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A COSMOLOGICAL KINETIC THEORY FOR THE EVOLUTION OF COLD DARK MATTERHALOS WITH SUBSTRUCTURE: QUASI-LINEAR THEORY

Chung-Pei Ma

Department of Astronomy, University of California, 601 Campbell Hall, Berkeley, CA 94720; [email protected]

and

Edmund Bertschinger

Department of Physics, Massachusetts Institute of Technology, Room 37-602A, 77 Massachusetts Avenue, Cambridge, MA 02139; [email protected]

Received 2003 November 26; accepted 2004 April 9

ABSTRACT

We present a kinetic theory for the evolution of the phase-space distribution of dark matter particles in galaxyhalos in the presence of a cosmological spectrum of fluctuations. This theory introduces a new way to model theformation and evolution of halos, which traditionally have been investigated by analytic gravitational infallmodels or numerical N-body methods. Unlike the collisionless Boltzmann equation, our kinetic equation containsnonzero terms on the right-hand side arising from stochastic fluctuations in the gravitational potential due tosubstructures in the dark matter mass distribution. Using statistics for constrained Gaussian random fields instandard cosmological models, we show that our kinetic equation to second order in perturbation theory is of theFokker-Planck form, with one scattering term representing drift and the other representing diffusion in velocityspace. The drift is radial, and the drift and diffusion coefficients depend only on positions and not velocities; ourrelaxation process in the quasi-linear regime is therefore different from the standard two-body relaxation. Weprovide explicit expressions relating these coefficients to the linear power spectrum of mass fluctuations andpresent results for the currently favored cold dark matter model with a nonzero cosmological constant. Solutionsto this kinetic equation will provide a complete description of the cold dark matter spatial and velocity dis-tributions for the average halo during the early phases of galaxy halo formation.

Subject headings: cosmology: theory — dark matter — galaxies: halos — gravitation —large-scale structure of universe

Online material: color figures

1. INTRODUCTION

Most mass in the universe is in the form of dark matter, andmost dark matter is gravitationally clustered in the form ofhalos. Dark matter halos are therefore the building blocks ofthe universe. Knowledge of the formation and evolution his-tory of dark matter halos is essential for an understanding ofthe halos’ structure and dynamics, as well as the environmentthat harbors galaxies and clusters. Designs for dark mattersearch experiments also rely on the predicted spatial and ve-locity distributions of dark matter particles in our own Galaxy.

According to the theory of structure formation in currentcosmological models, dark matter halos arise from tiny pri-mordial fluctuations seeded in the matter and radiation fieldsin the early universe at, for example, the end of an inflationaryera. These small inhomogeneities grow via gravitational in-stability, with the slightly overdense regions becoming denserand the slightly underdense regions becoming emptier. At agiven time, there exists a spectrum of density fluctuations overa wide range of length and mass scales, whose spatial distri-bution is characterized by the power density in Fourier spaceP(k; t), the power spectrum. Dark matter halos therefore rarelyform and evolve in isolation, growing instead by frequentaccretion of smaller mass halos and occasional major mergerswith other halos of comparable mass.

The earliest theoretical insight into cosmological halo for-mation came from the spherical radial infall model , whichconsiders an isolated spherical overdense volume and de-scribes the secondary infall of bound but initially expanding

collisionless matter onto this region (Gunn & Gott 1972). Ifthe initial density perturbation spectrum is scale-free and thevelocity distribution follows the Hubble law, the infall solu-tion is found to approach a self-similar form in an Einstein–de Sitter universe. The resulting density profile asymptoticallyapproaches � / r�� with � � 2 (Fillmore & Goldreich 1984;Bertschinger 1985). The exact value of � depends on theindex of the initial perturbation spectrum P(k) / kn: it is iso-thermal (� ¼ 2) for n � �1 and for n > �1 steepens to � �3(nþ 3)=(nþ 4) (Hoffman & Shaham 1985) or 3(nþ 3)=(nþ5) (Syer & White 1998; Nusser & Sheth 1999). The collapseof halos therefore retains memory of the initial conditions inthe radial infall model.By contrast, modern cosmological N-body simulations find

dark matter halos over a wide range of mass scales to follow adensity profile that is nearly independent of the initial powerspectrum and is shallower near the halo centers (with � �1)than that predicted by the radial infall model (Navarro et al.1996, 1997; Moore et al. 1998). Furthermore, the dark matterin the simulated halos is not entirely smoothly distributedspatially; instead, at least 10% of the halo mass is in the formof hundreds to thousands of smaller, dense satellite halos ofvarying mass (Ghigna et al. 1998; Klypin et al. 1999; Mooreet al. 1999). Thus, even though the analytic similarity solu-tions derived from one-dimensional infall onto a smoothoverdensity have provided valuable insight into the evolu-tion of self-gravitating systems, realistic halo formation isclearly more complicated. Numerical simulation has beenthe tool most frequently used in cosmology to study structure

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# 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.

formation. It has provided powerful insight and occasionallysurprising results (e.g., universal density profile) unexplain-able by the simulations themselves. Our understanding ofstructure formation can certainly be enhanced if a comple-mentary tool is developed. Complementary approaches arealso especially needed at this time to help us better understandand perhaps resolve the ‘‘cuspy halo’’ and ‘‘dwarf satellite’’problems of the cold dark matter (CDM) model.

In this paper we offer a new perspective on the subject ofdark matter halo formation by proposing an approach differentfrom both the radial infall model and N-body simulations. Wefocus on the phase-space distribution of dark matter in galaxyhalos instead of on the motion of a mass shell (as in infallmodels) or the orbit of a simulation particle (as in N-body).We attempt to understand the evolution of the phase-spacedistribution with a statistical description based on kinetictheory. Our method is to be contrasted with the standardmethod where the individual dark matter particles are de-scribed by the collisionless Boltzmann equation; N-body sim-ulations are in essence a Monte Carlo method for solving thisequation. Instead, we consider the phase space of dark matterparticles in an average halo and develop a cosmological ki-netic theory for halo evolution in the presence of a spectrumof density fluctuations.

The essential features of our approach are represented bythe fluctuation-dissipation theorem for time-dependent sto-chastic processes. To elaborate on this concept, we recallthat stochastic modeling was first developed to explain theBrownian motion of pollen particles suspended in a fluid (seex 5.3 for a more detailed discussion). This phenomenon is nowunderstood to arise from a fluctuating force on a Brownianparticle due to many rapid collisions with the molecules in thesurrounding fluid. These fluctuations cause successive smallchanges in the particle velocity, hence to a random walk. Theconsequence of many such random walks is diffusion, a formof dissipation.

Because of the large number of molecules, it is impossibleto solve the coupled equations of motion for the trajectories ofall molecules and the pollen particle in the fluid. Fortunately, astochastic treatment suffices. One mathematical prescription isprovided by the Langevin equation (Langevin 1908), whichmodels the forces arising from the fluctuations as a stochasticterm that is added to all other deterministic forces in theequation of motion for a particle. An alternative prescription isgiven by the Fokker-Planck equation (Fokker 1914; Planck1917), which is an equation of motion for the probabilitydensity or distribution function of particles undergoing ran-dom walks. In this case, the collective changes in the veloc-ities due to the fluctuations in the system are represented bya drift coefficient and a diffusion coefficient arising from thecollision terms in the collisional Boltzmann equation.

Stochastic phenomena occur beyond molecular scales. Inastrophysics, the most studied example is the evolution ofdense stellar systems such as globular clusters (Chandrasekhar1943; Spitzer 1987; Binney & Tremaine 1988). A star in aglobular cluster experiences a fluctuating force from manygravitational two-body encounters and suffers a succession ofsmall velocity changes. The fluctuations in these systems arisefrom granularity in the mass distribution at the locations of thestars. The Fokker-Planck approximation is valid for these weakencounters. We explore this example in greater detail and con-trast it with our model of galaxy halo evolution in x 5.4.

The key feature common to all of these systems is thepresence of stochastic fluctuations that result in dissipation.

In the cosmological setting addressed by this paper, the fluc-tuations arise from cosmological perturbations produced inthe early universe. At early times, these perturbations areGaussian and fully characterized by their power spectrum; atlate times, the substructures in halos seen in recent simulationsare the nonlinear counterparts of this spectrum of early ripples.These fluctuations scatter particles. If one considers an en-semble of halos, the masses and locations of the substructures(or their Gaussian predecessors) are random variables. Thesubstructures thereby introduce ‘‘noise’’ into the evolution ofdark matter halos. As we show in this paper, the result isdissipation in the form of drift and velocity space diffusion.

Recent work has explored the use of the Fokker-Planckequation in cosmological problems. Evans & Collett (1997),for example, searched for steady state solutions of the Fokker-Planck equation assuming that the fluctuations arise from two-body encounters. This paper also assumed a simple isotropicphase-space distribution dependent only on the total energyand a radial power-law form for the potential and density.Under these assumptions, Evans & Collett (1997) found that� / r�4=3 was a steady state solution under the effects of en-counters. There was no dynamics in this work , however: thesystem was assumed to be in collisionless equilibrium and thenature of the clumps producing the collisions was left arbi-trary. It also remains to be determined whether galaxy halos ,either dark matter or stellar, actually evolve to � ¼ 4=3 cusp.Weinberg (2001a, 2001b) examined the dynamics of halos dueto stochastic fluctuations from three different noise processesfor the Fokker-Planck scattering term: the merging clumpswere assumed to fly by with constant velocity, to followdecaying orbits due to dynamical friction, or to orbit insidehalos. The results of this study were promising and supportedthe notion that the density profiles of halos were affected bymergers and substructures. The fluctuations in this work ,however, were constructed by hand instead of being self-consistently calculated, and the results about a central cuspwere inconclusive.

Our objective here is to construct a kinetic equation inwhich the drift and diffusion terms are derived from realisticdensity fluctuations in current cosmological models. We alsoretain the full phase-space information by, in effect, usingf (E; J ; vr; t) without averaging over the angular momentumor radial velocity. We also do not assume collisionless equi-librium; instead, we base our approach on exact dynamics.

The organization of this paper is as follows. In x 2 we derivethe kinetic equation relevant for the evolution of an averagehalo, starting with the collisionless Boltzmann equation forindividual dark matter particles. This procedure gives rise to acollision term on the right-hand side of the kinetic equation.This term represents a correlated force density and arises fromfluctuations in the ensemble-averaged gravitational potentialdue to clustering of the matter distribution.

In x 3 we introduce the concepts and physical variablesneeded to apply this equation to cosmological problems. Thekey step is in relating the phase-space density f (r; v; t) to theprobability densities of two standard variables in cosmology:the mass density � and velocity v. We then focus on the quasi-linear regime in which the fluctuations about the mean densityand Hubble flow of an isotropic and homogeneous universeare small. This focus allows us to relate � and v to the densityand displacement fields � and y, which are Gaussian randomfields in standard cosmological models.

We leave the most mathematical part of the paper toAppendix A, where we describe how to use the statistics of

KINETIC THEORY FOR EVOLUTION OF CDM HALOS 29

constrained Gaussian random fields to express the probabilitydensities of x 3 in terms of the (constrained) means andcovariances of the density and displacement fields. The idea ofa constrained field is central to this study because our aim is toconstruct a kinetic theory for dark matter particles that willlater reside in a collapsed halo rather than at any randomlocation in the universe. The relevant perturbations are for adensity field constrained to have certain properties, e.g., aspecified height or gradient at some point in space. Thespectrum for such constrained fields is related but not identicalto the familiar linear power spectrum P(k) for unconstrainedGaussian density fluctuations.

In x 4 we put the final expression for the correlated forcederived in Appendix A back into the kinetic equation andshow that the resulting equation is of the Fokker-Planck formwhere one term represents a drift force and the other termrepresents diffusion in velocity space. We work out these dif-fusion coefficients for the currently favored cold dark matterplus a cosmological constant (�CDM) model and provideexplicit expressions for these coefficients in terms of the linearmatter fluctuation power spectrum. We also present the resultsfor the coefficients computed for this model.

In x 5 we discuss further the physical meaning of the resultsin x 4. We interpret the diffusion terms by taking velocitymoments of the kinetic equation. To place our cosmologicalkinetic equation in a broader perspective, we discuss in moredetail the classical examples of Brownian motion and globularclusters and point out the similarities and differences amongthe three cases. In x 6 we provide a summary and conclusions.In Appendix B we investigate the relaxation to stationarysolutions of our cosmological Fokker-Planck equation.

2. DERIVATION OF THE KINETIC EQUATION

We derive the kinetic equation for galaxy halo evolutionfollowing the methods presented by Bertschinger (1993) forthe derivation of the BBGKY hierarchy. The starting point isthe one-particle phase-space density for dark matter particlesin a single halo, the Klimontovich (or Klimontovich-Dupree)density (Klimontovich 1967; Dupree 1967)

fK(r; v; t) ¼ mXi

�D r� ri(t)½ ��D v� vi(t)½ �; ð1Þ

where �D is the Dirac delta function and the subscript Kdenotes the use of the Klimontovich density. We suppose thateach halo is made of equal-mass particles of mass m, for ex-ample, the elementary particles of nonbaryonic dark mattermodels. For convenience, we use velocity rather than mo-mentum and choose units so that f d 3v is a mass density.We use proper coordinates except in x 3.2 and Appendix A,where we switch to the appropriate cosmological (comoving)coordinates.

The Klimontovich density is simply a way of casting theexact trajectories of all N particles in a system into the form ofa phase-space density. The number of particles in d 3r d 3vabout (r, v) is fK d 3r d 3v, which equals either 0 or 1. TheKlimontovich density consists of delta functions because,when one is dealing with a single system (as opposed to anensemble), every particle has a well-defined position and ve-locity. The reason for introducing this quantity is that it gives aparticularly clean way of going from exact N-body dynamicsto a statistical description by taking expectation values of fKor products of more than one fK over an ensemble of N-body

systems. This approach to kinetic theory is different from theusual method that begins with the N-particle distribution foran ensemble.The Klimontovich density obeys the Klimontovich-Dupree

equation

@fK@t

þ v =@fK@r

þ ggK =@fK@v

¼ 0; ð2Þ

where

ggK(r; t) ¼ �GmXi

r� ri

r� rij j3¼ �G

Zd6w0 fK w0; tð Þ r� r0ð Þ

r� r0j j3:

ð3Þ

For notational convenience we have grouped together all sixphase-space coordinates into w � fr; vg. Equation (2) is exact,as may be verified by substituting equation (1) and using theequations of motion for individual particles,

dr

dt¼ v;

dv

dt¼ ggK: ð4Þ

In practice, the gravitational force may be softened by modi-fying equation (3) to reduce artificial two-body relaxation,as is done in N-body simulations. Equations (1) and (2) areentirely equivalent to equation (4) for one system. TheKlimontovich phase-space density retains all informationabout a particular halo because it specifies the trajectories ofall particles. Equations (1)–(4) are formally equivalent to aperfect N-body simulation. That is not what we want. Ratherthan giving a perfect description of a single halo, we averageover halos to obtain a statistical description of halo evolution.While it is impossible to completely describe a single galactichalo of dark matter particles, by using statistical mechanicsmethods we can describe the average of infinitely many halos!Before proceeding further, we make one important modifi-

cation to the usual procedure in kinetic theory. Equation (4)assumes that we work in an inertial frame. This is satisfactoryfor an isolated halo but not for a halo formed in a cosmo-logical context where neighbors cause it to orbit in the netgravitational field produced by all matter. The center of massof a halo typically moves much farther in a Hubble time thanthe size of the halo itself, even in an inertial frame in which thehalo begins at rest. For example, for the CDM family ofmodels, the displacement of a mass element from its initialposition (in comoving coordinates) is typically several mega-parsecs today. If we statistically average over an ensemble ofhalos, each of which wanders this far in a Hubble time, thephase-space density will be artificially smeared out, makingthe resulting average ‘‘halo’’ much larger than any realistichalo. We avoid this by doing the same thing that simulators dowhen they construct halo profiles: we work in the center-of-mass frame of each halo.Indeed, at each moment the halo center of mass must be at

rest in our reference frame. This frame is noninertial becauseof the gravitational forces acting on the halo. Thus, we mustmodify the kinetic equation to apply to an accelerating frame.Fortunately, the accelerations are small and the frame is ex-tended only a few megaparsecs, so we do not have to worryabout relativistic effects. We simply include the inertial(‘‘fictitious’’) forces associated with working in an acceler-ating frame.We choose to define r ¼ 0 as the center of mass of each

halo. Transforming equation (4) to the center-of-mass frame

MA & BERTSCHINGER30 Vol. 612

is very simple: we replace gg by the tidal gravitational fieldrelative to the center of each halo, gg(r)� gg(0), and then sub-tract the center-of-mass velocity from the initial velocity ofeach particle. Equation (2) is unchanged except that gg isreplaced by

ggKT (r; t) ¼ �GmXi

r� ri

r� rij j3þ ri

rij j3

!

¼ �G

Zd6w 0 fK w 0; tð Þ r� r0

r� r0j j3þ r0

r0j j3

!: ð5Þ

The subscript T is a reminder that this is the tidal gravitationalfield.

Now we return to the usual procedure for deriving a kineticequation. The Klimontovich density is the phase-space densityfor a single realization of a halo. We might imagine per-forming many N-body simulations of different halos, eachwith slightly different initial conditions. If we average theKlimontovich density over this ensemble of halos, the result(at least in the limit where the size of the ensemble becomesinfinite) is no longer a sum of delta functions but instead is asmooth one-particle distribution function:

f (w; t) � fK(w; t)h i; ð6Þ

where angle brackets denote the ensemble average. Recall thatfor notational convenience we have grouped all six phase-space variables into w.

Our goal is to derive a tractable kinetic equation for f. Theobvious next step is to take the ensemble average of equa-tion (2). To proceed, however, we will need the two-particledistribution function f2(w1; w2; t), which follows from aver-aging the product of Klimontovich densities at two points:

fK w1; tð Þ fK w2; tð Þh i � �D w1 � w2ð Þ f w1; tð Þ þ f2 w1; w2; tð Þ:ð7Þ

The delta-function term �D(w1 � w2) � �D(r1 � r2)�D(v1 � v2)corresponds to the case in which the two points lie on topof one particle. Two distinct particles give the f2 term, whichmay always be written as a product term plus a two-particlecorrelation:

f2 w1; w2; tð Þ � f w1; tð Þ f w2; tð Þ þ f2c w1; w2; tð Þ: ð8Þ

This equation defines the two-point correlation function inphase space, f2c.

The averaging we perform is similar to the averaging used indefining ordinary clustering correlation functions in cosmol-ogy. The differences here are that we are not averaging overarbitrary systems but only over those that lead to formationof a halo at r ¼ 0 and that we retain velocity information.Had we averaged over all space and integrated over velocities,we would have gotten

Rd3v f ¼ � and

Rd3v1 d

3v2 f2c ¼ �2�,where � is the cosmic mean mass density and � is the usualspatial two-point correlation function.

Now, by taking the ensemble average of equation (2), wearrive at a kinetic equation for the evolution of the averagehalo phase-space density (Bertschinger 1996):

@f

@tþ v =

@f

@rþ ggT =

@f

@v¼ � @

@v=F; ð9Þ

where we define the ensemble-averaged tidal field

ggT (r; t)� ggKT (r; t)h i ¼�G

Zd6w0 f w0; tð Þ r� r0

r� r0j j3þ r0

r0j j3

!

ð10Þ

and the correlated force density

F(w; t) � Cov ggKT (r; t); fK(w; t)½ �

¼ �G

Zd6w0 f2c w; w0; tð Þ r� r0

r� r0j j3þ r0

r0j j3

!: ð11Þ

The covariance of two random variables is denoted Cov½A;B� �h(A� hAi)(B� hBi)i ¼ hABi � hAihBi. The product of ggKTand fK involves a product of two Klimontovich densities, forwhich we have used equations (7) and (8). The one-particlediscreteness term in equation (7) makes no contribution be-cause the particle has no self-force.

Equation (9) is the first BBGKY hierarchy equation( Ichimaru 1973; Davis & Peebles 1977; Peebles 1980). In cos-mological applications this equation is usually derived startingfrom the Liouville equation for the N-particle distributionfunction. In order to clarify the averaging that is done here, wehave started instead from the one-particle Klimontovich den-sity. The result, equation (9), is an evolution equation thatlooks very much like equation (2). By averaging over halos,however, we have introduced a correlation integral term on theright-hand side. The average halo therefore does not evolveaccording to the Vlasov equation; instead, it evolves accord-ing to a kinetic equation with an effective collision term.Equations (9)–(11) are exact. No approximation has beenmade yet.

Equation (9) is a continuity equation for density in phasespace. Without the right-hand side, it says that phase-spacedensity is conserved along trajectories given by equation (4)modified for the tidal force. The right-hand side gives a cor-rection due to the fact that fluctuations about the ensembleaverage lead to correlated forces. As we will see, the effects ofthese fluctuations lead to dissipation. Because the correctionterm is a divergence, the mass density �(r; t) ¼

Rf d 3v is not

dissipated but instead is conserved.In most applications of kinetic theory, the right-hand side

of equation (9) arises from particle collisions. In the usualtreatment of a gas of particles with short-range forces, oneassumes f2c ¼ 0 always except during particle collisions. Thisis the assumption of molecular chaos that Boltzmann intro-duced to derive his celebrated kinetic equation. Our situation,however, is different: we cannot neglect f2c because it can bemuch larger than the single-particle term in equation (8) as aresult of strong gravitational clustering.

Heuristically, f2c describes the substructure within a galaxyhalo at the two-point level. (Higher order correlation functionswould be needed for a complete description.) Current cos-mological models and observations imply that the initialdensity field has fluctuations that cause the formation of manysmall halos that subsequently merge into present-day halos.The lumpiness of the matter distribution represents a fluctu-ation about the ensemble-average density field. Through thecorrelated force density F, these fluctuations cause changes inthe energy and angular momentum of individual particle orbitsthat are important for the evolution of the one-particle distri-bution function.

KINETIC THEORY FOR EVOLUTION OF CDM HALOS 31No. 1, 2004

Equation (9) is useful only if we find an expression for f2c.One way to proceed would be to derive a kinetic equationfor f2c by taking the ensemble average of equation (2) multi-plied by the Klimontovich density. This leads to the secondBBGKY equation, which depends on the three-point correla-tion in phase space, which obeys the third BBGKY equation,and so on. Thus, we either have an infinite chain of coupledequations of increasing dimensionality, or we must close thesystem. Davis & Peebles (1977) handled this problem in theirstudy of gravitational clustering by writing the three-pointfunction in terms of products of two-point functions. Weproceed differently, by seeking to close the hierarchy at thefirst level, equation (9).

To summarize this section, we have derived a formallyexact evolution equation for the average halo. The one-particlephase-space density is conserved up to a fluctuating force termarising from the two-point correlation function in phase space.Physically this term represents the effects of substructurewithin and around halos, which scatter particles to new orbits.In the next two sections we show that we can in fact expressf2c in terms of the one-particle phase-space density f using arelation that is exact to second order in cosmological pertur-bation theory. This relation will enable us to close the firstBBGKY hierarchy.

3. DISTRIBUTION FUNCTIONS FROMPROBABILITY THEORY

Solving our evolution equation in equations (9)–(11) re-quires specifying the phase-space density f at an initial timeand the two-particle correlation f2c at all times. In this sectionwe show how to compute these quantities from the probabilitydistributions of mass density and velocity at one and twopoints in space.

3.1. Probability Density vversus Phase-SpaceDistribution Function

We wish to derive expressions that would relate f (w1; t)and f2c(w1; w2; t) to probability distributions of mass densityand velocity because the latter quantities are naturally spec-ified by a cosmological model for structure formation. Themethod is to examine the mass and momentum in a smallvolume of space and to construct ensemble averages ofthe Klimontovich density using the probability distributionof density and velocity.

We use the symbol p(x; y; : : :) to refer to the proba-bility density with respect to its arguments, where the jointprobability distribution for (x; y; : : :) is dP(x; y; : : :) ¼p(x; y; : : :) dx dy � � �. We assume that these probability dis-tributions are always normalized to unit total probability.The variables relevant for this paper are the densities � andvelocities v at two points r1; r2 and at one time t : �1 ��(r1; t), v1 � v(r1; t), �2 � �(r2; t), and v2 � v(r2; t). Initiallywe assume that the velocity field in a given realizationis single valued in space. We use the following probabilitydistributions:

dP �1ð Þ¼p �1ð Þ d�1;dP �1; v1ð Þ¼p �1; v1ð Þ d�1 d3v1;

dP �1; v1; �2ð Þ¼p �1; v1; �2ð Þ d�1 d3v1 d�2;

dP �1; v1; �2; v2ð Þ¼p �1; v1; �2; v2ð Þ d�1 d3v1 d�2 d3v 2; ð12Þ

where the different probability densities are related to oneanother by

p �1; v1; �2ð Þ ¼Z

d3v2 p �1; v1; �2; v2ð Þ;

p �1; v1ð Þ ¼Z

d�2 p �1; v1; �2ð Þ;

p �1ð Þ ¼Z

d3v1 p �1; v1ð Þ;

1 ¼Z

d�1 p �1ð Þ: ð13Þ

These probability distributions depend on time, but we sup-press the t-dependence for clarity.For a small volume in phase space, f (w1; t) d

3r1 d3v1 equals

the mean mass contained in d 3r1 d3v1. The same quantity can

be calculated by averaging �1 d3r1 over density at fixed ve-

locity using dP(�1; v1), i.e.,R�1 d

3r1 dP(�1; v1). Equating theresults gives

f w1; tð Þ ¼Z 1

0

d�1 �1p �1; v1ð Þ ¼Z 1

0

d�1 �1p �1jv1ð Þp v1ð Þ

¼ �1jv1h ip v1ð Þ: ð14Þ

We have used the conditional probability density p(�1jv1) todefine the conditional mean h�1jv1i. Note that the probabilitydensities depend on (r1; t) through �1 ¼ �(r1; t) and v1 ¼v(r1; t), so equation (14) yields the desired one-particle phase-space dependence on w1 ¼ fr1; v1g.This argument is easily extended to the two-point phase-

space density, using the joint probability distribution of den-sity and velocity at two points. As equation (11) shows, weneed f2c(w1; r2; t), which depends on v1 but not v2; i.e., weneed the velocity at only one point. Recalling that f2c(w1;w2; t) ¼ f2(w1; w2; t)� f (w1; t) f (w2; t), we obtain

f2c w1; r2; tð Þ �Z

d 3v2 f2c w1; w2; tð Þ

¼Z 1

0

d�1 �1

Z 1

0

d�2 �2½ p �1; v1; �2ð Þ

� p �1; v1ð Þp �2ð Þ�

¼Z 1

0

d�1 �1

Z 1

0

d�2 �2½ p �1; �2jv1ð Þ

� p �1jv1ð Þp �2ð Þ�p v1ð Þ¼ �1�2jv1h i � �1jv1h i �2h i½ �p v1ð Þ: ð15Þ

Although we have derived equations (14) and (15) assuming asingle-valued velocity field in each realization, they are alsovalid when there is a distribution of velocities at each point. Onesimply interprets �1 as the total mass density while v1 is a single-particle velocity. Therefore, these equations are fully general.

3.2. Cosmologgical Variables

At high redshift, before dark matter halos formed, the den-sity and velocity fields were only slightly perturbed from auniformly expanding cosmological model. It is thought that thefluctuations of density and velocity responsible for all cosmicstructure were Gaussian random fields produced during theinflationary era of the very early universe. These fluctuationsprovide us with both the initial conditions and the random


element leading to a probabilistic description for halo forma-tion. Our goal now is to obtain expressions for equations (14)and (15) at early times while the density fluctuations are small.

Before proceeding further, we review the description ofdensity and velocity fields at high redshift in an almost uni-formly expanding cosmological model. First, the mean ex-pansion is described by the cosmic expansion scale factor a(t),which is related to the redshift by 1þ z ¼ 1=a(t). The Hubbleexpansion rate is H(t) ¼ a=a. The matter contributes a fraction�m(t) to the closure density. We account for the mean ex-pansion by defining new coordinates conventionally calledcomoving coordinates: x � r=a(t).

Adopting the standard cosmological model, we assumethat the dark matter is cold. This assumption means that thevelocity field is single valued in space at early times beforehalos form. We can then write the density field � and velocityfield v in terms of perturbations � and y around a uniformlyexpanding background:

�(r; t)¼ �(t) 1þ �(x; t)½ �; v(r; t)¼H(t) rþ b(t)y(x; t)½ �: ð16Þ

Here b(t) ¼ a(d lnD=d ln a) � a½�m(t)�0:6 is the growth ratefor density fluctuations with time dependence �(x; t) / D(a).We emphasize that equation (16) does not assume that thedensity fluctuations are small; it only assumes that the velocityfield is single valued. We are implicitly assuming that all ofthe matter may be described as a single cold noninteractingfluid (i.e., we assume that baryons move like CDM over thelength scales of interest).

The computation of the functions a(t), H(t), �(t), D(t), andb(t) is standard in cosmology and is not described here. Atearly times when �2T1, mass conservation relates the den-sity and velocity perturbation fields by � ¼ �(@=@x) =y, or

y(x; t) ¼ �Z

d3x0

4�

x� x0ð Þx� x0j j3

� x0; tð Þ: ð17Þ

Now we make a crucial step by changing variables from(r; v; �) used previously to (x; y; �). The reason for doing so isthat the description in comoving coordinates x and perturba-tion variables y and � is much simpler in the small pertur-bation limit. Following the notation in x 3.1, we write �1 ��(x1; t), �2 � �(x2; t), and y1 � y(x1; t). These variables havenormalized joint probability densities p(�1), p(�1; y1), andp(�1; y1; �2) in analogy with equations (12) and (13).

To change variables from (v; �) to (y; � ), we use d� ¼ � d�,d3v ¼ (Hb)3 d3 , and d� d3v p(�; v) ¼ d� d3 p(�; y). Equa-tion (14) then becomes

f w1; tð Þ ¼ � 1þ �1jy1h ið Þp y1ð Þ Hbð Þ�3; ð18Þ

where

h�1jy1i �Z

d�1 p �1jy1ð Þ�1; p �1jy1ð Þ ¼ p �1; y1ð Þp y1ð Þ ; ð19Þ

and p(�1jy1) is the conditional distribution of �1. Similarly,equation (15) becomes

f2c w1; x2; tð Þ ¼ �2( �2jy1h i � �2h i þ �1�2jy1h i� �1jy1h i �2h i)p y1ð Þ Hbð Þ�3: ð20Þ

Although the derivation of equations (18) and (20) assumedthat the velocity field is single valued, they are valid in generalprovided that one interprets �1 and �2 as total mass densityperturbations (not necessarily small) while y1 is proportionalto the single-particle velocity. Equations (18) and (20) areequivalent to equations (14) and (15) expressed in differentvariables. Their utility becomes evident in Appendix A, wherewe evaluate the probability distributions of density and ve-locity in cosmological models.

All that remains is to calculate the expectation values inequations (18) and (20) and to substitute the latter into equa-tion (11). This calculation is somewhat complicated and ispresented in Appendix A. Part of the complication arises be-cause of an important detail we have so far neglected: halosform preferentially in regions that are initially overdense. Ourcalculations assume that a halo forms at r ¼ 0. Consequently,we should not evaluate the expectation values for arbitrarypoints in space but rather for those that satisfy appropriateconstraints.

Strictly speaking, it is impossible to specify in advancethe regions that will form halos in a cosmological modelwith random initial fluctuations because the dynamics ofgravitational clustering is too complicated. A simple model,however, has been found to give reasonable agreement withnumerical simulations: halos form at maxima of the smoothedinitial density field (Bardeen et al. 1986). These constraints arefound to modify the expectation values in equations (18) and(20). To emphasize the application of constraints, we rewritethese expectation values with a subscript C:

f w1; tð Þ ¼ � 1þ �1jy1h iC� �

p y1ð Þ Hbð Þ�3 ð21Þ

and

f2c w1; x2; tð Þ¼ �2( �2jy1h iC � �2h iC þ �1�2jy1h iC� �1jy1h iC �2h iC)p y1ð Þ Hbð Þ�3: ð22Þ

Equations (21) and (22) are the main results of this section.We use them to specify the initial conditions on f and to de-termine the correlated force density in equation (11).

The next step is to evaluate the necessary constrained ex-pectation values using the standard cosmological model. Wepresent the details of this calculation in Appendix A, wherewe choose to err on the side of simplicity by followingBardeen et al. (1986) and using only the zeroth and firstderivatives of the smoothed initial density field �(x) as con-straints. Detailed comparison has shown that this simplemodel does not predict perfectly all halo formation sites (Katzet al. 1993), and more complicated constraints work better(e.g., Monaco et al. 2002). The procedure of Appendix A canpresumably be extended to incorporate more complicatedconstraints.

4. FOKKER-PLANCK EQUATION

After having devoted x 3 and Appendix A to probabilityand statistics, we are now ready to return to the kineticequation in x 2 and write down an explicit expression forthe correlated force density F(w; t) on the right-hand side ofthe kinetic equation (9). The approach we have followed isexact rather than phenomenological: our starting point was thefirst BBGKY hierarchy equation, an exact kinetic equation.The only approximation we have made is to assume that the


density and velocity fields are Gaussian random fields. Theresulting two-particle correlation function, equation (A22),depends in a simple way on the one-particle distributionp(y1). As we show next, this simple dependence leads to theFokker-Planck equation. This is a remarkable and nontrivialresult. The Fokker-Planck equation is a great simplificationof the exact (and intractable) BBGKY hierarchy. Usually theFokker-Planck equation is assumed to hold a priori, and thenits diffusion coefficients are estimated. Here, on the otherhand, we have derived the Fokker-Planck equation. Alongwith it we obtain the diffusion coefficients in the quasi-linearregime, as follows.

4.1. Diffusion Coefficients

Substituting equation (A22) into equation (11) using equa-tion (15), converting to comoving coordinates, using equa-tion (17) to relate y to v, and using equation (A21), to thirdorder in � we get

F(w; t) ¼� 4�G�a

�hC �; y0ð Þ � C(�; y) =C�1(y; y)

=C y; y0ð Þif þ (Hb)

hC(y; y)� C y0; yð Þ

� yh iCC(�; y)i=@f

@v

�þ O �4

� �; ð23Þ

where the covariance matrices C of variables � and y aregiven in equations (A14a)–(A14f ) and (A15). Here y �y(x; t) and y0 � y(0; t). Note that all the terms with sub-script 0 arise from our use of the tidal field rather than the totalgravitational acceleration.

Substituting equation (23) into equation (9), in propercoordinates our kinetic equation in the quasi-linear regimereduces to the Fokker-Planck equation

@f

@tþ v =

@f

@rþ ggT =

@f

@v¼ � @

@v= Af � D =

@f

@v

� �; ð24Þ

specified by a vector field A and a tensor field D. Together, Aand D are called diffusion coefficients. Heuristically, A rep-resents a drift force (A has units of acceleration) while Drepresents the diffusive effects of fluctuating forces. The fluxdensity in velocity space is Af � D =@f =@v. Keeping onlyterms up to second order in the density perturbations, thesequantities are given by

A(r; t) � �4�G�a C �; y0ð Þ � C(�; y) =C�1(y; y) =C y; y0ð Þ� �

¼ �Cov �; gg0jyð Þ ¼ Cov �; ggT jvð Þ ð25Þ

and

D(r; t) � 4�G�aHbC y�y0; yð Þ ¼ Cov(ggT ; v): ð26Þ

In equations (25) and (26) we have also used the fact that inthe quasi-linear regime the peculiar gravity and displacementare related by gg ¼ 4�G�ay (Zel’dovich 1970), and the tidalfield is ggT � gg(r; t)� gg0, where gg0 � gg(0; t). In writing thesecond line of equation (25) we have used equation (A7) withYB ¼ f�; y0g and YA ¼ y� hyiC . Thus, Cov(�; gg0jv) shouldbe understood to be subject to the same extremum or otherconstraints such as the constrained covariance functionC(�; y). The displacement and velocity are related by the

second formula in equation (16). Thus, in linear theory gg / vso that conditioning on v makes the fluctuation in gg vanish,allowing us to replace �Cov(�; gg0jy) by Cov(�; ggT jv) inequation (25). In all cases, the covariances are taken subjectto the extremum constraints discussed in Appendix A.It is worth noting that had we used the total gravitational

acceleration gg(r; t) instead of the tidal field ggT (r; t), the driftvector A would have been zero and D ¼ Cov(gg; v) wouldhave been larger. We conclude that tidal effects are crucial fora proper description of halo evolution (Dekel et al. 2003).This is evident in the final expressions for A and D. After thelengthy derivations, these expressions are remarkably simple,suggesting that they may have greater validity than the caseof small-amplitude Gaussian fluctuations that we have ana-lyzed here.In going from equations (A21) and (23) to equations (25)

and (26) we have dropped the contributions made by h�iC andhyiC because they do not contribute at second order in �. Atthis order, A and D are independent of the value of thesmoothed density �0 used for the constraint. However, theconstraint enters in third order.Carrying out the derivation to one higher order in � requires

using the following result for Gaussian random variables:Cov(AB; C ) ¼ hAiCov(B; C )þ hBiCov(A; C ). Applying thiswith B ¼ (1þ �)�1 ¼ 1� � þ O(�2), after some algebra wefind

A(r; t) ¼ Cov �;ggT

1þ �

v� �þ O �4

� �;

D(r; t) ¼ CovggT

1þ �; v

� �þ O �4

� �: ð27Þ

Although these results are correct for Gaussian fluctuations tothird order in the density perturbations, nonlinear evolutionwill generate nonvanishing irreducible three-point correlations(i.e., non-Gaussianity) that have been neglected here. Forexample, nonlinear evolution generates corrections to theZel’dovich approximation so that velocity and gravity are nolonger proportional to each other as we have assumed. For thisreason, equation (27) should not be regarded as being moreaccurate than equations (25) and (26).Despite its limitations, equation (27) is useful in showing

the qualitative effect of applying an initial constraint. Choos-ing the protohalo to have �0 > 0 means that typically � > 0 inthe protohalo, which decreases A and D compared withequations (25) and (26). Conversely, the diffusion coefficientsare enhanced in void regions. Note also that the constraintvalue appears explicitly in the initial conditions for f in thelinear regime through the term h�iC in equation (A21).Equations (25) and (26) imply that, in the quasi-linear re-

gime, the drift and diffusivity are independent of the velocitywithin the halo. (Although A is equivalent to a covariance atfixed v, it is independent of the particular value of v.) Thisfollows because f2c depends on y1 ¼ (v1 � Hx1)=(Hb) onlythrough p(y1) and @p=@y1 in equation (A22). As notedabove, at second order in perturbation theory the drift anddiffusivity are also independent of the initial overdensity ofthe peak �0 (but they depend on the smoothing scale usedto find extrema). Thus, both the Fokker-Planck equationand the diffusion coefficients appear to be remarkably robustquantities.To make further use of equations (25) and (26), we need

to write out the coefficients explicitly. Using equations


(A14a)–(A14f ), (25), and (26), we first factor out the depen-dence on cosmology by writing

A(r; t) ¼ 4�G�aAr(r; t) r; ð28aÞD(r; t) ¼ 4�G�aHb Dr(r; t) r rþ Dt(r; t) I� r rð Þ½ �; ð28bÞ

where Dr and Dt are scaled radial and tangential diffusion co-efficients. The scaled variables are

Ar(r; t) ¼� r�(r)� �(0)

�21

d�(r)

drþ 1

cr(r)

d�(r)

dr

�r�(r)�(r)

�20

þ 1

�21

d�(r)

dr�(r)� 2�(r)� ��

; ð29aÞ

Dr(r; t) ¼ cr(r)�d�(r)

dr

¼ �2 �

d�(r)

dr� r�(r)

�0

�2� �(r)� 2�(r)

�1

�2; ð29bÞ

Dt(r; t) ¼ ct(r)��(r)

r¼ �2

��(r)

r� �(r)

�1

�2: ð29cÞ

Here r is the comoving distance from the center of thehalo. The time dependence arises only because the functionsappearing in equations (29a)–(29c) depend on the time-dependent power spectrum of density fluctuations. These func-tions are summarized here for convenience:

�(r) ¼Z

d3k

2�ð Þ3P(k)

j1(kr)

kr;

�(r) ¼Z

d3k

2�ð Þ3P(k)

j1(kr)

k3; ð30aÞ

�(r) ¼Z

d3k

2�ð Þ3P(k)WR(k) j0(kr);

�(r) ¼Z

d3k

2�ð Þ3P(k)WR(k)

j1(kr)

kr; ð30bÞ

�20 ¼Z

d3k

2�ð Þ3P(k)W 2

R (k);

�21 ¼1

3

Zd3k

2�ð Þ3k2P(k)W 2

R (k);

�2 ¼ 1

3

Zd3k

(2�)3k�2P(k); ð30cÞ

where we recall that �20 and �2

1 are the covariances of theconstraints �0 and #0 defined in equations (A8a)–(A8c) andWR(k) is a smoothing window defined in equations (A3) and(A4). We use the Gaussian window WR(k) ¼ exp (� k2R2=2).

To gain some insight into the significance of these results,let us consider a protohalo at high redshift. If we suppose thatthe gravity field is dominated by its small-scale components,we may replace ggT by gg. If we further ignore the effects ofinitial constraints, in linear theory Dr � Dt � 3=2ð ÞH�2

v , where�v is the one-dimensional velocity dispersion. In one realiza-tion, the forces are deterministic, but taken across the en-semble they are random and fluctuating. A diffusivity D �H�2

v implies that these fluctuations cause the typical particle tochange its velocity by order itself in about a Hubble time.There is also a net drift force A with a dissipative timescale ofabout ��1 dynamical times, where � is a typical magnitude of

the density fluctuations. As a result, friction and diffusion willsignificantly alter halos within a Hubble time of their collapseat high redshift. We conclude that it is not valid to model halosusing the spherically symmetric Vlasov equation. This con-clusion has also been reached previously by Antonuccio-Delogu & Colafrancesco (1994).

To summarize, equations (24)–(26), as well as their moredetailed version, equations (28a)–(30c), are the main resultsof this subsection. Equation (24) has the standard form of aFokker-Planck equation. The Fokker-Planck form arises nat-urally and equations (25)–(26) are exact for Gaussian fluctu-ations up to second order in perturbation theory. The diffusioncoefficients A and D are due to fluctuations in the gravitationalfield over the ensemble of halos. The drift vector A representsa source of gravitational acceleration in addition to the ac-celeration ggT produced by the mean matter distribution. Thediffusion term �D =@f =@v causes the temperature (i.e., ve-locity dispersion) to increase (for a positive-definite D).

4.2. Results for �CDM Model

Having derived analytic expressions for the diffusioncoefficients in the previous subsection, we now evaluate themfor the currently favored cosmological model at high redshift.Figures 1 and 2 show Ar, Dr , and Dt as a function of radius r(measured from the halo center) for a range of Gaussiansmoothing lengths R in WR(k). We obtain these results bynumerically integrating equations (29a)–(30c) with the linearpower spectrum P(k) for a flat �CDM cosmological modelwith (�m; ��; �b; h) ¼ (0:3; 0:7; 0:05; 0:7).

The figures show a number of interesting features. First,the diffusion coefficients are negative over some regions ofradius for some parameters. As we show below, A is the

Fig. 1.—Radial drift coefficient Ar plotted as a function of halo radius rfor Gaussian smoothing lengths R ¼ 1; 0.5, 0.25, 0.1, and 0.05 h�1 Mpc. Itis computed from eq. (29a) for the �CDM model with (�m; ��; �b; h) ¼(0:3; 0:7; 0:05; 0:7). For comparison, the dashed curve shows Ar for anunconstrained field. The vertical axis shows the dimensionless Ar/� , i.e., thedrift normalized by the rms of the peculiar gravitational acceleration.


gravitational acceleration on particles in the halo caused bydrift. If the friction were produced by the Chandrasekharmechanism of tidal wakes, one would expect A k �v (seex 5.4). However, here A is independent of v and the particlemass. The source of friction is not the tidal wake of a flowpast a moving massive particle. Instead, it is the fluctuationsof the gravitational field of a Gaussian random process, andthese fluctuations can lead to a radially outward or inwardforce. We discuss this further in x 5.

More surprising at first is the fact that the diffusivities Dr

and Dt can be negative. Negative diffusivity causes the ve-locity dispersion (or temperature) to decrease, indicating aninstability. Such behavior is possible for gravitational systems.In fact, this result is to be expected in the quasi-linear regimeof gravitational instability because second-order perturbationsenhance gravitational collapse of dark matter halos (Peebles1980; Jain & Bertschinger 1994). In the strongly nonlinearregime, after halo virialization, we expect the diffusivities tobecome positive.

To elucidate the origin of the negative diffusivity, we showin Figures 3 and 4 that once the extremum constraint #0 �:�(0) ¼ 0 is removed (by setting ��2

1 ¼ 0), we obtain Ar �0 and Dr; Dt � 0 for all radii. The extremum constraint istherefore crucial to generating positive Ar and negative dif-fusivity seen in Figures 1 and 2. This effect is easy to un-derstand: if one stands at a random place in a Gaussian field,nonlinear effects can speed up or slow down gravitationalcollapse. Gravitational instability, however, speeds up theevolution in the vicinity of a density extremum.

Fig. 2.—Diffusion coefficient D in the radial (r) and tangential (t) directionsplotted as a function of halo radius r for Gaussian smoothing lengths R ¼ 1,0.5, and 0.05 h�1 Mpc. It is computed from eqs. (29b) and (29c) for the samecosmological model as Fig. 1. The dashed curve in each panel is for anunconstrained field. The vertical axis shows the dimensionless Dij=�

2 , i.e., the

diffusivity normalized by the mean squared peculiar gravitational acceleration.

Fig. 3.—Same as Fig. 1, except that no extremum constraint [i.e., #0 �:�(x ¼ 0) ¼ 0] is applied.

Fig. 4.—Same as Fig. 2, except that no extremum constraint is applied.


Another obvious systematic effect with the diffusion co-efficients in Figures 1 and 2 is their dependence on thesmoothing radius R used to set the initial constraints. For thepower spectrum assumed here, the maximum of Ar occurs atabout 2R. In the limit R ! 0, the variances �2

0 and �21 diverge,

causing the constraint terms in the diffusion coefficients (i.e.,those terms proportional to ��2

0 and ��21 ) to vanish.

Figure 2 shows that the diffusivity is isotropic at the halocenter but shows radial anisotropy outside the center. At smallradii the tangential diffusivity is larger in absolute value thanthe radial diffusivity. Tangential diffusion gives the infallingmatter angular momentum. Our treatment therefore provides anew statistical description of the growth of angular momentumby tidal torques (White 1984) and may enable, in principle, anexplanation for the universal angular momentum profile foundrecently by Bullock et al. (2001).

4.3. Asymptotic Behavvior

It is instructive to work out analytically the asymptoticbehavior of the functions in equations (29a)–(29c) for smalland large r. For small r we find

Ar(r; t)¼ r

�� 1

3�2� þ ��(0)

þ 1

1� �2(0)=�21�2

3�2(0)

�20� ��(0)

��þ O r3

� �;

ð31aÞ

Dr(r; t) ¼� �(0)

�1

�2þ r2

1

10�2� þ

3

5��(0)� �(0)

�0

�2( )

þ O r4� �

; ð31bÞ

Dt(r; t) ¼� �(0)

�1

�2þ r2

1

30�2� þ

1

5��(0)

�þO r4

� �; ð31cÞ

where we have defined

�2� �Z

d3k

2�ð Þ3P(k); ð32aÞ

�(0) ¼ 3�(0) �Z

d3k

2�ð Þ3P(k)WR(k); ð32bÞ

� �Rd3k k2P(k)WR(k)Rd3k k2P(k)W 2

R (k)¼ 1

3�21

Zd3k

2�ð Þ3k2P(k)WR(k):

ð32cÞ

Equations (31a)–(31c) show that if �2� is finite, then as

r ! 0, Ar drops to zero while Dr and Dt approach a negativeconstant as discussed above, ��2(0)=�21. This central value isdecreased as the smoothing scale is decreased (i.e., higher �0and �1) as shown in Figure 2. As R ! 0, we have �0 ! ��and �1 ! 1 as long as d ln P=d ln k > �5 as k ! 1. In thelimit �1 ! 1 (no extremum constraint), Dr and Dt are non-negative everywhere (cf. Figs. 2 and 4). Thus, negative dif-fusivity (instability) is produced by constraining the slope ofthe density field. Constraints have the opposite effect on Ar: asthe comparison of Figures 1 and 3 confirms, the contributionsto Ar from constraint terms are positive.

For large r, both cr(r) and ct(r) approach �2 , and �(r) / 1=r

(assuming P / k as k ! 0), so

Dr(r; t) ! �2 ; Dt(r; t) ! �2 : ð33Þ

Figure 2 shows that Dr rises slightly above �2 at a few hun-dred megaparsecs (for the standard model power spectrum)before reaching �2 . This is becauseDr ¼ cr � d�=dr and d�/dris negative for this range of r. The drift Ar(r; t) approaches zeroat large r. In practice, the large-r limit is unimportant becausehalo infall occurs only from within about 10 Mpc.

4.4. Power-Law Models

Another way to gain insight into the diffusion coefficients isto examine their behavior for scale-free spectra P(k) / kn.Many of the integrals are analytic in this case.

We define a set of functions

Gl(a; b) �Z 1

0

xa�1jl(bx) exp �12x2

� �dx: ð34Þ

The integral converges for aþ l > 0. Using properties of thespherical Bessel functions jl (x), we obtain the following usefulrelations:

bGlþ1(a; b) ¼ (aþ l � 1)Gl(a� 1; b)� Gl(aþ 1; b); ð35aÞbGl�1(a; b) ¼ (2þ l � a)Gl(a� 1; b)þ Gl(aþ 1; b); ð35bÞ

G0(a; 0) ¼ 2(a�2)=2�a

2

� : ð35cÞ

We also use the following relation, valid for aþ l > 0:

gl(a) � lim�!1

Z 1

0

xa�1jl(x) exp � x2

2�2

� �dx

¼ 2a�2ffiffiffi�

p�

aþ l

2

� ��

3þ l � a

2

� � ��1

: ð36Þ

For Gaussian-tapered power-law spectrum P(k) ¼ 2�2Bkn

exp (� k2R20) and Gaussian smoothing WR(k)¼ exp(�k2R2=2),

equations (30a)–(30c) give, for R0 ! 0 and n > �3,

r�(r) ¼ Bg1(nþ 2)r�(nþ2); ð37aÞ�(r) ¼ Bg1(n)r

�n for n > �1; ð37bÞ

�2 �d�

dr¼ B

ng1(n)r�(nþ1) þ 1

6�

nþ 1

2

� �R�(nþ1)0 ; n 6¼ �1;

� 1

9þ 1

6C þ 1

3log

r

R0

� �; n ¼ �1;

8>>><>>>:

ð37cÞ

�2 ��

r¼ B

�g1(n)r�(nþ1) þ 1

6�

nþ 1

2

� �R�(nþ1)0 ; n 6¼ �1;

� 4

9þ 1

6C þ 1

3log

r

R0

� �n ¼ �1;

8>>><>>>:

ð37dÞ�(r) ¼ BG0(nþ 3; r=R)R�(nþ3);

r�(r) ¼ BG1(nþ 2; r=R)R�(nþ2); ð37eÞ

�20 ¼

1

2B�

nþ 3

2

� �R�(nþ3);

�21 ¼

1

6B�

nþ 5

2

� �R�(nþ5); ð37f Þ

�2 ¼ 1

6B�

nþ 1

2

� �R�(nþ1)0 for n > �1: ð37gÞ

Here C ¼ 0:5772156649 � � � is Euler’s constant. For a purepower-law spectrum with n � �1, �(r) and �2

diverge at long


wavelength. However, equations (37c) and (37d) are valid forn > �3. For n � �1, c�1

r d�=dr ¼ 1 in equation (29a). Forn > �1, �2 converges at long wavelength but diverges atshort wavelength with �2 / R�(nþ1)

0as R0 ! 0.

The limiting behaviors of �(r) and �(r) are given by

�(r) �

B

1

2�

nþ 3

2

� �Rffiffiffi2

p� ��(nþ3)

1� (nþ 3)

6

r2

R2þ O r4

� � �; rTR;

�ng1(nþ 2)r�(nþ3); r3R;

8><>:

ð38aÞ�(r) �

B

1

6�

nþ 3

2

� �Rffiffiffi2

p� ��(nþ3)

1� (nþ 3)

10

r2

R2þ O r4

� � �; rTR;

g1(nþ 2)r�(nþ3); r3R:

8><>:

ð38bÞ

For all r/R, these functions obey the relation �þ(R2=r) d�=dr ¼�n�(r).

The drift coefficient is plotted in Figure 5 for n2f�2; �1;0; 1g. As expected from the previous section, constraintsmake Ar more positive. For power-law spectra, however, �2�diverges so one cannot use equations (31a)–(31c) to obtainthe behavior as r ! 0. The exact results here show that for�2 < n < 4, Ar ! �1 as r ! 0. For n < �2, Ar ! 0 asr ! 1.

These results show that models with a lot of small-scalepower and substructure (n > �2 as k ! 1) have a stronginward drift force, while models that are smoother on smallscales (n < �2) have negligible drift force as r ! 0. This iseasy to understand qualitatively in the context of halo for-mation. Density fields with a lot of substructure have a non-spherical mass distribution, with mass concentrated intosubhalos. For a given halo, the mass between r/2 and r, forexample, is concentrated into overdense substructures whosegravity field on infalling particles is stronger, on average, than

if this mass were uniformly distributed over a spherical shellas it is for the average halo. Given a density correlationfunction that rises with decreasing r, there is a greater con-centration of such clumps interior to r than outside of r,leading to a net inward drift force.Figure 6 shows the radial and tangential diffusivity for

power-law spectra with n � �1. For n > �1 the diffusivitydiverges because the velocity dispersion �2 diverges ask ! 1. For n ¼ �1 the divergence is logarithmic, resultingin a logarithmic dependence of Dr and Dt on radius. Figure 6shows that the effect of the constraints is to increase the dif-fusivity at radii within a few smoothing lengths of the peak.As we noted above, constraints on the density gradient makethe diffusivities negative as r ! 0. For n � �1 the dif-fusivities continue to rise with increasing r because �2 diverges as k ! 0. For the physical power spectrum used inFigures 1 and 2, �2 converges at both small and large k, so theasymptotic behavior is different.

5. DISCUSSION OF THE FOKKER-PLANCK EQUATION

The use of the Fokker-Planck equation to describe darkmatter clustering is novel. In order to gain insight into whatthis approach may reveal, in this section we examine theFokker-Planck equation from several different perspectives.

5.1. Velocity Moments

The Fokker-Planck equation describes relaxation processesin a weakly collisional fluid. In this regard it is similar to theBoltzmann equation, in which the right-hand side of equation(9) is replaced by a two-body collision integral that is bilinearin the one-particle distribution function. The present situationis physically very different, however: ‘‘collisions’’ are not dueto individual particles but rather to the gravity field of densityfluctuations in a Gaussian random field. Nevertheless, we canborrow one of the techniques used to gain insight into theBoltzmann equation, viz., velocity moments, to obtain a clearphysical interpretation of the meaning of the Fokker-Planckequation and the diffusion coefficients A and D.

Fig. 5.—Radial drift coefficient Ar(r) for power-law spectra, P(k) / kn,plotted as a function of halo radius r scaled to the Gaussian smoothing radiusR. Results are shown for four spectral indices, n ¼ �2, �1, 0, and 1, and forconstrained and unconstrained fields. The amplitude of Ar corresponds toP(k) ¼ 2�2Bkn normalized to B ¼ 1. [See the electronic edition of the Journalfor a color version of this figure.]

Fig. 6.—Diffusion coefficient D in the radial (r) and tangential (t) directionsfor power-law spectra, P(k) / kn, plotted as a function of halo radius r scaledto the Gaussian smoothing radius R. Results are shown for two spectral in-dices, n ¼ �1 and �2, and for constrained and unconstrained fields. Forn < �1, constraints cause the diffusivity to approach a negative constant forrTR. The amplitude of D corresponds to P(k) ¼ 2�2Bkn normalized toB ¼ 1. [See the electronic edition of the Journal for a color version of thisfigure.]


Moment equations are obtained by multiplying equation(24) by powers of vi and integrating over velocity. The threelowest moments give the mass density �, fluid velocity u, andvelocity dispersion tensor ij:

�(r; t) ¼Z

d3v f (r; v; t); ð39aÞ

ui(r; t) ¼ ��1

Zd3v f (r; v; t)vi; ð39bÞ

ij(r; t) ¼ ��1

Zd3v f (r; v; t) vi � uið Þ vj � uj

� �: ð39cÞ

Here we are using physical (proper) spatial coordinates ratherthan comoving coordinates. The density, velocity, and velocitydispersion tensor obey the following equations:

@�

@tþ @

@ri�uið Þ ¼ 0; ð40aÞ

@

@tþ uk

@

@rk

� �ui þ

1

�

@

@rj�ij� �

¼ gTi þ Ai; ð40bÞ

@

@tþ uk

@

@rk

� �ij þ

@ui@rk

jk

þ @uj@rk

ik þ1

�

@

@rk�ijk� �

¼ 2Dij: ð40cÞ

The first equation is the usual fluid continuity equation. Thesecond is the usual Euler equation, except that the pressure isgeneralized to a matrix �ij because the spatial stress need notbe diagonal for a collisionless fluid. The right-hand side hastwo force terms: the gravitational tidal field ggT produced by�(r; t) and the drift acceleration A(r; t) caused by the correlatedfluctuations of the mass density field. The third equationintroduces a tensor ijk , which is defined exactly like ij exceptthe integrand in equation (39c) acquires an additional factor(vk � uk). Equation (40c) is a generalized heat equation, whichrequires a little more discussion.

The velocity dispersion tensor ij generalizes the tempera-ture of an isotropic gas to the case of an anisotropic velocitydistribution. For an isotropic gas with ij ¼ T�ij (here T hasunits of v 2) and Dij ¼ D�ij, this equation takes the more fa-miliar form

3

2

dT

dtþ (:=u)T þ 1

�:= q ¼ 3D; ð41Þ

where qi ¼ 12�P3

j¼1 ijj is the heat flux. The velocity gradientterms in equation (40c) generalize the p dV term (i.e., theT: =u term) of equation (41) to an anisotropic velocity dis-tribution. For a perfect gas, the heat flux term is absent be-cause the Maxwellian velocity distribution has vanishingskewness. For an imperfect or collisionless gas, however, thethird moment of the velocity distribution generates a gener-alized heat flux �ijk. Notice that the heating rate is given notby the heat flux but instead by its divergence. For a weaklyimperfect gas, the heat flux can be approximated (e.g., in theChapman-Enskog expansion approach) by a conductive fluxq ¼ �:T , where is the conductivity, but for a collisionlessgas, ijk cannot be obtained simply from �, u, and ij. Thus,equations (40a)–(40c) do not form a closed system. This isprecisely why we must use a phase-space description for a

collisionless gas instead of trying to modify the fluid equa-tions. Equations (40a)–(40c) and (41), however, are peda-gogically useful in showing us how to interpret A and D.

The velocity diffusivity tensor Dij appears as a new term inthe anisotropic heat equation, with no familiar counterpart inthe dynamics of a collisional gas. Although Dij comes from adiffusion (i.e., gradient) term in phase space, it appears as adirect local heating term in real space (i.e., coordinate spacer). Diffusion in velocity space is equivalent to local heating:the temperature increases when particles spread out in velocityat a given point in space. The diffusion occurs in velocityspace, not in real space. The presence of this term begs thequestion, where does the energy come from? From the exactkinetic equation (9), it follows that the heating term is givenby the symmetrized first velocity moment of the correlatedforce density F produced by fluctuating gravitational fields.From this we see that the heating comes from gravitationalenergy. For this reason, it is possible for D to be positive ornegative: gravitational fluctuations can either locally heat orcool a gas. One should not try to draw conclusions aboutenergy conservation from these results because our calcu-lations are performed in the accelerating frame centered on ahalo. However, the equations of motion used in deriving ourkinetic theory imply local energy conservation.

5.2. Langgevvin Equation

The Fokker-Planck equation describes the evolution of theprobability distribution for particles undergoing randomwalks. Instead of describing the system by a distributionfunction, we can specify equations of motion for individualparticle trajectories using the Langevin equation (Langevin1908). Although this is reversing the logic we used in derivingthe Fokker-Planck equation, it is useful for providing both anunderstanding and a Monte Carlo method for numericallysolving the Fokker-Planck equation.

The Langevin equation is a stochastic differential equationfor the individual particle trajectories. The Langevin equa-tion corresponding to equation (24) is given by the pair ofequations

dr

dt¼ v;

dv

dt¼ ggT (r; t)þ A(r; t)þ &(r; t); ð42Þ

where & is a force due to a zero-mean Gaussian randomprocess with covariance

�i(r; t)�j r; t0ð Þ

� �¼ 2Dij(r; t)�D t � t 0ð Þ: ð43Þ

The velocity of a particle at time t þ dt, v(t þ dt), is thereforegiven by a deterministic term v(t)þ ½ggT (r; t)þ A(r; t)� dt, plusa random term drawn from a Gaussian distribution with var-iance /Ddt. Subtle differences exist between the cases inwhich the variance D is evaluated at [r(t); t] (‘‘Ito calculus’’)versus at [r(t þ dt=2); t þ dt] (‘‘Stratonovich calculus’’). De-tails can be found in Risken (1989). Note that the Langevinequation requires D > 0.

The interpretation of the Langevin equation is given by thefollowing result (Risken 1989; Gardiner 2002): if a suffi-ciently large sample of particles is drawn at random at time t0from the phase-space distribution f (r; v; t0) and their orbits areintegrated using equation (42), then at any later time t theirpositions and velocities are a sample from the phase-spacedensity f (r; v; t) obtained by integrating equation (24).


Equation (42) is similar to the exact equations of motion forindividual particles given by equation (4), but there are severalimportant differences. The gravity field ggK of one realizationis replaced by the spherically symmetric gravity field of theensemble-average density profile. The effects of spatially varyingforces in ggK are represented by the drift term A and the sto-chastic acceleration &. These additional terms have the effectthat, at least in the quasi-linear regime (which we assumed inour derivation of the Fokker-Planck equation), the phase-spaceevolution implied by the Langevin dynamics is equivalentto a Monte Carlo sample from the phase-space density.

This does not mean that we have come full circle. For in-dividual realizations of the ensemble, the density field isclustered and not spherically symmetric and the phase space issix-dimensional. The Langevin equations give orbits in asmooth, spherically symmetric matter distribution. The latteris far easier to model numerically, with much higher resolutionpossible, than the original dynamics. By taking advantage ofthe symmetries of the average halo, we have reduced the phasespace from six dimensions to three.

5.3. Comparison with Classical Brownian Motion

The Fokker-Planck equation was first written almost acentury ago to describe Brownian motion (Fokker 1914;Planck 1917). In this case, a small macroscopic particle (aBrownian particle) undergoes a random walk as a result of itscollisions with individual molecules in a liquid. Instead ofthe phase-space density of molecules, the relevant quantity isthe probability density W (v; t) for the Brownian particle tohave velocity v at time t. Assuming a spatially homogeneousand isotropic medium, this probability distribution obeys theFokker-Planck equation

@W

@t¼ @

@v= �vW þ �

kBT

m

@W

@v

� �; ð44Þ

where � is a constant (given, for a small spherical body ofmass m and radius a immersed in a fluid of viscosity �, by theStokes formula � ¼ 6��a=m), T is the temperature, and kB isBoltzmann’s constant. Equation (44) has the same form asequation (24) if we identify A ¼ ��v for the drag term andDij ¼ �(kBT=m)�ij for the diffusion term.

We note that the Brownian motion can be equally describedby the Langevin equations in equations (42) and (43) withggT ¼ 0, A ¼ ��v, and Dij ¼ �(kBT=m)�ij.

Brownian motion describes a very different physical situa-tion than dark matter halo formation. Why are both describedby a Fokker-Planck equation? The reason is that both prob-lems involve random walks. Let us examine this more closely.In the case of Brownian motion, the random walks arise fromimpulsive collisions that cause the Brownian particle’s ve-locity to change significantly on a timescale ��1. During thistime interval, the particle moves a distance �v=�. After anelapsed time t, the particle undergoes �t steps and thereforemoves a distance �(v2t=�)1=2. In thermal equilibrium, v2 �kBT=m so the particle moves a typical distance ��1(Dt)1=2,where D ¼ �kBT=m is the velocity diffusivity. A random walkin space leads naturally to a diffusion equation for the prob-ability distribution (Risken 1989; Gardiner 2002).

The darkmatter case at first appears to be very different. Insteadof a large particle being buffeted by small ones, we are studyingsmall particles being buffeted by large-mass fluctuations (i.e.,large in mass compared with the dark matter particles themselves,

whose mass never enters our discussion). Moreover, our deriva-tion of the Fokker-Planck equation has assumed that the motion isin the quasi-linear regime described by equation (16) with auniversal time dependence for y(x; t). Therefore, each particlehas moved in a straight line with (in appropriate units) a constantvelocity, which is not a random walk in space. However, thevelocity of a given particle is a randomwalk sincey is obtained byadding up the gravitational accelerations produced by all the massfluctuations in aGaussian randomfield. The net force on a particleis a random walk at fixed time.Although cosmic Gaussian random fields lead to a Fokker-

Planck equation, the diffusion coefficients are very differentfrom Brownian motion. First, the drift acceleration A is in-dependent of velocity, in contrast with the viscous drag ac-celeration ��v for Brownian motion. In the quasi-linearregime of gravitational clustering, the drift is not a friction atall. Instead, it is radially directed (because of spherical sym-metry for the average halo), either inward or outward de-pending on the density correlations of fluctuating substructure.Second, the velocity diffusivity tensor Dij is not proportionalto the temperature; the dark matter is treated as being com-pletely cold with vanishing temperature. In both cases thediffusivity is proportional to the mean squared displacement.In the cosmological case this depends simply on the powerspectrum of density fluctuations and not on the thermal ve-locity of the particles.In the Brownian case A and D are both proportional to the

Stokes drag coefficient �. The proportionality between A andD is no accident but is a consequence of the fluctuation-dissipation theorem, which states that these coefficients have aproportionality given by the condition of thermal equilibrium.It is instructive to see this in a generalized case of equa-tion (44). Suppose that the Brownian particle is moving in atime-independent potential �(r). The Fokker-Planck equationthen becomes

@W

@tþ v =

@W

@r� @�

@r=@W

@v¼ @

@v= �vWþ� kBT

m

@W

@v

� �: ð45Þ

The stationary solution (for suitable boundary conditions) isgiven by the Boltzmann distribution

W ¼ N exp � E

kBT

� �; E ¼ 1

2mv2 þ m�(r); ð46Þ

where N is a normalization constant. With this solution notonly does the left-hand side of equation (45) vanish, but sodoes the velocity flux on the right-hand side. Drag and dif-fusion balance each other in thermal equilibrium. Note that themean squared velocity at fixed r equals 3kBT/m. For a distri-bution of masses of Brownian particles, thermal equilibriumresults in an equipartition of energy with the heaviest particlesmoving most slowly. This is a natural consequence of having(D/�)1/2 equal the thermal speed.In thermodynamic systems, the Fokker-Planck equation

governs the relaxation to thermal equilibrium. In the cosmo-logical case of quasi-linear clustering considered here, it isnot thermal equilibrium but rather the statistics of Gaussianrandom fields that relate A and D. In Appendix B we inves-tigate our cosmological Fokker-Planck equation for the re-laxation to stationary solutions.

5.4. Comparison with Globular Cluster Dynamics

Globular cluster evolution is the most studied applicationof the Fokker-Planck equation in astrophysics. Unlike the


Brownian case, gravity plays the dominant role here. A star ina globular cluster can be approximated as experiencing twotypes of gravitational forces: a smoothly varying potential�(r; t) due to the smoothed matter distribution in the system,and a fluctuating force due to many two-body interactionswith other stars. The phase-space density of the stars obeysthe Fokker-Planck equation (Spitzer 1987; Binney & Tremaine1988)

@f

@tþ v =

@f

@r� @�

@r=@f

@v¼ � @

@v= A(v) f½ � þ 1

2

@2

@vi@vjDij(v) f� �

:

ð47Þ

This looks very similar to the cosmological kinetic equa-tion (24) in the quasi-linear regime, aside from a factor of 2in the diffusivity that is purely a matter of differing con-ventions, as well as the velocity dependence of the diffusioncoefficients. However, the physics of A and Dij is very dif-ferent here. The relaxation process in globular cluster dy-namics is two-body relaxation (Chandrasekhar 1943), i.e., thedissipation arising from the fluctuating forces of discreteNewtonian gravitating point masses. Examining this case willshed light on the relaxation processes that can affect galaxyhalo evolution.

The calculation of diffusion coefficients for two-body re-laxation is described in x 8.3 of Binney & Tremaine (1988).To understand the results, it is helpful to consider the pro-cess of Chandrasekhar dynamical friction acting on a testparticle with mass mt and velocity vt moving through a spa-tially homogeneous sea of background particles of massmb and velocity distribution f (vb) with mass density �b �mb

Rf (vb) d

3vb. For an isotropic Maxwellian distributionf (vb) / exp (� v2b=2�

2b), the test body feels an acceleration

dvtdt

¼� 4�G2�b mt þ mbð Þ ln�v3t

erf (X )� 2Xffiffiffi�

p e�X 2

�vt; ð48Þ

where X � vt=(ffiffiffi2

p�b).

Now treat each star within the globular cluster one at a timeas a test particle, with all the other stars serving as backgroundparticles. If the velocity distribution is Maxwellian and spa-tial gradients are ignored, dynamical friction will contributeA ¼ �dv=dt given by equation (48) with vt replaced by v.More generally, for a spatially homogeneous distribution onscales larger than the interparticle spacing, two-body relax-ation gives (Binney & Tremaine 1988)

A ¼ 4�G2mb mt þ mbð Þ ln� @

@vh(v); ð49aÞ

Dij ¼ 4�G2m2b ln �

@2

@vi@vjg(v); ð49bÞ

where ln� is the Coulomb logarithm, and the Rosenbluthpotentials h and g are given by (Rosenbluth et al. 1957)

h(v) �Z

f vbð Þ d3vbv� vbj j ; ð50aÞ

g(v) �Z

f vbð Þ v� vbj j d3vb: ð50bÞ

For the isotropic Maxwellian distribution f (vb) / exp(� v2b=2�

2b), A ¼ ��(v)v is a drag force and the needed coeffi-

cients are

� ¼ 4�G2�b mt þ mbð Þ ln�v 3

erf (X )� 2Xffiffiffi�

p e�X 2

�; ð51aÞ

Dk ¼ 8�G2�bmb

�2bv3

ln� erf (X )� 2Xffiffiffi�

p e�X 2

�; ð51bÞ

D? ¼ 8�G2�bmb

1

vln� 1� 1

2X 2

� �erf (X )þ 1ffiffiffi

�p

Xe�X 2

�;

ð51cÞ

where X is defined as before (with vt ! v) and Dij is decom-posed into two components:

Dij ¼vivjv2

Dk þ1

2�ij �

vivjv2

� D?: ð52Þ

We note that the drift A contains two terms: the first term isproportional to the test particle mass mt , while the second isindependent of mt . The first term represents a ‘‘polarizationcloud’’ effect, which arises from the fact that a test parti-cle movingparticle movingrelative to a background deflects more upstreamthan downstream background particles (Gilbert 1968; Mulder1983). This process results in a density gradient and hence adrag force on the test particle, whose amplitude increaseslinearly with mt since the gravitational deflection is propor-tional to mt. In contrast, the second term in A and the diffu-sivity tensor D in equations (51a)–(51c) are both /�bmb andindependent of mt . These two terms are due to fluctuations inthe gravitational potential caused by granularity in the back-ground particle distribution.

The derivation of the Fokker-Planck diffusion coefficientsfor two-body relaxation is completely different from our der-ivation based on quasi-linear cosmological density fluctua-tions. In both cases, fluctuating forces lead to random walksin velocity space and hence to the Fokker-Planck equation.The nature of the fluctuating forces and their consequences,however, are very different in the two cases.

Our cosmological Fokker-Planck derivation is exact (insecond-order cosmological perturbation theory); it yields nodrag, and the resulting drift and diffusion coefficients dependon position but are independent of velocity, as shown inequations (28a)–(30c). In contrast, the two-body relaxationcalculation is based on a Taylor series expansion of the phe-nomenological (not exact) master equation; it yields no radialdrift, and the drag and diffusion coefficients depend on ve-locity but are (in the usual derivation) independent of position(Rosenbluth et al. 1957). Furthermore, the two-body relaxa-tion rates are proportional to G2, while the rates we computeare proportional to G from equations (28a) and (28b). In thecosmology case, the phase-space correlations are built into theinitial power spectrum of density fluctuations, whose ampli-tude is given independently of G. With two-body relaxation,however, correlations are gravitationally induced by fluctua-tions instead of being present ab initio; hence, the rates pickup another factor of G.

The differences in the A term (radial drift vs. drag) have animportant consequence for equipartition of energy. As shownpreviously in the discussion of Brownian motion, drag anddiffusion work together to drive a system to thermal equilib-rium in which the mean squared velocity is proportional to


kT/m. Binney & Tremaine (1988, p. 513) show that this alsohappens with two-body relaxation. Because equilibrium sys-tems with only gravitational forces generally have negativespecific heat, there is no stable thermal equilibrium. None-theless, two-body relaxation tends to drive the velocity dis-tribution toward the Maxwellian form, with equipartition ofdifferent mass species.

In the quasi-linear cosmological case, the drift and diffusioncoefficients are independent of both velocity and particlemass. Therefore, the equilibrium velocity distribution cannotdepend on particle mass: there is no equipartition of energy.This result is similar to violent relaxation, a process arisingduring the initial collapse and virialization of a dark matterhalo when the gravitational potential is rapidly varying in time(Lynden-Bell 1967). Our relaxation process, however, arisesin the quasi-linear regime and is present even if @�=@t ¼ 0 asit is (at fixed comoving position) if �m ¼ 1. Moreover, violentrelaxation (as well as its partner, phase mixing) is generallyunderstood as arising from the collisionless dynamics repre-sented by the left-hand side of equation (47), while both two-body relaxation and our cosmological relaxation process arerepresented by the ‘‘collision’’ terms on the right-hand side.

Why did our cosmological calculation not yield a dragterm (and a corresponding diffusion term linked to it by thefluctuation-dissipation theorem)? We suspect that it is becauseour calculation was limited to small-amplitude perturbationsabout a homogeneous and isotropic expanding cosmologicalmodel. The induced effects of substructure would only comein at higher order in perturbation theory. In the nonlinear re-gime we expect two types of drift terms to be present: A ¼� r� �v, where � is the radial drift and � is the drag coeffi-cient. The Chandrasekhar calculation suggests that the dragand its accompanying diffusivity will depend on both position(through �b) and velocity. The radial drift is absent in theChandrasekhar calculation because the background there isassumed to be spatially homogeneous; with an inhomogeneousdistribution in a virialized halo there should be a radial drift(and corresponding diffusivity) that could depend on bothposition and velocity. In the nonlinear regime drag should arisefrom dynamical friction just as with the globular clusters, ex-cept that the ‘‘background’’ particles whose discretenesscauses the mb terms in equations (51a)–(51c) are now the sub-halos and substructures that rain on a halo. Because we have notyet extended our derivation to the fully nonlinear regime, theseexpectations, while plausible, have yet to be demonstrated.

6. SUMMARY AND CONCLUSIONS

In this paper we have developed a cosmological kinetictheory, valid to second order in perturbation theory, to de-scribe the evolution of the phase-space distribution of darkmatter particles in galaxy halos. This theory introduces a newway to model the early phases of galaxy halo formation,which has traditionally been studied by analytic infall modelsor numerical N-body methods.

The key physical ingredients behind our kinetic descriptionare stochastic fluctuations and dissipation caused by sub-structures arising from a spectrum of cosmological densityperturbations. The kinetic equation that we have obtained atthe end of our derivation, equations (24)–(26), has the stan-dard form of a Fokker-Planck equation. The diffusion coef-ficients A and D represent acceleration due to a drift forceand velocity space diffusion, respectively. To second orderin perturbation theory, these coefficients are related to var-ious covariance matrices of the cosmological density and

velocity fields given by equations (25) and (26). Theseexpressions can in turn be written in terms of the familiarlinear power spectrum P(k) of matter fluctuations, as shownin equations (29a)–(30c).The results for A and D from second-order perturbation

theory are shown in Figures 1–4 for the currently favored�CDM model and in Figures 5 and 6 for various scale-freemodels with power law P(k) / kn. Our results indicate thatdissipative processes are important during the initial collapse ofa dark matter halo, and it is not valid to model halos using thespherically symmetric Vlasov (collisionless Boltzmann) equa-tion. Furthermore, we find that the diffusivity is initially neg-ative close to the center of a halo, indicating a thermodynamicinstability. The source of this instability is gravity: perturba-tions enhance gravitational collapse of dark matter halos.We emphasize that our derivation leading to the Fokker-

Planck equation given by equations (24)–(26) is exact tosecond order in cosmological perturbation theory. We do notfollow the frequently used approach (e.g., Binney & Tremaine1988) that begins by expanding the collisional terms of thenonexact master equation in an infinite series of changes inphase-space coordinates (i.e., �w) and arrives at the Fokker-Planck equation by assuming weak encounters (i.e., small�wj j) and truncating the series after the second-order terms.Instead, our starting point is equation (9), the first BBGKYhierarchy equation for a smooth one-particle distributionfunction. This is an exact kinetic equation; solving it requiresspecifying the one-particle distribution at an initial time andthe two-particle correlation function at all times. We haveshown in x 3 and Appendix A that if the cosmological densityfield � and displacement (or velocity) field are Gaussianrandom fields, the resulting two-particle correlation functiondepends on the one-particle distribution in a simple way givenby equation (A22). Equation (A22) is exact to second order inperturbation theory and provides the closure relation for theBBGKY hierarchy in equation (9). Combining equation (A22)with equation (9) results in equation (24), which is a Fokker-Planck equation.The cosmological dissipative processes identified in this

paper are quite different from the standard two-body relaxation,which also leads to a Fokker-Planck equation.We find that thereis no dynamical friction (i.e., terms proportional to �v in thediffusion coefficients) in second-order cosmological perturba-tion theory. Instead, there is a radial drift toward (or away from)the center of a halo as a result of the clustering of matter withinthe halo. The usual treatment of two-body relaxation has nosuch drift term because it assumes the medium to be homoge-neous. Thus, we have identified a new relaxation process thatmust affect the phase-space structure of dark matter halos.Although our derivation is exact to second order in the

density fluctuations, it is valid only when the fluctuations aresmall. We are therefore describing only the early stages ofdark matter halo formation. The expressions we obtain forthe diffusion coefficients are plainly invalid in the nonlinearregime, where we expect dynamical friction to be present,as well as radial drift. In a later paper we will investigate thenonlinear generalizations of the exact second-order resultsderived in this paper.The Fokker-Planck description should still apply in the

nonlinear regime provided that the fluctuating force on aparticle can be modeled as a Markov process consisting of arandom walk of many steps. We conjecture that this descrip-tion will be approximately valid when the matter distribu-tion is modeled as a set of clumps (i.e., the halo model) that


scatter individual dark matter particles away from the orbitsthey would have in a smooth, spherical potential. TheFokker-Planck description should apply to not only the initialcollapse and virialization of a dark matter halo containingsubstructure but also the subsequent evolution under minormergers. Our inability to solve exactly the nonlinear dynam-ics, however, means that the diffusion coefficients will haveto be calibrated using the nonlinear halo model or numericalN-body simulations.

Calibrating the diffusion coefficients is not equivalent toreproducing the N-body simulations. N-body simulationsprovide a Monte Carlo solution of the BBGKY hierarchy,which is much more general than the Fokker-Planck equation.If the simulation results are describable by Fokker-Planckevolution, this already represents a significant new result.Moreover, even if functional forms for the radial dependenceof the diffusion coefficients have to be calibrated with simu-lations, the solution of the Fokker-Planck equation containsfar more information because it gives the complete phase-space density distribution, not simply radial profiles.

Another important step will therefore be to actually solvethe Fokker-Planck equation. In deriving it we have identifiedand quantified the relaxation processes affecting halo formationand evolution, but this is only the first step. In Appendix Bwe presented solutions in unrealistically simple cases just tohighlight the roles played by radial drift, drag, and diffusion.An outstanding question is whether these processes actuallyare rapid enough to erase the memory of cosmological initial

conditions sufficiently so that dark matter halos relax to auniversal profile. Once we have a nonlinear model for thediffusion coefficients, we will address this important question.

The work presented in this paper has a number of impli-cations. First, we have highlighted the importance of thephase-space density for understanding dark matter dynamicsand have developed a method for calculating its evolutionduring the early phases of structure formation. Second, wehave identified a dissipative process that may erase memoriesof initial conditions during the early phases of galaxy haloevolution. This erasure is necessary for universal halo densityprofiles, although further investigation is needed to quantifyhow this dissipative process affects the density profiles. Third,the statistical description of the potential fluctuations causedby substructure that we have developed can also be added toinvestigate related problems such as the heating of galacticdisks (Benson et al. 2004) and the merging of central blackholes when galaxy halos merge (Kauffmann & Haehnelt 2000;Haehnelt & Kauffmann 2002; Hughes & Blandford 2003).

We thank Jon Arons, Avishai Dekel, and Martin Weinbergfor useful conversations. The Aspen Center for Physics pro-vided a stimulating environment where a portion of the re-search was carried out. C.-P. M. is partially supported by anAlfred P. Sloan Fellowship, a Cottrell Scholars Award fromthe Research Corporation, and NASA grant NAG5-12173.

APPENDIX A

STATISTICS OF CONSTRAINED GAUSSIAN RANDOM FIELDS

In this appendix we derive expressions for the probability distributions of displacement and density perturbation, p(y1)and p(�2; y1), that are needed in equations (21) and (22). We then use them to evaluate equations (21) and (22) at high redshiftwhen the matter distribution was described by a Gaussian random field of small-amplitude density perturbations. We use comovingspatial coordinates x ¼ r=a(t).

In accordance with the standard cosmological model, we assume that �(x; t) is a Gaussian random field with power spectrumP(k; t) and that the displacement and density are related by equation (17). These are the appropriate assumptions at high redshift instandard models of structure formation. Strictly speaking, at high redshift, when linear theory applies, the evolution of thedistribution function is trivial because the density and velocity fields evolve by spatially homogeneous amplification. Here we usethe statistics of Gaussian random fields to compute quantities driving the lowest order corrections to linear evolution. We find thatthe fluctuating force is formally second order in perturbation theory.

A Gaussian random field is described most simply by its Fourier transform, which we define by

�(x) ¼Z

d3k

2�ð Þ3�(k)eik = x: ðA1Þ

We suppress the time dependence for convenience. Before we apply a constraint, both �(x) and �(k) have zero mean. Thecovariance of �(k) gives the power spectrum,

� k1ð Þ� k2ð Þh i ¼ 2�ð Þ3P k1ð Þ�D k1 þ k2ð Þ; ðA2Þ

where �D is the Dirac delta function. In the linear regime, y(x) is a linear functional of �(x). The multivariate distribution of anylinear functional of a Gaussian random field is itself Gaussian (i.e., multivariate normal) and is therefore described completely byits mean and covariance matrix. This remains true even for a constrained Gaussian random field (Bardeen et al. 1986).

In our case, we wish to constrain the initial density field to represent a protohalo. The constraints are applied to the smootheddensity field

�(x) ¼Z

d3x0 WR x� x0ð Þ� x0ð Þ ¼Z

d3k

2�ð Þ3WR(k)�(k)e

ik = x; ðA3Þ


where

WR(x) ¼Z

d3k

2�ð Þ3WR(k)e

ik = x ðA4Þ

is a smoothing window with unit volume integral,Rd3xWR(x) ¼ 1. The subscript R indicates the smoothing length. We

assume that the smoothing window is spherically symmetric so that WR(k) is real and depends only on the magnitude ofthe wavevector. Throughout the paper we will use the Gaussian window WR(x) ¼ (2�R2)�3=2 exp (� x2=2R2) and WR(k) ¼exp (� k2R2=2).

Unfortunately, there is no analytic theory to tell us where halos will form. However, there are a variety of plausible approaches.For example, following Bardeen et al. (1986), we might constrain � to have a maximum of specified height, orientation, and shapeat x ¼ 0. This peak constraint involves 10 variables (� and the components of its first and second derivatives with respect to eachof the coordinates), presenting a formidable challenge to evaluating the covariance matrix of the four variables (�2; y1). The peakconstraint is appealing but is difficult to calculate and in practice has not been found to predict well the actual formation sites ofdark matter halos (Katz et al. 1993).

Instead of trying to impose complicated constraints for halo formation (e.g., Monaco et al. 2002), we err on the side of simplicityby choosing to use only the zeroth and first derivatives of the smoothed initial density field �(x). We require that x ¼ 0 bean extremum of the smoothed density field by requiring :�(0) ¼ 0. Of course, an extremum may be a maximum, a minimum, ora saddle point. We therefore keep track also of the smoothed density �0 � �(0). Provided that �0 is greater than 2 �, the extremaof �(x) are close to the peaks (Bardeen et al. 1986).

How will we know if our results depend strongly on the choice of constraint? We will do this in two ways. First, we will varythe height of the extremum, �0. Second, we will drop the gradient constraint and use the single constraint �0. While these testswill not prove that we have a good model for the sites of halo formation, they do provide control cases for us to assess thesensitivity of our results to the details of the initial constraints.

Now we must calculate the mean and covariances of (�2; y1) subject to constraints (�0; #0), where

�0 � �(x ¼ 0); #0 � :�(x ¼ 0): ðA5Þ

The method for this calculation is based on the theorem presented in Appendix D of Bardeen et al. (1986), which states that ifYA and YB are zero-mean Gaussian variables (more generally, vectors of any length), then the conditional distribution for YB givenYA, p(YBjYA) ¼ p(YA; YB)=p(YA), is Gaussian with mean

YBjYAh i �Z

dYB p YBjYAð ÞYB ¼ YB YAh i YA YAh i�1YA ðA6Þ

and covariance matrix

C YB; YBð Þ �Z

dYB p YBjYAð Þ�YB �YB ¼ YB YBh i � YB YAh i YA YAh i�1 YA YBh i; ðA7Þ

where �YB ¼ YB � YBjYAh i. The symbol denotes a tensor product and angle brackets denote mean values. Thus, hYA YAiis the covariance matrix of the constraints, while C(YB; YB) ¼ h�YB �YBiC ¼ h�YB �YBjYAi is the covariance matrix ofYB subject to the constraints. (The symbol C, used either as a function or as a subscript, implies that a constraint is applied.) Thejuxtaposition of two matrices implies matrix multiplication.

A1. COVARIANCE OF CONSTRAINTS AND VARIABLES

To apply these results, first we compute the covariance matrix of constraints YA ¼ (�0;#0). Using equations (A1)–(A4), we getthe covariances

�20

� �¼ �20 �

Zd3k

2�ð Þ3P(k)W 2

R (k); ðA8aÞ

�0#0h i ¼ 0; ðA8bÞ

#0 #0h i ¼ �21I; �21 �1

3

Zd3k

2�ð Þ3k2P(k)W 2

R (k); ðA8cÞ

where I is the unit tensor.


Next we consider the covariance matrix of the variables YB ¼ (�1; �2; y1). Since � ¼ �(@=@x) =y, the displacement field y hasan additional factor ik=k2 in the integrand of equation (A1). For the off-diagonal elements of the covariance matrix of YB we obtain

�1�2h i ¼ �(r) �Z

d3k

2�ð Þ3P(k) j0(kr); r � x1 � x2; ðA9aÞ

�1y1h i ¼ 0; ðA9bÞ

�2y1h i ¼ ��(r)r; �(r) �Z

d3k

2�ð Þ3P(k)

j1(kr)

kr; ðA9cÞ

where the spherical Bessel functions are j0(x) ¼ x�1 sin x and j1(x) ¼ x�2(sin x� x cos x) ¼ �dj0=dx. The diagonal variances are

�21� �

¼ �22� �

¼ �(0); ðA10aÞ

y1 y1h i ¼ �2 I; �2

¼ 1

3

Zd3k

2�ð Þ3k�2P(k): ðA10bÞ

Note that in equations (A9a)–(A9c) (and in the following) we are using r as a comoving displacement vector and not as the radiusvector in proper coordinates. Proper coordinates are not used in this appendix.

A2. CROSS COVARIANCE BETWEEN VARIABLES AND CONSTRAINTS

Finally, we have the cross covariances between our primary variables and the constraints, i.e., the components of hYB YAi.These follow from equations (A1)–(A3):

�(x)�0h i ¼ �(r) �Z

d3k

2�ð Þ3P(k)WR(k) j0(kr); r ¼ jxj; ðA11aÞ

y(x)�0h i ¼ ��(r)x; �(r) �Z

d3k

2�ð Þ3P(k)WR(k)

j1(kr)

kr; ðA11bÞ

�(x)#0h i ¼ � d�

drr; r ¼ x=r; ðA11cÞ

y(x)#0h i ¼ rd�

drr rþ � I: ðA11dÞ

To arrive at the last expression above, we have used the identity

Zd�

4�e�ik = xn n ¼ r

@

@r

j1(kr)

kr

�r rþ j1(kr)

krI; ðA12Þ

where k ¼ kn, x ¼ rr, and I is the unit tensor.

A3. MEAN OF VARIABLES SUBJECT TO CONSTRAINTS

Using equations (A6) and (A8a)–(A11d), we get the mean values subject to the extremum constraints (�0;#0 ¼ 0),

�1h iC� �1j�0;#0h i ¼ �0

�20� r1ð Þ; r1 ¼ x1j j; ðA13aÞ

�2h iC� �2j�0;#0h i ¼ �0

�20� r2ð Þ; r2 ¼ x2j j; ðA13bÞ

y1h iC� y1j�0;#0h i ¼ ��0

�20� r1ð Þx1: ðA13cÞ

The subscript C denotes ‘‘subject to constraints.’’Equations (A13a)–(A13c) would be unchanged had we dropped the extremum condition and imposed instead the single

constraint �(0) ¼ �0; the constraint #0 ¼ 0 has no effect on the mean values. This follows because h�0#0i ¼ 0 so that the onlyway that #0 can enter the mean values is through terms proportional to #0=�

21, which vanish when the extremum constraint is

imposed.


A4. COVARIANCE OF VARIABLES SUBJECT TO CONSTRAINTS

We now compute the covariance matrix of our variables YB ¼ (�1; �2; y1) subject to the extremum constraints YA ¼ (�0; #0).The covariances follow from equations (A7)–(A11d). Recalling the definition C(YB;YB) � h�YB �YBjYAi, we obtain

C �1; �2ð Þ ¼ �(r)� �1�2�20

� 1

�21

d�

dr1

d�

dr2r1 = r2; r � x1 � x2; r1 �

x1r1; r2 �

x2r2; ðA14aÞ

C �2; y1ð Þ ¼ ��(r)rþ �2�1x1�20

þ 1

�21

d�

dr2�1 � 3�1� �

r1 = r2ð Þr1 þ �1r2� �

; ðA14bÞ

C �1; y1ð Þ ¼ �1�1x1�20

þ 1

�21

d�

dr1�1 � 2�1� �

x1; ðA14cÞ

C y1; y1ð Þ ¼ cr r1 r1 þ ct I� r1 r1ð Þ; ðA14dÞ

cr r1ð Þ � �2 �r1�1�0

� �2

� �1 � 2�1�1

� �2

; ðA14eÞ

ct r1ð Þ � �2 ��1�1

� �2

; ðA14f Þ

where �i � �(ri) and �1 � �(r1), and we have used the relation r1(d�1=dr1) ¼ �1 � 3�1. Note that C(�i; y1) is a vector andC(y1; y1) is a second-rank tensor. Later we will need the tensor C�1(y1; y1), which is the matrix inverse of C(y1; y1).

The role of constraints is easy to see in equations (A14a)–(A14f ). The constraint on �0 is responsible for the termsproportional to ��2

0 , while the constraint on #0 gives rise to the terms proportional to ��21 . Therefore, the constraints are easily

removed by setting ��21 ! 0 (to eliminate the constraint on #0) or �

�20 (to eliminate the constraint on �0). As expected, in the

absence of any constraints, the covariance matrix recovers the same expressions as the unconstrained covariance in equations (A9a)–(A10b).

We will need one more covariance, C(y1; y0) subject to the extremum constraints where y0 � y(0). Using equations (A7),(A11a)–(A11d), and (A12), we get

C y1; y0ð Þ ¼ d�

dr1r1 r1 þ

�

r1I� r1 r1ð Þ; �(r1) �

Zd3k

2�ð Þ3P(k)k�3j1 kr1ð Þ: ðA15Þ

A5. PROBABILITY DISTRIBUTIONS

Now we are ready to calculate the probability distribution functions and means of various quantities needed in x 3. Specifically,we need p(y1), h�1jy1iC , h�2jy1iC, h�2iC , and h�1�2jy1iC that appear in the key equations (21) and (22) for f (w1; t) andf2c(w1; x2; t). Equation (A13b) already gives h�2iC , but we need to calculate the other quantities.

First, the probability density distribution for the displacement p(y1) (subject to the extremum constraints) is a three-dimensionalGaussian whose mean and covariance are given above, implying

p(y1) ¼ (2�)�3=2 crc2t

� ��1=2exp � 1

2y1 � y1h iC� �

=C�1 y1; y1ð Þ = y1 � y1h iC� � �

: ðA16Þ

Note that the argument of the exponential is a scalar: the dot denotes a contraction (dot product).The joint distribution of displacement and density (subject to the constraints) is a little more complicated to work out since this

requires inverting the 4 ; 4 covariance matrix of (�2; y1). The distribution factors into conditional and marginal distributions,p(�2; y1) ¼ p(y1)p(�2jy1). Straightforward algebra gives the conditional distribution,

p �2jy1ð Þ ¼ 2�Qð Þ�1=2exp � 1

2Q�2 � �2h iC�C �2; y1ð Þ =C�1 y1; y1ð Þ = y1 � y1h iC

� �� 2� �; ðA17Þ

where

Q � C �2; �2ð Þ � C �2; y1ð Þ =C�1 y1; y1ð Þ =C y1; �2ð Þ: ðA18Þ

Note that the second term in Q (and the similar term in eq. [A17]) is a scalar. Writing it out with indices and using the summationconvention, C(�2; �2)� Q ¼ Ci(�2; y1)C

�1ij (y1; y1)Cj(y1; �2).


Equations (A17) and (A18) could also have been derived by noting that hYBjYAiC ¼ h�2jy1iC � h�2iC for YA ¼ y1 � hy1iC andYB ¼ �2 � h�2iC . Then using equation (A6) and relations hYBYAiC ¼ h(�2 � h�2iC)(y1 � hy1iC)iC ¼ C(�2; y1) and hYAYAiC ¼C(y1; y1), we obtain

�2jy1h iC ¼ �2h iC þ C �2; y1ð Þ =C�1 y1; y1ð Þ = y1 � y1h iC� �

¼ �2h iC � C �2; y1ð Þ = @

@y1

log p y1ð Þ; ðA19Þ

which is identical to the mean in equation (A17). The constrained mean h�1j 1iC follows simply by replacing h�2iC with h�1iC andC(�2; y1) with C(�1; y1).

The only remaining constrained mean that we need to evaluate is h�1�2jy1iC . Direct evaluation of the conditional distributionwould require inverting a 5 ; 5 matrix. It is much simpler to use equations (A6) and (A7) with YB ¼ f(�1 � h�1iC); (�2 � h�2iC)gand YA ¼ y1 � hy1iC . From the off-diagonal element of h�YB �YBjYAi, we get

�1�2jy1h iC ¼ �1h iC �2h iC þ C �1; �2ð Þ � C �1; y1ð Þ =C�1 y1; y1ð Þ =C y1; �2ð Þ: ðA20Þ

At last we are ready to evaluate the one- and two-particle distribution functions for small-amplitude Gaussian fluctuations.Substituting h�1jy1iC into equation (21), to second order in � we obtain

f (w; t) ¼ � 1þ h�iC� �

p y� C(�; y)½ � Hbð Þ�3þO �3� �

; ðA21Þ

where h�iC is given by equation (A13a) with r1 ¼ r and C(�;y) is a vector equal to the covariance of the constrained � and y givenin equation (A14c). Without an initial constraint we would have h�iC ¼ 0 and C(�; y) ¼ 0. An initial constraint on the protohaloshifts the mean density and causes the density and velocity to be correlated; hence, it changes the phase-space density. Note thatequation (A21) implies that in linear theory the spatial and velocity distributions decouple. As expected, the mean spatial density is�(1þ h�iC). The velocity distribution at each point in the halo is simply the underlying Gaussian of equation (A16) with the meanshifted by C(�; y). Note that the value of the protohalo smoothed density constraint �0 affects the density profile, h�iC / �0, but itdoes not affect the velocity distribution up to second order in perturbation theory. The imposition of the extremum constraint#0 ¼ 0 affects the phase-space density only through the term in C(�; y) / ��2

1 . Setting ��21 ! 0 everywhere removes the

constraint on #0.The two-particle correlation is found by substituting equations (A19) and (A20) into equation (22). The result is

f2c w1; x2; tð Þ¼ �2 C �1; �2ð Þ � C �1;y1ð Þ =C�1 y1;y1ð Þ =C y1; �2ð Þ � C �2;y1ð Þ � C �1;y1ð Þ �2h iC� �

=@

@y1

log p y1ð Þ� �

p y1ð Þ Hbð Þ�3:

ðA22Þ

This equation is the main result of this appendix. It is exact for Gaussian random fields.

APPENDIX B

STATIONARY SOLUTIONS AND RELAXATION TO EQUILIBRIUM

Stationary solutions of the Fokker-Planck equation are relevant for systems that relax on a timescale faster than external changestake place. The stationary solution for classical Brownian motion was given in x 5.3. Stationary solutions may also exist forcosmological systems where the constant-temperature heat bath is replaced by small-scale structure that rains on dark matter halos.

Stationary solutions have @f =@t ¼ 0 and F � Af � D =@f =@v ¼ 0 so that the flux vanishes in velocity space. We considerspherically symmetric solutions as appropriate for the average halo. From Jeans’ Theorem, the distribution function is a function ofthe integrals of motion (Binney & Tremaine 1988). For spherical systems it is often assumed that stationary solutions have theform f ¼ f (E; L), where

E ¼ 12v2 þ �(r); L ¼ r < vj j: ðB1Þ

(The particle masses are irrelevant, so we use the specific energy and angular momentum.) The collisionless Boltzmann equa-tion givesequation givesno constraint on f (E; L). The Fokker-Planck equation, however, constrains f because stationarity requires thatD =@ ln f =@v ¼ A. Assuming spherical symmetry, we write

A ¼ � r� �v; ðB2Þ

introducing the radial drift coefficient � and drag coefficient �. We assume isotropic diffusivity D. For the globular cluster casewith Chandrasekhar’s dynamical friction, � ¼ 0. For the case of quasi-linear cosmological fluctuations arising from Gaussian


random fields, we have seen that � ¼ 0. (The cosmological case also allows for different diffusivities in the radial and tangentialdirections, but that is an unnecessary complication here.)

Requiring the phase-space flux to vanish now gives the following constraints on the distribution function f (E; L):

@ ln f

@E¼ � �

Dþ �

vrD;

r2

L

@ ln f

@L¼ � �

vrD: ðB3Þ

It is interesting to see what conditions lead to thermal equilibrium with a Maxwellian distribution of velocities, i.e., equation (46)with f ¼ W . Equation (B3) shows that the conditions are � ¼ 0 and �=D ¼ m=(kBT ). Thus, the drift must be velocity dependentand behave like a drag, A / �v. Friction converts shear into heat in such a way as to drive the system toward thermal equilibrium.Chandrasekhar dynamical friction satisfies these conditions; therefore, two-body relaxation drives the velocity distribution towardthe Maxwell-Boltzmann form. Quasi-linear cosmological fluctuations, however, are quite different, having � ¼ 0 and � 6¼ 0,implying @ ln f =@E ¼ �(r2=L)@ ln f =@L ¼ �=(vrD). While these equations do not have simple general solutions, it is clear thatcosmological fluctuations do not drive a system toward thermal equilibrium with equipartition of energies. Our cosmologicalrelaxation process is similar to violent relaxation in this respect (Lynden-Bell 1967).

Although the general cosmological case of relaxation in a dark matter halo with � 6¼ 0 is complicated and cannot be fully solvedhere, there is a special case worthy of closer examination, namely, the evolution of an unconstrained cosmological Gaussian randomfield, where fluctuations generate peculiar velocities. That is, we drop the constraint of having a halo centered at r ¼ 0 and insteadconsider the velocity distribution of dark matter particles at a randomly selected point in space. To consider the cosmological case, weshould use the appropriate comoving variables to factor out the Hubble expansion. In place of r and v, we use the following variables:

x ¼ a�1r; u ¼ a(v� Hr); ðB4Þ

where a(t) is the cosmic scale factor and H ¼ d ln a=dt. With this change of variables, the Fokker-Planck equation (24) becomes

@f

@tþ u

a2=@f

@xþ a gg � 1

a

d2a

dt2r

� �=@f

@u¼ a

@

@u= �Af þ aD =

@f

@u

� �: ðB5Þ

If the velocities are measured in the comoving frame, ggT is replaced by gg. When using comoving coordinates, the gravitational fieldis reduced by the Hubble acceleration, (d2a=dt2)x. If the mean density field is homogeneous, then all particles move with theunperturbed Hubble expansion, r / a(t), implying gg ¼ (d2a=dt2)x. Thus, both the velocity and acceleration terms on the left-handside vanish when @f =@x ¼ 0. If we apply no initial constraint on the density field, then the one-point distribution function f is afunction only of time and peculiar velocity v� Hr. We use the variable u for the velocity because v� Hr decreases in proportion toa�1(t) for a freely falling body in an unperturbed homogeneous and isotropic expanding universe. If not for the right-hand side, thesolution of equation (B5) for a homogeneous and isotropic model would be f (x; u; t) ¼ f (x ¼ 0; u; t ¼ 0); i.e., the initial peculiarvelocity distribution would simply redshift with cosmic expansion and remain spatially homogeneous.

The Fokker-Planck equation does not, however, describe an unperturbed homogeneous and isotropic universe. Rather, itdescribes the statistical average of an ensemble of universes with fluctuations. In the case of unconstrained initial conditions,f ¼ f (u; t) and equation (B5) reduces to

1

a

@f

@t¼ 1

u2@

@uu2 �u f þ aD

@f

@u

� � �; ðB6Þ

where �u ¼ �u(u; t) and D ¼ D(u; t) in general. We have used statistical isotropy to require A ¼ ��uu, where u is a unit vector inthe direction of u. In second-order perturbation theory (x 4.1), �u ¼ 0 (no drag) and D ¼ 4�G�aHb�2

(assuming that we measurevelocities in the comoving frame). The Fokker-Planck equation then takes the form of a diffusion equation with D independent ofu, and the general solution is

f (u; t) ¼Z

exp � u� u0j j2

4R t0a2Ddt

!f (u; 0) d3u0

4�R t0a2Ddt

� �3=2 : ðB7Þ

The effect of fluctuations is simply to spread the initial velocity distribution (if D > 0) while retaining the Gaussian form.Physically, the gravitational fluctuations of substructure cause local heating.

Contrary to equation (B7), cosmological simulations show that the distribution of peculiar velocities in the nonlinear regime iscloser to exponential rather than Gaussian, at least in the tails (Sheth & Diaferio 2001). It is interesting to ask whether the Fokker-Planck equation can explain such a result. With the drift term retained, equation (B6) has the stationary solution

f (u; t) ¼ exp �Z u �u(u; t) du

a(t)D(u; t)

�: ðB8Þ

If �u/D is positive and independent of u, the result is indeed an exponential distribution of peculiar velocities u. In second-orderperturbation theory we found that �u ¼ 0 and D is independent of u; however, Chandrasekhar’s dynamical friction gives A=D / u.


Evidently, a third case is needed to explain the exponential velocity distribution: there must be a drag such that �u/D is independentof u, quite unlike dynamical friction from two-body relaxation. Determining whether such a drag term arises from nonlineargravitational clustering is a subject for further work.

As one final example, we consider the effects of both drag and radial drift in a simplified model illustrating the effects ofcollisions on a system for which the relaxation time is much longer than the dynamical time (e.g., the crossing time for a particle ina halo). Under these conditions, the system at all times is nearly in collisionless equilibrium with the phase-space density being afunction of the actions alone (and not the angles, using action angle variables). Because the actions are conserved in the absence ofcollisions, for quasi-static evolution equation (24) simplifies to

@f

@t¼ � @

@v= (a� �v) f � D =

@f

@v

�; ðB9Þ

where we have generalized the radial drift to an arbitrary vector field a, which we assume to be independent of v. We can solve thisequation exactly for the simple case in which a, �, and D are all constants. The general solution is the Green’s function solution

f (r; v; t) ¼Z

d3v0 G r; v� v0; tð Þ f r; v0; 0ð Þ; ðB10Þ

with Green’s function

G(r; v; t) ¼Z

d3s

2�ð Þ3exp is = v� 1

�1� e��tð Þa

�� 1

2�s =D = s 1� e�2�t

� �� : ðB11Þ

The parameter ��1 is a relaxation time. Equation (B11) simplifies to a normalized Gaussian in two limits,

G(r; v; t) ¼N (at; 2Dt); �tT1;

N (a=�; D=�); �t31;

�ðB12Þ

where N (u; M) is a multivariate normal with mean u and covariance matrix M. The solution for a ¼ � ¼ 0 was given previouslyin equation (B7). Now we have a more general solution showing that the equilibrium (for constant diffusion coefficients) is aMaxwellian with velocity dispersion tensor D=� just in equation (B8) if �u ¼ �u, but now the drift vector a produces an offset ofthe mean velocity hvi ¼ �=�. Thus, velocity-independent drift � and velocity-dependent drag ��v play completely different rolesin collisional relaxation. In later papers we will investigate these roles for more realistic models of nonlinear halo evolution.

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