sigma_{11/2^{+}}(^{183}{\rm W}^{m}, 5.2~{\rm s}) = 0.177(18)$ b; $\sigma_{0}(^{183}{\rm W}) = 9.37(38)$ b and $\sigma_{5^{-}}(^{184}{\rm W}^{m}, 8.33~\mu{\rm s}) = 0.0247(55)$ b; $\sigma_{0}(^{184}{\rm W}) = 1.43(10)$ b and $\sigma_{11/2^{+}}(^{185}{\rm W}^{m}, 1.67~{\rm min}) = 0.0062(16)$ b; and, $\sigma_{0}(^{186}{\rm W}) = 33.33(62)$ b and $\sigma_{9/2^{+}}(^{187}{\rm W}^{m}, 1.38~\mu{\rm s}) = 0.400(16)$ b. These results are consistent with earlier measurements in the literature. The $^{186}$W cross section was also independently confirmed from an activation measurement, following the decay of $^{187}$W, yielding values for $\sigma_{0}(^{186}{\rm W})$ that are consistent with our prompt $\gamma$-ray measurement. The cross-section measurements were found to be insensitive to choice of level density or photon strength model, and only weakly dependent on E$_{\rm crit}$. Total radiative-capture widths calculated with DICEBOX showed much greater model dependence, however, the recommended values could be reproduced with selected model choices. The decay schemes for all tungsten isotopes were improved in these analyses. We were also able to determine new neutron separation energies from our primary $\gamma$-ray measurements for the respective (n,$\gamma$) compounds: $^{183}$W ($S_{\rm n} = 6190.88(6)$ keV); $^{184}$W ($S_{\rm n} = 7411.11(13)$ keV); $^{185}$W ($S_{\rm n} = 5753.74(5)$ keV); and, $^{187}$W ($S_{\rm n} = 5466.62(7)$ keV).'
author:
- 'A. M. Hurst'
- 'R. B. Firestone'
- 'B. W. Sleaford'
- 'N. C. Summers'
- 'Zs. R[é]{}vay'
- 'L. Szentmikl[ó]{}si'
- 'M. S. Basunia'
- 'T. Belgya'
- 'J. E. Escher'
- 'M. Krti[č]{}ka'
bibliography:
- 'w\_prc\_2col\_r.bib'
title: '**An investigation of the tungsten isotopes via thermal neutron capture**'
---
\[sec:level1\]Introduction\
===========================
Neutron-capture decay-scheme data from the Reference Input Parameter Library (RIPL) [@capote:09] are required for nuclear-reaction calculations that are used to generate the Evaluated Nuclear Data File (ENDF) [@chadwick:11]. These data play a valuable role for both nuclear applications and basic research into the statistical properties of the nucleus including level densities and photon strengths. They also provide a wealth of structural information including discrete level spins and parities $J^{\pi}$ and $\gamma$-ray branching ratios. In addition, information on neutron-capture cross sections may also be obtained. Preliminary capture $\gamma$-ray cross sections were previously measured on natural elemental targets and published in the Evaluated Gamma-ray Activation File (EGAF) [@firestone:06]. For many elements only data for the isotopes with the largest cross sections and/or abundances could be obtained with natural targets. This paper describes a new campaign to improve the EGAF database by measuring isotopically-enriched targets.
Traditional methods for determining the total radiative thermal neutron-capture cross section, $\sigma_0$, include neutron-transmission and pile-oscillator measurements, both of which require precise knowledge of the neutron flux, and activation measurements which require an accurate decay-scheme normalization. Large corrections due to epithermal (1 eV to 10 keV), fast ($\gtrsim 10$ keV), and high-energy neutrons ($\gtrsim 1$ MeV) are typically necessary to determine thermal-capture cross sections. In this work we apply a newer method to determine the total radiative thermal neutron-capture cross sections for the tungsten isotopes using partial thermal neutron-capture $\gamma$-ray cross sections, $\sigma_{\gamma}$, measured with a guided thermal-neutron beam, combined with statistical-model calculations to account for unresolved continuum $\gamma$-rays, as described previously for the palladium [@krticka:08], potassium [@firestone:13], and gadolinium isotopes [@choi:13]. The prompt neutron-capture $\gamma$-rays were measured using both isotopically-enriched $^{182,183,186}$W targets and a natural elemental sample, to determine neutron-capture decay schemes for the compound tungsten nuclides $^{183,184,185,187}$W. This information was then used to normalize Monte Carlo simulations for the corresponding neutron-capture decay schemes calculated with the statistical-decay code [DICEBOX]{} [@becvar:98]. The neutron-capture $\gamma$-ray cross sections directly populating the ground state (GS) from low-lying levels were summed with the smaller, calculated, quasi-continuum contribution feeding the GS from higher levels to determine $\sigma_{0}$ for each tungsten isotope. Comparison of the simulated and experimental neutron-capture $\gamma$-ray cross sections populating and depopulating each excited state was also used to improve the tungsten decay schemes with the augmentation of more-complete data: determination of accurate $\gamma$-ray branchings, assessment of multipolarity and $\gamma$-ray mixing ratios ($\delta_{\gamma}$), placements of new $\gamma$-ray transitions, resolution of ambiguous (or tentative) energy-level and $J^{\pi}$ assignments, and neutron-separation energies ($S_{\rm n}$) determined from the observed primary $\gamma$-ray data for $^{183,184,185,187}$W. Also, as a validation of the current approach, the $\gamma$-decay emission probabilities, $P_{\gamma}$, were determined from the activation $\gamma$-ray cross sections corresponding to $^{187}$W $\beta^{-}$ decay. These measurements were found to be consistent with the adopted values, reported in the Evaluated Nuclear Structure Data File (ENSDF) [@ensdf], that are based on the work of Marnada *et al*. [@marnada:99].
Sample Mass \[mg\] $^{182}$W \[%\] $^{183}$W \[%\] $^{184}$W \[%\] $^{186}$W \[%\]
--------------- ------------- ----------------- ----------------- ----------------- -----------------
$^{\rm nat}$W 240 26.50(16) 14.31(4) 30.64(2) 28.43(19)
$^{182}$W 274 92.7(9) 2.0(3) 4.8(9) 0.5(1)
$^{183}$W 180 9.0(8) 74.9(3) 13.7(5) 2.4(3)
$^{186}$W 169 0.35(3) $\sim 0$ $\sim 0$ 99.65(3)
: \[tab:isocomp\] Isotopic composition of natural [@berglund:11] and enriched tungsten samples used in this work. The left-most column refers to the principal-enriched component in the sample.
\[sec:level2\]Experiment and Data Analysis\
===========================================
Isotopically-enriched stable and natural tungsten targets were irradiated with a supermirror-guided [*near-thermal*]{} neutron beam ($T \sim 120$ K; $E_{\rm beam} \sim 4.2$ meV) at the 10-MW Budapest Research Reactor [@rosta:97; @rosta:02]. The isotopic compositions of the enriched samples are shown in Table \[tab:isocomp\] and were determined by comparison with the ratios of peak intensities of strong, well-resolved transitions from the different tungsten isotopes in an elemental sample after accounting for their natural abundances. All enriched samples were oxide powders (WO$_{2}$) that were suspended in the evacuated neutron beam line in Teflon bags. During bombardment the thermal-neutron flux at the Prompt Gamma Activation Analysis (PGAA) target station was approximately $2.3 \times 10^{6}$ ${\rm n} \cdot {\rm cm}^{-2} \cdot {\rm s}^{-1}$. The PGAA facility is located $\sim 35$ m from the reactor wall in a low-background environment. The observed deexcitation $\gamma$ rays from the $^{A}$W(n,$\gamma$)$^{A+1}$W reactions were recorded in a single Compton-suppressed $n$-type high-purity germanium (HPGe) detector with a closed-end coaxial-type geometry, positioned $\sim 23.5$ cm from the target location. The PGAA facility is described in detail in Refs. [@revay:04; @szentmiklosi:10]. Energy and counting-efficiency calibrations of the HPGe detector were accomplished using standard radioactive and reaction sources covering an energy range from approximately $0.05-11$ MeV . The non-linearity and efficiency curves were generated using the $\gamma$-ray spectroscopy software package [HYPERMET-PC]{} [@hypermet], which was also used to perform peak-fitting analysis of the complex capture-$\gamma$ spectra.
Singles $\gamma$-ray data were collected in these (n,$\gamma$) measurements and peak areas for unresolved doublets, and higher-order multiplets, were divided based on branching ratios reported in the ENSDF [@ensdf]. Internal conversion coefficients for all transitions were calculated with the [BRICC]{} calculator, which is based on the Band Raman prescription [@kibedi:08].
\[sec:standard\]Standardization Procedure\
------------------------------------------
Partial neutron-capture $\gamma$-ray cross sections were derived from the measured peak intensities of the tungsten capture-$\gamma$ lines using an internal-standardization procedure where the observed $\gamma$-ray intensities are normalized by scaling to well-known comparator lines [@revay:03]. Here we used tungstic acid (H$_{2}$WO$_{4}$) for standardization [@szentmiklosi:pc] where hydrogen was used as the comparator with $\sigma_{\gamma}(2223~{\rm keV})=0.3326(7)$ b [@revay:03] with a stoichiometric $2:1$ H to W atomic ratio. The cross sections of the standardized tungsten transitions are listed in Table \[tab:tungstic\]. Cross sections for the more intense tungsten $\gamma$-ray transitions were measured with a natural elemental WO$_{2}$ target and then normalized to the standardized, strong, well-resolved cross sections from the standardization measurement using the well-known natural abundances [@berglund:11]. Weaker $\gamma$-ray transitions were measured in irradiations of enriched targets and similarly standardized. Since the tungsten isotopes and the calibration standard cross sections have a pure *1/v* dependence near thermal neutron energies i.e. increasing cross section with lower incident-neutron energy, no correction was necessary for the neutron-beam temperature.
Compound $E_{\gamma}$ \[keV\] $\sigma_{\gamma}$ \[b\]
----------- ---------------------- -------------------------
$^{187}$W 77.39(3) 0.234(4)
$^{187}$W 145.79(3) 1.344(13)
$^{187}$W 273.10(5) 0.380(4)
$^{187}$W 5261.68(6) 0.653(9)
$^{183}$W 6190.78(3) 0.726(10)
: \[tab:tungstic\] Elemental cross sections corresponding to strong lines observed in the tungsten compounds following an internal-standardization (n,$\gamma$) measurement with H$_{2}$WO$_{4}$ [@szentmiklosi:pc] comprising natural elemental tungsten.
\[sec:thickness\]Determination of the Effective Thickness\
----------------------------------------------------------
Since the WO$_{2}$ powders used in these measurements have a density of 10.8 g/cm$^{3}$, the intensity of low energy $\gamma$-rays must be corrected for self attenuation within the sample. To make this correction it is necessary to determine the [*effective*]{} sample thickness and calculate the intensity-attenuation coefficients as a function of $\gamma$-ray energy based on the prescription outlined in Ref. [@hubbel:95] using data from [XMUDAT]{} [@xmudat]. For irregular-shaped targets with non-uniform surfaces, such as the oxide powders used here, it is difficult to measure the sample thickness directly. Thus, to determine the effective WO$_2$ target thicknesses we compared the thin, lower-density (5.6 g/cm$^{3}$), attenuation-corrected tungstic acid target standardization-cross-section data, listed in Table \[tab:tungstic\], to the attenuated cross sections of these same transitions in the WO$_{2}$ targets. We then iteratively varied the sample thickness of the WO$_{2}$ targets until the calculated attenuation converged with the observed values for all transitions. An attenuation correction was then applied to all $\gamma$-rays in the spectrum.
\[sec:level3\]Statistical Model Calculations\
=============================================
The Monte Carlo statistical-decay code [DICEBOX]{} [@becvar:98] was used to simulate the thermal neutron-capture $\gamma$-ray cascade. [DICEBOX]{} assumes a generalization of the extreme statistical model, proposed by Bohr [@bohr:37] in the description of compound-nucleus formation and its subsequent decay. In thermal neutron capture the compound nucleus is formed with an excitation energy slightly above the neutron-separation energy threshold where particle evaporation is negligible. Within this theoretical framework, the [DICEBOX]{} calculation is constrained by the experimental decay scheme known up to a cut-off energy referred to as the critical energy, $E_{\rm crit}$, where all energies, spins and parities, and $\gamma$-ray deexcitations of the levels are regarded as complete and accurate. The code generates a random set of levels between $E_{\rm crit}$ and the neutron-separation energy according to an [*a priori*]{} assumed level density (LD) model $\rho(E,J^{\pi})$. Transitions to and from the quasi continuum to low-lying levels are then determined according to a choice of an [*a priori*]{} assumed photon strength function (PSF), $f^{(XL)}$, where $XL$ denotes the multipolarity of the transition. Selection rules are used to determine allowed transitions between all possible permutations of pairs of initial ($E_{i}$) and final ($E_{f}$) states given by $E_{\gamma} = E_{i} - E_{f}$. The partial radiation widths, $\Gamma_{if}^{XL}$, of the corresponding transition probabilities for non-forbidden transitions are assumed to follow a Porter-Thomas distribution [@porter:56], centered on a mean value according to the expression $$\langle \Gamma_{if}^{(XL)} \rangle = \frac{f^{(XL)}(E_{\gamma}) E_{\gamma}^{2L+1}}{\rho(E_{i}, J_{i}^{\pi_{i}})}.
\label{eq:2}$$ Internal conversion is accounted for using [BRICC]{} [@kibedi:08]. The corresponding simulated decay schemes are called [*nuclear realizations*]{}. Statistical fluctuations in the Porter-Thomas distributions are reflected in the variations between nuclear realizations and provide the uncertainty in the simulation inherent in the Porter-Thomas assumption. In these calculations we performed 50 separate nuclear realizations, with each realization comprising 100,000 capture-state $\gamma$-ray cascades.
The experimental $\gamma$-ray cross sections depopulating the low-lying levels below $E_{\rm crit}$, can then be used to renormalize the simulated population per neutron capture, from [DICEBOX]{}, to absolute cross sections feeding these levels. The total radiative thermal neutron-capture cross section $\sigma_{0}$ is determined as $$\sigma_{0}=\sum \sigma_{\gamma}^{\rm exp}({\rm GS})+\sum \sigma_{\gamma}^{\rm sim}({\rm GS}) = \frac{\sum \sigma_{\gamma}^{\rm exp}({\rm GS})}{1-P({\rm GS})},
\label{eq:3}$$ where $\sum \sigma_{\gamma}^{\rm exp}({\rm GS})$ represents the sum of experimental $\gamma$-ray cross sections feeding the ground state in direct single-step transitions, either via a primary GS transition or secondary transition from a level below $E_{\rm crit}$. The simulated contribution from the quasi continuum above $E_{\rm crit}$ feeding the ground state, $\sum \sigma_{\gamma}^{\rm sim}({\rm GS})$, may also be written as the product of $\sigma_{0}$ and the simulated ground-state population per neutron capture, $P({\rm GS})$, given by [DICEBOX]{} as shown in Equation \[eq:3\].
\[sec:level3.A\]Adopted Models\
===============================
Compound $T$ \[MeV\] $E_{0}$ \[MeV\] $a$ \[${\rm MeV}^{-1}$\] $E_{1}$ \[MeV\] $\Delta$ \[MeV\] $D_{0}$ \[eV\]
----------- ------------- ----------------- -------------------------- ----------------- ------------------ ----------------
$^{183}$W 0.55(2) $-0.92(17)$ 19.22(30) $-0.24(10)$ 0 59.9(61)
$^{184}$W 0.58(2) $-0.64(21)$ 18.76(30) 0.08(14) 0.763 12.0(10)
$^{185}$W 0.56(1) $-1.30(14)$ 19.45(28) $-0.50(8)$ 0 69.9(69)
$^{187}$W 0.57(2) $-1.63(22)$ 19.14(36) $-0.81(13)$ 0 84.8(79)
The simulated population of the levels below $E_{\rm crit}$ depends upon the assumed experimental decay scheme, the capture-state spin composition, $J = 1/2^{+}$ for even-even targets and $J = J_{\rm gs}(\rm target)\pm 1/2$ for odd-odd and odd-$A$ targets, and the choice of adopted phenomenological LD and PSF models.
\[sec:levelLD\]Level Densities\
-------------------------------
The constant temperature formula (CTF) [@gilbert:65] and the back-shifted Fermi gas (BSFG) [@newton:56; @gilbert:65] models were considered in this work. Both models embody a statistical procedure describing the increasing cumulative number density of levels $N(E)$ with increasing excitation energy such that, $$N(E) = \int \rho(E) d(E),
\label{eq:ld_insert1}$$ where $\rho(E)$ represents the level density at an excitation energy $E$. In the CTF model, a constant temperature is assumed over the entire range of nuclear excitation energy that may be explicitly stated as $$\rho(E,J) = \frac{f(J)}{T} \exp{\left( \frac{E-E_{0}}{T} \right)}.
\label{eq:ld_insert2}$$ The nuclear temperature $T$ may be interpreted as the critical temperature necessary for breaking nucleon pairs. The energy backshift related to proton- and neutron-pairing energies is given by $E_{0}$. The temperature and backshift-energy parametrizations used in this work are taken from von Egidy and Bucurescu [@vonegidy:05] and listed in Table \[tab:2\]. A spin-distribution factor $f(J)$ [@gilbert:65] is introduced in Equation \[eq:ld\_insert2\] and assumed to have the separable form of Ref. [@gilbert:65] $$f(J) = \frac{2J+1}{2 \sigma_{c}^{2}} \exp \left( - \frac{(J+1/2)^{2}}{2 \sigma_{c}^{2}} \right),
\label{eq:5}$$ where $\sigma_{c} = 0.98 \cdot A^{0.29}$ denotes the spin cut-off factor [@vonegidy:88].
The BSFG level density model is based on the assumption that the nucleus behaves like a fluid of fermions and may be written as $$\rho(E,J) = f(J) \frac{ \exp (2\sqrt{a(E-E_{1})})}{12\sqrt{2}\sigma_{c}a^{1/4}(E-E_{1})^{5/4} }.
\label{eq:4}$$ Here, the spin cut-off factor $\sigma_{c}$ is defined with an energy dependence given by $$\sigma_{c}^{2} = 0.0146 \cdot A^{5/3} \cdot \frac{ 1+\sqrt{1+4a(E-E_{1})} }{2a}.
\label{eq:6}$$ Since fermions exhibit a tendency to form pairs, the extra amount of energy required to separate them is accounted for by the introduction of the level density parameter, $E_{1}$, in Equation \[eq:4\], above. This parameter corresponds to the back-shift in excitation energy, while $a$ represents the shell-model level density parameter that varies approximately with $0.21\cdot A^{0.87}$ MeV$^{-1}$ [@dobaczewski:01]. As with the CTF, the adopted BSFG parameters used in this work have also been taken from von Egidy and Bucurescu [@vonegidy:05] and are presented in Table \[tab:2\]. In that work, the level density parameters were treated as adjustable and determined by fitting the functional forms of Equations \[eq:ld\_insert2\] and \[eq:4\], above, to experimentally-observed neutron resonance spacings in the region of the capture state above the neutron-separation energy.
\[sec:levelPSF\]Photon Strength Functions\
------------------------------------------
The dominant decay following thermal neutron capture is by $E1$ primary $\gamma$-ray transitions. The $E1$ photon strength is dominated by the low-energy tail of the giant dipole electric resonance (GDER). Theoretical models of the PSF describing the GDER are typically based on parametrizations of the corresponding giant resonance, observed in photonuclear reactions, whose transition probabilities are well described as a function of $\gamma$-ray energy [@krticka:08]. Total photonuclear cross-section data derived from $^{186}$W photoabsorption measurements [@berman:69] can be used to test the validity for a variety of PSFs near the GDER. These data [@berman:69] can be transformed to experimental PSF values $f^{(E1)}(E_{\gamma})$ using the empirical relationship of Ref. [@kopecky:87] $$f^{(E1)}(E_{\gamma}) = \frac{1}{3(\pi \hbar c)^{2}} \cdot \frac{\sigma_{\rm abs}}{E_{\gamma}},
\label{eq:insert2}$$ where the constant $\frac{1}{3(\pi \hbar c)^{2}} = 8.68 \times 10^{-8}$ ${\rm mb} \cdot {\rm MeV}^{-2}$, the photoabsorption cross section $\sigma_{\rm abs}$ is in units of \[mb\], and the $\gamma$-ray energy is in \[MeV\]. The results of this transformation for $^{186}$W are shown in Fig. \[fig:2\].
Isotope Resonance $E_{G_{1}}$ \[MeV\] $\Gamma_{G_{1}}$ \[MeV\] $\sigma_{G_{1}}$ \[mb\] $E_{G_{2}}$ \[MeV\] $\Gamma_{G_{2}}$ \[MeV\] $\sigma_{G_{2}}$ \[mb\]
----------- ----------- --------------------- -------------------------- ------------------------- --------------------- -------------------------- -------------------------
$^{183}$W GDER 12.68 2.71 268.0 14.68 3.62 395.0
GQER 11.10 3.91 4.55 $-$ $-$ $-$
$^{184}$W GDER 12.59 2.29 211.0 14.88 5.18 334.0
GQER 11.08 3.90 4.54 $-$ $-$ $-$
$^{185}$W GDER 12.68 2.71 268.0 14.68 3.62 395.0
GQER 11.06 3.89 4.53 $-$ $-$ $-$
$^{187}$W GDER 12.68 2.71 268.0 14.68 3.62 395.0
GQER 11.02 3.87 4.51 $-$ $-$ $-$
The Brink-Axel (BA) model [@brink:55; @axel:62] and the enhanced generalized Lorentzian (EGLO) model [@kopecky:91; @kopecky:93; @kopecky:98] were used in these calculations to compare with experimental data. The BA model is a form of the standard Lorentzian given by $$f^{(E1)}_{\rm BA}(E_{\gamma}) = \frac{1}{3(\pi \hbar c)^{2}} \cdot \sum_{i=1}^{i=2} \frac{\sigma_{G_{i}} E_{\gamma} \Gamma_{G_{i}}^{2}}{(E_{\gamma}^{2}-E_{G_{i}}^{2})^{2} + E_{\gamma}^{2} \Gamma_{G_{i}}^{2}}.
\label{eq:BA}$$ The resonance shape-driving parameters in Equation \[eq:BA\] are represented by the terms $E_{G_{i}}$ \[MeV\], the centroid of the GDER resonance, $\Gamma_{G_{i}}$ \[MeV\], the width of the resonance, and $\sigma_{G_{i}}$ \[mb\], the cross section of the resonance. The adopted experimental parametrizations for the tungsten isotopes were taken from RIPL [@capote:09] and are listed in Table \[tab:3\]. The corresponding BA PSF based on this parametrization is also shown in Fig. \[fig:2\] where it is compared to the experimental photoabsorption data. Although these data are only available above $E_{\gamma} \gtrsim 9$ MeV, they demonstrate excellent agreement with the Brink hypothesis [@brink:55] in this region.
The EGLO model is derived from the idea of the generalized Lorentzian (GLO) model and was originally proposed by Kopecky and Uhl [@kopecky:90], with the analytic form $$\begin{aligned}
f^{(E1)}_{\rm GLO} (E_{\gamma}, \Theta) &=& \sum_{i=1}^{i=2} \frac{\sigma_{G_{i}} \Gamma_{G_{i}}}{3 (\pi \hbar c)^{2}} \left[ F_{K} \frac{4 \pi^{2} \Theta^{2} \Gamma_{G_{i}}}{E_{G_{i}}^{5}} \right. \nonumber \\
&& \left.+\frac{E_{\gamma} \Gamma_{G_{i}}(E_{\gamma}, \Theta)}{(E^{2}_{\gamma}-E^{2}_{G_{i}})^{2} + E^{2}_{\gamma} \Gamma^{2}_{G_{i}}(E_{\gamma}, \Theta)} \right].
\label{eq:7}\end{aligned}$$ In this model a value of 0.7 has been used for the Fermi-liquid parameter $F_{K}$ [@kadmenski:83]. This factor, together with the remaining terms of the first quotient in the parentheses of Equation \[eq:7\], represent a correction to the Lorentzian function in describing the electric dipole operator in the limit of zero energy (as $E_{\gamma} \rightarrow 0$). This form of the PSF is a violation of the Brink hypothesis since there is an additional dependence on the nuclear temperature $\Theta$, which may be written as a function of excitation energy $$\Theta = \sqrt{(E_{\rm ex} - \Delta)/a},
\label{eq:8}$$ where $E_{\rm ex}$ is the excitation energy of a final state, and $\Delta$ is the pairing energy. The pairing correction has been determined according to the following convention: for even-even nuclei $\Delta = +0.5 \cdot |P_{d}|$ = 0.763 ($^{184}$W); for odd-$A$ nuclei $\Delta = 0$ ($^{183,185,187}$W); and for odd-odd nuclei $\Delta = -0.5 \cdot |P_{d}|$. The deuteron-pairing energy, $P_{d}$ is tabulated in Ref. [@vonegidy:05]. Consequently, GDERs built on excited states may differ vastly in both shape and size to those built on the ground state since the width of the resonance is also a function of the nuclear temperature according to $$\Gamma_{G_{i}} (E_{\gamma}, \Theta) = \frac{\Gamma_{G_{i}}}{E_{G_{i}}^{2}} (E_{\gamma}^{2} + 4 \pi^{2} \Theta^{2}).
\label{eq:9}$$ In the EGLO version of this model, the term $\Gamma_{G_{i}}(E_{\gamma}, \Theta)$ has been modified by an enhancement factor given by an empirical generalization of the width [@kopecky:91; @kopecky:93; @kopecky:98] $$\Gamma_{G_{i}}'(E_{\gamma},\Theta) = \left[ k_{0} + (1-k_{0}) \frac{(E_{\gamma} - E_{0})}{(E_{G_{i}} - E_{0})} \right] \Gamma_{G_{i}}(E_{\gamma},\Theta),
\label{eq:10}$$ where $\Gamma_{G_{i}}'(E_{\gamma}, \Theta)$ is substituted for $\Gamma_{G_{i}}(E_{\gamma}, \Theta)$ in Equation \[eq:7\] to evaluate $f^{(E1)}_{\rm EGLO}(E_{\gamma}, \Theta)$. A fixed value of $E_{0} = 4.5$ MeV has been adopted for the reference-energy [@kopecky:93; @kopecky:98] and is found to have only a weak influence on the overall enhancement. The parameter $k_{0}$ was then varied to optimize agreement with the absorption data of Ref. [@berman:69]. Figure \[fig:2\] shows that for $k_{0} = 3.5$ the EGLO PSF follows closely the experimental data for $E_{\gamma} \lesssim 17$ MeV. Beyond this regime the PSF is heavily damped, however, these $\gamma$-ray energies are not of interest in thermal capture. The GLO model is also plotted in Fig. \[fig:2\] along with an EGLO PSF using the empirically-determined value of $k_{0}$ from the mass-dependent model of Ref. [@kopecky:98] where $k_{0} = 1+ \left[(0.09(A-148) \cdot {\rm exp}(-0.180(A-148))\right]$. The plot illustrates very little difference in overall behavior between the GLO model and EGLO model with the mass-modeled-$k_{0}$ value. Both PSFs fail to reproduce the experimental data at low energy and can only adequately describe the data in the double-humped resonance region.
For the magnetic-dipole transitions, $M1$, a PSF based on the single-particle (SP) model was adopted. The value of $f^{(M1)}_{SP}$ was treated as an adjustable parameter in the [DICEBOX]{} calculations to obtain good agreement between statistical-model predictions and experimental-decay data in addition to the derived value of the total radiative capture width. For the even-odd $^{183,185,187}$W compounds a value of $f^{(M1)}_{SP} = 1 \times 10^{-9}$ MeV$^{-3}$ was used, while a higher value of $f^{(M1)}_{SP} = 3 \times 10^{-9}$ MeV$^{-3}$ was found to reproduce the data better for the even-even $^{184}$W. Other models, such as the scissors [@richter:90] and spin-flip models, were also be considered, however a lack of experimental evidence for a giant dipole magnetic resonance (GDMR) in the tungsten isotopes and the relative insignificance of these transitions in the calculations [@bolinger:70], make the SP model a practical approach.
A giant quadrupole electric resonance (GQER) model has been used to describe the PSF for $E2$ multipoles. This model is represented by a single-humped Lorentzian (cf. the standard Lorentzian in Equation \[eq:BA\]) to describe an isoscalar-isovector quadrupole-type vibration. A global parametrization has been used to determine the set of resonance parameters, listed in Table \[tab:3\]. The following convention was adopted in determining this parametrization: $E_{G} = 63 \cdot A^{-1/3}$ MeV [@speth:81], $\Gamma_{G} = 6.11 - 0.012A$ MeV [@prestwitch:84], and $\sigma_{G} = 1.5 \times 10^{-4} \cdot \frac{Z^{2}E_{G}^{2}A^{-1/3}}{\Gamma_{G}}$ mb [@prestwitch:84]. Quadrupole strength contributes far less than the dipole strengths. Transitions corresponding to higher multipoles, including $M2$, are not considered in modeling capture-state decay in this work.
\[sec:level4\]Results\
======================
Thermal neutron-capture (n,$\gamma$) $\gamma$-ray cross sections depopulating levels in the $^{183,184,185,187}$W compounds, from irradiations of the isotopically-enriched $^{182,183,186}$W targets and a natural tungsten target for $^{184}$W(n,$\gamma$), are discussed below. Only the primary $\gamma$ rays from the capture state or secondary $\gamma$ rays depopulating levels below $E_{\rm crit}$ are included in this paper. The complete decay scheme determined in these measurements will be available in the EGAF database.
$\sigma_{0}$ \[b\] Reference
-------------------- -------------------------------------
[**20.5(14)**]{} [**This work**]{}
19.2(19) H. Pomerance [@pomerance:52]
20.7(5) S. J. Friesenhahn [@friesenhahn:66]
19.6(3) K. Knopf [@knopf:87]
20.0(6) V. Bondarenko [@bondarenko:11]
19.9(3) Atlas [@mughabghab:06]
: \[tab:W182ng\] Summary of $\sigma_{0}$ measurements for $^{182}$W(n,$\gamma$).
All combinations of PSF and LD models described earlier, were used in the [DICEBOX]{} calculations and compared to experimental data by plotting the simulated population against the experimental depopulation for each level below $E_{\rm crit}$ in population-depopulation plots. For model combinations invoking the EGLO PSF we assumed a $k_{0} = 3.5$ enhancement factor. Uncertainties in the population along the vertical axis correspond to Porter-Thomas fluctuations from independent nuclear realizations, while those along the horizontal axis are due to the experimental uncertainty in the measured cross sections depopulating the levels. The vertical axis shows the calculated population per neutron capture to a given level, determined by [DICEBOX]{}, and the experimental depopulation of the corresponding level along the horizontal axis is normalized to the total radiative thermal-capture cross section according to $$P_{L}^{\rm exp} = \sum\limits_{i=1}^{N} \frac{\sigma_{\gamma_{i}}(1+\alpha_{i})}{\sigma_{0}},
\label{eq:14}$$ where $N$ denotes the number of $\gamma$ rays depopulating the level.
The population-depopulation plots compare the intensity balance through all states up to $E_{\rm crit}$. Scatter around the ${\rm population} = {\rm depopulation}$ line is a measure of the quality and completeness of the experimental data and provides a test of the ability of the statistical model to simulate the experimental decay scheme. Model dependence in the population-depopulation plot is indicated by either smooth or spin dependent deviations, and isolated deviations for individual levels are indications of problems with the experimental $J^{\pi}$ assignments or other decay-scheme data.
In this work, we also investigated the parity dependence $\pi (E)$ on the overall LD assuming its separable form $\rho (E,J,\pi) = \rho(E) \cdot f(J) \cdot \pi(E)$. The $\pi(E)$ dependence may be described by a Fermi-Dirac distribution parametrized according to Ref. [@quraishi:03]. In this framework, at large excitation energies $\pi(E) = 0.5$. As $E \rightarrow 0$: $\pi(E) \rightarrow 1$ for even-even nuclei; $\pi(E) \rightarrow 0(1)$ for odd-$A$ nuclei for which the odd nucleon is in an odd-parity (even-parity) orbit; and, $\pi(E) \approx 0.5$ for odd-odd and odd-$A$ nuclei if the Fermi level is occupied by nearly degenerate positive- and negative-parity orbits. Adopting an additional parity dependence in the LD models, $\rho (E,J,\pi) = \rho(E) \cdot f(J) \cdot \pi(E)$, the simulated populations for the odd-A isotopes $^{183,185,187}$W and even-even $^{184}$W were found to yield statistically consistent results with the parity-independent LD models, $\rho (E,J) = \rho(E) \cdot f(J)$; a representative comparison is illustrated in Fig. \[fig:LDPDEP\]. A parity-independent approach was, therefore, considered adequate for modeling the LD in these analyses.
\[sec:level4.A\]$^{182}$W(n,$\gamma$)$^{183}$W\
-----------------------------------------------
A $^{182}$WO$_2$ target was irradiated for a 2.46-h period. The current analysis and previous information in ENSDF [@firestone:92] implies that for $^{183}$W the level scheme is complete up to a level at 485.1 keV and we have set $E_{\rm crit} = 490.0$ keV, which includes an additional level over the value given in RIPL [@capote:09]. A total of 12 levels in $^{183}$W are below $E_{\rm crit}$ with spins ranging from $1/2 \leq J \leq 13/2$, deexcited by 33 $\gamma$ rays and fed by four primary $\gamma$ rays, shown in Table \[tab:183Wg\]. Transition intensities have been corrected for absorption in the source, as discussed earlier. The multipolarities in Table \[tab:183Wg\] are taken from ENSDF [@firestone:92] where available, or assumed based on angular-momentum selection rules, and the conversion coefficients were recalculated with [BRICC]{} [@kibedi:08].
Figure \[w183:PD\] shows the population-depopulation balance for $^{183}$W using the corresponding $\sigma_{\gamma}$ information from Table \[tab:183Wg\] calculated with various LD and PSF models. These plots show little statistical-model dependence in the population of most excited states except for the high-spin $11/2^{+}$, $11/2^{-}$, and $13/2^{+}$ states at 309.5, 475.2, and 485.1 keV, respectively, that appear to be better reproduced using the EGLO PSF. This is also shown in Fig. \[fig:deviation\] where the difference in the [DICEBOX]{}-modeled population ($P_{L}^{\rm sim}$) for a variety of PSF/LD combinations and the experimental depopulation ($P_{L}^{\rm exp}$) is model independent and insensitive to cut-off energies, $E_{\rm c}$, above 300 keV. Figure \[fig:deviation\] shows excellent consistency between the models at each value of $E_{\rm c}$.
The total-capture cross section, $\sigma_{0}$, determined for the different PSF/LD combinations, is also independent of $E_{\rm crit}$ for various model combinations as seen in Fig. \[fig:w183sigma0\]. For $E_{\rm crit} = 100$ keV, with only three low-lying levels, $\sigma_{0}$ remains nearly constant although the systematic uncertainty is larger. This rapid convergence is due to the dominant ground-state feeding from experimental transitions deexciting low-lying levels that dominates the calculation. We adopt the value $\sigma_{0} = 20.5(14)$ b corresponding to the EGLO/CTF combination. Of the $\sim 7$ % uncertainty on our value, the systematic uncertainty from the simulated cross section is 4.3 % and $\gamma$-ray self attenuation accounts for 3.2 %. The statistical and normalization errors are far less significant with each only contributing $\lesssim 2$ %. The result for the total radiative thermal-capture cross section for $^{182}$W(n,$\gamma$)$^{183}$W is consistent with the recommended value of 19.9(3) b [@mughabghab:06] and previous experimental investigations [@pomerance:52; @friesenhahn:66; @knopf:87; @bondarenko:11] listed in Table \[tab:W182ng\].
The choice of PSF and LD combination has a pronounced effect on the calculated capture-state total radiative width. The EGLO/CTF result, $\Gamma_{0} = 0.040(3)$ eV, agrees best with the recommended value of $\langle \Gamma_{0} \rangle = 0.051(4)$ eV. For the EGLO/BSFG and BA/CTF combinations somewhat poorer agreement is obtained with $\Gamma_{0}$ values of 0.071(3) and 0.076(6) eV respectively. The BA/BSFG combination gives much poorer agreement with $\Gamma_{0}=0.138(7)$ eV. Fortunately, the choice of PSF/LD model has only a small effect on the derived cross section.
The $11/2^{+}$ ($T_{1/2}=5.2$ s) isomer at 309.49 keV [@firestone:92] decays by a highly-converted 102.48-keV [@firestone:92] $M2$ transition that was not resolved from the 101.93-keV transition deexciting the 308.95-keV level and the 101.80-keV transition deexciting the 302.35-keV level in $^{187}$W which also contributes to the observed intensity due to a 0.5(1) % $^{186}$W impurity (Table \[tab:isocomp\]) in the measured sample. The total intensity of the triplet is $\sim 15(2)$ % of the 209.69-keV $\gamma$-ray intensity deexciting the 308.95-keV level, which is significantly larger than 7.4(4) % observed from the same level in $^{183}$Ta $\beta^{-}$ decay [@firestone:92]. Assuming the excess intensity, after the additional correction for the $^{186}$W impurity (see Section \[sec:level4.D\]), comes from the isomer transition, we get $\sigma_{\gamma}(102.48)=0.0049(19)$ b. Accounting for internal conversion this gives an experimental depopulation of 0.197(76) b which is consistent with the observed total $\gamma$-ray intensity feeding the metastable isomer, $\sigma_{11/2^{+}}(^{183}{\rm W}^{m}) = 0.177(18)$ b, from the 485.72- and 622.22-keV levels which are deexcited by transitions at 175.89 and 312.72 keV, respectively. The combined intensity of these transitions yields $\sum\sigma_{\gamma}^{\rm exp}(11/2^{+}) = 0.177(18)$ b and the DICEBOX-modeled population of the 309.49-keV isomer is $P(11/2^{+})=0.00154(97)$. The experimental depopulation of the 309.49-keV level is consistent with the simulated population from our [DICEBOX]{} calculations to within 3 $\sigma$ as indicated in the log-log space of Fig. \[w183:PD\]. The current measurement supports the proposed $J^{\pi} = 13/2^{+}$ assignment for the 485.72-keV level that was previously reported in reaction experiments [@saitoh:00]. Our simulations also support the inclusion of a new, highly-converted, 17.2-keV $E2$ transition deexciting the 308.95-keV level with a total intensity of $\sim 180$ mb feeding the 291.72-keV level that improves the agreement between population and depopulation for both levels. The 17.2-keV transition is below the detection threshold of our HPGe detector.
The next level above $E_{\rm crit}$ at 533 keV is reported in ENSDF [@firestone:92] with $J^{\pi} = (1/2,3/2)$. The 533-keV level was only reported as populated by primary $\gamma$-rays in a resonance (n,$\gamma$) experiment [@casten:73] and not seen in our work or later (n,$\gamma$) or reaction experiments. The existence of this level is considered doubtful; certainly the proposed $J^{\pi}$ assignment is highly questionable since these states are expected to be strongly populated in $s$-wave capture on $^{182}$W (see Fig. \[w183:PD\]). Raising the cut-off energy to 625-keV and including the next three levels at 551.1, 595.3, and 622.22 keV leads to poorer agreement in the population-depopulation balance for several levels as shown in Fig. \[w183:PD\](c). We observe the transitions from these three levels, but since the statistical model gives better agreement for $E_{\rm crit} = 490$ keV, it is likely that the decay-scheme information is incomplete between the 490 and 622.22 keV.
\[sec:level4.B\]$^{183}$W(n,$\gamma$)$^{184}$W\
-----------------------------------------------
$\sigma_{0}$ \[b\] Reference
---------------------- -------------------------------------
[**[9.37(38)]{}**]{} [**This work**]{}
10.9(11) H. Pomerance [@pomerance:52]
10.0(3) S. J. Friesenhahn [@friesenhahn:66]
10.5(2) K. Knopf [@knopf:87]
10.4(2) Atlas [@mughabghab:06]
: \[tab:183Wr\] Summary of $^{183}$W(n,$\gamma$) $\sigma_{0}$ measurements.
A $^{183}$WO$_2$ target was irradiated for 2.24 h. Comparison of the DICEBOX-population calculations with the experimental depopulation data for $^{184}$W sets $E_{\rm crit} = 1370.0$ keV. This value is higher than in RIPL where $E_{\rm crit} = 1252.2$ keV and includes 12 levels. There are 18 levels below our cut-off energy including one tentative level assignment. The $^{184}$W decay scheme consists of seven primary $\gamma$-rays and 47 secondary $^{184}$W $\gamma$-rays that are listed in Table \[tab:184Wg\]. The experimental multipolarities and mixing ratios are taken from ENSDF [@baglin:10] where available or assumed based on selection rules. The ground state of the $^{183}$W target nucleus is $J^{\pi} = 1/2^{-}$, allowing $s$-wave neutron capture to populate resonances with $J^{\pi} = 0^{-},1^{-}$. The *Atlas of Neutron Resonances* [@mughabghab:06] indicates that $1^{-}$ capture-states account for 78.3 % of the observed total-capture cross section, 7.4 % is from $0^{-}$ capture states, and the remaining 14.3 % of the cross section is attributed to a negative-parity [*bound*]{} resonance at $E_{0} = -26.58$ eV (with respect to the separation energy) with unknown spin.
The population-depopulation plots in Figs. \[w184:PD\](a) and (b) show that $\sigma_0$ is insensitive to both the $0^{-}/1^{-}$ composition of the capture state and the choice of PSF and LD combinations. Figure \[fig:sigma0Ec\_w184\] shows the dependence of the derived cross section on $E_{\rm c}$. For $E_{\rm c} \leq 900$ keV there are only four levels and $\sigma_0 = 8.65(64)$ b. Adding the level at 903.31 keV, which feeds the ground state with $\sigma_{\gamma} = 1.185(52)$ b, increases the derived cross section significantly, demonstrating the necessity to include as many experimentally known low-lying levels as possible in the simulation. For $E_{\rm crit}$ = 1370.0 keV, with a total of 17 levels (not including the tentative 1282.7-keV level, see later), we get $\sigma_{0} = 9.37(38)$ b, which is comparable at 2 $\sigma$ with the recommended value of 10.4(2) b [@mughabghab:06] and previous measurements shown in Table \[tab:183Wr\]. We also find that the total thermal-capture cross section is statistically insensitive to the $J^{\pi}$ composition of the capture state as illustrated in Fig. \[fig:s0\_varyCS\]. The overall uncertainty on our adopted value for $\sigma_{0}$ of 4.0 % is dominated by the 3.4 % systematic uncertainty in the simulation and the 1.7 % statistical uncertainty. Uncertainties due to $\gamma$-ray self attenuation and normalization are much lower, each contributing $ < 1.0$ %.
The capture-state width, $\Gamma_{0}$, is strongly dependent on the choice of PSF/LD combination, but is only weakly influenced by the capture-state spin composition, as shown in Fig. \[fig:Gamma0\_CS\]: $\Gamma_{0}$ is nearly constant up to $\sim 65$-% $0^{-}$ contribution, and only gradually increases up to $\sim 80~\%$. The EGLO/CTF model combination, with a 78.3-% $1^-$ capture-state composition (Fig. \[w184:PD\](b)), gives $\Gamma_0$=0.066(2) eV, in agreement with the adopted value of 0.073(6) eV [@mughabghab:06]. For the model combinations: EGLO/BSFG, $\Gamma_{0} = 0.129(3)$; BA/CTF, $\Gamma_{0} = 0.121(3)$; and BA/BSFG, $\Gamma_{0}=0.242(6)$; all are substantially higher than the adopted value. The effect of the capture-state composition is most sensitive to the modeled population of the $0^{+}$ and $J \geq 4$ low-lying levels. For $0^{-}$ capture-state compositions of 7.4 % (Fig. \[w184:PD\](a)) and 21.7 % (Fig. \[w184:PD\](b)), the EGLO results give excellent agreement with experiment. If the $0^{-}$ capture-state composition increases to 85 % (Fig. \[w184:PD\](c)), the predicted population of $0^{+}$ and high-spin states is much poorer. The 85-% $0^{-}$ composition also gives $\Gamma_{0}$ values of 0.348(8) for the EGLO/BSFG model combination and 0.178(5) eV for the EGLO/CTF combination that are considerably higher than the adopted value. To determine the most likely $J^{\pi}$ capture-state composition we varied this parameter and calculated the corresponding reduced $\chi^{2}$, using the population-depopulation data for the weakly populated states (circled in Fig. \[w184:PD\]), as $$\chi^{2}/{\rm ndf} = \sum \frac{(P_{L}^{\rm exp} - P_{L}^{\rm sim})^{2}}{(dP_{L}^{\rm sim})^{2}},
\label{eq:chi2_highJ}$$ where $P_{L}^{\rm exp}$ is the expectation value. Figure \[fig:chi2\_JPi\] shows that $\chi^{2}$ approaches 1.0 for capture-state compositions with $J^{\pi}(0^{-}) < 10$ %. Indeed, the simulated populations to these levels is more than $3~\sigma$ away from the expectation value assuming $J^{\pi}(0^{-}) \approx 22$ %. This result implies a likely capture-state composition $J^{\pi}(0^{-}) \lesssim 7$ %, and hence, $J^{\pi} = 1^{-}$ is the most probable assignment for the bound resonance at $-26.58$ eV [@mughabghab:06]. Thus, an overall fractional distribution of $J^{\pi} = 0^{-}(7.4~\%) + 1^{-}(92.6~\%)$ is consistent with the capture-state composition of Ref. [@mughabghab:06].
Our analysis confirms the decay scheme for $^{184}$W reported in ENSDF [@baglin:10] except for the 161.3-keV $\gamma$ ray depopulating the 1282.71-keV $(1,2)^{-}$ level, which we did not observe. This level assignment was tentative and the 161.3-keV $\gamma$-ray was placed twice in the level scheme (also depopulating the $6^{-}$ level at 1446.27 keV). Since this level is expected to be strongly populated, we conclude that it most likely does not exist (or has a considerably different $J^{\pi}$) and have removed it from our analysis. We have also assigned a new $\gamma$ ray at 65.36(19) keV, depopulating the 1360.38-keV level. Another 9.94-keV $\gamma$ ray depopulating the 1294.94-keV level is proposed based on the population-depopulation balance. The 504.03-keV $\gamma$ ray deexciting the 1252.20-keV $8^{+}$ level was not firmly identified although we can set an experimental limit of $\sigma_{\gamma} < 0.16$ mb which is consistent with statistical-model predictions of 0.1(1) mb.
Some $\gamma$ rays from levels below $E_{\rm crit}$ were not observed in our data and their relative cross sections were taken from ENSDF [@baglin:10], normalized to the cross sections of (observed) stronger transitions from those levels, as indicated in Table \[tab:184Wg\]. An unresolved doublet centered at 769 keV $\gamma$-ray deexcites the 1133.85- and 1775.34-keV levels and was resolved using the ENSDF-adopted branching intensities from both levels. Doublets centered around 215 and 996 keV, depopulating levels at 1221.31 and 1360.38 keV, respectively, were also resolved in a similar manner, as indicated in Table \[tab:184Wg\]. The 1285.00-keV level is an 8.33-$\mu$s isomer with $J^{\pi}=5^{-}$, and is populated with a cross section $\sigma_{5^{-}} = 24.7(55)$ mb from beneath $E_{\rm crit}$; transitions from above $E_{\rm crit}$ known to feed the isomer were not observed in this work.
\[sec:level4.C\]$^{184}$W(n,$\gamma$)$^{185}$W\
-----------------------------------------------
A $^{\rm nat}$WO$_{2}$ target was irradiated for 11.52 h. Comparison of the DICEBOX-population calculations with the experimental-depopulation data for the $^{185}$W compound sets $E_{\rm crit} = 392.0$ keV. This value is higher than in RIPL where $E_{\rm crit} = 243.4$ keV which includes eight levels. Table \[tab:185Wg\] lists 11 levels beneath the cut-off energy, deexcited by 25 secondary $\gamma$ rays, and populated by three primary $\gamma$ rays. These data were measured with a natural tungsten sample and supplemented with data from Bondarenko *et al*. [@bondarenko:05] that was renormalized to our cross sections. Ten levels below $E_{\rm crit}$ have negative parity with spins ranging from $1/2^{-}$ to $9/2^{-}$, and there are two positive-parity levels at 197.43 (11/2$^{+}$, $T_{1/2}=1.67$ min) [@wu:05] and 381.70 keV (13/2$^{+}$) [@bondarenko:05] that are high-spin with no $\gamma$ rays observed deexciting them. We have used the total cross section populating the 197.43-keV level from higher-lying levels in $^{185}$W from Ref. [@bondarenko:05], $\sigma_{11/2^{+}} = 6.2(16)$ mb, to determine the $\gamma$-ray cross sections deexciting this isomer. This cross section is substantially larger that than the recommended value, $\sigma_0 = 2(1)$ mb [@mughabghab:06]. The positive-parity levels below $E_{\rm crit}$ play only a small role in our simulations and do not limit the choice of $E_{\rm crit}$. The mixing ratios and multipolarities in Table \[tab:185Wg\] were taken from ENSDF [@wu:05] where available or assumed based on selection rules associated with the $\Delta J$ transitions.
We determined the thermal-capture cross section, $\sigma_{0} = 1.43(10)$ b, for $^{184}$W(n,$\gamma$). The result is largely insensitive with respect to PSF/LD combinations and comparable to the adopted value $\sigma_{0} = 1.7(1)$ b [@mughabghab:06]. Table \[tab:184Wr\] shows the comparison of our value with other reported measurements. For the EGLO/BSFG model combination, shown in Fig. \[fig:sigma0Ec\_w185\], $\sigma_{0}$ is statistically independent of $E_{\rm crit}$. The uncertainty in $\sigma_{0}$ is 7 %. Several low-energy $\gamma$ rays contribute significantly to $\sigma_{0}$ but were not observed by experiment and were, instead, estimated from statistical-model calculations. The systematic uncertainty in the ground-state feeding from the simulation is 4.7 %. A statistical uncertainty of 3.2 % and an uncertainty of 2.4 % in the normalization also contribute. The data from Ref. [@bondarenko:05] were measured with a very thin target so no correction due to $\gamma$-ray self attenuation was required.
The total radiative width of the capture state in $^{185}$W varies widely depending on the choice of PSF/LD models. The EGLO/BSFG combination generates a total width $\Gamma_{0} = 0.052(3)$ eV that is in excellent agreement with the adopted value, $\langle \Gamma_{0} \rangle = 0.052(4)$ eV [@mughabghab:06]. Other combinations show poorer agreement: $\Gamma_{0} = 0.034(3)$ eV for EGLO/CTF; $\Gamma_{0} = 0.069(6)$ eV for BA/CTF; and, $\Gamma_{0} = 0.108(7)$ eV for the BA/BSFG combination.
$\sigma_{0}$ \[b\] Reference
-------------------- -------------------------------------
[**1.43(10)**]{} [**This work**]{}
2.12(42) L. Seren [@seren:47]
1.97(30) H. Pomerance [@pomerance:52]
2.28(23) W. S. Lyon [@lyon:60]
1.70(10) S. J. Friesenhahn [@friesenhahn:66]
1.70(10) K. Knopf [@knopf:87]
1.76(9) V. Bondarenko [@bondarenko:05]
2.40(10) V. A. Anufriev [@anufriev:83]
1.70(10) Atlas [@mughabghab:06]
: \[tab:184Wr\] Summary of $^{184}$W(n,$\gamma$) $\sigma_{0}$ measurements.
Here we report more precise energies for the 301.13 and 332.11-keV levels than are in ENSDF [@wu:05]. No $\gamma$ rays were previously reported deexciting these levels. Our [DICEBOX]{} calculations support the results of Bondarenko *et al*. [@bondarenko:05] where six new $\gamma$ rays were identified depopulating these levels. Two new, low-energy $\gamma$ rays are proposed deexciting levels at 187.88 ($E_{\gamma} \approx 14$ keV) and 390.92 keV ($E_{\gamma} \approx 58$ keV) based on the population-depopulation intensity balance. The $\sim 58$-keV $\gamma$-ray transition is highly
[cccccccccccc]{}
\
$E_{\rm L}$ \[keV\] & $J^{\pi}$ & $E_{\gamma}$ \[keV\] & $\sigma_{\gamma}^{\rm exp}$ \[b\] & $\alpha$ & $XL$ & $E_{\rm L}$ \[keV\] & $J^{\pi}$ & $E_{\gamma}$ \[keV\] & $\sigma_{\gamma}^{\rm exp}$ \[b\] & $\alpha$ & $XL$\
\
$E_{\rm L}$ \[keV\] & $J^{\pi}$ & $E_{\gamma}$ \[keV\] & $\sigma_{\gamma}^{\rm exp}$ \[b\] & $\alpha$ & $XL$ & $E_{\rm L}$ \[keV\] & $J^{\pi}$ & $E_{\gamma}$ \[keV\] & $\sigma_{\gamma}^{\rm exp}$ \[b\] & $\alpha$ & $XL$\
0 & $3/2^{-}$ & & & & & (493.4)& $(9/2^{-})$& (143.2(1))& 0.0222(58)& 1.7 & $[M1]$\
77.29 & $5/2^{-}$ & 77.30(5) & 0.823(14) & 10.17 & $M1+E2$ & 510.00 & $11/2^{-}$& (145.8(1))& 0.0052(21) & 1.62 & $[M1]$\
145.85 & $1/2^{-}$ & 145.84(5) & 4.727(46) & 1.65 & $M1$ & 522.15 & $9/2^{-}$& 171.70(6) & 0.0526(32) & 0.71 & $[M1+E2]$\
201.45 & $7/2^{-}$ & 124.18(5) & 0.282(16) & 2.01 & $M1+E2$ & 538.45 & $11/2^{-}$& 337.18(19) & 0.0096(18) & 0.0604 &$[E2]$\
& & 201.51(5) & 1.515(76) & 0.303 & $[E2]$ & 574.05 & $11/2^{-}$& 209.59(33) & 0.0042(16) & 0.59 & $[M1]$\
204.90 & $3/2^{-}$ & 59.30(5) & 1.048(43) & 3.73 & $M1$ & 597.24 & $11/2^{+}$& - & - & - &\
& & 127.55(5) & 0.646(37) & 1.87 & $M1+E2$ & 613.38 & $9/2^{-}$& (16.20(13))& (0.00293(43)) & 10.08 & $[E1]$\
& & 204.87(5) & 0.666(33) & 0.631 & $[M1]$ & & & 282.86(19) & 0.0054(18) & 0.259 & $[M1]$\
303.35 & $5/2^{-}$ & 98.51(8) & 0.0261(27) & 4.97 & $[M1]$ & & & 310.52(12) & 0.0119(17) & 0.0771 & $[E2]$\
& & 101.80(5) & 0.234(16) & 4.61 & $M1$ & & & 410.8(5)& 0.00031(5) & 0.0944 & $[M1]$\
& & 157.47(5) & 0.1474(81) & 0.713 & $[E2]$ & 640.49 & $5/2^{-}$ & 276.19(6) & 0.0635(48) & 0.109 & $[E2]$\
& & 226.02(5) & 0.379(19) & 0.243 & $M1+E2$ & & & 289.98(6) & 0.300(15) & 0.17 & $M1+E2$\
& & 303.31(6) & 0.248(13) & 0.213 & $[M1]$ & & & 438.91(10) & 0.0174(24) & 0.0794 & $[M1]$\
330.78 & $9/2^{-}$& 129.1(2)& 0.0051(38) & 2.34 & $[M1]$ & & & 563.33(13) & 0.0321(49) & 0.0415 & $[M1]$\
& & 253.51(5)& 0.1268(92) & 0.143 & $[E2]$ & & & 640.55(10) & 0.085(10) & 0.0298 & $[M1]$\
350.43 & $7/2^{-}$ & (19.60(5))& 0.00051(18) & 97.66 & $[M1]$& 710.78 & $13/2^{-}$& (380.0(2))& (0.00030(30)) & 0.0431 & $[E2]$\
& & 148.89(5) & 0.204(11) & 1.55 & $[M1]$ & 727.86 & $11/2^{-}$& 205.7(1)& 0.0016(4) & 0.6166 & $[M1]$\
& & 273.12(5) & 1.337(14) & 0.283 & $[M1]$ & & & 377.0(2)& 0.0014(2) & 0.0444 & $[E2]$\
& & 350.34(9) & 0.0219(25) & 0.0542 & $[E2]$ & 741.08 & $7/2^{+}$& 218.81(7) & 0.0220(27) & 0.0503 & $[E1]$\
364.22 & $9/2^{-}$ & (13.80(4))& 0.00293(22) & 275.2 & $[M1]$& & & 330.97(6) & 0.0775(45) & 0.1682 & $[M1]$\
& & 162.59(12) & 0.0100(15) & 1.2 & $[M1]$ & & & 376.80(5)& 0.184(21) & 0.0134 & $[E1]$\
& & 286.79(7) & 0.0314(26) & 0.0981 & $[E2]$ & & & 390.56(10) & 0.0661(42) & 0.0123 & $[E1]$\
410.06 & $9/2^{+}$& 45.8(3)& 0.301(12) & 0.5941 & $[E1]$ & & & 539.58(14)& 0.0092(21) & 0.00605 &$[E1]$\
432.28 & $7/2^{-}$ & 128.93(6)& 0.1064(82) & 2.34 & $[M1]$ & & & 663.91(8) & 0.0764(62) & 0.00394 & $[E1]$\
& & 227.37(10) & 0.0506(37) & 0.203 & $[E2]$ & 762.15 & $1/2^{-}$& 557.24(5)& 0.572(33)& 0.0427 &$(M1+E2)$\
& & 230.56(14) & 0.0148(33) & 0.453 & $[M1]$ & & & 616.33(5) & 0.304(16) & 0.0329 & $[M1]$\
& & 354.92(7) & 0.1814(95) & 0.14 & $[M1]$ & & & 762.0(5)& 0.0286(60) & 0.0191 & $[M1]$\
& & 432.4(5)& 0.0098(28) & 0.0305 & $[E2]$ & 775.60 & $7/2^{-}$& (135.1(5))& 0.0478(71) & 2.01 & $[M1]$\
& & 253.50(16)& 0.0205(47) & 0.349 & $[M1]$ & & & 659.18(9)& 0.0738(91) & 0.0109 & $[E2]$\
& & 411.28(9) & 0.0164(23) & 0.0944 & $[M1]$ & & & 783.74(13) & 0.0836(82) & 0.0179 & $[M1]$\
782.29 & $1/2^{-}$ & 577.36(5)& 0.921(46) & 0.031 &$(M1+E2)$ & & & 860.77(12) & 0.1058(82) & 0.0141 & $[M1]$\
& & 636.64(35)& 0.0396(42) & 0.0303 & $[M1]$& 863.29 & $5/2^{-}$& 513.0(5)& 0.024(10) & 0.0531 & $[M1]$\
& & 704.9(4)& 0.0138(20) & 0.0094 & $[E2]$ & & & 532.41(7)& 0.039(12) & 0.0180 & $[E2]$\
& & 782.25(5) & 0.606(31) & 0.0179 & $[M1]$ & & & 559.79(9) & 0.0283(32) & 0.0423 & $[M1]$\
797.03 & $11/2^{-}$& 364.7(1)& 0.0013(5) & 0.0482 & $[E2]$ & & & 658.0(3)& 0.0168(84) & 0.0279 & $[M1]$\
& & 466.3(1)& 0.0017(5) & 0.0680 & $[M1]$ & & & 661.9(3)& 0.038(15) & 0.0275 & $[M1]$\
798.22 & $(9/2^{+})$ & - & - & - & - & & & 717.36(14) & 0.0300(64) & 0.00905 & $[E2]$\
803.37 & $3/2^{-}$& 500.02(6)& 0.115(17) & 0.0565 & $(M1)$ & & & 785.73(11) & 0.0850(81) & 0.0177 & $[M1]$\
& & 598.55(15)& 0.0608(88) & 0.0355 & $[M1]$ & & & 862.96(10)& 0.099(11) & 0.014 & $[M1]$\
& & 657.50(7)& 0.320(32) & 0.0279 & $[M1]$ & 866.68 & $3/2^{-}$& 563.51(6)& 0.023(13) & 0.0157 & $[E2]$\
& & 726.03(5) & 0.1118(74) & 0.0216 & $[M1]$ & & & 661.65(7)& 0.068(24) & 0.0275 & $[M1]$\
& & 803.25(8)& 0.1043(70) & 0.0168 & $[M1]$ & & & 789.38(10) & 0.234(52) & 0.00735 & $[E2]$\
809.79 & $(13/2^{-})$ & - & - & - & - & & & 866.37(13) & 0.278(16) & 0.0139 & $[M1]$\
811.7 & $(15/2^{+})$ & - & - & - & - & 881.77 & $5/2^{+}$& 140.47(13) & 0.0260(55) & 1.82 & $[M1]$\
815.51 & $13/2^{+}$& - & - & - & - & & & 449.58(11)& 0.0086(43) & 0.00899 & $[E1]$\
816.26 & $3/2^{-}$ & 176.6(6)& 0.0087(46) & 0.9436 & $[M1]$ & & & 531.29(5)& 0.201(26) & 0.00624 & $[E1]$\
& & 383.87(8) & 0.0217(22) & 0.0422 &$[E2]$& & & 676.79(8) & 0.0475(50) & 0.00379 & $[E1]$\
& & 465.54(8)& 0.0464(34) & 0.0252 & $[E2]$ & & & 679.97(14)& 0.0105(45) & 0.00375 &$[E1]$\
& & 512.52(14)& 0.065(13) & 0.0531 &$[M1]$& & & 803.7(4)& 0.0171(15) & 0.0027 & $[E1]$\
& & 611.34(5)& 0.167(21) & 0.0336 & $(M1)$ & & & 881.58(6) & 0.214(12) & 0.00226 & $[E1]$\
& & 670.37(5) & 0.227(12) & 0.0265 & $[M1]$ & 884.13 & $(5/2^{+})$& 143.15(6)& 0.0414(36) & 1.71 & $[M1]$\
& & 738.84(6) & 0.185(10) & 0.0208 & $[M1]$ & & & 243.63(37)& 0.00296(15) & 0.0381 & $[E1]$\
& & 816.20(20) & 0.436(67) & 0.0161 & $[M1]$ & & & 451.29(19)& 0.0065(21) & 0.0089 & $[E1]$\
840.21 & $1/2^{-}$& 537.21(23) & 0.0114(41) & 0.0176 &$[E2]$& & & 474.02(6) & 0.296(15) & 0.0240 & $[E2]$\
& & 635.37(8)& 0.1059(86) & 0.0304 & $[M1]$ & & & 533.63(6) & 0.0934(64) & 0.00619 & $[E1]$\
& & 694.33(5) & 0.235(13) & 0.0243 & $[M1]$ & 891.93 & $3/2^{-}$& 460.1(8)& 0.0069(25) & 0.0259 & $[E2]$\
& & 762.82(7) & 0.172(14) & 0.00792 & $[E2]$ & & & 541.46(7) & 0.0848(64) & 0.0173 & $[E2]$\
& & 840.17(5) & 0.662(34) & 0.015 & $[M1]$ & & & 588.55(6) & 0.0971(62) & 0.0371 & $[M1]$\
852.41 & $3/2^{-}$ & 502.0(6)& 0.0137(60) & 0.0209 & $[E2]$ & & & 690.15(16)& 0.0082(41) & 0.00985 & $[E2]$\
& & 549.0(5)& 0.0195(80) & 0.0443 & $[M1]$& & & 745.88(5) & 0.236(13) & 0.0203 & $[M1]$\
& & 647.41(8) & 0.1065(73) & 0.029 & $[M1]$ & & & 814.03(19)& 0.122(10) & 0.0162 & $[M1]$\
& & 650.88(14) & 0.0212(41) & 0.0113 & $[E2]$ & & & 891.89(5)& 0.408(22) & 0.0129 & $[M1]$\
& & 706.59(6)& 0.195(16) & 0.0232 & $[M1]$ & 5466.62 & $1/2^{+}$ & 4574.67(7) & 0.397(21) & 0 & $[E1]$\
& & 774.92(6)& 0.128(13) & 0.0184 & $[M1]$ & & & 4585.7(6)& 0.0052(20) & 0 & $[E2]$\
& & 852.18(6) & 0.160(11) & 0.0144 & $[M1]$ & & & 4602.6(15)& 0.024(12) & 0 & $[E1]$\
860.76 & $3/2^{-}$& 428.48(8) & 0.0701(48) & 0.0313 & $[E2]$ & & & 4606.6(11)& 0.0159(60) & 0 & $[E1]$\
& & 655.87(7) & 0.227(14) & 0.0281 & $[M1]$ & & & 4615.3(7)& 0.0052(12) & 0 & $[E1]$\
& & 4626.40(7) & 0.627(33) & 0 & $[E1]$ & & & 5163.5(4)& 0.0135(20) & 0 & $[M2]$\
& & 4650.27(8) & 0.207(12) & 0 & $[E1]$ & & & 5388.85(26)& 0.0143(12) & 0 & $[M2]$\
& & 4662.94(27) & 0.0197(30) & 0 & $[E1]$ & & & 5261.67(9) & 2.297(32) & 0 & $[E1]$\
& & 4684.31(7) & 0.765(40) & 0 & $[E1]$ & & & 5466.47(12) & 0.0675(50) & 0 & $[E1]$\
& & 4704.8(4)& 0.0091(12) & 0 & $[E1]$ & & & 5320.65(8) & 1.625(83) & 0 & $[E1]$\
& & 4826.0(10)& 0.0048(12) & 0 & $[M2]$ & & & & & &\
converted and obscured by a strong tungsten X ray at 57.98 keV, making a $\gamma$ ray of this energy difficult to observe. Both new transitions were assumed to have $M1$ multipolarity. The improvement by including these transitions is shown in Fig. \[w185:PD\]. The $^{185}$W $\gamma$ rays deexciting the first three excited states at 23.55, 65.85, and 93.30 keV were not observed in either this work or that of Bondarenko *et al*. [@bondarenko:05]. The transition cross sections depopulating these levels were determined from the simulated cross section populating those levels, using the EGLO/BSFG model combination and the branching ratios from ENSDF [@wu:05], as shown in Fig. \[w185:PD\]. Our [DICEBOX]{}-simulated population per neutron capture to each of these levels is: 23.55 keV, 0.178(28); 65.85 keV, 0.254(33); and, 93.30 keV, 0.201(32). These values can be compared to those of Bondarenko *et al*. [@bondarenko:05]: 23.55 keV, 0.168(16); 65.85 keV, 0.126(14); and, 93.30 keV, 0.201(17). The difference between simulation and Ref. [@bondarenko:05] for the 65.85-keV level implies there is a substantial contribution from the quasi continuum that is not observed experimentally. Four levels were previously reported with tentative $J^{\pi}$ assignments [@wu:05]. For three of these levels, our simulations are consistent with the assignments of $9/2^{-}$, $7/2^{-}$, and $9/2^{-}$ to the 301.13-, 332.11-, and 390.4-keV levels, respectively. The agreement between modeled population and experimental depopulation by assuming these $J^{\pi}$-level assignments is illustrated in the population-depopulation plot of Fig. \[w185:PD\](b). Those assignments are also consistent with the distorted-wave Born approximation (DWBA) calculations described in Ref. [@bondarenko:05].
\[sec:level4.D\]$^{186}$W(n,$\gamma$)$^{187}$W\
-----------------------------------------------
A $^{186}$WO$_2$ target was irradiated for 2.03 h. Comparison of the DICEBOX-population calculations with the experimental-depopulation data for $^{187}$W sets $E_{\rm crit} = 900.0$ keV. This value is substantially higher than in RIPL where $E_{\rm crit} = 145.9$ keV and includes only three levels. Table \[tab:187Wg\] lists 40 levels below $E_{\rm crit} = 900.0$ keV, deexcited by 121 secondary $\gamma$ rays and populated by 16 primary $\gamma$ rays, with a range of spins from $1/2 \leq J \leq 15/2$. The capture state has $J^{\pi} = 1/2^{+}$. Multipolarities and mixing ratios are taken from ENSDF [@basunia:09] where available or assumed according to $\Delta J$ and $\Delta \pi $ selection rules.
As was the case for the other tungsten isotopes investigated in this study, $\Gamma_{0}$ shows a strong dependence on PSF/LD. The EGLO/BSFG models give $\Gamma_{0} = 0.058(3)$ eV, which compares well with the adopted value of $\langle \Gamma_{0} \rangle = 0.051(5)$ eV [@mughabghab:06]. For the EGLO/CTF combination $\Gamma_{0} = 0.038(2)$ eV, BA/CTF gives $\Gamma_{0} = 0.083(6)$ eV, and BA/BSFG gives $\Gamma_{0} = 0.127(7)$ eV.
A total thermal-capture cross section $\sigma_{0} = 33.33(62)$ b was determined for the $^{186}$W(n,$\gamma$) reaction. Figure \[fig:sigma0Ec\_w187\] shows the stability of this value with increasing cut-off energy, where $\sigma_{0}$ is nearly insensitive to $E_{\rm crit}$ even when as few as three levels are included. For three levels and $E_{\rm crit} = 200$ keV ,we get $\sigma_{0} = 34.7(32)$ b. Adopting $E_{\rm crit} = 900$ keV, with 40 levels in the decay scheme, $\sigma_{0}$ barely changes although the uncertainty is reduced by a factor of five. The overall uncertainty of 1.9 % is dominated by a 1.7 % uncertainty in the simulated cross section with all other errors contributing less than 1 %. In Table \[tab:187Wr\] we compare our result with other measurements in the literature and the value adopted by Mughabghab of $\sigma_{0} = 38.1(5)$ b [@mughabghab:06]. That value was based on an older activation decay-scheme normalization. The literature values in Table \[tab:187Wr\] have been corrected for the decay-scheme normalization from our activation measurement, described in Section \[sec:level4.E\], where possible.
$\sigma_{0}$ \[b\] Reference
-------------------- ----------------------------------------
[**33.33(62)**]{} [**This work (prompt)**]{}
34.2(70) L. Seren [@seren:47]
34.1(27) H. Pomerance [@pomerance:52]
41.3, 51 W. S. Lyon [@lyon:60]
33 J. H. Gillette [@gillette:66]
37.8(12) S. J. Friesenhahn [@friesenhahn:66]
35.4(8) P. P. Damle [@damle:67]
40.0(15) C. H. Hogg [@hogg:70]
33.6(16) G. Gleason [@gleason:77; @exfor]
33.3(11) R. E. Heft [@heft:78]
37.0(30) V. A. Anufriev [@anufriev:81]
38.5(8) K. Knopf [@knopf:87]
34.8(3) M. R. Beitins [@beitins:92]
34.7(15), 37.9(20) S. I. Kafala [@kafala:97]
32.7(10) N. Marnada [@marnada:99]
32.8(10) F. De Corte [@decorte:03-2]
30.6(19) M. Karadag [@karadag:04]
33.4(11) L. Szentmikl[ó]{}si [@szentmiklosi:06]
35.9(11) V. Bondarenko [@bondarenko:08]
38.7(23) M. S. Uddin [@uddin:08]
28.9(18) N. Van Do [@vando:08]
29.8(32) A. El Abd [@elabd:10]
38.1(5) Atlas [@mughabghab:06]
: \[tab:187Wr\] Summary of $^{186}$W(n,$\gamma$) $\sigma_{0}$ measurements.
Figure \[w187:PD\](a) shows excellent agreement between modeled population and experimental depopulation data for all levels except the 364.22-keV level. This level was reported in ENSDF to be deexcited by 162.7- and 286.9-keV $\gamma$ rays [@basunia:09]. The [DICEBOX]{}-simulated population is much larger than the experimentally observed depopulation of this level. Since the experimental data for all other levels compares well with their modeled populations over a range of five orders of magnitude, it is evident that the statistical model is an accurate simulation tool for the $^{187}$W capture-$\gamma$ decay scheme and discrepancies with the experimental intensity suggest incomplete experimental level or transition data. The $J^{\pi} = 9/2^{-}$ assignment is firmly established for this level [@basunia:09], so new $\gamma$ rays depopulating the 364.22-keV level were sought. In Fig. \[w187:PD\](b) we show that including a $\sim14$-keV transition populating the 350.43-keV level considerably improves agreement between experiment and theory. An additional low-energy $\gamma$ ray at 19.6 keV depopulating the 350.43-keV level is also suggested based on the statistical-model calculation. These newly proposed $\gamma$-ray transitions were also inferred from the coincidence data of Bondarenko *et al*. [@bondarenko:08].
In an earlier ENSDF evaluation of $^{187}$W [@firestone:91] two additional levels were reported at 493.41 and 551 keV that were removed in the latest evaluation [@basunia:09]. We see tentative evidence for the 143.2-keV $\gamma$ ray depopulating the 493.41-keV level. The statistical model simulations imply a $J^{\pi} = 9/2^{-}$ assignment for this state. There is insufficient evidence to support a level at around 551 keV, although there is a strong transition at 551.6 keV in the prompt capture-$\gamma$ spectrum. This transition is also present in the delayed $^{187}{\rm W} \rightarrow ^{187}{\rm Re} + \beta^{-}$ beta-decay spectrum and can be attributed to the decay of $^{187}$Re. We propose an additional 135.1-keV $\gamma$ ray depopulating the 775.60-keV level from the observed spectrum and consistency with statistical-model predictions. An additional low-energy transition at 16.20 keV, with likely $E1$ multipolarity, is proposed to depopulate the 613.38-keV level based on statistical-model calculations. The statistical model has also been used to estimate the intensity of the known 380.0-keV transition depopulating the $13/2^{-}$ level at 710.78-keV. A doublet centered on 380.22 keV is observed in our data and we have resolved the intensity of the known 380.0-keV component by determining the intensity limit consistent with model predictions for a transition decaying out of this high-spin state.
The statistical-model simulations were also used to test uncertain $J^{\pi}$ assignments for levels in $^{187}$W. The majority of the tentative $J^{\pi}$ assignments, for energy levels beneath $E_{\rm crit}$, were found to be consistent with the current ENSDF assignments, and 19 $J^{\pi}$ assignments for $^{187}$W [@basunia:09] could be confirmed in our analysis (see Table \[tab:187Wg\]). A recent investigation of the $J^{\pi}$ assignments in $^{187}$W using polarized deuterons incident upon a natural tungsten foil to measure the (d,p) reaction [@bondarenko:08] compared the observed particle angular distribution with DWBA calculations and determined $J$ and $l$-transfer values utilizing the [CHUCK3]{} code [@kunz:CHUCK3]. Our results are consistent with most of the $J^{\pi}$ assignments from (d,p) analysis except for an excited state at 884.13 keV. The (d,p) analysis suggests a value of $J^{\pi} = 7/2^{+}$ for this state, but we find that $J^{\pi} = 5/2^{+}$ is in agreement with our (n,$\gamma$) data, as illustrated in the population-depopulation plots in Fig. \[w187:PD\_2\]. The 884.13-keV state decays by a 474.02-keV transition, an assumed $E2$ quadrupole, to the 1.38-$\mu$s isomer at 410.06 keV, implying a likely $J^{\pi}=9/2^{+}$ assignment for this bandhead. Consequently, all other members of the rotational sequence built on this level will have spin values increased by one unit of angular momentum, as shown in Fig. \[w187:PD\_2\]. The previous $J^{\pi} = (11/2^{+})$ [@basunia:09] assignment for the 410.06-keV isomer was based on the systematics of neighboring odd-$A$ tungsten isotopes. Since only a few DWBA fits have been published, it would be instructive to see how well DWBA calculations for the lower-spin sequence would compare with the (d,p) data, as the shapes of experimental angular distributions are often well described by more than one set calculations, especially where counting statistics may be poor.
$E_{\gamma}$ \[keV\] $\sigma_{\gamma}^{(P)}$ $\sigma_{\gamma}^{(D)}$ $P_{\gamma}$ $P_{\gamma}$ $\sigma_{\gamma}$ $\sigma_{0}$ $\sigma_{\gamma}$ $\sigma_{0}$
---------------------- ------------------------- ------------------------- -------------- -------------- ------------------- -------------- ------------------- --------------
134.34(7) 3.60(12) 3.66(12) 0.110(4) 0.104(2) 3.65(7) 33.2(14) 3.50(2) 31.9(12)
479.47(5) 9.55(16) 9.65(22) 0.289(9) 0.266(4) 9.29(14) 32.1(11) 9.19(9) 31.7(10)
551.22(9) 2.16(19) 2.20(4) 0.0661(17) 0.0614(10) 2.16(4) 32.6(10) 2.14(1) 32.37(85)
617.96(6) 3.12(11) 2.54(5) 0.0762(21) 0.0757(12) 2.66(5) 35.0(11) 2.68(1) 35.18(98)
625.03(10) 0.35(11) 0.419(19) 0.0126(6) 0.0131(2) 0.47(1) 37.2(20) - -
685.74(5) 11.85(21) 11.74(20) 0.352(9) 0.332(5) 11.78(21) 33.5(10) 11.48(6) 32.60(84)
772.99(10) 1.606(95) 1.771(57) 0.053(2) 0.0502(8) 1.75(3) 33.0(13) 1.74(1) 32.8(12)
We did not observe the 45.8(3) keV, presumed $E1$ transition [@basunia:09], deexciting 410.06-keV 1.38-$\mu$s isomer, that was reported by Bondarenko [*et al*]{}. [@bondarenko:08] on the basis of delayed coincidences with the 474.02-keV $\gamma$-ray deexciting the 884.13-keV level. Bondarenko [*et al*]{}. also postulated a second, $\sim 59$-keV transition, based on delayed coincidences with $\gamma$ rays deexciting the 350.43-keV, $7/2^{-}$ level. This transition is of the same energy as the strong tungsten $K_{\alpha_{1}}$ X rays that obscure it in the spectrum. Bondarenko [*et al*]{}. speculated the existence of the 59-keV $\gamma$-ray as unlikely since it required an $M2$ multipolarity assuming an $11/2^{+}$ assignment for the 410.06-keV level. Our new $J^{\pi} = 9/2^{+}$ assignment for the 410.06-keV level implies an acceptable $E1$ transition for this 59-keV $\gamma$ ray. However, the existence of the 59-keV $\gamma$-ray still remains in doubt since the proposed 13.80-keV transition deexciting the 364.22-keV level would also explain the coincidence results. We observed two $\gamma$-rays populating the 410.06-keV isomer from higher levels below $E_{\rm crit}$. The experimental intensity feeding the isomer, $\sum \sigma_{\gamma}^{\rm exp}(9/2^{+};410.06~{\rm keV}) = 0.394(16)$ b, together with the [DICEBOX]{}-modeled contribution from the quasi continuum, $P(9/2^{+};410.06~{\rm keV}) = 0.0145(14)$, yields a radiative thermal-capture cross section for the isomer $\sigma_{9/2^{+}} = 0.400(16)$ b. This lower limit is consistent with our simulated population for $J^{\pi} =9/2^{+}$ and inconsistent with $J^{\pi} = 11/2^{+}$ (Fig. \[w187:PD\_2\]). Based on our analysis we propose new $J^{\pi}$ assignments for the five levels at: 410.06 keV ($9/2^{+}$); 493.4 keV ($9/2^{-}$); 597.24 keV ($11/2^{+}$); 815.51 keV ($13/2^{+}$); and, 884.13 keV ($5/2^{+}$).
\[sec:level4.E\]Activation cross sections for $^{187}$W ($T_{1/2} = 24.000(4)$ h)\
----------------------------------------------------------------------------------
The same $^{186}$W target used in the prompt $\gamma$-ray measurements was later analyzed, offline, to determine the activation cross sections, $\sigma_{\gamma}$, for $\gamma$ rays emitted following $^{187}$W decay. Since this measurement was performed in the same experiment, the decay $\gamma$-ray cross sections could be determined proportionally to the cross sections of the prompt $\gamma$ rays. These activation $\gamma$-ray cross sections, together with their $\gamma$-decay emission probabilities, $P_\gamma$, independently determine the total radiative neutron-capture cross section, $\sigma_{0}$.
The decay $\gamma$ rays were observed in both the prompt spectrum, where the background from prompt $\gamma$ rays was high, and after bombardment, when the background was much lower. To determine the activation $\gamma$-ray cross sections, they must be corrected for saturation during bombardment, decay following bombardment and before counting begins, and decay during the counting interval. The decay $\gamma$ rays, measured in the prompt spectrum, can be corrected with an in-beam saturation factor ($B$) defined as $$B = 1 - \left( \frac{1-{\rm exp}(-\lambda t_{S})}{\lambda t_{S}} \right),
\label{pg2}$$ where $\lambda=\ln(2)/T_{1/2}$ is the decay constant and $t_{S}$ is the irradiation period. This expression is valid assuming a constant neutron flux. Monitoring showed little power variation at the Budapest Research Reactor [@belgya:pc] during our measurements. The corrected activation $\gamma$-ray cross sections, measured in the prompt spectrum, are then given by $$\sigma_{\gamma}^{(P)} = \frac{\sigma_{\gamma}}{B},
\label{pg3}$$ where $\sigma_{\gamma}$ is the uncorrected cross section observed during bombardment.
When the sample is analyzed offline the $\gamma$-ray cross sections in the delayed spectrum must also be corrected for saturation corresponding to in-beam exposure according to the factor $S = 1 - \exp(-\lambda t_{S})$. The decay time $t_D$ following bombardment until analysis commences, introduces a further correction factor $D = \exp(-\lambda t_{D})$. In addition, decay during the counting interval $t_C$ is corrected by a factor $C = [1 - \exp(-\lambda t_{C})]/(\lambda t_{C})$. The overall correction factor accounting for saturation, decay, and counting intervals can then be applied to the cross sections of the decay $\gamma$-rays observed in the delayed spectrum as $$\sigma_{\gamma}^{(D)} = \frac{\sigma_{\gamma}}{S \cdot D \cdot C}.
\label{pg7}$$
In this work the irradiation time was $t_{S} = 7536$ s, and the source decayed for a time $t_{D} = 64859$ s before being counted for $t_{C} = 11645$ s. The activation $\gamma$-ray cross sections for the most intense transitions in the prompt and delayed spectra are shown in Table \[tab:8\]. The prompt and delayed $\gamma$-ray cross sections were consistent. We can then determine the $\gamma$-ray emission probabilities, $P_{\gamma} = \sigma_{\gamma} / \sigma_{0}$, using $\sigma_{0} =33.33(62)$ b from our prompt $\gamma$-ray measurement. These probabilities are also listed in Table \[tab:8\] and are consistent with the $P_{\gamma}$ values from ENSDF [@basunia:09], based on the decay scheme normalization of Marnada *et al*. [@marnada:99]. Using the $P_{\gamma}$ values from our activation data, we can then find independent total radiative thermal neutron-capture cross sections, $\sigma_{0} = \sigma_{\gamma} / P_{\gamma}$, based on the delayed-transition cross sections reported in the activation measurements of Szentmikl[ó]{}si *et al*. [@szentmiklosi:06] and De Corte and Simonits [@decorte:03-2]. In this approach, we find that our prompt measurement, $\sigma_{0} = 33.33(62)$ b, compares well with the weighted average of Szentmikl[ó]{}si *et al*. [@szentmiklosi:06], $\sigma_{0} = 33.4(11)$ b, and also, with that of De Corte and Simonits [@decorte:03-2], $\sigma_{0} = 32.8(10)$.
\[sec:level7\]Neutron separation energies\
==========================================
A byproduct of our analysis is the determination of neutron separation energies, $S_{\rm n}$, for $^{183,184,185,187}$W from the (n,$\gamma$) primary $\gamma$-ray energy measurements and the final-level energies taken from ENSDF. These results, corrected for recoil, are shown in Table \[tab:Sn\] where they are compared with the recommended values of Wang *et al*. [@Wang:AME]. We present more precise determinations of $S_{\rm n}$ for $^{184,185}$W.
Nuclide $E_{\gamma}$ $E_{f}$ $S_{\rm n}$
----------- -------------- -------------- -------------
$^{183}$W 6190.78(6) 0.0 6190.88(6)
6144.28(6) 46.4839(4) 6190.87(6)
Average 6190.88(6)
Adopted 6190.81(5)
$^{184}$W 7410.99(14) 0.0 7411.14(14)
7299.69(16) 111.2174(4) 7411.03(16)
6507.63(16) 903.307(9) 7411.05(16)
6408.60(12) 1002.49(4) 7411.20(13)
6289.51(13) 1121.440(14) 7411.06(13)
Average 7411.11(13)
Adopted 7411.66(25)
$^{185}$W 5753.65(5) 0.0 5753.74(5)
Adopted 5753.71(30)
$^{187}$W 5466.47(12) 0.0 5466.55(12)
5320.65(8) 145.848(9) 5466.57(8)
5261.67(9) 204.902(9) 5466.65(9)
4684.31(7) 782.290(19) 5466.66(7)
4662.94(27) 803.369(22) 5466.37(27)
4650.27(8) 816.256(19) 5466.58(8)
4626.40(7) 840.205(16) 5466.66(7)
4574.67(7) 891.93(4) 5466.66(8)
Average 5466.62(7)
Adopted 5466.79(5)
: \[tab:Sn\] Neutron-separation energies determined from (n,$\gamma$) reactions: $S_{\rm n} = E_{\gamma} + E_{f} + E_{r}$, where $E_{f}$ is the energy of the final level and $E_{r} = E_{\gamma}^{2}/2A$ is the recoil energy. The weighted average for each nuclide is compared to the adopted value of Wang *et al*. [@Wang:AME].
\[sec:level6\]Summary\
======================
The total radiative thermal neutron-capture $\gamma$-ray cross sections, $\sigma_{0}$, for the four major tungsten isotopes are summarized in Table \[tab:summary\]. The cutoff energies, $E_{\rm crit}$, partial $\gamma$-ray cross sections, $\sum \sigma^{\rm exp}_\gamma$, simulated continuum GS feedings, $P({\rm GS})$, and simulated cross sections, $\sum \sigma^{\rm sim}_\gamma$, and an error budget are also given in Table \[tab:summary\]. Our new cutoff energies exceed the RIPL-suggested $E_{\rm crit}$ values [@capote:09] for all isotopes. These analyses have established that $\sigma_{0}$ is nearly independent of the assumed value of $E_{\rm crit}$, which is consistent with our earlier results for the palladium isotopes [@krticka:08].
Several combinations of photon strength function and level density formalisms were compared to the experimental data. Total radiative widths of the capture state were found to be very model dependent. For the compound $^{183,184}$W capture states, we could best reproduce the mean-adopted width $\langle \Gamma_{0} \rangle$ [@mughabghab:06] with the EGLO/CTF model combination. In the cases of $^{185,187}$W, $\Gamma_{0}$ was best reproduced assuming the EGLO/BSFG combination. All combinations involving BA gave much poorer agreement with the adopted $\Gamma_{0}$.
This analysis proposes several changes to the decay schemes for the compound tungsten isotopes $^{183,184,185,187}$W. For $^{183}$W, one new $\gamma$ ray below $E_{\rm crit}$ is proposed, based on statistical-model simulations, and a tentative $J^{\pi}$ assignment is confirmed. The 309.49-keV, 5.2(3)-s, $11/2^{+}$ isomer in $^{183}$W was populated with a cross section of 0.177(18) b. For $^{184}$W, one new $\gamma$ ray was placed in the decay scheme, based on our experiments, an additional low-energy transition is proposed from simulations, and four tentative $J^{\pi}$ assignments are confirmed. Our analysis also indicates that the capture state in $^{184}$W is consistent with the composition $J^{\pi}_{\rm CS} = 1^{-}(\gtrsim 80~\%)$, $J^{\pi}_{\rm CS} = 0^{-}(\lesssim 20~\%)$, which is also consistent with the *Atlas of Neutron Resonances* [@mughabghab:06]. We find $J^{\pi} = 1^{-}$ the most likely assignment for the bound resonance at $-26.58$ eV, implying a likely capture-state spin composition of $J^{\pi}_{\rm CS} = 0^{-}(7.4~\%) + 1^{-}(92.6~\%)$. The 1285.00-keV, 8.33(18)-$\mu$s, $5^-$ isomer in $^{184}$W was populated with a cross section of 0.0246(55) b. In $^{185}$W two new low-energy $\gamma$-ray transitions are proposed based on simulations, and three previous tentative $J^{\pi}$ assignments have been validated. The 197.38-keV, 1.67(3)-min, 11/2$^+$ isomer in $^{185}$W was populated with a cross section of 0.0062(16) b. For $^{187}$W, 19 of the previous $J^{\pi}$ assignments are confirmed and new $J^{\pi}$ assignments are proposed for five levels, including a new $9/2^{+}$ bandhead assignment at 410.06 keV that was previously assigned $(11/2^{+})$. In addition, we reintroduced the 493.4-keV level, from an earlier ENSDF evaluation [@firestone:91], and a new $\gamma$ ray depopulating this level based on tentative evidence in the capture-$\gamma$ spectrum. There is also tentative evidence for a new transition at around 135.1 keV, depopulating the 775.60-keV level. Our $^{187}$W simulations support inclusion of four new low-energy $\gamma$ rays, three of which were previously inferred in the work of Bondarenko [*et al*]{}. [@bondarenko:08]. The 410.06-keV, 1.38(7)-$\mu$s, $11/2^{+}$ isomer in $^{187}$W was populated with a cross section of $0.400(16)$ b. An analysis of the $\beta^{-}$-delayed $\gamma$-ray spectrum provided an independent decay-scheme normalization based on a new set of $P_{\gamma}$ measurements that compare well to the ENSDF decay-scheme normalization [@basunia:09], adopted from the earlier work of Marnada *et al*. [@marnada:99]. Independent values of $\sigma_{0}$, consistent with our prompt measurement, were then determined based on our activation-data decay-scheme normlaization, thus providing confirmation of our approach.
The decay-scheme improvements suggested in this work will be used to improve the ENSDF nuclear-structure evaluations [@ensdf], that contribute to the RIPL nuclear-reaction database [@capote:09]. The new thermal-capture (n,$\gamma$) data will be added to the EGAF database [@firestone:06]. These new data will also be used to help produce a more extensive and complete thermal-capture $\gamma$-ray library for the ENDF [@chadwick:11] neutron-data library. Additional measurements of capture $\gamma$-rays from the rare isotope $^{180}$W(n,$\gamma$) are in progress and will complete our knowledge of the tungsten isotopes and resolve discrepancies in the measured $\sigma_{0}$ for this nucleus.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was performed under the auspices of the University of California, supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U. S. Department of Energy at the Lawrence Berkeley National Laboratory under Contract DE-AC02-05CH11231, and by the U. S. Department of Energy by the Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. The access to the Budapest PGAA facility was financially supported by the NAP VENEUS08 grant under Contract OMFB-00184/2006. Additional support was received through the research plan MSM 002 162 0859 supplied by the Ministry of Education of the Czech Republic. The operations staff at the Budapest Research Reactor are gratefully acknowledged.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This paper develops a new family of estimators, the minimum density power divergence estimators (MDPDEs), for the parameters of the one-shot device model as well as a new family of test statistics, Z-type test statistics based on MDPDEs, for testing the corresponding model parameters. The family of MDPDEs contains as a particular case the maximum likelihood estimator (MLE) considered in Balakrishnan and Ling (2012). Through a simulation study, it is shown that some MDPDEs have a better behavior than the MLE in relation to robustness. At the same time, it can be seen that some Z-type tests based on MDPDEs have a better behavior than the classical Z-test statistic also in terms of robustness.'
author:
- 'N. Balakrishnan, E. Castilla, N. Martín, and L.Pardo'
title: '**Robust Estimators and Test-Statistics for One-Shot Device Testing Under the Exponential Distribution**'
---
Introduction\[sec1\]
====================
The reliability of a product, system, weapon, or piece of equipment can be defined as the ability of the device to perform as designed, or, more simply, as the probability that the device does not fail when used. Engineers’s assess reliability by repeatedly testing the device and observing its failure rate. Certain products, called one-shot devices, make this approach challenging. One-shot devices can only be used once and after use the device is either destroyed or must be rebuilt. Consequently, one can only know whether the failure time is either before or after the test time. The outcomes from each of the devices are therefore binary, either left-censored (failure) or right-censored (success). Some examples of one-shot devices are nuclear weapons, space shuttles, automobile air bags, fuel injectors, disposable napkins, heat detectors, and fuses. In survival analysis, these data are called current status data. For instance, in animal carcinogenicity experiments, one observes whether a tumor occurs at the examination time for each subject.
Due to the advances in manufacturing design and technology, products have now become highly reliable with long lifetimes. This fact would pose a problem in the analysis if only few or no failures are observed. For this reason, accelerated life tests are often used by adjusting a controllable factor such as temperature in order to have more failures in the experiment. On the other hand, accelerated life testing would shorten the experimental time and also help to reduce the experimental cost. In this paper, we shall assume that the failure times of devices follow an exponential distribution. In this context, Balakrishnan and Ling (2012) developed the EM algorithm for finding the maximum likelihood estimators of the model parameters. Fan et al. (2009) studied a Bayesian approach for one-shot device testing along with an accelerating factor, in which the failure times of devices is assumed to follows once again an exponential distribution. Rodrigues et al. (1993) presented two approaches based on the likelihood ratio statistics and the posterior Bayes factor for comparing several exponential accelerated life models. Chimitova and Balakrishnan (2015) made a comparison of several goodness-of-fit tests for one-shot device testing.
In Section \[sec2\], we present a description of the one-shot device model as well as the maximum likelihood estimators for the model parameters. Section \[sec3\] develops the minimum density power divergence estimator as a natural extension of the maximum likelihood estimator, as well as its asymptotic distribution. In Section \[sec4\], $Z$-type test statistics are introduced in order to test some hypotheses about the parameters of the one-shot device model. Some numerical examples are presented in Section \[sec5\], with one of them relating to a reliability situation and the other two are real applications to tumorigenicity experiments. In Section \[sec6\], an extensive simulation study is presented in order to analyze the robustness of the MDPDEs, as well as the $Z$-type test introduced earlier. Finally, some concluding remarks are made in Section \[sec7\].
Model formulation and maximum likelihood estimator\[sec2\]
==========================================================
Consider a reliability testing experiment in which at each time, $t_{j}$, $j=1,2,...,J$, $K$ devices are placed in total under temperatures $w_{i}$, $i=1,...,I$. Therefore, $IJK$ devices are tested in total at temperatures $w_{i}$, $i=1,...,I$, at times $t_{j},$ $j=1,...,J$. It is worth noting that a successful detonation occurs if the lifetime is beyond the inspection time, whereas the lifetime will be before the inspection time if the detonation is a failure. For each temperature $w_{i}$ and at each inspection time $t_{j}$, the number of failures, $n_{ij}$, is then recorded.
In Balakrishnan and Ling (2012), an example is illustrated, in which $30$ devices were tested at temperatures $w_{i}\in\{35,$ $45,$ $55\}$, each with $10$ units being detonated at times $t_{j}\in\{10,$ $20,$ $30\}$, respectively.$\ $In this example, we have $I=3$, $J=3$ and $K=10$. The number of failures observed is summarized in the $3\times3$ table given in Table \[table1\]. In this one-shot device testing experiment, there were in all $48$ failures out a total of $90$ tested devices.
$\begin{tabular}
[c]{l|l|l|l|}\cline{2-4}
& $t\_[1]{}=10$ & $t\_[2]{}=20$ & $t\_[3]{}=30$\\\hline
\multicolumn{1}{|l|}{$w_{1}=35$} & \multicolumn{1}{|c|}{$3$} &
\multicolumn{1}{|c|}{$3$} & \multicolumn{1}{|c|}{$7$}\\\hline
\multicolumn{1}{|l|}{$w_{2}=45$} & \multicolumn{1}{|c|}{$1$} &
\multicolumn{1}{|c|}{$5$} & \multicolumn{1}{|c|}{$7$}\\\hline
\multicolumn{1}{|l|}{$w_{3}=55$} & \multicolumn{1}{|c|}{$6$} &
\multicolumn{1}{|c|}{$7$} & \multicolumn{1}{|c|}{$9$}\\\hline
\end{tabular}
\ \ \ \ \ $
We shall assume here, in accordance Balakrishnan and Ling$\ $(2012), that the true lifetimes $T_{ijk}$, where $i=1,2,...,I$, $j=1,2,...,J$, $k=1,...,K$, are independent and identically distributed exponential random variables with probability density function $$f(t|\lambda)=\lambda\exp\left( -\lambda t\right) ,$$ where $\lambda>0$ is the unknown failure rate. In practice, we consider inspection times $t_{j}$, $j=1,...,J$, rather than $t>0$, and we relate the parameter $\lambda$ to an accelerating factor of temperature $w_{i}>0$ through a log-linear link function as $$\lambda_{w_{i}}(\boldsymbol{\alpha})=\alpha_{0}\exp\left\{ \alpha_{1}w_{i}\right\} ,$$ where $\alpha_{0}>0$ and $\alpha_{1}\in\mathbb{R}$ are unknown parameters. Therefore, the corresponding distribution function is$$\begin{aligned}
F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))&=1-\exp\left\{ -\lambda_{w_{i}}(\boldsymbol{\alpha})t_{j}\right\} \nonumber \\
& =1-\exp\left\{ -\alpha_{0}\exp\left\{
\alpha_{1}w_{i}\right\} t_{j}\right\} \label{eq:expo}$$ and the density function$$f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))=\alpha_{0}\exp\left\{
\alpha_{1}w_{i}\right\} \exp\left\{ -\alpha_{0}\exp\left\{ \alpha_{1}w_{i}\right\} t_{j}\right\} . \label{eq:expo2}$$ The data are completely described on $K$ devices, through the contingency table of failures $\boldsymbol{n}=(n_{11},...,n_{1J},...,\allowbreak
n_{I1},...,n_{IJ})^{T}$, collected at the temperatures $\boldsymbol{w}=(w_{1},...,w_{I})^{T}$ and the inspection times $\boldsymbol{t}=(t_{1},...,t_{J})^{T}$.
We shall consider the theoretical probability vector $\boldsymbol{p}(\boldsymbol{\alpha})$ defined by $$\begin{aligned}
\boldsymbol{p}(\boldsymbol{\alpha})=&\left( \tfrac{F(t_{1}|\lambda_{w_{1}}(\boldsymbol{\alpha}))}{IJ},\tfrac{1-F(t_{1}|\lambda_{w_{1}}(\boldsymbol{\alpha}))}{IJ}, \right. \\
&\left....,\tfrac{F(t_{J}|\lambda_{w_{I}}(\boldsymbol{\alpha}))}{IJ},\tfrac{1-F(t_{J}|\lambda_{w_{I}}(\boldsymbol{\alpha}))}{IJ}\right) ^{T},\end{aligned}$$ as well as the observed probability vector$$\widehat{\boldsymbol{p}}=\left( \tfrac{n_{11}}{IJK},\tfrac{K-n_{11}}{IJK},...,\tfrac{n_{IJ}}{IJK},\tfrac{K-n_{IJ}}{IJK}\right) ^{T},$$ both of dimension $2IJ$. Then the Kullback-Leibler divergence between the probability vectors $\widehat{\boldsymbol{p}}$ and $\boldsymbol{p}(\boldsymbol{\alpha})$ is given by $$\begin{aligned}
d_{KL}\left( \widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha
})\right) &=\frac{1}{IJ}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\left(
\frac{n_{ij}}{K}\log\frac{n_{ij}}{KF(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))}\right.\\
&+\left. \frac{K-n_{ij}}{K}\log\frac{K-n_{ij}}{K\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right) }\right) .\end{aligned}$$ It is not difficult to establish the following result.
The likelihood function$$\begin{aligned}
\mathcal{L}\left( \boldsymbol{\alpha}\left\vert K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right) =\prod\limits_{i=1}^{I}\prod\limits_{j=1}^{J}F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))^{n_{ij}}\\
\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right) ^{K-n_{ij}},\end{aligned}$$ where $F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))$ is given by (\[eq:expo\]), is related to the Kullback-Leibler divergence between the probability vectors $\widehat{\boldsymbol{p}}$ and $\boldsymbol{p}(\boldsymbol{\alpha})$ through$$\begin{aligned}
d_{KL}\left( \widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha
})\right) =\frac{1}{IJK}\left( s-\log\mathcal{L}\left( \boldsymbol{\alpha
}\left\vert K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right)
\right) , \label{1}$$ with $s$ being a constant not dependent on $\boldsymbol{\alpha}$.
Based on the previous result, we have the following definition for the maximum likelihood estimators of $\alpha_{0}$ and $\alpha_{1}.$
We consider the data given by $K$, $\boldsymbol{n}$, $\boldsymbol{t}$, $\boldsymbol{w}$ for the one-shot device model. Then, the maximum likelihood estimator of $\boldsymbol{\alpha}=(\alpha_{0},\alpha_{1})^{T}$, $\widehat{\boldsymbol{\alpha}}=(\widehat{\alpha}_{0},\widehat{\alpha}_{1})^{T}$, can be defined as $$\widehat{\boldsymbol{\alpha}}=\arg\min_{\boldsymbol{\alpha}\in\Theta}d_{KL}\left( \widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha
})\right) , \label{2}$$ where $\Theta=(\mathbb{R}^{+},\mathbb{R})^{T}$.
Minimum density power divergence estimator\[sec3\]
==================================================
Based on expression (\[2\]), we can think of defining an estimator minimizing any distance or divergence between the probability vectors $\widehat{\boldsymbol{p}}$ and $\boldsymbol{p}(\boldsymbol{\alpha})$. There are many different divergence measures (or distances) known in the lierature, see, for instance, Pardo (2006) and Basu et al. (2011), and the natural question is if all of them are valid to define estimators with good properties. Initially the answer is yes, but we must think in terms of efficiency as well as robustness of the defined estimators. From an asymptotic point of view, it is well-known that the maximum likelihood estimator is a BAN (Best Asymptotically Normal) estimator, but at the same time we know that the maximum likelihood estimator has a very poor behavior, in general, in relation to robustness. It is well-known that a gain in robustness leads to a loss of efficiency. Therefore, the distances (divergence measures) that we must use are those which result in estimators having good properties in terms of robustness with low loss of efficiency. The density power divergence measure introduced by Basu et al. (1998) has the required properties and has been studied for many different problems until now. For more details, see Ghosh et al. (2016), Basu et al. (2016) and the references therein.
Based on Ghosh and Basu (2013), the MDPDE of $\boldsymbol{\alpha}$ is first introduced, and later in Result \[Th4\] it is shown that this estimator can be considered as a natural extension of (\[2\]).
\[def1\]Let $y_{ijk}$ , $i=1,2,...,I$, $j=1,2,...,J$, $k=1,...,K$, be a sequence of independent Bernoulli random variables, $y_{ijk}\overset{ind}{\sim
}Ber(\pi_{ij}(\boldsymbol{\alpha}))$, such that $\pi_{ij}(\boldsymbol{\alpha
})=F\left( t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})\right) $ and $n_{ij}={\textstyle\sum\nolimits_{k=1}^{K}}
y_{ijk}$. The MDPDE of $\boldsymbol{\alpha}$, with tuning parameter $\beta
\geq0$, is given by $$\widehat{\boldsymbol{\alpha}}_{\beta}=\arg\min_{\boldsymbol{\alpha}\in\Theta
}\frac{1}{IJK}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\sum\limits_{k=1}^{K}V_{ij}\left( y_{ijk},\beta\right), \label{4}$$ where $$\begin{aligned}
V_{ij}\left( y_{ijk},\beta\right) &=\pi_{ij}^{\beta+1}(\boldsymbol{\alpha })+(1-\pi_{ij}(\boldsymbol{\alpha}))^{\beta+1}\\
&-\frac{1+\beta}{\beta}\left(\pi_{ij}^{y_{ijk}}(\boldsymbol{\alpha})(1-\pi_{ij}(\boldsymbol{\alpha }))^{1-y_{ijk}}\right) ^{\beta}.\end{aligned}$$
For more details about the interpretation of Definition 3, see formula 2.3 in Ghosh and Basu (2013), in which $\pi_{ij}^{y_{ijk}}(\boldsymbol{\alpha})(1-\pi_{ij}(\boldsymbol{\alpha}))^{1-y_{ijk}}$ plays the role of the density in our context. Notice that the expression to be minimized in (\[4\]) can be simplified as $$\begin{aligned}
& \frac{1}{IJK}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\sum\limits_{k=1}^{K}\left\{ \pi_{ij}^{\beta+1}(\boldsymbol{\alpha})+(1-\pi_{ij}(\boldsymbol{\alpha}))^{\beta+1}\right. \nonumber \\
&\left. -\frac{1+\beta}{\beta}\left( \pi_{ij}^{y_{ijk}}(\boldsymbol{\alpha})(1-\pi_{ij}(\boldsymbol{\alpha
}))^{1-y_{ijk}}\right) ^{\beta}\right\} \nonumber\\
& =\frac{1}{IJ}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\left\{ \pi
_{ij}^{\beta+1}(\boldsymbol{\alpha})+(1-\pi_{ij}(\boldsymbol{\alpha}))^{\beta+1}\right. \nonumber \\
&\left. -\frac{1+\beta}{\beta}\frac{n_{ij}}{K}\pi_{ij}^{\beta
}(\boldsymbol{\alpha})-\frac{1+\beta}{\beta}\frac{K-n_{ij}}{K}(1-\pi
_{ij}(\boldsymbol{\alpha}))^{\beta}\right\} . \label{4b}$$
The following result provides an alternative expression for $\widehat{\boldsymbol{\alpha}}_{\beta}$, given in Definition \[def1\], which is closer to (\[2\]) in its expression, since only a divergence measure between two probabilities is involved. Given two probability vectors $\boldsymbol{p}=\left( p_{1},...,p_{M}\right) ^{T}$ and $\boldsymbol{q}=\left( q_{1},...,q_{M}\right) ^{T}$, the power density divergence measure between $\boldsymbol{p}$ and $\boldsymbol{q}$, with tuning parameter $\beta
>0$, is given by$$d_{\beta}\left( \boldsymbol{p},\boldsymbol{q}\right) =\sum\limits_{j=1}^{M}\left\{ q_{j}^{\beta+1}-(1+\tfrac{1}{\beta})q_{j}^{\beta}p_{j}+\tfrac
{1}{\beta}p_{j}^{1+\beta}\right\} ,$$ and for $\beta=0$,$$d_{0}\left( \boldsymbol{p},\boldsymbol{q}\right) =\lim_{\beta\rightarrow
0^{+}}d_{\beta}\left( \boldsymbol{p},\boldsymbol{q}\right) =d_{KL}\left(
\boldsymbol{p},\boldsymbol{q}\right) .$$ Therefore, the density power divergence measure between the probability vectors $\widehat{\boldsymbol{p}}$ and $\boldsymbol{p}(\boldsymbol{\alpha})$, with tuning parameter $\beta>0$, has the expression $$\begin{aligned}
d_{\beta}\left( \widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha
})\right) & =\frac{1}{(IJ)^{\beta+1}}\sum\limits_{i=1}^{I}\sum
\limits_{j=1}^{J}\left\{ \pi_{ij}^{1+\beta}(\boldsymbol{\alpha})\right. \nonumber\\
&\left.-\tfrac{\beta+1}{\beta}\pi_{ij}^{\beta}(\boldsymbol{\alpha})\frac{n_{ij}}{K}+\tfrac{1}{\beta}\left( \frac{n_{ij}}{K}\right) ^{1+\beta}\right.
\nonumber\\
& \left. +\left( 1-\pi_{ij}(\boldsymbol{\alpha})\right) ^{1+\beta}-\tfrac{\beta+1}{\beta}\left( 1-\pi_{ij}(\boldsymbol{\alpha})\right)
^{\beta}\tfrac{K-n_{ij}}{K} \right. \nonumber\\
&\left.+\tfrac{1}{\beta}\left( \frac{K-n_{ij}}{K}\right)
^{1+\beta}\right\} , \label{3}$$ and for $\beta=0$ $$d_{\beta=0}\left( \widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha
})\right) =\lim_{\beta\rightarrow0^{+}}d_{\beta}\left(
\widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha})\right)
=d_{KL}\left( \widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha
})\right) .$$
\[Th4\]The MDPDE of $\boldsymbol{\alpha}$, with tuning parameter $\beta\geq0$, given in Definition \[def1\], can be alternatively defined as $$\widehat{\boldsymbol{\alpha}}_{\beta}=\arg\min_{\boldsymbol{\alpha}\in\Theta
}d_{\beta}\left( \widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha
})\right) , \label{3b}$$ where $d_{\beta}\left( \widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha})\right) $ is as in (\[3\]).\
In the following result, the estimating equations needed to get the MDPDEs are presented.
\[Th3\]The MDPDE of $\boldsymbol{\alpha}$ with tuning parameter $\beta
\geq0$, $\widehat{\boldsymbol{\alpha}}_{\beta}$, can be obtained as the solution of equations (\[5\]) and (\[5B\]).
$$\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\left( K\ F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))-n_{ij}\right) f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))t_{j}\left[ F^{\beta-1}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))+\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\right) ^{\beta-1}\right] =0 \label{5}$$
$$\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\left( K\ F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))-n_{ij}\right) f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))t_{j}w_{i}\left[ F^{\beta-1}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))+\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\right) ^{\beta-1}\right] =0. \label{5B}$$
In the following results, the asymptotic distribution of the MDPDE of $\boldsymbol{\alpha}$, $\widehat{\boldsymbol{\alpha}}_{\beta}$, for the one-shot device model is presented.
\[Th5\]The asymptotic distribution of the MDPDE $\widehat{\boldsymbol{\alpha}}_{\beta}$ is given by $$\sqrt{K}\left( \widehat{\boldsymbol{\alpha}}_{\beta}-\boldsymbol{\alpha}_{0}\right) \overset{\mathcal{L}}{\underset{K\mathcal{\rightarrow}\infty}{\longrightarrow}}\mathcal{N}\left( \boldsymbol{0},\boldsymbol{\bar
{J}}_{\beta}^{-1}(\boldsymbol{\alpha}_{0})\boldsymbol{\bar{K}}_{\beta
}(\boldsymbol{\alpha}_{0})\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha
}_{0})\right) ,$$ where $$\begin{aligned}
\boldsymbol{\bar{J}}_{\beta}(\boldsymbol{\alpha}) & =\sum\limits_{i=1}^{I}\begin{pmatrix}
\frac{1}{\alpha_{0}^{2}} & \frac{w_{i}}{\alpha_{0}} \nonumber \\
\frac{w_{i}}{\alpha_{0}} & w_{i}^{2}\end{pmatrix}
\sum\limits_{j=1}^{J}t_{j}^{2}f^{2}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\\
&\times \left[ F^{\beta-1}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))+(1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})))^{\beta-1}\right]
,\label{Jbar}\\
\boldsymbol{\bar{K}}_{\beta}(\boldsymbol{\alpha}) & =\sum\limits_{i=1}^{I}\begin{pmatrix}
\frac{1}{\alpha_{0}^{2}} & \frac{w_{i}}{\alpha_{0}}\\
\frac{w_{i}}{\alpha_{0}} & w_{i}^{2}\end{pmatrix}
\sum\limits_{j=1}^{J}t_{j}^{2}f^{2}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\nonumber\\
& \times\left\{ \left[ F^{2\beta-1}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))+(1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
})))^{2\beta-1}\right] \right.\nonumber \\
& \left. -\left[ F^{\beta}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))-(1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
})))^{\beta}\right] ^{2}\right\}, \label{Kbar}$$ and $F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))$ and $f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))$ are given by (\[eq:expo\]) and (\[eq:expo2\]), respectively.
Since $\widehat{\boldsymbol{\alpha}}_{\beta=0}$ is the MLE of $\boldsymbol{\alpha}$, obtained by maximizing $\log\mathcal{L}\left(
\boldsymbol{\alpha}\left\vert K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right)$, or equivalently by minimizing$$\begin{aligned}
d_{\beta=0}\left( \widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha
})\right) &=\lim_{\beta\rightarrow0^{-}}d_{\beta}\left(
\widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha})\right)
=d_{KL}\left( \widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha
})\right) \\
&=\frac{1}{IJK}\left( s-\log\mathcal{L}\left( \boldsymbol{\alpha
}\left\vert K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right)
\right) ,\end{aligned}$$ the following result relates the asymptotic distribution of $\widehat{\boldsymbol{\alpha}}_{\beta=0}$ given previously in terms of $\boldsymbol{\bar{J}}_{\beta=0}(\boldsymbol{\alpha}_{0})$ and $\boldsymbol{\bar{K}}_{\beta=0}(\boldsymbol{\alpha}_{0})$, with respect to the Fisher information matrix, well-known in the classical asymptotic theory of the MLEs.
The asymptotic distribution of the MLE of $\boldsymbol{\alpha}$, $\widehat{\boldsymbol{\alpha}}_{\beta=0}$, is$$\sqrt{K}\left( \widehat{\boldsymbol{\alpha}}_{\beta}-\boldsymbol{\alpha}_{0}\right) \overset{\mathcal{L}}{\underset{K\mathcal{\rightarrow}\infty}{\longrightarrow}}\mathcal{N}\left( \boldsymbol{0},\tfrac{1}{IJ}\boldsymbol{I}_{F}^{-1}\left( \boldsymbol{\alpha}_{0}\right) \right) ,$$ where
$$\boldsymbol{I}_{F}\left( \boldsymbol{\alpha}\right) =\frac{1}{IJ}\sum\limits_{i=1}^{I}\begin{pmatrix}
\frac{1}{\alpha_{0}^{2}} & \frac{w_{i}}{\alpha_{0}}\\
\frac{w_{i}}{\alpha_{0}} & w_{i}^{2}\end{pmatrix}
\sum\limits_{j=1}^{J}t_{j}^{2}\frac{f^{2}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))}{F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))(1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})))}$$
is the Fisher Information matrix for the one-shot device model. In addition, relating the theory of MDPDEs with the Fisher Information matrix, we have $$\boldsymbol{J}_{\beta=0}(\boldsymbol{\alpha})=\boldsymbol{K}_{\beta
=0}(\boldsymbol{\alpha})=\boldsymbol{I}_{F}(\boldsymbol{\alpha}).$$
Robust Z-type tests\[sec4\]
===========================
For testing the null hypothesis of a linear combination of $\boldsymbol{\alpha
}=(\alpha_{0},\alpha_{1})^{T}$, $H_{0}$: $m_{0}\alpha_{0}+m_{1}\alpha_{1}=d$, or equivalently$$H_{0}\text{: }\boldsymbol{m}^{T}\boldsymbol{\alpha}=d, \label{W1}$$ where $\boldsymbol{m}^{T}=(m_{0},m_{1})$, it is important to know the asymptotic distribution of the MDPDE of $\boldsymbol{\alpha}$. In particular, in case we wish to test if the different temperatures do not affect the model of the one-shot devices, $\boldsymbol{m}^{T}=(m_{0},m_{1})=(0,1)$ and $d=0$ must be considered. In the following definition, we present $Z$-type test statistics based on $\widehat{\boldsymbol{\alpha}}_{\beta}$. Since $Z$-type test statistics are a particular case of the Wald-type test, we can say that this type of robust test statistics have been considered previously in Basu et al. (2016) and Ghosh et al. (2016).
Let $\widehat{\boldsymbol{\alpha}}_{\beta}=(\widehat{\alpha}_{0,\beta},\widehat{\alpha}_{1,\beta})^{T}$ be the MDPDE of $\boldsymbol{\alpha}=(\alpha_{0},\alpha_{1})^{T}$. The family of $Z$-type test statistics for testing (\[W1\]) is given by $$Z_{K}(\widehat{\boldsymbol{\alpha}}_{\beta})=\sqrt{\frac{K}{\boldsymbol{m}^{T}\boldsymbol{\bar{J}}_{\beta}^{-1}(\widehat{\boldsymbol{\alpha}}_{\beta
})\boldsymbol{\bar{K}}_{\beta}(\widehat{\boldsymbol{\alpha}}_{\beta
})\boldsymbol{\bar{J}}_{\beta}^{-1}(\widehat{\boldsymbol{\alpha}}_{\beta
})\boldsymbol{m}}}(\boldsymbol{m}^{T}\widehat{\boldsymbol{\alpha}}_{\beta}-d).
\label{W2}$$
In the following theorem, the asymptotic distribution of $Z_{K}(\widehat{\boldsymbol{\alpha}}_{\beta})$ is presented.
The asymptotic distribution of $Z$-type test statistics, $Z_{K} (\widehat{\boldsymbol{\alpha}}_{\beta})$, defined in (\[W2\]), is standard normal.
Based on the previous result, the null hypothesis given in (\[W1\]) will be rejected, with significance level $\alpha$, if $\left\vert Z_{K}(\widehat{\boldsymbol{\alpha}}_{\beta})\right\vert >z_{\frac{\alpha}{2}}$, where $z_{\frac{\alpha}{2}}$ is a right hand side quantile of order $\frac{\alpha}{2}$ of a normal distribution. Now we are going to present a result in order to provide an approximation for the test statistic defined in (\[W2\]).
Let $\boldsymbol{\alpha}^{\ast}\in\Theta$ be the true value of the parameter $\boldsymbol{\alpha}$ so that $$\widehat{\boldsymbol{\alpha}}_{\beta}\overset{\mathcal{P}
}{\underset{K\mathcal{\rightarrow}\infty}{\longrightarrow}}\boldsymbol{\alpha
}^{\ast}\in\Theta,$$ and $\boldsymbol{m}^{T}\boldsymbol{\alpha}^{\ast}\neq d$. Then, the approximated power function of the test statistic in (\[W2\]) at $\boldsymbol{\alpha}^{\ast}$ is given by equation (\[W31\]), where $\Phi(\cdot)$ is the standard normal distribution function.
$$\pi\left( \boldsymbol{\alpha}^{\ast}\right) \simeq2\left( 1-\Phi\left(
z_{\frac{\alpha}{2}}-\sqrt{\frac{K}{\boldsymbol{m}^{T}\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha}^{\ast})\boldsymbol{\bar{K}}_{\beta
}(\boldsymbol{\alpha}^{\ast})\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha}^{\ast})\boldsymbol{m}}}(\boldsymbol{m}^{T}\boldsymbol{\alpha}^{\ast}-d)\right) \right) \label{W31}$$
Based on the previous results, it is possible to establish an explicit expression of the number of devices
$$K=\left[ \frac{\boldsymbol{m}^{T}\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha}^{\ast})\boldsymbol{\bar{K}}_{\beta}(\boldsymbol{\alpha
}^{\ast})\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha}^{\ast
})\boldsymbol{m}}{\boldsymbol{m}^{T}\boldsymbol{\alpha}^{\ast}-d}\left(
z_{\frac{\alpha}{2}}-\Phi^{-1}(1-\tfrac{\pi^{\ast}}{2})\right) ^{2}\right]
+1,$$
placed under temperatures $w_{i}$, $i=1,...,I$, at each time, $t_{j}$, $j=1,2,...,J$, necessary in order to get a fixed power $\pi^{\ast}$ for a specific significance level $\alpha$. Here, $[m]$ denotes $\left[ \cdot\right] $ the largest integer less than or equal to $m$.
Real data examples\[sec5\]
==========================
In this section, we present some numerical examples to illustrate the inferential results developed in the precedings sections. The first one is an application to the reliability example considered in Section \[sec2\], and the other two are real applications to tumorigenicity experiments considered earlier by other authors.
Example 1 (Reliability experiment)
----------------------------------
Based on the example introduced in Section \[sec2\], in this section, the MDPDEs of the parameters of the one-shot device model are considered. As tuning parameter, $\beta\in\{0,$ $0.1,$ $0.2,$ $0.3,$ $0.4,$ $0.5,$ $0.6,$ $0.7,$ $0.8,$ $0.9,$ $1,\;2,\;3,\;4\}$ are taken. In Table \[table2\], apart from the MDPDEs of $\boldsymbol{\alpha}$, the MDPDEs of the reliability function$$R(t|\lambda_{w_{0}}(\boldsymbol{\alpha}))=1-F(t|\lambda_{w_{0}}(\boldsymbol{\alpha}))=e^{-\lambda_{w_{0}}t}=\exp(-\alpha_{0}e^{\alpha
_{1}w_{0}}t)$$ are also presented at mission times (time points in the future at which we are interested in the reliability of the unit) $t\in\{10,20,30\}$, namely $R(10|\lambda_{w_{0}}(\widehat{\boldsymbol{\alpha}}_{\beta}))$, $R(20|\lambda_{w_{0}}(\widehat{\boldsymbol{\alpha}}_{\beta}))$, $R(30|\lambda_{w_{0}}(\widehat{\boldsymbol{\alpha}}_{\beta}))$, as well as the MDPDEs of the mean of the lifetime $T(\lambda_{w_{0}}(\boldsymbol{\alpha}))$, namely, $$E[T(\lambda_{w_{0}}(\boldsymbol{\alpha}))]=\frac{1}{\lambda_{w_{0}}(\boldsymbol{\alpha})}=\frac{1}{\alpha_{0}e^{\alpha_{1}w_{0}}},$$ under the normal operating temperature $w_{0}=25$. Table \[table2\] shows that the mean lifetime obtained by the maximum likelihood estimator ($\beta=0$) is greater than that obtained from the alternative MPDPDEs.
\[c\][l||cc|cccc]{}$\beta$ & $\qquad\widehat{\alpha}_{0,\beta}\qquad$ & $\qquad\widehat{\alpha
}_{1,\beta}\qquad$ & $R(10|\lambda_{25}(\widehat{\boldsymbol{\alpha}}_{\beta
}))$ & $R(20|\lambda_{25}(\widehat{\boldsymbol{\alpha}}_{\beta}))$ & $R(30|\lambda_{25}(\widehat{\boldsymbol{\alpha}}_{\beta}))$ & $E[T(\lambda
_{25}(\widehat{\boldsymbol{\alpha}}_{\beta}))]$\
0 & 0.00487 & 0.04732 & 0.85300 & 0.72761 & 0.62065 & 62.89490\
0.1 & 0.00489 & 0.04722 & 0.85288 & 0.72741 & 0.62039 & 62.83953\
0.2 & 0.00490 & 0.04714 & 0.85277 & 0.72722 & 0.62016 & 62.79031\
0.3 & 0.00491 & 0.04706 & 0.85268 & 0.72706 & 0.61995 & 62.74654\
0.4 & 0.00492 & 0.04700 & 0.85260 & 0.72693 & 0.61978 & 62.70965\
0.5 & 0.00493 & 0.04695 & 0.85253 & 0.72681 & 0.61963 & 62.67944\
0.6 & 0.00494 & 0.04690 & 0.85247 & 0.72671 & 0.61950 & 62.65188\
0.7 & 0.00495 & 0.04687 & 0.85246 & 0.72669 & 0.61947 & 62.64457\
0.8 & 0.00495 & 0.04683 & 0.85236 & 0.72651 & 0.61925 & 62.59732\
0.9 & 0.00496 & 0.04681 & 0.85233 & 0.72646 & 0.61918 & 62.58398\
1 & 0.00496 & 0.04681 & 0.85239 & 0.72656 & 0.61931 & 62.61131\
2 & 0.00496 & 0.04679 & 0.85231 & 0.72644 & 0.61915 & 62.57739\
3 & 0.00494 & 0.04687 & 0.85255 & 0.72684 & 0.61966 & 62.68584\
4 & 0.00491 & 0.04700 & 0.85292 & 0.72748 & 0.62048 & 62.85869\
$\ \ \ \ \ $
Example 2 (ED01 Data)
---------------------
In 1974, the National Center for Toxicological Research made an experiment on 24000 female mice randomized to a control group or one of seven dose levels of a known carcinogen, called 2-Acetylaminofluorene (2-AAF). Table 1 in Lyndsey and Ryan (1993) shows the results obtained when the highest dose level ($150$ parts per million) was administered. The original study considered four different outcomes: Number of animals dying tumour free (DNT) and with tumour (DWT), and sacrified without tumour (SNT) and with tumour (SWT), summarized over three time intervals at $12$, $18$ and $33$ months. A total of $3355$ mice were involved in the experiment.
[ llccc ]{}\
& & $r=0$ & $r=1$& $r=2$\
& $w=0$ & 115 & 22 &8\
& $w=1$ & 110 & 49& 16\
& $w=0$ & 780 & 42 & 8\
& $w=1$ & 540 & 54 & 26\
& $w=0$ & 675 & 200 & 85\
& $w=1$ & 510 & 64 & 51\
Balakrishnan et al. (2016a) made an analysis combining SNT and SWT as the sacrificed group ($r=0$); and denoting the cause of DNT as natural death ($r=1$), and the cause of DWT as death due to cancer ($r=2$). This modified data are presented in Table \[table:ED01\], while MDPDEs of the model parameters and the corresponding estimates of mean lifetimes are presented in Table \[table:MPDE\_ED01\]. Here $w=0$ refers to control group and $w=1$ is the test group, while $E(T_1)$ and $E(T_2)$ are the estimated mean lifetimes for sacrifice or nature death ($r=0,1$) and death due to cancer ($r=2$), respectively.
$\beta$ $\widehat{\alpha}_{10}$ $\widehat{\alpha}_{11}$ E$_{w=0}$($T_1$) E$_{w=1}$($T_1$) $\widehat{\alpha}_{20}$ $\widehat{\alpha}_{21}$ E$_{w=0}$($T_2$) E$_{w=1}$($T_2$) E$_{w=0}$($T$) E$_{w=1}$($T$)
--------- ------------------------- ------------------------- ------------------ ------------------ ------------------------- ------------------------- ------------------ ------------------ ---------------- ----------------
0 0.00617 $-$0.12790 162.233 184.165 0.00236 0.25620 426.425 331.582 117.447 118.299
0.1 0.00702 0.09355 142.352 129.639 0.00250 0.32870 399.794 287.795 104.988 89.392
0.2 0.00698 0.06495 143.302 134.290 0.00250 0.31173 400.433 293.189 105.504 92.072
0.3 0.00703 0.00999 142.253 140.840 0.00249 0.29613 401.393 298.513 105.045 95.708
0.4 0.00690 0.00998 145.019 143.578 0.00249 0.27957 401.602 303.655 106.545 97.484
0.5 0.00677 0.00998 147.662 146.195 0.00249 0.26421 401.839 308.537 107.965 99.175
0.6 0.00666 0.00998 150.085 148.594 0.00283 0.00997 353.925 350.414 105.342 104.296
0.7 0.00682 $-$0.06678 146.635 156.763 0.00249 0.23702 401.985 317.157 107.415 104.876
0.8 0.00680 $-$0.08753 147.020 160.468 0.00279 0.00997 358.642 355.083 104.256 110.508
0.9 0.00679 $-$0.10530 147.321 163.680 0.00278 0.00997 360.357 356.781 104.516 112.141
1 0.00678 $-$0.11980 147.546 166.324 0.00277 0.00995 361.607 358.028 104.739 113.506
: MDPDEs of the parameters and the mean lifetimes of the ED01 experiment[]{data-label="table:MPDE_ED01"}
From Table \[table:MPDE\_ED01\], some MDPDEs of $\alpha_{11}$ are seen to be negative. As pointed out in Balakrishnan et al. (2016), this can be due to the fact that the true value of it may be quite close to zero. In fact, for the values of the tuning parameter $\beta \in \{0.1,0.2,0.3,0.4,0.5,0.6,0.7 \}$, the estimators of $\alpha_{11}$ are very close to zero, meaning that the drug will not increase the hazard rate of the natural death outcome. Furthermore, if we look at the estimates of mean lifetimes, these last estimators show a reduction when the carcinogenic drug is administered, but the other ones, $\beta \in \{0,0.8,0.9,1 \}$, do not show this behavior (see Figure \[fig: lifetimeT\_ED01\]). Thus, in this case, we observe that the MDPDEs with tuning parameter $\beta \in \{0.1,0.2,0.3,0.4,0.5,0.6,0.7 \}$ give a more meaningful result in the context of the laboratory experiment than, in particular, the maximum likelihood estimator ($\beta=0$). The simulation study presented in this paper will prove how, in a general case, MDPDEs with these tuning parameters will also present a better behaviour in terms of robustness.
![ MDPDEs of the mean lifetimes, for different values of the tuning parameter $\beta$, from the ED01 experiment []{data-label="fig: lifetimeT_ED01"}](lifetimeT_ED01)
Example 3 (Benzidine Dihydrochloride Data)
------------------------------------------
[ llccc ]{}\
& & $r=0$ & $r=1$& $r=2$\
& $w=1$ & 70 & 2 &0\
& $w=2$ & 22 & 3 & 0\
& $w=1$ &48 & 1 & 0\
& $w=2$ & 14 & 4 & 17\
& $w=1$ & 35 & 4 & 7\
& $w=2$ & 1 & 1 & 9\
The benzidine dihydrochloride experiment was also conducted at the National Center for Toxicological Research to examine the incidence in mice of liver tumours induced by the drug, and studied by Lyndsey and Ryan (1993) and Balakrishnan et al. (2016b). The inspection times used on the mice were $9.37$, $14.07$ and $18.7$ months. In Table \[table:MPDE\_Benzidine\], the numbers of mice sacrified ($r=0$), died without tumour ($r=1$) and died with tumour ($r=2$), are shown, for two different doses of drug: $60$ parts per million ($w=1$) and $400$ parts per million ($w=2$). As in the previous example, we consider as “failures” the mice died due to cancer.
Table \[table:MPDE\_Benzidine\] shows the MDPDEs of the model parameters and the corresponding estimates of mean lifetimes. Although some differences are observed in the results for different values of the tuning parameter, in all the cases, the mean lifetime shows a reduction when the carcinogenic drug is administered.
$\beta$ $\widehat{\alpha}_{10}$ $\widehat{\alpha}_{11}$ E$_{w=0}$($T_1$) E$_{w=1}$($T_1$) $\widehat{\alpha}_{20}$ $\widehat{\alpha}_{21}$ E$_{w=1}$($T_2$) E$_{w=2}$($T_2$) E$_{w=1}$($T$) E$_{w=2}$($T$)
--------- ------------------------- ------------------------- ------------------ ------------------ ------------------------- ------------------------- ------------------ ------------------ ---------------- ----------------
0 0.00074 1.08665 1342.580 452.912 0.00018 2.49999 5472.201 449.190 1081.274 227.233
0.1 0.00093 0.87121 1072.790 448.905 0.00022 2.45781 4459.410 381.825 867.943 208.460
0.2 0.00097 0.84038 1032.863 445.729 0.00024 2.42125 4110.686 365.071 827.690 202.187
0.3 0.00101 0.81098 994.958 442.182 0.00026 2.39084 3867.836 354.112 790.471 196.024
0.4 0.00104 0.78168 958.766 438.766 0.00029 2.34614 3507.841 335.834 750.183 188.387
0.5 0.00109 0.75071 920.459 434.483 0.00029 2.33901 3449.648 332.624 726.525 188.353
0.6 0.00112 0.72656 893.899 432.261 0.00032 2.29717 3168.017 318.521 695.074 181.946
0.7 0.00115 0.70252 866.492 429.206 0.00032 2.28271 3078.308 314.009 678.390 182.918
0.8 0.00118 0.68232 845.366 427.285 0.00033 2.27346 3011.326 310.030 660.973 180.322
0.9 0.00121 0.66476 826.449 425.122 0.00034 2.25372 2902.649 304.799 645.163 178.887
1 0.00124 0.64796 807.541 422.432 0.00035 2.23942 2823.643 300.774 629.593 176.897
: MDPDEs of the parameters and the mean lifetimes of the Benzidine Dihydrochlorid experiment[]{data-label="table:MPDE_Benzidine"}
In order to have an idea of the behavior of the different MDPDEs, in relation to the efficiency as well as the robustness, we carry out an extensive simulation study in the next section.
Simulation study\[sec6\]
========================
In this section, a simulation study is carried out to examine the behavior of the MDPDEs of the parameters of the one-shot device model, studied in Section \[sec3\], as well as the $Z$-type tests, based on MDPDEs, detailed in Section \[sec4\]. We pay special attention to the robustness issue. It is interesting to note, in this context, the following. For each fixed time, $t_{j}$, under a fixed temperature, $w_{i}$, $K$ devices are tested. In this sense, we can identify our data as a $I\times J$ contingency table with $K$ observations in each cell. Hence, under this setting, we must consider outlying cells rather than outlying observations. A cell which does not follow the one-shot device model will be called an outlying cell or outlier. The strong outliers may lead to reject a model fitting even if the rest of the cells fit the model properly. In other cases, even though the cells seem to fit reasonably well the model, the outlying cells contribute to an increase in the values of the residuals as well as the divergence measure between the data and the fitted values according to the one-shot device model considered. Therefore, it is very important to have robust estimators as well as robust test statistics in order to avoid the undesirable effects of the outliers in the data. The main purpose of this simulation study is to show that inside the family of MDPDEs, developed here, there are estimators with better robust properties than the MLE, and the $Z$-type tests constructed from them are at the same time more robust than the classical $Z$-type test, constructed through the MLEs.
The MDPDEs\[sec6.1\]
--------------------
In this section, we carry out a simulation study to compare the behavior of some MDPDEs with respect to the MLEs of the parameters in the one-shot device model under the exponential distribution. In order to evaluate th performance of the proposed MDPDEs, as well as the MLEs, we consider the root of the mean square errors (RMSEs). We have considered a model in which, $I=J=3$, $w\in\{35,$ $45,$ $55\}$, $t\in\{10,$ $20,$ $30\}$ and $K=20$, as in the example in Table \[table1\], and the simulation experiment proposed by Ling (2012). This model has been examined under three choices of $(\alpha_{0},\alpha_{1})=(0.005,0.05)$, $(\alpha_{0},\alpha_{1})=(0.004,0.05)$ and $(\alpha_{0},\alpha_{1})=(0.003,0.05)$ for low-moderate, moderate and moderate-high reliability, respectively.
To evaluate the robustness of the MDPDEs, we have studied the behavior of this model under the consideration of an outlying cell for $(w_{1},t_{1})$ in our contingency table, with $10,000$ replications and estimators corresponding to the tuning parameter $\beta\in\{0,0.1,0.2,0.4,0.6,0.8,1\}$. The reduction of each one of the parameters of the outlying cell, denoted by $\tilde{\alpha}_{0}$ or $\tilde{\alpha}_{1}$ ($\alpha_{0}\geq\tilde{\alpha}_{0}$ or $\alpha_{1}\geq\tilde{\alpha}_{1}$) increases the mean of its lifetime distribution function in (\[eq:expo\]). The results obtained by decreasing parameter $\alpha_{0}$ are shown in Figure \[fig:MPDE\_1\], while the results obtained by decreasing parameter $\alpha_{1}$ are shown in Figure \[fig:MPDE\_2\]. In all the cases, we can see how the MLEs and the MDPDEs with small values of tuning parameter $\beta$ present the smallest RMSEs for weak outliers, i.e., when $\tilde{\alpha}_{0}$ is close to $\alpha_{0}$ ($1-\tilde{\alpha}_{0}/\alpha_{0}$ is close to $0$) or $\tilde{\alpha}_{1}$ is close to $\alpha_{1}$ ($1-\tilde{\alpha}_{1}/\alpha_{1}$ is close to $0$). On the other hand, large values of tuning parameter $\beta$ turn the MDPDEs to present the smallest RMSEs, for medium and strong outliers, i.e., when $\tilde{\alpha}_{0}$ is not close to $\alpha_{0}$ ($1-\tilde{\alpha}_{0}/\alpha_{0}$ is not close to $0$) or $\tilde{\alpha}_{1}$ is not close to $\alpha_{1}$ ($1-\tilde{\alpha}_{1}/\alpha_{1}$ is not close to $0$). Therefore, the MLE of $(\alpha_{0},\alpha_{1})$ is very efficient when there are no outliers, but highly non-robust when there are outliers. On the other hand, the MDPDEs with moderate values of the tuning parameter $\beta$ exhibit a little loss of efficiency without outliers but at the same time a considerable improvement of robustness with outliers. Actually, these values of the tuning parameter $\beta$ are the most appropriate ones for the estimators of the parameters in the one-shot device model according to robustness theory: To improve in a considerable way the robustness of the estimators, a small amount of efficiency needs to be compromised.
The Z-type tests based on MDPDEs\[sec6.2\]
------------------------------------------
We will study the performance, with respect to robustness, through simulation of the one-shot device model defined in Section \[sec2\] with the same values of $I,J,t,w$ of the example of Balakrishnan and Ling (2012) given in Table \[table1\] and for the same tuning parameter, $\beta$, as in Section \[sec6.1\]. We are interested in testing the null hypothesis $H_{0}:\alpha_{1}=0.05$ against the alternative $H_{1}:\alpha_{1}\neq0.05$, through the $Z$-type test statistics based on MDPDEs. Under the null hypothesis, we consider as true parameters $(\alpha_{0},\alpha_{1})=(0.004,0.05)$, while under the alternative we consider as true parameters $(\alpha_{0},\alpha_{1})=(0.004,0.02)$. In Figure \[fig:LP\_1\], we present the empirical significance level (measured as the proportions of test statistics exceeding in absolute value the standard normal quantile critical value) with $10,000$ replications. The empirical power (obtained in a similar manner) is also presented in the right hand side of Figure \[fig:LP\_1\]. Notice that in all the cases the observed levels are quite close to the nominal level of $0.05$. The empirical power is similar for the different values of the tuning parameters $\beta$, a bit lower for large values of $\beta$, and closer to one as $K$ or the sample size ($n=IJK$) increases.
-- --
-- --
[0.5]{}
[c]{}\
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[0.5]{}
\[c\][c]{}\
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\[c\][cc]{} &\
&
\[c\][cc]{} &\
&
To evaluate the robustness of the level and the power of the $Z$-type tests based on MDPDEs with an outlier placed on the first-row first-column cell, we perform the simulation for the same test and the same true values for the null and alternative hypotheses, in two different scenarios depending on the way the outlying cell is considered. In the first scenario, we keep $\alpha_{1}$ the same and modify the true value of $\alpha_{0}$ to be $\tilde{\alpha}_{0}\leq\alpha_{0}$, and in the second one, we keep $\alpha_{0}$ the same and modify the true value of $\alpha_{1}$ to be $\tilde{\alpha}_{1}\leq\alpha_{1}$. Both cases have been analyzed for different values of $K$ and decreasing $\tilde{\alpha}_{0}$ in the first scenario (increasing $1-\tilde{\alpha}_{0}/\alpha_{0}$) or decreasing $\tilde{\alpha}_{1}$ in the second scenario (increasing $1-\tilde{\alpha}_{1}/\alpha_{1}$).
The results for the first scenario are presented in Figure \[fig:LP\_2\]. The empirical level for the one-shot device model with $K$ from $10$ to $150$, true value $(\alpha_{0},\alpha_{1})=(0.004,0.05)$ and $\tilde{\alpha}_{0}=0.001$ for the outlying cell is presented on the left and top panel. Similarly, the empirical power for the one-shot device model with $K$ from $10$ to $150$, true parameter $(\alpha_{0},\alpha_{1})=(0.004,0.02)$ and $\tilde{\alpha}_{0}=0.001$ for the outlying cell is presented on the right top panel. In addition, the empirical level for the one-shot device model with $1-\tilde{\alpha}_{0}/\alpha_{0}$ from $0$ to $1$ for the outlying cell and true value $(\alpha_{0},\alpha_{1})=(0.004,0.05)$ and $K=20$ is presented on the left bottom panel. Similarly, the empirical power for the one-shot device model with $1-\tilde{\alpha}_{0}/\alpha_{0}$ from $0$ to $1$ for the outlying cell and true value and true parameter $(\alpha_{0},\alpha_{1})=(0.004,0.02)$ is presented on the right bottom panel.
Notice that the outlying cell represents 1/9 of the total observations in the last plots. For large values of $K$ (very large sample sizes, since $n=9K$), there is a large inflation in the empirical level and shrinkage of the empirical power, but for the $Z$-type test statistic based on the MDPDEs with large values of the tuning parameter $\beta$, the effect of the outlying cell is weaker in comparison to those of smaller values of $\beta$, included the MLEs ($\beta=0$). If $\tilde{\alpha}_{0}$ is separated from $\alpha_{0}$ ($1-\tilde{\alpha}_{0}/\alpha_{0}$ increases from $0$ to $1$), the empirical level of the $Z$-type test statistics based on the MDPDEs is not stable around the nominal level, being however closer as the tuning parameter $\beta$ becomes larger. If $\tilde{\alpha}_{0}$ is separated from $\alpha_{0}$ ($1-\tilde{\alpha}_{0}/\alpha_{0}$ increases from $0$ to $1$), the empirical power of the $Z$-type test statistics based on the MDPDEs decreases, being however more slowly as the tuning parameter $\beta$ becomes larger.
Figure \[fig:LP\_3\] presents the results for the second scenario, in which $\tilde{\alpha}_{1}=0.01$ for the outlying cell on the left top panel and $\tilde{\alpha}_{1}=-0.01$ for the outlying cell on the right top panel. Even though the outliers are, in the current scenario, slightly more pronunced with respect to the previous scenario, in general terms, we arrive at the same conclusions as in the previous scenario.
The results of the tests statistics presented here show again the poor behavior in robustness of the $Z$-type tests based on the MLE of the parameters of the one-shot device model. Furthermore, the robustness properties of the $Z$-type test statistics based on the MDPDEs with large values of the tuning parameter $\beta$ are often better as they maintain both level and power in a stable manner. Moreover, the comments made at the end of Section \[sec6.1\] for the MDPDEs regarding moderate values of the tuning parameter $\beta$ are valid for the $Z$-type test statistics based on the MDPDEs as well.
$$\mathcal{T}_{1,\beta}(\boldsymbol{\alpha})=\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\left\{ \left( \frac{F\left( t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})\right) }{IJ}\right) ^{1+\beta}-(1+\tfrac{1}{\beta
})\left( \frac{F\left( t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})\right)
}{IJ}\right) ^{\beta}\frac{n_{ij}}{IJK}+\tfrac{1}{\beta}\left( \frac{n_{ij}}{IJK}\right) ^{1+\beta}\right\}
\label{eq:reult5_a}$$
$$\mathcal{T}_{2,\beta}(\boldsymbol{\alpha})=\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\left\{ \left( \frac{1-F\left( t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})\right) }{IJ}\right) ^{1+\beta}-(1+\tfrac{1}{\beta
})\left( \frac{1-F\left( t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})\right)
}{IJ}\right) ^{\beta}\frac{K-n_{ij}}{IJK}+\tfrac{1}{\beta}\left(
\frac{K-n_{ij}}{IJK}\right) ^{1+\beta}\right\}
\label{eq:reult5_b}$$
Concluding Remarks\[sec7\]
==========================
In this paper, we have introduced and studied the minimum density power divergence estimators for one-shot device testing with an accelerating factor of temperature. Based on these estimators, we have also introduced a Wald-type test statistic family. Since the maximum likelihood estimator is a particular estimator in the family of minimum density power divergence estimators developed here, the classical Wald test is also taken into account for comparison. The results obtained in the simulation study suggest that some minimum density power divergence estimators are considerably better for the estimation of the model parameters when outliers are present in the data and at the same time not facing much loss of efficiency when outliers are not present. Similar results are obtained for some Wald-type test statistics in terms of stability with respect to level and power. These proposed estimators also give a more meaningful result in the case of ED01 tumorigenicity experiment data than the maximum likelihood estimators.
Proofs of Results
=================
Proof of Result 1:
------------------
We have $$\begin{aligned}
d_{KL}\left( \widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha
})\right) & =\frac{1}{IJK}\left( s-\sum\limits_{i=1}^{I}\sum
\limits_{j=1}^{J}\log\left( F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\right) ^{n_{ij}} \right.\\
& \left.+\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\log\left(
\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right) \right)
^{K-n_{ij}}\right) \\
& =\frac{1}{IJK}\left( s-\log\prod\limits_{i=1}^{I}\prod\limits_{j=1}^{J}F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))^{n_{ij}}\right.\\
& \times \left.\left(1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right) ^{K-n_{ij}}\right.\Bigg) \\
& =\frac{1}{IJK}\left( s-\log\mathcal{L}\left( \boldsymbol{\alpha
}\left\vert K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right)
\right) ,\end{aligned}$$ with $$s=\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}n_{ij}\log\frac{n_{ij}}{K}+\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}(K-n_{ij})\log\frac{K-n_{ij}}{K},$$ as required.
Proof of Result 4:
------------------
The relationship between (\[4b\]) and $d_{\beta}\left(
\widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha})\right) $ defined in (\[3\]) is given by $$\begin{aligned}
& \frac{1}{IJ}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\left\{ \pi
_{ij}^{\beta+1}(\boldsymbol{\alpha})+(1-\pi_{ij}(\boldsymbol{\alpha}))^{\beta+1} \right.\\
&\left.-\frac{1+\beta}{\beta}\frac{n_{ij}}{K}\pi_{ij}^{\beta
}(\boldsymbol{\alpha})-\frac{1+\beta}{\beta}\frac{K-n_{ij}}{K}(1-\pi
_{ij}(\boldsymbol{\alpha}))^{\beta}\right\} \\
& =(IJ)^{\beta+1}d_{\beta}\left( \widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha})\right) +c,\end{aligned}$$ where $c$ is a constant not dependent on $\boldsymbol{\alpha}$, and so $\widehat{\boldsymbol{\alpha}}_{\beta}$ is the same for both cases. Hence, the result.
Proof of Result 5:
------------------
We have $$\begin{aligned}
\frac{\partial F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))}{\partial
\alpha_{0}}&=\exp\left\{ -\alpha_{0}\exp\left( \alpha_{1}w_{i}\right)
t_{j}\right\} \exp\left\{ \alpha_{1}w_{i}\right\} t_{j} \nonumber \\
&=f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\frac{t_{j}}{\alpha_{0}} \label{6}$$ and $$\begin{aligned}
\frac{\partial F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))}{\partial
\alpha_{1}}=&\exp\left\{ -\alpha_{0}\exp\left( \alpha_{1}w_{i}\right)
t_{j}\right\} \nonumber \\
& \times \exp\left\{ \alpha_{1}w_{i}\right\} \alpha_{0}t_{j}w_{i} \nonumber \\
=&f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))t_{j}w_{i}. \label{7}$$ We denote $$d_{\beta}\left( \widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha
})\right) =\mathcal{T}_{1,\beta}(\boldsymbol{\alpha})+\mathcal{T}_{2,\beta
}(\boldsymbol{\alpha}),$$ where $\mathcal{T}_{1,\beta}(\boldsymbol{\alpha})$ and $\mathcal{T}_{2,\beta}(\boldsymbol{\alpha})$ are given by (\[eq:reult5\_a\]) and (\[eq:reult5\_b\]), respectively, for $\beta>0$.
Based on (\[6\]), we have $$\begin{aligned}
\frac{\partial\mathcal{T}_{1,\beta}(\boldsymbol{\alpha})}{\partial\alpha_{0}}=&\frac{\beta+1}{\left( IJ\right) ^{\beta+1}}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\left( F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))-\frac{n_{ij}}{K}\right)\\
& \times f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\frac{t_{j}}{\alpha_{0}}F^{^{\beta-1}}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\end{aligned}$$ and $$\begin{aligned}
\frac{\partial\mathcal{T}_{2,\beta}(\boldsymbol{\alpha})}{\partial\alpha_{0}}=&\frac{\beta+1}{\left( IJ\right) ^{\beta+1}}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\left( F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))-\frac{n_{ij}}{K}\right)\\
&\times f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\frac{t_{j}}{\alpha_{0}}\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right) ^{^{\beta-1}}.\end{aligned}$$ On the other hand, by (\[7\]), we have $$\begin{aligned}
\frac{\partial\mathcal{T}_{1,\beta}(\boldsymbol{\alpha})}{\partial\alpha_{1}}=&\frac{\beta+1}{\left( IJ\right) ^{\beta+1}}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\left( F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))-\frac{n_{ij}}{K}\right)\\
&\times f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))t_{j}w_{i}F^{^{\beta-1}}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\end{aligned}$$ and $$\begin{aligned}
\frac{\partial\mathcal{T}_{2,\beta}(\boldsymbol{\alpha})}{\partial\alpha_{1}}=&\frac{\beta+1}{\left( IJ\right) ^{\beta+1}}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\left( F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))-\frac{n_{ij}}{K}\right) \\
&\times f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))t_{j}w_{i}\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right)
^{^{\beta-1}}.\end{aligned}$$ Finally, the system of equations is given by $$\begin{aligned}
\frac{\left( IJ\right) ^{\beta+1}}{\beta+1}\left( \frac{\partial
\mathcal{T}_{1,\beta}(\boldsymbol{\alpha})}{\partial\alpha_{0}}+\frac
{\partial\mathcal{T}_{2,\beta}(\boldsymbol{\alpha})}{\partial\alpha_{0}}\right) & =0,\\
\frac{\left( IJ\right) ^{\beta+1}}{\beta+1}\left( \frac{\partial
\mathcal{T}_{1,\beta}(\boldsymbol{\alpha})}{\partial\alpha_{1}}+\frac
{\partial\mathcal{T}_{2,\beta}(\boldsymbol{\alpha})}{\partial\alpha_{1}}\right) & =0.\end{aligned}$$ If we consider $\beta=0$ in (\[5\]) and (\[5B\]), we get the system needed to solve in order to get the maximum likelihood estimator (MLE). Hence, the previous system of equations is valid not only for tuning parameters $\beta
>0$, but also for $\beta=0$.
Proof of Result 6:
------------------
Based on Ghosh and Basu (2013) and also on Definition \[def1\], we have$$\sqrt{IJK}\left( \widehat{\boldsymbol{\alpha}}_{\beta}-\boldsymbol{\alpha
}_{0}\right) \overset{\mathcal{L}}{\underset{IJK\mathcal{\rightarrow}\infty}{\longrightarrow}}\mathcal{N}\left( \boldsymbol{0},\boldsymbol{J}_{\beta}^{-1}(\boldsymbol{\alpha}_{0})\boldsymbol{K}_{\beta}(\boldsymbol{\alpha}_{0})\boldsymbol{J}_{\beta}^{-1}(\boldsymbol{\alpha}_{0})\right) ,$$ where $$\begin{aligned}
\boldsymbol{J}_{\beta}(\boldsymbol{\alpha}) = &\frac{1}{IJK}\sum
\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\sum\limits_{k=1}^{K}\boldsymbol{J}_{ij,\beta}(\boldsymbol{\alpha})\\
=&\frac{1}{IJ}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\boldsymbol{J}_{ij,\beta}(\boldsymbol{\alpha}),\\
\boldsymbol{J}_{ij,\beta}(\boldsymbol{\alpha}) =&\boldsymbol{u}_{ij}(\boldsymbol{\alpha})\boldsymbol{u}_{ij}^{T}(\boldsymbol{\alpha})F^{\beta+1}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\\
&+\boldsymbol{v}_{ij}(\boldsymbol{\alpha})\boldsymbol{v}_{ij}^{T}(\boldsymbol{\alpha
})(1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})))^{\beta+1}\\
= &t_{j}^{2}f^{2}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\begin{pmatrix}
\frac{1}{\alpha_{0}^{2}} & \frac{w_{i}}{\alpha_{0}}\\
\frac{w_{i}}{\alpha_{0}} & w_{i}^{2}\end{pmatrix}
\left[ F^{\beta-1}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})) \right.\\
& \left. +(1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})))^{\beta-1}\right],\end{aligned}$$
$$\begin{aligned}
\boldsymbol{u}_{ij}(\boldsymbol{\alpha}) = & \frac{\partial\log
F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))}{\partial\boldsymbol{\alpha}}\\
=&\frac{1}{F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))}\frac{\partial
}{\partial\boldsymbol{\alpha}}F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})),\\
\boldsymbol{v}_{ij}(\boldsymbol{\alpha}) = & \frac{\partial\log\left[
1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right] }{\partial
\boldsymbol{\alpha}}\\
=&-\frac{1}{1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))}\frac{\partial}{\partial\boldsymbol{\alpha}}F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})),\\
\frac{\partial}{\partial\boldsymbol{\alpha}}F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})) = & -\frac{\partial}{\partial\boldsymbol{\alpha}}\exp\left\{ -\alpha_{0}\exp\left\{ \alpha_{1}w_{i}\right\} t_{j}\right\}\\
=&
\begin{pmatrix}
\frac{1}{\alpha_{0}}\\
w_{i}\end{pmatrix}
t_{j}f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})),\end{aligned}$$
and$$\begin{aligned}
\boldsymbol{K}_{\beta}(\boldsymbol{\alpha}) = &\frac{1}{IJK}\sum
\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\sum\limits_{k=1}^{K}\boldsymbol{K}_{ij,\beta}(\boldsymbol{\alpha})\\
=&\frac{1}{IJ}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\boldsymbol{K}_{ij,\beta}(\boldsymbol{\alpha}),\\
\boldsymbol{K}_{ij,\beta}(\boldsymbol{\alpha}) =& \boldsymbol{S}_{ij,\boldsymbol{\beta}}(\boldsymbol{\alpha})-\boldsymbol{\xi}_{ij,\beta
}(\boldsymbol{\alpha})\boldsymbol{\xi}_{ij,\beta}^{T}(\boldsymbol{\alpha}),\\
\boldsymbol{S}_{ij,\boldsymbol{\beta}}(\boldsymbol{\alpha})
= &\boldsymbol{u}_{ij}(\boldsymbol{\alpha})\boldsymbol{u}_{ij}^{T}(\boldsymbol{\alpha})F^{2\beta+1}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\\
&+\boldsymbol{v}_{ij}(\boldsymbol{\alpha})\boldsymbol{v}_{ij}^{T}(\boldsymbol{\alpha})(1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})))^{2\beta+1}\\
= & t_{j}^{2}f^{2}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})))\begin{pmatrix}
\frac{1}{\alpha_{0}^{2}} & \frac{w_{i}}{\alpha_{0}}\\
\frac{w_{i}}{\alpha_{0}} & w_{i}^{2}\end{pmatrix}
\left[ F^{2\beta-1}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})) \right.\\
&\left.+(1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})))^{2\beta-1}\right] ,\\
\boldsymbol{\xi}_{ij,\beta}(\boldsymbol{\alpha}) =&\boldsymbol{u}_{ij}(\boldsymbol{\alpha})F^{\beta+1}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\\
&+\boldsymbol{v}_{ij}(\boldsymbol{\alpha})(1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})))^{\beta+1}\\
=&
\begin{pmatrix}
\frac{1}{\alpha_{0}}\\
w_{i}\end{pmatrix}
t_{j}f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\left[ F^{\beta}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right.\\
&\left.-(1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})))^{\beta}\right] .\end{aligned}$$ Since $I$, $J$ are fixed and $IJK\mathcal{\rightarrow}\infty$, it follows that $K\mathcal{\rightarrow}\infty$ and$$\sqrt{K}\left( \widehat{\boldsymbol{\alpha}}_{\beta}-\boldsymbol{\alpha}_{0}\right) \overset{\mathcal{L}}{\underset{K\mathcal{\rightarrow}\infty}{\longrightarrow}}\mathcal{N}\left(
\boldsymbol{0},\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha}_{0})\boldsymbol{\bar{K}}_{\beta}(\boldsymbol{\alpha}_{0})\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha}_{0})\right)$$ where $$\begin{aligned}
\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha}_{0})\boldsymbol{\bar{K}}_{\beta}(\boldsymbol{\alpha}_{0})\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha}_{0}) & =\frac{1}{IJ}\boldsymbol{J}_{\beta}^{-1}(\boldsymbol{\alpha}_{0})\boldsymbol{K}_{\beta}(\boldsymbol{\alpha}_{0})\boldsymbol{J}_{\beta}^{-1}(\boldsymbol{\alpha}_{0}),\\
\boldsymbol{\bar{J}}_{\beta}(\boldsymbol{\alpha}_{0}) & =(IJ)\boldsymbol{J}_{\beta}(\boldsymbol{\alpha}_{0}),\\
\boldsymbol{\bar{K}}_{\beta}(\boldsymbol{\alpha}_{0}) & =(IJ)\boldsymbol{K}_{\beta}(\boldsymbol{\alpha}_{0}).\end{aligned}$$
Proof of Result 7:
------------------
The Fisher information matrix for $IJK$ observations is $$\boldsymbol{I}_{IJK,F}\left( \boldsymbol{\alpha}\right) =E\left[
-\frac{\partial\boldsymbol{v}^{T}\left( \boldsymbol{\alpha}\left\vert
K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right) }{\partial\boldsymbol{\alpha}}\right] ,$$ where$$\boldsymbol{v}\left( \boldsymbol{\alpha}\left\vert K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right) =\frac{\partial\log
\mathcal{L}\left( \boldsymbol{\alpha}\left\vert K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right) }{\partial\boldsymbol{\alpha}}.$$ From (\[1\]), $$\begin{aligned}
\boldsymbol{I}_{IJK,F}\left( \boldsymbol{\alpha}\right) & =IJK\;E\left[
\frac{\partial^{2}d_{KL}\left( \widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha})\right) }{\partial\boldsymbol{\alpha}\partial
\boldsymbol{\alpha}^{T}}\right] \\
& =IJK\;E\left[ \frac{\partial\boldsymbol{u}^{T}\left( \boldsymbol{\alpha}\left\vert K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right) }{\partial\boldsymbol{\alpha}}\right] ,\end{aligned}$$ where$$\begin{aligned}
\boldsymbol{u}\left( \boldsymbol{\alpha}\left\vert K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right) &=\frac{\partial d_{KL}\left(
\widehat{\boldsymbol{p}},\boldsymbol{p}(\boldsymbol{\alpha})\right)
}{\partial\boldsymbol{\alpha}}\\
&=\frac{\partial\mathcal{T}_{1,\beta
=0}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}}+\frac{\partial
\mathcal{T}_{2,\beta=0}(\boldsymbol{\alpha})}{\partial\boldsymbol{\alpha}}.\end{aligned}$$ The Fisher information matrix for a single observation, i.e., the Fisher information matrix for the one-shot device model is$$\boldsymbol{I}_{F}\left( \boldsymbol{\alpha}\right) =\frac{1}{IJK}\boldsymbol{I}_{IJK,F}\left( \boldsymbol{\alpha}\right) =E\left[
\frac{\partial\boldsymbol{u}^{T}\left( \boldsymbol{\alpha}\left\vert
K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right) }{\partial\boldsymbol{\alpha}}\right] .$$ From Result \[Th5\], the first and second components of $\boldsymbol{u}\left( \boldsymbol{\alpha}\left\vert K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right) $ are
$$\begin{aligned}
& u_{1}\left( \boldsymbol{\alpha}\left\vert K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right) =\frac{\partial\mathcal{T}_{1,\beta
=0}(\boldsymbol{\alpha})}{\partial\alpha_{0}}+\frac{\partial\mathcal{T}_{2,\beta=0}(\boldsymbol{\alpha})}{\partial\alpha_{0}}\\
& =\frac{1}{IJK}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\left(
K\ F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))-n_{ij}\right)\\
& \ \ \times f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\frac{t_{j}}{\alpha_{0}}\left[
F^{-1}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))+\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right) ^{-1}\right] \\
& =\frac{1}{IJK}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\frac
{K\ F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))-n_{ij}}{F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right) }f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\frac{t_{j}}{\alpha_{0}}$$
and
$$\begin{aligned}
& u_{2}\left( \boldsymbol{\alpha}\left\vert K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right) =\frac{\partial\mathcal{T}_{1,\beta
=0}(\boldsymbol{\alpha})}{\partial\alpha_{1}}+\frac{\partial\mathcal{T}_{2,\beta=0}(\boldsymbol{\alpha})}{\partial\alpha_{1}}\\
& =\frac{1}{IJK}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\left(
K\ F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))-n_{ij}\right)\\
& \ \ \times f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))t_{j}w_{i}\left[ F^{-1}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))+\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right) ^{-1}\right] \\
& =\frac{1}{IJK}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\frac
{K\ F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))-n_{ij}}{F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right) }f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))t_{j}w_{i},\end{aligned}$$
respectively. The $(1,1)$th term of $\boldsymbol{I}_{F}\left( \boldsymbol{\alpha}\right) $ is the expectation of
$$\begin{aligned}
&\frac{\partial u_{1}\left( \boldsymbol{\alpha}\left\vert K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right) }{\partial\alpha_{0}} \\
&
=\frac{1}{IJK}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\left\{ -\frac{t_{j}}{\alpha_{0}^{2}}\frac{K\ F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))-n_{ij}}{F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right) }f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right. \\
& \ \ +\left. \frac{\partial f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))}{\partial\alpha_{0}}\frac{t_{j}}{\alpha_{0}}\frac{K\ F(t_{j}|\lambda
_{w_{i}}(\boldsymbol{\alpha}))-n_{ij}}{F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\right) }\right. \\
& \ \ +\left. \frac{\partial}{\partial\alpha_{0}}\left( \frac{K\ F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))-n_{ij}}{F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\right) }\right) \frac{t_{j}}{\alpha_{0}}f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right\} .\end{aligned}$$
Since the expectation of the first two summands of $\partial u_{1}\left(
\boldsymbol{\alpha}\left\vert K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right) /\partial\alpha_{0}$ are zero, the interest is on the expectation of $L_{ij} $ which is given in (\[eq:result7\]).
$$\begin{aligned}
L_{ij} & =\frac{\partial}{\partial\alpha_{0}}\left( \frac{K\ F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))-n_{ij}}{F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\right) }\right) \frac{t_{j}}{\alpha_{0}}f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\nonumber \\
& =\frac{K\frac{t_{j}}{\alpha_{0}}f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right) }{F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))^{2}\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right) ^{2}}\frac{t_{j}}{\alpha_{0}}f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})) \nonumber \\
& -\frac{\frac{\partial}{\partial\alpha_{0}}\left[ F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\right) \right] \left( K\ F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))-n_{ij}\right) }{F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))^{2}\left(
1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right) ^{2}}\frac{t_{j}}{\alpha_{0}}f(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha})).
\label{eq:result7}\end{aligned}$$
The expectation of the second summand of $L_{ij}$ is zero and hence$$E[L_{ij}]=\frac{K\left( \frac{t_{j}}{\alpha_{0}}\right) ^{2}f^{2}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))}{F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\right) }.$$ These finally yield the $(1,1)$th term of $\boldsymbol{I}_{F}\left( \boldsymbol{\alpha
}\right) $ as $$\begin{aligned}
&E\left[ \frac{\partial u_{1}\left( \boldsymbol{\alpha}\left\vert
K,\boldsymbol{n},\boldsymbol{t},\boldsymbol{w}\right. \right) }{\partial\alpha_{0}}\right] \\ & =\frac{K}{IJK}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\frac{\left( \frac{t_{j}}{\alpha_{0}}\right) ^{2}f^{2}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))}{F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\left( 1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha
}))\right) }\\
& =\frac{1}{IJ}\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\frac{\left(
\frac{t_{j}}{\alpha_{0}}\right) ^{2}f^{2}(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))}{F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\left(
1-F(t_{j}|\lambda_{w_{i}}(\boldsymbol{\alpha}))\right) }.\end{aligned}$$ The rest of the terms of $\boldsymbol{I}_{F}\left( \boldsymbol{\alpha
}\right) $ can be obtained in a similar manner. On the other hand, from Theorem \[Th5\], sustituting $\beta=0$ into $\boldsymbol{J}_{\beta}(\boldsymbol{\alpha
})=\frac{1}{IJ}\boldsymbol{\bar{J}}_{\beta}(\boldsymbol{\alpha})=$ and $\boldsymbol{K}_{\beta}(\boldsymbol{\alpha})=\frac{1}{IJ}\boldsymbol{\bar{K}}_{\beta}(\boldsymbol{\alpha})$, we simply obtain $\boldsymbol{J}_{\beta
=0}(\boldsymbol{\alpha})=\boldsymbol{K}_{\beta=0}(\boldsymbol{\alpha
})=\boldsymbol{I}_{F}(\boldsymbol{\alpha})$.
Proof of Result 9:
------------------
Let $\boldsymbol{\alpha}_{0}$ be the true value of parameter $\boldsymbol{\alpha}.$ It is clear that under (\[W1\]) $$\boldsymbol{m}^{T}\widehat{\boldsymbol{\alpha}}_{\beta}-d=\boldsymbol{m}^{T}(\widehat{\boldsymbol{\alpha}}_{\beta}-\boldsymbol{\alpha}_{0})$$ and we know $$\sqrt{K}(\widehat{\boldsymbol{\alpha}}_{\beta}-\boldsymbol{\alpha}_{0})\overset{\mathcal{L}}{\underset{K\mathcal{\rightarrow}\infty
}{\longrightarrow}}\mathcal{N}(\boldsymbol{0},\boldsymbol{\bar{J}}_{\beta
}^{-1}(\boldsymbol{\alpha}_{0})\boldsymbol{\bar{K}}_{\beta}(\boldsymbol{\alpha
}_{0})\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha}_{0})),$$ from which it follows that $$\sqrt{K}(\boldsymbol{m}^{T}\widehat{\boldsymbol{\alpha}}_{\beta}-d)\overset{\mathcal{L}}{\underset{K\mathcal{\rightarrow}\infty
}{\longrightarrow}}\mathcal{N}(0,\boldsymbol{m}^{T}\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha}_{0})\boldsymbol{\bar{K}}_{\beta
}(\boldsymbol{\alpha}_{0})\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha
}_{0})\boldsymbol{m}).$$ Dividing the left hand side by $$\sqrt{\boldsymbol{m}^{T}\boldsymbol{\bar{J}}_{\beta}^{-1}(\widehat{\boldsymbol{\alpha}}_{\beta})\boldsymbol{\bar{K}}_{\beta}(\widehat{\boldsymbol{\alpha}}_{\beta})\boldsymbol{\bar{J}}_{\beta
}^{-1}(\widehat{\boldsymbol{\alpha}}_{\beta})\boldsymbol{m}},$$ since $\boldsymbol{m}^{T}\boldsymbol{\bar{J}}_{\beta}^{-1}(\widehat{\boldsymbol{\alpha}}_{\beta})\boldsymbol{\bar{K}}_{\beta
}(\widehat{\boldsymbol{\alpha}}_{\beta})\boldsymbol{\bar{J}}_{\beta}^{-1}(\widehat{\boldsymbol{\alpha}}_{\beta})\boldsymbol{m}$ is a consistent estimator of $\boldsymbol{m}^{T}\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha}_{0})\boldsymbol{\bar{K}}_{\beta}(\boldsymbol{\alpha}_{0})\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha}_{0})\boldsymbol{m}$, the desired result is obtained.
Proof of Result 10:
-------------------
The power function at $\boldsymbol{\alpha}^{\ast}$ of $Z_{K}(\widehat{\boldsymbol{\alpha}}_{\beta})$ is given by equation (\[eq:reuslt10\]).
$$\begin{aligned}
\pi\left( \boldsymbol{\alpha}^{\ast}\right) & =\Pr\left( \left\vert
Z_{K}(\widehat{\boldsymbol{\alpha}}_{\beta})\right\vert >z_{\frac{\alpha}{2}}|\boldsymbol{\alpha}=\boldsymbol{\alpha}^{\ast}\right) \nonumber \\
& =2\Pr\left( Z_{K}(\widehat{\boldsymbol{\alpha}}_{\beta})>z_{\frac{\alpha
}{2}}|\boldsymbol{\alpha}=\boldsymbol{\alpha}^{\ast}\right) \nonumber \\
& =2\Pr\left( \sqrt{\frac{K}{\boldsymbol{m}^{T}\boldsymbol{\bar{J}}_{\beta
}^{-1}(\widehat{\boldsymbol{\alpha}}_{\beta})\boldsymbol{\bar{K}}_{\beta
}(\widehat{\boldsymbol{\alpha}}_{\beta})\boldsymbol{\bar{J}}_{\beta}^{-1}(\widehat{\boldsymbol{\alpha}}_{\beta})\boldsymbol{m}}}(\boldsymbol{m}^{T}\widehat{\boldsymbol{\alpha}}_{\beta}-d)>z_{\frac{\alpha}{2}}|\boldsymbol{\alpha}=\boldsymbol{\alpha}^{\ast}\right) \nonumber \\
& =2\Pr\left( \sqrt{\frac{K}{\boldsymbol{m}^{T}\boldsymbol{\bar{J}}_{\beta
}^{-1}(\widehat{\boldsymbol{\alpha}}_{\beta})\boldsymbol{\bar{K}}_{\beta
}(\widehat{\boldsymbol{\alpha}}_{\beta})\boldsymbol{\bar{J}}_{\beta}^{-1}(\widehat{\boldsymbol{\alpha}}_{\beta})\boldsymbol{m}}}\boldsymbol{m}^{T}(\widehat{\boldsymbol{\alpha}}_{\beta}-\boldsymbol{\alpha}^{\ast})>\right.
\nonumber \\
& \hspace*{3cm}\left. z_{\frac{\alpha}{2}}-\sqrt{\frac{K}{\boldsymbol{m}^{T}\boldsymbol{\bar{J}}_{\beta}^{-1}(\widehat{\boldsymbol{\alpha}}_{\beta
})\boldsymbol{\bar{K}}_{\beta}(\widehat{\boldsymbol{\alpha}}_{\beta
})\boldsymbol{\bar{J}}_{\beta}^{-1}(\widehat{\boldsymbol{\alpha}}_{\beta
})\boldsymbol{m}}}(\boldsymbol{m}^{T}\boldsymbol{\alpha}^{\ast}-d)\right) .
\label{eq:reuslt10}\end{aligned}$$
Finally, since $\boldsymbol{m}^{T}\boldsymbol{\bar{J}}_{\beta}^{-1}(\widehat{\boldsymbol{\alpha}}_{\beta})\boldsymbol{\bar{K}}_{\beta
}(\widehat{\boldsymbol{\alpha}}_{\beta})\boldsymbol{\bar{J}}_{\beta}^{-1}(\widehat{\boldsymbol{\alpha}}_{\beta})\boldsymbol{m}$ is a consistent estimator of $\boldsymbol{m}^{T}\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha}^{\ast})\boldsymbol{\bar{K}}_{\beta}(\boldsymbol{\alpha
}^{\ast})\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha}^{\ast
})\boldsymbol{m}$ and
$$\boldsymbol{m}^{T}\sqrt{K}(\widehat{\boldsymbol{\alpha}}_{\beta}-\boldsymbol{\alpha}^{\ast})\overset{\mathcal{L}}{\underset{K\mathcal{\rightarrow}\infty}{\longrightarrow}}\mathcal{N}(0,\boldsymbol{m}^{T}\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha
}^{\ast})\boldsymbol{\bar{K}}_{\beta}(\boldsymbol{\alpha}^{\ast})\boldsymbol{\bar{J}}_{\beta}^{-1}(\boldsymbol{\alpha}^{\ast})\boldsymbol{m}),$$
the desired result follows.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In the standard supernova picture, type Ib/c and type II supernovae are powered by the potential energy released in the collapse of the core of a massive star. In studying supernovae, we primarily focus on the ejecta that makes it beyond the potential well of the collapsed core. But, as we shall show in this paper, in most supernova explosions, a tenth of a solar mass or more of the ejecta is decelerated enough that it does not escape the potential well of that compact object. This material falls back onto the proto-neutron star within the first 10-15 seconds after the launch of the explosion, releasing more than $10^{52}$erg of additional potential energy. Most of this energy is emitted in the form of neutrinos and we must understand this fallback neutrino emission if we are to use neutrino observations to study the behavior of matter at high densities. Here we present both a 1-dimensional study of fallback using energy-injected, supernova explosions and a first study of neutrino emission from fallback using a suite of 2-dimensional simulations.'
author:
- 'Chris L. Fryer'
title: Neutrinos from Fallback onto Newly Formed Neutron Stars
---
Introduction
============
The collapse of a massive star down to a neutron star releases over $10^{53}$erg of potential energy. Just one percent of this energy is required to power the observed type Ib/c and type II supernovae. Most of the energy is emitted in the form of neutrinos. How this small fraction of the energy is converted into explosion energy is still a matter of debate, but it is believed by most that neutrinos play a role in depositing energy above the collapsed core to drive this explosion (see Fryer 2003 for a review). Whether or not neutrinos are important for the explosion, they do provide astronomers a window into the explosion mechanism behind supernovae.
Neutrinos also provide a window into the material conditions at the core of a supernova explosion. With temperatures above 10MeV and densities above nuclear densities, core-collapse supernovae make ideal laboratories for nuclear physics. To extract information about nuclear physics from such laboratories, we must understand the systematics in our supernova experiment. When the explosion engine is active, convective motions in the engine (Herant et al. 1994) make it very difficult to interpret the neutrino signal. An observed elevated neutrino luminosity could be a change in the neutrino opacity or it could be a difference in the convective engine. One approach to eliminate issues with convection would be to wait until all convective activity ceases. A number of studies have now been presented following the evolution of a collapsing star over 0.5-1.5s after bounce (Fryer & Heger 2000; Burrows et al. 2006; Scheck et al. 2006; and, in 3-dimensions, Fryer & Young 2007). In all these calculations, hydrodynamic motions near the proto-neutron star continue to dramatically affect the neutrino emission through the end of the simulations. To achieve a clean neutrino signal, we must wait until after the launch of the explosion. We must also wait a few seconds after the launch of the explosion because the proto-neutron star may experience deep convection (Keil et al. 1996). After this time (roughly a few seconds to 10-20s), the supernova finally seems to have achieved an ideal condition as a physics laboratory (Reddy et al. 1999). After 10-20s, the neutrino signal will be too weak to do much experimental science, so we truly are limited to this narrow time window.
Unfortunately, even at these late times, Nature does not allow completely pristine conditions. Material falling back from the supernova explosion (“fallback”) may well produce a new round of convection and confusion to our neutrino signal. This fallback has been studied both in its important role in calculating the initial mass of the neutron star formed in a supernova explosion (e.g. Fryer & Kalogera 2001) and in estimating the r-rpocess yields in supernovae Fryer et al. (2006). In this paper, we study its role in determining the neutrino luminosity arising after the first few seconds of a supernova. In §2, we review the history of supernova fallback and present new calculations of supernova explosions estimating the fallback for a range of explosion energies and stellar masses. §3 describes the 2-dimensional code used to model the neutrino emission from the this fallback and shows results for the suite of simulations run for this paper. Different than the multi-dimensional simulations modeling stellar collapse, these simulations do not start at the onset of collapse (or bounce) and end at the launch of the explosion. Instead, they start a 2-10s (depending on the explosion energy, etc. from §2) after the launch of the explosion when material begins to fall back onto the neutron star. We conclude with a brief discussion on how neutrino signals can be used to help better understand the supernova explosion, neutron star birth masses and neutrino cross-sections.
Supernova Fallback
==================
The idea of fallback was first discussed by Colgate (1971) to overcome nucleosynthesis issues arising from the supernova ejection of neutron rich material produced in stellar cores (Arnett 1971, Young et al. 2006). Colgate argued that the inner layers of the ejected material would deposit its energy to the stellar material above it, ultimately reducing its energy below that needed to escape the neutron star, and it would fall back onto the neutron star. In such a scenario, one would expect the inner material to fall back quickly (within the first few to ten seconds). It was argued that this material (the neutron rich material from the initial explosion) would accrete onto the neutron star, alleviating any nucleosynthesis issues.
Since this work by Colgate, supernova explosion calculations have confirmed that fallback does occur (Bisnovatyi-Kogan & Lamzin 1984; Woosley 1989; Fryer et al. 1999; MacFadyen et al. 2001) and new arguments for the cause of this fallback were suggested. For example, Woosley (1989) argued that when the supernova shock decelerates in the hydrogen layers of the star, it sends a reverse shock that drives fallback. Such a model argues that the fallback will happen at late times, long after it can affect the neutrino luminosity from the cooling proto-neutron star. It also suggested that close binary systems (where a star’s hydrogen envelope was removed prior its collapse) might experience a very different amount of fallback than the amount of fallback in the collapse of a single star.
Which is the true cause of the fallback? And more to the point for neutrino emission, when does fallback occur? This confusion has mostly exists because, without a quantitatively reliable explosion mechanism, scientists have artificially driven supernova explosions to be able to study the results of these explosions (e.g. supernova light curves, nucleosynthetic yields, and fallback). Much of the past work (e.g. Woosley 1989; Fryer et al. 1999; MacFadyen et al. 2001) used piston driven explosions. By moving the piston out, scientists artificially lowered the amount of fallback and delayed this fallback considerably. These calculations all predicted that fallback would occur more than 100s after the launch of the explosion.
But the piston-driven explosion mechanism may not accurately model the nature of the fallback. Young & Fryer (2007) found that piston-driven explosions produced both different fallback rates and different nucleosynthetic yields than energy-driven explosions of the same final energy. The errors in the yields or light-curves are on par with slight changes in the explosion energy. Such small errors seemed unimportant in matching the observations. For fallback, the differences are much more dramatic. Not only does the amount of fallback change, but the timescale at which the fallback occurs can change by more than an order of magnitude, moving a falback time of a few hundred seconds down to just 3-15s. This changes the fallback accretion rate by over an order of magnitude and ultimately determines whether or not fallback is important in estimating the observed neutrino signal. It also means that the the amount of fallback will not change between binary and single stars (unless binary interactions change the internal structure of the star).
In this paper, we focus on fallback calculations from energy-injected explosions. Energy-injection is much better at mimicking the currently-favored supernova mechnanisms. Let’s take the convection-enhanced, neutrino-driven supernova engine as an example (see Fryer 2003 for a review). In this mechanism, the basic energy source is neutrinos (either diffusing out of the proto-neutron star core or from newly accreting material) that heat the atmosphere above the neutron star. This atmosphere is topped by the accretion shock of the infalling star. When enough energy is deposited in this region, the accretion shock will be pushed outward and an explosion occurs. Convection aids this mechanism by both allowing heated material to rise (cooling by adiabatic expansion instead of forcing it to continue to heat until it can cool by neutrino emission) and allowing shocked material at the top of the convective region to flow down to the neutron star surface to accrete onto the neutron star (emitting neutrinos) or be heating to be part of the rising bubble.
An energy injection method, although not mimicking the effects of convection directly, can mimick the basic tenets of this model - heating just above the neutron star surface to blow off the accretion shock. Ideally, once the region above the neutron star becomes more rarefied, the energy injection will essentially halt (aside from a weak, by supernova standards, neutrino-driven wind). Some groups have gone so far as to only inject this energy through a neutrino flux (Frölich et al. 2006). In this manner, once the region becomes rarefied, the energy injection drops naturally. A piston-driven explosion can not mimic this effect well unless very specific attention is paid to the input of the piston. The very different fallback results from piston explosions demonstrate just how far off such explosions are from the energy drive of the standard neutrino model.
This does not mean that by using energy-injection, we can produce a definitive fallback estimate for a supernova explosion. The explosion energy is one of our primary uncertainties in estimating the fallback rate. Supernova scientists have yet to agree on the exact mechanism behind the supernovae and we are far from achieving quantitative predictions of these explosions. With accurate light-curve and spectra calculations, we may be able to estimate the explosion energy for a particular supernova based on its observations. By studying a range of explosion energies for a given progenitor, we can provide a template for the neutrino flux (from fallback) arising from these systems, allowing us to possibly extract this effect and once more focus a study on the physics of dense, hot matter.
Calculating Fallback
--------------------
Our fallback calculations will all be based on fallback estimates from energy-driven explosions. We use the same multi-step technique employed in Young et al. (2006) and Young & Fryer (2007). We start with progenitor stars modeled to collapse (either from Heger et al. 2000 or Young et al. 2008). These progenitors are mapped into the 1-dimensional core-collapse code from Herant et al. (1994). This code includes equations of state valid from stellar densities up to nuclear densities, a 3-flavor flux-limited diffusion neutrino transport scheme, and a simple nuclear network. With this code, we follow the collapse and formation of the proto-neutron star and the propogation of the shock produced when the collapse halts due to nuclear forces.
After this shock stalls (as it loses its energy via neutrino losses), we have a structure defined by a shocked “atmosphere” produced by the now-stalled bounce shock above a dense proto-neutron star. The proto-neutron star typically has a baryonic mass between 1.1 and 1.3M$_\odot$. The edge of the proto-neutron star is determined by the mass where the density drops below $10^{10} {\rm g cm^{-3}}$. At these densities, there is typically a well-defined edge where the density drops from $10^{12} {\rm g cm^{-3}}$ to $10^8 {\rm g cm^{-3}}$ over a very narrow mass cut. The exact location of this edge depends upon the progenitor mass and probably the exact code used to model this collapse phase (e.g. 1-dimensional versus multi-dimensional results).
In general, 1-dimensional calculations have not produced supernova explosions. With the proto-neutron star removed, we have also removed the energy source for any explosion. To induce an explosion in 1-dimension, we must source in energy. The simulations here source the energy directly into the inner 15 cells (roughly 0.1M$_\odot$) of the star. We keep this energy source on for a limited time (between 50 and 300ms), varying the energy injection rate and time to produce a range of explosion energies. Young & Fryer (2007) found that such energy sourcing was more flexible than a simple neutrino enhancement to modeling the full range of proposed explosion mechanisms. In our calculations, we use the shorter (50ms) injection duration for the lowest energy explosions and the lowest mass stars and the longer (300ms) injection duration for the more energetic explosions.
We follow this explosion for 400s. This allows us to follow the shock as it moves well into the star (and, for binary systems, out of the star surface). Our simulation space includes enough matter to ensure that for this 400s duration, the shock is well within the simulation space (there are at least 100 zones between the final shock postion and the outer zone of our star). In the case of the binary systems, we have included a mass loss estimated from the stellar models. Typically, we model this star with roughly 2000 zones. Young et al. (2008) have done a resolution study, comparing 1000,2000 and 4000 zones. They do not find an appreciable difference in the ejected (and hence fallback) mass based on this resolution. It will make a large difference on how the fallback mass tapers off. However, note that with the Lagrangian code in this paper, we can not accurately calculate low fallback rates. But, as we shall see in our 2-dimensional models, it may well be that fallback powered explosions might cut off this low-rate fallback.
We use the same 1-dimensional code to model the fallback. As material falls back onto the edge of our proto-neutron star and its density rises above $10^9 {\rm g cm^{-3}}$, we remove the particle and add its mass to our proto-neutron star. Although we could follow the accretion and neutrino cooling of this material to higher densities with our code, as the density rises, the sound speed increases and the cell size decreases, both of which cause the time step to decrease. To make this problem tractable, we choose a low enough density that the material accretes before dropping the timestep below 1 microsecond. Even so, these simulations typically take between 1 million to 10 million timesteps. As we shall find in §3, the modeling the true behavior of this matter requires modeling the accretion in multi-dimensions. Since these simulations are focused on calculating the infall rate (not true accretion rate), our assumptions do not introduce large errors in our analysis.
In this paper, we present the results from a suite of 1-dimensional explosion models models using 3 different progenitor masses and explosion energies (for a summary, see Table \[tab:fallback\]). Table \[tab:fallback\] also shows the peak accretion rates and total mass accreted for these models. Note that we predict a range of neutron star masses based on both progenitor mass and explosion energy, in agreement with Fryer & Kalogera (2001). But remember that the neutron star masses assume that all of the fallback remains on the neutron star. As we shall see in §3, some of this matter is re-ejected. For normal explosion energies, it is likely that 12-15M$_\odot$ stars produce neutron stars with gravitational masses in the 1.3-1.5M$_\odot$ range. If the explosions are stronger, the gravitational masses may well be as low as 1.2M$_\odot$. For weak explosions, or more-massive stars, the remnant masses may be so large that the neutron star collapses to a black hole.
Figure \[fig:macc\] shows the mass accreted in the first 15s for our models. Note that in all casses, fallback occurs almost immediately (with some delays of 2-7 seconds). Unless large amounts of fallback occur (above 1M$_\odot$), the fallback is likely to be mostly over after 10-15s. With over 0.1M$_\odot$ falling back in 10-15s, accretion rates above 0.01M$_\odot$s$^{-1}$ are expected, corresponding to neutrino luminosities in excess of $10^{51} {\rm ergs
\, s^{-1}}$. The bottom line is that, for normal supernovae, fallback occurs, at least for stars with initial masses at or above 12M$_\odot$. This fallback occurs early, so it will definitely play some role in the neutrino emission at the 3-15s timescale.
Angular Momentum
----------------
An additional feature of the progenitor that affects the fallback and the neutrino luminosity is the angular momentum in the progenitor. Figure \[fig:1drot\] shows the angular momentum of the inner 4M$_\odot$ of a star for a variety of stars with and without magnetic braking (Heger et al. 2000,2005). For neutron star accretion, the relevant angular momentum is that within the inner 1.4-2.0M$_\odot$. Such mass zones have low-angular momenta: a few$\times 10^{15} {\rm cm^2 s^{-1}}$ for stars with magnetic braking, a few$\times 10^{16} {\rm cm^2 s^{-1}}$ for stars without. Our typical simulations use a value of $10^{16} {\rm cm^2 s^{-1}}$. These low angular momenta had little effect on our results, and we ran some simulations with twice that amount. For our black hole systems, we ran even higher angular momenta. For neutron stars, the angular momentum is not enough for this material to form a true disk in the star, but it can alter the downflow and it is this effect that we would like to study in this paper.
Note that although there is quite a bit of structure in the angular momenta (caused by incomplete angular momentum transport across elemental boundaries), in general, the angular momenta of the stars increases as one moves to higher and higher mass shell. This is one reason why black-hole forming systems are more likely to produce asymmetric explosions than typical neutron-star forming systems. The increased angular momentum means that angular momentum plays a bigger role in shaping the explosion, possibly producing larger asymmetries.
Neutrinos from Fallback\[results\]
==================================
Our 1-dimensional calculations provide us with a fallback rate. If we assume that any fallback material emits all of the potential energy released from its downfall at the moment it hits the proto-neutron star, we can estimate the neutrino luminosity from the fallback. The accretion of 0.1M$_\odot$ masses onto a 10km, 1.4M$_\odot$ neutron star over 10s would correspond to a neutrino luminosity of $3.7\times 10^{51} {\rm ergs \, s^{-1}}$. For our more massive stars, this fallback can be ten times higher, corresponding to a neutrino luminosity of $4\times 10^{52} {\rm ergs \, s^{-1}}$ for over 10s after the launch of the explosion. Assuming pair-annihilation dominates the neutrino emission, this neutrino luminosity would be nearly 50% electron and 50% anti-electron neutrinos. Typical neutron star luminosities after 1s are below $10^{52} {\rm ergs \,
s^{-1}}$ and can be as low as a few $\times 10^{51} {\rm ergs \,
s^{-1}}$ (e.g. Bruenn 1987, Keil & Janka 1996). Even at 1s, fallback can dominate the neutrino emission if the fallback is heavy. After $\sim$5s our fallback estimates argue that fallback neutrinos dominate the neutrino emission.
But we have made several assumptions in this estimate for the neutrino luminosity. First, we have assumed that all of the potential energy released is immediately emitted in neutrinos. This energy can go into heating the proto-neutron star which will then cool on a longer timescale. This energy also may go into ejecting other infalling material. As the material falls onto the proto-neutron star, it is shock heated and can rise. Fryer et al. (1996,2006) found that these rising shocked bubbles can accelerate above the escape velocity and actually be re-ejected. So it is quite possible that a fraction of the “fallback” material does not end up on the neutron star if we account for these multi-dimensional effects. This re-ejected material both does not contribute to the total energy available to produce neutrinos, but it takes some fraction of the potential energy released to power its explosion. Finally, $\mu$ and $\tau$ neutrinos may also be released and we must include these neutrinos in our energy budget. In this section, we study the effects of the above assumptions to get a more accurate neutrino luminosity from fallback.
Code Description
----------------
Our primary concern with our simple estimate of fallback neutrino emission is the fact that we ignore multi-dimensional effects. We have known for some time that if a compact object is accreting mass with considerable angular momentum (and inefficient cooling), outflows occur (e.g. Blandford & Begelman 1999). But even if the angular momentum is minimal, if the compact object is a neutron star, a sizable fraction of the infalling material can be re-ejected (Fryer et al. 2006). To study the fate of supernova fallback, we want to understand a number of effects. For example, how do the results vary with accretion rate? But we would like to also understand the role of angular momentum and the differences between hot or cold neutron stars. Finally, numerical effects, such as boundary conditions, are bound to play a role. Before we discuss the results of our simulations, let’s discuss the code used for these calculations and our tests of the physical and numerical effects.
Our code must model the physics of downflows and allow us to answer the questions in the preceding paragraph. First and foremost, we must follow the evolution in a multi-dimensional manner. As a first step, we use the two-dimensional smooth particle hydrodynamics code described in Fryer et al. (1996,2006). We model the region from 10,000km above the proto-neutron star down to the proto-neutron star surface. Typical runs range from an initial set of $\sim 11,000$ particles moving up to 50-70,000 particles by the end of the simulation. The code includes an equation of state valid from densities below 1gcm$^{-3}$ up to nuclear densities (including an estimate of nuclear statistical equilibrium). Neutrino transport is followed using flux-limited diffusion neutrino scheme for three neutrino species (Herant et al. 1994). The neutrino emission and cross-sections are also outlined in Herant et al. (1994).
Our simulations are set to mimic the conditions 2-10s after the launch of the explosion when material begins falling onto the proto-neutron star. The initial condition begins with a set of particles ranging from 1000km up to 10,000km. The material is given velocities set to the free-fall velocity (In our 1-dimensional simulations, the infall velocity is within 1-10velocity) at their radial position with densities set to give the desired accretion rate (recall, $\rho=\dot{M}/4 \pi r^2 v_{\rm
infall}$. With $v_{\rm infall}$ within 1-10% of the free-fall velocity, we can accurately set up our initial conditions: our 3 separate rates correspond to 0.001,0.01,0.1M$_\odot$s$^{-1}$ (Table \[tab:neut\]). For these simulations, we model only constant fallback rates. Note that in Nature, the fallback rate varies quite a bit. But we are focusing our study on the neutrino emission at peak fallback rates, and this suite of simulations will definitely bracket the range of results. The entropy of the infalling material is generally set to a few. The results are fairly insensitive to this initial entropy as the shock resets the entropy. Boundary conditions are probably the biggest uncertainty in our calculations. Particles are fed in through the outer boundary and are allowed to accrete through the inner boundary and be ejected out of the outer boundary. The infall through the outer boundary is determined by a fixed infall rate. This infall is only altered if material is flowing out of this outer boundary. At any point where an outflow occurs, the inflow is temporarily halted. This models the effect of outflows choking off the accretion.
The inner boundary is more difficult. Ideally, we would model the matter until its density reaches neutron star densities ($\sim 10^{14}
{\rm g cm^{-3}}$) and it is mostly deleptonized. At such time, the matter has lost most of its energy and we can be sure that we have accounted for the total neutrino luminosity. However, the sound speeds at such densities and the size of our Lagrangian particles near the proto-neutron star would decrease the timestep to fractions of a microsecond. Such small timesteps prohibit us from following the evolution of the fallback for more than a fraction of a second. Instead we opt to use slightly less demanding criteria for the removal of particles on the inner boundary. Our standard set of models accretes particles whose density rises above $\sim 10^{10} {\rm g
cm^{-3}}$ with electron fractions below 0.3. This means that we are assuming any further energy released by the matter as it accretes goes into heating the neutron star which will cool on longer timescales. In our suite of models, we include a test of this boundary condition and find that the total neutrino luminosity[^1] is not too sensitive to our assumption for accretion.
We assume the axis of rotation lies along our axis of symmetry in our 2-dimensional caclulation. Each particle is given an angular momentum and it retains that angular momentum for the duration of the calculation. The angular momentum for each particle is determined to fit the angular velocities of our progenitor stars (Fig. 2), so material along the axis has a low angular momentum whereas material in the equator has the highest angular momentum. The specific angular momentum $j$ is given by $j=(x/10,000\,km)^2 \omega$ where $\omega$ is the angular velocity taken from the stellar models. The inner material starts below the angular momentum given by $\omega$, but by the end of our simulations, most of the material near the neutron star surface has the full angular momentum set by this $\omega$ value. There is no angular momentum transport and hence, technically, no heating from this transport. However, the angular momentum will slow the materials inflow, breaking the symmetry in the downflow. Fryer & Heger (2000) also found that the angular momentum alters the instability criterion, preventing convection in the equatorial region where the angular momentum gradient is highest.
To study the effect of a hot neutron star, we introduced a non-zero neutrino flux arising from our inner, neutron-star, boundary. In particular, we are interested in how the neutrinos from a hot neutron star affect the hydrodynamics, so we have chosen rather large luminosities ($10^{52},2\times10^{52} {\rm ergs \, s^{-1}}$ electron neutrino luminosities with energies of 10MeV with corresponding $8\times10^{51},1.6\times10^{52} {\rm ergs \, s^{-1}}$ anti-electron neutrino luminosities with energies of 15MeV). These neutrinos are a boundary source for our flux-limited diffusion transport scheme and transported out of the system, heating the inflowing material. However, in most of our simulations, the neutrino optical depth is fairly low, and the total energy deposited is also minimal (see §3.2).
Finally, there is always concern that the artificial viscosity used in SPH is introducing spurious effects into our calculations. For the most part, our studies have found this numerical artifact to play a small role in results studying core-collapse supernovae, and we do not expect it to play a large role in these calcualtions. Nonetheless, we have included a simulation where we have increased both the bulk and von Neumann-Richtmyer viscosities by a factor of 2 (3.0,6.0 respectively versus our standard values of 1.5,3.0).
With both numerical and physical effects to study, we have run a suite of simulations to test the dependence of the neutrino luminosity on the initial conditions and on the numerics. A summary of this suite of models, along with their basic results, is summarized in table \[tab:neut\]. Our base model has an accretion rate of 0.01M$_\odot$ s$^{-1}$ with an angular momentum equal to that shown in the circle in Figure \[fig:1drot\]. We also include higher rotating runs with angular momenta set by equating the angular momenta to the value denoted by the square in Figure \[fig:1drot\]. In both cases, the angular momentum is low, so we don’t expect the formation of a full accretion disk. We study the inner boundary in a number of ways. We include a simulation where the criteria for accretion is more strict: $\sim 10^{11} {\rm g cm^{-3}}$ with electron fractions below 0.1. We also include a set of simulations with an absorbing boundary at 100km (a black hole boundary condition). We study accretion rates 10 times higher and ten times lower than our canonical rate. Finally, since the accretion occurs at early times, we have also included a few simulations where the neutron star itself is still emitting neutrinos. In § \[sec:dynamics\], we compare the results on the dynamics for this suite of calculations. In § \[sec:neutrinos\] we compare the resulting neutrino luminosities.
Accretion Dynamics\[sec:dynamics\]
----------------------------------
Before we discuss the neutrino fluxes from this suite of calculations, let’s compare the dynamics in the accretion. We expect material to shock as it hits the proto-neutron star, and some of this shocked material will begin to rise, driving convection. These simulations are 2-dimensional. Even without angular momentum, perturbations along our symmetry axis would allow the instability to develop stronger along this axis. But we have studied this in some detail in core-collapse calculations and have minimized this effect (see, for example, Fryer & Heger 2000). However, the fact that the angular momentum axis also lies along the symmetry axis drives convection along our axis of symmetry. The growth of convective instabilities is stabilized by angular momentum gradients and the deceleration of material (along with the fact that our initial condition has an angular momentum gradient which is strongest along the equator), the convection initially grows strongest in the poles. Figure \[fig:comp-nsvbh\] shows the results of our standard calculation showing this outflow. This figure shows the evolution at 0.15s. The right panel shows the evolution of the corresponding “black hole” simulation with the absorptive boundary. The true innermost stable circular orbit for a slowly rotatign 3M$_\odot$ black hole is closer to 20km, so our 100km absorption radius understimates the activity from a real black hole. However, we can see that although the angular momentum alters the inflow, it is insufficient to stop it, and the material continues to accrete directly onto the black hole.
As our neutron star model evolves the outflow expands, constraining the accretion to a funnel roughly $45^{\circ}$ from the plane. Figure \[fig:comp-late\] shows the evolution of our standard model to 0.3 and 0.45s. Over half of the inflowing material is ultimately ejected. Not only does the ejected material not contribute to the energy available for neutrino emission, some of the energy of that accreted material goes toward accelerating this ejecta.
Such outflows have been studied for over a decade in supermassive black hole systems such as active galactic nuclei (see Blandford & Begelman 1999) for a review. The amount of potential energy released in the accretion of material onto a compact object is enormous. If this energy is tapped (either by viscous forces or neutrino emission), it can drive an explosion. Rockefeller et al. (2007) and Fryer et al. (2006) have recently applied this physics to stellar-massed compact objects to better understand their simulations of accreting systems. In rotating black hole systems, material hangs up in a disk. Viscous forces that transport out angular momentum also transport energy, driving outflows. What prevents this ejection is cooling (either via photon radiation in the case of supermassive black holes or neutrinos in the case of most collapsing systems). Of course, for black hole systems that do not have a hard surface preventing the inflow of material, high angular momentum is required to prevent the energy from all flowing directly into the black hole. In neutron star systems, outflows (or at least vigorous convection) have been expected for more than a decade (Chevalier 1989; Fryer et al. 1996).
If the angular momentum were high enough, the infalling material would hang up in a disk. This is most important in the black hole systems. We ran a series of models increasing the specific angular momentum ($j$) a factor of 3 and, for our black hole models, a factor of 10. MacFadyen & Woosley (1999) found that the specific angular momentum must be at least $\sim 10^{17} {\rm cm^2 s^{-1}}$ (a factor of 1000 higher than our standard $j^2$ value). As we can see from Figure \[fig:comp-rot\], our factor of 10 increase in $j$ is not enough to change the fate of material falling back on our “black hole” simulation with its large absorptive boundary. In a true black hole system, the angular momentum available would increase as we move beyond 3M$_\odot$ (well above the angular momenta shown in Fig. \[fig:1drot\]). This is one reason why the collapsar GRB model argues for systems where the black hole mass exceeds 3M$_\odot$ when the angular momentum in the star is sufficient to produce an accretion disk. Neutrinos from these systems have been considered in detail elsewhere (MacFadyen & Woosley 1999, Hungerford et al. 2006, Rockefeller et al. 2007) and we do not study them in this paper.
If the neutron star is still hot and emitting neutrinos, it can also alter the inflow of fallback. Figure \[fig:comp-hot\] shows two simulations with varying amounts of neutrino energy (18 and 36$\times10^{51} {\rm ergs \, s^{-1}}$) arising off the neutron star surface. The effect on the dynamics is minimal, although a slight increase in the velocity (and hence position at a given time) can be seen. As we shall see below, the neutrinos emanating from the proto-neutron star surface will dominate the total neutrino flux in such cases. But we don’t expect these high luminosities at 10s and the accretion luminosity will dominate at these later times. But at early times, the modification of the downflow on the neutrino opacities is the dominant effect.
Finally, we have varied the infall rate from 0.001 to 0.1M$_\odot$s$^{-1}$. The effect this has on the dynamics is shown in Figure \[fig:comp-mdot\]. As with many of our models, the nature of the dynamics is not altered significantly by these changes. But we shall see that the neutrino flux very much depends on the accretion rate.
Neutrino Emission\[sec:neutrinos\]
----------------------------------
We found in the last section that the dynamical behavior of the fallback was relatively insensitive to both uncertainties in the numerics as well as uncertainties initial conditions: e.g. fallback rate and rotation. But what about the neutrino luminosity? First let’s study the uncertainties in the numerics. Fig \[fig:nunum\] shows the neutrino luminosity and mean energy for 3 different models studying the numerical effects (both the artificial viscosity and the treatment of the inner boundary) on the calculation. The variations in the viscosity and the inner boundary lead to differences that are less than a factor of 2 in the neutrino luminosity and 5% variations in the neutrino mean energy. Many of these errors could be dominated by the explicit transport scheme used in these calculations and it is possible that these errors can be significantly diminished with implicit schemes that are more stable.
Figure \[fig:nunum\] also shows the results for a fast rotating NS model (NS-Rot2). With the low angular momenta in our calculations based on the rotation velocities in the inner core material of massive stars (Fig. \[fig:1drot\]), rotation does not alter the neutrino luminosity noticeably. Black hole systems depend more sensitively on the rotation because it is the angular momentum that prevents the material from accreting directly into our black hole. But as we expected from the fact that our angular momentum is too low to produce accretion disks, the neutrino emission from our black hole systems is negligible (Fig. \[fig:nubh\]). In such low-angular momentum systems, we expect essentially no emission after the collapse of the neutron star down to a black hole. Contrast this to typical collapsar conditions, where the material falling back onto the black hole has enough angular momentum to hang up in a disk and its falback accretion rate is high enough to produce high-density, high-temperature structures. In this case, the neutrino emission from a black hole can be quite large - with luminosities on par with the neutrino burst of the original collapse (MacFadyen & Woosley 1999; Hungerford et al. 2006; Rockefeller et al. 2007).
In our models, the $\mu$ and $\tau$ neutrinos tend to be an order of magnitude lower than the corresponding electron neutrino fluxes. So our assumption that most of the neutrinos are emitted as electron or anti-electron neutrinos is reasonably valid. Hungerford et al. (2006) and Rockefeller et al. (2007) found that for collapsar models, the $\mu$ and $\tau$ neutrinos make up a sizable fraction of the total emission. At higher accretion rates (and higher angular momentum in the infalling material), the fraction of the luminosity emitted in $\mu$ and $\tau$ neutrinos will likely increase.
The strongest dependency on our initial conditions in the mass accretion rate. Figure \[fig:numdot\] shows the neutrino emission from 3 different accretion rates onto our neutron star surface. Our basic potential energy released estimate would argue that the neutrino luminosity scales with the accretion rate. And this trend is basically true for our simulations. However, note that the electron and anti-electron neutrinos for our highest accretion rate (0.1M$_\odot$s$^{-1}$) case are not quite an order of magnitude higher than our standard accretion rate (0.01M$_\odot$s$^{-1}$). This is because the electron neutrinos become trapped in this highest accretion rate case, lowering the escaping luminosity. The $\mu$ and $\tau$ remain nearly an order of magnitude higher as they are not trapped in any of our simulations.
Figure \[fig:nuhot\] shows the neutrino emission for 3 different “hot” neutron star models. Fallback contributes an additional 10-20% of the luminosity on average for the NS2-hot2 model, 20-40% to the NS2-hot1 model, and over 50% to the NS1-hot model. The neutrino energy is also altered by an amount comparable to the change in the luminosity. Although a hot neutron star may dominate the neutrino luminosity, fallback clearly can contribute a sizable fraction of the observed neutrino flux.
Finally, note that in general, our predicted fallback neutrino energies are high. The most notable exception is that the mean neutrino energy is 5-10MeV cooler for our low accretion-rate simulations. As the accretion rate decreases, so too will the mean energy of the emitted neutrinos.
Conclusions
===========
Fallback (of at least 0.1M$_\odot$) is likely to occur in normal ($10^{51}$erg) supernova explosions from stars with initial masses of 12M$_\odot$ or greater. With energy-injected (more realistic than piston-driven) explosion calculations, this fallback occurs quickly, generally in the first 15s and peak accretion rates in the 0.01-0.1M$_\odot$s$^{-1}$ range should be expected. If the engine stays active long after the launch of the supernova shock, the total fallback will be lower. But neutrino engines weaken significantly quickly after the shock is launched and the material above the neutrinosphere (absorbing the energy) is ejected. Magnetic-driven explosions could be very different. The mechanism for this fallback is essentially that proposed by Colgate (1971) whereby the outflowing material decelerates as it pushes against the material above it, ultimately decreasing its velocity below the escape velocity and causing it to fall back onto the proto-neutron star. This means that the bulk of the fallback is fairly insensitive to the structure on the outer envelope of the star (so fallback is roughly the same whether or not the star is in a binary).
Our multi-dimensional simulations of this fallback suggest that not all of this fallback material is actually incorporated into the proto-neutron star. Some of it flows out of the system. The corresponding neutrino luminosity is also lower, both because less material is accreted and some of the energy released goes toward driving outflows. But the neutrinos from fallback could still dominate the neutrino emission from a proto-neutron star 10s into the explosion.
At this time, observations of neutrinos from supernovae are limited to those of supernova 1987A (Hirata et al. 1987, Bionta et al. 1987). With only 20 neutrinos over $\sim$15s, it is difficult to place many constraints on the fallback. Some authors have claimed that claimed that the late-time neutrinos could not be easily explained by a cooling neutron star (e.g. Suzuki & Sato 1987), but others have argued that the observed neutrino signal is consistent with neutrinos diffusing out of a hot proto-neutron star (Burrows & Lattimer 1987; Bruenn 1987) . In our neutron-star forming models, fallback ends within the first 10-15s, comparable to the duration of the observed neutrino burst and the flux is consistent with our more standard accretion rates. The observations are consistent with fallback, but could easily be explained by a neutron star without any fallback. Given that the progenitor star is believed to be greater than 15M$_\odot$, fallback is likely to have occured. But with the current errors in the time-dependent luminosity, it is difficult to determine whether or not accretion is taking place.
If we assume fallback accretion did occur, the fact that the mean neutrino energy appears to be dropping with time in the observations suggests that the accretion rate is dropping at 10s. But neutrinos are still observed at 10s. Unless the fallback material has considerable angular momentum, the compact remnant is not a black hole at 10s. In addition, because the accretion rate is dropping dramatically at this time, it is unlikely that the remnant will accrete much additional mass. It will remain a neutron star. In our fallback calculations, systems that accrete enough material to form a black hole are either already a black hole at 10s or still accreting rapidly at 10s, neither of which is supported by the observations. But we really need a better neutrino signal to say anything definitive.
With such high accretion rates, fallback can easily dominate the neutrino luminosity after a few seconds, more than doubling the emission from the neutron star at thearly times. If we wish to study neutrino opacities in the supernova explosion, we will have to be able to calculate this fallback. It may be possible to estimate the fallback rate from detailed study of supernova light-curves. Fortunately, any system that is detected in neutrinos will have a wealth of data in photons of all wavelengths.
If the fallback rate is high, the infalling material will alter the position of the neutrinosphere, and the problem becomes nearly as complex as the supernova explosion modeling itself. We have only touched the surface on the difficulties in modeling such systems. In this paper, we have not addressed the production or feedback of magnetic fields, which may definitely alter the fate of the fallback. We have also not discussed the nuclear yields from the ejecta (a first paper on this is by Fryer et al. 2007 and we plan future projects studying this nucleosynthesis). However, note that the old “wind” r-process picture will not work in any system with fallback (stars more massive than $\sim 12$M$_\odot$). The matter trajectories just can’t be explained by the wind solution in the cases where fallback dominates the motion near the neutron star. We also have not studied the actual accretion onto the neutron star in detail. Note that the final neutron star masses in Table \[tab:fallback\] assumed no mass ejecta. But as we have found in this study, over half of the fallback material may be ejected, leading to smaller neutron star masses and a narrower mass range for these neutron stars.
It is a pleasure to thank Patrick Young, Frank Timmes and Aimee Hungerford for useful conversations on this project. This project was funded in part under the auspices of the U.S. Dept. of Energy, and supported by its contract W-7405-ENG-36 to Los Alamos National Laboratory, and by a NASA grant SWIF03-0047.
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[llcccc]{}
12 & & & &\
& 0.55 & 0.33 & 1.64 & $>$0.02 & 120\
& 0.92 & 0.23 & 1.54 & $>$0.01 & 86\
& 2.4 & 0.030 & 1.34 & $>$0.003 & 11\
& 3.6 & 0.015 & 1.33 & $>$0.001 & 5.6\
15 & & & &\
& 0.9 & 0.34 & 1.75 & $>$0.05 & 130\
& 1.65 & 0.25 & 1.66 & $>$0.05 & 93\
& 7.0 & 0.11 & 1.52 & $>$0.02 & 41\
& 17 & 0.08 & 1.49 & $>$0.01 & 30\
23 & & & &\
& 1.25 & 2.2 & 3.9 & $>$0.15 & 820\
& 2.0 & 1.75 & 3.45 & $>$0.1 & 650\
& 2.5 & 0.86 & 2.56 & $>$0.1 & 320\
& 6.6 & 0.02 & 1.72 & $>$0.01 & 7.5\
[lccccc]{}
NS3 & $10^{-3}$ & $\rho>10^{10},Y_e<0.3$ & $3\times10^{15}{\rm cm^2 s^{-1}}$ & 0. &\
BH3 & $10^{-3}$ & $r<100$km & $3\times10^{15}{\rm cm^2 s^{-1}}$ & 0. &\
NS2 & $10^{-2}$ & $\rho>10^{10},Y_e<0.3$ & $3\times10^{15}{\rm cm^2 s^{-1}}$ & 0. &\
NS2-Rot2 & $10^{-2}$ & $\rho>10^{10},Y_e<0.3$ & $10^{16}{\rm cm^2 s^{-1}}$ & 0. &\
NS2-Hot1 & $10^{-2}$ & $\rho>10^{10},Y_e<0.3$ & $3\times10^{15}{\rm cm^2 s^{-1}}$ & 18. &\
NS2-Hot2 & $10^{-2}$ & $\rho>10^{10},Y_e<0.3$ & $3\times10^{15}{\rm cm^2 s^{-1}}$ & 36. &\
NS2alpha & $10^{-2}$ & $\rho>10^{10},Y_e<0.3$ & $3\times10^{15}{\rm cm^2 s^{-1}}$ & 0. & $2\times \alpha$\
NS2bound & $10^{-2}$ & $\rho>10^{11},Y_e<0.1$ & $3\times10^{15}{\rm cm^2 s^{-1}}$ & 0. & $2\times \alpha$\
BH2 & $10^{-2}$ & $r<100\,km$ & $3\times10^{15}{\rm cm^2 s^{-1}}$ & 0. &\
BH2-Rot0 & $10^{-2}$ & $r<100\,km$ & 0. & 0. &\
BH2-Rot2 & $10^{-2}$ & $r<100\,km$ & $10^{16}{\rm cm^2 s^{-1}}$ & 0. &\
BH2-Rot10 & $10^{-2}$ & $r<100\,km$ & $3\times10^{16}{\rm cm^2 s^{-1}}$ & 0. &\
NS1 & $10^{-1}$ & $\rho>10^{10},Y_e<0.3$ & $3\times10^{15}{\rm cm^2 s^{-1}}$ & 0. &\
NS1hot & $10^{-1}$ & $\rho>10^{10},Y_e<0.3$ & $3\times10^{15}{\rm cm^2 s^{-1}}$ & 36. &\
BH1 & $10^{-1}$ & $r<100\,km$ & $3\times10^{15}{\rm cm^2 s^{-1}}$ & 0. &\
[^1]: Note, that except for our low accretion rates, our inner boundary at an neutrino depth of more than a few (above 10 in the highest accretion rates). It gets close to, and at times is below, 2/3 for low accretion rates.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'It has been widely observed that capitalization-weighted indexes can be beaten by surprisingly simple, systematic investment strategies. Indeed, in the U.S. stock market, equal-weighted portfolios, random-weighted portfolios, and other naïve, non-optimized portfolios tend to outperform a capitalization-weighted index over the long term. This outperformance is generally attributed to beneficial factor exposures. Here, we provide a deeper, more general explanation of this phenomenon by decomposing portfolio log-returns into an average growth and an excess growth component. Using a rank-based empirical study we argue that the excess growth component plays the major role in explaining the outperformance of naïve portfolios. In particular, individual stock growth rates are not as critical as is traditionally assumed.'
author:
- 'Adrian Banner[^1]'
- 'Robert Fernholz$^*$'
- 'Vassilios Papathanakos$^*$'
- 'Johannes Ruf[^2]'
- 'David Schofield[^3]'
bibliography:
- 'aa\_bib.bib'
title: '**Diversification, Volatility, and Surprising Alpha**'
---
Introduction
============
In the Summer of 2013 a paper published in the Journal of Portfolio Management entitled ‘The Surprising Alpha From Malkiel’s Monkey and Upside-Down Strategies’ by Rob Arnott, Jason Hsu, Vitali Kalesnik and Phil Tindall observed that in the US and Global stock markets equal-weighted portfolios, random-weighted portfolios and other naïve, non-optimised portfolios tend to outperform a capitalization-weighted index in the long run. This was a prominent paper which attracted a good deal of attention at the time in both trade and popular press, and won the 2013 Bernstein Fabozzi/Jacobs Levy Award for Outstanding Paper in the Journal of Portfolio Management.
The apparent fact that the cap-weighted index could so easily be beaten was characterised by the authors as ‘surprising’, ‘perplexing’, ‘paradoxical’ and a ‘puzzle’, and the paper offered by way of explanation two main lessons to be learnt: 1.) the investment thesis underlying the various different portfolios examined was not responsible for the observed outperformance; and 2.) all the portfolios displayed significant size and value factor biases which were credited with explaining most of the outperformance. In the small number of cases for which the extended four-factor risk model was not sufficient to explain all of the observed outperformance, the call was raised to discover other factors to explain it: ‘Let the quest for the missing risk factor(s) begin!’
Risk factors, especially the ‘big 4’ (market, size, value and momentum) have been adopted by many investment practitioners and finance academics as the basic principal components used to explain portfolio performance. Once it has been established that a portfolio’s relative performance is explained by, say, the presence of size and value factors, then no further explanation is thought to be necessary, or even possible as the factors cannot be further broken down.
So prevalent has this mind-set become that any portfolio of which the performance cannot be explained by these 4 factors is thought to indicate the presence of some yet-to-be discovered factor, or the similarly elusive dark-matter of manager skill. Factors are the ‘atoms’ of attribution, the ultimate particles of portfolio performance.
But of course it is well-known that scientists of the late nineteenth and early twentieth centuries demonstrated that the atom was not the ultimate, indivisible particle of matter – it could be further decomposed providing one had the right detection equipment.
This paper does not set out to discover the ‘missing’ factor sought by [@Arnott:2013] but we will instead propose an alternative, scientific decomposition for the results observed in that paper. Furthermore this decomposition is universally applicable to all portfolios. We repeat a representative sample of the experiments conducted in [@Arnott:2013] and we explain the results using simple methods first introduced by [@Fernholz:Shay:1982]. The detection equipment used in this case is just mathematics, and the particular lens applied is that of Stochastic Portfolio Theory.
It has long been known in this field that the cap-weighted portfolio is relatively easy to outperform. Based on these same methods, precisely structured ‘naïve’ portfolios that systematically outperform capitalization-weighted benchmarks were introduced in the 1990s with [@Fernholz:Garvy:Hannon]. The theory behind all these methods was reviewed in Fernholz (2002) and more current presentations can be found in [@FK_survey], and [@Karatzas:Ruf:2016].
A review of some basic concepts {#S:2_new}
===============================
Before describing our experiment and its results, it will be important to review and define various basic concepts that will be crucial to understanding and interpreting these results.
As we will be examining and attempting to account for the returns of various different portfolios it is important first of all to know exactly what we mean when we talk about return, and what kind of return we are talking about. This may seem trivial but it will ultimately bring to light an important aspect of the long term returns of portfolios.
The classical definition of the return on an investment is simply the difference between the final value and the initial value, divided by the initial value: $$\text{return} \eqdef \frac{\text{final value }-\text{ initial value}}{\text{initial value}}.$$
This calculation is fine for a single-period return but suppose we wish to calculate the average annual return of an investment over several years. Suppose that over $N$ years, a stock has annual returns of $r_1,r_2,\ldots,r_N$. There are several common methods for calculating this and they all have different characteristics:
1. [*Arithmetic average return:*]{} This is simply calculated as the sum of all the annual returns, divided by the number of years: $$\frac{1}{N}\Big((1+r_1)+\cdots+(1+r_N)\Big)-1.$$ This form of return is widely used in Modern Portfolio Theory and is compatible with the linear models used to calculate the Sharpe ratio and beta. It is, however, upward-biased as an estimator of expected long-term growth and can lead to absurd estimates in some cases. For example consider the case when a +100% return one year is followed by a -50% return the following year. Here the average arithmetic return over the two-year period is 25%, whereas in reality such an investment would have zero growth over the two years.
2. [*Geometric average return:*]{} For a period of $N$ years, this is calculated as the $N$-th root of the product of the annual returns: $$\sqrt[N]{(1+r_1)\times\cdots\times(1+r_N)}-1.$$ This form of return may be the most common method in practice. It helps to avoid the absurd results apparent in the example of arithmetic average returns given above, and this gives the method a somewhat scientific gloss. Unfortunately the geometric return is awkward to work with, compatible with neither the Sharpe ratio nor beta, and it too is upward-biased as an estimator of expected long-term growth.
3. [*Logarithmic average return:*]{} This is calculated simply as the sum of the logarithms of the annual returns, divided by the number of years: $$\displaystyle \frac{1}{N}\Big(\log(1+r_1)+\cdots+\log(1+r_N)\Big).$$ Logarithmic return is used in Stochastic Portfolio Theory and is the only one of these three alternative measures of average return that is *unbiased* as an estimator of expected long term growth.
It can be seen from these definitions that[^4] $$\text{ arithmetic return } \ge \text{ geometric return } \ge \text{ logarithmic return}.$$
For the remainder of this paper we shall concentrate on arithmetic and logarithmic returns, and shall refer to arithmetic return simply as ‘return’ and logarithmic return as ‘log-return’.
The relationship between return and log-return {#S:3_new}
==============================================
For any single stock there is a now well-known relationship between the return of the stock and its log-return as follows:[^5] $$\text{log-return of stock} \approx \text{return of stock} - \frac{\text{variance of return}}{2}.$$ In other words the log-return of the stock is approximately equal to its arithmetic return less half its variance. This latter term is often referred to as the ‘volatility drag’, the negative impact on a stock’s long term compound growth arising from its volatility. This was noticed in [@Fernholz:Shay:1982].
Portfolio return and log-return {#S:4_new}
===============================
Until now we have been considering the relationship between different measures of return for single stocks. We shall now apply this to portfolios.
As one might expect, the return of a portfolio over a single period is simply the weighted average return of all the stocks making up the portfolio. This was first formalised in [@Markowitz:1952], however the same does not apply for a portfolio’s log-return.[^6]
When applying the relationship between return and log-return for a single stock, given in the previous section, to a portfolio comprised of multiple stocks, it emerges that the log-return of a portfolio is not simply the weighted average log return of its constituents – remarkably it is actually greater than that: $$\text{portfolio log-return} = \text{weighted average stock log-return} + \text{excess growth rate}.$$
The amount by which a portfolio’s log-return exceeds that of its stocks is known in the literature of Stochastic Portfolio Theory as a portfolio’s excess growth rate, and was first noted in [@Fernholz:Shay:1982].
The excess growth rate (EGR) itself is simply defined as follows: $$\text{EGR} = \frac{\text{weighted average stock variance} - \text{portfolio variance}}{2}.$$
For practical purposes, given the above definitions, it can be seen that the excess growth rate is an important component of a portfolio’s log-return. It measures the positive boost to a portfolio’s long-term return that arises from the extent to which the volatility of the portfolio is less than that of its constituent stocks. That is to say, it represents the boost to return that arises from the efficacy of diversification.
Importantly it can even be shown that this quantity cannot be negative for a long-only portfolio (see [@Fe]). It is also clear to see that the excess growth rate will be higher for portfolios of volatile stocks with low correlations. *If all else is equal, a higher excess growth rate will increase the long-term growth of a portfolio.*
Estimation of expected portfolio log-return with a rank-based stock analysis {#S:3}
============================================================================
To recap, a portfolio’s log-return can be decomposed into two key components: $$\text{portfolio log-return} = \text{weighted average stock log-return} + \text{excess growth rate}.$$ We can now use this natural decomposition to estimate the expected log-returns of portfolios. For convenience we shall refer to the weighted average stock log-return as the average growth component, and the excess growth rate as the excess growth component.
The excess growth component can be estimated relatively easily, since its value depends only on variances, or relative variances, which are not difficult to determine in practice.
The average growth component, however, is more difficult to estimate. As most aspiring stock-pickers will testify, the expected returns or log-returns of individual stocks are difficult to estimate with any accuracy, and this has been known in the literature since at least [@Sharpe:1964]. Fortunately, in the case of our proposed experiment, we do not need to estimate the individual expected log-returns of the individual stocks.
Since we are considering naïve strategies for the top 1000 stocks, where the portfolio weights are assigned essentially at random and not picked by a stock-picker, the stock’s rank, in terms of its market capitalization, is more important to us than its name. We can therefore use rank-based methods to determine the value of the average growth component. We do this by measuring the average rank-based log-return (the average log-return over time associated with whichever stock is occupying a given rank in the largest 1000 stocks), without considering stocks individually by name (see [@Fe]).[^7]
In order to perform this calculation, we selected the 1000 largest stocks by market capitalization on every trading day for the period from 1964 to 2012, and ranked them in order of size, largest to smallest. We then measured the log-returns of the stock at each rank on each trading day and computed the daily average for each of the 1000 ranks, which was finally annualised by multiplying by 250.
The results of this analysis are demonstrated in Figure \[f1\].
The slightly downward-sloping line in Figure \[f1\] is a least-squares fit of all the points. Its slope is around $-.00001 \pm .00001$ (2 standard errors). This would seem to indicate that there is not much difference between the ranked stock growth rates, which implies that the portfolio’s expected average growth component should be about the same for all naïve portfolios. If it were the case, for example, that smaller stocks do indeed have higher long term returns then we would expect the line to be upward-sloping, which it manifestly is not.
Given this result it follows that any variation between the log-returns of different portfolios will depend largely on the differences in their respective excess growth components. Furthermore, since the excess growth component depends only on variances, we can conclude that the differences between the log-returns of naïve portfolios will depend only on variances and covariances.
If we now look at stocks’ variances by rank, rather than returns, we see a very different picture. Figure \[f2\] confirms that smaller stocks tend to have a larger variance, and since we have just established that a higher variance leads to a higher excess growth component, this would lead us to expect that portfolios which are more diversified into smaller stocks will have a higher return.
However this explanation of the outperformance of smaller stock portfolios is very much at odds with conventional wisdom. In traditional finance, given the assumed positive relationship between risk and return, smaller stocks are expected to have higher returns as compensation for their increased riskiness. These higher expected returns for smaller stocks translate into higher expected returns for portfolios comprised of these smaller stocks.
This may be true for single period arithmetic returns, but as long-term investors we should care about log-returns. We have already demonstrated in Figure \[f1\] that the log-returns of the smaller stocks are not, in fact, higher, and so when viewed through the lens of stochastic portfolio theory it becomes clear that the observed outperformance of small stock portfolios is not due to higher long-term returns of the small stocks themselves but to a higher portfolio excess growth rate.
The experiments
===============
[@Arnott:2013] test several naïve, non-optimized portfolio strategies versus a capitalization-weighted benchmark of the largest 1000 U.S. stocks over the period 1964-2012. All these strategies have a higher return than the benchmark, and most have a higher Sharpe ratio. It is well-known that capitalization-weighted portfolios are not particularly well-diversified – there is too much weight concentrated into the largest stocks. All of the naïve strategies have more diversification into the smaller stocks than the capitalization-weighted index.
Importantly, this greater diversification into the smaller stocks is not likely to affect the average growth component much. As we have seen in Figure \[f1\], the growth rates are about the same for the top 1000 stocks. In other words, the naïve strategies’ outperformance is not due to higher long-term returns of the smaller stocks. However the greater diversification is likely to increase the portfolio’s excess growth component, since both improved *diversification* and higher stock *volatility* increase excess growth.
To see what actually happens we run an experiment on the largest 1000 U.S. stocks using overlapping one-year periods starting each month from 1964-2012, quite similar in spirit to the experiment of [@Arnott:2013].
More precisely, at the beginning of each month we choose the largest 1000 U.S. stocks and use their one-year returns over the following year to compute the returns of the various strategies described below. Altogether there are 5384 different stocks which were, at the beginning of some month during this 49-year period, among the top 1000 stocks by market capitalization in the U.S.
We do not replicate all the strategies tested by [@Arnott:2013] but instead choose 5 representative naïve strategies. These 5 buy-and-hold strategies begin each one-year period with the following weights:
1. [**Capitalization-weighted (CW):**]{} stock weights proportional to their market capitalization.
2. [**Equal-weighted (EW):**]{} weight of each stock = $1/1000$.
3. [**Large-overweighted (LO):**]{} stock weights proportional to the square of their market capitalization.
4. [**Random-weighted (RW):**]{} weights proportional to $[0,1]$–uniformly distributed random variables.
5. [**Inverse-random-weighted (IRW):**]{} weights proportional to the reciprocals of $[0,1]$–uniformly distributed random variables.
All weights are always normalized to sum to 1.
The capitalization-weighted strategy corresponds to holding the market.
The equal-weighted strategy splits the capital at the beginning of each one-year period equally among whatever the top 1000 names are at that point in time.
The large-overweighted strategy is not tested by [@Arnott:2013]. This strategy puts a higher proportion than the index into the larger stocks and a smaller proportion into the smaller stocks.[^8] This portfolio is even less diversified than the capitalization-weighted portfolio, and according to our thesis we would therefore expect it to underperform the market, that is to say, to have negative excess return by virtue of its lower excess growth component.
The random-weighted and inverse-random-weighted strategies are our version of the ‘monkey’ and ‘upside-down’ portfolios in [@Arnott:2013].
To avoid exposure to random draws we simulate 1000 such portfolios and report the median values in Table \[T:1\] below.
Each of the five columns in the table corresponds to one of the five strategies described above. We show the total logarithmic return for each strategy, and then decompose the total return into the two components we have discussed extensively in this paper: the average growth component and the excess growth component. In all cases we show both the absolute values as well as the relative values compared to those of the cap-weighted portfolio. Finally we show the arithmetic absolute and relative returns, the standard deviation of the arithmetic returns, and the associated Sharpe ratios.
----------------------------------------------------------------------------------------------- -- -- -- -- --
**& [**CW**]{}(%) & [**EW**]{}(%) & [**LO**]{}(%) &[**RW**]{}(%) & [**IRW**]{}(%)\
& [9.12]{} & [10.98]{} & [7.46]{} &[10.98 ]{} &[10.46]{}\
relative to cap-weighted index & &1.86 &-1.66 &1.86 &1.34\
&[5.57]{}& [5.64]{} &[5.36]{} &[5.65]{} &[5.67]{}\
relative to cap-weighted index &&.07 &-.21 &.08 &.10\
**[Excess growth component]{} & **[3.87]{} & **[5.82]{} & **[2.19]{} &**[5.82]{} &**[5.18]{}\
relative to cap-weighted index & &1.95 &-1.68 &1.95 &1.31\
Total arithmetic return & 10.97&13.33 & 9.15 &13.33 &13.34\
relative to cap-weighted index & &2.36 & -1.82 &2.36 &2.37\
Standard deviation& 17.07&19.14 &16.90 &19.07 &22.35\
Sharpe ratio&.29&.38 &.18 &.38 &.32\
**************
----------------------------------------------------------------------------------------------- -- -- -- -- --
\[T:1\]
The equal-weighted, random-weighted and inverse-random-weighted portfolios all outperform the capitalization-weighted portfolio. The large-overweighted strategy, on the other hand, underperforms. These results are consistent with our understanding that that the first three strategies have greater diversification than the index, while the latter is less well-diversified.
Importantly we also note that most of the differences in the strategies’ returns can be explained by differences in the excess growth component. Indeed the differences in the average stock growth component are of a much lesser magnitude, which is consistent with the analysis in Figure \[f1\] in which we demonstrated that the average individual stock growth rates were essentially the same.
Conclusion
===========
The outperformance of a range of different strategies of the type presented in [@Arnott:2013] can be explained using concepts from Stochastic Portfolio Theory.
The logarithmic return of a portfolio can be decomposed into two elements. The first term represents the weighted average of the logarithmic returns of the stocks. The second term measures the excess growth, the additional component of portfolio return arising from the benefits of diversification. This term only depends on the variances and covariances of the portfolio’s constituents, and is larger for more diversified portfolios.
Taking a rank-based view of stock returns we argued empirically that the contribution of the weighted average of the logarithmic stock returns is approximately the same for all portfolios. What varies much more from portfolio to portfolio is the excess growth component, which depends only on stocks’ variances and covariances.
Studying the performance of several different trading strategies, some more diversified and some less diversified than the capitalization-weighted portfolio, confirmed these insights. In general, the more diversified portfolios outperform and the single less diversified portfolio underperforms, because the more diversified portfolios have a higher excess growth rate. This arises from the higher variances associated with the smaller stock exposure in these more diversified portfolios, and not because such stocks have inherently higher returns. This higher excess growth rate, in turn, increases the portfolios’ logarithmic returns.
All in all, this helps to explain the ‘surprising’ alpha found by [@Arnott:2013] in a variety of strategies, without the need to invoke factors.
Appendices {#appendices .unnumbered}
==========
The dynamics of arithmetic return and log-return {#S:2.2}
================================================
Let $X(t)$ represent the price of a stock at time $t$. The standard continuous-time model for the behavior of this price is an process that satisfies $$\frac{dX(t)}{X(t)}=\a(t) \d t+\s(t) \d W(t),$$ where $\a(t)$ is the [*rate of return process*]{} of $X$ at time $t$, $\s^2(t)>0$ is the [*variance rate process,*]{} and $W$ is a Brownian motion process. For simplicity we assume that $\s^2(t)$ is bounded.
With $X$ as above, ’s rule (see [@KS1]) implies that $$\begin{aligned}
\d\log X(t)
&= \frac{\d X(t)}{X(t)}-\half\s^2(t) \d t\\
&= \Big(\a(t)-\half\s^2(t)\Big)\d t+ \s(t)\,\d W(t)\\
&= \g(t)\,\d t + \s(t)\,\d W(t).\phantom{\half}\end{aligned}$$ The process $\displaystyle{\g(t)=\a(t)-\half\s^2(t)}
$ is called the [*growth rate process*]{} (see [@Fernholz:Shay:1982]) or the [*log-return process*]{} of $X$. This explains the display in Section \[S:3\_new\].
Under mild regularity conditions it can be shown that $$\limT{1}\Big( \log X(T)- \intT\g(t)\d t\Big)=0,\as$$ This confirms the claim made in Section \[S:2\_new\] that logarithmic return is an unbiased estimator of long-term growth. Moreover, the fact that $\gamma(t) \leq \alpha(t)$ is consistent with the fact that logarithmic return $\leq$ arithmetic return.
Portfolio return and log-return – the mathematics {#A:B}
=================================================
In this appendix, we provide the mathematical formulas for the statements in Section \[S:3\_new\].
Suppose we have stocks $X_1,\ldots,X_n$ and a portfolio $\p$ with weights $\p_1(t)+\cdots+\p_n(t)=1$ and value $Z_\p(t)$ at time $t$. Then the portfolio return satisfies $$\frac{\d Z_\p(t)}{Z_\p(t)}= \sumi\p_i(t)\,\frac{\d X_i(t)}{X_i(t)}$$ according to [@Markowitz:1952]. The analogous equation for the portfolio log-return is $$\label{eq:171103.1}
\d\log Z_\p(t)=\sumi\p_i(t)\d\log X_i(t) + \g^*_\p(t)\d t,$$ where $\g_\p^*(t)$ denotes the [*excess growth rate (EGR) process*]{} of the portfolio. More precisely, if we denote the portfolio variance rate process by $\s^2_\pi(t)$, then we have $$\begin{aligned}
\d \log Z_\p(t) &=\frac{\d Z_\p(t)}{Z_\p(t)}-\half\s^2_\p(t)\d t\\
& = \sumi\p_i(t)\,\frac{\d X_i(t)}{X_i(t)}-\half\s^2_\p(t)\d t\\
& = \sumi\p_i(t)\Big(\d\log X_i(t)+\half\s^2_i(t)\,dt\Big)-\half\s^2_\p(t)\d t\\
&= \sumi\p_i(t)\d\log X_i(t)+\half\Big(\sumi\p_i(t)\s^2_i(t) - \s^2_\p(t)\Big)\d t\\
&= \sumi\p_i(t)\d\log X_i(t)+\g^*_\p(t)\d t;\end{aligned}$$ hence $$\label{eq:180512}
\g^*_\p(t)=\half\Big(\sumi\p_i(t)\s^2_i(t) - \s^2_\p(t)\Big).$$ This equality corresponds to the last display in Section \[S:3\_new\].
Estimation of expected portfolio log-return with a rank-based stock analysis – the mathematics {#A:C}
==============================================================================================
We shall use the notation of Appendix \[A:B\]. For the interval $[0,T]$, yields $$\begin{aligned}
\text{portfolio log-return } &=\int_{0}^{T} \sumi \p_i(t)\, \d\log X_i(t)+ \int_{0}^{T} \g^*_\p(t) \d t = \text{A}_\p(T) +\Gamma_\p(T) \end{aligned}$$ where $$\text{A}_\p(T) = \int_{0}^{T} \sumi \p_i(t)\, \d\log X_i(t)$$ is called the [*average growth*]{} component, representing the weighted average growth rate of the stocks in the portfolio, and $$\Gamma_\p(T) = \int_{0}^{T} \g^*_\p(t) \d t$$ is called the [*excess growth*]{} component.
To describe the rank-based method mathematically, used to determine the value of the average growth component, let $r_t(i)$ be the rank of $X_i(t)$. That is, if $i$ corresponds to the company with the largest capitalization at time $t$, then $r_t(i) = 1$. Similarly, if $i$ is the $k$-largest company at time $t$, we have $r_t(i) = k$. This notation allows us to define the [*average rank-based growth rates*]{} $\bg_k$ over $[0,T]$ by $$\label{3}
\bg_k = \frac{1}{T} \int_{0}^{T}\sumi \1_{\{r_t(i)=k\}}\d\log X_i(t).$$ In a stable system the time-averaged value is equal to the expected value, so that $$\E\big[\d\log X_i(t) \big| r_t(i)=k\big] \,=\, \bg_k\d t,$$ or $$\E\big[\d\log X_i(t) \big]\,=\, \E\big[\bg_{r_t(i)}]\d t.$$ The definition in can be used directly to estimate the values of the $\bg_k$, and these estimated values for the period from 1964 to 2012 appear in Figure \[f1\] above.
Since the values seem to be roughly the same we shall assume, for the moment, that $\bg_k = \bg$, which then yields $$\E\big[\d\log X_i(t) \big]\,=\, \bg \d t.$$
We can now use these values of $\bg_k$ to estimate the expected average growth component over the period studied. Indeed, $$\begin{aligned}
\E\big[\text{A}_\p(T)\big]
&= \E\Bigg[ \int_{0}^{T} \sumi \p_i(t)\d\log X_i(t)\Bigg]\notag\\
&= \int_{0}^{T} \sumi \E\big[ \p_i(t)\d\log X_i(t)\big]\notag\\
& \simeq \int_{0}^{T}\sumi \E\big[\p_i(t)\big]\E\big[\d\log X_i(t)\big]\label{4}\\
&= \int_{0}^{T}\sumi\E\big[\p_i(t)\big]\bg \d t,\label{5}
\end{aligned}$$ where the approximate equality in is justified by the fact that $\p$ is naïve, and hence the the weight $\p_i(t)$ and the return on the weight $\d\log X_i(t)$ should be independent. If this were a portfolio constructed by a skilled stock picker, then we would expect a positive correlation between these two quantities. From Figure \[f1\] it appears that there is not much difference among the ranked stock growth rates, so implies that the portfolio’s expected average growth component should be about the same for all naïve portfolios.
Data sources {#A:data}
============
The stock price data for the figures and the backtesting of the strategies are provided by CRSP. In order to be as close as possible to the experimental setup of [@Arnott:2013], we use data only from 1964 to 2012. However, we have tested all results with data up to 2017, and they are robust.
There are two returns of -100% in the dataset. For the charts of Figures \[f1\] and \[f2\] and for computing $A_\pi$ in Table \[T:1\], these two returns are changed to -95%. Otherwise, the corresponding log-returns would be $-\infty$ and the corresponding ranks would not have finite sample averages and variances. The results of this paper are robust with respect to the choice of the number -95%; other choices would lead to basically the same results.
For a few data points, returns were missing due to incomplete delisting information. We tested the results with different inputs, and the results were robust.
For computing the Sharpe ratio in Table \[T:1\] and the necessary excess returns, we used the one-year U.S. Treasury yields, publicly available on <https://www.federalreserve.gov/pubs/feds/2006/200628/200628abs.html>; see also [@GSW].
[^1]: Intech, 1 Palmer Square, Princeton, NJ 08542.
[^2]: Department of Mathematics, London School of Economics and Political Science, London WC2A 2AE.
[^3]: Intech, 201 Bishopsgate, London EC2M 3AE.
[^4]: Mathematically, these inequalities can be proven via an application of Jensen’s inequality.
[^5]: For a more detailed discussion and derivation of this relationship see Appendix \[S:2.2\], ‘The dynamics of arithmetic return and log-return’.
[^6]: For a more mathematical discussion of the results in this section, see Appendix \[A:B\], ‘Portfolio return and log-return – the mathematics’.
[^7]: For a more mathematical discussion of these procedure, see Appendix \[A:C\], ‘Estimation of expected portfolio log-return with a rank-based stock analysis – the mathematics’.
[^8]: To see this, consider a market with two stocks with relative market capitalizations $\mu$ and $\nu$, where $\nu < \mu$ and $\mu + \nu = 1$. Then $\nu^2 < \mu \nu$; hence the large-overweighted strategy puts the proportion $$\begin{aligned}
\frac{\mu^2}{\mu^2 + \nu^2} > \frac{\mu^2}{\mu^2 + \mu \nu} = \frac{\mu}{\mu + \nu} = \mu\end{aligned}$$ of the wealth into the larger stock.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'An elementary proof of a theorem of Montanucci and Zini on the automorphism group of generalized Aritn–Schreier–Mumford curves is presented, with the argument of Korchmáros and Montanucci for Artin–Schreier–Mumford curves being improved. Although the characteristic of a ground field is assumed to be [*odd*]{} in the article of Montanucci and Zini, the proof in the present article is applicable to the case of characteristic two also. As an application of the theorem of Montanucci and Zini, the arrangement of Galois points or Galois lines for the generalized Artin–Schreier–Mumford curve is determined.'
address: 'Department of Mathematical Sciences, Faculty of Science, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata 990-8560, Japan'
author:
- Satoru Fukasawa
title: 'A theorem of Montanucci and Zini for generalized Artin–Mumford curves and its application to Galois points'
---
Introduction
============
The Artin–Schreier–Mumford (ASM) curve over an algebraically closed field $k$ of characteristic $p>0$ is the smooth model of the plane curve defined by $$(x^{p^e}-x)(y^{p^e}-y)=c,$$ where $e>0$, $p^e >2$ and $c \in k \setminus \{0\}$. This curve is important in the study of the automorphism groups of algebraic curves, since this is an ordinary curve and its automorphism group is large compared to its genus (see [@subrao]). The ASM curve is generalized as the smooth model $X$ of the curve $C$ defined by $$L_1 (x)\cdot L_2(y)+c=0,$$ where $c \in k \setminus \{0\}$ and $L_1$ and $L_2$ are linearlized polynomials of degree $p^e$, that is, $$L_i=\alpha_{i e} x^{p^e}+\alpha_{i e-1}x^{p^{e-1}}+\cdots+\alpha_{i 0} x$$ for some $\alpha_{i e}, \alpha_{i e-1}, \ldots, \alpha_{i 0} \in k$ with $\alpha_{i e}\alpha_{i 0} \ne 0$, for $i=1, 2$. We can assume that $\alpha_{1 e}=\alpha_{2 e}=1$ for a suitable system of coordinates. This curve was studied by the present author [@fukasawa1; @fukasawa2] (for the case $L_1=L_2$), and by Montanucci and Zini [@montanucci-zini]. The curve $X$ is called a [*generalized Artin–Mumford curve*]{} in [@montanucci-zini]. The automorphism group ${\rm Aut}(X)$ of $X$ is completely determined by Montanucci and Zini [@montanucci-zini Theorems 1.1 and 1.2], as follows.
\[montanucci and zini\] Assume that $p >2$. Let $\mathbb{F}_{p^{k}}=\bigcap_{i >0, \alpha_{1 i} \ne 0}\mathbb{F}_{p^i} \cap \bigcap_{j>0, \alpha_{2 j} \ne 0}\mathbb{F}_{p^j}$.
- If $L_1 = L_2$, then ${\rm Aut}(X) \cong \Sigma \rtimes D_{p^{k}-1}$, where $\Sigma$ is an elementary abelian $p$-group of order $p^{2e}$ and $D_{p^k-1}$ is the dihedral group of order $2(p^k-1)$.
- If $L_1 \ne L_2$, then ${\rm Aut}(X) \cong \Sigma \rtimes \mathbb{F}_{p^{k}}^*$.
It is assumed that the characteristic is [*odd*]{} in [@montanucci-zini]. One key point to prove is [@montanucci-zini Lemma 3.1 v) and Corollary 3.2], which asserts that a Sylow $p$-subgroup of ${\rm Aut}(X)$ is linear and acts on $\Omega_1 \cup \Omega_2$, where the set $\Omega_1$ (resp. $\Omega_2$) consists of all poles of $x$ (resp. of $y$). This assertion relies on a theorem of Nakajima [@nakajima Theorem 1] on relations between the $p$-rank and Sylow $p$-subgroups of the automorphism group of algebraic curves. Another key point is that the genus $(p^e-1)^2$ of $X$ is [*even*]{} if $p>2$, because Montanucci and Zini used some group-theoretic lemmas by a work of Giulietti and Korchmáros [@giulietti-korchmaros] for algebraic curves of [*even*]{} genus.
An alternative proof of Fact \[montanucci and zini\] for the ASM curve was obtained by Korchmáros and Montanucci [@korchmaros-montanucci]. It was proved that the linear system induced by some embedding into $\mathbb{P}^3$ is complete, and asserted that ${\rm Aut}(X)$ acts on $\Omega_1 \cup \Omega_2$, by using its completeness. We will prove the same things for generalized Artin–Schreier–Mumford curves in a different order. It was pointed out by Garcia [@garcia2] (see also [@boseck; @garcia1]) that points of $\Omega_1 \cup \Omega_2$ are Weierstrass points (see Lemma \[weierstrass\] for a more precise statement), and this implies that ${\rm Aut}(X)$ acts on $\Omega_1 \cup \Omega_2$. We reprove it. We also present an elementary proof of the completeness of the linear system for generalized ASM curves (Lemma \[complete\]). With these two results combined, an inclusion ${\rm Aut}(X) \hookrightarrow PGL(4, k)$ is obtained (Corollary \[extendable in P\^3\]). More strongly:
\[extendable in P\^2\] There exists an injective homomorphism $${\rm Aut}(X) \cong {\rm Bir}(C) \hookrightarrow PGL(3, k).$$
This is very close to the theorem of Montanucci and Zini. Therefore:
\[char 2\] The same assertion as Fact \[montanucci and zini\] holds for the case where $p=2$.
As an application of the theorem of Montanucci and Zini, the arrangement of Galois points or Galois lines for the generalized Artin–Schreier–Mumford curve is determined in Sections 3 and 4.
Proof of Theorems \[extendable in P\^2\] and \[char 2\]
=======================================================
The system of homogeneous coordinates on $\mathbb{P}^2$ is denoted by $(X:Y:Z)$ and the system of affine coordinates of $\mathbb{A}^2$ is denoted by $(x, y)$ with $x=X/Z$ and $y=Y/Z$.
Let $q=p^e$. The set of all poles of $x$ (resp. of $y$) is denoted by $\Omega_1$ (resp. by $\Omega_2$), which coincides with the set of all zeros of $L_2(y)$ (resp. of $L_1(x)$). The sets $\Omega_1$ and $\Omega_2$ consist of $q$ points. The pole of $x$ (resp. of $y$) corresponding to $y=\beta$ (resp. $x=\alpha$) for $L_2(\beta)=0$ (resp. $L_1(\alpha)=0$) is denoted by $P_{\beta}$ (resp. by $Q_\alpha$). For the point $P_{\beta}$, $t=\frac{1}{x}$ is a local parameter. Let $P'=(1:0:0)$ and $Q'=(0:1:0) \in \mathbb{P}^2$. Then, ${\rm Sing}(C)=\{P', Q'\}$, and the point $P'$ (resp. the point $Q'$) is the image of $\Omega_1$ (resp. of $\Omega_2$) under the normalization.
Let $D$ be the divisor given by the line $\{Z=0\}$ of the plane model. It is known that the genus $g$ of $X$ is equal to $(q-1)^2$ (see [@montanucci-zini Lemma 3.1], [@stichtenoth III. 7.10]). Therefore, the degree of the canonical divisor is $2g-2=2q(q-2)$. We consider the linear space $\mathcal{L}((q-2)D)$ associated with the divisor $(q-2)D$. The following two lemmas were proved by Boseck [@boseck] and Garcia [@garcia2] in a more general setting (see also [@garcia1]). We reprove them, for the convenience of the readers.
\[canonical\] A divisor $(q-2)D$ is a canonical divisor, and $$\mathcal{L}((q-2)D)=\langle x^i y^j \ | \ 0 \le i, j \le q-2 \rangle.$$
Each element of the linear space $\mathcal{L}((q-2)D)$ is represented by a polynomial of $x$ and $y$, since each function $g \in \mathcal{L}((q-2)D)$ is regular on the affine open set $C \cap \{Z \ne 0\}$. Using the defining polynomial, since $$x^q y^q=\sum_{i \le q, j \le q, (i, j) \ne (q, q)} a_{ij} x^i y^j$$ in $k(X)$, it follows that any $g \in \mathcal{L}((q-2)D)$ is represented as a linear combination of monomials $$x^i y^j \ \mbox{ with } i<q \ \mbox{ or } \ j <q.$$
Assume that there exists an element $g=\sum_{i=0}^m a_i(y) x^i \in \mathcal{L}((q-2)D)$ with $m \ge q$ and $a_m(y) \ne 0$. Then, $\deg a_m(y) <q$. This implies that there exists a pole $P$ of $x$ such that ${\rm ord}_P a_m(y)=0$. Then, ${\rm ord}_P g=-m \le -q< -(q-2)$. This is a contradiction to $g \in \mathcal{L}((q-2)D)$. It follows that any $g \in \mathcal{L}((q-2)D)$ is represented as a linear combination of monomials $$x^i y^j \ \mbox{ with } i<q \ \mbox{ and } \ j <q.$$
Let $g=\sum_{i=0}^m a_i(y) x^i$ with $m < q$. Since $\deg a_m(y) <q$, there exists a pole $P$ of $x$ such that ${\rm ord}_P a_m(y)=0$. Then, ${\rm ord}_P g=-m$. By the condition $g \in \mathcal{L}((q-2)D)$, $-m ={\rm ord}_P g \ge -(q-2)$. It follows that any $g \in \mathcal{L}((q-2)D)$ is represented as a linear combination of monomials $$x^i y^j \ \mbox{ with } i \le q-2 \ \mbox{ and } \ j \le q-2.$$ These monomials are linearly independent, since $\deg C=2q$. Since $\deg (q-2)D=2g-2$ and $\dim \mathcal{L}((q-2)D) =g$, it follows from [@stichtenoth I.6.2] that the assertion follows.
\[weierstrass\] The set $\Omega_1 \cup \Omega_2$ coincides with $\{P \in X \ | \ q \in H(P)\}$, where $H(P)$ is the Weierstrass semigroup of $P$. In particular, all points of $\Omega_1 \cup \Omega_2$ are Weierstrass points.
We consider the embedding $\psi$ induced by the canonical linear system $|(q-2)D|$. Let $P \in \Omega_1$ and let $t=(1/x)$. Then, $t$ is a local parameter at $P$. Note that ${\rm ord}_P (y-\beta)=q$ for some $\beta \in k$. Considering the functions $t^k y^j \in \mathcal{L}((x^{q-2})+(q-2)D)$, it follows that the orders ${\rm ord}_P\psi^*H$ for hyperplanes $H \ni P$ are $$1, 2, \ldots, q-2, q, \ldots,$$ namely, $q-1+1$ is a [*non-gap*]{} (of pole numbers). On the other hand, ${\rm ord}_{(a, b)}(x-a)^{q-2}(y-b)=q-1$ for each point $(x, y)=(a, b)$ of $X \setminus (\Omega_1 \cup \Omega_2)$, since functions $x-a$ and $y-b$ are local parameters.
\[action\] The automorphism group ${\rm Aut}(X)$ preserves $\Omega_1 \cup \Omega_2$.
We consider the morphism $$\varphi: X \rightarrow \mathbb{P}^3; \ (x:y:1:x y),$$ similar to the case of the ASM curve (see [@korchmaros-montanucci]). For the point $P_{\beta} \in \Omega_1$ defined by $L_2(\beta)=0$, $t=\frac{1}{x}$ is a local parameter at $P_{\beta}$. It follows that $$\varphi=(x:y:1:x y)=(t x: t y: t: t x y)=(1:t y: t: y),$$ and $\varphi(P_{\beta})=(1:0:0:\beta)$. Therefore, $q$ points of $\varphi(\Omega_1)$ are contained in the line $Y=Z=0$ in $\mathbb{P}^3$ with a system of coordinates $(X:Y:Z:W)$. Similarly, $q$ points of $\varphi(\Omega_2)$ are contained in the line $X=Z=0$.
\[complete\] The morphism $\varphi$ is an embedding, and the linear system induced by $\varphi$ is complete.
The former assertion is derived from the fact that the set $\varphi(\Omega_1 \cup \Omega_2)=\varphi(X)\cap \{Z=0\}$ consists of $2q$ points (this proof is similar to [@korchmaros-montanucci]). We consider the latter assertion. It follows that $1, x, y, x y \in \mathcal{L}(D)$. Each element of the linear space $\mathcal{L}(D)$ is represented by a polynomial of $x$ and $y$, since each function $g \in \mathcal{L}(D)$ is regular on the affine open set $C \cap \{Z \ne 0\}$. Using the defining polynomial, since $$x^q y^q=\sum_{i \le q, j \le q, (i, j) \ne (q, q)} a_{ij} x^i y^j$$ in $k(X)$, it follows that any $g \in \mathcal{L}(D)$ is represented as a linear combination of monomials $$x^i y^j \ \mbox{ with } i<q \ \mbox{ or } \ j <q.$$
Assume that $g=\sum_{i=0}^m a_i(y) x^i$ with $m \ge q$ and $a_m(y) \ne 0$. Then, $\deg a_m(y) <q$. This implies that there exists a pole $P$ of $x$ such that ${\rm ord}_P a_m(y)=0$. Then, ${\rm ord}_P g=-m \le -q< -1$. This is a contradiction to $g \in \mathcal{L}(D)$. It follows that any $g \in \mathcal{L}((D)$ is represented as a linear combination of monomials $$x^i y^j \ \mbox{ with } i<q \ \mbox{ and } \ j <q.$$
Let $g=\sum_{i=0}^m a_i(y) x^i$ with $m < q$ and $a_m(y) \ne 0$. Since $\deg a_m(y) <q$, there exists a pole $P$ of $x$ such that ${\rm ord}_P a_m(y)=0$. Then, ${\rm ord}_P g=-m$. By the condition $g \in \mathcal{L}(D)$, $-m ={\rm ord}_P g \ge -1$. It follows that any $g \in \mathcal{L}(D)$ is represented as a linear combination of monomials $ 1, x, y, x y$.
\[extendable in P\^3\] There exists an injective homomorphism $${\rm Aut}(X) \hookrightarrow PGL(4, k).$$
By Corollary \[action\], $\sigma^*D=D$ for each $\sigma \in {\rm Aut}(X)$. By Lemma \[complete\], $\dim |D|=3$. The assertion follows.
Using Corollaries \[action\] and \[extendable in P\^3\], we prove Theorem \[extendable in P\^2\].
Note that $\varphi(\Omega_1)$ and $\varphi(\Omega_2) \subset \mathbb{P}^3$ are contained in lines $Y=Z=0$ and $X=Z=0$ respectively. The point $R=(0:0:0:1)$ given by the intersection of such lines is fixed by each element of ${\rm Aut}(X)$. Then, ${\rm Aut}(X)$ acts on the linear subspace $\langle x, y, 1 \rangle \subset \mathcal{L}(D)$.
The image of the injective homomorphism described in Theorem \[extendable in P\^2\] is denoted by ${\rm Lin}(X)$. Let $$\Sigma:=\{\sigma_{\alpha, \beta}: (x, y) \mapsto (x+\alpha, y+\beta) \ | \ L_1(\alpha)=0, L_2(\beta)=0\} \subset PGL(3, k),$$ $$\Gamma:=\{\theta_\lambda: (x, y) \mapsto (\lambda x, \lambda^{-1} y) \ | \ \lambda \in \mathbb{F}_{p^k}^*\} \subset PGL(3, k),$$ and let $\tau \in PGL(3, k)$ be defined by $\tau(x, y)=(y, x)$. It follows that $\langle \Sigma, \Gamma \rangle \subset {\rm Lin}(X)$. If $L_1 =L_2$, then $\langle \Sigma, \Gamma, \tau \rangle \subset {\rm Lin}(X)$.
We prove that $\langle \Sigma, \Gamma \rangle= {\rm Lin}(X)$ if $L_1 \ne L_2$, and that $\langle \Sigma, \Gamma, \tau \rangle={\rm Lin}(X)$ if $L_1=L_2$. Let $P'=(1:0:0)$ and let $Q'=(0:1:0)$. Then, ${\rm Lin}(X)$ acts on ${\rm Sing}(C)=\{P', Q'\}$. Note that all tangent lines at $P'$ (resp. at $Q'$) are defined by $Y-\beta Z=0$ (resp. $X-\alpha Z=0$) for some $\beta \in k$ with $L_2(\beta)=0$ ($\alpha \in k$ with $L_1(\alpha)=0$). Since ${\rm Lin}(X)$ acts on the set of tangent lines at $P'$ or at $Q'$, it follows that there exists $\tau' \in {\rm Lin}(X)$ such that $\tau'(P')=Q'$ and $\tau'(Q')=P'$ if and only if $L_1=L_2$. Therefore, we prove that if $\sigma \in {\rm Lin}(X)$, $\sigma(P')=P'$ and $\sigma(Q')=Q'$, then $\sigma \in \langle \Sigma, \Gamma \rangle$.
Assume that $\sigma \in {\rm Lin}(X)$, $\sigma(P')=P'$ and $\sigma(Q')=Q'$. Then, $\sigma$ is represented by a matrix $$A_{\sigma}=\left(
\begin{array}{ccc}
a & 0 & c \\
0 & b & d \\
0 & 0 & 1
\end{array} \right)$$ for some $a, b, c, d \in k$. Let $\beta \in k$ (resp. $\alpha \in k$) be a root of $L_2$ (resp. $L_1$). Then, the image of the tangent line $Y-\beta Z=0$ (resp. $X-\alpha Z=0$) under $\sigma$ is some tangent line $Y-\beta'Z=0$ (resp. $X-\alpha' Z=0$). Then, an automorphism $\sigma_1:=\sigma_{\alpha-\alpha', \beta-\beta'} \circ \sigma$ fixes the tangent lines $Y-\beta Z=0$ and $X-\alpha Z=0$. Therefore, this automorphism represented by $$A_{\sigma_1}=\left(
\begin{array}{ccc}
a & 0 & \alpha (1-a) \\
0 & b & \beta (1-b) \\
0 & 0 & 1
\end{array} \right).$$ Since $$\sigma_1^* (L_1(x)\cdot L_2(y)+c)=\left(\sum_{i}a^{p^i}\alpha_{1 i}x^{p^i}+c_1\right)\left(\sum_{j}b^{p^j}\alpha_{2 j}y^{p^j}+c_2\right)+c=L_1(x)\cdot L_2(y)+c$$ up to a constant for some $c_1, c_2 \in k$, it follows that $c_1=c_2=0$, $ab=1$ and $a,b \in \mathbb{F}_{p^k}^*$. For an automorphism $\theta_{a^{-1}} \in \Gamma$, $\sigma_2:=\theta_{a^{-1}}\circ\sigma_{\alpha-\alpha', \beta-\beta'} \circ \sigma$ is represented by $$A_{\sigma_2}=\left(
\begin{array}{ccc}
1 & 0 & \alpha(a^{-1}-1) \\
0 & 1 & \beta (a-1) \\
0 & 0 & 1
\end{array} \right).$$ This implies that $\theta_{a^{-1}}\circ\sigma_{\alpha-\alpha', \beta-\beta'} \circ \sigma \in \Sigma$, namely, $\sigma \in \langle \Sigma, \Gamma \rangle$.
Application to the arrangement of Galois points
===============================================
A point $R \in \mathbb{P}^2 \setminus C$ is called an outer Galois point for a plane curve $C \subset \mathbb{P}^2$ if the function field extension $k(C)/\pi_R^*k(\mathbb{P}^1)$ induced by the projection $\pi_R$ from $R$ is Galois (see [@miura-yoshihara; @yoshihara]). Furthermore, an outer Galois point is said to be extendable if each element of the Galois group is the restriction of some linear transformation of $\mathbb{P}^2$ (see [@fukasawa1]). The number of outer Galois points (resp. of extendable outer Galois points) is denoted by $\delta(C)$ (resp. by $\delta_0'(C)$).
In this section, we consider outer Galois points for the plane model $C$ of the genelazied Artin–Schreier–Mumford curve $X$. According to Theorem \[extendable in P\^2\], $\delta(C)=\delta_0'(C)$. It was proved by the present author that for the case where $L_1=L_2$, $\delta_0'(C) \ge p^k-1$ and the equality holds if $p=2$ (see [@fukasawa1; @fukasawa2]). Therefore, it has been proved that $\delta(C)=\delta_0'(C)=p^k-1$ if $p=2$. The same holds for the case where $p>2$.
\[outer Galois\] If $L_1=L_2$, then $\delta'(C)=\delta_0'(C)=p^k-1$.
Let $R \in \mathbb{P}^2 \setminus C$ be an outer Galois point. Note that the line $\overline{RP'}$ corresponds to the fiber of the projection $\pi_R$, where $P'=(1:0:0) \in {\rm Sing}(C)$. If $R \not \in \{Z=0\}$, then $\pi_R$ is ramified at each point of $\Omega_1$, since the Galois group $G_R$ acts on $\Omega_1 \cup \Omega_2$ (see [@stichtenoth III.7.1, III.7.2]). However, the directions of the tangent lines at $P'$ are different. This is a contradiction. Therefore, $R \in \{Z=0\}$.
We can assume that $p>2$. Since $|G_R|=2q$, there exists an involution $\tau' \in G_R \subset {\rm Lin}(X)$. If $\tau'(P')=(P')$, then $\tau'$ fixes some point of $\Omega_1$. This is a contradiction to the transitivity of $G_R$ on fibers (see [@stichtenoth III.7.1]). Therefore, $\tau'(P')=Q'$ and $\tau'(Q')=P'$, where $Q'=(0:1:0) \in {\rm Sing}(C)$. Considering the elements of ${\rm Lin}(X)=\langle \Sigma, \Gamma, \tau \rangle$ described in the previous section, $\tau'$ is given by $$(X: Y: Z) \mapsto (\lambda Y: \lambda^{-1}X: Z)$$ for some $\lambda \in \mathbb{F}_{p^{k}}^*$. Then, fixed points of $\tau'$ on $\{Z=0\}$ are $(\lambda:1:0)$ and $(-\lambda:1:0)$. Note that any element of $G_R$ fixes $R$, since $G_R$ preserves any line passing through $R$. Therefore, $R=(\lambda: 1: 0)$ or $(-\lambda:1:0)$. The claim follows.
Let $R_1, \ldots, R_{p^k-1}$ be all outer Galois points for $C$ and let $G_{R_1}, \ldots, G_{R_{p^k-1}}$ be their Galois groups. Then, ${\rm Aut}(X)=\langle G_{R_1}, \ldots, G_{R_{p^k-1}} \rangle$.
For the case where $L_1 \ne L_2$, the following holds.
\[outer Galois 2\] If $L_1 \ne L_2$, then $\delta'(C)=0$.
Let $R \in \mathbb{P}^2 \setminus C$ be an outer Galois point. Since the Galois group $G_{R}$ acts on $\Omega_1$ and $|G_R|=2q$, the projection $\pi_R$ is ramified at each point of $\Omega_1$ (see [@stichtenoth III.7.1, III.7.2]). However, the directions of the tangent lines at $P'$ are different. This is a contradiction.
The generalized Artin–Schreier–Mumford curve with $L_1 \ne L_2$ does not belong to families studied by the present author in [@fukasawa1; @fukasawa2], since some subgroup of the automorphism group of the families acts on the set defined by $Z=0$ transitively.
Application to the arrangement of Galois lines
==============================================
A line $\ell \subset \mathbb{P}^3$ is called a Galois line for a space curve $X \subset \mathbb{P}^3$ if the function field extension $k(X)/\pi_\ell^*k(\mathbb{P}^1)$ induced by the projection $\pi_{\ell}$ from $\ell$ is Galois (see [@duyaguit-yoshihara; @yoshihara2]). In this section, we consider Galois lines for a space model $\varphi(X) \subset \mathbb{P}^3$ of the generalized Aritn–Schreier–Mumford curve $X$, where $$\varphi: X \rightarrow \mathbb{P}^3; \ (x: y: 1: x y).$$ For the case where $\mathbb{F}_{p^k}=\mathbb{F}_{p^e}$, that is, $L_1=L_2=x^q+x$ (for a suitable system of coordinates), the arrangement of Galois lines was determined in [@fukasawa3]. We can assume that $k <e$.
Assume that $L_1 \ne L_2$. Let $\ell \subset \mathbb{P}^3$ be a line. Then, $\ell$ is a Galois line for $\varphi(X)$ if and only if $\ell$ is defined by $a W-b X=a Y-b Z=0$ or $a W-b Y=a X-b Z=0$ for some $a, b \in k$ with $(a, b) \ne (0, 0)$.
The proof of the if-part is similar to [@fukasawa3], and is easily verified by a direct computation. Assume that $\ell \subset \mathbb{P}^3$ is a Galois line. If there exists a hyperplane $H \supset \ell \cup \Omega_1$, the claim follows, similar to [@fukasawa3 Lemma 1 (b)]. Assume that $H \not\supset \Omega_1$ and $H \not\supset \Omega_2$ for each hyperplane $H \supset \ell$. Note that the Galois group $G_{\ell}$ acts on each fiber transitively (see [@stichtenoth III.7.1]). Since $\sigma$ acts on $\Omega_1$ and $\Omega_2$, it follows that a fiber containing a point of $P \in \Omega_1$ does not contain a point of $\Omega_2$. Therefore, each hyperplane $H \supset \ell$ with $H \cap (\Omega_1 \cup \Omega_2) \ne \emptyset$, the corresponding fiber contains $P$ only, namely, $G_\ell$ fixes each point of $\Omega_1 \cup \Omega_2$. This is a contradiction to the fact that the action of ${\rm Aut}(X)$ on $\Omega_1 \cup \Omega_2$ is faithful.
Assume that $L_1=L_2$ and $k<e$. Let $\ell \subset \mathbb{P}^3$ be a line. Then, $\ell$ is a Galois line for $\varphi(X)$ if and only if $\ell$ is one of the following:
- an $\mathbb{F}_{p^k}$-line contained in $\{Z=0\}$ and passing through $(0:0:0:1)$, or
- the line defined by $W-a X=Y-a Z=0$ or $W-a Y=X-a Z=0$ for some $a\in k$.
The proof of the if-part is easily verified by a direct computation. Assume that $\ell$ is a Galois line. The proof for the case where $\ell \cap \varphi(X)=\emptyset$ and $\ell \not\ni (0:0:0:1)$ is similar to [@fukasawa3]. If $\ell \cap \varphi(X)=\emptyset $ and $\ell \ni (0:0:0:1)$, then such Galois lines correspond to Galois points in $\mathbb{P}^2$. According to Theorem \[outer Galois\], $\ell$ is an $\mathbb{F}_{p^k}$-line. If the degree of the projection from $\ell$ is $2q-1$, then $\ell$ is not a Galois line, by considering the orders $|G_{\ell}|$ and $|{\rm Aut}(X)|$.
Assume that $\ell$ is a tangent line or $\ell \cap \varphi(X)$ consists of at least two points. If $\ell \subset \{Z=0\}$, then there exists $P \in \Omega_1$ and $Q \in \Omega_2$ such that $\ell=\overline{PQ}$, where $\overline{PQ} \subset \mathbb{P}^3$ is a line passing through $P$ and $Q$. Then, $\deg \pi_{\ell}=2q-2=2(q-1)$. Since $|{\rm Aut}(X)|=2q^2(p^k-1)$, $\ell$ must not become a Galois line. Assume that $\ell \cap \{Z=0\}=\{R\}$. If $R$ is contained in the line spanned by $\Omega_1$ or $\Omega_2$, then the claim follows, by [@fukasawa3 Lemma 1 (b)]. Assume that $R$ is not contained in the lines spanned by $\Omega_1$ or $\Omega_2$. If there does not exist a pair of points $P, Q \in \Omega_1 \cup \Omega_2$ such that $P, Q \in H$, then $G_{\ell}$ fixes $\Omega_1 \cup \Omega_2$ pointwise. This is a contradiction. Therefore, there exist points $P \in \Omega_1$ and $Q \in \Omega_2$ such that $R \in \overline{PQ}$. Note that for any tangent hyperplane $H$ at $P$, ${\rm ord}_PH \ge q$. The same holds for $Q$. If $H$ contains the tangent lines at $P$ and at $Q$, then $\ell \cap \varphi(X)=\emptyset$. If $H$ is a tangent hyperplane at $P$, then $H$ is a tangent hyperplane at $Q$ also, by [@stichtenoth III.7.2]. We can assume that $(H \setminus \ell) \cap \varphi(X)=\{P, Q\}$. It follows that $\deg \pi_\ell=2$. According to Lemma \[canonical\], $X$ is not hyperelliptic. This is a contradiction.
[**Acknowledgments**]{}
The author is grateful to Doctor Kazuki Higashine for helpful discussions.
[100]{} H. Boseck, Zur Theorie der Weierstraßpunkte, Math. Nachr. [**19**]{} (1958), 29–63. C. Duyaguit and H. Yoshihara, Galois lines for normal elliptic space curves, Algebra Colloq. [**12**]{} (2005), 205–212. S. Fukasawa, Galois points for a plane curve in characteristic two, J. Pure Appl. Algebra [**218**]{} (2014), 343–353. S. Fukasawa, A family of plane curves with two or more Galois points in positive characteristic, Contemporary Developments in Finite Fields and Applications, 62–73, World Sci. Publ., 2016. S. Fukasawa, Galois lines for the Artin–Schreier–Mumford curve, preprint, arXiv:2005.10073. A. Garcia, On Weierstrass points on Artin–Schreier extensions of $k(x)$, Math. Nachr. [**144**]{} (1989), 233–239. A. Garcia, On Weierstrass points on certain elementary abelian extensions of $k(x)$, Comm. Algebra [**17**]{} (1989), 3025–3032. M. Giulietti and G. Korchmáros, Algebraic curves with many automorphisms, Adv. Math. [**349**]{} (2019), 162–211. G. Korchmáros and M. Montanucci, The geometry of the Artin–Schreier–Mumford curves over an algebraically closed field, Acta Sci. Math. (Szeged) [**83**]{} (2017), 673–681. K. Miura and H. Yoshihara, Field theory for function fields of plane quartic curves, J. Algebra [**226**]{} (2000), 283–294. M. Montanucci and G. Zini, Generalized Artin–Mumford curves over finite fields, J. Algebra [**485**]{} (2017), 310–331. S. Nakajima, $p$-ranks and automorphism groups of algebraic curves, Trans. Amer. Math. Soc. [**303**]{} (1987), 595–607. H. Stichtenoth, Algebraic Function Fields and Codes, Universitext, Springer-Verlag, Berlin (1993). D. Subrao, The $p$-rank of Artin–Schreier curves, manuscripta math. [**16**]{} (1975), 169–193. H. Yoshihara, Function field theory of plane curves by dual curves, J. Algebra [**239**]{} (2001), 340–355. H. Yoshihara, Galois lines for space curves, Algebra Colloq. [**13**]{} (2006), 455–469.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study associated production of prompt photon and two jets at high energies in the framework of the parton Reggeization approach, which is based on multi-Regge factorization of hard processes and Lipatov’s effective theory of Reggeized gluons and quarks. In this approach, initial-state off-shell effects and transverse momenta of initial partons are included in a gauge-invariant way. We compute azimuthal angle difference spectra in $\gamma + 2 jets$ events and compare results with data from D0 Collaboration at the Tevatron. It was found that agreement between predictions and data can be achieved only under assumption on strong violation in transverse momentum ordering during initial-state QCD evolution which corresponds multi-Regge kinematical regime instead of DGLAP picture.'
address:
- 'Department of Physics, Samara National Research University, Moskovskoe Shosse, 34, Samara, 443086, Russia'
- |
Department of Physics, Samara National Research University, Moskovskoe Shosse, 34, Samara, 443086, Russia\
and\
Joint Institute for Nuclear Research, Dubna 141980, Russia
author:
- 'Anton Karpishkov [^1], Vladimir Saleev [^2]'
- 'Alexandra Shipilova [^3]'
title: 'Angular decorrelations in $\gamma + 2 jets$ events at high energies in the Parton Reggeization Approach'
---
Introduction {#introduction .unnumbered}
============
Theoretical and experimental study of associated production of prompt photon and jets with large transverse momenta in high-energy hadronic collisions is a very important task for various reasons. First, this is a challenging test of our understanding of higher-order corrections in quantum chromodynamics (QCD). In general, it is a nontrivial task to provide reliable predictions for multiscale and correlational observables, based on the conventional Collinear Parton Model (CPM) of QCD. Second, correlational observables in photon plus jets production are primary tools in experimental searches of Double Parton Scattering(DPS) mechanism manifestations in hadron-hadron collisions at high energies [@DPS].
In the present paper we discuss associated production of prompt photon and two jets in $p\bar p$ collisions at the $\sqrt{S} = 1.96$ TeV which has been observed by the D0 Collaboration [@D0gamma2]. A comparison of theoretical predictions obtained in the next-to-leading order (NLO) approximation of CPM with the measured cross sections and the azimuthal angle difference distributions pointed towards DPS as the reliable source of events in the region of small $\Delta\phi$ [@D0gamma2].
However, there are many evidences that $k_T-$factorization approach of high-energy QCD is more adequate for description of multi-jet production processes, specially for decorrelation observables such as azimuthal angle difference spectra or rapidity difference spectra [@NSS2013; @gammaHERA]. In the $k_T-$factorization approach the main part of high-order corrections from hard real partons emission is included using the unintegrated Parton Distribution Functions (unPDFs). The presence of non-vanishing transverse momenta of initial-state partons before hard interaction provides a flat behavior of photon-jets azimuthal angle difference spectra in the region far from back-to-back configuration, together with DPS contribution.
In Ref. [@D0gamma2], the study of azimuthal correlations in $\gamma+2 jets$ events is introduced to be an extremely sensitive tool for DPS signals search. They measured azimuthal angle difference $\Delta \phi_A$ between sum of photon and leading jet transverse momenta ${\bf p}_{T}^A={\bf p}_T^{\gamma}+{\bf
p}_{T}^{jet1}$ and transverse momentum of second (sub-leading) jet ${\bf p}_{T}^{jet2}$. Such a way, in the LO CPM we should expect a strong peak near $\Delta \phi=\pi$, and broadening of this peak should be explained by emission of additional hard jet in the NLO approximation of the CPM. Such calculations, performed using SHERPA [@SHERPA] program (see Figs. 9-11, in Ref. [@D0gamma2]), fail to describe data from D0 Collaboration at $\Delta\phi<\pi/4$ without inclusion of sufficient contribution from DPS mechanism.
Here, we examine the Single Parton Scattering (SPS) mechanism to describe data on $\gamma +2 jets$ associated production, when higher-order QCD corrections are partially taken into account using the parton Reggeization approach (PRA) [@NSS2013; @NKS2017].
Parton Reggeization Approach {#PRA}
============================
The master formulas of Leading-Order (LO) approximation of PRA are presented below. The more detailed description can be found in [@NKS2017], while the development of PRA in the NLO approximation is further discussed in [@PRANLO1]. The main ingredients of PRA are the factorization formula for hard processes in the Multi-Regge Kinematics (MRK), the Kimber-Martin-Ryskin (KMR) unPDFs [@KMR] and the gauge-invariant amplitudes of hard processes with off-shell initial-state partons, derived using the Lipatov’s Effective Field Theory (EFT) of Reggeized gluons [@Lipatov95] and Reggeized quarks [@LipatovVyazovsky].
The factorization formula of PRA in LO approximation can be obtained from the relevant factorization formula of the CPM [@NKS2017] for the auxiliary hard subprocess with two additional final-state partons using the modified Multi-Regge Kinematics (mMRK) approximation. Last one correctly reproduces the multi-Regge and collinear limits of corresponding QCD amplitude. In this mMRK-approximation one has: $$d\sigma = \int \frac{dx_1}{x_1} \int \frac{d^2{\bf q}_{T1}}{\pi} {\Phi}_1(x_1,t_1,\mu^2)
\int \frac{dx_2}{x_2} \int \frac{d^2{\bf q}_{T2}}{\pi}
{\Phi}_2(x_2,t_2,\mu^2)\cdot d\hat{\sigma}^{\rm PRA},
\label{eqI:kT_fact}$$ where $t_{1,2}={\bf q}_{T1,2}^2$, the partonic cross-section $\hat\sigma^{\rm PRA}$ in PRA is determined by squared PRA amplitude, $\overline{|{\cal A}_{PRA}|^2}$. Despite the fact that four-momenta ($q_{1,2}$) of partons in the initial state of ${\cal
A}_{PRA}$ are off-shell ($q_{1,2}^2=-t_{1,2}<0$), the PRA hard-scattering amplitude is gauge-invariant because the initial-state off-shell partons are treated as Reggeized gluons ($R$) or Reggeized quarks ($Q$) in a sense of gauge-invariant EFT for QCD processes in MRK, introduced by L.N. Lipatov in [@Lipatov95]. The Feynman rules of this EFT are written down in Ref. [@AntonovLipatov; @LipatovVyazovsky].
The unPDF in PRA formally coincides with Kimber-Martin-Ryskin (KMR) unPDF [@KMR] and can be presented as follows: $$\Phi_i(x,t,\mu^2) = \frac{T_i(t,\mu^2)}{t} \frac{\alpha_s(t)}{2\pi} \sum_{j=q,\bar{q},g}
\int\limits_x^{1-\Delta_{KMR}} dz\ P_{ij}(z)\cdot \frac{x}{z}f_{j}\left(\frac{x}{z},t \right)
, \label{eqI:KMR}$$ where $\Delta_{KMR}(t,\mu^2)=\sqrt{t}/(\sqrt{\mu^2}+\sqrt{t})$ is the KMR cutoff function [@KMR], which regularizes infrared (IR) divergence at $z_{1,2}\to 1$ and introduces rapidity-ordering between initial-state and final-state partons. The collinear singularity in KMR unPDF is regularized by the Sudakov formfactor: $$T_i(t,\mu^2)=\exp\left[ - \int\limits_t^{\mu^2} \frac{dt'}{t'} \frac{\alpha_s(t')}{2\pi}
\sum\limits_{j=q,\bar{q},g} \int\limits_0^{1-\Delta_{KMR}} dz\
z\cdot P_{ji}(z) \right], \label{eq:Sudakov}$$ which resums doubly-logarithmic corrections $\sim\log^2 (t/\mu^2)$ in the leading-logarithmic approximation.
In contrast to most of studies in the $k_T$-factorization, the gauge-invariant matrix elements with off-shell initial-state partons (Reggeized quarks and Reggeized gluons) of Lipatov’s EFT [@Lipatov95; @LipatovVyazovsky] allow one to study arbitrary processes involving non-Abelian structure of QCD without violation of Slavnov-Taylor identities due to the nonzero virtuality of initial-state partons. This approach, together with KMR unPDFs gives stable and consistent results in a wide range of phenomenological applications, which include the description of the different spectra of single jet and prompt-photon inclusive production [@gamma; @gamma-jet], two jets [@NSS2013] or two photons [@Diphotons] in $pp$ and $p\bar p$ collisions, and photon plus jet in $\gamma p$ collisions at HERA Collider [@gammaHERA].
Associated production of photon and two jets in PRA {#spectra}
===================================================
To describe experimental data of prompt photon spectra we should take into account two production mechanisms. The first one is a direct production, when photons are produced in hard quark-gluon collisions. The second one is a fragmentation production, when quarks or gluons produced in hard collisions emit collinear photons. Fragmentation production of prompt photons can be strongly suppressed by experimental cuts, producing only a few percents from total number of events at small photon transverse momentum. This fact allows us to neglect here the small fragmentation contribution.
The LO PRA processes which contribute in direct $\gamma+2 jets$ events are the following: $$\begin{aligned}
Q+R &\to& q + g +\gamma, \label{QR}\\
R+R &\to& q + \bar q+ \gamma, \label{RR}\\
Q + \bar Q &\to& q(q') + \bar q (\bar q')+ \gamma,\\
Q + \bar Q &\to&
g + g + \gamma,\\
Q + Q&\to& q + q +\gamma, \label{QQ}\\
Q + Q'&\to& q + q' +\gamma \label{QQ1} \label{QaQ}.\end{aligned}$$ We omit the processes, which contributions are smaller than 1 % in total cross section. The main contribution comes from the process (\[QR\]), which is described by the set of Feynman diagrams of Lipatov’s EFT presented in the Fig. \[fig-0\].
The amplitudes of all [above-mentioned]{} LO PRA processes can be obtained in analytical form using model-file **ReggeQCD** [@ReggeQCD], which implements the Feynman rules of Lipatov’s EFT in **FeynArts** [@FeynArts] at tree level. To generate the gluon, $\Phi_g(x,t,\mu^2)$, and quark, $\Phi_q(x,t,\mu^2)$, unPDFs, according to the Eq. (\[eqI:KMR\]) we use the LO PDFs from the Martin-Roberts-Stirling-Thorne (MRST) set [@MRST].
We set the renormalization and factorization scales equal to the transverse momentum of leading jet, $p_{1T}$: $\mu_R=\mu_F=\xi
p_{1T}$, where $\xi=1$ for the central lines of our predictions, and we vary $1/2<\xi<2$ to estimate the scale uncertainty of our prediction, which is shown in the figures by the gray band.
After our numerical calculations, based on analytical amplitudes obtained from the Lipatov’s EFT, have been completed, we have got an opportunity to perform a cross-check of our results with MC generator [*Katie*]{} [@Katie].
The last one uses gauge-invariant scattering amplitudes with off-shell initial-state partons, obtained using the spinor-helicity techniques and BCFW-like recursion relations for such amplitudes [@Hameren1; @Hameren2]. This formalism for numerical generation of off-shell amplitudes is equivalent to the Lipatov’s EFT at the tree level.
DGLAP and MRK regimes
=====================
Now, we come back to the factorization formula (\[eqI:kT\_fact\]) where upper limits of integrals over $t_{1,2}$ should be defined. In MRK of hard processes [@QMRK], when initial-state radiation is not ordered in transverse momenta of emitted partons, transverse momentum of initial-state parton in hard collision can be arbitrary large up to some maximum value following from general kinematical conditions, $t_{\infty}$.
When we study production of jets or associated production of photon and jets, as in the case of present task, the upper limit for the squares of the Reggeized parton’s transverse momenta $t_1$ and $t_2$ should be truncated by the condition $t_1,t_2 < p_{2T}^2$, where $p_{2T}^2$ is the smaller transverse momentum of a jet from the pair of two leading jets. The above-mentioned condition arises from the constraints of the jet-production experiment: one can measure an azimuthal angle difference $\Delta\phi$ between the two most energetic jets but it is impossible to separate final-state partons produced in the hard parton scattering phase from the ones generated during the QCD evolution of unPDFs [@NKS2017]. The MRK evolution suggests a strong ordering in rapidity but the transverse momenta of partons in the QCD ladder keep similar values. This means that the transverse momenta of partons generated in the initial-state evolution described via the unPDF must be smaller than the transverse momenta of both measured leading jets, $\sqrt{t_{1,2}}<p_{2T}<p_{1T}$. However, as in the relevant experiment the leading and subleading jets are produced in some central region of rapidity $|y^{1,2}|<Y$, there is a probability to find partons with larger transverse momenta ($p_T>p_{2T}$), originated from evolution of unPDFs outside this region of rapidity and they can not be considered as leading jets, which should be in central region of rapidity. Formally, we can rewrite integrals over $t_{1,2}$ as sum of “DGLAP” (first term) and additional “MRK” (second term) contributions: $$\int\limits_0^{t_{\infty}}dt \Phi(x,t,\mu^2) \Rightarrow
\int\limits_{0}^{p_{2T}^2}dt \Phi(x,t,\mu^2) +
\int\limits_{p_{2T}^2}^{t_\infty}dt
w(x,t,\mu^2)\Phi(x,t,\mu^2),\label{eq:mrk}$$ where $w(x,t,\mu^2)$ can be considered as damping function which is independent on details of hard process but it can depend on QCD-evolution of unPDF. In the first approximation, we can interpret the average value $\overline{w}=\overline{w}(Y,p_{2T})$ as a probability to find more energetic partons with rapidities $|y|>Y$ than ones defined as leading jets in the rapidity region $|y^{1,2}|<Y$. After this we rewrite formulae (\[eq:mrk\]) as follows $$\int\limits_0^{t_{\infty}}dt \Phi(x,t,\mu^2) \Rightarrow
\int\limits_{0}^{p_{2T}^2}dt \Phi(x,t,\mu^2) +
\overline{w}(Y,p_{2T})\times \int\limits_{p_{2T}^2}^{t_\infty}dt
\Phi(x,t,\mu^2).$$ Further, we will consider $\overline{w}(Y,p_{2T})$ as a free parameter of our model with two boundary conditions, which correspond either DGLAP regime $(\overline{w}=0)$ or asymptotic MRK regime $(\overline{w}=1)$. Such a way, the factorization formula (\[eqI:kT\_fact\]) reads as $$d\sigma=d\sigma^{DLAP}+\overline{w} d\sigma^{MRK1}+\overline{w}
d\sigma^{MRK2}+\overline{w}^2 d\sigma^{MRK12},$$ where $$\begin{aligned}
d\sigma^{DGLAP} &=& \int\limits_0^1 \frac{dx_1}{x_1} \int \frac{d\phi_1}{2\pi}\int\limits_0^{p_{2T}^2} dt_1
{\Phi}_1(x_1,t_1,\mu^2) \times \nonumber \\
& \times &\int\limits_0^1 \frac{dx_2}{x_2} \int
\frac{d\phi}{2\pi}\int\limits_0^{p_{2T}^2} dt_2
{\Phi}_2(x_2,t_2,\mu^2)\cdot d\hat{\sigma}^{\rm PRA},
\label{eq:DGLAP}
\end{aligned}$$
$$\begin{aligned}
d\sigma^{MRK1} &=& \int\limits_0^1 \frac{dx_1}{x_1} \int \frac{d\phi_1}{2\pi}\int\limits_{p_{2T}^2}^{t_\infty} dt_1
{\Phi}_1(x_1,t_1,\mu^2) \times \nonumber \\
& \times &\int\limits_0^1 \frac{dx_2}{x_2} \int
\frac{d\phi}{2\pi}\int\limits_0^{p_{2T}^2} dt_2
{\Phi}_2(x_2,t_2,\mu^2)\cdot d\hat{\sigma}^{\rm PRA},
\label{eq:MRK1}
\end{aligned}$$
$$\begin{aligned}
d\sigma^{MRK2} &=& \int\limits_0^1 \frac{dx_1}{x_1} \int \frac{d\phi_1}{2\pi}\int\limits_0^{p_{2T}^2} dt_1
{\Phi}_1(x_1,t_1,\mu^2) \times \nonumber \\
& \times &\int\limits_0^1 \frac{dx_2}{x_2} \int
\frac{d\phi}{2\pi}\int\limits_{p_{2T}^2}^{t_\infty} dt_2
{\Phi}_2(x_2,t_2,\mu^2)\cdot d\hat{\sigma}^{\rm PRA},
\label{eq:MRK2}
\end{aligned}$$
$$\begin{aligned}
d\sigma^{MRK12} &=& \int\limits_0^1 \frac{dx_1}{x_1} \int \frac{d\phi_1}{2\pi}\int\limits_{p_{2T}^2}^{t_\infty} dt_1
{\Phi}_1(x_1,t_1,\mu^2) \times \nonumber \\
& \times &\int\limits_0^1 \frac{dx_2}{x_2} \int
\frac{d\phi}{2\pi}\int\limits_{p_{2T}^2}^{t_\infty} dt_2
{\Phi}_2(x_2,t_2,\mu^2)\cdot d\hat{\sigma}^{\rm PRA}.
\label{eq:MRK12}
\end{aligned}$$
Numerical results and discussion
================================
To study a relative role of MRK effects in azimuthal angle decorrelation effects, we compute normalized $\Delta\phi$ spectra for $\gamma+2jets$ events, which were measured by D0 Collaboration [@D0gamma2] at the $\sqrt{S}=1.96$ TeV. Leading and subleading jets are measured in the rapidity region $|y^{1,2}|<Y=3.5$ while the photon rapidity belongs to the interval $|y^\gamma|<2.5$, excluding subinterval $1.0<|y^\gamma|<1.5$. Transverse momentum of photon is restricted by the condition $50<p_T^{\gamma}<90$ GeV, and the leading jet has a transverse momentum $p_{1T}>30$ GeV. The data were collected in three sets with different conditions for subleading jet: set 1 – $15<p_{2T}<20$ GeV, set 2 – $20<p_{2T}<25$ GeV, set 3 – $25<p_{2T}<30$ GeV.
The results of calculation in DGLAP approximation (\[eq:DGLAP\]) for azimuthal angle difference spectra are shown in Fig. \[fig-1\] - \[fig-3\] as dashed green lines. The same as NLO calculations in CPM [@SHERPA], they strongly underestimate data at the small $\Delta\phi$ and the inclusion of DPS contributions is needed. Following Ref.[@D0gamma2], we define $\beta$ as a fraction of DPS events: $$\frac{1}{\sigma}\frac{d\sigma}{d\Delta
\phi}=(1-\beta)\frac{1}{\sigma^{SPS}}\frac{d\sigma^{SPS}}{d\Delta
\phi} +\beta \frac{1}{\sigma^{DPS}}\frac{d\sigma^{DPS}}{d\Delta
\phi}$$ Our estimations for parameters $\beta^{PRA}$ in calculations using DGLAP approximation at the different values of $p_{2T}$ are presented in the Table \[TableI\] as well as the values of $\beta^{SHERPA}$ obtained in NLO CPM calculations. Both parameters are extracted by a fit of experimental data in assumption that DPS contribution is constant throughout the $\Delta\phi$-spectra. We find a good agreement between NLO collinear and LO $k_T$-factorized schemes of calculations based on DGLAP picture of initial-state QCD evolution, proving the earlier obtained results for many other processes.
At very large energy in the MRK limit, when QCD evolution of $t-$channel partons should be described by BFKL equations [@BFKL] instead of DGLAP equations [@DGLAP], the transverse momenta of Reggeized partons connecting gauge-invariant clusters can be arbitrary large and independent on transverse momenta of final particles in clusters.
At the present energy such MRK picture can be realized in some part and we suggest to control a signal of MRK regime adding to LO PRA cross section the $d\sigma^{MRK1,2}$ and $d\sigma^{MRK12}$ terms. The phenomenological parameter $\overline{w}$ can be extracted from the data, see Table \[TableII\]. In the Figs. \[fig-1\]-\[fig-3\], the normalized azimuthal angle difference spectra at the different cuts on transverse momentum of subleading jet $p_{2T}$ are shown. We found the MRK contribution to the total cross section to be small, $$R(p_{2T})=\frac{\sigma^{MRK}}{\sigma^{DGLAP}+\sigma^{MRK}}=0.046
\div 0.163.\label{Rpt2}$$ We see that probability $\overline{w}$ degenerates with growth of $p_{2T}$ following our guess, as well as MRK contribution to the total cross section, see Table \[TableII\]. As it can be expected, initial-state partons with large transverse momenta contribute to the region of small $\Delta\phi$ enhancing decorrelation effect.
Finally, we would like to note that a theoretical calculation of modified unPDFs, which should be dependent on transverse momenta of leading hard jets and the rapidity region $Y$ where they are measured, can be implemented principally, providing an additional test for the already known models of unPDFs, such as used here KMR [@KMR] as well as the ones collected in TMD-library [@TMDlib], and also recently suggested unPDFs based on Parton Branching method [@PBM].
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to Maxim Nefedov for help in the stage of analytical calculations with model-file ReggeQCD. Authors thank the Ministry of Education and Science of the Russian Federation for financial support in the framework of the Samara University Competitiveness Improvement Program among the world’s leading research and educational centers for 2013-2020, the task number 3.5093.2017/8.9. Authors (A.K. and V.S.) thank the Foundation for the Advancement of Theoretical Physics and Mathematics BASIS, grant No. 18-1-1-30-1.
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\[TableI\]
\[TableII\]
![Feynman diagrams for process $RQ\to g q \gamma$ in Lipatov’s EFT.[]{data-label="fig-0"}](QRqgy){width="80.00000%"}
![Azimuthal angle difference $(\Delta\phi)$ spectrum at the $15<p_{T}^{jet2}<20$ GeV. The dashed green line is DGLAP contribution, the dash-dotted blue line is MRK contribution with optimized $\overline{w}$ from the Table \[TableII\] and the solid red line is their sum. The data are from D0 Collaboration [@D0gamma2].[]{data-label="fig-1"}](gamma+2jets_Wprob_15_20){width="80.00000%"}
![Azimuthal angle difference $(\Delta\phi)$ spectrum at the $20<p_{T}^{jet2}<25$ GeV. The curves are defined as in the Fig. \[fig-1\]. The data are from D0 Collaboration [@D0gamma2].[]{data-label="fig-2"}](gamma+2jets_Wprob_20_25){width="80.00000%"}
![Azimuthal angle difference $(\Delta\phi)$ spectrum at the $25<p_{T}^{jet2}<30$ GeV. The curves are defined as in the Fig. \[fig-1\]. The data are from D0 Collaboration [@D0gamma2].[]{data-label="fig-3"}](gamma+2jets_Wprob_25_30){width="80.00000%"}
[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Only a very small fraction of the asteroid population at size scales comparable to the object that exploded over Chelyabinsk, Russia has been discovered to date, and physical properties are poorly characterized. We present previously unreported detections of 106 close approaching near-Earth objects (NEOs) by the Wide-field Infrared Survey Explorer mission’s NEOWISE project. These infrared observations constrain physical properties such as diameter and albedo for these objects, many of which are found to be smaller than 100 m. Because these objects are intrinsically faint, they were detected by WISE during very close approaches to the Earth, often at large apparent on-sky velocities. We observe a trend of increasing albedo with decreasing size, but as this sample of NEOs was discovered by visible light surveys, it is likely that selection biases against finding small, dark NEOs influence this finding.'
author:
- 'A. Mainzer, J. Bauer, T. Grav, J. Masiero, R. M. Cutri, E. Wright, C. R. Nugent, R. Stevenson, E. Clyne, G. Cukrov, F. Masci'
title: 'The Population of Tiny Near-Earth Objects Observed by NEOWISE'
---
Introduction
============
As the products of collisional processes, small near-Earth objects (NEOs), defined as minor planets with perihelia less than 1.3 AU, are far more numerous than larger ones. Discovering, tracking, and characterizing these objects allows us to better understand the impact hazard they pose to Earth, as well as their origins and subsequent evolution. Because their small sizes usually make them undetectable until they are very nearby the Earth, it is often difficult for the current suite of asteroid surveys and follow-up telescopes to track them for very long. Consequently, the fraction of the total population at small sizes that has been discovered to date remains very low. While about 90% of NEOs with effective spherical diameters larger than 1 km have now been discovered, the integral survey completeness drops to $\sim$25% at 100 m, and it is likely to be $<$1% at sizes comparable to the 17-20 m diameter NEO that exploded over Chelyabinsk, Russia on February 15, 2013 [@Mainzer.2011a; @Harris.2008a].
Approximately 10,000 NEOs have been discovered to date, $\sim$900 of which are 1 km or larger. The current catalog includes $\sim$3500 objects with absolute magnitude $H>22.75$ mag. This corresponds to a diameter of 100 m or smaller assuming a geometric visible albedo of 14%, the average albedo for the infrared-selected NEO sample from @Mainzer.2011a, using the relationship $$D = \left[\frac{1329\cdot 10^{-0.2H}}{p_{v}^{1/2}}\right],$$ where $D$ is the effective spherical diameter [@Fowler.1992a; @Bowell.1989a]. For NEOs with diameters less than 20 m, corresponding to $H>$26.25 mag for =14%, there are $\sim$720 objects that have been discovered to date. However, the true number is quite uncertain, since NEO albedos are known to range from $\sim$1% to $>$50% [@Mainzer.2011a; @Trilling.2010a; @Stuart.2004a]. For NEOs 10 m and smaller, it is reasonable to assume that somewhere between 50 - 150 have been discovered. The true numbers discovered at different size ranges depend on the objects’ albedos, of course, and it is not possible to extrapolate the sample of discovered objects to the entire population without careful accounting for survey biases due to instrument sensitivity, survey cadence, weather, seeing, availability of follow-up assets, etc. [c.f. @Jedicke.1998a; @Spahr.1998a; @Bottke.2002a; @Mainzer.2011a; @Mainzer.2012a; @Grav.2011b; @Grav.2012a]. At these very small sizes, the survey completeness is very low.
Given recent interest in NEOs down to even very small sizes, it is useful to compute the average albedo for tiny NEOs discovered by visible light surveys and see how it compares to the average albedo determined for NEOs larger than 100 m by the sample returned by the *Wide-field Infrared Survey Explorer’s* NEOWISE project [@Wright.2010a; @Mainzer.2011a; @Mainzer.2011b]. Because NEOWISE detected and discovered NEOs using thermal infrared wavelengths, and because new discoveries received timely ground-based follow-up, its sample was shown to be essentially albedo-insensitive [@Mainzer.2011a]. The average albedo found by that sample was 14% for NEOs 100 m and larger. Few objects smaller than 100 m were detected by the NEOWISE automated minor planet detection software, known as the WISE Moving Object Processing System (WMOPS), which required five or more detections of objects moving at apparent on-sky velocities slower than $\sim$3.2$^{\circ}$/day. In general, NEOs smaller than $\sim$100 m are only detected when they are quite close, resulting in significantly higher angular velocities; the smallest objects, with sizes $<$10 m, were detected when they were only 2-3 lunar distances away from Earth. Such small objects often passed through the WISE field of view fewer than the five times required for WMOPS to detect them. Alternately, they may have passed through the field of view more than five times, but were too faint to have been detected at least five times because of their small size or because they were trailed in the individual exposures.
A total of 429 NEOs detected by WMOPS were reported from the fully cryogenic portion of the survey, and 116 NEOs were detected and/or discovered following the partial and complete depletion of the spacecraft’s cryogen [@Mainzer.2011a; @Mainzer.2012b]. All but a handful were larger than 100 m. Here, we report the detection by NEOWISE of an additional 106 NEOs that were discovered by ground-based visible light surveys and made very close approaches to the Earth while WISE was observing. These objects tend to be small and fast-moving. This sample represents a pilot study for a future effort to conduct a wholesale search of the NEOWISE databases and images for the entire set of known minor planets; this effort will be carried out by the NEOWISE project in the near future.
Methods
=======
A combination of methods were used to identify known objects that were not previously identified by WMOPS in the single-exposure images. Images were searched using the Infrared Science Archive (IRSA)/WISE Image Service[^1] as well as by searching the WISE single-exposure source database using the Known Solar System Object Possible Association List [KSSOPAL; @Cutri.2012a]. Both of these tools compute an object’s ephemeris and predict where it would have intersected with a WISE observation. While WMOPS actively rejects stationary objects such as stars and galaxies before linking transient detections, neither the Image Service nor KSSOPAL discriminates between stationary objects and the asteroid, so the probability of a chance association is greater. We rejected stationary objects by examining the WISE Atlas Images and Catalogs, which combine all possible exposures at a given location to produce a deeper image, following the methods described in @Mainzer.2011a [@Mainzer.2011c].
Of the currently known NEOs with $H>26.25$ mag, the median observational arc length spans only $\sim$3 days, and only a few dozen exceed 20 days. The vast majority of these objects are therefore lost, frustrating efforts to locate them in the NEOWISE data. However, many of the objects that were discovered by ground-based surveys during the WISE survey phase did pass through the WISE field of view near 90$^{\circ}$ solar elongation and were bright enough to be detected, albeit often with fewer than five observations. Only those objects whose astrometric uncertainties were very small (less than a few arc seconds) at the time of their passage through the WISE field of view were included in this analysis, with the exception of a handful of extremely close-approaching NEOs with larger uncertainties that were identifiable by color and morphology (the images were slightly trailed). Astrometric uncertainties were taken from the MPC’s ephemeris service. For objects that were stacked, only objects with very low astrometric uncertainty (less than $\sim$3 arcsec) were used. Future work will extend to identifying and stacking objects with larger astrometric uncertainties.
The WISE instrument used three beamsplitters to collect images in all four bands simultaneously (3.4, 4.6, 12 and 22 $\mu$m, hereafter W1, W2, W3, and W4). The exposure time in all four WISE bands was set to 8.8 sec in bands W3 and W4 and 7.7 sec in bands W1 and W2. An NEO moving faster than $\sim 17^{\circ}$/day could thus be trailed in the 6.5 arcsec W3 beam. All candidate images were visually inspected for evidence of trailing. If the images were trailed, aperture magnitudes reported by the data reduction pipeline (denoted w1mag\_x, w2mag\_x, w3mag\_x, and w4mag\_x, where 1-4 represents the wavelength band, 1-4, and x represents the number of the aperture that encircled the source, 1-8) were used instead of photometry derived using point-spread function fits; see Section IV.4.c of the WISE Explanatory Supplement [@Cutri.2012a] for a detailed description of the WISE pipeline photometry. Aperture curves-of-growth were constructed for the trailed objects, and the aperture at which the curve converged was selected.
For objects that were too faint to be detected reliably in single exposures, the single exposure images were coadded in the moving frame of the asteroid, aligning the frames using each object’s ephemeris, with the ICORE (Image Co-addition with Optional Resolution Enhancement) stacking algorithm [@Masci.2009a; @Masci.2013a]. The ICORE algorithm includes outlier rejection and resamples the stacked image to a pixel scale of 1 arcsec/pixel. The algorithm requires a minimum of five images to ensure adequate pixel outlier rejection; all coadded objects in this analysis exceeded this number of images. An example of an object recovered in the NEOWISE data by stacking with ICORE is shown in Figure \[fig:stack\], demonstrating the utility of this technique for obtaining infrared detections of minor planets that fell just below the single frame detection threshold; similar success with the extended object comet 17P/Holmes was shown in @Stevenson.2013a. For the stacked objects, aperture photometry was performed using radii of 11, 11, 11, and 22 arcsec, respectively, in the four WISE bands. Table 1 lists the magnitudes and apertures used for each object.
Thermal Modeling and Error Analysis
-----------------------------------
The Near-Earth Asteroid Thermal Model [NEATM; @Harris.1998a] was used to determine diameters and albedos for most objects. The color corrections and modifications that account for the observed discrepancy between red and blue calibrators noted in @Wright.2010a were applied following the methods described in @Mainzer.2011c. Table 2 shows the thermal fit results for the 106 NEOs that were recovered here; Figure \[fig:K10G07H\] shows an example of a small ($\sim$8 m) NEO identified in the NEOWISE data using the IRSA/WISE Image Service and KSSOPAL.
As described in @Mainzer.2011a and @Mainzer.2011c, the so-called beaming parameter $\eta$ employed by the NEATM was fit when two or more thermally-dominated infrared bands were available. Since an NEO’s flux at 3.4 and 4.6 $\mu$m generally consists of a mix of thermal emission and reflected sunlight, it was necessary to fit for the reflectivity at these wavelengths. Albedo was assumed to be the same for both 3.4 and 4.6 $\mu$m. This simplifying assumption may not always be valid. @Grav.2012a and @Grav.2012b have shown that albedo varies between 3.4 and 4.6 $\mu$m for Hilda group asteroids and Jovian Trojans, although the NEOs’ smaller heliocentric distances and warmer temperatures means that the 4.6 $\mu$m band is often thermally-dominated, lessening the effect of albedo at 4.6 $\mu$m on the total flux. The infrared flux, $p_{IR}=p_{3.4 \mu m}=p_{4.6 \mu m}$, could only be fit when the flux at 3.4 $\mu$m was dominated by reflected sunlight, which depends on both reflectivity and heliocentric distance.
The best-fit values for diameter, visible geometric albedo , infrared albedo , and beaming parameter $\eta$ were determined using a least-squares optimization that accounted for the measurement uncertainties for bands W1, W2, W3, W4, absolute magnitude $H$, and phase curve slope parameter $G$. $G$ was generally taken to be 0.15$\pm$0.1 unless a measurement was available [@Bowell.1989a]. Errors on $H$ ($\sigma _{H}$) were assumed to be 0.3 mag, although in some cases, examination of the MPC observations file revealed the uncertainty in $H$ to be considerably larger; in these cases, $H$ was assumed to be unknown and was not used in the thermal fit. Poorly known $H$ and $G$ values continue to be a persistent difficulty that sometimes inhibits precise determination of albedo, although this problem does not much affect the determination of diameter when using radiometric methods such as NEATM.
Although there is evidence that systematic biases in $H$ magnitudes may be present in the existing asteroid databases such as MPC and *Horizons* that derive $H$ using measurements made primarily for improvements to astrometry rather than accurate photometry, targeted measurements of $H$ [e.g. @Pravec.2012a] have concentrated on objects with $H\leq$21 mag. We therefore chose not to adopt a blanket offset to all $H$ values and instead bounded the errors by assuming $\sigma _{H}=0.3$ mag. The errors in diameter, , , and $\eta$ were determined through the use of Monte Carlo trials that varied the measured values of W1, W2, W3, W4, $H$, and $G$ by their respective errors. If could not be fit, the default value was taken to be the average value determined in @Mainzer.2011a for NEOs, or 1.6$\pm$1.0.
Another source of error is that the slope parameter $G$ is known to vary with taxonomic type; C-complex asteroids typically have lower $G$ values, closer to 0.05-0.10, whereas S-complex asteroids have $G$ closer to 0.2, and the highest albedo classes such as E-types have $G\sim$0.4 [@Harris.1989a; @Harris.1989b; @Lagerkvist.1990a; @Oszkiewicz.2012a]. Our generic assumption of $G=0.15$ can therefore yield $H$ values that are too high for C-type objects, which in turn result in overly low values of . Similarly, our assumed $G$ value of 0.15 is likely to be too low for S-complex objects, particularly E- and V-types, resulting in $H$ values that are erroneously low. These effects will be more pronounced at the high phase angles at which NEOWISE typically observed small NEOs and Hungarias. Erroneous values of $G$ and $H$ do not much affect diameters derived from thermal measurements, but rather albedo. For example, assuming that 2010 TN$_{4}$ has $G=0.05$ instead of $G=0.15$ results in an offset to $H$ of -0.28 mag at its NEOWISE-observed phase angle of 80.6$^{\circ}$. The derived diameter (18 m) is unchanged, but =0.071 instead of 0.054.
However, without knowledge of an object’s spectral type, we cannot know for certain which value of $G$ to choose if it has not been directly measured. In our previous works [e.g. @Mainzer.2011d; @Mainzer.2012c], we found that there is not a perfect correlation between albedo and taxonomic type. We therefore chose not to apply offsets to $G$ based on assumed spectral type (which in most cases we could only assume based on albedo - a circular argument). Instead, we model and bound the errors caused by imperfect knowledge of $G$ by assuming that $G$ varies by $\pm$0.10 in the Monte Carlo trial fits for each object. It is important to note that errors in $G$ and $H$ could cause systematic offsets in albedo, with lower albedo objects’ albedos sometimes being too low, and higher albedo objects’ albedos coming out too high. For this reason, direct measurements of $H$, $G$, and spectral type for individual objects would be useful, particularly when determining albedos of objects observed at high phase angles. Nevertheless, a major benefit of radiometric fits to infrared observations is that while errors in $H$ and $G$ can affect albedo, they cause little change to diameter.
Results
-------
Figure \[fig:beaming\] shows the beaming distribution for the small, close-approaching NEOs recovered from the NEOWISE data compared with the sample of 429 NEOs detected by WMOPS during the fully cryogenic portion of the mission. The close-approaching NEOs tend to have been observed near 90$^{\circ}$ phase, because the WISE spacecraft only observed near 90$^{\circ}$ solar elongation [@Wright.2010a]. Thus, the objects were typically observed near their local terminators. The NEATM assumes zero flux contribution from the night side of the asteroids. Because small NEOs are frequently rapidly rotating with rotation periods much less than the cooling timescale [@Pravec.2008a; @Hergenrother.2011a; @Statler.2013a], the approximation of zero night side flux is likely inappropriate in many cases. Indeed, the beaming parameter $\eta$, which is used by the NEATM to compensate for the “beaming" effect of radiation observed near zero phase angle, converges to its theoretical maximum value of $\pi$ in many cases when more than one thermally dominated band is available. The average value found for the small NEOs reported in this work was 2.0$\pm$0.5, higher than the 1.4$\pm$0.5 reported for all NEOs over a wide range of phase angles in @Mainzer.2011b. Therefore, if $\eta$ could not be fit, a default value of 2.0$\pm$0.5 was used for objects observed close to 90$^{\circ}$ phase angle. However, we note that while a fit yielding the theoretical maximum value of $\eta=\pi$ may indeed indicate high thermal inertia and/or rapid rotation, it could also be that the NEATM’s assumption of zero nightside flux is inappropriate for asteroids observed at very high phase angles. At high phase angles, a substantial portion of the nightside hemisphere is visible, so the approximation of zero nightside flux contribution may be poor. Furthermore, for the very smallest objects, the heat conduction length scale begins to approach the object size.
In cases where $\eta = \pi$ resulted from the NEATM fit, the Fast Rotating Model [FRM; @Lebofsky.1978a; @Lebofsky.1989a; @Veeder.1978a] was used instead. In the FRM, the asteroid is assumed to be rotating rapidly compared to its cooling timescale, so that the temperature is isotropic with respect to longitude and varies only with local latitude, assuming that the object’s rotational axis is perpendicular to the Earth-Sun line. The FRM uses fast rotation to smooth out longitudinal temperature variations, although it still has latitudinal temperature variations. The timescale for these to change is seasonal, so thermal conduction (especially in a solid boulder) could lead to uniform temperature in longitude. When using the FRM, we made the assumption that the rotational pole is perpendicular to the plane defined by the Sun, Earth, and object. This assumption may be reasonable given that there is evidence that the non-gravitational thermal pressure torques exerted by the YORP effect are thought to drive small asteroids to this state. For objects in the 10 - 100 m size range, the mean time between spin axis reorientation due to collisions is thought to be much longer than the timescale over which YORP will reshape spin states [@Farinella.1998a; @Rossi.2009a]. Using the FRM instead of the NEATM usually has the effect of shrinking an object’s best-fit effective spherical diameter, since the flux that was distributed across only one hemisphere using the NEATM is now spread across the entire visible area; diameter must therefore shrink to conserve the emitted energy.
With so few observations (in some cases only one), the rotational lightcurves of many NEOs in this sample were not well-sampled. For the WMOPS-detected sample, the WISE observational cadence typically resulted in 10-12 observations of each object collected over a $\sim$36 hour span. The question is the extent to which diameters determined using only sparse detections are representative of the actual effective spherical diameter. As shown in figure 5 in @Mainzer.2011a, most NEOs detected by NEOWISE had lightcurve amplitudes of $\sim$0.4 mag or less.
Two effects come into play when one attempts to use sparse detections near the sensitivity limit to investigate a population. First, there is a systematic bias in flux measurements that are made very close to the noise limit of a detector. Random noise can scatter flux above the detection threshold, making a source appear brighter. This manifestation of the Eddington bias was observed with WISE’s predecessor, the *Infrared Astronomical Satellite*[^2]. Second, asteroids rotate, and they can be elongated. The tendency would be to detect a rotating, elongated body when it is closer to presenting its maximum surface area to the observer. Both of these effects combine to produce an overestimate of fluxes for small bodies observed by WISE only a handful of times.
In an effort to better understand the effects of sparsely sampled infrared lightcurves on thermal model outputs, a comparison was made between fits using the entire set of NEOWISE thermal measurements for each object and fits derived using only the single brightest point per object. Because WISE observes in all four bands simultaneously using beamsplitters, we selected the time at the maximum of the lightcurve in the band with the highest signal-to-noise ratio, then used all bands available at that time. Figure \[fig:diam\] (top) shows $\Delta D = (D_{all} - D_{max})/D_{all}$, where $D_{all}$ is the diameter derived using all points, and $D_{max}$ are the diameters resulting from the maximum brightnesses, as a function of $D_{all}$. Over most size ranges, the dispersion between $D_{max}$ and $D_{all}$ is typically a factor of $\sim$1.3, although it worsens at smaller sizes. In most cases, the tendency is unsurprisingly to overestimate the diameter when using only the observations at the maximum of a lightcurve. However, at smaller sizes in particular, $D_{max}$ can be observed in Figure \[fig:diam\] to be approximately double $D_{all}$.
Unfortunately, this analysis cannot account for the possibility of real variations in the shapes of NEOs at sizes smaller than the WMOPS-selected sample shown in Figure \[fig:diam\]. Very small NEOs might have shapes that are either more round or more irregular than larger objects, depending on their origins and subsequent evolution. The observed amplitude of an asteroid’s light curve depends on the interplay between the shape of the object, the orientation of the rotation pole with respect to the line of sight, and the phase angle of the object at the time of observation when using reflected light. To first order, the larger the axial ratios of the object, the larger the amplitude of the light curve will be. However, if the sub-observer point is near the rotation pole, the light curve amplitude will be reduced.
Amplitudes larger than two magnitudes have been seen at optical wavelengths for small NEOs that are a result of very elongated shapes [e.g. 2009 UU$_{1}$; @Warner.2009a]. However, small amplitudes are frequently observed by WISE, implying either a shape close to spherical or an observing geometry looking along the rotation pole. Nevertheless, the analysis described above suggests that the sizes derived from sparse observations can serve as valuable upper limits to NEO sizes, and it is therefore quite likely that many of the objects presented in Table 2 are somewhat smaller than their derived sizes. The error bars reported in Table 2 are the formal errors resulting from the Monte Carlo trials that vary the WISE magnitudes, $H$, and $G$ by their respective error bars, and they do not include the systematic errors that may be associated with the sparseness of some of the lightcurve sampling. A typical systematic error for the sizes given in Table 2 is likely to be of order the $\sim$30% shown in Figure \[fig:diam\], although in some cases, it is possible that the derived sizes are quite different.
The structure of asteroids smaller than 100 m is uncertain. @Pravec.2008a have shown that while asteroids larger than a few hundred meters show a spin limit coincident with the theoretical maximum rotation rate of a gravitationally bound rubble pile, smaller asteroids typically rotate well above this limit, which was interpreted as evidence that these objects were monolithic. @Scheeres.2010a showed that cohesive forces on small grains ($<1~$cm) can be sufficient to allow small rubble piles to rotate at rates above the spin barrier without disrupting. However, in the process of forming these objects, any monolithic components of the original body could be ejected onto independent orbits, becoming NEOs in their own right. Thus, there may be different kinds of small NEOs. Monolithic fragments created by the breakup of larger bodies will have shape distributions determined by the impact and fracture processes of the collisions that generated them, which may vary among collisions. Objects held together by cohesive forces may have a range of shapes allowed by the force balance, from nearly spherical to oblong. While it is beyond the scope of this work to constrain these shape distributions and the division between monolithic and cohesive objects, we can regard the diameters derived from sparsely sampled lightcurves as, at worst, useful upper limits.
*2009 BD.* One of the objects shown in Table 2, 2009 BD, passed through the WISE fields of view but was not detected even when the images were stacked. However, it is possible to use the images to set an upper limit on its size from the non-detection. We extracted magnitudes from 13 images coadded using ICORE, taken on June 13 and 14 of 2010 (during the fully cryogenic mission phase) at a phase angle of 88$^{\circ}$, using aperture photometry with a sample radius of 11 arcsec. The counts yielded a $5-\sigma$ magnitude limit of 11.77 for W3, our most sensitive band for NEOs, and 7.59 mag for W4. Given that 2009 BD was discovered by a visible light telescope (the Catalina Sky Survey), Figure \[fig:diam\_alb\] suggests that it is more likely to have a higher albedo than a lower albedo. Using the NEATM fit code, this magnitude limit corresponds to a 5-$\sigma$ effective spherical diameter upper limit of $\sim$14.5 m, assuming =0.2 and $\eta$=2.0; using =0.04 produces an upper limit of 14 m. The FRM results in a 5-$\sigma$ upper limit of 8 m.
Discussion
----------
Figure \[fig:diam\_alb\] shows the diameters and albedos of the optically-selected sample of small NEOs recovered from the NEOWISE data compared with the infrared-selected sample drawn using WMOPS from @Mainzer.2011a. All recoveries were discovered by visible light telescopes, which are preferentially biased against discovering small, dark NEOs. The infrared-selected sample’s albedo distribution is essentially flat with diameter, whereas the optically-selected sample’s albedos increase with decreasing diameter. While there may be real physical changes in albedo with diameter, the visible light survey biases against discovering small, dark NEOs complicate efforts to determine the true albedo distribution of the population of small NEOs. It is also apparent that the current suite of visible light survey telescopes is not efficient at discovering very tiny low albedo NEOs.
While one might be tempted to use the albedo distribution for tiny NEOs shown in Figure \[fig:diam\_alb\] to extrapolate to the total number of NEOs in the population [c.f. @Mainzer.2011a; @Mainzer.2012a], we caution that this is not an infrared-selected sample with well-determined survey biases. All objects presented here were discovered by ground-based visible light surveys that are subject to the effects of weather and seeing variations, in addition to the NEOWISE detection biases. Debiasing this sample requires careful modeling of all of these selection effects, and for objects in this size range, infrasound measurements of small meteors and cratering studies should be included [c.f. the lunar impact flash studies such as @Oberst.2012a; @Suggs.2008a] to provide a realistic assessment of their true numbers, sizes, and albedos.
Conclusions
===========
We have reported sizes, albedos, and thermal model parameters such as beaming for 106 NEOs, roughly half of which are smaller than 100 m in effective spherical diameter, and the smallest of which are $\sim$8 m. These objects generally were not detected by the automated WMOPS survey because they were too faint in the individual images or were not detected the required five times for the pipeline to trigger on them. Many of the objects were extremely close to Earth when they were observed by WISE, with $\sim$10 m NEOs being detectable only when they approached within several lunar distances. Because close NEOs tend to move with large apparent velocities, many of the objects were only detected a handful of times; approximately one-third of the sample reported here was detected only once. While radiometric thermal models are typically able to constrain the sizes of asteroids when lightcurves are well-sampled, the results of models using these sparsely-sampled lightcurves must be regarded with caution. We have attempted to estimate the degree to which thermal models of NEOs using sparse data will err in their predictions of effective spherical diameter. We find that in most cases, typical errors are $\sim$30% if one assumes that the distribution of shapes for NEOs smaller than $\sim$100 m is similar to those larger than 100 m. However, in some cases, the size is overestimated by factors of several. Nonetheless, sparse infrared data can still provide useful estimates of effective spherical diameters.
We have shown that our sample of small NEOs displays a marked trend of increasing albedo with decreasing diameter, but given that the optical surveys that discovered this sample are biased against finding small, dark, faint NEOs, this is not surprising. These data alone are insufficient to determine whether or not this result represents a real physical trend or merely selection effects, as the optical surveys are highly incomplete at these small sizes. Selection biases must be carefully disentangled using models of survey performance. It is, however, clear that much work remains to be done to discover and characterize the population of very small NEOs.
Acknowledgments
===============
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![\[fig:stack\] Left: The NEO 2010 FK (center of image) was only detected in W3 by creating a moving coadd that combined 13 exposures, as it fell just below the detection threshold in the single-frame images (right, center of image). ](K10F00K.png){width="6in"}
![\[fig:K10G07H\] The NEO 2010 GH$_{7}$ (circled in green) was detected only once by NEOWISE due to its high apparent on-sky velocity in bands W2, W3, and W4. This object is on the list of accessible targets for potential human exploration; it makes close approaches to Earth every $\sim$5 years. Blue = band W1; green = W2; red = W3. ](K10G07H.png){width="3in"}
![\[fig:beaming\] Beaming parameter vs. phase angle for objects observed in two or more thermal wavelengths. The small, close-approaching NEOs that were detected by NEOWISE using the KSSOPAL and WISE Image Service tools (black squares) were usually observed at very high phase angles; the beaming parameter $\eta$ required was significantly larger than the average value for NEOs observed at lower phase angles (cyan points; cyan line shows running median). Cyan points taken from @Mainzer.2011a. Only objects with more than one thermally-dominated band available are plotted as these are the objects for which $\eta$ can be fit; the maximum value of $\eta=\pi$.](beaming.png){width="6in"}
![\[fig:diam\] Top: Comparison between NEATM diameter fits derived using the full set of NEOWISE measurements for each NEO vs. fits derived using only the brightest measurement in W3 for each object; the red line shows a running median. Bottom: Histogram of diameter differences between full lightcurve fits and maximum brightness fits; a Gaussian is overplotted (red dashed line).](diam.png){width="6in"}
![\[fig:diam\_alb\] The sample of objects recovered from the NEOWISE dataset using the KSSOPAL and WISE Image Service tools were all discovered by visible light ground-based surveys (black squares). The albedo distribution of these objects is distinctly different from the sample of NEOs that were selected using the WMOPS algorithm working on 12 $\mu$m NEOWISE data (cyan circles). Because the WMOPS algorithm treated new discoveries the same way as recoveries of previously known objects, and because asteroids’ thermal fluxes depend only weakly on albedo, the 12 $\mu$m sample is albedo-insensitive. The optically-selected sample’s albedo distribution increases sharply with decreasing size (black line), whereas the albedo distribution of the infrared-selected sample (cyan line) remains essentially unchanged with decreasing size. ](albedo.png){width="6in"}
[lllllll]{} 85989 & 55310.732616 & $>$16.882 & $>$15.004 & 9.264 $\pm$ 0.081 & 7.098 $\pm$ 0.260 & 0\
85989 & 55310.86492 & $>$16.703 & 14.323 $\pm$ 0.175 & 8.765 $\pm$ 0.055 & 6.734 $\pm$ 0.198 & 0\
85989 & 55310.997225 & $>$16.695 & 14.525 $\pm$ 0.266 & 8.247 $\pm$ 0.035 & 6.260 $\pm$ 0.124 & 0\
85989 & 55311.129529 & 16.623 $\pm$ 0.431 & 14.170 $\pm$ 0.162 & 8.195 $\pm$ 0.035 & 5.864 $\pm$ 0.106 & 0\
85989 & 55311.261833 & 16.237 $\pm$ 0.300 & 14.507 $\pm$ 0.212 & 8.598 $\pm$ 0.052 & 6.514 $\pm$ 0.161 & 0\
85989 & 55311.327922 & $>$16.373 & 14.234 $\pm$ 0.171 & 8.491 $\pm$ 0.044 & 6.320 $\pm$ 0.142 & 0\
85989 & 55311.394137 & $>$17.032 & $>$15.588 & 8.842 $\pm$ 0.056 & 7.073 $\pm$ 0.294 & 0\
85989 & 55311.460226 & $>$16.506 & 13.843 $\pm$ 0.126 & 8.190 $\pm$ 0.037 & 6.145 $\pm$ 0.113 & 0\
85989 & 55311.526442 & $>$16.304 & 15.161 $\pm$ 0.375 & 9.434 $\pm$ 0.094 & 7.656 $\pm$ 0.426 & 0\
85989 & 55311.59253 & $>$16.498 & 13.954 $\pm$ 0.120 & 8.425 $\pm$ 0.040 & 6.197 $\pm$ 0.114 & 0\
85989 & 55311.592657 & $>$16.636 & 14.295 $\pm$ 0.168 & 8.395 $\pm$ 0.041 & 6.429 $\pm$ 0.154 & 0\
85989 & 55311.658746 & 16.835 $\pm$ 0.465 & 14.376 $\pm$ 0.182 & 8.774 $\pm$ 0.051 & 6.872 $\pm$ 0.223 & 0\
85989 & 55311.724834 & 16.324 $\pm$ 0.288 & 14.618 $\pm$ 0.207 & 8.643 $\pm$ 0.049 & 6.711 $\pm$ 0.173 & 0\
85989 & 55311.857139 & $>$16.664 & 14.961 $\pm$ 0.291 & 9.144 $\pm$ 0.075 & 6.715 $\pm$ 0.183 & 0\
85989 & 55311.857266 & $>$17.049 & 15.356 $\pm$ 0.434 & 9.274 $\pm$ 0.082 & 6.908 $\pm$ 0.233 & 0\
85989 & 55311.989443 & 16.923 $\pm$ 0.479 & 14.216 $\pm$ 0.156 & 8.871 $\pm$ 0.058 & 6.758 $\pm$ 0.185 & 0\
85989 & 55311.98957 & $>$16.486 & 14.833 $\pm$ 0.286 & 9.028 $\pm$ 0.065 & 7.277 $\pm$ 0.308 & 0\
85989 & 55312.121747 & 16.674 $\pm$ 0.409 & 13.984 $\pm$ 0.129 & 8.267 $\pm$ 0.035 & 6.370 $\pm$ 0.135 & 0\
85989 & 55312.254051 & 15.991 $\pm$ 0.233 & 14.235 $\pm$ 0.160 & 8.105 $\pm$ 0.038 & 6.027 $\pm$ 0.099 & 0\
85989 & 55312.254051 & 15.991 $\pm$ 0.233 & 14.235 $\pm$ 0.160 & 8.105 $\pm$ 0.038 & 6.027 $\pm$ 0.099 & 0\
85989 & 55322.044695 & $>$16.852 & 13.933 $\pm$ 0.162 & 8.645 $\pm$ 0.047 & 6.643 $\pm$ 0.174 & 0\
85989 & 55406.690608 & 12.304 $\pm$ 0.026 & 9.050 $\pm$ 0.018 & 3.689 $\pm$ 0.012 & 1.755 $\pm$ 0.017 & 0\
85989 & 55406.756696 & 11.754 $\pm$ 0.023 & 8.649 $\pm$ 0.016 & 3.753 $\pm$ 0.011 & 2.008 $\pm$ 0.030 & 0\
85989 & 55406.822912 & 11.803 $\pm$ 0.030 & 8.484 $\pm$ 0.019 & 3.370 $\pm$ 0.014 & 1.590 $\pm$ 0.021 & 0\
85989 & 55406.889 & 12.697 $\pm$ 0.027 & 9.565 $\pm$ 0.021 & 4.298 $\pm$ 0.010 & 2.359 $\pm$ 0.021 & 0\
85989 & 55310.732616 & $>$16.882 & $>$15.004 & 9.264 $\pm$ 0.081 & 7.098 $\pm$ 0.260 & 0\
85989 & 55310.86492 & $>$16.703 & 14.323 $\pm$ 0.175 & 8.765 $\pm$ 0.055 & 6.734 $\pm$ 0.198 & 0\
85989 & 55310.997225 & $>$16.695 & 14.525 $\pm$ 0.266 & 8.247 $\pm$ 0.035 & 6.260 $\pm$ 0.124 & 0\
85989 & 55311.129529 & 16.623 $\pm$ 0.431 & 14.170 $\pm$ 0.162 & 8.195 $\pm$ 0.035 & 5.864 $\pm$ 0.106 & 0\
85989 & 55311.261833 & 16.237 $\pm$ 0.300 & 14.507 $\pm$ 0.212 & 8.598 $\pm$ 0.052 & 6.514 $\pm$ 0.161 & 0\
85989 & 55311.327922 & $>$16.373 & 14.234 $\pm$ 0.171 & 8.491 $\pm$ 0.044 & 6.320 $\pm$ 0.142 & 0\
85989 & 55311.394137 & $>$17.032 & $>$15.588 & 8.842 $\pm$ 0.056 & 7.073 $\pm$ 0.294 & 0\
85989 & 55311.460226 & $>$16.506 & 13.843 $\pm$ 0.126 & 8.190 $\pm$ 0.037 & 6.145 $\pm$ 0.113 & 0\
85989 & 55311.526442 & $>$16.304 & 15.161 $\pm$ 0.375 & 9.434 $\pm$ 0.094 & 7.656 $\pm$ 0.426 & 0\
85989 & 55311.59253 & $>$16.498 & 13.954 $\pm$ 0.120 & 8.425 $\pm$ 0.040 & 6.197 $\pm$ 0.114 & 0\
85989 & 55311.592657 & $>$16.636 & 14.295 $\pm$ 0.168 & 8.395 $\pm$ 0.041 & 6.429 $\pm$ 0.154 & 0\
85989 & 55311.658746 & 16.835 $\pm$ 0.465 & 14.376 $\pm$ 0.182 & 8.774 $\pm$ 0.051 & 6.872 $\pm$ 0.223 & 0\
85989 & 55311.724834 & 16.324 $\pm$ 0.288 & 14.618 $\pm$ 0.207 & 8.643 $\pm$ 0.049 & 6.711 $\pm$ 0.173 & 0\
85989 & 55311.857139 & $>$16.664 & 14.961 $\pm$ 0.291 & 9.144 $\pm$ 0.075 & 6.715 $\pm$ 0.183 & 0\
85989 & 55311.857266 & $>$17.049 & 15.356 $\pm$ 0.434 & 9.274 $\pm$ 0.082 & 6.908 $\pm$ 0.233 & 0\
85989 & 55311.989443 & 16.923 $\pm$ 0.479 & 14.216 $\pm$ 0.156 & 8.871 $\pm$ 0.058 & 6.758 $\pm$ 0.185 & 0\
85989 & 55311.98957 & $>$16.486 & 14.833 $\pm$ 0.286 & 9.028 $\pm$ 0.065 & 7.277 $\pm$ 0.308 & 0\
85989 & 55312.121747 & 16.674 $\pm$ 0.409 & 13.984 $\pm$ 0.129 & 8.267 $\pm$ 0.035 & 6.370 $\pm$ 0.135 & 0\
85989 & 55312.254051 & 15.991 $\pm$ 0.233 & 14.235 $\pm$ 0.160 & 8.105 $\pm$ 0.038 & 6.027 $\pm$ 0.099 & 0\
85989 & 55312.254051 & 15.991 $\pm$ 0.233 & 14.235 $\pm$ 0.160 & 8.105 $\pm$ 0.038 & 6.027 $\pm$ 0.099 & 0\
85989 & 55322.044695 & $>$16.852 & 13.933 $\pm$ 0.162 & 8.645 $\pm$ 0.047 & 6.643 $\pm$ 0.174 & 0\
85989 & 55406.690608 & 12.304 $\pm$ 0.026 & 9.050 $\pm$ 0.018 & 3.689 $\pm$ 0.012 & 1.755 $\pm$ 0.017 & 0\
85989 & 55406.756696 & 11.754 $\pm$ 0.023 & 8.649 $\pm$ 0.016 & 3.753 $\pm$ 0.011 & 2.008 $\pm$ 0.030 & 0\
85989 & 55406.822912 & 11.803 $\pm$ 0.030 & 8.484 $\pm$ 0.019 & 3.370 $\pm$ 0.014 & 1.590 $\pm$ 0.021 & 0\
85989 & 55406.889 & 12.697 $\pm$ 0.027 & 9.565 $\pm$ 0.021 & 4.298 $\pm$ 0.010 & 2.359 $\pm$ 0.021 & 0\
88254 & 55272.46093 & 12.569 $\pm$ 0.031 & 9.723 $\pm$ 0.025 & 4.990 $\pm$ 0.011 & 3.197 $\pm$ 0.026 & 0\
88254 & 55272.527146 & 12.450 $\pm$ 0.025 & 9.715 $\pm$ 0.020 & 4.971 $\pm$ 0.010 & 3.223 $\pm$ 0.026 & 0\
D7032 & 55473.298128 & 11.269 $\pm$ 0.027 & 8.454 $\pm$ 0.020 & – & – & 0\
D7032 & 55473.298256 & 11.372 $\pm$ 0.029 & 8.334 $\pm$ 0.019 & – & – & 0\
D7032 & 55473.364344 & 11.359 $\pm$ 0.025 & 8.296 $\pm$ 0.021 & – & – & 0\
F2742 & 55225.802684 & 14.567 $\pm$ 0.070 & 11.976 $\pm$ 0.032 & 6.167 $\pm$ 0.014 & 4.092 $\pm$ 0.037 & 0\
F2742 & 55225.868773 & 15.267 $\pm$ 0.118 & 12.286 $\pm$ 0.040 & 6.317 $\pm$ 0.015 & 4.134 $\pm$ 0.036 & 0\
F2742 & 55225.8689 & 15.185 $\pm$ 0.117 & 12.346 $\pm$ 0.038 & 6.249 $\pm$ 0.014 & 4.179 $\pm$ 0.040 & 0\
F4590 & 55373.377953 & 16.816 $\pm$ 0.435 & 14.859 $\pm$ 0.264 & 8.848 $\pm$ 0.061 & 6.480 $\pm$ 0.153 & 0\
F4590 & 55373.444041 & 16.723 $\pm$ 0.457 & $>$14.936 & 8.869 $\pm$ 0.059 & 6.789 $\pm$ 0.209 & 0\
M6554 & 55203.412309 & 15.326 $\pm$ 0.127 & 12.919 $\pm$ 0.057 & 7.121 $\pm$ 0.020 & 5.047 $\pm$ 0.053 & 0\
M6554 & 55203.80926 & 15.295 $\pm$ 0.120 & 13.179 $\pm$ 0.090 & 7.213 $\pm$ 0.021 & 5.091 $\pm$ 0.047 & 0\
M6554 & 55203.875476 & 15.556 $\pm$ 0.188 & 13.170 $\pm$ 0.069 & 7.330 $\pm$ 0.021 & 5.239 $\pm$ 0.067 & 0\
M6554 & 55204.073869 & 15.400 $\pm$ 0.140 & 13.045 $\pm$ 0.064 & 7.074 $\pm$ 0.022 & 5.040 $\pm$ 0.045 & 0\
M6554 & 55386.281957 & $>$12.860 & $>$14.954 & $>$10.718 & $>$7.573 & 0\
M6554 & 55386.41426 & $>$17.115 & $>$15.676 & 10.731 $\pm$ 0.304 & 7.157 $\pm$ 0.304 & 0\
M6554 & 55386.546564 & $>$17.023 & $>$15.444 & 10.518 $\pm$ 0.260 & $>$7.863 & 0\
M6554 & 55386.678868 & $>$16.901 & $>$15.096 & 10.410 $\pm$ 0.239 & $>$7.129 & 0\
M6554 & 55386.811171 & $>$16.550 & 14.962 $\pm$ 0.318 & 10.662 $\pm$ 0.293 & $>$7.407 & 0\
M6554 & 55386.877387 & 16.712 $\pm$ 0.419 & $>$15.551 & $>$10.911 & 7.257 $\pm$ 0.308 & 0\
M6554 & 55387.009691 & $>$16.977 & $>$14.912 & 10.847 $\pm$ 0.347 & 7.713 $\pm$ 0.469 & 0\
M6554 & 55387.141994 & 16.700 $\pm$ 0.407 & 15.178 $\pm$ 0.382 & 10.995 $\pm$ 0.387 & $>$7.727 & 0\
M6554 & 55387.274298 & $>$17.072 & 14.688 $\pm$ 0.237 & $>$10.711 & 7.375 $\pm$ 0.367 & 0\
M6554 & 55387.406602 & $>$17.026 & $>$15.627 & 10.396 $\pm$ 0.228 & $>$7.706 & 0\
M6554 & 55387.803386 & $>$17.049 & 15.149 $\pm$ 0.365 & 10.767 $\pm$ 0.321 & $>$7.198 & 0\
M6554 & 55387.803513 & $>$16.527 & $>$15.521 & 10.429 $\pm$ 0.233 & $>$7.700 & 0\
N0549 & 55213.307566 & 16.328 $\pm$ 0.290 & 14.227 $\pm$ 0.173 & 8.375 $\pm$ 0.040 & 6.201 $\pm$ 0.127 & 0\
N0549 & 55213.439998 & 15.747 $\pm$ 0.186 & 13.509 $\pm$ 0.101 & 8.018 $\pm$ 0.035 & 5.806 $\pm$ 0.094 & 0\
N0549 & 55213.506087 & 16.799 $\pm$ 0.434 & 14.208 $\pm$ 0.169 & 8.320 $\pm$ 0.039 & 6.157 $\pm$ 0.110 & 0\
N0549 & 55213.572303 & 15.969 $\pm$ 0.221 & 13.807 $\pm$ 0.134 & 8.185 $\pm$ 0.036 & 6.283 $\pm$ 0.125 & 0\
N0549 & 55213.638519 & 16.095 $\pm$ 0.235 & 13.792 $\pm$ 0.134 & 8.258 $\pm$ 0.041 & 6.021 $\pm$ 0.111 & 0\
N0549 & 55213.770951 & 15.813 $\pm$ 0.178 & 13.267 $\pm$ 0.078 & 7.899 $\pm$ 0.031 & 5.875 $\pm$ 0.089 & 0\
N4145 & 55276.662954 & 14.682 $\pm$ 0.145 & 11.154 $\pm$ 0.027 & 6.172 $\pm$ 0.015 & 4.179 $\pm$ 0.032 & 0\
N4145 & 55276.729043 & 11.760 $\pm$ 0.021 & 10.817 $\pm$ 0.026 & 6.680 $\pm$ 0.025 & 4.465 $\pm$ 0.050 & 0\
O7517 & 55405.053303 & 15.460 $\pm$ 0.140 & 13.243 $\pm$ 0.068 & 7.717 $\pm$ 0.026 & 5.443 $\pm$ 0.060 & 0\
O7517 & 55405.119519 & 15.423 $\pm$ 0.142 & 9.734 $\pm$ 0.009 & 7.927 $\pm$ 0.031 & 5.837 $\pm$ 0.083 & 0\
O7517 & 55405.185607 & 16.315 $\pm$ 0.312 & 13.888 $\pm$ 0.123 & 8.056 $\pm$ 0.033 & 5.926 $\pm$ 0.098 & 0\
O7517 & 55405.251695 & 16.410 $\pm$ 0.302 & 14.083 $\pm$ 0.128 & 7.922 $\pm$ 0.027 & 5.791 $\pm$ 0.114 & 0\
Q4357 & 55543.555716 & 16.689 $\pm$ 0.399 & 14.680 $\pm$ 0.259 & – & – & 0\
Q4357 & 55543.820325 & 16.880 $\pm$ 0.474 & 14.565 $\pm$ 0.256 & – & – & 0\
Q4357 & 55543.886414 & $>$16.733 & 14.489 $\pm$ 0.232 & – & – & 0\
Q4357 & 55543.952502 & 16.812 $\pm$ 0.474 & 14.326 $\pm$ 0.224 & – & – & 0\
Q4357 & 55544.151022 & 16.689 $\pm$ 0.387 & 14.485 $\pm$ 0.207 & – & – & 0\
W3179 & 55332.780675 & 12.511 $\pm$ 0.027 & 10.264 $\pm$ 0.021 & 5.138 $\pm$ 0.013 & 3.186 $\pm$ 0.032 & 0\
W3179 & 55332.847018 & 13.656 $\pm$ 0.047 & 10.448 $\pm$ 0.020 & 5.116 $\pm$ 0.015 & 3.136 $\pm$ 0.033 & 0\
J94C00B & 55393.643342 & 15.560 $\pm$ 0.188 & 14.105 $\pm$ 0.168 & 8.835 $\pm$ 0.052 & 7.039 $\pm$ 0.237 & 0\
J94C00B & 55393.775646 & 16.355 $\pm$ 0.392 & 14.469 $\pm$ 0.192 & 9.203 $\pm$ 0.078 & 7.080 $\pm$ 0.281 & 0\
J94C00B & 55393.841862 & 15.908 $\pm$ 0.199 & 14.145 $\pm$ 0.150 & 8.837 $\pm$ 0.056 & 6.730 $\pm$ 0.173 & 0\
J94C00B & 55393.974166 & 15.303 $\pm$ 0.200 & 14.453 $\pm$ 0.232 & 8.969 $\pm$ 0.065 & 6.985 $\pm$ 0.247 & 0\
J94C00B & 55394.040254 & 14.479 $\pm$ 0.078 & 14.054 $\pm$ 0.169 & 9.025 $\pm$ 0.061 & 6.466 $\pm$ 0.144 & 0\
J94C00B & 55394.238773 & $>$16.131 & 15.034 $\pm$ 0.337 & 8.908 $\pm$ 0.054 & 6.755 $\pm$ 0.178 & 0\
J98K03N & 55260.749926 & 12.270 $\pm$ 0.026 & 9.421 $\pm$ 0.020 & 4.826 $\pm$ 0.014 & 3.156 $\pm$ 0.026 & 0\
J98W02B & 55292.050593 & 16.337 $\pm$ 0.305 & 13.985 $\pm$ 0.130 & 8.107 $\pm$ 0.032 & 6.323 $\pm$ 0.140 & 0\
J98W02B & 55292.116808 & 16.483 $\pm$ 0.403 & 14.078 $\pm$ 0.145 & 8.246 $\pm$ 0.038 & 6.232 $\pm$ 0.129 & 0\
J99XD6K & 55542.071462 & 14.141 $\pm$ 0.058 & 11.241 $\pm$ 0.026 & – & – & 0\
K00A06A & 55403.870464 & 13.560 $\pm$ 0.041 & 10.367 $\pm$ 0.022 & 5.193 $\pm$ 0.014 & 3.271 $\pm$ 0.023 & 0\
K00A06A & 55403.870591 & 13.554 $\pm$ 0.042 & 10.371 $\pm$ 0.027 & 5.225 $\pm$ 0.014 & 3.316 $\pm$ 0.032 & 0\
K00A06C & 55525.179159 & 16.414 $\pm$ 0.331 & 13.736 $\pm$ 0.116 & – & – & 0\
K00A06C & 55525.245375 & 16.965 $\pm$ 0.490 & 13.646 $\pm$ 0.104 & – & – & 0\
K00A06C & 55525.311463 & 16.479 $\pm$ 0.317 & 13.530 $\pm$ 0.090 & – & – & 0\
K00A06C & 55525.377551 & 15.894 $\pm$ 0.193 & 13.640 $\pm$ 0.101 & – & – & 0\
K00A06C & 55525.377551 & 15.894 $\pm$ 0.193 & 13.640 $\pm$ 0.101 & – & – & 0\
K00R52E & 55269.275193 & 15.638 $\pm$ 0.144 & 12.815 $\pm$ 0.055 & 7.329 $\pm$ 0.019 & 5.542 $\pm$ 0.058 & 0\
K00R52E & 55269.341408 & 16.020 $\pm$ 0.206 & 13.452 $\pm$ 0.080 & 7.559 $\pm$ 0.023 & 5.651 $\pm$ 0.083 & 0\
K00R52E & 55269.407497 & $>$17.108 & 13.699 $\pm$ 0.100 & 7.970 $\pm$ 0.026 & 6.018 $\pm$ 0.117 & 0\
K02B25F & 55393.366511 & 15.824 $\pm$ 0.213 & 12.907 $\pm$ 0.070 & 7.008 $\pm$ 0.021 & 4.928 $\pm$ 0.043 & 0\
K02B25F & 55393.366638 & 15.831 $\pm$ 0.187 & 12.801 $\pm$ 0.062 & 7.019 $\pm$ 0.019 & 4.981 $\pm$ 0.053 & 0\
K02JA0R & 55316.413042 & $>$16.829 & $>$15.589 & 10.043 $\pm$ 0.128 & 7.732 $\pm$ 0.457 & 0\
K02X38Y & 55251.356346 & 15.556 $\pm$ 0.192 & 12.894 $\pm$ 0.078 & 7.673 $\pm$ 0.026 & 5.920 $\pm$ 0.097 & 0\
K02X38Y & 55251.422434 & 16.038 $\pm$ 0.256 & 13.548 $\pm$ 0.144 & 8.000 $\pm$ 0.034 & 6.244 $\pm$ 0.124 & 0\
K02X38Y & 55251.422562 & 16.597 $\pm$ 0.448 & 13.612 $\pm$ 0.131 & 7.949 $\pm$ 0.034 & 6.078 $\pm$ 0.121 & 0\
K03U11V & 55499.353352 & 10.816 $\pm$ 0.020 & 7.966 $\pm$ 0.019 & – & – & 0\
K04F11G & 55295.775994 & 15.355 $\pm$ 0.140 & 12.881 $\pm$ 0.054 & 7.733 $\pm$ 0.026 & 5.768 $\pm$ 0.077 & 0\
K04F11G & 55295.776121 & 15.588 $\pm$ 0.208 & 12.841 $\pm$ 0.057 & 7.720 $\pm$ 0.027 & 5.655 $\pm$ 0.082 & 0\
K04K17H & 55359.062532 & 16.962 $\pm$ 0.516 & 14.314 $\pm$ 0.182 & 8.427 $\pm$ 0.042 & 6.464 $\pm$ 0.154 & 0\
K04K17H & 55359.12862 & $>$16.453 & 13.902 $\pm$ 0.115 & 8.537 $\pm$ 0.044 & 6.653 $\pm$ 0.181 & 0\
K04K17H & 55359.194708 & $>$15.533 & 13.872 $\pm$ 0.122 & 8.260 $\pm$ 0.038 & 6.450 $\pm$ 0.163 & 0\
K04K17H & 55359.260797 & $>$16.426 & 13.637 $\pm$ 0.096 & 8.063 $\pm$ 0.034 & 6.265 $\pm$ 0.124 & 0\
K04K17H & 55359.260924 & $>$16.571 & 13.719 $\pm$ 0.116 & 8.144 $\pm$ 0.037 & 6.152 $\pm$ 0.124 & 0\
K04K17H & 55359.3931 & $>$16.406 & 14.569 $\pm$ 0.203 & 8.451 $\pm$ 0.043 & 6.567 $\pm$ 0.176 & 0\
K04S56C & 55416.036814 & $>$16.635 & $>$15.048 & 9.756 $\pm$ 0.122 & – & 0\
K04S56C & 55416.16899 & 13.643 $\pm$ 0.062 & 13.425 $\pm$ 0.094 & 10.840 $\pm$ 0.372 & – & 0\
K04S56C & 55416.367382 & 16.373 $\pm$ 0.413 & $>$15.258 & 10.356 $\pm$ 0.222 & – & 0\
K04S56C & 55416.367382 & 15.689 $\pm$ 0.206 & $>$15.384 & 10.495 $\pm$ 0.254 & – & 0\
K05GC0C & 55532.148373 & 13.180 $\pm$ 0.038 & 10.531 $\pm$ 0.024 & – & – & 0\
K05U00O & 55265.148428 & $>$17.014 & 14.885 $\pm$ 0.265 & 9.463 $\pm$ 0.098 & 7.619 $\pm$ 0.456 & 0\
K05U00O & 55265.214644 & $>$16.971 & 15.190 $\pm$ 0.390 & 9.507 $\pm$ 0.104 & $>$6.858 & 0\
K05X01B & 55305.354108 & $>$17.074 & $>$15.772 & 10.651 $\pm$ 0.262 & $>$7.230 & 0\
K05X01B & 55305.420196 & $>$17.229 & $>$15.407 & 10.504 $\pm$ 0.210 & 7.341 $\pm$ 0.327 & 0\
K05X01B & 55305.486412 & $>$17.113 & $>$14.983 & 10.437 $\pm$ 0.214 & $>$7.641 & 0\
K05X01B & 55305.5525 & $>$16.888 & $>$14.939 & 10.695 $\pm$ 0.257 & $>$7.978 & 0\
K07E00F & 55256.742863 & 15.018 $\pm$ 0.129 & 11.903 $\pm$ 0.034 & 6.476 $\pm$ 0.017 & 4.648 $\pm$ 0.044 & 0\
K07E00F & 55256.808952 & 14.948 $\pm$ 0.098 & 11.836 $\pm$ 0.032 & 6.446 $\pm$ 0.017 & 4.588 $\pm$ 0.033 & 0\
K08N03P & 55398.084496 & 16.307 $\pm$ 0.331 & 12.470 $\pm$ 0.052 & 7.210 $\pm$ 0.019 & 5.198 $\pm$ 0.055 & 0\
K08N03P & 55398.150584 & 16.095 $\pm$ 0.256 & 12.847 $\pm$ 0.065 & 7.480 $\pm$ 0.023 & 5.503 $\pm$ 0.070 & 0\
K08X00M & 55214.618135 & 15.190 $\pm$ 0.107 & 12.786 $\pm$ 0.056 & 6.877 $\pm$ 0.020 & 4.922 $\pm$ 0.182 & 0\
K08X00M & 55214.618262 & 15.185 $\pm$ 0.109 & 12.737 $\pm$ 0.052 & 6.928 $\pm$ 0.018 & 4.768 $\pm$ 0.128 & 0\
K09B00D\* & 55360.417023 & $>$16.738 & $>$15.505 & $>$11.776 & $>$7.587 & 11, 11, 11, 12\
K09F00D & 55493.292455 & 14.786 $\pm$ 0.082 & 11.401 $\pm$ 0.036 & – & – & 0\
K09F00D & 55493.358543 & 15.101 $\pm$ 0.105 & 11.453 $\pm$ 0.032 & – & – & 0\
K09F00D & 55493.358671 & 14.893 $\pm$ 0.096 & 11.408 $\pm$ 0.031 & – & – & 0\
K09F04Y & 55245.376313 & $>$15.811 & 15.222 $\pm$ 0.419 & 9.254 $\pm$ 0.087 & 7.373 $\pm$ 0.368 & 0\
K09U19Y & 55278.763519 & $>$17.235 & 14.870 $\pm$ 0.271 & 9.418 $\pm$ 0.082 & 7.533 $\pm$ 0.346 & 0\
K09U19Y & 55278.895824 & $>$16.511 & $>$15.076 & 9.405 $\pm$ 0.079 & 7.774 $\pm$ 0.441 & 0\
K09U19Y & 55279.689648 & $>$17.143 & 15.374 $\pm$ 0.406 & 9.263 $\pm$ 0.068 & 7.098 $\pm$ 0.230 & 0\
K09U19Y & 55279.689775 & $>$16.527 & 15.212 $\pm$ 0.333 & 9.248 $\pm$ 0.073 & 6.444 $\pm$ 0.134 & 0\
K09U19Y & 55279.821952 & $>$16.501 & 15.004 $\pm$ 0.288 & 9.188 $\pm$ 0.067 & 7.033 $\pm$ 0.220 & 0\
K09U19Y & 55279.822079 & $>$16.702 & 14.555 $\pm$ 0.203 & 9.230 $\pm$ 0.072 & 7.358 $\pm$ 0.309 & 0\
K09U19Y & 55279.954383 & $>$16.442 & 15.162 $\pm$ 0.337 & 9.189 $\pm$ 0.065 & 6.978 $\pm$ 0.242 & 0\
K09U19Y & 55280.218991 & $>$16.082 & 14.538 $\pm$ 0.181 & 9.182 $\pm$ 0.067 & 7.726 $\pm$ 0.432 & 0\
K09U19Y & 55280.351295 & $>$16.632 & 14.727 $\pm$ 0.215 & 9.378 $\pm$ 0.080 & 7.044 $\pm$ 0.222 & 0\
K09U19Y & 55280.748209 & $>$16.964 & 15.407 $\pm$ 0.400 & 9.237 $\pm$ 0.071 & 7.537 $\pm$ 0.355 & 0\
K09U19Y & 55280.880513 & $>$16.555 & 14.721 $\pm$ 0.238 & 9.188 $\pm$ 0.068 & 7.192 $\pm$ 0.266 & 0\
K09U19Y & 55281.012817 & $>$17.067 & 14.691 $\pm$ 0.211 & 9.159 $\pm$ 0.065 & 7.079 $\pm$ 0.255 & 0\
K09U19Y & 55281.145121 & 16.812 $\pm$ 0.414 & 14.512 $\pm$ 0.180 & 9.191 $\pm$ 0.072 & 6.953 $\pm$ 0.219 & 0\
K09U19Y & 55281.277425 & 16.580 $\pm$ 0.420 & 14.732 $\pm$ 0.227 & 9.070 $\pm$ 0.061 & $>$7.336 & 0\
K09U19Y & 55281.542034 & $>$17.217 & 15.122 $\pm$ 0.303 & 9.308 $\pm$ 0.077 & 7.120 $\pm$ 0.265 & 0\
K09U19Y & 55281.674338 & $>$17.046 & 14.963 $\pm$ 0.277 & 9.359 $\pm$ 0.079 & 7.065 $\pm$ 0.234 & 0\
K09U19Y & 55281.806642 & $>$17.167 & 15.254 $\pm$ 0.395 & 9.310 $\pm$ 0.076 & 6.602 $\pm$ 0.154 & 0\
K09U19Y & 55281.938946 & $>$16.666 & 15.011 $\pm$ 0.259 & 9.318 $\pm$ 0.076 & 6.804 $\pm$ 0.188 & 0\
K09U19Y & 55281.939073 & 16.850 $\pm$ 0.439 & 14.908 $\pm$ 0.241 & 9.333 $\pm$ 0.075 & 6.835 $\pm$ 0.196 & 0\
K09U19Y & 55282.07125 & 16.503 $\pm$ 0.318 & 14.572 $\pm$ 0.198 & 9.186 $\pm$ 0.068 & 6.891 $\pm$ 0.202 & 0\
K09U19Y & 55282.071377 & $>$17.038 & 15.316 $\pm$ 0.396 & 9.128 $\pm$ 0.064 & 7.006 $\pm$ 0.220 & 0\
K09U19Y & 55282.335985 & 14.923 $\pm$ 0.095 & 13.620 $\pm$ 0.088 & 8.926 $\pm$ 0.054 & 6.706 $\pm$ 0.198 & 0\
K09U19Y & 55282.600593 & $>$16.455 & 14.354 $\pm$ 0.154 & – & – & 0\
K09U19Y & 55282.732897 & $>$17.212 & 14.941 $\pm$ 0.251 & 8.945 $\pm$ 0.054 & 6.911 $\pm$ 0.220 & 0\
K10A02N & 55244.206332 & $>$16.811 & 14.802 $\pm$ 0.229 & 9.884 $\pm$ 0.133 & $>$7.847 & 0\
K10A02N & 55244.338636 & $>$17.032 & 14.822 $\pm$ 0.255 & 9.873 $\pm$ 0.132 & $>$7.496 & 0\
K10A02N & 55244.404724 & $>$16.554 & 15.125 $\pm$ 0.324 & 10.055 $\pm$ 0.156 & 7.659 $\pm$ 0.450 & 0\
K10A02N & 55244.537028 & $>$16.716 & 15.603 $\pm$ 0.519 & 10.006 $\pm$ 0.136 & $>$7.672 & 0\
K10A30E & 55214.760372 & $>$17.101 & 15.572 $\pm$ 0.515 & 9.608 $\pm$ 0.095 & 7.778 $\pm$ 0.494 & 0\
K10A30E & 55214.760372 & $>$16.228 & $>$15.693 & 9.971 $\pm$ 0.177 & $>$7.345 & 0\
K10A30E & 55214.760372 & $>$17.152 & $>$15.588 & 10.732 $\pm$ 0.275 & $>$7.461 & 0\
K10A40G & 55220.615821 & 15.638 $\pm$ 0.151 & 12.708 $\pm$ 0.058 & 7.519 $\pm$ 0.024 & 5.759 $\pm$ 0.078 & 0\
K10B00C & 55221.475481 & 14.694 $\pm$ 0.075 & 12.457 $\pm$ 0.045 & 6.435 $\pm$ 0.016 & 4.497 $\pm$ 0.037 & 0\
K10B02K & 55216.413863 & 16.131 $\pm$ 0.248 & 13.081 $\pm$ 0.065 & 7.304 $\pm$ 0.022 & 5.268 $\pm$ 0.051 & 0\
K10C01K & 55252.229502 & 16.686 $\pm$ 0.377 & $>$15.159 & 10.294 $\pm$ 0.169 & $>$7.384 & 0\
K10C01K & 55252.229629 & $>$16.508 & $>$14.927 & 10.397 $\pm$ 0.192 & $>$7.519 & 0\
K10C01K & 55252.229502 & 16.686 $\pm$ 0.377 & $>$15.159 & 10.294 $\pm$ 0.169 & $>$7.384 & 0\
K10C19C & 55247.394099 & $>$16.478 & 14.248 $\pm$ 0.158 & 9.296 $\pm$ 0.091 & 7.454 $\pm$ 0.353 & 0\
K10C19C & 55247.394226 & $>$16.830 & 14.356 $\pm$ 0.173 & 9.226 $\pm$ 0.075 & 7.301 $\pm$ 0.319 & 0\
K10C19C & 55247.460315 & $>$16.600 & 14.481 $\pm$ 0.229 & 9.225 $\pm$ 0.081 & 7.758 $\pm$ 0.498 & 0\
K10C19F & 55225.611168 & $>$17.173 & 15.114 $\pm$ 0.438 & 10.664 $\pm$ 0.244 & $>$7.254 & 0\
K10C19F & 55225.677384 & 16.800 $\pm$ 0.470 & 15.251 $\pm$ 0.370 & 10.645 $\pm$ 0.234 & $>$7.382 & 0\
K10C19F & 55225.7436 & $>$17.204 & 15.669 $\pm$ 0.495 & 10.035 $\pm$ 0.147 & $>$7.374 & 0\
K10C44O & 55220.21114 & $>$17.200 & $>$15.141 & 10.278 $\pm$ 0.193 & $>$7.922 & 0\
K10C55D & 55251.619171 & 15.958 $\pm$ 0.254 & 13.434 $\pm$ 0.116 & 8.073 $\pm$ 0.032 & 6.334 $\pm$ 0.152 & 0\
K10C55D & 55251.68526 & $>$16.906 & 14.267 $\pm$ 0.286 & 8.409 $\pm$ 0.048 & 6.725 $\pm$ 0.228 & 0\
K10C55F & 55272.262029 & $>$16.712 & 14.721 $\pm$ 0.251 & 9.711 $\pm$ 0.115 & $>$7.155 & 0\
K10D00L & 55224.83708 & $>$16.476 & 13.903 $\pm$ 0.120 & 9.229 $\pm$ 0.077 & $>$6.882 & 0\
K10D00L & 55224.903296 & $>$16.299 & 13.521 $\pm$ 0.092 & 9.144 $\pm$ 0.079 & $>$7.258 & 0\
K10D00O & 55249.932962 & 16.739 $\pm$ 0.481 & 13.657 $\pm$ 0.103 & 7.925 $\pm$ 0.032 & 5.966 $\pm$ 0.105 & 0\
K10D01F & 55287.22576 & $>$17.163 & $>$15.668 & 10.142 $\pm$ 0.161 & $>$7.320 & 0\
K10D01F & 55287.291976 & $>$16.830 & $>$15.321 & 10.440 $\pm$ 0.196 & 7.725 $\pm$ 0.496 & 0\
K10D01F & 55287.358064 & $>$17.214 & 15.697 $\pm$ 0.538 & 10.666 $\pm$ 0.256 & 7.897 $\pm$ 0.514 & 0\
K10D01F & 55287.42428 & $>$17.017 & 14.891 $\pm$ 0.290 & 10.490 $\pm$ 0.221 & 7.873 $\pm$ 0.505 & 0\
K10D01F & 55287.490496 & $>$16.888 & $>$15.135 & 10.214 $\pm$ 0.162 & 7.728 $\pm$ 0.476 & 0\
K10D01F & 55287.556584 & $>$17.143 & $>$15.007 & 10.832 $\pm$ 0.293 & $>$7.326 & 0\
K10D01X & 55238.600531 & 15.791 $\pm$ 0.171 & 13.216 $\pm$ 0.068 & 8.210 $\pm$ 0.032 & 6.161 $\pm$ 0.109 & 0\
K10D01X & 55238.600658 & 15.832 $\pm$ 0.199 & 13.115 $\pm$ 0.066 & 8.381 $\pm$ 0.042 & 6.314 $\pm$ 0.142 & 0\
K10D01X & 55238.666746 & $>$16.330 & 13.737 $\pm$ 0.108 & 8.565 $\pm$ 0.044 & 6.411 $\pm$ 0.131 & 0\
K10E21G\* & 55296.4873036 & $>$17.061 & $>$13.603 & 11.586 $\pm$ 0.161 & $>$7.682 & 11, 11, 11, 12\
K10E43C & 55282.188656 & $>$17.014 & $>$15.274 & 9.976 $\pm$ 0.158 & $>$7.219 & 0\
K10E43C & 55282.585568 & $>$17.018 & $>$15.498 & – & 7.165 $\pm$ 0.281 & 0\
K10E43C & 55282.651784 & $>$16.900 & $>$15.235 & 10.709 $\pm$ 0.346 & $>$7.158 & 0\
K10E43C & 55282.784088 & $>$16.960 & $>$15.462 & 10.613 $\pm$ 0.276 & $>$6.939 & 0\
K10E43K & 55254.097037 & $>$16.463 & 14.365 $\pm$ 0.181 & 8.700 $\pm$ 0.052 & 6.621 $\pm$ 0.185 & 0\
K10E43K & 55254.163125 & 16.566 $\pm$ 0.341 & 14.107 $\pm$ 0.139 & 8.708 $\pm$ 0.053 & 6.765 $\pm$ 0.224 & 0\
K10F00C\* & 55306.8863662 & $>$17.061 & $>$13.603 & 12.254 $\pm$ 0.175 & $>$8.253 & 11, 11, 11, 12\
K10F00K\* & 55293.0473 & $>$17.061 & $>$13.603 & 11.422 $\pm$ 0.133 & $>$7.762 & 11, 11, 11, 12\
K10F00S & 55271.941138 & $>$16.908 & $>$15.191 & 9.927 $\pm$ 0.135 & $>$7.898 & 0\
K10F00S & 55272.007226 & $>$17.171 & $>$15.792 & 9.767 $\pm$ 0.115 & $>$7.273 & 0\
K10F00S & 55272.007353 & $>$17.060 & $>$15.696 & 9.981 $\pm$ 0.143 & $>$7.149 & 0\
K10F00T & 55280.951568 & $>$16.639 & $>$15.228 & 8.881 $\pm$ 0.051 & 6.377 $\pm$ 0.130 & 0\
K10F06D & 55267.728293 & 16.184 $\pm$ 0.268 & 15.167 $\pm$ 0.345 & 8.717 $\pm$ 0.053 & 6.713 $\pm$ 0.181 & 0\
K10F06D & 55267.72842 & $>$16.968 & $>$14.921 & 8.612 $\pm$ 0.049 & 6.391 $\pm$ 0.160 & 0\
K10F07D & 55302.871016 & $>$16.552 & 14.559 $\pm$ 0.188 & 8.576 $\pm$ 0.044 & 6.666 $\pm$ 0.184 & 0\
K10F07D & 55302.937232 & $>$16.507 & 14.730 $\pm$ 0.235 & 8.687 $\pm$ 0.046 & 6.747 $\pm$ 0.173 & 0\
K10F07E & 55241.261768 & $>$17.031 & $>$15.039 & 9.726 $\pm$ 0.118 & 7.340 $\pm$ 0.339 & 0\
K10F07E & 55241.327983 & $>$17.161 & 15.454 $\pm$ 0.487 & 9.326 $\pm$ 0.083 & $>$7.238 & 0\
K10F09R & 55288.3173 & $>$17.117 & $>$15.582 & 10.326 $\pm$ 0.194 & $>$7.471 & 0\
K10F09R & 55288.383389 & $>$17.150 & $>$15.178 & 10.276 $\pm$ 0.182 & $>$7.332 & 0\
K10F09W & 55282.31905 & 15.821 $\pm$ 0.214 & 13.149 $\pm$ 0.072 & 8.357 $\pm$ 0.039 & 6.496 $\pm$ 0.158 & 0\
K10F09W & 55282.319177 & 15.122 $\pm$ 0.146 & 13.109 $\pm$ 0.092 & 8.212 $\pm$ 0.039 & 6.381 $\pm$ 0.157 & 0\
K10F09X & 55272.489454 & $>$17.159 & $>$15.787 & 10.673 $\pm$ 0.255 & 7.103 $\pm$ 0.243 & 0\
K10F10F & 55318.265811 & $>$17.157 & 14.946 $\pm$ 0.300 & 9.399 $\pm$ 0.079 & 7.992 $\pm$ 0.513 & 0\
K10G05Z & 55307.946455 & $>$16.446 & 14.622 $\pm$ 0.200 & 8.574 $\pm$ 0.043 & 6.244 $\pm$ 0.120 & 0\
K10G06B\* & 55261.791 & $>$16.927 & $>$15.794 & 11.888 $\pm$ 0.235 & $>$7.973 & 11, 11, 11, 12\
K10G06T & 55318.700544 & $>$17.166 & 15.292 $\pm$ 0.366 & 10.456 $\pm$ 0.205 & 7.824 $\pm$ 0.463 & 0\
K10G06T & 55318.700671 & $>$17.247 & 15.609 $\pm$ 0.526 & 10.462 $\pm$ 0.222 & $>$7.390 & 0\
K10G06T & 55318.76676 & $>$17.089 & 14.897 $\pm$ 0.276 & 9.825 $\pm$ 0.121 & $>$7.573 & 0\
K10G06U & 55321.642434 & $>$16.454 & 14.069 $\pm$ 0.189 & 8.378 $\pm$ 0.055 & 6.088 $\pm$ 0.150 & 0\
K10G06U & 55321.70865 & $>$16.679 & 14.664 $\pm$ 0.291 & 8.380 $\pm$ 0.047 & 6.455 $\pm$ 0.158 & 0\
K10G06U & 55321.774738 & $>$16.315 & 13.743 $\pm$ 0.115 & 8.329 $\pm$ 0.038 & 6.486 $\pm$ 0.191 & 0\
K10G07H & 55288.141829 & $>$16.582 & 14.325 $\pm$ 0.183 & 8.567 $\pm$ 0.046 & 6.380 $\pm$ 0.153 & 0\
K10G23L & 55273.164982 & $>$16.997 & $>$15.638 & 11.493 $\pm$ 0.523 & $>$7.641 & 0\
K10G23L & 55273.164982 & $>$16.939 & $>$15.625 & 9.989 $\pm$ 0.143 & $>$7.388 & 0\
K10G23L & 55273.231197 & $>$16.915 & 15.398 $\pm$ 0.414 & 10.253 $\pm$ 0.174 & $>$7.152 & 0\
K10G23L & 55273.231197 & 13.837 $\pm$ 0.055 & 13.993 $\pm$ 0.124 & $>$10.847 & $>$7.775 & 0\
K10G23L & 55273.297286 & $>$16.793 & $>$15.526 & 9.979 $\pm$ 0.135 & $>$7.598 & 0\
K10G23L & 55273.297286 & 13.154 $\pm$ 0.037 & 13.087 $\pm$ 0.067 & $>$11.496 & $>$7.975 & 0\
K10G23L & 55273.363501 & 14.061 $\pm$ 0.058 & 13.951 $\pm$ 0.125 & 10.667 $\pm$ 0.264 & $>$7.793 & 0\
K10G23L & 55273.363501 & 11.909 $\pm$ 0.025 & 11.984 $\pm$ 0.036 & 11.300 $\pm$ 0.462 & $>$7.965 & 0\
K10G23L & 55273.363501 & 15.795 $\pm$ 0.186 & $>$14.787 & $>$11.463 & $>$7.669 & 0\
K10G23M & 55299.095057 & 16.852 $\pm$ 0.479 & 12.864 $\pm$ 0.060 & 6.363 $\pm$ 0.017 & 4.285 $\pm$ 0.033 & 0\
K10G23X & 55282.677505 & $>$17.093 & 14.521 $\pm$ 0.218 & 8.477 $\pm$ 0.041 & 6.404 $\pm$ 0.134 & 0\
K10G23X & 55282.743721 & $>$16.873 & 14.644 $\pm$ 0.240 & 8.511 $\pm$ 0.048 & 6.427 $\pm$ 0.151 & 0\
K10G24A & 55330.6605 & 14.919 $\pm$ 0.090 & 12.578 $\pm$ 0.047 & 7.599 $\pm$ 0.023 & 5.722 $\pm$ 0.077 & 0\
K10G24A & 55330.726716 & 15.176 $\pm$ 0.102 & 12.702 $\pm$ 0.049 & 7.646 $\pm$ 0.023 & 5.635 $\pm$ 0.076 & 0\
K10H00A & 55285.077825 & $>$17.064 & $>$15.007 & 10.495 $\pm$ 0.208 & $>$7.507 & 0\
K10H20V & 55319.519836 & 15.660 $\pm$ 0.151 & 12.061 $\pm$ 0.036 & 6.381 $\pm$ 0.019 & 4.535 $\pm$ 0.032 & 0\
K10H20V & 55319.519964 & 15.833 $\pm$ 0.176 & 11.990 $\pm$ 0.037 & 6.415 $\pm$ 0.015 & 4.550 $\pm$ 0.035 & 0\
K10H20V & 55319.586052 & 14.486 $\pm$ 0.069 & 11.294 $\pm$ 0.026 & 6.238 $\pm$ 0.013 & 4.542 $\pm$ 0.123 & 0\
K10J03H & 55328.677603 & $>$17.081 & 15.557 $\pm$ 0.421 & 9.483 $\pm$ 0.088 & 7.270 $\pm$ 0.275 & 0\
K10J03J & 55305.935151 & $>$17.097 & 15.331 $\pm$ 0.454 & 9.090 $\pm$ 0.066 & 7.251 $\pm$ 0.342 & 0\
K10J03J & 55306.001239 & $>$16.496 & 14.761 $\pm$ 0.246 & 9.230 $\pm$ 0.081 & 7.106 $\pm$ 0.250 & 0\
K10J34V & 55341.608746 & 14.392 $\pm$ 0.063 & 11.585 $\pm$ 0.027 & 6.261 $\pm$ 0.014 & 4.168 $\pm$ 0.031 & 0\
K10J39W & 55307.8032 & $>$16.769 & $>$15.647 & 10.610 $\pm$ 0.239 & $>$7.988 & 0\
K10J39W & 55307.869416 & $>$16.657 & $>$15.377 & 10.324 $\pm$ 0.208 & $>$7.250 & 0\
K10J41L & 55335.655835 & $>$16.305 & 13.066 $\pm$ 0.059 & 7.537 $\pm$ 0.022 & 5.485 $\pm$ 0.061 & 0\
K10J41L & 55335.721924 & 16.795 $\pm$ 0.403 & 13.748 $\pm$ 0.103 & 7.949 $\pm$ 0.028 & 5.786 $\pm$ 0.073 & 0\
K10J71O & 55343.927944 & $>$16.688 & $>$15.660 & 10.462 $\pm$ 0.202 & $>$7.427 & 0\
K10K07V & 55341.066415 & $>$16.441 & 15.270 $\pm$ 0.498 & 10.240 $\pm$ 0.200 & 7.521 $\pm$ 0.403 & 0\
K10K07V & 55341.132631 & $>$17.066 & 15.266 $\pm$ 0.430 & 10.348 $\pm$ 0.214 & 7.123 $\pm$ 0.320 & 0\
K10K08A & 55305.657172 & $>$16.512 & 14.148 $\pm$ 0.139 & 8.721 $\pm$ 0.051 & 6.624 $\pm$ 0.172 & 0\
K10K08A & 55305.789476 & 15.988 $\pm$ 0.206 & 14.027 $\pm$ 0.140 & 8.820 $\pm$ 0.048 & 6.416 $\pm$ 0.148 & 0\
K10K10P & 55302.942835 & $>$17.152 & $>$15.425 & 10.511 $\pm$ 0.198 & 7.632 $\pm$ 0.370 & 0\
K10K10P & 55303.009051 & $>$17.211 & $>$15.776 & 11.151 $\pm$ 0.340 & $>$7.586 & 0\
K10K10P & 55303.009051 & 16.011 $\pm$ 0.204 & 15.407 $\pm$ 0.416 & $>$11.233 & 7.098 $\pm$ 0.260 & 0\
K10K10P & 55303.207443 & 15.800 $\pm$ 0.175 & 14.804 $\pm$ 0.226 & $>$11.111 & $>$7.276 & 0\
K10K10P & 55303.273659 & $>$16.880 & $>$15.467 & 10.368 $\pm$ 0.186 & 7.231 $\pm$ 0.297 & 0\
K10K10P & 55303.538267 & $>$16.984 & 15.272 $\pm$ 0.330 & 10.359 $\pm$ 0.182 & $>$7.632 & 0\
K10L34K & 55363.962739 & $>$17.273 & 15.292 $\pm$ 0.382 & 9.510 $\pm$ 0.084 & 6.959 $\pm$ 0.236 & 0\
K10L34K & 55364.028828 & $>$16.940 & 15.264 $\pm$ 0.355 & 9.146 $\pm$ 0.069 & 6.873 $\pm$ 0.195 & 0\
K10L34K & 55364.028955 & $>$17.104 & $>$15.645 & 9.130 $\pm$ 0.069 & 6.969 $\pm$ 0.231 & 0\
K10L61K & 55344.064577 & $>$17.003 & 14.878 $\pm$ 0.261 & 9.004 $\pm$ 0.058 & 6.900 $\pm$ 0.252 & 0\
K10L61K & 55344.130665 & 16.252 $\pm$ 0.321 & 15.338 $\pm$ 0.372 & 9.314 $\pm$ 0.084 & 7.026 $\pm$ 0.219 & 0\
K10L61K & 55344.130793 & 15.474 $\pm$ 0.143 & 14.273 $\pm$ 0.199 & 9.302 $\pm$ 0.071 & 7.803 $\pm$ 0.512 & 0\
K10L61K & 55344.196881 & $>$17.073 & $>$15.640 & 9.235 $\pm$ 0.071 & 6.921 $\pm$ 0.262 & 0\
K10L61K & 55344.262969 & $>$17.049 & $>$15.141 & 9.218 $\pm$ 0.068 & 7.788 $\pm$ 0.459 & 0\
K10L63Z & 55374.316813 & 13.701 $\pm$ 0.045 & 10.485 $\pm$ 0.024 & 5.564 $\pm$ 0.012 & 3.634 $\pm$ 0.047 & 0\
K10L63Z & 55374.382901 & 13.626 $\pm$ 0.041 & 10.432 $\pm$ 0.023 & 5.486 $\pm$ 0.011 & 3.551 $\pm$ 0.026 & 0\
K10L64B\* & 55381.0071 & $>$16.588 & $>$15.548 & 11.382 $\pm$ 0.135 & $>$7.747 & 11, 11, 11, 12\
K10M01G & 55383.93678 & $>$17.192 & $>$14.858 & 10.054 $\pm$ 0.154 & 7.681 $\pm$ 0.469 & 0\
K10M01G & 55384.002996 & $>$17.181 & $>$15.163 & 10.422 $\pm$ 0.202 & $>$7.130 & 0\
K10M01G & 55384.069211 & $>$17.141 & $>$15.594 & 10.175 $\pm$ 0.166 & 7.526 $\pm$ 0.392 & 0\
K10M01G & 55384.135299 & $>$17.196 & $>$15.096 & 10.256 $\pm$ 0.203 & $>$7.701 & 0\
K10M01G & 55384.26773 & $>$17.125 & $>$15.074 & 10.464 $\pm$ 0.211 & $>$7.595 & 0\
K10M01P & 55360.337436 & 16.862 $\pm$ 0.479 & $>$15.724 & 8.502 $\pm$ 0.039 & 6.079 $\pm$ 0.093 & 0\
K10M01P & 55360.403525 & $>$15.920 & $>$15.576 & 8.769 $\pm$ 0.048 & 6.655 $\pm$ 0.181 & 0\
K10M01Y & 55372.696314 & $>$16.615 & 14.617 $\pm$ 0.206 & 8.654 $\pm$ 0.043 & 6.658 $\pm$ 0.166 & 0\
K10M01Y & 55372.76253 & 16.335 $\pm$ 0.266 & $>$15.636 & 8.769 $\pm$ 0.046 & 6.948 $\pm$ 0.226 & 0\
K10M01Y & 55372.828618 & 16.262 $\pm$ 0.274 & 14.392 $\pm$ 0.172 & 8.564 $\pm$ 0.041 & 6.624 $\pm$ 0.157 & 0\
K10M01Y & 55372.894706 & $>$16.486 & 15.195 $\pm$ 0.359 & 9.006 $\pm$ 0.059 & 6.840 $\pm$ 0.203 & 0\
K10N00A & 55381.237224 & 16.273 $\pm$ 0.271 & 13.678 $\pm$ 0.095 & 7.745 $\pm$ 0.026 & 5.711 $\pm$ 0.078 & 0\
K10N01K & 55382.945966 & 13.523 $\pm$ 0.039 & 10.452 $\pm$ 0.023 & 5.488 $\pm$ 0.011 & 3.512 $\pm$ 0.030 & 0\
K10N01K & 55383.012182 & 13.558 $\pm$ 0.041 & 10.422 $\pm$ 0.024 & 5.470 $\pm$ 0.016 & 3.563 $\pm$ 0.035 & 0\
K10N01K & 55383.144485 & 13.534 $\pm$ 0.047 & 10.396 $\pm$ 0.020 & 5.470 $\pm$ 0.018 & 3.541 $\pm$ 0.020 & 0\
K10O01A & 55398.91461 & 16.421 $\pm$ 0.383 & 13.596 $\pm$ 0.102 & 8.251 $\pm$ 0.038 & 6.915 $\pm$ 0.267 & 0\
K10O01A & 55398.980825 & 15.899 $\pm$ 0.196 & 13.627 $\pm$ 0.104 & 8.334 $\pm$ 0.041 & 6.391 $\pm$ 0.162 & 0\
K10P09K & 55395.805917 & 16.739 $\pm$ 0.438 & 14.737 $\pm$ 0.270 & 9.843 $\pm$ 0.146 & $>$7.050 & 0\
K10P09K & 55395.93822 & 16.515 $\pm$ 0.379 & 14.325 $\pm$ 0.179 & 9.119 $\pm$ 0.081 & 7.301 $\pm$ 0.350 & 0\
K10P09K & 55396.004309 & $>$16.592 & 14.660 $\pm$ 0.222 & 9.487 $\pm$ 0.109 & $>$7.048 & 0\
K10P09K & 55396.004436 & 16.877 $\pm$ 0.500 & 14.928 $\pm$ 0.307 & 9.321 $\pm$ 0.089 & 7.189 $\pm$ 0.307 & 0\
K10P66R & 55303.619254 & $>$17.055 & $>$15.662 & 10.442 $\pm$ 0.229 & $>$7.579 & 0\
K10P66R & 55304.016167 & $>$16.998 & $>$15.731 & 10.740 $\pm$ 0.310 & $>$7.769 & 0\
K10P66R & 55304.148471 & 14.998 $\pm$ 0.097 & 15.151 $\pm$ 0.313 & $>$11.172 & $>$7.563 & 0\
K10P66R & 55304.148599 & 15.248 $\pm$ 0.123 & 14.869 $\pm$ 0.252 & 10.897 $\pm$ 0.306 & $>$7.793 & 0\
K10P66R & 55304.280903 & $>$16.998 & $>$15.254 & 10.344 $\pm$ 0.191 & 7.164 $\pm$ 0.266 & 0\
K10P66R & 55304.346991 & $>$17.147 & $>$15.614 & 10.427 $\pm$ 0.223 & $>$7.836 & 0\
K10P66R & 55304.413207 & $>$16.841 & $>$15.133 & 10.232 $\pm$ 0.175 & $>$7.867 & 0\
K10P66R & 55304.6116 & $>$17.133 & $>$15.428 & 10.468 $\pm$ 0.214 & $>$7.867 & 0\
K10P66R & 55304.743904 & $>$17.045 & $>$14.979 & 10.479 $\pm$ 0.220 & $>$7.645 & 0\
K10P66R & 55308.250156 & $>$16.568 & 15.019 $\pm$ 0.390 & 10.197 $\pm$ 0.180 & $>$7.602 & 0\
K10P66R & 55399.411991 & 11.734 $\pm$ 0.022 & 8.794 $\pm$ 0.018 & 3.289 $\pm$ 0.015 & 1.475 $\pm$ 0.017 & 0\
K10P66R & 55303.619254 & $>$17.055 & $>$15.662 & 10.442 $\pm$ 0.229 & $>$7.579 & 0\
K10P66R & 55304.016167 & $>$16.998 & $>$15.731 & 10.740 $\pm$ 0.310 & $>$7.769 & 0\
K10P66R & 55304.148471 & 14.998 $\pm$ 0.097 & 15.151 $\pm$ 0.313 & $>$11.172 & $>$7.563 & 0\
K10P66R & 55304.148599 & 15.248 $\pm$ 0.123 & 14.869 $\pm$ 0.252 & 10.897 $\pm$ 0.306 & $>$7.793 & 0\
K10P66R & 55304.280903 & $>$16.998 & $>$15.254 & 10.344 $\pm$ 0.191 & 7.164 $\pm$ 0.266 & 0\
K10P66R & 55304.346991 & $>$17.147 & $>$15.614 & 10.427 $\pm$ 0.223 & $>$7.836 & 0\
K10P66R & 55304.413207 & $>$16.841 & $>$15.133 & 10.232 $\pm$ 0.175 & $>$7.867 & 0\
K10P66R & 55304.6116 & $>$17.133 & $>$15.428 & 10.468 $\pm$ 0.214 & $>$7.867 & 0\
K10P66R & 55304.743904 & $>$17.045 & $>$14.979 & 10.479 $\pm$ 0.220 & $>$7.645 & 0\
K10P66R & 55308.250156 & $>$16.568 & 15.019 $\pm$ 0.390 & 10.197 $\pm$ 0.180 & $>$7.602 & 0\
K10P66R & 55399.411991 & 11.734 $\pm$ 0.022 & 8.794 $\pm$ 0.018 & 3.289 $\pm$ 0.015 & 1.475 $\pm$ 0.017 & 0\
K10Q02G & 55442.835858 & 16.537 $\pm$ 0.418 & 12.840 $\pm$ 0.105 & $>$8.231 & – & 0\
K10R53J & 55446.26074 & 15.112 $\pm$ 0.117 & 11.506 $\pm$ 0.030 & 6.060 $\pm$ 0.063 & – & 0\
K10T04N & 55468.515321 & $>$17.061 & 13.603 $\pm$ 0.105 & $>$8.338 & – & 0\
K10TE9U & 55498.588434 & 15.247 $\pm$ 0.163 & 12.164 $\pm$ 0.054 & – & – & 0\
K10U07C & 55503.081294 & 16.543 $\pm$ 0.334 & 12.574 $\pm$ 0.058 & – & – & 0\
K10V11T & 55502.789563 & 16.112 $\pm$ 0.277 & 13.413 $\pm$ 0.084 & – & – & 0\
K10V11T & 55502.855652 & 15.868 $\pm$ 0.196 & 13.248 $\pm$ 0.072 & – & – & 0\
K10V72D & 55512.768409 & 15.951 $\pm$ 0.329 & 12.658 $\pm$ 0.054 & – & – & 0\
K10V72D & 55512.834497 & 15.294 $\pm$ 0.128 & 12.856 $\pm$ 0.063 & – & – & 0\
K10V72D & 55512.966674 & 15.962 $\pm$ 0.230 & 12.853 $\pm$ 0.065 & – & – & 0\
K10V72D & 55514.288315 & 15.823 $\pm$ 0.187 & 13.017 $\pm$ 0.076 & – & – & 0\
K10VD9K & 55511.208514 & $>$16.587 & 14.830 $\pm$ 0.281 & – & – & 0\
K10VD9K & 55511.274602 & 16.094 $\pm$ 0.220 & 14.143 $\pm$ 0.139 & – & – & 0\
K10VD9K & 55511.274729 & $>$16.534 & 14.486 $\pm$ 0.189 & – & – & 0\
K10VD9K & 55511.340818 & 16.511 $\pm$ 0.320 & 14.446 $\pm$ 0.174 & – & – & 0\
K10W00B & 55512.114019 & 14.332 $\pm$ 0.064 & 11.404 $\pm$ 0.027 & – & – & 0\
K10Y00D & 55566.173782 & $>$16.983 & 14.356 $\pm$ 0.240 & – & – & 0\
K11A55V & 55571.253941 & 16.601 $\pm$ 0.345 & 13.365 $\pm$ 0.086 & – & – & 0\
K11A55V & 55571.320157 & 16.208 $\pm$ 0.315 & 13.321 $\pm$ 0.080 & – & – & 0\
K11A55V & 55571.386372 & 16.075 $\pm$ 0.223 & 13.523 $\pm$ 0.094 & – & – & 0\
[lllllll]{} 85989 & 17.10 & 0.15 & 1.834 $\pm$ 0.018 & 0.076 $\pm$ 0.017 & 1.674 $\pm$ 0.025 & 0.072 $\pm$ 0.027\
85989 & 17.10 & 0.15 & 1.853 $\pm$ 0.077 & 0.074 $\pm$ 0.018 & 2.001 $\pm$ 0.151 & 0.994 $\pm$ 0.110\
88254 & 17.60 & 0.15 & 0.800 $\pm$ 0.012 & 0.252 $\pm$ 0.035 & 1.309 $\pm$ 0.046 & 0.352 $\pm$ 0.150\
D7032 & 16.60 & 0.15 & 1.055 $\pm$ 0.322 & 0.183 $\pm$ 0.325 & 2.000 $\pm$ 0.500 & 0.310 $\pm$ 0.288\
F2742 & 19.10 & 0.15 & 0.413 $\pm$ 0.005 & 0.264 $\pm$ 0.011 & 1.400 $\pm$ 0.500 & 0.276 $\pm$ 0.086\
F4590 & 21.70 & 0.15 & 0.086 $\pm$ 0.002 & 0.530 $\pm$ 0.076 & 1.400 $\pm$ 0.500 & 0.848 $\pm$ 0.114\
M6554 & 19.50 & 0.15 & 0.482 $\pm$ 0.007 & 0.125 $\pm$ 0.024 & 3.055 $\pm$ 0.060 & 0.200 $\pm$ 0.038\
N0549 & 20.60 & 0.15 & 0.217 $\pm$ 0.004 & 0.216 $\pm$ 0.046 & 2.402 $\pm$ 0.072 & 0.346 $\pm$ 0.073\
N4145 & 21.30 & 0.15 & 0.344 $\pm$ 0.006 & 0.045 $\pm$ 0.009 & 1.571 $\pm$ 0.044 & 0.010 $\pm$ 0.033\
O7517 & 19.70 & 0.15 & 0.269 $\pm$ 0.005 & 0.357 $\pm$ 0.055 & 1.400 $\pm$ 0.500 & 0.929 $\pm$ 0.142\
Q4357 & 21.00 & 0.15 & 0.439 $\pm$ 0.106 & 0.037 $\pm$ 0.024 & 2.000 $\pm$ 0.406 & 0.058 $\pm$ 0.039\
W3179 & 18.20 & 0.15 & 1.170 $\pm$ 0.022 & 0.068 $\pm$ 0.013 & 1.970 $\pm$ 0.037 & 0.387 $\pm$ 0.177\
J94C00B & 21.00 & 0.15 & 0.193 $\pm$ 0.010 & 0.195 $\pm$ 0.026 & 2.124 $\pm$ 0.229 & 0.311 $\pm$ 0.042\
J98K03N & 18.40 & 0.15 & 1.060 $\pm$ 0.020 & 0.074 $\pm$ 0.014 & 1.165 $\pm$ 0.029 & 0.118 $\pm$ 0.022\
J98W02B & 21.80 & 0.15 & 0.150 $\pm$ 0.007 & 0.151 $\pm$ 0.034 & 2.790 $\pm$ 0.224 & 0.192 $\pm$ 0.429\
J99XD6K & 20.30 & 0.15 & 0.811 $\pm$ 0.403 & 0.020 $\pm$ 0.031 & 0.900 $\pm$ 0.390 & 0.032 $\pm$ 0.063\
K00A06A & 22.10 & 0.15 & 0.316 $\pm$ 0.005 & 0.026 $\pm$ 0.005 & 1.753 $\pm$ 0.034 & 0.085 $\pm$ 0.025\
K00A06C & 21.50 & 0.15 & 0.176 $\pm$ 0.005 & 0.143 $\pm$ 0.021 & 2.300 $\pm$ 0.500 & 0.316 $\pm$ 0.133\
K00R52E & 22.30 & 0.15 & 0.149 $\pm$ 0.005 & 0.095 $\pm$ 0.020 & 2.397 $\pm$ 0.150 & 0.230 $\pm$ 0.244\
K02B25F & 22.20 & 0.15 & 0.152 $\pm$ 0.003 & 0.100 $\pm$ 0.021 & 2.619 $\pm$ 0.111 & 0.101 $\pm$ 0.182\
K02JA0R & 24.30 & 0.15 & 0.028 $\pm$ 0.002 & 0.441 $\pm$ 0.110 & 3.000 $\pm$ 0.390 & 0.705 $\pm$ 0.145\
K02X38Y & 22.90 & 0.15 & 0.096 $\pm$ 0.003 & 0.134 $\pm$ 0.026 & 1.972 $\pm$ 0.113 & 0.130 $\pm$ 0.084\
K03U11V & 19.50 & 0.15 & 0.260 $\pm$ 0.003 & 0.376 $\pm$ 0.075 & 1.400 $\pm$ 0.500 & 0.601 $\pm$ 0.121\
K04F11G & 21.00 & 0.15 & 0.152 $\pm$ 0.003 & 0.306 $\pm$ 0.050 & 1.730 $\pm$ 0.067 & 0.415 $\pm$ 0.115\
K04K17H & 22.00 & 0.15 & 0.197 $\pm$ 0.011 & 0.072 $\pm$ 0.012 & 2.318 $\pm$ 0.224 & 0.115 $\pm$ 0.019\
K04S56C & 22.70 & 0.15 & 0.291 $\pm$ 0.042 & 0.017 $\pm$ 0.008 & 2.000 $\pm$ 0.514 & 0.028 $\pm$ 0.012\
K05GC0C & 19.60 & 0.15 & 0.906 $\pm$ 0.072 & 0.031 $\pm$ 0.006 & 0.755 $\pm$ 0.056 & 0.050 $\pm$ 0.009\
K05U00O & 22.10 & 0.15 & 0.164 $\pm$ 0.019 & 0.094 $\pm$ 0.028 & 2.303 $\pm$ 0.400 & 0.151 $\pm$ 0.045\
K05X01B & 22.00 & 0.15 & 0.099 $\pm$ 0.012 & 0.287 $\pm$ 0.099 & 2.000 $\pm$ 0.440 & 0.459 $\pm$ 0.158\
K07E00F & 21.40 & 0.15 & 0.260 $\pm$ 0.005 & 0.072 $\pm$ 0.013 & 2.062 $\pm$ 0.049 & 0.101 $\pm$ 0.034\
K08N03P & 23.30 & 0.15 & 0.193 $\pm$ 0.003 & 0.023 $\pm$ 0.004 & 1.993 $\pm$ 0.054 & 0.036 $\pm$ 0.006\
K08X00M & 19.90 & 0.15 & 0.367 $\pm$ 0.009 & 0.128 $\pm$ 0.032 & 2.989 $\pm$ 0.095 & 0.204 $\pm$ 0.051\
K09F00D & 22.10 & 0.15 & 0.472 $\pm$ 0.045 & 0.010 $\pm$ 0.003 & 2.421 $\pm$ 0.176 & 0.010 $\pm$ 0.003\
K09F04Y & 21.00 & 0.15 & 0.129 $\pm$ 0.013 & 0.421 $\pm$ 0.091 & 2.000 $\pm$ 0.366 & 0.673 $\pm$ 0.145\
K09U19Y & 23.40 & 0.15 & 0.105 $\pm$ 0.001 & 0.063 $\pm$ 0.012 & 1.400 $\pm$ 0.500 & 0.101 $\pm$ 0.019\
K10A02N & 22.20 & 0.15 & 0.123 $\pm$ 0.010 & 0.155 $\pm$ 0.035 & 1.600 $\pm$ 0.212 & 0.247 $\pm$ 0.055\
K10A30E & 23.60 & 0.15 & 0.050 $\pm$ 0.008 & 0.259 $\pm$ 0.147 & 2.000 $\pm$ 0.597 & 0.415 $\pm$ 0.198\
K10A40G & 22.10 & 0.15 & 0.089 $\pm$ 0.004 & 0.319 $\pm$ 0.053 & 1.803 $\pm$ 0.141 & 0.144 $\pm$ 0.119\
K10B00C & 22.20 & 0.15 & 0.080 $\pm$ 0.002 & 0.433 $\pm$ 0.034 & 1.400 $\pm$ 0.500 & 0.440 $\pm$ 0.125\
K10B02K & 22.00 & 0.15 & 0.118 $\pm$ 0.002 & 0.205 $\pm$ 0.043 & 1.400 $\pm$ 0.500 & 0.329 $\pm$ 0.069\
K10C01K & 24.00 & 0.15 & 0.084 $\pm$ 0.011 & 0.063 $\pm$ 0.024 & 2.000 $\pm$ 0.432 & 0.101 $\pm$ 0.038\
K10C19C & 22.30 & 0.15 & 0.104 $\pm$ 0.005 & 0.196 $\pm$ 0.032 & 1.724 $\pm$ 0.126 & 0.313 $\pm$ 0.052\
K10C19F & 21.90 & 0.15 & 0.083 $\pm$ 0.006 & 0.448 $\pm$ 0.111 & 2.400 $\pm$ 0.298 & 0.717 $\pm$ 0.144\
K10C44O & 24.60 & 0.15 & 0.034 $\pm$ 0.005 & 0.218 $\pm$ 0.102 & 2.000 $\pm$ 0.431 & 0.348 $\pm$ 0.151\
K10C55D & 23.10 & 0.15 & 0.083 $\pm$ 0.012 & 0.145 $\pm$ 0.034 & 2.009 $\pm$ 0.539 & 0.232 $\pm$ 0.055\
K10C55F & 21.70 & 0.15 & 0.176 $\pm$ 0.022 & 0.119 $\pm$ 0.031 & 2.000 $\pm$ 0.453 & 0.190 $\pm$ 0.050\
K10D00L & 27.00 & 0.15 & 0.019 $\pm$ 0.001 & 0.080 $\pm$ 0.018 & 1.102 $\pm$ 0.090 & 0.128 $\pm$ 0.029\
K10D00O & 21.70 & 0.15 & 0.149 $\pm$ 0.002 & 0.176 $\pm$ 0.036 & 1.400 $\pm$ 0.500 & 0.282 $\pm$ 0.058\
K10D01F & 21.90 & 0.15 & 0.159 $\pm$ 0.020 & 0.121 $\pm$ 0.060 & 2.000 $\pm$ 0.474 & 0.193 $\pm$ 0.097\
K10D01X & 21.60 & 0.15 & 0.167 $\pm$ 0.004 & 0.146 $\pm$ 0.023 & 1.606 $\pm$ 0.055 & 0.632 $\pm$ 0.164\
K10E21G & 24.40 & 0.15 & 0.090 $\pm$ 0.013 & 0.038 $\pm$ 0.016 & 2.000 $\pm$ 0.428 & 0.060 $\pm$ 0.025\
K10E43C & 23.00 & 0.15 & 0.051 $\pm$ 0.005 & 0.434 $\pm$ 0.095 & 2.400 $\pm$ 0.403 & 0.695 $\pm$ 0.127\
K10E43K & 21.70 & 0.15 & 0.233 $\pm$ 0.023 & 0.068 $\pm$ 0.014 & 2.346 $\pm$ 0.390 & 0.109 $\pm$ 0.023\
K10F00C & 24.00 & 0.15 & 0.093 $\pm$ 0.015 & 0.051 $\pm$ 0.017 & 2.000 $\pm$ 0.410 & 0.082 $\pm$ 0.027\
K10F00K & 23.80 & 0.15 & 0.087 $\pm$ 0.009 & 0.071 $\pm$ 0.020 & 2.000 $\pm$ 0.419 & 0.113 $\pm$ 0.032\
K10F00S & 24.50 & 0.15 & 0.023 $\pm$ 0.003 & 0.548 $\pm$ 0.164 & 2.000 $\pm$ 0.550 & 0.877 $\pm$ 0.156\
K10F00T & 25.70 & 0.15 & 0.018 $\pm$ 0.001 & 0.294 $\pm$ 0.062 & 1.400 $\pm$ 0.500 & 0.471 $\pm$ 0.099\
K10F06D & 26.80 & 0.15 & 0.008 $\pm$ 0.000 & 0.497 $\pm$ 0.081 & 1.400 $\pm$ 0.500 & 0.796 $\pm$ 0.117\
K10F07D & 22.10 & 0.15 & 0.075 $\pm$ 0.002 & 0.448 $\pm$ 0.070 & 1.400 $\pm$ 0.500 & 0.716 $\pm$ 0.112\
K10F07E & 23.70 & 0.15 & 0.045 $\pm$ 0.002 & 0.288 $\pm$ 0.043 & 1.400 $\pm$ 0.500 & 0.460 $\pm$ 0.068\
K10F09R & 26.10 & 0.15 & 0.015 $\pm$ 0.001 & 0.287 $\pm$ 0.053 & 1.400 $\pm$ 0.500 & 0.459 $\pm$ 0.085\
K10F09W & 26.70 & 0.15 & 0.024 $\pm$ 0.001 & 0.075 $\pm$ 0.014 & 1.226 $\pm$ 0.057 & 0.120 $\pm$ 0.022\
K10F09X & 24.20 & 0.15 & 0.030 $\pm$ 0.002 & 0.425 $\pm$ 0.068 & 3.142 $\pm$ 0.500 & 0.681 $\pm$ 0.109\
K10F10F & 21.70 & 0.15 & 0.095 $\pm$ 0.010 & 0.407 $\pm$ 0.093 & 2.000 $\pm$ 0.441 & 0.650 $\pm$ 0.146\
K10G05Z & 22.80 & 0.15 & 0.071 $\pm$ 0.002 & 0.281 $\pm$ 0.056 & 1.400 $\pm$ 0.500 & 0.450 $\pm$ 0.090\
K10G06B & 22.10 & 0.15 & 0.134 $\pm$ 0.021 & 0.142 $\pm$ 0.055 & 2.000 $\pm$ 0.433 & 0.227 $\pm$ 0.088\
K10G06T & 21.60 & 0.15 & 0.116 $\pm$ 0.015 & 0.300 $\pm$ 0.080 & 2.000 $\pm$ 0.470 & 0.480 $\pm$ 0.129\
K10G06U & 20.70 & 0.15 & 0.252 $\pm$ 0.022 & 0.147 $\pm$ 0.030 & 2.336 $\pm$ 0.335 & 0.235 $\pm$ 0.048\
K10G07H & 27.30 & 0.15 & 0.008 $\pm$ 0.000 & 0.326 $\pm$ 0.043 & 1.400 $\pm$ 0.500 & 0.521 $\pm$ 0.069\
K10G23L & 22.80 & 0.15 & 0.092 $\pm$ 0.014 & 0.159 $\pm$ 0.099 & 2.000 $\pm$ 0.557 & 0.255 $\pm$ 0.155\
K10G23M & 24.70 & 0.15 & 0.341 $\pm$ 0.000 & 0.010 $\pm$ 0.000 & 0.860 $\pm$ 0.500 & 0.010 $\pm$ 0.000\
K10G23X & 23.00 & 0.15 & 0.060 $\pm$ 0.001 & 0.334 $\pm$ 0.051 & 1.400 $\pm$ 0.500 & 0.535 $\pm$ 0.081\
K10G24A & 21.10 & 0.15 & 0.150 $\pm$ 0.005 & 0.285 $\pm$ 0.080 & 1.532 $\pm$ 0.110 & 0.708 $\pm$ 0.202\
K10H00A & 23.90 & 0.15 & 0.032 $\pm$ 0.004 & 0.482 $\pm$ 0.130 & 2.100 $\pm$ 0.357 & 0.772 $\pm$ 0.141\
K10H20V & 21.90 & 0.15 & 0.465 $\pm$ 0.007 & 0.016 $\pm$ 0.003 & 1.912 $\pm$ 0.035 & 0.025 $\pm$ 0.005\
K10J03H & 25.30 & 0.15 & 0.039 $\pm$ 0.005 & 0.087 $\pm$ 0.026 & 2.000 $\pm$ 0.414 & 0.139 $\pm$ 0.042\
K10J03J & 23.60 & 0.15 & 0.032 $\pm$ 0.001 & 0.609 $\pm$ 0.076 & 1.400 $\pm$ 0.500 & 0.678 $\pm$ 0.273\
K10J34V & 20.80 & 0.15 & 0.226 $\pm$ 0.005 & 0.165 $\pm$ 0.041 & 1.988 $\pm$ 0.080 & 0.239 $\pm$ 0.069\
K10J39W & 24.50 & 0.15 & 0.039 $\pm$ 0.005 & 0.187 $\pm$ 0.063 & 2.000 $\pm$ 0.511 & 0.300 $\pm$ 0.100\
K10J41L & 20.70 & 0.15 & 0.359 $\pm$ 0.023 & 0.073 $\pm$ 0.014 & 2.382 $\pm$ 0.255 & 0.117 $\pm$ 0.022\
K10J71O & 23.90 & 0.15 & 0.037 $\pm$ 0.004 & 0.349 $\pm$ 0.114 & 2.000 $\pm$ 0.440 & 0.558 $\pm$ 0.182\
K10K07V & 26.00 & 0.15 & 0.013 $\pm$ 0.001 & 0.397 $\pm$ 0.070 & 1.400 $\pm$ 0.500 & 0.635 $\pm$ 0.112\
K10K08A & 21.40 & 0.15 & 0.183 $\pm$ 0.019 & 0.147 $\pm$ 0.034 & 2.298 $\pm$ 0.487 & 0.235 $\pm$ 0.054\
K10K10P & 23.40 & 0.15 & 0.087 $\pm$ 0.010 & 0.101 $\pm$ 0.036 & 2.000 $\pm$ 0.390 & 0.161 $\pm$ 0.057\
K10L34K & 21.90 & 0.15 & 0.108 $\pm$ 0.002 & 0.271 $\pm$ 0.048 & 1.400 $\pm$ 0.500 & 0.434 $\pm$ 0.078\
K10L61K & 21.90 & 0.15 & 0.191 $\pm$ 0.019 & 0.084 $\pm$ 0.022 & 2.221 $\pm$ 0.420 & 0.135 $\pm$ 0.035\
K10L63Z & 20.10 & 0.15 & 0.877 $\pm$ 0.015 & 0.021 $\pm$ 0.007 & 1.610 $\pm$ 0.041 & 0.062 $\pm$ 0.033\
K10L64B & 21.70 & 0.15 & 0.088 $\pm$ 0.008 & 0.477 $\pm$ 0.165 & 2.300 $\pm$ 0.386 & 0.763 $\pm$ 0.177\
K10M01G & 23.30 & 0.15 & 0.053 $\pm$ 0.005 & 0.301 $\pm$ 0.107 & 2.000 $\pm$ 0.384 & 0.482 $\pm$ 0.149\
K10M01P & 21.40 & 0.15 & 0.104 $\pm$ 0.004 & 0.518 $\pm$ 0.091 & 1.400 $\pm$ 0.500 & 0.830 $\pm$ 0.135\
K10M01Y & 23.90 & 0.15 & 0.043 $\pm$ 0.001 & 0.265 $\pm$ 0.029 & 1.400 $\pm$ 0.500 & 0.424 $\pm$ 0.046\
K10N00A & 20.50 & 0.15 & 0.320 $\pm$ 0.011 & 0.109 $\pm$ 0.026 & 2.862 $\pm$ 0.191 & 0.144 $\pm$ 0.426\
K10N01K & 22.40 & 0.15 & 0.219 $\pm$ 0.002 & 0.040 $\pm$ 0.006 & 1.624 $\pm$ 0.025 & 0.109 $\pm$ 0.031\
K10O01A & – & 0.15 & 0.663 $\pm$ 0.035 & – & 2.058 $\pm$ 0.186 & 0.418 $\pm$ 0.274\
K10P09K & 22.00 & 0.15 & 0.143 $\pm$ 0.008 & 0.138 $\pm$ 0.040 & 1.798 $\pm$ 0.159 & 0.220 $\pm$ 0.064\
K10P66R & 19.40 & 0.15 & 0.695 $\pm$ 0.018 & 0.063 $\pm$ 0.012 & 1.992 $\pm$ 0.046 & 0.178 $\pm$ 0.044\
K10P66R & 19.40 & 0.15 & 0.488 $\pm$ 0.061 & 0.129 $\pm$ 0.041 & 2.000 $\pm$ 0.463 & 0.362 $\pm$ 0.115\
K10Q02G & 24.30 & 0.15 & 0.038 $\pm$ 0.006 & 0.228 $\pm$ 0.064 & 2.300 $\pm$ 0.315 & 0.228 $\pm$ 0.064\
K10R53J & 24.00 & 0.15 & 0.774 $\pm$ 0.020 & 0.010 $\pm$ 0.000 & 2.186 $\pm$ 0.064 & 0.010 $\pm$ 0.000\
K10T04N & 27.36 & 0.15 & 0.018 $\pm$ 0.006 & 0.062 $\pm$ 0.098 & 2.000 $\pm$ 0.582 & 0.100 $\pm$ 0.156\
K10TE9U & 20.70 & 0.15 & 0.603 $\pm$ 0.195 & 0.025 $\pm$ 0.015 & 1.388 $\pm$ 0.374 & 0.041 $\pm$ 0.025\
K10U07C & 24.70 & 0.15 & 0.043 $\pm$ 0.011 & 0.129 $\pm$ 0.066 & 2.000 $\pm$ 0.461 & 0.206 $\pm$ 0.105\
K10V11T & 21.50 & 0.15 & 0.152 $\pm$ 0.044 & 0.189 $\pm$ 0.162 & 2.000 $\pm$ 0.500 & 0.302 $\pm$ 0.209\
K10V72D & 21.50 & 0.15 & 0.124 $\pm$ 0.005 & 0.290 $\pm$ 0.035 & 1.400 $\pm$ 0.500 & 0.434 $\pm$ 0.150\
K10VD9K & 23.70 & 0.15 & 0.055 $\pm$ 0.010 & 0.196 $\pm$ 0.114 & 2.000 $\pm$ 0.377 & 0.313 $\pm$ 0.182\
K10W00B & 24.10 & 0.15 & 0.057 $\pm$ 0.018 & 0.126 $\pm$ 0.049 & 1.582 $\pm$ 0.447 & 0.202 $\pm$ 0.079\
K10Y00D & 26.60 & 0.15 & 0.026 $\pm$ 0.009 & 0.060 $\pm$ 0.068 & 2.000 $\pm$ 0.602 & 0.096 $\pm$ 0.108\
K11A55V & 23.50 & 0.15 & 0.063 $\pm$ 0.024 & 0.176 $\pm$ 0.156 & 1.128 $\pm$ 0.415 & 0.282 $\pm$ 0.193\
[^1]: https://irsa.ipac.caltech.edu/applications/wise/
[^2]: http://irsa.ipac.caltech.edu/IRASdocs/exp.sup/ch11/J.html
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Benito Hernández–Bermejo $^1$'
title: |
Characterization and global analysis\
of a family of Poisson structures
---
[*Escuela Superior de Ciencias Experimentales y Tecnología. Edificio Departamental II.*]{}\
[*Universidad Rey Juan Carlos. Calle Tulipán S/N. 28933–Móstoles–Madrid. Spain.*]{}
[**Abstract**]{}
A three-dimensional family of solutions of the Jacobi equations for Poisson systems is characterized. In spite of its general form it is possible the explicit and global determination of its main features, such as the symplectic structure and the construction of the Darboux canonical form. Examples are given.
[**PACS codes:**]{} 45.20.-d, 45.20.Jj, 02.30.Jr.
[**Keywords:**]{} Finite-dimensional Poisson systems — Jacobi identities — 3-d systems — PDEs.
$^1$ Telephone: (+34) 91 488 73 91. Fax: (+34) 91 488 73 38. E-mail: [[email protected] ]{}
[**1. Introduction**]{}
Finite-dimensional Poisson structures (see [@olv1] and references therein) have a significant presence in all fields of mathematical physics, such as mechanics [@haa1], electromagnetism [@7], plasma physics [@25], optics [@17; @dht3], dynamical systems theory [@9]-[@28], etc. Describing a given physical system in terms of a Poisson structure allows the obtainment of a wide range of information which may be in the form of perturbative solutions [@cyl1], invariants [@byv3], stability analysis [@hyct], bifurcation properties and characterization of chaotic behaviour [@dht3], efficient numerical integration [@mac1] or integrability results [@mag1], to cite a sample.
Mathematically, a finite-dimensional dynamical system defined on $I \! \! R^n$ is said to have a Poisson structure if it can be written in terms of a set of ODEs of the form: $$\label{plnnham}
\frac{\mbox{d}x_i}{{\mbox{d}t}} = \sum_{j=1}^n J_{ij} \partial _j H
\; , \;\:\; i = 1, \ldots , n,$$ where $ \partial_j \equiv \partial / \partial x_j$ and function $H$, which is usually taken to be a time-independent first integral, plays the role of Hamiltonian. The $J_{ij}(x)$ are the entries of an $n \times n$ structure matrix ${\cal J}$ (which can be degenerate in rank) and they have the property of being solutions of the Jacobi identities: $$\label{plnjac}
\sum_{l=1}^n ( J_{li} \partial_l J_{jk} + J_{lj} \partial_l J_{ki} +
J_{lk} \partial_l J_{ij} ) = 0 \:\; , \;\:\;\: i,j,k=1, \ldots ,n$$ The $J_{ij}$ must also verify an additional skew-symmetry condition: $$\label{plnsksym}
J_{ij} = - J_{ji} \;\:\:\:\: \mbox{for all} \:\; i,j$$
One of the reasons justifying the importance and flexibility of the Poisson representation is the (at least) local equivalence bewteen Poisson systems and classical Hamiltonian systems, as stated by Darboux Theorem [@olv1]. This explains that Poisson systems can be seen, to a large extent, as a generalization of classical Hamiltonian systems (for instance, allowing for odd-dimensional flows). The issue of describing a given vector field not explicitly written in the form (\[plnnham\]) in terms of a Poisson structure is a fundamental question in this context, which still remains as an open problem [@9]-[@hyg1],[@5]-[@21]. This is a nontrivial decomposition to which important efforts have been devoted in past years in a variety of approaches. The source of the difficulty is obviously twofold: First, a known constant of motion of the system able to play the role of the Hamiltonian is required. And second, it is necessary to find a suitable structure matrix for the vector field. Consequently, finding a solution of the Jacobi identities (\[plnjac\]) complying also with conditions (\[plnsksym\]) is unavoidable. This explains the attention deserved in the literature by the obtainment and classification of skew-symmetric solutions of the Jacobi equations [@olv1]-[@byv2],[@nut1]-[@cyl1],[@5]-[@29].
As far as the Jacobi identities constitute a set of nonlinear coupled PDEs, the characterization of $n$-dimensional solutions has followed, roughly speaking, a sequence of increasing nonlinearity. Thus we can speak of constant structure matrices (including the classical symplectic matrices), Lie-Poisson structures [@olv1; @29], affine-linear structures [@bha1], and quadratic structures [@byv2; @pla1; @byr1; @lyx1], as well as some families of structure matrices which may contain functions of arbitrary nonlinearity [@byv4]. However, the set of solutions of system (\[plnjac\]-\[plnsksym\]) seems to be still mostly unexplored. Perhaps the only exception to this situation is that of three-dimensional (3-d in what follows) vector fields, which constitute an important case which has been repeatedly considered in the literature and is the best understood at present [@7; @17],[@9]-[@hyg1],[@nut1; @nut2; @27; @byv1; @bs1; @bs2]. In dimension three, the strategy for finding suitable skew-symmetric solutions of the Jacobi equations has often been problem-dependent. In this sense, we can find recipes based on the use of either convenient [*ansatzs*]{} for the solution [@gyn1; @nut1; @nut2; @27], or symmetry considerations [@7; @17], or the knowledge of additional information about the system, such as the existence of a constant of motion [@hyg1; @byv1; @8]. Additionally, in the 3-d situation it is also possible to recast the problem (\[plnjac\]-\[plnsksym\]) in equivalent forms which may be more suitable for the determination of the desired solutions [@9]-[@hyg1]. This is certainly a more elaborate state of affairs than the one existing in the general $n$-dimensional case. Moreover, it is worth recalling that the 3-d scenario is particularly relevant for several reasons. First, a large number of 3-d systems arising in very diverse fields have a Poisson structure [@7; @17],[@9]-[@hyg1],[@nut1; @nut2; @27; @8; @bs2]. Therefore 3-d Poisson structures are the natural framework for the analysis of such systems. In second place, dimension three corresponds to the first nontrivial case where a Poisson structure does not imply a symplectic structure. In other words, it is the simplest meaningful kind of Poisson structures which is not symplectic. Finally, three is the lowest dimension for which the Jacobi identities are not always identically verified (recall that every skew-symmetric $2 \times 2$ matrix is a structure matrix). Since the complexity of equations (\[plnjac\]-\[plnsksym\]) is increasing with the dimension $n$, the 3-d case is the simplest nontrivial one as well as a natural first approach to the full problem of analyzing the solutions of (\[plnjac\]-\[plnsksym\]).
In this work a new family of skew-symmetric solutions of the three-dimensional Jacobi equations (\[plnjac\]-\[plnsksym\]) is considered. Such family is very general, and in particular it is defined in terms of functions of arbitrary nonlinearity. This explains that well-known three-dimensional Poisson structures and systems now happen to appear embraced as particular cases, as it will be seen. Moreover, this unification is not only conceptual. In fact, the new familiy is amenable to explicit and detailed analysis. In particular, it is possible to explicitly determine features such as the symplectic structure and the construction of the Darboux canonical form. The advantage of these common strategies is that they are simultaneously valid for all the particular cases which can now be analyzed in a unified and more economic way, instead of using a case-by-case approach. In addition, the methods developed are valid globally in phase space, thus ameliorating the usual scope of Darboux theorem which does only guarantee, in principle, a local reduction. The possibility of constructing the Darboux canonical form is also remarkable in view that the practical determination of Darboux coordinates is a complicated task in general, which has been carried out only for a very limited sample of systems.
The structure of the article is as follows. In Section 2 the results enumerated above, leading to the explicit determination and analysis of a new family of skew-symmetric solutions of the 3-d Jacobi equations, are presented. Section 3 contains several examples which illustrate the theory. The work is concluded in Section 4 with some final remarks.
[**2. A family of Poisson structures and its global analysis**]{}
The first result to be presented is the following one:
[**Theorem 1.**]{} [*Let $\{ \eta (x), \phi_1(x_1), \phi_2(x_2), \phi_3(x_3) \}$ be a set of functions defined in an open set $\Omega \subset I \!\! R^3$, all of which are $C^1(\Omega)$ and nonvanishing in $\Omega$. In addition, let $\kappa _{ij}$, $i,j = 1,2,3$, be arbitrary real constants that constitute a skew-symmetric matrix $$\label{plnkappa1}
\kappa _{ij} + \kappa _{ji}=0 \: , \:\:\: \mbox{\rm for all} \:\:\: i,j$$ and satisfy the zero-sum condition $$\label{plnkappa2}
\kappa _{12} + \kappa _{23} + \kappa _{31} = 0$$ Then ${\cal J}=(J_{ij})$ is a family 3-d Poisson structures which is globally defined in $\Omega$, with* ]{} $$\label{plnsol1}
J_{ij}(x)= \eta (x) \left( \psi_i(x_i) - \psi_j(x_j) + \kappa_{ij} \right)
\sum_{k=1}^3 (\epsilon _{ijk})^2 \phi_k(x_k)
\:\: , \:\:\:\:\: i,j = 1, 2, 3 \: ,$$ [ *where $\epsilon _{ijk}$ denotes the Levi-Civita symbol, and for every $i=1,2,3$, function $\psi_i(x_i) $ denotes one of the primitive functions of $\phi_i(x_i)$.* ]{}
[**Proof.**]{} Note that for $n=3$, system (\[plnjac\]-\[plnsksym\]) actually consists of the following independent nonlinear equation: $$\label{plnjac3df}
J_{12} \partial_1 J_{31} - J_{31} \partial_1 J_{12} +
J_{23} \partial_2 J_{12} - J_{12} \partial_2 J_{23} +
J_{31} \partial_3 J_{23} - J_{23} \partial_3 J_{31} = 0$$ Consider first family (\[plnsol1\]) in the particular case $\eta (x) =1$. For this, let $J^*_{ij}(x) \equiv J_{ij}(x)/ \eta (x)$ in (\[plnsol1\]). Then, substitution in (\[plnjac3df\]) produces after some algebra: $$J^*_{12} \partial_1 J^*_{31} - J^*_{31} \partial_1 J^*_{12} + J^*_{23} \partial_2 J^*_{12} -
J^*_{12} \partial_2 J^*_{23} + J^*_{31} \partial_3 J^*_{23} - J^*_{23} \partial_3 J^*_{31} =
-2 \phi_1 \phi_2 \phi_3 ( \kappa_{12} + \kappa_{23} + \kappa_{31})$$ This demonstrates the result for the case $\eta = 1$. For general $\eta$ it suffices to recall [@gyn1; @bs2] that in the 3-d case $\eta \cdot {\cal J}$ is a structure matrix for every arbitrary nonvanishing $C^1$ function $\eta (x)$ and for every structure matrix ${\cal J}$. This completes the proof.
Now some remarks are in order. In first place, it is useful for what is to follow to give the explicit form of the components of ${\cal J}$ for family (\[plnsol1\]), which are: $$\left\{ \begin{array}{c}
J_{12}(x) = \eta (x) \left( \psi_1(x_1) - \psi_2(x_2) + \kappa_{12} \right)
\phi_3(x_3) \\
J_{23}(x) = \eta (x) \left( \psi_2(x_2) - \psi_3(x_3) + \kappa_{23} \right)
\phi_1(x_1) \\
J_{31}(x) = \eta (x) \left( \psi_3(x_3) - \psi_1(x_1) + \kappa_{31} \right)
\phi_2(x_2)
\end{array} \right.$$
As indicated in the theorem, for every $i$ the primitive $\psi_i(x_i)$ of $\phi_i(x_i)$ must be chosen to be one and the same for all the entries of ${\cal J}$. However, the specific choice is actually arbitrary. To see this it suffices to notice that if a different integration constant is selected, for instance after replacing $\psi_i(x_i)$ by $\psi_i(x_i) + k_i$ for every $i$, then the outcome is also a member of the solution family, this time with constants $\tilde{\kappa}_{ij} = \kappa_{ij} + k_i - k_j$, which also verify (\[plnkappa1\]-\[plnkappa2\]). Thus conditions (\[plnkappa1\]-\[plnkappa2\]) express in a generalized form this degree of freedom associated with the choice of the primitives of functions $\phi_i(x_i)$.
Secondly, notice that the form of the Poisson structures we are dealing with is such that only two possibilities exist regarding the vanishing of the independent entries $(J_{12},J_{23},J_{31})$ at a given point, namely: [*(i)*]{} either none or one of them vanishes (case of rank two), or [*(ii)*]{} all of them vanish (case of zero rank). To see this, it is convenient to define the functions: $$\chi_{ij}(x_i,x_j) \equiv \psi_i(x_i) - \psi_j(x_j) + \kappa_{ij}
\:\: , \:\:\:\:\: i,j = 1, 2, 3$$ Thus it is clearly not possible that only two of such entries $(J_{12},J_{23},J_{31})$ vanish at the same point, as a consequence of the zero-sum relation $\chi_{12}(x_1,x_2)+\chi_{23}(x_2,x_3)+\chi_{31}(x_3,x_1)=0$.
To conclude, it is interesting for what is to come to recall the physical interpretation of the degree of freedom corresponding to the factor $\eta (x)$, namely the fact that in the 3-d case $\eta \cdot {\cal J}$ is a structure matrix if and only if ${\cal J}$ is [@gyn1; @bs2]. Such result is not generally valid for dimension $n \geq 4$, as it can be easily verified. The interpretation of such three-dimensional feature is naturally associated to time reparametrizations [@gyn1; @bs2], which are transformations of the form $$\label{plnntt}
\mbox{d}\tau = \frac{1}{\eta (x)}\mbox{d}t$$ where $t$ is the initial time variable, $\tau$ is the new time and $\eta (x) : \Omega
\rightarrow I \!\! R$ is a $C^1(\Omega)$ function which does not vanish in $\Omega$. Thus, if $$\label{pln3dpos}
\frac{\mbox{d}x}{\mbox{d}t} = {\cal J} \cdot \nabla H$$ is an arbitrary three-dimensional Poisson system defined in $\Omega$, then every time reparametrization (\[plnntt\]) leads from (\[pln3dpos\]) to the system: $$\label{pln3dposntt}
\frac{\mbox{d}x}{\mbox{d} \tau} = \eta {\cal J} \cdot \nabla H$$ Therefore, in the 3-d case time reparametrizations (\[plnntt\]) preserve the Poisson structure, this time with structure matrix $\eta {\cal J}$ in (\[pln3dposntt\]). This is not the case in general for $n \geq 4$, as mentioned.
We can now characterize some of the properties of the family identified in Theorem 1:
[**Theorem 2.**]{} [*Let ${\cal J}= (J_{ij})$ be a Poisson structure of the form (\[plnsol1\]) characterized in Theorem 1, which is defined in an open domain $\Omega \subset I \!\! R^3$ and such that for a given pair $(i,j)$ one has $\chi_{ij}(x_i,x_j) \neq 0$ everywhere in $\Omega$. Then Rank(${\cal J}$)$=2$ in $\Omega$ and a Casimir invariant for ${\cal J}$ is* ]{} $$\label{plncas}
C_{k}(x) = \frac{\psi_j(x_j) - \psi_k(x_k) + \kappa_{jk}}{\psi_i(x_i) - \psi_j(x_j)
+ \kappa_{ij}} = \frac{\chi_{jk}(x_j,x_k)}{\chi_{ij}(x_i,x_j)}$$ [ *where $(i,j,k)$ is a cyclic permutation of $(1,2,3)$. Moreover, every Casimir invariant (\[plncas\]) is globally defined in $\Omega$ and belongs to $C^2(\Omega)$.* ]{}
[**Proof.**]{} After some algebra it is not difficult to demonstrate that $\partial _i C_a = -( \eta \chi^2_{bc})^{-1}J_{jk}$, where both $(a,b,c)$ and $(i,j,k)$ are cyclic permutations of $(1,2,3)$. With the help of this property the result can be directly shown through the verification of the fact that ${\cal J} \cdot \nabla C_k = 0$ for each of the three cases $k=1,2,3$ indicated. The statement is completed taking into account the $C^1( \Omega )$ property of the $\phi_i(x_i)$ .
Therefore it is possible to give the explicit list of Casimir invariants corresponding to the three complementary cases just analyzed: $$C_{1}(x) = \frac{\psi_3(x_3) - \psi_1(x_1) + \kappa_{31}}{\psi_2(x_2) - \psi_3(x_3) + \kappa_{23}} = \frac{\chi_{31}(x_3,x_1)}{\chi_{23}(x_2,x_3)}
\:\:\;\;\; \mbox{\rm if } \:\: \chi_{23}(x_2,x_3) \neq 0 \:\;\; \mbox{\rm in } \:\:
\Omega .$$ $$C_{2}(x) = \frac{\psi_1(x_1) - \psi_2(x_2) + \kappa_{12}}{\psi_3(x_3) - \psi_1(x_1) + \kappa_{31}} = \frac{\chi_{12}(x_1,x_2)}{\chi_{31}(x_3,x_1)}
\:\:\;\;\; \mbox{\rm if } \:\: \chi_{31}(x_3,x_1) \neq 0 \:\;\; \mbox{\rm in } \:\:
\Omega .$$ $$C_{3}(x) = \frac{\psi_2(x_2) - \psi_3(x_3) + \kappa_{23}}{\psi_1(x_1) - \psi_2(x_2) + \kappa_{12}} = \frac{\chi_{23}(x_2,x_3)}{\chi_{12}(x_1,x_2)}
\:\:\;\;\; \mbox{\rm if } \:\: \chi_{12}(x_1,x_2) \neq 0 \:\;\; \mbox{\rm in } \:\:
\Omega .$$ Notice the symmetry of such a choice, since $C_1C_2C_3=1$ when all of them are defined in $\Omega$. The previous results allow the constructive and global determination of the Darboux canonical form for this kind of Poisson structures:
[**Theorem 3.**]{} [*Let $\Omega \subset I \!\! R^3$ be an open domain where a Poisson system (\[plnnham\]) with $n=3$ is defined everywhere, for which ${\cal J} = (J_{ij})$ is a structure matrix of the form (\[plnsol1\]) characterized in Theorem 1, and such that for a given pair $(i,j)$ one has $\chi_{ij}(x_i,x_j) \neq 0$ everywhere in $\Omega$. Then such Poisson system can be globally reduced in $\Omega$ to a one degree of freedom Hamiltonian system and the Darboux canonical form is accomplished globally in $\Omega$ in the new coordinate system $\{y_1,y_2,y_3\}$ and the new time $\tau$, where $\{y_1,y_2,y_3\}$ are given by the diffeomorphism globally defined in $\Omega$ $$\label{plndarbco}
y_i (x) = x_i \;\: , \;\:\;\:
y_j (x) = x_j \;\: , \;\:\;\:
y_k (x) = -C_k(x) \;\: ,$$ in which $(i,j,k)$ is a cyclic permutation of $(1,2,3)$ and $C_k(x)$ is the Casimir invariant (\[plncas\]); while the new time $\tau$ is defined by the time reparametrization of the form (\[plnntt\]):* ]{} $$\label{plndarbntt}
\mbox{d} \tau = J_{ij}(x(y)) \mbox{d} t$$
[**Proof.**]{} Only the case $\chi_{12}(x_1,x_2) \neq 0$ will be considered here, since the analysis of the other two cases is analogous. Note that, according to Theorem 2, the Darboux theorem is applicable because ${\cal J}$ has constant rank 2 everywhere in $\Omega$. Recall also that, after a general diffeomorphism $y = y(x)$, an arbitrary structure matrix ${\cal J}(x)$ is transformed into another one ${\cal J'}(y)$ as: $$\label{plnjdiff}
J'_{ij}(y) = \sum_{k,l=1}^n \frac{\partial y_i}{\partial x_k} J_{kl}(x)
\frac{\partial y_j}{\partial x_l} \;\; , \;\:\; i,j = 1, \ldots , n$$ The reduction can be carried out in two steps. We first perform the change of variables (\[plndarbco\]), which in this case is $$\label{plnd12}
y_1 = x_1 \;\: , \;\:\;\:
y_2 = x_2 \;\: , \;\:\;\:
y_3 = -C_3(x)$$ where $C_3(x)$ is given by (\[plncas\]). For what is to come it is necessary to explicitly write the transformation inverse of (\[plnd12\]) which is: $$\label{plninvd12}
x_1 = y_1 \; , \;\:
x_2 = y_2 \; , \;\:
x_3 = \zeta_3 \left( \psi_2(y_2) + \kappa_{23} + \left( \psi_1(y_1) - \psi_2(y_2) +
\kappa_{12} \right) y_3 \right)$$ where function $\zeta_3$ is the inverse function of $\psi_3(x_3)$. Note that $\zeta_3$ exists and is differentiable in $\tilde{\Omega} = \psi_3( \Omega )$. The examination of (\[plnd12\]-\[plninvd12\]) easily shows that the variable transformation (\[plnd12\]) to be performed exists and is a diffeomorphism everywhere in $\Omega$ as a consequence that by hipothesis we have $\chi_{12}(x_1,x_2) \neq 0$ and $\phi _3(x_3) \neq 0$ in $\Omega$. Then, according to (\[plncas\]) and (\[plnd12\]), and taking (\[plnjdiff\]) into account, after some algebra we are led to $$\label{plnjdarb1}
{\cal J'}(y) = J_{12}(x(y)) \left( \begin{array}{ccc}
0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)$$ where from equations (\[plnsol1\]) and (\[plninvd12\]) we have $$\label{plnj12ntt}
J_{12}(x(y)) = \eta (y_1,y_2,x_3(y)) \left( \psi_1(y_1) - \psi_2(y_2) + \kappa_{12}
\right) \phi_3(x_3(y))$$ The explicit dependence of $x_3(y)$ is obviously the one given in (\[plninvd12\]) and was not displayed in (\[plnj12ntt\]) for the sake of clarity. Note that $J_{12}(x(y))$ is nonvanishing in $\Omega ' = y(\Omega)$ and $C^1(\Omega ')$. These properties allow the accomplisment of the second step of the reduction which is a reparametrization of time. Thus, making use of (\[plnj12ntt\]) in equation (\[plndarbntt\]), the transformation $\mbox{d} \tau = J_{12}(x(y)) \mbox{d} t$ is performed. According to (\[plnntt\]-\[pln3dposntt\]) this leads from the structure matrix (\[plnjdarb1\]) to the Darboux one: $$\label{plnjdarb}
{\cal J}_D (y) = \left( \begin{array}{ccc}
0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)$$ The reduction is thus globally completed.
The previous results can be now illustrated by means of some examples. This is the aim of the next section.
[**3. Examples**]{}
[**Example 1.**]{} Poisson structures for the Halphen equations and the system of circle maps.
Let us first consider the following Poisson structure which has deserved some attention regarding the analysis of the Halphen system [@gyn1]: $$\label{plnjhalp1}
{\cal J} = \eta (x) \left( \begin{array}{ccc}
0 & x_1 - x_2 & x_1 - x_3 \\
x_2 - x_1 & 0 & x_2 - x_3 \\
x_3 - x_1 & x_3 - x_2 & 0
\end{array} \right)$$ with $$\label{plnjhalp2}
\eta (x) = (2(x_1-x_2)(x_2-x_3)(x_3-x_1))^{-1}$$ It can be seen that the structure matrix (\[plnjhalp1\]-\[plnjhalp2\]) belongs to the family (\[plnsol1\]) with $\psi_i(x_i)=x_i$ and $\kappa_{ij} = 0$ for all $i,j=1,2,3$, provided $x_i \neq x_j$ in $\Omega$ for every pair $i \neq j$. If this is the case, function $\eta (x)$ is $C^1 ( \Omega )$ and nonvanishing in $\Omega$. Note that this condition also implies $\chi _{ij}(x_i,x_j) \neq 0$ (and therefore $J_{ij}(x) \neq 0$) in $\Omega$ for every pair $i \neq j$. In order to perform the Darboux reduction it should be noted that every Casimir invariant (\[plncas\]) is now defined in $\Omega$ and can thus be employed. For instance, we can focus on $C_3(x)$: $$\label{plnhcas}
C_3(x) = \frac{x_2-x_3}{x_1-x_2}$$ Therefore the reduction to Darboux form now makes use of the following diffeomorphism $$y_1 = x_1 \;\: , \;\:\;\:
y_2 = x_2 \;\: , \;\:\;\:
y_3 = -C_3(x)$$ with $C_3(x)$ given by (\[plnhcas\]). The inverse of this transformation is then: $$x_1 = y_1 \;\: , \;\:\;\:
x_2 = y_2 \;\: , \;\:\;\:
x_3 = y_2 + (y_1 - y_2) y_3$$ After applying (\[plnjdiff\]) the outcome is that ${\cal J}$ in (\[plnjhalp1\]-\[plnjhalp2\]) is transformed into: $${\cal J'} = (y_1 - y_2) \eta (y_1,y_2,y_2 + y_3 (y_1 - y_2)) \left( \begin{array}{ccc}
0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) \equiv
\tilde{J}_{12}(y) \left( \begin{array}{ccc}
0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)$$ with $\tilde{J}_{12}(y)=(2(y_1-y_2)^2y_3(1-y_3))^{-1}$. The reduction is then completed by means of the time reparametrization $\mbox{d} \tau = \tilde{J}_{12}(y) \mbox{d} t$, which finally leads to the Darboux canonical form (\[plnjdarb\]) with $y_3$ acting as the decoupled Casimir and $(y_1,y_2)$ as classical conjugate Hamiltonian variables.
To conclude, it is worth mentioning in this context the Poisson structure appearing in the study of the system of circle maps [@gyn1]. Such structure is of the form (\[plnjhalp1\]), but now having $$\eta (x) = -((x_1-x_2)(x_2-x_3)(x_3-x_1))^{-1}$$ Thus the conditions for the regularity of the functions are exactly the same, the functions $\phi_i(x_i)$ and $\psi_i(x_i)$ retain their definitions, and the constants $\kappa _{ij}$ have the same zero values, than in the case of the Poisson structure for the Halphen system. The difference existing in $\eta (x)$ does not induce variations in the form of the Casimir invariants, in the diffeomorphic changes of variables leading to the Darboux reduction, or in the conditions indicating when all of them are properly defined. Consequently these results also remain valid in the context of the Poisson structure for circle maps.
[**Example 2.**]{} Euler top.
As a second example, the following cubic and homogeneous Poisson structure appearing [@gyn1] in the analysis of the Euler equations for a triaxial top will be considered: $$\label{plnjtop}
J_{12} = ( \alpha_2 x_1^2 - \alpha_1 x_2^2) x_3 \;\; , \;\;\;\;
J_{23} = ( \alpha_3 x_2^2 - \alpha_2 x_3^2) x_1 \;\; , \;\;\;\;
J_{31} = ( \alpha_1 x_3^2 - \alpha_3 x_1^2) x_2$$ where the $\alpha_i$ are real constants related to the principal moments of inertia $I_i$ of the top according to the expressions: $$\alpha_1 = \frac{I_2-I_3}{I_2 I_3} \;\; , \;\;\;\;
\alpha_2 = \frac{I_3-I_1}{I_1 I_3} \;\; , \;\;\;\;
\alpha_3 = \frac{I_1-I_2}{I_1 I_2}$$ Assuming that $\alpha _1 \alpha _2 \alpha _3 \neq 0$, equations (\[plnjtop\]) can be equivalently written as: $$\label{plnjtopf}
\left\{ \begin{array}{c}
J_{12}= {\displaystyle \frac{1}{2 \alpha _1 \alpha _2 \alpha _3}}
( \alpha_2 \alpha_3 x_1^2 - \alpha_1 \alpha_3 x_2^2)(2 \alpha_1 \alpha_2 x_3) \\
\mbox{} \\
J_{23}= {\displaystyle \frac{1}{2 \alpha _1 \alpha _2 \alpha _3}}
( \alpha_1 \alpha_3 x_2^2 - \alpha_1 \alpha_2 x_3^2)(2 \alpha_2 \alpha_3 x_1) \\
\mbox{} \\
J_{31}= {\displaystyle \frac{1}{2 \alpha _1 \alpha _2 \alpha _3}}
( \alpha_1 \alpha_2 x_3^2 - \alpha_2 \alpha_3 x_1^2)(2 \alpha_1 \alpha_3 x_2)
\end{array} \right.$$ Expressed in this way, the structure matrix (\[plnjtopf\]) can be recognized as a member of family (\[plnsol1\]) with $$\eta = (2 \alpha _1 \alpha _2 \alpha _3)^{-1} \;\; , \;\;\;\;
\psi_1(x_1) = \alpha_2 \alpha_3 x_1^2 \;\; , \;\;\;\;
\psi_2(x_2) = \alpha_1 \alpha_3 x_2^2 \;\; , \;\;\;\;
\psi_3(x_3) = \alpha_1 \alpha_2 x_3^2$$ and $\kappa_{ij} = 0$ for all $i,j=1,2,3$. Since functions $\phi_i(x_i)$ must be nonvanishing in $\Omega$, this implies that in what follows the Poisson structure (\[plnjtopf\]) is to be analyzed in an open subset of $\{ (x_1,x_2,x_3) \in I \!\! R^3 : x_1x_2x_3 \neq 0 \}$. In addition, according to (\[plncas\]) we can employ different forms for the Casimir invariant. For instance, if $\chi_{12}(x_1,x_2) = \alpha_2 \alpha_3 x_1^2 - \alpha_1 \alpha_3 x_2^2 \neq 0$ in $\Omega$, we have: $$\label{plntopc3}
C_{3}(x) = \frac{\alpha_1 \alpha_3 x_2^2 - \alpha_1 \alpha_2 x_3^2}{\alpha_2 \alpha_3 x_1^2 - \alpha_1 \alpha_3 x_2^2}$$ Then, in this case a transformation leading to the Darboux canonical form is defined by (\[plnd12\]) and (\[plntopc3\]), and its inverse is a diffeomorphism in $y( \Omega )$ given by: $$x_1 = y_1 \;\; , \;\;\;\; x_2 = y_2 \;\; , \;\;\;\;
x_3 = \sigma_3 \left( \frac{\alpha_3}{\alpha_2}y_2^2 + \left(
\frac{\alpha_3}{\alpha_1}y_1^2 - \frac{\alpha_3}{\alpha_2}y_2^2 \right) y_3 \right)^{1/2}$$ where $\sigma_3 \equiv$ sign($x_3$) denotes the usual sign function, namely $(+1)$ if $x_3 > 0$ and $(-1)$ if $x_3 < 0$ (recall that $x_3 \neq 0$ in $\Omega$). The rest of the Darboux reduction does not present special features apart from the ones indicated in the proof of Theorem 3, and therefore is omitted for the sake of conciseness.
[**4. Final remarks**]{}
The study of skew-symmetric solutions of the Jacobi equations provides an increasingly rich perspective of finite-dimensional Poisson structures. In this letter a new family of skew-symmetric solutions of the Jacobi identities has been presented and analyzed. Interestingly, the resulting solutions are not limited to a given degree of nonlinearity. This generality implies that already known Poisson structures and systems can now be seen as particular cases and become amenable to analysis in a common framework. Therefore, this unification allows the development of general methods simultaneously valid for every particular Poisson structure considered. Specifically, it is possible to determine in an algorithmic and explicit way the Casimir invariants and the Darboux canonical form. This is interesting, as far as the determination of the Darboux coordinates is in general a nontrivial task only known for a limited sample of systems. Moreover, in this work such coordinates have been charazerized globally in phase space, therefore beyond the usual scope of Darboux theorem, which only ensures a local reduction. Together with the fact that the obtainment, classification and analysis of solutions of (\[plnjac\]-\[plnsksym\]) are unavoidable steps in order to recast a given differential flow as a Poisson system, whenever possible, the previous results reinforce the conclusion that the direct investigation of the Jacobi equations provides fruitful perspectives for a better understanding of finite-dimensional Poisson structures, and therefore of the scope of Hamiltonian dynamics.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We find that a sunspot with positive polarity had an obvious counter-clockwise rotation and resulted in the formation and eruption of an inverse S-shaped filament in NOAA active region (AR) 08858 from 2000 February 9 to 10. The sunspot had two umbrae which rotated around each other by 195 degrees within about twenty-four hours. The average rotation rate was nearly 8 degrees per hour. The fastest rotation in the photosphere took place during 14:00UT to 22:01UT on February 9, with the rotation rate of nearly 16 degrees per hour. The fastest rotation in the chromosphere and the corona took place during 15:28UT to 19:00UT on February 9, with the rotation rate of nearly 20 degrees per hour. Interestingly, the rapid increase of the positive magnetic flux just occurred during the fastest rotation of the rotating sunspot, the bright loop-shaped structure and the filament. During the sunspot rotation, the inverse S-shaped filament gradually formed in the EUV filament channel. The filament experienced two eruptions. In the first eruption, the filament rose quickly and then the filament loops carrying the cool and the hot material were seen to spiral into the sunspot counterclockwise. About ten minutes later, the filament became active and finally erupted. The filament eruption was accompanied with a C-class flare and a halo coronal mass ejection (CME). These results provide evidence that sunspot rotation plays an important role in the formation and eruption of the sigmoidal active-region filament.'
author:
- 'X. L. Yan, Z. Q. Qu, D. F. Kong, C.L. Xu'
title: 'Sunspot rotation, sigmoidal filament, flare, and coronal mass ejection: The event on 2000 February 10'
---
Introduction
============
Sunspot rotational motions have been observed by many authors for many decades (Evershed, 1910; Maltby, 1964; Gopasyuk, 1965). Stenflo (1969) and Barnes & Sturrock (1972) suggested that the rotational motion of a sunspot may be involved with energy build-up and the build-up energy is released by a flare later.
With the high spatial and temporal resolution of recent satellite-borne telescopes, the observations of rotating sunspots are easily obtained (Nightingale et al. 2002). Using white-light images from TRACE, Brown et al. (2003) analyzed the rotation speed of the umbrae and penumbrae of several rotating sunspots. They found that the average rotation speed of the penumbrae of the rotating sunspots was larger than that of the umbra of the rotating sunspots. Through the method of time-distance helioseismology, Zhao & Kosovichev (2003) found the evidence of structural twist beneath the visible surface of a rotating sunspot. The rotating sunspots related to other magnetic structures were also identified by many authors. R$\acute{e}$gnier & Canfield (2006) found that the slow rotation of the sunspot in NOAA AR 8210 enabled the storage of magnetic energy and allowed for the release of magnetic energy as C-class flares. Tian & Alexander (2006) found that the sunspot and the sunspot group exhibited a counterclockwise rotation. The twist of the active-region magnetic fields was dominantly left handed. The vertical current and the current helicity were predominantly negative. Later, Yan & Qu (2007) presented that sunspot rotation resulted in the appearance of the $\Omega$ magnetic loop in the corona and finally the $\Omega$ magnetic loop erupted as a M-class flare. Zhang, Li & Song (2007) reported that a flare was caused by the interaction between a fast rotating sunspot and ephemeral regions. In addition, Schrijver et al. (2008) used non-linear force-free modelling to show the evolution of the coronal field associated with a rotating sunspot, and suggested that the flare energy comes from an emerging twisted flux rope. The detailed information about the polarities, rotation directions and helicities of rotating sunspots in cycle 23 was presented by Yan, Qu & Xu (2008). The active regions with rotating sunspots were classified into six types by Yan, Qu & Kong (2008). They also found that several types have higher flare productivity. Using multi-wavelength observations of Hinode, Yan et al. (2009) and Min & Chae (2009) studied the rapid rotation of a sunspot in NOAA active region 10930 in detail. They found extraordinary counterclockwise rotation of the sunspot with positive polarity before an X3.4 flare. Moreover, the sheared loops and an inverse S-shaped magnetic loop in the corona formed gradually after the sunspot rotation. From a series of vector magnetograms, Yan et al. (2009) found that magnetic force lines are highly sheared along the neutral line accompanying the sunspot rotation. Through analyzing the buildup of the energy and the helicity associated with the eruptive flare on 2005 May 13, Kazachenko et al. (2009) found that sunspot rotation alone can store sufficient energy to power a very large flare. Sunspot rotation may be the primary driver of helicity production and injection into the corona (Zhang, Flyer, & Low 2006; Zhang, Liu, & Zhang 2008; Kumar, Manoharan, & Uddin 2010; Park et al. 2010; Ravindra, Yoshimura, & Dasso 2011).
The sigmoid structure wes often observed to be precursors to CMEs (Sterling & Hudson 1997; Sterling et al. 2000; Pevtsov 2002; Liu et al. 2007; Jiang et al. 2007; Green & kliem 2009; Bi et al. 2011) and statistically more likely to erupt (Hudson et al. 1998; Canfield, Hudson & McKenzie 1999; Canfield et al. 2007). The eruptions of the sigmoid structures or filaments are usually involved with flares and CMEs (Jing et al. 2004; Wang et al. 2007; Yan, Qu & Kong 2011). Amari et al. (2000) presented that the shearing motion resulted in the formation of an S-shaped flux rope by MHD simulation. The emergence of the flux tube can also exhibit a sigmoid structure (Magara & Longcope 2001; Fan 2001; Gibson et al. 2004). A double-J loop pattern can be merged into full S-shaped loops by a slip-running tether-cutting reconnection in the coronal hyperbolic flux tube (Moore et al. 2001; Aulanier et al. 2010). Tripathi et al. (2009) found the coexistence of a pair of J-shaped hot arcs at temperature T $>$ 2 MK with an S-shaped structure at somewhat lower temperature (T $\approx$ 1-1.3 MK). Some observational findings provide strong evidence to support the bald-patch separatrix surface model (Titov & D$\acute{e}$moulin 1999) for the sigmoid (McKenzie & Canfield 2008). Other observations and simulations supposed that the X-ray sigmoid appears at the quasi-separatrix layer between the flux rope and external fields (Gibson et al. 2002; Low & Berger 2003, Kliem, Titov, & T$\ddot{o}$r$\ddot{o}$k 2004; Savcheva & van Ballegooijen 2009). Liu et al. (2002) reported that the sigmoid structure was formed by the reconnection of the emerging flux and the pre-existing field. Green, Kliem, & Wallace (2011) exhibited that the flux cancellation at the internal polarity inversion line resulted in the formation of a soft X-ray sigmoid along the inversion line and a coronal mass ejection. By using reconstructed 3D coronal magnetic field, R$\acute{e}$gnier & Amari (2004) found that the sigmoid was higher than the filament in the corona, while the filament and the sigmoid had the same orientation. Consequently, the formation of the sigmoid structure remains an interesting open question.
In this paper, we present a clear case of the S-shaped active-region filament formation and eruption caused by the sunspot rotation in NOAA active region 08858 on Febuary 10, 2000.
Observations
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The NOAA AR 08858 was observed by several spacecrafts from 2000 February 9 to 10. The active region was located at N28E01 with $\beta$ field configuration of the sunspot group on February 9, 2000. This active region was a very productive active region. It produced 13 C-class, 3 M-class and 1 X-class flares during its journey over the whole solar disk.
The observation of Transition Region and Coronal Explorer (TRACE) covered the whole process of this event from white-light to EUV wavelength. The data of TRACE white-light, 1600 Å and Fe IX/X 171 Å images have a cadence of about 30 seconds - 1 minute and a pixel size of 0.$^\prime$$^\prime$5 (Handy et al. 1999). Full-disk line-of-sight magnetograms are used to show the magnetic fields in the photosphere. The magnetograms were taken by the Michelson Doppler Imager (MDI) on board the Solar and Heliospheric Observatory (SOHO) (Scherrer et al. 1995) with a 96-min cadence and a spatial resolution of 2$^\prime$$^\prime$ per pixel. In addition, we also use the data of soft X-ray flux observed by Geostationary Operational Environmental Satellite (GOES) to identify flare occurrence. The data from Large Angle and Spectrometric Coronagraph (LASCO; Brueckner et al. 1995) C2 on board SOHO (Domingo et al. 1995) are used to identify the coronal mass ejection (CME).
Sunspot rotation and the magnetic field evolution
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Sunspot rotation
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Figure 1 shows the whole NOAA AR 08858 observed by TRACE white-light (left panel) and SOHO/MDI magnetogram (right panel). The rotating sunspot is marked by the red box and the black arrows in Fig. 1. The area of the red box and the yellow box is used to calculate the positive and the negative magnetic flux. The rotating sunspot with positive polarity had two umbrae signed by umbra 1 and umbra 2. This active region contains twelve sunspots. The rotating sunspot was the largest one and located in the southeast of the active region.
Figure 2 shows the evolution of the rotating sunspot acquired at white-light, 1600 Å and 171 Å by TRACE. The left column of Fig. 2 shows the white-light images observed by TRACE. We mark the two umbrae as “U1" and “U2". From 00:00:22UT to 02:40:36UT on February 9, the rotating sunspot was almost quiet. Later, the two umbrae began to rotate counterclockwise. The detailed motion in the photosphere can be seen from the change of the positions of “U1" and “U2". Following the sunspot rotation, the loop-shaped structure first appeared in the chromosphere and then formed an arch-shaped structure. In the corona, the filament was gradually formed in the filament channel. The filament connecting the rotating sunspot was also found to rotate around the center of rotating sunspot counterclockwise.
Figure 3 shows the three images acquired at white light, 1600Å, 171Å on February 9. The circles in the images contain two umbrae of the rotating sunspot and are used to calculate the rotational angle. The white brackets denote the rotational angles. The arrows denote the features which are used to calculate the rotational angle. We calculated the rotational angle of umbra “U2" around the center of circle. The front of umbra “U2" (see the arrow in the left panel of Fig. 3) that moved along the circle is used to evaluate the rotational angle (see the left panel of Fig. 3). From a series of the TRACE images, we can get the coordinates of the center of the circle and the points. We adopt the average values of three repeated measurements of the angles. The measurement uncertainty is about one degree. Moreover, the rotational angles of both the bright loop-shaped structure marked by the arrows in TRACE 1600 Å images and the filament marked by the arrows in TRACE 171 Å images were also calculated. The emitting structure is identified as filament whereas the absorptive dark structure is identified as filament channel in the right panel of Fig. 2. We use the part of the bright loop-shaped structure (see the arrow in the middle panel of Fig. 3) and the filament that connected the umbrae of the rotating sunspot (see the arrow in the right panel of Fig. 3) to calculate the rotational angle. We trace the evolution of the features from a series of TRACE 1600 Å and 171 Å images to determine the positions of the features. Note that the angle is defined as the angle between the line connecting the point where the bright loop-shaped structure is situated on the circle with the center of the circle and the radius of the circle at 0 degree. Because the bright features have a certain width, we adopt the center point of the bright features to do the measurement, which is located on the circle. The radius of the circle is 5 arcseconds. The coordinates can be seen from Fig. 3. It is worth pointing out that the bright loop-shaped structure in the chromosphere and the filament are 3-dimensional and the projection effect has to be taken into account when measuring apparent motion of a feature in general. It is hard to reconstruct the real shape of the loops by the single spacecraft observations. Our observations are based on the evolution of the magnetic loop topology from two-dimensional data.
Figure 4 shows the rotational angle of umbra “U2" (red line), the bright loop-shaped structure in TRACE 1600 Å (blue line), the filament in 171 Å images (green line) and the evolution of the magnetic flux (negative: dashed line; positive: dotted line). The umbra “U2“ rotated by 195 degrees. The average rotation rate was about 8 degrees per hour for 24 hours. The fastest rotation in the photosphere took place during 14:00UT to 22:01UT on February 9, with a rotation rate of nearly 16 degrees per hour. The bright loop-shaped structure in the chromosphere and the filament in the corona rotated by 142 degrees and 116 degrees. From 15:28UT to 19:00UT, the bright loop-shaped structure in the chromosphere and the filament in the corona rotated by 65 degrees and 85 degrees, with a rotation rate of nearly 19 degrees and 24 degrees per hour. Diamond (green line), Asterisk (blue line) and Plus (red line) respectively denote the rotational angles of the filament in the corona, the bright loop-shaped structure in the chromosphere, and umbra ”U2” in the photosphere. The rotational angle decreased from the photosphere to the corona. It is evidenced that the sunspot rotation transfers the magnetic twist from the sub-surface to the corona.
The magnetic field evolution
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The dashed line and the dotted line in Fig. 4 show the evolution of magnetic flux (right axis) calculated from the region marked by the yellow (negative magnetic flux) box and the red (positive magnetic flux) box. From the evolution of the magnetic flux, there was a slow decrease of negative magnetic flux from 11:15UT to 20:47UT on February 9 and then the negative magnetic flux increased a little. For the positive magnetic flux, there was a slow increase from 23:59UT on February 8 to 14:27UT on February 9 and then the positive magnetic flux increased rapidly from 16:03UT to 19:11UT on February 9. Interestingly, the rapid increase of the positive magnetic flux just occurred during the fastest rotation of the rotating sunspot, the bright loop-shaped structure and the filament. At the beginning of the sunspot rotation, the magnetic flux was very stable. In addition, there was no eruption within about five hours from GOES observation before the sunspot rotation. The disturbance from the eruptions in this active region can also be excluded.
The formation and eruption process of the filament
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The formation process of the filament
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From 02:49:36UT on Feb. 9 to 02:49:45UT on Feb. 10, 2000, the two umbrae rotated counterclockwise by 195 degrees. The middle column of Fig. 2 shows the 1600 Å images observed by TRACE. There was a small bright loop-shaped structure marked by the white arrows at 02:40:32UT in TRACE 1600 Å images. The loop-shaped structure was followed by the sunspot rotation and rotated counterclockwise around the center of the rotating sunspot. From 16:19:54UT to 20:35:48UT on Feb. 9, the loop-shaped structure formed an arch shape. Finally, it disappeared after the flare. The right column of Fig. 2 shows the 171 Å images observed by TRACE. The dotted lines in the first two images of the right column denote the EUV filament channel. The red dotted lines indicate the filament. Until 12:17:46UT on Feb. 9, a curve loop-shaped filament marked by the white arrows and outlined by the red dotted line appeared. The filament also rotated counterclockwise around the center of the sunspot. The filament was formed as a dark structure initially, and then part of it was brightened. This brightened part connecting the rotating sunspot was identified and measured. The change of the filament can be seen from the positions marked by the white arrows in the right column. Following the sunspot rotation, the part of the filament that connected the umbra of the rotating sunspot met the left part of the filament channel (see the position marked by the black arrow in the right panel of Fig. 2), then the rotational motion stopped and the filament finally erupted. The field of view of the left and the middle column images is 50$^\prime$$^\prime$ $\times$ 50$^\prime$$^\prime$. In order to show the formation process of the active-region filament, the field of view of the right column images is adjusted to 150$^\prime$$^\prime$ $\times$ 150$^\prime$$^\prime$. The detailed formation process of the filament can be seen from the movie (filamentformation.mpg) linked to Fig. 2.
The first failed eruption of the filament
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Figure 5 shows a sequence of 171 Å images during the first failed filament eruption on February 10, 2000. The dashed line in Fig. 5a indicates the filament channel and the white line in Fig. 5a denotes the position of the time slice of Fig. 6. From a sequence of TRACE 171Å images, one can see that the filament gradually rose from the EUV filament channel after the sunspot underwent tens of hours of rotation motion. The filament was marked by the dotted lines in Figs. 5b and 5c. Moreover, the filament exhibited a swirling shape. The white arrows in Fig. 5b and 5c point to the hot material of the filament. Before the filament eruption, the filament loops carrying the hot material can be seen to be moving counterclockwise. After about ninety seconds, the hot plasma moved to the position signed by the white arrow in Fig. 5c. At 01:14:38UT, the filament rose rapidly and formed a fan-shaped structure. The two dotted lines outline the outer and the inner boundary of the filament in Fig. 5d. Note that the filament is composed by many bright loops. The white arrow and the black arrow in Figs. 5d and 5e denote the lower and the upper part of the filament. During the rise of the filament, we observe apparent counterclockwise motion of hot and cool materials along the filament loops. The black arrows in Figs. 5f-5i denote the change of the position of the cool material. The white arrows denote the hot material which gradually fell into the umbrae of the sunspot. It is worth pointing out that the movement of the cool and the hot material is not the true movement of the material. In fact, the movement of the filament loops carried the cool and the hot material. At 01:26:48UT on February 10, 2000, the loops of the whole filament were contracted and later were seen to spiral into the sunspot umbrae. The dotted lines in Figs. 5e-5l outline the outer boundary of the filament. From 01:14:38UT to 01:26:48UT on February 10, the filament loops gradually contracted (see the dotted lines in Fig. 5). The detailed process of the first filament eruption can be seen from the movie (firsteruption.mpg) linked to Fig. 5.
Figure 6 shows the time slice at the position marked by the white line in Fig. 5a. The bright structure shows the trajectory of the filament. The two dotted lines denote the lower and the upper boundary of the filament. From the evolution of the filament intensity, one can see the filament first expanded outward and then fell down.
The second successful eruption of the filament
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After the first failed eruption, the filament gradually became active. At 01:36:16UT on February 10, a small part of the filament began to erupt. From the observation of TRACE 1600 Å (see Fig. 9), the two flare ribbons began to form at 01:40UT as signature of magnetic reconnection upon the filament eruption. Figure 7 shows a sequence of 171 Å images from 01:38:24UT to 02:39:05UT on February 10, 2000. At 01:38:24UT on February 10, the right part of the filament became active again. The white arrow points to the same loop of the filament in Figs. 7a-7c. At 01:39:59UT, another bright loop marked by the black arrow in Fig. 7b appeared. Subsequently, the bright loop marked by the black arrow first disappeared and then the other bright loop marked by the white arrow vanished. The disappearance of the features may be temperature effect. After the eruption of the two bright loops, the bright material of the filament was found to flow from right to left. The white arrows in Figs. 7d-7g indicate the positions of the hot plasma from 01:43:39UT to 01:48:23UT on February 10. The filament loops carrying the hot plasma gradually moved along the loop from west to east. At 01:48:56UT, another part of the filament enclosing the rotating sunspot also erupted. Next, the filament exhibited clearly an inverse S-shaped structure marked by the dotted lines in Figs. 7i and 7j. There was a data gap from 02:08:05UT to 02:35:22UT on February 10. However, comparing the change of the magnetic structure, it is easy to find that the inverse S-shaped filament disappeared. The post-flare loops marked by the white arrows in Figs. 7k and 7l can be seen clearly. The detailed process of the second filament eruption can be seen from the movie (seconderuption.mpg) linked to Fig. 7.
The associated flare and CME
----------------------------
Figure 8 shows the evolution of GOES soft X-ray emission for the C7.3 flare on February 10, 2000. The C7.3 flare started at 01:40UT, peaked at 02:08UT, and ended at 02:39UT. Figure 9 shows the evolution of the flare ribbons from 01:41:39UT to 02:35:28UT on February 10. The two white arrows in Fig. 9a and 9b indicate the two flare ribbons. The two flare ribbons gradually became brightening. The left flare ribbon along the dotted lines in Figs. 9c-9g expanded toward the southwest of the following sunspot. The flare ribbons swept across the umbra of the following sunspot while this did not happen to the leading rotating sunspot. Finally, the flare ribbon swept completely the following sunspot. Li & Zhang (2009) suggested that the emergence, the rotation, and the shear motion of the following sunspot and leading sunspot caused flare ribbons to sweep across sunspots completely. In this event, the flare ribbon did not sweep the rotating sunspot unlike those examples that Li & Zhang (2009) investigated. After the inverse S-shaped filament erupted, the SOHO/LASCO observed a halo CME.
Conclusion and Discussion
=========================
We investigate the relationship between the sunspot rotation and the formation and eruption of an active-region filament associated with a C7.3 flare and a halo CME in NOAA AR 08858 on Feb 10, 2000 using the GOES12 soft X-ray flux, TRACE WL, 1600 Åand 171 Å images, SOHO/MDI 96-min magnetograms, and SOHO/LASCO C2 images. We find that the formation of the active-region filament in EUV filament channel was followed by sunspot rotation. The sunspot rotated counter-clockwise and the active-region filament exhibited an inverse S-shaped structure. The filament experienced two eruptions. In the first eruption, a part of the filament rose and much of the material warmed up (becoming bright). The filament loops carrying the material were seen to spiral into the sunspot counterclockwise in the middle as it fell back towards the solar surface. In the second eruption, the inverse S-shaped filament fully erupted and produced a C-class flare and a halo CME. Before the second eruption, the filament loops carrying the hot material moved clockwise along the magnetic loop.
This event is a clear case of the formation of the sigmoidal active-region filament caused by sunspot rotation. According to the sunspot rotational direction (counterclockwise) and the shape of the filament (the inverse S-shaped filament), we can determine that the sunspot had negative helicity. The inverse S-shaped filament followed the hemisphere helicity rule. From the topology evolution of the magnetic loops in the corona, one can see that the sunspot rotation resulted in the upper magnetic field rotation and made the magnetic fields trend to non-potential field. It is evidenced that sunspot rotation is a means of magnetic energy storage. The energy was released via flares and CME later.
During the observation, no obvious magnetic flux emergence was found before the sunspot rotation. But there was a slow increase of the positive magnetic flux from 23:59UT on February 8 to 14:27UT on February 9 and then the positive magnetic flux increased rapidly from 16:03UT to 19:11UT on February 9. It is interesting that the rapid increase of the positive magnetic flux just occurred during the fastest rotation of the rotating sunspot, the bright loop-shaped structure and the filament. The observation provides evidence that the sunspot rotation could be regarded as a result of the transfer of additional magnetic twist from the sub-surface to the corona. The investigation of Zhao & Kosovichev (2003) also evidenced that there was a strong subsurface vortical flow below a rotating sunspot. Magara & Longcope (2001) and Fan (2009) presented a simulation on the emergence of a twisted flux tube into the solar atmosphere. During the emergence, the opposite polarity regions separated and rotated toward a more axial orientation. Fan (2009) concluded that the rotation in the two polarities is a result of propagation of nonlinear torsional Alfven waves along the flux tube, which transports significant twist from the tube’s interior portion to its expanded coronal portion. In some events, the sunspot rotation was obviously accompanied with the emergence flux and polarity separation (Zhang, Li & Song 2007; Jiang et al. 2011). However, in this event, no these characteristics were found. We assume that the sunspot rotation can originate in two ways. It needs more observations to confirm these results.
When the total twist of the field exceeds a little over one turn, or 2.5$\pi$ (Hood 1991; $Vr\check{s}nak$, $Ru\check{z}djak$ & Rompolt 1991; Rust et al. 1994; T$\ddot{o}$r$\ddot{o}$k, & Kliem 2003), the flux rope becomes unstable. Leamon et al. (2003) measured the total twist of 191 X-ray sigmoids and found that most of the sigmoids have a total twist less than one turn. In this event, the sunspot rotated by 195 degrees and the twist was less than the critical value obtained by former authors. However, the filament also erupted finally. We assume that the eruption of the flux rope is relative to not only the twist caused by sunspot rotation but also self-twist before sunspot rotation.
The authors thank the referee for very constructive comments and suggestions. The authors thank the TRACE, SOHO, and GOES consortia for their data. SOHO is a project of international cooperation between ESA and NASA. This work is sponsored by National Science Foundation of China (NSFC) under the grant numbers 10903027, 11078005, 10943002, Yunnan Science Foundation of China under number 2009CD120, China’s 973 project under the grant number G2011CB811400.
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abstract: 'Delsarte, Goethals, and Seidel (1977) used the linear programming method in order to find bounds for the size of spherical codes endowed with prescribed inner products between distinct points in the code. In this paper, we develop the linear programming method to obtain bounds for the number of vertices of connected regular graphs endowed with given distinct eigenvalues. This method is proved by some “dual” technique of the spherical case, motivated from the theory of association scheme. As an application of this bound, we prove that a connected $k$-regular graph satisfying $g>2d-1$ has the minimum second-largest eigenvalue of all $k$-regular graphs of the same size, where $d$ is the number of distinct non-trivial eigenvalues, and $g$ is the girth. The known graphs satisfying $g>2d-1$ are Moore graphs, incidence graphs of regular generalized polygons of order $(s,s)$, triangle-free strongly regular graphs, and the odd graph of degree $4$.'
author:
- Hiroshi Nozaki
title: Linear programming bounds for regular graphs
---
**Key words**: linear programming bound, graph spectrum, expander graph, Ramanujan graph, distance-regular graph, Moore graph.
Introduction
============
Delsarte [@D73_2] has introduced the linear programming method to find bounds for the size of codes with prescribed distances over finite field. This is called Delsarte’s method, and he stated it for codes in certain special association schemes, so called $Q$-polynomial schemes, including the Johnson scheme and the Hamming scheme. Delsarte, Goethals, and Seidel [@DGS77] gave the linear programming method on the Euclidean sphere. This is naturally generalized to the compact two-point homogeneous spaces [@KL78]. Delsarte’s method is also extended to various situations like the permutation codes [@T99], the Grassmannian codes [@B06], or the ordered codes [@BP09]. The linear programming is very powerful to solve optimization problems, for instance maximizing the size of codes for given distances [@OS79; @M08], or maximizing the minimum distance for a fixed cardinality [@L92; @CK07]. In the present paper, we develop the linear programming method to find bounds for the order of connected regular graphs with given distinct eigenvalues. This method is not based on Delsarte’s but a kind of “dual” technique of the spherical case inspired from the theory of association schemes.
Let $X$ be a finite set, and $R_0, \ldots, R_d$ symmetric binary relations on $X$. The [*$i$-th adjacency matrix*]{} $\boldsymbol{A}_i$ is defined to be the matrix indexed by $X$ whose $(x,y)$-entry is 1 if $(x,y) \in R_i$, 0 otherwise. A configuration $\mathfrak{X}=(X,\{R_i\}^{d}_{i=0})$ is called a [*symmetric association scheme*]{} of class $d$ if $\{\boldsymbol{A}_i \}_{i=0}^d$ satisfies the following: (1) $\boldsymbol{A}_0=\boldsymbol{I}$ (identity matrix), (2) $\sum_{i=0}^d \boldsymbol{A}_i=\boldsymbol{J}$ (all-ones matrix), (3) there exist real numbers $p_{ij}^k$ such that $\boldsymbol{A}_i\boldsymbol{A}_j =\sum_{k=0}^d p_{ij}^k\boldsymbol{A}_k$ for all $i,j \in \{0,1,\ldots,d\}$. The vector space $\mathfrak{A}$ spanned by $\{\boldsymbol{A}_i \}_{i=0}^d$ over $\mathbb{C}$ forms a commutative algebra, and it is called the [*Bose–Mesner algebra*]{} of $\mathfrak{X}$. It is well known that $\mathfrak{A}$ is semi-simple [@BIb Section 2.3, II], hence it has the primitive idempotents $\boldsymbol{E}_0=(1/|X|)\boldsymbol{J}, \boldsymbol{E}_1, \ldots, \boldsymbol{E}_d$, which form a basis of $\mathfrak{A}$.
We have two remarkable classes of association schemes so called $P$-polynomial association schemes and $Q$-polynomial association schemes. An association scheme is said to be [*$P$-polynomial*]{} if for each $i\in \{0,1, \ldots, d\}$ there exists a polynomial $v_i$ of degree $i$ such that $\boldsymbol{A}_i=v_i(\boldsymbol{A}_1)$. A $P$-polynomial scheme has the relations as the path distances of the graph $(X,R_1)$, and $(X,R_1)$ becomes a distance-regular graph [@BCNb]. An association scheme is said to be [*$Q$-polynomial*]{} if for each $i\in \{0,1, \ldots, d\}$ there exists a polynomial $v^{*}_i$ of degree $i$ such that $|X|\boldsymbol{E}_i=v_i^*(|X|\boldsymbol{E}_1^\circ)$, where $\circ$ means the multiplication is the entry-wise product. Roughly speaking, the $P$-polynomial schemes and the $Q$-polynomial schemes correspond to discrete cases of the concepts of two-point homogeneous spaces and rank 1 symmetric spaces, respectively [@BIb Section 3.6, III], [@CSb Chapter 9].
By swapping the matrix multiplication $\cdot$ and the entry-wise multiplication $\circ$, the bases $\{\boldsymbol{A}_i\}_{i=0}^d$ and $\{\boldsymbol{E}_i\}_{i=0}^d$ very similarly behave in the Bose–Mesner algebra. The following are basic equations for the bases [@BIb Section 2.2, 2.3, II]: $$\begin{aligned}
&\sum_{i=0}^d \boldsymbol{A}_i =\boldsymbol{J}=|X|\boldsymbol{E}_0, & &\sum_{i=0}^d \boldsymbol{E}_i=\boldsymbol{I}=\boldsymbol{A}_0, \label{1.1}\\
&\boldsymbol{A}_i \circ \boldsymbol{A}_j = \delta_{ij}\boldsymbol{A}_i, &
& \boldsymbol{E}_i \cdot \boldsymbol{E}_j = \delta_{ij}\boldsymbol{E}_i, \label{1.2} \\
&\boldsymbol{A}_i \cdot \boldsymbol{A}_j = \sum_{k=0}^d p_{ij}^k \boldsymbol{A}_k, &
&\boldsymbol{E}_i \circ \boldsymbol{E}_j = \frac{1}{|X|}\sum_{k=0}^d q_{ij}^k \boldsymbol{E}_k, \label{1.3} \\
&\boldsymbol{A}_i= \sum_{j=0}^d P_i(j)\boldsymbol{E}_j,& &\boldsymbol{E}_i=\frac{1}{|X|}\sum_{j=0}^d Q_i(j) \boldsymbol{A}_j, \label{1.4} \end{aligned}$$ $$\begin{aligned}
&\boldsymbol{A}_i\cdot \boldsymbol{J} =k_i \boldsymbol{J}, & &|X|\boldsymbol{E}_i \circ \boldsymbol{I}= m_i \boldsymbol{I}, \label{1.5} \\
&\boldsymbol{A}_i \circ \boldsymbol{I}=0\ (i\ne 0), & & \boldsymbol{E}_i \cdot \boldsymbol{J}=0\ (i\ne 0), \label{1.9} \\
&\tau(\boldsymbol{A}_i)=|X|k_i, & &{\rm tr}(\boldsymbol{E}_i)=m_i, \label{1.6}\\
&{\rm tr} (\boldsymbol{A}_i)=0\ (i\ne 0), & &\tau(\boldsymbol{E}_i)=0\ (i \ne 0), \label{1.8}\\
&k_0=1, & &m_0=1, \label{1.7}\end{aligned}$$ where $\delta_{ij}$ denotes the Kronecker delta, $\tau(\boldsymbol{M})$ denotes the summation of all entries in $\boldsymbol{M}$, $k_i$ is the degree of the graph $(X,R_i)$, and $m_i$ is the rank of $\boldsymbol{E}_i$. Here $p_{ij}^k$ is called the [*intersection number*]{}, and it is equal to the size of $\{z \in X \mid (x,z) \in R_i, (z,y) \in R_j\}$ with $(x,y) \in R_k$. Naturally $p_{ij}^k$ is a non-negative integer. On the other hand, $q_{ij}^k$ is called the [*Krein number*]{}, it can be proved that it is a non-negative real number [@S73]. Such a kind of similar properties obtained by swapping $\boldsymbol{A}_i$, $\boldsymbol{E}_i$, multiplications, and corresponding parameters is called a [*dual property*]{}. It is obviously seen that the dual concept of $P$-polynomial scheme is $Q$-polynomial scheme. There are a number of non-trivial dual properties between $P$-polynomial schemes and $Q$-polynomial schemes [@MT09; @BIb], and several conjectures are still left [@MT09].
The matrix $\boldsymbol{A}_i$ is regarded as a regular graph. The matrix $\boldsymbol{E}_i$ is positive semidefinite with equal diagonals, and it is interpreted as a spherical set. We can observe the dual relationship of $\boldsymbol{A}_i$ and $\boldsymbol{E}_i$ in the Bose–Mesner algebra, and it shows how properties of graphs dually correspond to those of spherical sets. For example, says that eigenvalues of graphs dually correspond to inner products in spherical sets. Table \[tb:1\] shows the dual correspondence of the properties of graphs and spherical sets.
regular graph spherical set reason
------------------------------------------------------------------ ---------------------------------------------------------------------------- ----------------
$\boldsymbol{A}$: adjacency matrix $\boldsymbol{E}$: Gram matrix bases
$k$: degree $m$: dimension ,
regular spherical
eigenvalues inner products
connected constant weight
no loop spherical 1-design
no multiple edge spherical 2-design
$\tau$ ${\rm tr}$ ,
Moore graph tight spherical design tightness
$F_i^{(k)}(x)$ $\mathcal{Q}_i^{(m)}(x)$
${\rm tr}(F_i^{(k)}(\boldsymbol{A})) \geq 0$ $\tau(\mathcal{Q}_i^{(m)}(\boldsymbol{E}^\circ))\geq 0$ [@S66; @DGS77]
${\rm tr}(F_i^{(k)}(\boldsymbol{A})) = 0$ for $1\leq i \leq g-1$ $\tau(\mathcal{Q}_i^{(m)}(\boldsymbol{E}^\circ)) = 0$ for $1\leq i \leq t$
$\Leftrightarrow$ girth $g$ $\Leftrightarrow$ spherical $t$-design [@S66; @DGS77]
: Dual properties between regular graph and spherical set[]{data-label="tb:1"}
We can interpret the Euclidean sphere $S^{m-1}$ as a continuous case of $Q$-polynomial scheme. The polynomial $v_i^*$ on a $Q$-polynomial scheme corresponds to the Gegenbauer polynomial $\mathcal{Q}^{(m)}_i$ on $S^{m-1}$ [@DGS77]. We have fundamental parameters $s$ and $t$ for a finite subset $X$ in $S^{m-1}$. The parameter $s$ is just the number of the Euclidean distances between distinct points in $X$. If $X$ has only $s$ distances, then we have $$\label{eq:s-dis}
|X| \leq \binom{m+s-1}{s}+\binom{m+s-2}{s-1}.$$ The other parameter $t$ is the strength in the sense of spherical design. We call $X$ a spherical $t$-design if for any polynomial $f$ in $m$ variables of degree at most $t$ the following equation holds: $$\frac{1}{|S^{m-1}|}\int_{S^{m-1}} f(x) dx= \frac{1}{|X|} \sum_{x \in X} f(x),$$ where $|S^{m-1}|$ is the volume of $S^{m-1}$. One of unexpected results is that if $t\geq 2s-2$ holds, then $X$ has the structure of a $Q$-polynomial scheme with the relations of distances [@DGS77]. For a spherical $2e$-design $X$ in $S^{m-1}$, we have an absolute bound [@DGS77]: $$|X| \geq \binom{m+e-1}{e} + \binom{m+e-2}{e-1}.$$ A spherical design is said to be [*tight*]{} if it attains this equality. A tight design satisfies $t=2s$ [@DGS77], and hence it becomes a $Q$-polynomial scheme. A tight design also attains the bound . Moreover the polynomial $v_i^{\ast}$ of a tight design coincides with $\mathcal{Q}_i^{(m)}$.
For connected regular graph, we have a very similar situation to the above argument on the sphere. Let $G$ be a connected $k$-regular graph with $v$ vertices. Throughout this paper, we assume a graph is simple. Since a graph with $d+1$ distinct eigenvalues is of diameter at most $d$, we can change the assumption of the Moore bound to the number of eigenvalues. Namely if $G$ has only $d+1$ distinct eigenvalues, then we have $$v \leq 1+k\sum_{j=0}^{d-1}(k-1)^j.$$ If equality holds, then $G$ is called a [*Moore graph*]{}. Tutte [@T66] showed that if $G$ is of girth $2e+1$, then we have $$v \geq 1+k\sum_{j=0}^{e-1}(k-1)^j.$$ The graph that attains this equality becomes a Moore graph. It is well known that a Moore graph is distance-regular. Actually we can show that if $g\geq 2d-1$ holds, then $G$ is distance-regular (Theorem \[thm:girth\]). Let $F_i^{(k)}$ be the polynomial of degree $i$ defined by and in Section \[sec:2\]. The polynomial $v_i$ of a $k$-regular Moore graph coincides with $F_i^{(k)}$. Apparently the dual concept of tight spherical design is Moore graph, and the polynomial $F_i^{(k)}$ dually corresponds to the Gegenbauer polynomial $\mathcal{Q}_i^{(m)}$.
The linear programming method for spherical codes is essentially based on the positive definiteness of the Gegenbauer polynomials, namely $\tau(\mathcal{Q}_i^{(m)}(\boldsymbol{E}^\circ)) \geq 0$, where $\boldsymbol{E}$ is the Gram matrix. In this paper, we dually show the linear programming method for connected regular graphs by using the property ${\rm tr}(F_i^{(k)}(\boldsymbol{A})) \geq 0$, where $\boldsymbol{A}$ is the adjacency matrix.
We can apply the linear programming method for determining the graph maximizing the spectral gap. The [*spectral gap*]{} of a graph is the difference between the first and second largest eigenvalues of the graph. The [*edge expansion ratio*]{} $h(G)$ of a $k$-regular graph $G=(V,E)$ is defined as $$h(G)=\min_{S\subset V, |S|\leq |V|/2} \frac{|\partial S|}{|S|},$$ where $\partial S=\{\{u,v\} \mid u \in S, v\in V\setminus S, \{u,v\} \in E \}$. By the spectral gap $\tau$ of $G$, we have $\tau/2 \leq h(G) \leq \sqrt{2k\tau}$ [@A86; @AM85; @D84]. This implies that a graph with large spectral gap has high connectivity. The second-largest eigenvalue cannot be much smaller than $2\sqrt{k-1}$ [@A86]. Ramanujan graphs have an asymptotically smallest possible second-largest eigenvalue (see [@HLW06]). Several regular graphs with very small second-largest eigenvalues are determined (see [@KS13]).
The dual concept of the graphs maximizing the spectral gap is well known as optimal spherical code in the sense of maximizing the minimum distance. The optimal configurations of $n$ points on $S^2$ are known only for $n\leq 13$, and $n=24$ [@EZ01 Chapter 3], [@MT12]. For higher dimensions, the linear or semidefinite programming bound determined many optimal codes [@L92; @CK07; @BV09]. In particular, we have a strong theorem using the parameters $s$ and $t$, namely if $t\geq 2s-1$ holds, then the set is optimal [@L92; @CK07]. In the present paper, as the dual theorem of it, we prove that a connected $k$-regular graph satisfying $g\geq 2d$ has the minimum second-largest eigenvalue of all $k$-regular graphs of the same size, where $d$ is the number of distinct non-trivial eigenvalues, and $g$ is the girth.
Linear programming method {#sec:2}
=========================
In the present section, we give the linear programming bounds for connected regular graphs. First let us introduce certain polynomials $F_i^{(k)}(x)$ which play a key role in the linear programming method. Indeed $F_i^{(k)}(x)$ is the polynomial attached to the homogeneous tree of degree $k$, which is an infinite distance-regular graph.
A graph $G=(V,E)$ is said to be [*locally finite*]{} if the degree of any vertex is finite. We also consider an infinite graph here. A [*path*]{} in a graph is a sequence of vertices, where any two consecutive vertices are connected. Let $d(x,y)$ be the shortest path distance from $x\in V$ to $y \in V$. The [*$i$-th distance matrix*]{} $\boldsymbol{A}_i$ of $G$ is defined to be the matrix indexed by $V$ whose $(x,y)$-entry is 1 if $d(x,y)=i$, 0 otherwise. In particular $\boldsymbol{A}_1$ is called the [*adjacency matrix*]{} of $G$.
A locally finite graph $G=(V,E)$ is called a [*distance-regular graph*]{} if for any choice of $x,y \in V$ with $d(x,y)=k$, the number of vertices $z\in V$ such that $d(x,z)=i$, $d(z,y)=j$ is independent of the choice $x, y$. For $x,y \in V$ with $d(x,y)=k$, the number $
p_{ij}^k=|\{z \in V | d(x,z)=i, d(z,y)=j\}|
$ is called the [*intersection number*]{} of a distance-regular graph. We use the notation $a_i=p_{1,i}^i$, $b_i=p_{1,i+1}^i$, and $c_i=p_{1,i-1}^i$. The [*intersection array*]{} of a distance-regular graph is defined to be $$\begin{pmatrix}
\ast & c_1& c_2& \cdots\\
a_0&a_1 &a_2 & \cdots \\
b_0&b_1 &b_2 & \cdots
\end{pmatrix}.$$ The matrix $\boldsymbol{A}_i$ of a distance-regular graph can be written as the polynomial $v_i$ in $\boldsymbol{A}_1$ of degree $i$ [@BCNb page 127], where $v_i$ is defined by $$\begin{aligned}
&v_0(x)=1, \qquad v_1(x)=x,\\
&c_{i+1}v_{i+1}(x)=(x-a_i)v_i(x)- b_{i-1}v_{i-1}(x) \qquad (i=1,2,\ldots). \end{aligned}$$
A homogeneous tree of degree $k$ is an infinite distance-regular graph with intersection numbers $$b_0=k, \quad
b_i=k-1 (i=1,2,\ldots), \quad
c_i=1 (i=1,2,\ldots), \quad
a_i=0 (i=0,1,2,\ldots).$$ Let $F_i^{(k)}$ denote a polynomial of degree $i$ defined by: $$\label{eq:F_012}
F_0^{(k)}(x)=1, \qquad F_1^{(k)}(x)=x, \qquad F_2^{(k)}(x)=x^2 - k,$$ and $$\label{eq:F_i}
F_i^{(k)}(x)=x F_{i-1}^{(k)}(x)- (k-1) F_{i-2}^{(k)}(x)$$ for $i\geq 3$. Let $q=\sqrt{k-1}$. The polynomials $F_i^{(k)}$ form a sequence of orthogonal polynomials with respect to the weight $$\label{eq:w(x)}
w(x)=\frac{\sqrt{4q^2-x^2}}{k^2-x^2}$$ on the interval $[-2q, 2q]$ (see [@HOb Section 4]). Note that $F_i^{(k)}(k)=k(k-1)^{i-1}$ for any $i\geq 1$.
A path $u_0 \sim u_1 \sim \cdots \sim u_p$ is said to be [*reducible*]{} if any sequence $u_i \sim u_{j} \sim u_i$ appears [@S66]. A path is said to be [*irreducible*]{} if the path is not reducible.
\[thm:red\_path\] Let $G$ be a connected $k$-regular graph with adjacency matrix $\boldsymbol{A}$. Then the $(u,v)$-entry of $F_i^{(k)}(\boldsymbol{A})$ is the number of irreducible paths of length $i$ from $u$ to $v$.
By Theorem \[thm:red\_path\], the following is obvious.
\[coro:girth\] Let $G$ be a connected $k$-regular graph with adjacency matrix $\boldsymbol{A}$. Then the following are equivalent.
1. ${\rm tr}( F_i^{(k)}(\boldsymbol{A}))=0$ for each $1\leq i \leq g-1$, and ${\rm tr}( F_{g}^{(k)}(\boldsymbol{A}))\ne 0$.
2. $G$ is of girth $g$.
The following is the linear programming bound for connected regular graphs.
\[thm:lp\_bound\] Let $G$ be a connected $k$-regular graph with $v$ vertices. Let $\tau_0=k, \tau_1, \ldots, \tau_d $ be the distinct eigenvalues of $G$. Suppose there exists a polynomial $f(x)=\sum_{i\geq 0} f_i F_i^{(k)}(x)$ such that $f(k)>0$, $f(\tau_i) \leq 0$ for any $i\geq 1$, $f_0>0$, and $f_i \geq 0$ for any $i\geq 1$. Then we have $$\label{eq:lp}
v \leq \frac{f(k)}{f_0}.$$
Let $\boldsymbol{A}$ be the adjacency matrix of $G$. From the spectral decomposition $\boldsymbol{A}=\sum_{i=0}^d \tau_i \boldsymbol{E}_i$, we have $$\label{eq:lp_1}
\sum_{i=0}^d f(\tau_i)\boldsymbol{E}_i =f(\boldsymbol{A})=\sum_{i\geq 0} f_i F_i^{(k)}(\boldsymbol{A})
=f_0 \boldsymbol{I} + \sum_{i\geq 1} f_i F_i^{(k)}(\boldsymbol{A}).$$ Taking the traces in , we have $$f(k)={\rm tr} (f(k)\boldsymbol{E}_0)
\geq {\rm tr}( \sum_{i=0}^d f(\tau_i)\boldsymbol{E}_i)
={\rm tr}(f_0 \boldsymbol{I} + \sum_{i\geq 1} f_i F_i^{(k)}(\boldsymbol{A}))
\geq {\rm tr}(f_0 \boldsymbol{I})=vf_0.$$ Therefore we have $
v \leq f(k)/f_0
$.
We can normalize $f_0=1$ in Theorem \[thm:lp\_bound\].
\[rem:attain\] Let $f$ be a polynomial which satisfies the condition in Theorem \[thm:lp\_bound\]. The equality holds in if and only if $f_i{\rm tr} (F_i^{(k)}(\boldsymbol{A}))=0$ for any $i=1,\ldots, {\rm deg}(f)$, and $f(\tau_i)=0$ for any $i=1,\ldots, d$. In particular, if $f_i>0$ for any $i$, then the girth of $G$ is at least ${\rm deg}(f)+1$ by Corollary \[coro:girth\].
Theorem \[thm:lp\_bound\] can be expressed as the following linear programming problem and its dual. [$$v \leq \max_{m_i} \left\{1+m_1+\cdots+m_d \mid
\begin{array}{cc}
-\sum_{i=1}^d m_i F_j^{(k)}(\tau_i)\leq F_j^{(k)}(k), & j=1,\ldots,u, \\
m_i\geq 0, & i=1,\ldots ,d
\end{array}
\right\},$$ $$v \leq \min_{f_j} \left\{ 1+f_1F_1^{(k)}(k)+\cdots+f_uF_u^{(k)}(k) \mid
\begin{array}{cc}
-\sum_{j=1}^u f_j F_j^{(k)}(\tau_i) \geq 1, &
i=1,\ldots,d,\\
f_j\geq 0, &j=1,\ldots, u
\end{array}
\right\},$$ ]{} where $u$ is the degree of $f$, $m_i$ is the multiplicity of $\tau_i$ and $f_0=1$.
Delsarte, Goethals, and Seidel [@DGS77] gave the linear programming bounds for spherical codes by using inner products and Gegenbauer polynomials, instead of eigenvalues and $F_i^{(k)}$. This is the dual version of Theorem \[thm:lp\_bound\].
Minimizing the second-largest eigenvalue
========================================
For fixed $v$ and $k$, a graph $G$ is said to be [*extremal expander*]{} if $G$ has the minimum second-largest eigenvalue in all $k$-regular graphs of order $v$. A disconnected graph is not extremal expander, because the first and second largest eigenvalues are equal. In the present section, we obtain extremal expander graphs for several $v$ and $k$ by applying the linear programming method. First we give several results related to $F_i^{(k)}(x)$.
\[thm:posi\_coef\] Let $F_i^{(k)}(x)F_j^{(k)}(x)=\sum_{l=0}^{i+j} p_l(i,j)F_l^{(k)}(x)$ for real numbers $p_l(i,j)$. Then we have $
p_0(i,j)=F_i^{(k)}(k)\delta_{ij}$ and $p_l(i,j) \geq 0
$ for all $l,i,j$. Moreover $p_l(i,j)>0$ if and only if $|i-j|\leq l \leq i+j$ and $l \equiv i+j \pmod{2}$.
Let $T_k$ be a homogeneous tree of degree $k$. Let $\boldsymbol{A}_i$ be the $i$-th distance matrix of $T_k$, and $p_{ij}^k$ the intersection number of $T_k$. Since $F_i^{(k)}(x)$ is the polynomial attached to $T_k$, we have $$\label{eq:AsAt}
\sum_{l=0}^{i+j} p_l(i,j)\boldsymbol{A}_l=\sum_{l=0}^{i+j} p_l(i,j)F_l^{(k)}(\boldsymbol{A}_1)=F_i^{(k)}(\boldsymbol{A}_1)F_j^{(k)}(\boldsymbol{A}_1)=\boldsymbol{A}_i\boldsymbol{A}_j=
\sum_{l=0}^{i+j} p_{ij}^l\boldsymbol{A}_l.$$ Clearly $p_l(i,j)=p_{ij}^l$ holds. This theorem now follows by a counting argument.
Since $F_{i+1}^{(k)}(k)-(k-1)F_i^{(k)}(k)=0$ holds, let $G_i^{(k)}(x)$ denote the polynomial of degree $i$ $$G_i^{(k)}(x)=\frac{F_{i+1}^{(k)}(x)-(k-1)F_i^{(k)}(x)}{x-k}$$ for any $i\geq 1$, and $G_0^{(k)}(x)=1$. By the three-term recurrence relation , it holds that $$G_i^{(k)}(x)=\sum_{j=0}^iF_j^{(k)}(x).$$ From Lemmas 3.3, 3.5 in [@CK07], $G_0,G_1,\ldots$ are monic orthogonal polynomials with respect to the positive weight $
u(x)=(k-x)w(x)
$ on the interval $[-2q, 2q]$, where $w(x)$ is defined in .
\[thm:CK\] Let $p_0,p_1, \ldots$ be monic orthogonal polynomials with ${\rm deg}(p_i)=i$. Then for any $\alpha\in \mathbb{R}$, the polynomial $p_n+\alpha p_{n-1}$ has $n$ distinct real roots $
r_1<\cdots<r_n$. Moreover for $k<n$, $
\prod_{i=1}^k (x-r_i)
$ has positive coefficients in terms of $p_0(x),p_1(x), \ldots, p_k(x)$.
The following is a key theorem.
\[thm:expander\] Let $G$ be a connected $k$-regular graph of girth $g$. Assume the number of distinct eigenvalues of $G$ is $d+1$. If $g \geq 2d$ holds, then $G$ is an extremal expander graph.
Let $\tau_0=k>\tau_1>\ldots>\tau_{d}$ be the distinct eigenvalues of $G$. We show the polynomial $$f(x)=(x-\tau_1)\prod_{i=2}^d(x-\tau_i)^2=\sum_{i=0}^{2d-1}f_i F_i^{(k)}(x)$$ satisfies the condition in Theorem \[thm:lp\_bound\]. It trivially holds that $f(k)>0$, and $f(\tau_i)=0$ for any $i=1,\ldots,d$.
Let $\boldsymbol{A}$ be the adjacency matrix of $G$. If the diameter of $G$ is greater than $d$, then the number of distinct eigenvalues is greater than $d+1$. Thus the diameter of $G$ is at most $d$. Since $g \geq 2d$ holds, the diameter is exactly $d$. Then $G$ partially has the structure of a homogeneous tree around any vertex, namely $F_i^{(k)}(\boldsymbol{A})=\boldsymbol{A}_i$ for any $i=0,1,\ldots,d-1$. Because the Hoffman polynomial [@H63] of $G$ is of degree $d$, there exists a natural number $e$ such that $$\sum_{i=0}^{d-1}F_i^{(k)}(\boldsymbol{A})+\frac{1}{e} F_d^{(k)}(\boldsymbol{A})=\boldsymbol{J},$$ where $\boldsymbol{J}$ is the all-ones matrix. Note that the roots of the Hoffman polynomial $P(x)=\sum_{i=0}^{d-1}F_i^{(k)}(x)+(1/e) F_d^{(k)}(x)$ are the non-trivial distinct eigenvalues of $G$ [@H63].
For some positive constant number $c$, the polynomial $f(x)$ can be expressed as $$f(x)= \frac{c P(x)^2}{x-\tau_1}
= \frac{c}{e} \frac{G_d^{(k)}(x)-(1-e) G_{d-1}^{(k)}(x)}{x-\tau_1}P(x).$$ By Theorem \[thm:CK\], $g(x)=(G_d^{(k)}(x)-(1-e) G_{d-1}^{(k)}(x))/(x-\tau_1)$ has positive coefficients in terms of $G_{0}^{(k)}(x),G_1^{(k)}(x),\ldots, G_{d-1}^{(k)}(x)$. This implies that $g(x)$ has positive coefficients in terms of $F_{0}^{(k)}(x),F_1^{(k)}(x),\ldots, F_{d-1}^{(k)}(x)$. By Theorem \[thm:posi\_coef\], it is shown that $f(x)$ has positive coefficients in terms of $F_{0}^{(k)}(x),F_1^{(k)}(x),\ldots, F_{2d-1}^{(k)}(x)$. Thus $f(x)$ satisfies the condition in Theorem \[thm:lp\_bound\].
By Remark \[rem:attain\], $G$ attains the linear programming bound obtained from $f(x)$. Assume there exists a graph $G'$ such that its second-largest eigenvalue is smaller than $\tau_1$, and it has the same number of vertices as $G$. Then $G'$ also attains the linear programming bound obtained from $f(x)$. By Remark \[rem:attain\], $G'$ has only $d$ distinct eigenvalues, and the girth of $G'$ is at least $2d$. Therefore $G'$ is of diameter at least $d$, it contradicts that the number of distinct eigenvalues of $G'$ is greater than $d$. Thus $G$ is an extremal expander graph.
Levenshtein [@L92] proved that a spherical $s$-distance set of strength $t$ satisfying $t\geq 2s-1$ is an optimal spherical code in the sense of maximizing the minimum distance. This result is the dual version of Theorem \[thm:expander\]. Cohn and Kumar [@CK07] extended this result to universally optimal codes.
We characterize connected regular graphs satisfying $g \geq 2d$ as follows.
\[thm:girth\] Let $G$ be a connected $k$-regular graph of girth $g$, and with only $d+1$ distinct eigenvalues. If $g \geq 2d-1$ holds, then $G$ is a distance-regular graph of diameter $d$.
Brouwer and Haemers [@BH93] proved that a graph with the spectrum of a distance-regular graph with diameter $D$ and girth at least $2D-1$, is such a graph. The proof in [@BH93] used the fact that regularity, connectedness, girth, and diameter of a graph are determined by the spectrum. Therefore the theorem of Brouwer and Haemers is interpreted as that a connected regular graph with $g\geq 2D-1$ is distance-regular. In general we have $d\geq D$. Therefore in our condition, $g\geq 2d-1 \geq 2D-1$ holds, and $G$ is distance-regular. Since $G$ is distance-regular, $d$ is equal to the diameter [@BCNb Section 4.1].
Abiad, Van Dam, and Fiol [@AVFx] proved Theorem \[thm:girth\] independently.
Delsarte, Goethals, and Seidel [@DGS77] proved that a spherical $s$-distance set of strength $t$ satisfying $t\geq 2s-2$ has the structure of a $Q$-polynomial association scheme. This result is the dual version of Theorem \[thm:girth\].
The distance-regular graph with $g\geq 2d$ is called a [*Moore polygon*]{} [@DG81] and it has the following intersection array: $$\begin{pmatrix}
\ast & 1 & 1 & \cdots & 1 &c \\
0 & 0 & 0 & \cdots &0 & k-c \\
k & k-1 &k-1& \cdots & k-1 & \ast
\end{pmatrix},$$ where $c$ is a natural number. If $c=1$, then the graph is a Moore graph and it does not exist for $d\geq 3$ (with $k \geq 3$) [@BI73; @D73]. If $c=k$, then the graph is an incidence graph of a regular generalized polygon of order $(s,s)$ [@BCNb Section 6.9], and it does not exist for $d\geq 7$ (with $k \geq 3$) [@FH64]. If $c\ne 1,k$, then the graph is called a [*non-trivial*]{} Moore polygon, and it does not exist for $d\geq 6$ [@DG81]. Strongly regular graphs of girth $4$ are non-trivial Moore polygons.
$v$ $k$ $g$ Eigenvalues Name
---------------------- ------- ------ ----------------------------------------- ---------------------------------------
$g$ $2$ $g$ $2\cos(2k\pi/n); 1 \leq k \leq n-1$ $g$-cycle $C_g$
$k+1$ $k$ $3$ $0$ complete $K_{k+1}$
$2k$ $k$ $4$ $0,-k$ comp. bipartite $K_{k,k}$
$2\sum_{i=0}^2 q^i$ $q+1$ $6$ $\pm \sqrt{q},-(q+1)$ inc. graph of $PG(2,q)$ [@S66; @BCNb]
$2\sum_{i=0}^3 q^i$ $q+1$ $8$ $\pm \sqrt{2q},0,-(q+1)$ inc. graph of $GQ(q,q)$ [@B66; @BCNb]
$2\sum_{i=0}^5 q^i $ $q+1$ $12$ $\pm\sqrt{3q}, \pm \sqrt{q}, 0, -(q+1)$ inc. graph of $GH(q,q)$ [@B66; @BCNb]
$10$ 3 5 $1^5$, $(-2)^4$ Petersen [@HS60]
$50$ 7 5 $2^{28}$, $(-3)^{21}$ Hoffman–Singleton [@HS60]
$35$ 4 6 $2^{14}$, $(-1)^{14}$, $(-3)^{6}$ Odd graph [@M82]
$16$ 5 4 $1^{10}$, $(-3)^5$ Clebsch [@S68; @G95]
$56$ 10 4 $2^{35}$, $(-4)^{20}$ Gewirtz [@BH93; @G95]
$77$ 16 4 $2^{55}$, $(-6)^{21}$ $M_{22}$ [@HS68; @G95]
$100$ 22 4 $2^{77}$, $(-8)^{22}$ Higman–Sims [@HS68; @G95]
: Extremal expander graphs[]{data-label="tab:2"}
\
$PG(2,q)$: projective plane, $GQ(q,q)$: generalized quadrangle,\
$GH(q,q)$: generalized hexagon, $q$: prime power
Table \[tab:2\] shows known examples of extremal expander graphs satisfying $g \geq 2d$. The following graphs are unique: Petersen graph [@HS60], Hoffman–Singleton graph [@HS60], Odd graph [@M82], Clebsch graph [@GRb Theorem 10.6.4], Gewirtz graph [@G69], $M_{22}$ graph [@B83], Higman–Sims graph [@G69], $PG(2,q)$ for $q \leq 8$ [@M07; @H53; @HSW56], $GQ(q,q)$ for $q \leq 4$ [@P75; @P77], and $GH(2,2)$ [@CT85]. For $PG(2,9)$, there are four non-isomorphic graphs [@HSK59; @VM07]. The uniqueness of other examples in Table \[tab:2\] is open.
**Acknowledgments.** The author would like to thank Noga Alon, Eiichi Bannai, Tatsuro Ito, Jack Koolen, Hajime Tanaka, Sho Suda, and Masato Mimura for useful information and comments. The author is supported by JSPS KAKENHI Grant Numbers 25800011, 26400003.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We give an analytical formula for the vacuum polarization of a massless minimally coupled scalar field at the horizon of a rotating black hole with subtracted geometry. This is the first example of an exact, analytical result for a four-dimensional rotating black hole.'
author:
- Mirjam Cvetič
- 'Zain H. Saleem'
- Alejandro Satz
title: An analytical formula for the vacuum polarization of rotating black holes
---
=by 60 => 59 by -60
UPR-1272-T
Quantum field theory in curved spacetime can be used to understand a lot of interesting features of black holes in a semiclassical approximation, most notably particle production near the black hole horizon [@Hawking:1974sw]. The calculation of vacuum polarization or $\langle\phi^2\rangle$ (for a scalar field) is the simplest standard probe of quantum fluctuations in a black hole background, and can also be used to understand the symmetry breaking and Casimir effects near a black hole. Computation of $\langle\phi^2\rangle$ is also a preliminary step in evaluating the stress energy tensor $\langle T_{\mu\nu}\rangle$, which contributes to the backreaction through the semiclassical Einstein equation.
Candelas studied the vacuum polarization of a scalar field in the Schwarzschild black hole [@Candelas:1980zt] and was able to obtain an analytical expression for $\langle\phi^2\rangle $ at the horizon. Candelas’ methods extend easily to charged static black holes; there have also been numerical studies of vacuum polarization of scalar fields on general static black hole backgrounds beyond the event horizon (e.g. [@Anderson:1990jh] for asymptotically flat solutions and [@Flachi:2008sr] for the asymptotically anti-de Sitter case), and analytical computations at the horizon of a black hole threaded with a cosmic string [@Ottewill:2010hr]. The case of rotating black holes is much more challenging. Frolov [@Frolov:1982pi] was able to calculate the analytical expression for $\langle\phi^2\rangle $ only at the pole ($\theta=0$) of the event horizon, and Ottewill and Duffy [@Duffy:2005mz] have provided a numerical evaluation throughout the black hole horizon. However so far no one has been able to give an analytical formula for $\langle\phi^2\rangle$ throughout the horizon of a four-dimensional rotating black hole. (An analytic approximation good for fields with large mass is available, however [@Belokogne:2014ysa], and exact results are obtainable in $d=3$ with AdS asymptotics [@Krishnan; @Louko].)
In this case we will be studying a particular example of rotating black holes that exist in “subtracted geometry” [@Cvetic:2011hp; @Cvetic:2011dn; @Cvetic:2012tr; @Cvetic:2014sxa]. Subtracted geometry black holes are non extremal solutions of the bosonic sector of N=2 STU supergravity coupled to three vector multiplets. These black holes are obtained by subtracting some terms in the “warp factor” of the original black hole metric in such a way that the wave equation for a massless minimally coupled scalar field becomes separable and analytical solutions are obtainable. This subtracted black hole metric effectively places the black hole in an asymptotically conical box and mimics the “hidden conformal symmetry” [@Castro:2010fd] of the wave equation on rotating black holes in the near-horizon, near-extremal, and/or low energy regimes, which is a key motivator for the Kerr/CFT conjecture (see e.g. [@Compere:2012jk]). The energy density of the matter fields in this new geometry falls off as second power of radial distance, thus confining thermal radiation. The classical near horizon properties of the subtracted black hole are the same as the original black hole ones; in particular, the classical thermodynamics of the subtracted black hole is analogous to the standard one [@Cvetic:2014nta], although loop corrections to the horizon entropy differ [@Cvetic:2014tka]).
The horizon vacuum polarization in the static subtracted metric was studied in [@Cvetic:2014eka]. In this letter we shalll consider the subtracted geometry of the uncharged rotating Kerr black hole. We shall see that the special features of the subtracted rotating metric, in particular the well-defined nature of the thermal vacuum and the solvability of the wave equation, allow us to obtain analytical results that are unavailable for the standard Kerr black hole.
The subtracted Kerr metric is given by:
$$\begin{aligned}
d s^2 &= - \Delta^{-1/2} G \,
( d{ t}+{ {\cal A\, \mathrm{d}\tilde{\varphi}}})^2 \nonumber\\
&+ { \Delta}^{1/2}
\left(\frac{d r^2} { X} +
d\theta^2 + \frac{ X}{ G} \sin^2\theta\, d\tilde{\varphi}^2 \right)\,.\label{metric4d}\end{aligned}$$
with $$\begin{aligned}
{ X} & =& { r}^2 - 2{ M }{ r} + { a}^2~,\;\;\; { G} = { r}^2 - 2{ M}{ r} + { a}^2 \cos^2\theta\, \cr
{ {\cal A}} &=&{2{ M} { a}r \sin^2\theta \over { G}}, \;\;\;\; \Delta = 8 M^3 r - 4M^2 a^2 \cos^2 \theta\,.\end{aligned}$$ (The only difference between this metric and the standard Kerr metric is the form of the “warp factor” $\Delta$. For the explicit form of gauge potentials and axio-dilatons of the STU model, supporting this geometry, see [@Cvetic:2012tr].) The horizons and their surface gravities and angular velocities are given by: $$\begin{aligned}
\label{param}
r_\pm &=& M\pm \sqrt{M^2-a^2}\,, \cr
\kappa_\pm &=& \frac{1}{2M}\left[\frac{M}{\sqrt{M^2-a^2}}\pm1\right]^{-1}\,,\cr
\Omega_\pm &=&\kappa_\pm\frac{a} {\sqrt{M^2-a^2}} \,.\end{aligned}$$
We switch to co-rotating coordinates $(t,r,\theta,\varphi)$, with the new angular variable being defined by: $$\varphi=\tilde{\varphi} - \Omega_+ t\,.$$ These are adapted to observers co-rotating with the black hole at the horizon. A noteworthy feature of subtracted geometry is that outside the horizon there is a globally defined timelike Killing vector, written as $\partial_t$ in the co-rotating coordinates [@Cvetic:2013lfa; @Cvetic:2014ina]. This guarantees that there are no superradiant modes and ensures the existence of a Hartle-Hawking-like vacuum state adapted to the co-rotating observers. This is different from the case of ordinary Kerr black hole, where there is no such Killing vector [@kaywald; @Ottewill:2000qh] and a physical co-rotating vacuum requires enclosing the black hole in a reflective box [@Frolov:1989jh; @Duffy:2005mz]. The subtracted Kerr resembles more in this respect the Kerr/AdS black hole [@Krishnan].
The general algorithm we follow for computing the horizon vacuum polarization in the Hartle-Hawking state starts by defining the Euclidean Green’s function $G_H(x,x')$ (in a state regular at the horizon and infinity, and where the modes are adapted to co-rotating coordinates). Then we will evaluate $-iG_H$ with radial point splitting, perform the mode sum, and subtract the covariant divergent counterterms.
After writing the metric in coordinates $(t,r,\theta,\varphi)$ we perform the Wick rotation setting $t=-i \tau$. The metric becomes: $$\begin{aligned}
d&s_E^2 = - \frac{G}{\Delta^{1/2} } \,
[{ {\cal A\, \mathrm{d}\varphi}}-i (1+{\cal A}\Omega_+)d\tau]^2 \nonumber\\
&+{ \Delta}^{1/2}
\left(\frac{d r^2} { X} +
d\theta^2 + \frac{ X}{ G} \sin^2\theta\, (d\varphi-i\Omega_+ d\tau)^2 \right)\,.\label{metric4d}\end{aligned}$$
On writing the massless minimally coupled wave equation and proposing a solution of the form $\mathrm{e}^{i n\kappa_+ \tau}\mathrm{e}^{im\varphi}P_l^m(\cos\theta)\chi_{lmn}(r)$, we obtain straightforwardly a radial equation which, in the re-scaled variable $x =(r - \frac{1}{2}(r_+ + r_-))/( r_+ - r_-)$, reads: $$\begin{aligned}
\,&\Big[ \frac{\partial}{\partial x} \left(x^2 -
\frac{1}{4}\right)\frac{\partial}{\partial x}-
\frac{n^2 }{4\left(x-\half \right)}\nonumber\\
& + \frac{\beta_{mn}}{4\left( x+\half\right)}
- l(l+1) \Big]\, \chi_{lmn}(x)= 0\,, \label{wavehom}\end{aligned}$$ where $$\beta_{mn} = \frac{2 M n^2 r_- - \,
a^2 (4 m^2 + n^2)-4 i {a} {m} {n} r_- }{r_+^2}\,.$$
Two independent solutions of the equation, respectively regular at the horizon and at infinity, are: $$\chi_{lmn}^{(1,2)} = \frac{\left( x-\half \right)^{\frac{n}{2}}}{\left(x+\half \right)^{\frac{n}{2 }+l+1}}
F \left(a_{lmn},b_{lmn}, c_{ln}^{(1,2)}
;z^{(1,2)}\right) \,,$$ where $$\begin{aligned}
\,&c_{ln}^{(1)} = n+1\,,\,\, c_{ln}^{(2)} = 2l+2\,,\,\,z^{(1)}=\frac{x-\half}{x+\half}\,,\,\,z^{(2)}=\frac{1}{x+\half}\,,\nonumber\\
&(a_{lmn},b_{lmn})=l+1+\frac{|n|}{2}\pm\frac{\sqrt{\beta_{mn}}}{2}\,,\end{aligned}$$ and the symmetry of the hypergeometric function makes irrelevant which branch of the square root is chosen. The full Green’s function is expanded as $$\begin{aligned}
\label{greenexpandrot}
\,&G_H( -i\tau , x , \theta , \varphi \; ; {-i \tau}' , {x}' , {\theta}' , {\varphi}')= \frac{ i \kappa}{ 2 \pi\,r_0} \sum_{n =- \infty}^ {\infty} e^{ i n \kappa( \tau -{\tau}')} \nonumber\\
&\sum_{l=0}^{\infty}
\sum_{m =- l}^ {l} Y_l^m(\theta,\varphi)Y_l^{m*}(\theta',\varphi') G _{mln}(x,x')\,,\end{aligned}$$ where $r_0=r_+-r_-=2\sqrt{M^2-a^2}$, $\kappa\equiv \kappa_+$ as defined in (\[param\]), and $$G_{mln} ( x,x') = \frac{ \Gamma( a_{mln}) \Gamma(b_{mln})} {\Gamma(2l+2)\Gamma\left(1 + \left| n \right| \right) }\chi_{mln}^{(1)}( x_<) \chi_{mln}^{(2)}( x_>) \,.$$ To evaluate the vacuum polarization at the horizon we set $x=1/2, x'=\epsilon+\frac{1}{2}$ (note that this is a dimensionless regulator $\epsilon=(r'-r)/r_0$) and join the points in the other directions, calling the resulting Green’s function $G_H(\epsilon,\theta)$. All the terms in the sum vanish except $n=0$, so we are reduced to: $$\begin{aligned}
\label{greensum}
-&iG_H(\epsilon,\theta)= \frac{ \kappa}{ 8 \pi^2\,r_0} \sum_{l=0}^{\infty}\sum_{m =- l}^ {l} \frac{(l-m)!}{(l+m)!}\left[P_l^m( \cos{\theta})\right]^2 \nonumber\\
&\times \frac{ \Gamma( l+1+i\alpha m) \Gamma( l+1-i\alpha m) } {\Gamma(2l+1)} \left(1+\epsilon\right)^{-(l+1)}\nonumber\\
&\times F\left(l+1+i\alpha m,l+1-i\alpha m,2l+2,\frac{1}{1+\epsilon}\right)\,,\end{aligned}$$ where the parameter $\alpha\equiv a/r_+$ takes values between 0 and 1. We replace the hypergeometric by an integral expression using formula 9.111 of [@Grad], leading to: $$\begin{aligned}
-&iG_H(\epsilon,\theta)
= \frac{ \kappa}{ 8 \pi^2\,r_0} \sum_{l=0}^{\infty}{(2l+1)}\sum_{m =- l}^ {l} \frac{(l-m)!}{(l+m)!}\left[P_l^m( \cos{\theta})\right]^2 \nonumber\\
&\times \int_0^1\mathrm{d}t\,\left(\frac{t(1-t)}{1+\epsilon -t}\right)^l \frac{1}{1+\epsilon-t} \cos\left(m \alpha \ln \lambda \right)\,,\end{aligned}$$ where $\lambda = \left(\frac{(1+\epsilon)(1-t)}{t(1+\epsilon-t)}\right)$.
The addition theorem for the associated Legendre polynomials is used to compute the sum over $m$, and formula III.4 from [@sansone] subsequently yields the sum over $l$. This leads, after a change of variables to $x=1-t$, to the integral expression $$-iG_H(\epsilon,\theta)= \frac{ \kappa}{ 8 \pi^2\,r_0} \int_0^1\mathrm{d}x\, f_\epsilon(x) \,;$$ $$f_\epsilon(x)=\frac{\frac{\epsilon^2+2\epsilon x+(2-x)x^3}{(x^2+\epsilon)^3}}{\left[
1+\frac{4x(1-x)(x+\epsilon)}{(x^2+\epsilon)^2}\sin^2\theta \sin^2\left(\frac{\alpha}{2} \ln \lambda\right)
\right]^{3/2}}\,,$$ with $\lambda=\lambda(t(x))$. It is easy to see from numerical evaluation that the leading divergences in the integral as $\epsilon\to 0$ match those provided by the standard counterterms [@Christensen:1976vb], $$\label{countersigma}
G_{div} = \frac{1+\frac{1}{12}R_{\mu\nu}\sigma^{,\mu}\sigma^{,\nu}}{8\pi^2\sigma} -\frac{1}{96\pi^2} R \ln (\mu^2 \sigma)\,,$$ where $\sigma$ is the halved geodesic distance between the points and $\mu$ is an arbitrary mass scale. It is more difficult, however, to obtain an explicit expression for the finite result of the subtraction. To make progress we perform the following sequence of changes of variables: $$u = \frac{1}{2}\ln\left(\frac{x(1+\epsilon)}{(1-x)(x+\epsilon)}\right)\,,\quad\quad w = \sinh u\,.$$ This leads to the more tractable expression for the integral $I_\epsilon\equiv\int_0^1\mathrm{d}x\,f_\epsilon(x)$: $$I_\epsilon =\int_0^\infty dw \frac{ \sqrt{1+\epsilon}}{\left[\epsilon + (1+\epsilon)w^2+v^2 \sin^2(\alpha \sinh^{-1} w)\right]^{3/2}}\,,$$ where $v\equiv\sin\theta$. The intermediate $u$-integral expression is also obtainable directly from dimensional reduction from the Euclidean Green’s function in AdS$^3\times$S$^2$, using the higher-dimensional embedding of subtracted geometry described in [@Cvetic:2011dn][^1].
To analyze the small $\epsilon$ limit and subtract explicitly the counterterms, we set aside momentarily the $\sqrt{1+\epsilon}$ prefactor and split the integral in two subintervals, $I_\epsilon^<$ over $(0, \epsilon^{1/6})$ and $I_\epsilon^>$ over $(\epsilon^{1/6},+\infty)$. In the second subinterval we can set $\epsilon$ to zero, at the expense of an error that vanishes as $\epsilon\to 0$. Then we can add and subtract terms compensating for the leading divergences at the lower limit, take $\epsilon\to 0$ safely in the subtraction, and integrate explicitly the added coutnerterms. This leads to: $$\begin{aligned}
I_\epsilon^>&\sim\int_{0}^\infty\mathrm{d}w\Bigg[\frac{1}{\left[w^2+v^2 \sin^2(\alpha \sinh^{-1} w)\right]^{3/2}} \nonumber\\
&-\left(\frac{1}{w^3(1+\alpha^2 v^2)^{3/2}}+\frac{v^2\alpha^2(1+\alpha^2)}{2w(1+w)(1+\alpha^2v^2)^{5/2}}\right)\Bigg]\nonumber\\
&+\frac{1}{2\epsilon^{1/3}(1+\alpha^2 v^2)^{3/2}}-\frac{v^2\alpha^2(1+\alpha^2)\ln \epsilon}{12(1+\alpha^2v^2)^{5/2}}\label{second}\,,\end{aligned}$$ where $\sim$ stands for equivalence as $\epsilon\to 0$. The second subintegral is thus reduced to a finite integral involving no regulator, that can be evaluated numerically, plus two explicit divergent terms.
In the first subinterval, we can show that: $$\begin{aligned}
&I_\epsilon^<=\int_0^{\epsilon^{1/6}}\frac{\mathrm{d}w}{\left[\epsilon + (1+\epsilon)w^2+v^2 \sin^2(\alpha \sinh^{-1} w)\right]^{3/2}}\nonumber\\
&\sim\int_0^{\epsilon^{1/6}}\frac{\mathrm{d}w}{\left[\epsilon + (1+\epsilon)w^2+v^2\left(\alpha^2 w^2-\frac{\alpha^2(\alpha^2+1)w^4}{3}\right) \right]^{3/2}} \,,\end{aligned}$$ which is expressible (formula 3.163.3 of [@Grad]) in terms of the incomplete elliptic integrals of first and second kind, $F(\gamma,k)$ and $E(\gamma,k)$. Here $$\gamma = \arcsin\left(\frac{\epsilon^{1/6}}{\sqrt{c_+}}\sqrt{\frac{c_- + c_+}{c_- + \epsilon^{1/3}}}\right)\,,\quad\quad k = \sqrt{\frac{c_+}{c_-+c_+}}\,,$$ and $c_{\pm}$ are the coefficients appearing in the denominator of the integrand when it is factored in a form proportional to $[(c_+^2-w^2)(c_-^2+w^2)]^{3/2}$. We need the expansions of the elliptic functions near $(\gamma,k)= (\frac{\pi}{2},1)$, which have been derived in [@gustafson]. In order to obtain all the divergent and finite contributions to $I_\epsilon^<$, we need $F$ accurately to order $1$ and $E$ accurately to order $\epsilon$. This in turns require obtaining the argument $k$ accurately to order $\epsilon$ and $\gamma$ to order $\epsilon^{4/3}$. The result of this expansion is the following expression for the divergent and finite pieces of $I_\epsilon^<$: $$\begin{aligned}
\label{first}
I&_\epsilon^<\sim-\frac{1}{2\epsilon^{1/3}(1+\alpha^2v^2)^{3/2}}+\frac{1}{\epsilon\sqrt{1+\alpha^2v^2}}\nonumber\\
&+\frac{1}{6(1+\alpha^2v^2)^{5/2}}
\times\Big(-3-\alpha^2(7+4\alpha^2)v^2\nonumber\\
&+\alpha^2(1+\alpha^2)v^2(\ln(8(1+\alpha^2v^2)^{3/2})-\ln\epsilon)\Big) \,.\end{aligned}$$ There is an additional finite contribution coming from the prefactor $\sqrt{1+\epsilon}$ to the integral, which yields when expanded a $1/2$ multiplied by the coefficient of the linear divergence of the integral. The complete result is thus expressed as: $$I_\epsilon=I_\epsilon^<+I_\epsilon^>+ \frac{1}{2\sqrt{1+\alpha^2v^2}}\,,$$ with the first to terms given by (\[first\]) and (\[second\]) respectively. We see that the $\epsilon^{-1/3}$ divergences cancel out, leaving only linear and logarithmic divergences that will match those of counterterms (\[countersigma\]), leaving a finite renormalized result.
This concludes the computation of the explicit divergent and finite portions of the Green’s function’s coincidence limit. The counterterms (\[countersigma\]) need now to be evaluated as a function of $\epsilon$ to the order $O(1)$. The form of $\sigma$ can be computed from the formulas expressing $\sigma$ in terms of coordinate separation: $$\begin{aligned}
\sigma&=\frac{1}{2}g_{ab}\Delta x^a\Delta x^b + A_{abc}\Delta x^a\Delta x^b\Delta x^c\nonumber\\
&+ B_{abcd}\Delta x^a\Delta x^b\Delta x^c\Delta x^d+\cdots\end{aligned}$$ where $A,B$ are obtained from symmetrized derivatives of the metric tensor, as described in [@Ottewill:2008uu].
These expressions are valid in a coordinate system in which the metric is regular. We use the Kruskal coordinates for the subtracted geometry that have been derived in [@Cvetic:2014ina], which take the form $(U,V,\theta,\varphi)$ with $(-UV)\propto( r-r_+)$ near the horizon. Our radial coordinate separation is therefore written as $\Delta x^a = (-\delta,\delta,0,0)$ (with $\delta\propto\sqrt{\epsilon}$). After computing $\sigma$ by this procedure (leading to an expression of the form $\sigma=\beta_1\epsilon+\beta_2\epsilon^2+O(\epsilon^3)$) it is easy to obtain the Ricci counterterm in (\[countersigma\]) because to the relevant order $O(\epsilon)$ we have $R_{\mu\nu}\sigma^{,\mu}\sigma^{,\nu} = R_{rr}\sigma^{,r}\sigma^{,r}$.
Once all the counterterms are computed by this procedure, when expressed in terms of the $\alpha$ parameter they take the relatively simple form: $$\begin{aligned}
&G_{div}=\frac{1+\alpha^2}{64\pi^2\,M^2}\Bigg[\frac{1}{\epsilon\sqrt{1+\alpha^2v^2}}-\frac{\alpha^2 v^2(1+\alpha^2)\ln \epsilon}{4(1+\alpha^2v^2)^{5/2}}\nonumber\\ &
+\frac{(-1+\alpha^2(-4+\alpha^2+(7+\alpha^2+\alpha^4)v^2+3\alpha^2v^4))}{12(1+\alpha^2v^2)^{5/2}} \Bigg]\,,\end{aligned}$$ (plus a term of the form $R(r_+,\theta)\ln\mu^2$). Then, absorbing some $R$-proportional terms into the arbitrary constant $\mu$, the final result for the vacuum polarization is: $$\begin{aligned}
\langle \phi^2\rangle&_{r_+}=R(r_+,\theta)\ln\mu^2+\frac{1+\alpha^2}{64\pi^2\,M^2}\Bigg\{\frac{1}{12(1+\alpha^2v^2)^{5/2}} \nonumber\\
\times&\Big[(1-\alpha^2(-4+\alpha^2(9+9\alpha^2+\alpha^4)v^2-3\alpha^2v^4))\nonumber\\
&-3\alpha^2(1+\alpha^2)^2\ln(1+\alpha^2v^2)\Big]\nonumber\\
&+\int_{0}^\infty\mathrm{d}w\Bigg[\frac{1}{\left[w^2+v^2 \sin^2(\alpha \sinh^{-1} w)\right]^{3/2}} \nonumber\\
&-\left(\frac{1}{w^3(1+\alpha^2 v^2)^{3/2}}+\frac{v^2\alpha^2(1+\alpha^2)}{2w(1+w)(1+\alpha^2v^2)^{5/2}}\right)\Bigg]\Bigg\}\,,\end{aligned}$$ where $$R(r_+,\theta)=\frac{3\alpha^2(1+\alpha^2)^2 v^2}{8\,M^2(1+\alpha^2v^2)^{5/2}}\,.$$
![Vacuum polarization (without $R$ term) at the horizon as a function of $v=\sin \theta$, for $\alpha\equiv{{a}/r_+} = 0$ (full line), $\alpha = 0.5$ (dashed line), $\alpha = 0.75$ (dotted line), and $\alpha = 1$ (dot-dashed line).[]{data-label="kerrpolots1"}](plots1.pdf){width="45.00000%"}
The angular profile for the vacuum polarization, neglecting the arbitrary term proportional to $R$, is depicted in Figure 1. Notice that in the absence of rotation spherical symmetry is recovered, with its value $\langle\phi^2\rangle_{r_+}^{Sch_{sub}}=(768\pi^2 M^2)^{-1}$ matching the result obtained in [@Cvetic:2014eka] for the subtracted Schwarzschild black hole. In addition, the result at the pole takes the form $\langle\phi^2\rangle_{r_+,\theta=0}=(768\pi^2 M^2)^{-1}(1+\alpha^2)(1+4\alpha^2-\alpha^4)$, agreeing with result found in [@Cvetic:2014eka] using a non-corotating vacuum state (at the pole, the distinction is irrelevant). The dot-dashed plot corresponds to the extremal case $a = M$. It would be interesting to compare our results with numerical computations of the vacuum polarization in the standard Kerr metric (with a mirror in place to define the vacuum). Our calculation holds for the minimally coupled field, and the numerical results in [@Duffy:2005mz] are for the conformal case, so a direct comparison is not yet available. We expect our calculations to be easily generalized to the case of fields with higher spins as well as to rotating charged black holes, including multi-charged solutions [@Cvetic:1996kv; @Cvetic:1996xz; @Chow:2013tia]. We also expect our methods to be applicable to the computation of the stress-energy tensor, which would open the possibility of using the subtracted approximation to study analytically the backreaction for rotating four-dimensional black holes.
0.1 in [**Acknowledgements**]{} We thank Finn Larsen and Gary Gibbons for valuable discussions and collaborations on related topics. MC would like to thank the organizers of the 2015 Mitchell Institute Workshop at the Great Brampton House for hospitality during the course of the work. The work is supported in part by the DOE (HEP) Award DE-SC0013528, the Fay R. and Eugene L. Langberg Endowed Chair (MC), and the Slovenian Research Agency (ARRS) (MC).
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[^1]: We thank Finn Larsen for bringing this point to our attention.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This is a survey article on Gorenstein initial complexes of extensively studied ideals in commutative algebra and algebraic geometry. These include defining ideals of Segre and Veronese varieties, toric deformations of flag varieties known as Hibi ideals, determinantal ideals of generic matrices of indeterminates, and ideals generated by Pfaffians of generic skew symmetric matrices. We give a summary of recent work on the construction of squarefree Gorenstein initial ideals of these ideals when the ideals are themselves Gorenstein. We also present our own independent results for the Segre, Veronese, and some determinantal cases.'
address:
- 'Dipartimento di Matematica, Universite degli Studi di Genova, Via Dodecaneso, 35 16146 Genova, Italy'
- 'Mathematics Department, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, USA'
- 'Department of Mathematics, University of Washington, Seattle, WA 98195,USA'
author:
- Aldo Conca
- Serkan Hoşten
- 'Rekha R. Thomas'
title: Nice Initial Complexes of some Classical Ideals
---
|
{
"pile_set_name": "ArXiv"
}
|
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