Statistical physics for cosmic structures
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec EPJ manuscript No. (will be inserted by the editor)
Statistical physics for cosmic structures
Francesco Sylos Labini and Luciano Pietronero “Enrico Fermi Center”, Piazzale del Viminale 2, 00184 Rome, Italy & “Istituto dei Sistemi Complessi” CNR, Via dei Taurini19, 00185 Rome, Italy Physics Department, University of Rome “Sapienza”, Piazzale A. Moro 2, 00185 Rome, ItalyReceived: date / Revised version: date
Abstract.
The recent observations of galaxy and dark matter clumpy distributions have provided newelements to the understanding of the problem of cosmological structure formation. The strong clumpingcharacterizing galaxy structures seems to be present in the overall mass distribution and its relationto the highly isotropic Cosmic Microwave Background Radiation represents a fundamental problem. Theextension of structures, the formation of power-law correlations characterizing the strongly clustered regimeand the relation between dark and visible matter are the key problems both from an observational anda theoretical point of view. We discuss recent progresses in the studies of structure formation by usingconcepts and methods of statistical physics.
PACS.
PACS-key 98.80.-k – PACS-key 05.45.-a
In contemporary cosmological models the structures ob-served today at large scales in the distribution of galaxiesin the universe (see Fig.1 — discovered by the projects,e.g., 2dF [1], SDSS [2,3]) are explained by the dynamicalevolution of purely self-gravitating matter (dark matter)from an initial state with low amplitude density fluctu-ations, the latter strongly constrained by satellite obser-vations of the fluctuations in the temperature of the cos-mic microwave background radiation (e.g. the satellitesCOBE[4] and WMAP[5]). Despite the apparent simplicityof the scheme, fundamental theoretical problems remainopen and the overall picture is based on the assumptionthat the main mass component is dark.In this theoretical framework one crucial element isrepresented by the initial conditions (IC) of the matterdensity field. Models of the early universe [6] predict cer-tain primordial fluctuations in the matter density field,defining their correlation properties and their relation tothe present day matter distribution. When gravity startto dominate the dynamical evolution of density fluctu-ations, which can generally be described by the Vlasovor “collision-less Boltzmann” equations coupled with thePoisson equation, perturbations are still of very low am-plitude. One of the most basic results (see e.g., [7]) aboutself-gravitating systems, treated using perturbative ap-proaches to the problem (i.e. the fluid limit), is that theamplitude of small fluctuations grows monotonically intime, in a way which is independent of the scale. This lin-earized treatment breaks down at any given scale when
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Fig. 1.
Latest progress in redshift surveys. SDSS Great Wall(2003) compared to CfA2 (1986) Great Wall at the same scale.Redshift distances cz are indicated. The small circle at the bot-tom has a diameter of 5 Mpc/ h , the clustering length accordingto the standard interpretation of galaxy correlation. The SDSSslice is 4 degrees wide, the CfA2 slice is 12 degrees wide to makeboth slices approximately the same physical width at the twowalls. (From [8]). the relative fluctuation at the same scale becomes of or-der unity, signaling the onset of the “non-linear” phase ofgravitational collapse of the mass in regions of the corre-sponding size. If the initial velocity dispersion of particlesis small, non-linear structures start to develop at small Francesco Sylos Labini, Luciano Pietronero: Statistical physics for cosmic structures scales first and then the evolution becomes “hierarchical”,i.e., structures build up at successively larger scales. Giventhe finite time from the IC to the present day, the devel-opment of non-linear structures is limited in space, i.e.,they can not be more extended than the scale at whichthe linear approach predicts that the density contrast be-comes of order unity at the present time. This scale isfixed by the initial amplitude of fluctuations, constrainedby the CMBR, by the hypothesized nature of the dominat-ing dark matter component and its correlation properties.Observations of large scale galaxy distributions pro-vide important tests for these models. On the one handthe first question concerns the extension of the regime ofnon-linear clustering and the intrinsic properties of galaxystructures. On the other hand according to this scenario,at some large scales where fluctuations are still of smallamplitude, the imprints of primordial correlations shouldbe preserved and their detection represents a key obser-vation for the validation of the model.In order to approach this complex problem, we usemethods and concepts of modern statistical physics [8] tomake a bridge between the primordial fluctuation field andthe development of large scale structure in the universe.The first issue we discuss in what follows concerns the cor-relation properties of the observed distribution of galaxiesand galaxy clusters, approaching this problem with theperspective of a statistical physicist, exposed to the de-velopments of the last decades in the description of in-trinsically irregular structures, and by using instrumentssuitable to describe strong irregularity, even if limited toa finite range of scales [9,10,11]. These methods offer awider framework in which to approach the problem ofhow to characterize the correlations in galaxy distribu-tions, without the a priori assumption of homogeneity.That is, without the assumption that the distribution in-side a given sample is already uniform enough to give to asufficiently good approximation, the true (non-zero) meandensity of the underlying distribution of galaxies. Whilethis is a simple and evident step for a statistical physicist,it can seem to be a radical one for a cosmologist. Afterall the whole theoretical framework of cosmology (i.e., theFriedmann-Robertson-Walker – FRW – solutions of gen-eral relativity) is built on the assumption of an homoge-neous and isotropic distribution of matter. The approachwe propose is thus an empirical one, which surely is ap-propriate when faced with the characterization of data.Further it is evidently important for the formulation oftheoretical explanations to understand and characterizethe data.The second question in which the use of methods andconcepts of statistical physics allow us to clarify an impor-tant issue, concerns the correlation properties of the initialmatter density fields in standard cosmological models. Inthese models the matter density field is described as hav-ing small fluctuations about a well defined mean densityand the initial conditions (i.e., very early in the historyof the universe) are specified by the so-called Harrison-Zeldovich condition. It is here that the concept of “super-homogeneity” introduced, for example, in the studies of plasma and glass distributions, is relevant, as these mod-els describe fluctuations which are in fact of this type.Standard type models are indeed characterized by surfacequadratic fluctuations (of the mass in spheres) and, for theparticular form of primordial cosmological spectra, by anegative power-law in the reduced correlation function atlarge separations [12,13]. The clarification of these prop-erties, which correspond to a global fine-tuning of positiveand negative correlations, allow us to define the strategyto measure such signals in real galaxy samples and to iden-tify several problems concerning, for example, the effectsrelated to sampling (galaxy distribution can be regardedas a sampling of the underlying dark matter density field).The third issue in which a statistical physics approachmaybe useful concerns the theoretical modeling of non lin-ear structure formation. Analytical solutions of the Poisson-Vlasov equations are very difficult to be formulated andthe only instrument beyond the linear regime is repre-sented by numerical simulations. N-body simulations solvenumerically for the evolution of a system of N particles in-teracting purely through gravity, with a softening at verysmall scales. The number of particles N in the very largestcurrent simulations [14] is ∼ , many more than twodecades ago, but still many orders of magnitude fewer thanthe number of real dark matter particles ( ∼ in acomparable volume for a typical candidate). While suchsimulations constitute a very powerful and essential tool,they lack the valuable guidance which a fuller analytic un-derstanding of the problem would provide. The questioninevitably arises of the extent to which such numericalsimulations of a finite number of particles, reproduce themean-field/Vlasov limit of the cosmological models. Thetheoretical questions concerns the validity of this collision-less limit and thus the crucial point is represented by theanalysis of the “discreteness effects” [15,16,17,18].As already mentioned, although dark matter is sup-posed to provide with more than 0.9 of the total fractionof the mass-energy in universe (see e.g. [19]), its amountand properties can only be defined a posteriori. In addi-tion the relation of dark matter to visible matter is stillnot clear and the distribution itself of visible matter re-quires more observations to be understood on the relevantscales (see e.g. [20,10]). More than twenty years ago it hasbeen surprisingly discovered that galaxy velocity rotationcurves remain flat at large distances from the galaxy cen-ter while the density profile of luminous matters rapidlydecays (see e.g. [22]). This is one of the strongest indica-tions of the need from dynamically dominant dark matterin the universe. Most attention has been focused on thefact that these bound gravitational systems contain largequantities of unseen matter and an intricate paradigm hasbeen developed in which non-baryonic dark matter playsa central role not only in accounting for the dynamicalmass of galaxies and galaxy clusters but also for provid-ing the initial seeds which have given rise to the formationof structure via gravitational collapse [7]. In current stan-dard cosmological models, various forms of dark matterare needed to explain a number of different phenomena,while baryons, which can be detected in the form of, for ex- rancesco Sylos Labini, Luciano Pietronero: Statistical physics for cosmic structures 3 ample, luminous objects such as stars and galaxies, wouldonly be the 5% of the total mass in the universe; the restis made of entities about which very little is understood:dark matter and dark energy. Very recently there havebeen developed observational techniques which, by mea-suring the effect of gravitational lensing in galaxy clusters[23], or by measuring the gravitational influence of struc-tures on the CMBR [24], are able to reconstruct the three-dimensional distribution of dark matter and thus allowa comprehension of the relative distribution of luminousand dark matters, whose theoretical modeling is still lack-ing. These observations have lead to surprising discoverieswhich rise new and crucial questions to the validity of thestandard interpretation of structure formation [25]. The most prominent feature of the IC in the early uni-verse, in standard theoretical models, derived from infla-tionary mechanisms, is that matter density field presentson large scale super-homogeneous features [12]. This meansthe following. If one considers the paradigm of uniform dis-tributions, the Poisson process where particles are placedcompletely randomly in space, the mass fluctuations in asphere of radius R growths as R , i.e., like the volumeof the sphere. A super-homogeneous distribution is a sys-tem where the average density is well defined (i.e., it isuniform) and where fluctuations in a sphere grow slowerthan in the Poisson case, e.g., like R : in this case thereare the so-called surface fluctuations to differentiate themfrom Poisson-like volume fluctuations.A well known system in statistical physics systems ofthis kind is the one component plasma [13] (OCP) whichis characterized by a dynamics which at thermal equilib-rium gives rise to such configurations. The OCP is simplya system of charged point particles interacting througha repulsive 1 /r potential, in a uniform background whichgives overall charge neutrality. Simple modifications of theOCP can produce equilibrium correlations of the kind as-sumed in the cosmological context [13].In terms of the normalized mass variance σ ( R ) = h M ( R ) i − h M ( R ) i / h M ( R ) i , where h M ( R ) i is the av-erage mass in a sphere of radius R and h M ( R ) i is theaverage of the square mass in the same volume. Thus fora Poisson distribution, where there are no correlation be-tween particles (or density fluctuations) at all, one sim-ply has σ ( R ) ∼ R − . For an ordered system character-ized by small-scale anti-correlation the variance behavesas σ ( R ) ∼ R − which is the fastest possible decay fordiscrete or continuous distributions [12].The reason for this peculiar behavior of primordialdensity fluctuations is the following. In a FRW cosmol-ogy there is a fundamental characteristic length scale, thehorizon scale R H ( t ). It is simply the distance light cantravel from the Big Bang singularity t = 0 until any giventime t in the evolution of the Universe, and it grows lin-early with time. The Harrison-Zeldovich (H-Z) criterioncan be written as σ M ( R = R H ( t )) = constant . This con-ditions states that the mass variance at the horizon scale is constant: this can be expressed more conveniently interms of the power spectrum of density fluctuations [12] P ( k ) = (cid:10) | δ ρ ( k ) | (cid:11) where δ ρ ( k ) is the Fourier Transform ofthe normalized fluctuation field ( ρ ( r ) − ρ ) /ρ , being ρ theaverage density. It is possible to show that H-Z criterionis equivalent to assume P ( k ) ∼ k : in this situation mat-ter distribution present fluctuations of super-homogeneoustype given [12].The H-Z condition is a consistency constraint in theframework of FRW cosmology. In fact the FRW is a cosmo-logical solution for a homogeneous Universe, about whichfluctuations represent an inhomogeneous perturbation: ifdensity fluctuations obey to a different condition than P ( k ) ∼ k , then the FRW description will always breakdown in the past or future, as the amplitude of the pertur-bations become arbitrarily large or small. For this reasonthe super-homogeneous nature of primordial density fieldis a fundamental property independently on the nature ofdark matter. This is a very strong condition to impose,and it excludes even Poisson processes ( P ( k ) = constantfor small k ) [12].Various models of primordial density fields differ forthe behavior of the power spectrum at large wave-lengths,i.e., at relatively small scales [6]. However at small k theyboth exhibit the H-Z tail P ( k ) ∼ k which is in fact thecommon feature of all density fluctuations compatible withFRW models. Thus theoretical models of primordial mat-ter density fields in the expanding universe are character-ized by a single well-defined length scale, which is an im-print of the physics of the early universe at the time of thedecoupling between matter and radiation [6]. The redshiftcharacterizing the decoupling is directly related to thescale at which the change of slope of the power-spectrumof matter density fluctuations P ( k ) occurs, i.e., it definesthe wave-number k c at which there is the turnover ofthe power-spectrum between a regime, at large enough k , where it behaves as a negative power-law of the wavenumber P ( k ) ∼ k m with − < m ≤ −
3, and a regime atsmall k where P ( k ) ∼ k as predicted by inflationary the-ories. Given the generality of this prediction, it is clearlyextremely important to look for this scale in the data. Asmentioned in the introduction the range of length-scalescorresponding to the regime of small fluctuations is lin-early amplified during the growth of gravitational insta-bilities. According to current models the scales at whichnon-linear clustering occurs at the present time (of order10 Mpc) are much smaller than the scale r c , correspondingto the wave-number k c , which is predicted to be r c ≈ . < h < r c the real space correlation function ξ ( r )(Fourier transform of the power spectrum) crosses zero,becoming negative at larger scales. In particular the cor-relation function presents a positive power-law behavior atscales r ≪ r c and a negative power-law behavior ( ξ ( r ) ∼− r − ) at scales r ≫ r c . Positive and negative correla- Francesco Sylos Labini, Luciano Pietronero: Statistical physics for cosmic structures tions are exactly balanced in way such that the integralover the whole space of the correlation function is equalto zero. This is a global condition on the system fluctua-tions which corresponds to the fact that the distributionis super-homogeneous.By considering the observational features of super ho-mogeneity one has to take into account that in standardmodels galaxies result from a sampling of the underlyingdark matter density field: for instance one selects (ob-servationally) only the highest fluctuations of the fieldwhich would represent the locations where galaxy willeventually form. It has been shown that sampling a super-homogeneous fluctuation field changes the nature of cor-relations [26], introducing a stochastic noise which makesthe system substantially Poisson (e.g. P ( k ) ∼ constant)at large scales. However one may show that the negative ξ ( r ) ∼ r − tail does not change under sampling: on largeenough scales, where in these models (anti) correlationsare small enough, the biased fluctuation field has a corre-lation function which is linearly amplified with respect tothe underlying dark matter correlation function. For thisreason the detection of such a negative tail would be themain confirmation of the super-homogeneous character ofprimordial density field [8].The scale r c marks the maximum extension of pos-itively correlated structures: beyond r c the distributionmust be anti-correlated since the beginning, as there wasno time to develop other correlations. The presence ofstructures, which mark long-range correlations, whetheror not of large amplitude, reported both by observationsof galaxy distributions (as those shown in Fig.1) and bythe indirect detection of dark matter [23,24] is alreadypointing toward the fact that positive correlations extendwell beyond r c . For example, in [24] it is shown that deepcounts of radio-galaxies present a dip of about 20 − ∼
140 Mpc radius completely empty void. Thisresult, if confirmed, shows that (i) there are large-scalestructures of all matter (dark and visible) extended wellbeyond the possible prediction of current models and that(ii) these structures are of very large amplitude. This re-sult must be tested in by the analysis of three-dimensionalgalaxy catalogs. Up to now, measurements of large sam-ples of galaxy redshifts are not extended enough to reachthis region, where it is expected that ξ ( r ) ∼ − r − , withthe appropriate and robust statistical properties. Futuresurveys, like the complete SDSS catalog [2], may samplethis range of scales, but a precise study of the crossover tohomogeneity, discretization effects, sampling effects andstatistical noise is still required. In the past twenty years observations have provided sev-eral three dimensional maps of galaxy distribution, fromwhich there is a growing evidence of existence large scalestructures. This important discovery has been possiblethanks to the advent of large redshift surveys: angulargalaxy catalogs, considered in the past, are in fact es-sentially smooth and structure-less. In the CfA2 catalog(1990) [27], which was one of the first maps surveying thelocal universe, it has been surprisingly observed the giant“Great Wall”, a filament linking several groups and clus-ters of galaxies of extension of about 200 Mpc/h. Recentlythe SDSS project [2] (2004—2009) has allowed to discoverthe “Sloan Great Wall” which is almost double longer thanthe Great Wall. Nowadays this is the most extended struc-ture ever observed, covering about 400 Mpc/h, and whosesize is again limited by the boundaries of the sample. Thesearch for the “maximum” size of galaxy structures andvoids, beyond which the distribution becomes essentiallysmooth is still one of main open problems. Instead it iswell established that galaxy structures are strongly irreg-ular and form complex patterns.The first question in this context concerns the studiesof galaxy correlation properties. Two-point properties areparticularly useful to determine correlations and their spa-tial extension. There are different ways of measuring two-point properties and, in general, the most suitable methoddepends on the type of correlation, strong or weak, charac-terizing a given point distribution in a sample. The earliestobservational studies, from angular catalogs, produced theprimary result [28] that the reduced two-point correlationfunction ξ ( r ) ≡ h n ( r ) n (0) ih n i − n is the density ofpoints) is well approximated, in the range of scales fromabout 0 . ξ ( r ) ≈ ( r/r ) − γ with γ ≈ . r ≈ . h . Thisresult was subsequently confirmed by numerous other au-thors in different redshift surveys (see e.g., [29]). However,while ξ ( r ) shows consistently a simple power-law behav-ior characterized by this exponent, there is very consider-able variation among samples, with different depths andluminosity cuts, in the measured amplitude of ξ ( r ). Thisvariation is usually ascribed a posteriori to an intrinsicdifference in the correlation properties of galaxies of dif-ferent luminosity (see e.g., [29]): brighter galaxies presentlarger values of r . Theoretically it is interpreted as a realphysical phenomenon, as a manifestation of “biasing” [30].Such a variation of the amplitude of the measured cor-relation function may, however, be explained, entirely orpartially, as a finite-size effect i.e., as an artifact of statis-tical analysis in finite samples. The explanation is as fol-lows (see [8]): The reduced correlation function ξ ( r ) canbe written as ξ ( r ) = h n ( r ) i p h n i −
1, where h n ( r ) i p is the condi-tional density of points, i.e., the mean density of points in aspherical shell of radius r centered on a galaxy. The latteris generally a very stable local quantity, the reliable esti-mation of which at a given scale r requires only a samplelarge enough to allow a reasonable number of independentestimates of the density in a shell. The mean density h n i , rancesco Sylos Labini, Luciano Pietronero: Statistical physics for cosmic structures 5 on the other hand, is a global quantity. The size of a sam-ple in which it is estimated reliably is not known a priori ,but depends on the properties of the underlying distribu-tion. Specifically the sample must be large enough so thatthe mean density estimated in it has a sufficiently smallfluctuation with respect to the true asymptotic averagedensity.It has been pointed out [31] that, when analyzing apoint distribution which, like the galaxy distribution, ischaracterized by large fluctuations, one should, in fact,first establish the existence of a well defined mean den-sity (and ultimately the scale at which it becomes welldefined and independent of the sample size, if it does)before a statistic like ξ ( r ), which measures fluctuationswith respect to such a mean density, is employed. Furtherthe existence of power-law correlations, which are clearlypresent in the galaxy distribution, is typical of fractal dis-tributions, which are asymptotically empty. In such dis-tributions the mean density is always strongly sample de-pendent, with an average value decreasing as a functionof sample size. Given the observation of such correlationsin the system, and the instability of the amplitude of thecorrelation function ξ ( r ) estimated in different samples,special care should be taken in establishing first the scale(if any) at which homogeneity becomes a good approx-imation. The simplest way to do this is in fact to mea-sure the conditional density h n ( r ) i p . These quantities aregenerally well defined, and give a characterization of thetwo-point correlation properties of the distribution, irre-spective of whether the underlying distribution has a welldefined mean density or not. A simple power law behavior h n ( r ) i p = Br − γ is characteristic of scale-invariant fractaldistributions, with the exponent γ < fractaldimension through D = 3 − γ . The pre-factor B is, in thiscase, simply related to the lower cut-off of the distribution[8]. If the distribution has a well defined mean density, onehas, asymptotically, h n ( r ) i p = constant > D = 3 inthe previous formula). Measurements of this quantity canthus both characterize (i) the regime of strong clusteringand (ii) the scale and nature of a transition to homogene-ity. Only once the existence of an average density withinthe sample size is established in this manner does it makesense to use ξ ( r ).Results in past catalogs (see [8] and references therein)and in preliminary samples of the SDSS [20,11] show (Fig.2)that in the range of scales [0.5, ∼
30] Mpc/h galaxy dis-tributions are characterized by power-law correlations inthe conditional density in redshift space, with an exponent γ = 1 . ± .
1. In the range of scales [ ∼ ∼ γ = 0 in Fig. 2.
Behavior of the conditional density (red dots) in apreliminary sample of the SDSS survey [20], together with thedetermination of the conditional density (blue dots) in a sampleof the CfA2 catalog reported in [21]. There is a substantialagreement between the two catalogs and that the new SDSSdata seem to show a flattening at about 70 Mpc/h. A moredetailed analysis is required to study this transition and tocharacterize possible finite size effects which may affect thisbehavior. (From [10]) the conditional density) occurs before 100 Mpc/h, or thatcorrelations extend to scales of order 100 Mpc/h (with asmaller exponent 0 < γ <
The understanding of the thermodynamics and dynamicsof systems of particles interacting only through their mu-tual Newtonian self gravity is of fundamental importancein cosmology and astrophysics. In statistical physics theproblem of the evolution of self gravitating classical bod-ies has been relatively neglected, primarily because of theintrinsic difficulties associated with the attractive long-range nature of gravity and its singular behavior at van-ishing separation. Long-range interacting systems (LRIS)present a series of peculiar properties which make themqualitatively different from systems in which the interac-tions between the component elements are short-range. Inthe case of LRIS every element is coupled to every otherelement in the system and not only with those located in a
Francesco Sylos Labini, Luciano Pietronero: Statistical physics for cosmic structures finite neighborhood around itself. For this reason some ofthe most basic concepts and instruments in physics, e.g.the framework of equilibrium statistical mechanics, whichhave been developed for short-range interacting systems,cannot be extended to treat LRIS. One of the main featureof these systems is that thermodynamical equilibrium isnot generally reached.Gravity is the paradigmatic example of LRIS and thepeculiar features of self gravitating systems have beenmainly considered in the context of astrophysics and cos-mology. More recently [32] primarily through the studyof various simplified toy models, it has been shown thatLRIS generally exhibit a whole set of new qualitative prop-erties and behaviors: ensemble in-equivalence (negativespecific heat, temperature jumps), long-time relaxation(quasi-stationary states), violations of ergodicity, subtletiesin the relation of the fluid (i.e., continuum) picture andthe particle (granular) picture, etc.. These are commonsto other physical laboratory systems such as systems withunscreened Coulomb interactions and wave-particle sys-tems relevant to plasma physics [32].With the aim of approaching the problem of gravita-tional clustering in the context of statistical mechanics itis natural to start by reducing as much as possible thecomplexity of the analogous cosmological problem and tofocus on the essential aspects of the problem. Thus weconsider clustering without the expansion of the universe,and starting from particularly simple initial conditions.Our recent results suggest that in simplifying we do notloose any essential elements which change the nature ofgravitational clustering [15,16,17,18].The problem of the evolution of self gravitating classi-cal bodies, initially distributed very uniformly in infinitespace, is as old as Newton. Modern cosmology poses es-sentially the same problem as the matter in the universeis now believed to consist predominantly of almost purelyself-gravitating particles which is, at early times, indeedvery close to uniformly distributed in the universe, and atdensities at which quantum effects are completely negli-gible. Despite the age of the problem and the impressiveadvances of modern cosmology in recent years, our un-derstanding of it remains, however, very incomplete. Inits essentials it is a simple well posed problem of classicalstatistical mechanics.
We have recently formulated [15,16] a perturbative theoryof the discrete N body problem which represents an use-ful approach to control the problem of discreteness evenin cosmological simulations in the regime of small fluctu-ations, i.e., in the linear regime (see Fig.3). This situationis obtained by using as initial conditions of the probleman infinite lattice of particles slightly displaced with smallor zero initial velocity dispersion. Thus up to a change insign in the force, the initial configuration is identical to theCoulomb lattice (or Wigner crystal) in solid state physics(see e.g. [33]), and we exploit this analogy to develop an approximation to the evolution, in the linear regime, ofthe gravitational problem.More specifically, the equation of motion of particlesmoving under their mutual self-gravity is [34] m i ¨ x i = − X i = j Gm i m j ( x i − x j ) | x i − x j | . (1)Here dots denote derivatives with respect to time t , x i isposition of the i th particle of mass m i . We treat a systemof N point particles, of equal mass m , initially placed ona Bravais lattice, with periodic boundary conditions. Per-turbations from the Coulomb lattice are described simplyby Eq. (1) with and Gm → − e (where e is the elec-tronic charge). As written in Eq. (1) the infinite sum giv-ing the force on a particle is not explicitly well defined.It is calculated by solving the Poisson equation for thepotential, with the mean mass density subtracted in thesource term. In the cosmological case this is appropriateas the effect of the mean density is absorbed in the Hubbleexpansion; in the case of the Coulomb lattice and of thegravitational static case (which we consider here) it corre-sponds to the assumed presence of an oppositely charged(negative mass for gravity) neutralizing background (seediscussion in [35]).We consider now perturbations about the perfect lat-tice. It is convenient to adopt the notation x i ( t ) = R + u ( R , t ) where R is the lattice vector of the i th particle,and u ( R , t ) is the displacement of the particle from R . Ex-panding to linear order in u ( R , t ) about the equilibriumlattice configuration (in which the force on each particleis exactly zero), we obtain ¨u ( R , t ) = X R ′ D ( R − R ′ ) u ( R ′ , t ) . (2)The matrix D is known in solid state physics, for any in-teraction, as the dynamical matrix (see e.g. [33]). It is pos-sible to compute the Fourier transform of D : diagonalizingit one can determine, for each k , three orthonormal eigen-vectors e n ( k ) and their eigenvalues ω n ( k ) ( n = 1 , , P n ω n ( k ) = − πGρ ,where ρ is the mean mass density.At this point one may solve Eq.2 by standard tech-niques, obtaining that ˜ δ ( k ) ∼ exp( p πGǫ n ( k ) t ) where˜ δ ( k ) is the Fourier mode k of the density contrast δ ( r ) =( ρ ( r ) − ρ ) /ρ and ǫ n ( k ) = − ω n ( k )4 πGρ . The eigenvalues arerepresented in Fig.3 (right panel) for the case of a simplecubic lattice: we note that this particular case presentsboth oscillating modes ( ǫ n ( k ) <
0) and modes which growfaster ( ǫ n ( k ) >
1) than in the fluid limit (which corre-sponds to ǫ n ( k ) = 1 ∀ k ).In the limit that the initial perturbations are restrictedto wavelengths much larger than the lattice spacing, theevolution corresponds exactly to that derived from ananalogous linearization of the dynamics of a pressure-lessself-gravitating fluid. Our less restricted approximation al-lows one to trace the evolution of the fully discrete distri-bution until the time when particles approach one another, rancesco Sylos Labini, Luciano Pietronero: Statistical physics for cosmic structures 7 Fig. 3.
Initial condition for a N-body simulation correspondingto a perturbed lattice (left). In this situation density pertur-bations are small and a linear analysis of the discrete problemallows one to identify a spectrum of eigen-values (right) cor-responding to different time scales of collapse for the variouswave-length of the perturbations. In the fluid limit the timescale is the same for all modes and, in these units, equal toone. (From [15]). with modifications of the fluid limit explicitly dependingon the lattice spacing. Thus one can understand exhaus-tively the modifications introduced, at a given time andlength scale, by the finiteness of N . In an infinite space, in which the initial fluctuations arenon-zero and finite at all scales, the collapse of larger andlarger scales will continue ad infinitum. The system cantherefore never reach a time independent state, thus neverreaching a thermodynamic equilibrium. One of the impor-tant results from numerical simulations of such systems inthe context of cosmology is that the system neverthelessreaches a kind of scaling regime, in which the temporalevolution is equivalent to a rescaling of the spatial vari-ables [34]. This spatio-temporal scaling relation is referredto as “self-similarity”.The evolution from above mentioned shuffled lattice(SL) initial conditions converges, after a sufficient time, toa “self-similar” behavior, in which the two-point correla-tion function obeys a simple spatio-temporal scaling rela-tion. The time dependence of the scaling is in good agree-ment with that inferred from the linearized fluid approx-imation. Between the time at which the first non-linearcorrelations emerge in a given SL and the convergence tothis “self-similar” behavior, there is a transient period ofsignificant duration. During this time, the two-point corre-lation function already approximates well, at the observednon-linear scales, a spatio-temporal scaling relation, butin which the temporal evolution is faster than the asymp-totic evolution. This behavior can be understood as aneffect of discreteness, which leads to an initial “lag” of thetemporal evolution at small scales. The non-linear corre-lations when they first develop are very well accountedfor solely in terms of two-body correlations. This is nat-urally explained in terms of the central role of nearestneighbor (NN) interaction in the build-up of these firstnon-linear correlations [36]. This two-body phase extendsto the time of onset of the spatio-temporal scaling, and thus the asymptotic form of the correlation function isalready established to a good approximation at this time.This situation has lead us to consider the comparisonof the evolution of such a system and that of “daugh-ter” coarse-grained (CG) particle distributions [18] (seeFig.4). These are sparser (i.e., lower density) particle dis-tributions, defined by a simple coarse-graining procedure,which share the same large-scale mass fluctuations. Inthe numerical simulations the CG particle distributionsare observed to evolve to give, after a sufficient time,two-point correlation properties which agree well, overthe range of scales simulated, with those in the origi-nal distribution. Indeed both the original system and itscoarse-grainings converge toward a simple dynamical scal-ing (“self-similar”) behavior with the same amplitude . Thecharacteristic time required for the CG system to begin toreproduce the clustering in the original particle distribu-tion at scales below the CG scale increases as the latterscale does. These observations are all very much in linewith the qualitative picture of the evolution of clusteringwidely accepted in cosmology: the CG distributions sharethe same fluctuations at large scales and it is these initialfluctuations alone, to a very good approximation, whichdetermine the correlations which develop at smaller scalesat later times.As discussed above once particles begin to fall on oneanother there is a phase in which very significant non-linear correlations develop due to interactions betweenNN pairs of particles. The form of the two-point corre-lation function which develops in this phase is very sim-ilar to that observed, in the same range of amplitude, inthe asymptotic scaling regime at later times [36]. Thusit appears that it is always possible to choose a CG ofthe original system, which reproduces quite well the non-linear correlations in the original system with this “earlytime”, explicitly discrete, dynamics of “macro-particles”of the CG distribution. This provides a simple physicalpicture/dynamical model for the generation of the non-linear correlation function in the relevant rangeThis finding is very different to any existing explana-tions of the dynamics giving rise to non-linear correlationsin N body simulations in cosmology. In this context the-oretical modeling invariably assumes that the non-linearcorrelations observed in simulations in this range shouldbe understood in the framework of a continuum Vlasovlimit, in which a mean-field approximation of the grav-itational field is appropriate. Indeed the fact that self-similarity is observed, with a behavior independent of theparticle density, is usually taken as an indication that sucha continuum description is appropriate. Our model is man-ifestly not of this type, a key element is the discrete NNdynamics, while also consistent with the amplitudes of thecorrelation function being independent of particle density.
The recent observations of galaxy and dark matter com-plex clumpy distributions have provided new elements for
Francesco Sylos Labini, Luciano Pietronero: Statistical physics for cosmic structures
Fig. 4.
Upper panels: Same initial conditions representing arandomly perturbed lattice, with different number of points.Bottom panels: gravitationally evolved systems. Despite thefact that the lower resolution simulation has much less points,it traces the same structures of the higher resolution one. Theidentification of the similarities and differences among thesesystems allows one to understand the effects related to thefiniteness of the number of points in the simulations. (From[18]). the understanding of the problem of cosmological struc-ture formation. The strong clumpiness characterizing galaxystructures seems to be present in the overall mass dis-tribution and its relation to the highly isotropic CMBRrepresents a fundamental problem. In contemporary cos-mological models the structures observed today at largescales in the distribution of galaxies are explained by thedynamical evolution of purely self-gravitating matter froman initial state with low amplitude density fluctuations.The extension of structures, the formation of power-lawcorrelations characterizing the strongly clustered regimeand the relation between dark and visible matter are thekey problems both from an observational and a theoreticalpoint of view.In this puzzle statistical physics plays an importantrole in various ways, which we have discussed above: (i)The complete characterization of the correlations of vis-ible and dark matter. (ii) The analysis of the very smallanisotropies of the CMBR and their implications on theinitial fluctuations which recall the super-homogeneousproperties similar to plasmas and glasses. (iii) The dynam-ical processes and theories for the formation of complexstructures from a very smooth initial distribution and ina relatively short time.It is a pleasure to thank Y. Baryshev, A. Gabrielli,M. Joyce, B. Marcos, N. Vasilyev for useful collaborationsand discussions.
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