The interplay between gravitational and dispersive forces in a multi-streamed
medium leads to an effect which is exposed in the present note as the genuine
driving force of stabilization of large-scale structure. The conception of
`adhesive gravitational clustering' is advanced to interlock the fairly
well-understood epoch of formation of large-scale structure and the onset of
virialization into objects that are dynamically in equilibrium with their
large-scale structure environment. The classical `adhesion model' is opposed to
a class of more general models traced from the physical origin of adhesion in
kinetic theory.Comment: LaTeX 8 pages, incl. 2 figures; uses paspconf.sty, epsf.sty, matches
  published version, `From Stars to the Universe', Workshop on Cosmology,
  Shanghai 199

Buchert, Thomas

English

arXiv

The interplay between gravitational and dispersive forces in a multi-streamed medium leads to an effect which is exposed in the present note as the genuine driving force of stabilization of large-scale structure. The conception of `adhesive gravitational clustering' is advanced to interlock the fairly well-understood epoch of formation of large-scale structure and the onset of virialization into objects that are dynamically in equilibrium with their large-scale structure environment. The classical `adhesion model' is opposed to a class of more general models traced from the physical origin of adhesion in kinetic theory

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arXiv:astro-ph/9901002v2  13 Aug 1999Stabilization of Large Scale Structureby Adhesive Gravitational ClusteringT. Buchert1Theory Division, CERN, CH–1211 Geneva 23, SwitzerlandAbstract. The interplay between gravitational and dispersive forcesin a multi–streamed medium leads to an effect which is exposed in thepresent note as the genuine driving force of stabilization of large–scalestructure. The conception of ‘adhesive gravitational clustering’ is ad-vanced to interlock the fairly well–understood epoch of formation of large–scale structure and the onset of virialization into objects that are dynam-ically in equilibrium with their large–scale structure environment. Theclassical ‘adhesion model’ is opposed to a class of more general modelstraced from the physical origin of adhesion in kinetic theory.1. Previrialization and Adhesion.‘Adhesive gravitational clustering’ touches on the basis of why structures stabi-lize after their formation. On the one hand, analytical understanding of structureformation is well advanced (see, e.g., the review by Sahni & Coles 1995). How-ever, models that evolve inhomogeneities into the nonlinear regime also predictstructure decay after their formation. On the other hand, the understanding ofvirialization is hosted in stellar systems theory of classically isolated bound ob-jects. What are the equilibrium structures that could be called ‘relaxed’ whilestill interacting gravitationally with their large–scale structure environment ?The fact that the transition to ‘virialized’ systems is not immediate is mirroredin expressions such as ‘previrialization’, invoked by Peebles and collaborators(Davis & Peebles 1977, Peebles 1990, Lokas et al. 1996). The ‘adhesion ap-proximation’ as invented by Gurbatov et al. (1989) takes its title from thephenomenological observation that structures ‘stick together’ after shells of coldmatter cross; numerical simulations predict multiple shell–crossings due to drag-ging forces that prevent the particles from escaping high–density regions. Weshall identify these forces in the framework of kinetic theory. It is here wherean important period of ‘adhesive clustering’ sets in, which eventually leads tothe type of bound objects that we observe in the Universe and that may finallypopulate fundamental planes in spaces spanned by integral properties of theseobjects (Fritsch & Buchert 1999).1on leave from: Theoretische Physik, Ludwig–Maximilians–Universita¨t, Theresienstr. 37, D–80333 Mu¨nchen, Germany; email: buchert@theorie.physik.uni-muenchen.de7772. The formation of large–scale structure.The understanding of large–scale structure formation may be centred on Zel’do-vich’s idea: just move the particles under inertia and follow them until they cross.For suitably scaled variables the exact solution of a force–free continuum repro-duces the linear theory of gravitational instability in the linear limit (Zel’dovich1970, Zel’dovich & Myshkis 1973, Shandarin & Zel’dovich 1989, Buchert 1989).Taking Zel’dovich’s simple view at face value, gravity is not important atall to form structures like curved sheets, filaments and clusters, their formationis merely an effect of focussing of trajectories we are familiar with in geometricaloptics of light rays. The resulting network of caustics is unstable: just increasethe depth (advance the time) of a water basin (the Universe) on the bottom ofwhich (at the time of shell–crossing) you see a network of high–intensity (high–density) structures, and you have this phenomenon. Indeed, Zel’dovich’s modelapplied to generic initial data, say powerlaw spectra with slopes in the range n <−3 at the high–frequency end, reproduces astonishingly well the density fieldspredicted by N–body runs of the same initial data (Melott et al. 1995). ‘Pancakemodels’, which don’t have structure on small scales in the initial conditions, fallin this category (Buchert et al. 1994). The spectral index n = −3 distinguishesbetween two different scenarios of structure formation: either structures formstarting from large scales (top–down), or they form starting hierarchically fromsmall scales (bottom–up); the total power in some wave number interval haslogarithmic dependence in the case n = −3 implying that structures on everyscale form at about the same instant, since all wave numbers were given similarinitial amplitudes. In the other cases n > −3, n < −3 the integrated power haspowerlaw shape; hence, n = −3 defines clearly distinct regimes.Truncation of high–frequency information in the sense of cutting down theamplitude of short–wavelength perturbations to the ‘critical spectrum’ n = −3results in optimized schemes which can compete with N–body runs down tothe scales of galaxy clusters (Coles et al. 1993, Melott et al. 1994). RefiningZel’dovich’s step is possible in the framework of the Lagrangian perturbationtheory, where it can be re–evaluated within the full gravitational context as afirst–order solution (Buchert 1989; 1996 and ref. therein). As a consequencethe performance can be improved (Melott et al. 1995); ‘optimized Lagrangianschemes’ work even for CDM–type intial data (Weiß et al. 1996). Also, othercharacteristics of the matter distribution are matching the expected (Bouchetet al. 1995).Figure 1 shows a comparison of N–body density fields with those predictedby various analytical schemes (some of them will be described below): the ‘adhe-sion approximation’ works in the right direction and shows better performancethan the Lagrangian schemes. Still, if the Lagrangian schemes are optimized,the ‘adhesion approximation’ falls short indicating that the model is not sat-isfactorily tailored for the gravitational multi–stream effect it was launched todescribe. The spectral index of a powerlaw spectrum was n = −1 which is an ex-treme test of the approximations: much power on small scales even results in thesuperiority of the first–order over the second–order Lagrangian approximation,since the latter accelerates structure decay after caustics have formed.In the following we shall encounter a broader range of possible models thatshow how to consolidate and generalize the adhesion idea.778Figure 1. Comparison of analytical approximation schemes: Dis-played is the cross–correlation coefficient S =<δnumδapproxσnumσapprox> thatcompairs the density contrast fields δ of a variety of analytical schemes(normalized to their r.m.s. values σ) and averaged over some scaleas a function of wavenumber; from top to bottom: Optimized La-grangian approximations (second–order (Melott et al. 1995) and first–order (Coles et al. 1993, Melott et al. 1994)), Adhesion model (Gur-batov et al. 1989, Weinberg & Gunn 1990), Zel’dovich approximation(Zel’dovich 1970), Second–order Lagrangian approximation (Buchert& Ehlers 1993), Frozen–Potential approximation (Bagla & Padmanab-han 1994, Brainerd et al. 1994), Linear theory (‘chopped’, i.e. withadapted average, and unchopped). S measures whether mass is movedto the right place, S = 1 means ideal congruence of both fields.7793. Adhesion: effective dynamics of multi–stream systems.Working towards a model that continues to be valid after shell–crossing singu-larities develop may be grounded on multi–stream hydrodynamics of interpene-trating streams of a cold medium. This way to go, although being logically themost adequate one, may not have the broad range of expression that we gain bymoving to a kinetic description. The conception to be presented below, however,is a phenomenological one. We keep the notion of a continuous (coarse–grained)fluid element, but we consider its motion in phase space. Thus we are deal-ing with a one–particle distribution function, or simply a phase space density,neglecting terms that would arise by coarse–graining the N–particle distribu-tion function. The basic system of equations therefore consists of the so–calledVlasov–Poisson system. By averaging out the velocity information we end upwith a set of equations that describe an effective dynamics of a possibly multi–streamed medium. (For details see: Buchert & Domı´nguez 1998). This effectivedynamics will in general entail a hierarchy of evolution equations for the velocitymoments of the phase space density. Here is what I shall call the “phenomeno-logical closure” condition: imagine the shape of a fluid element in velocity spacethat crosses a caustic in real space. The so–called velocity ellipsoid, centeredon a fluid element with a bulk velocity that originally coincides with the veloci-ties of individual particles, will expand in a highly anisotropic fashion; as soonas a three–stream system develops, two of the streams will head off in almostopposite directions and the third (decaying) stream will have small velocity, sothat overall the ellipsoid starts out to be a flattened disk. Only later, whenmore and more streams develop, the ellipsoid has a chance to become spherical(“virialized”). We are going to mask this ellipsoid with a sphere for all timesand look at the effective multi–stream force that is excerted on the bulk motionv¯ = 〈v〉 of the coarse–grained fluid element. Mathematically, we require thatthe velocity dispersion tensor be idealized by an isotropic tensor:Πij = ̺(〈vv〉 − v¯v¯) = pδij ; p = α(̺) > 0 . (1)The last equation requires in addition that the multi–stream pressure should begiven as a function of the density only. This is the ‘poor man’s way’ to close thehierarchy.In cosmology we study the evolution equations for the two dynamical fieldspeculiar–velocity u and peculiar–acceleration w, or the density ̺, respectively.The corresponding effective evolution equations for these fields with the assump-tions sketched above are (H := a˙a):∂t̺+ 3H̺+1a(̺ui),i = 0 , ∂tui +1aujui,j +Hui = wi −α′̺a̺,i , (2)subjected to the Newtonian field equations wi,i = −4πGa(̺ − ̺H), wi,j = wj,i.Note that u is the bulk velocity of a fluid element in coordinates that are comov-ing with the Hubble flow (the overbar is omitted for simplicity, and a commameans spatial derivative with respect to these coordinates). Figure 2 illustrateshow this bulk flow will give away its kinetic energy into internal kinetic energyof the fluid element; the elements acquire a “temperature” when they move intothe multi–stream region, while the amplitude of the bulk flow decays.780Figure 2. The coarse–graining idea is exemplified for a family ofcrossing trajectories from a 2D tree–code simulation: we appreciatethe oscillatory behavior of innermost trajectories that are kept backfrom escaping the high–density region due to multi–stream forces;within the multi–stream regions there are many streams (velocities)at a given Eulerian position. Two possible situations are highlightedfor a coarse–grained volume element with oscillatory bulk velocity (left)and smoothly decaying bulk velocity (right): the kinetic energy of thebulk motion is transformed into internal kinetic energy of the coarse–grained element as the element moves into the beast.7813.1. Old Adhesion.One of the most committed of all approaches to multi–streaming is furnished byBurgers’ equation. An exact solution in 3D is available. Still, numerical tech-niques await a highly resolved insight into the multi–stream regime. Instead, the“burgerlencing” effect, being quintessential for holding fluid elements togetherinside high–density peaks, is roughhewned into the phenomenological languageof ‘adhesion’. A smart way to employ Burgers’ equation on the cosmologicalstage was proposed by Gurbatov et al. (loc.cit.). Below, we give their formalarguments leading to the ‘adhesion approximation’, which we right after thatshall derive from kinetic theory following Buchert & Domı´nguez (loc.cit.).Certainly one of the simplest ways of stating the formal structure of the‘Zel’dovich approximation’ (Zel’dovich 1970, 1973) is to postulate a law of mo-tion of the formw = F (t)u . (3)If gravity drags into the direction of the peculiar–velocity field, then the gravi-tational field equations are not needed to close the system of equations (2): thepeculiar–velocity solvesu˙+ (H − F (t))u = 0 . (4)If the field is appropriately scaled and a new time–variable is introduced (seebelow), Zel’dovich’s approximation manifests itself as an essientially force–freedescription of the continuum.Gurbatov et al. (loc.cit.) proposed that one should add a forcing thatis directly proportional to the Laplacian of the peculiar–velocity field to thisequation which, in the scaled variables, attains the form of Burgers’ equation.To derive the ‘adhesion approximation’ from the effective kinetic equations (2)we just need to insert the law of motion (3) into the second of the equations (2),which involves the multi–stream “pressure”:u˙+ (H − F (t))u = ζF (t)∆qu ; ζ :=α′(̺)a214πG̺. (5)In the ‘adhesion approximation’ the function F (t) is determined as in Zel’dovich’sapproximation by the requirement of reproducing the linear solution of gravita-tional instability:F (t) = 4πG̺Hb(t)b˙(t), (6)where b(t) is identical to the growing density contrast mode solution of theEulerian linear theory of gravitational instability for “dust” (i.e., it solves theequation b¨+ 2Hb˙− 4πG̺Hb = 0).Changing the temporal variable from t to b and defining a rescaled velocityfield u˜ := u/ab˙, Equation (5) becomes the well–known key equation of the‘adhesion approximation’ where µ is assumed constant (Gurbatov et al. loc.cit.):du˜db= µ∆qu˜ ,ddb:=∂∂b+ u˜ · ∇q , µ :=ζF (t)b˙=α′(̺)a2̺H̺bb˙2. (7)7823.2. The Lagrangian Linear Regime.The ‘adhesion approximation’, as we saw above, can be derived from kineticroutes. Although this derivation is not a contrived experimenting way of formallygetting the Laplacian forcing, we may criticize it for its limited range of validityin the kinetic framework. Buchert & Domı´nguez (loc.cit.) have shown that thedescription has to be limited to small velocity dispersion which is a necessarycondition to still follow Zel’dovich’s trajectories for the bulk flow (the law ofmotion (3)). Starting out from the full system of effective kinetic equations wemay pursue a systematic way of constructing models of ‘adhesion’ by using theLagrangian perturbation theory. To the first order in the displacements from ahomogeneous–isotropic reference cosmology Adler & Buchert (1999) obtain forthe longitudinal part of the (Lagrangian) displacement field P(X, t):P¨+ 2HP˙− 4πG̺HP =Cαa2∆XP ; Cα := α′ = const. , X˙ = 0 . (8)The markedly familiar differential operator in this equation helps to constructsolutions of this Lagrangian linear equation from known solutions of the Eulerianlinear theory. The so constructed solutions may be employed as models foradhesive gravitational clustering in the weakly nonlinear regime.3.3. New Adhesion.The apparent disparity between the standard ‘adhesion model’ and the La-grangian perturbation approach can be made more transparent by reorganiz-ing the general equations into a single equation for the gravitational peculiar–acceleration:w¨ + 6Hw˙ + (2H˙ + 8H2 − 4πG̺H)w = 4πG̺Hζ∆qw + R˙+ 4HR . (9)R represents nonlinear residuals in the equations which are touched a littlefurther in a paper in preparation (Buchert 1999).Hence, we are led to suggesting the following model equation for adhesivegravitational clustering in the nonlinear regime, neglecting the residuals:w¨ + 6Hw˙ + (2H˙ + 8H2 − 4πG̺H)w = 4πG̺Hζ∆qw . (10)A beautiful feature of this equation is that it embodies two limiting cases: 1.the standard ‘adhesion approximation’ in the limit of small velocity dispersion,and 2. the Lagrangian linear model (8) being a solution ofw¨ + 6Hw˙ + (2H˙ + 8H2 − 4πG̺H)w =Cαa2∆Xw . (11)The proposed new models will have, by construction, an improved performancefor the modeling of large–scale structure, but they will hopefully also give moreinsight into the clustering properties at the stages of stabilization of large–scalestructure, previrialization and the onset to “virialized” systems. A further cluemight be inferred with regard to the possible emergence of N–soliton states(compare Go¨tz (1988) for a one–dimensional example).783Acknowledgments. It is a pleasure to thank the organizers of this work-shop for a pleasant stay at Hope Hotel, Shanghai, and for guided sightseeing.Thanks to Tom Abel, Thomas Boller, Karsten Jedamzik and Ravi Sheth for“deepening” discussions and worth–to–remember Shanghai tours. The Max–Planck–Society and the Chinese Academy of Sciences are acknowledged for theirfinancial support. 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Stabilization of Large Scale Structure by Adhesive Gravitational Clustering

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Stabilization of Large Scale Structure by Adhesive Gravitational Clustering

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