Page 1

arXiv:astro-ph/0208547v1 29 Aug 2002

To appear in “Galaxy Evolution: Theory and Observations (2002)”RevMexAA(SC)

A MODEL FOR THE FORMATION AND EVOLUTION OF

COSMOLOGICAL HALOS

Marcelo A. Alvarez,1,2Paul R. Shapiro,1and Hugo Martel1

1. INTRODUCTION

Adaptive SPH and N-body simulations were car-

ried out to study the collapse and evolution of dark

matter halos that result from the gravitational insta-

bility and fragmentation of cosmological pancakes.

Such halos resemble those formed by hierarchical

clustering from realistic initial conditions in a CDM

universe and, therefore, serve as a convenient test-

bed model for studying halo dynamics. Our halos

are in approximate virial equilibrium and roughly

isothermal, as in CDM simulations. Their density

profiles agree quite well with the fit to N-body re-

sults for CDM halos by Navarro, Frenk, & White

(1997; NFW).

This test-bed model enables us to study the evo-

lution of individual halos. The masses of our halos

evolve in three stages: an initial collapse involving

rapid mass assembly, an intermediate stage of con-

tinuous infall, and a final stage in which infall tapers

off as a result of finite mass supply. In the interme-

diate stage, halo mass grows at the rate expected for

self-similar spherical infall, with M(a) ∝ a. After

the end of initial collapse at (a ≡ a0), the concentra-

tion parameter grows linearly with the cosmic scale

factor a, c(a) ≃ 4(a/a0). The virial ratio 2T/|W|

just after virialization is about 1.35, a value close

to that of the N-body results for CDM halos, as

predicted by the truncated isothermal sphere model

(TIS) (Shapiro, Iliev, & Raga 1999) and consistent

with the value expected for a virialized halo in which

mass infall contributes an effective surface pressure.

Thereafter, the virial ratio evolves towards the value

expected for an isolated halo, 2T/|W| ≃ 1, as the

mass infall rate declines. This mass accretion his-

tory and evolution of concentration parameter are

very similar to those reported recently in N-body

simulations of CDM analyzed by following the evolu-

tion of individual halos. We therefore conclude that

the fundamental properties of halo collapse, virial-

ization, structure, and evolution are generic to the

formation of cosmological halos by gravitational in-

stability and are not limited to hierarchical collapse

1Department of Astronomy, University of Texas at Austin,

Austin, TX 78712

2DOE Computational Science Graduate Fellow

???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

Fig. 1. Dark matter particles at a/ac = 3.

scenarios or even to Gaussian-random-noise initial

conditions.

2. HALO FORMATION VIA PANCAKE

INSTABILITY

Test-bed Model:

modelled as single plane-wave density fluctuations

– are gravitationally unstable; density perturbations

transverse to the direction of pancake collapse cause

the pancake to fragment (Valinia et al. 1997). When

a pancake is perturbed by two transverse density

modes with wavevectors in the plane of pancake col-

lapse, a quasi-spherical halo forms at the intersec-

tion of two filaments in the pancake plane.

halo closely resembles those formed by hierarchical

clustering from initial conditions in a CDM universe.

ASPH/P3M Simulations: Two simulations

were run of the formation of a dark matter halo by

pancake instability for use as a test-bed model for

halo formation, with 643DM particles, both with

and without 643gas particles. The primary pancake

collapses (i.e. first forms accretion shocks and caus-

tics) at a scale factor ac. By a/ac= 3, a well-defined

pancake-filament-halo structure appears (Fig. 1).

Cosmological pancakes –

This

3. HALO PROFILES & EQUILIBRIUM

Density Profile: By a/ac = 3, the DM halo

(simulated with and without gas) has a spherically-

averaged density profile close to the fit by NFW to

N-body simulations of halos formed by hierarchical

clustering in CDM (Fig. 2).

1

Page 2

2 ALVAREZ, SHAPIRO, & MARTEL

Fig. 2. Density profiles: spherically-averaged simulation

of DM no gas (dots), best-fit NFW profiles (solid curves),

at a/ac = 3, 5, 7, 10; fractional deviations from NFW,

(ρNFW− ρ)/ρNFW, above.

Jeans Equilibrium & Anisotropy:

a/ac = 3, the halo is close to equilibrium, accord-

ing to the Jeans equation in spherical symmetry, and

is close to isothermal. Hereafter, we shall consider

the halo to form at a0 ≡ 3ac. The velocity distri-

bution is somewhat more anisotropic than found in

simulations of CDM, 0.2 < β < 0.8, whereas the

CDM simulations give 0.0 < βCDM < 0.6 , where

β = 1 − ?v2

r?).

After

t?/(2?v2

4. EVOLUTION

Concentration Parameter:

tion parameter of the best-fitting NFW density pro-

file at each epoch evolves linearly with scale factor

(Fig. 3). For a > a0, after the halo formation epoch,

we find c ≃ 4(a/a0), almost identical to that re-

ported by Wechsler et al. (2002) for N-body simula-

tions of CDM halos.

Mass Growth Rate:

M200 (mass within r200) grows rapidly, while for

The concentra-

For 2 < a/ac < 3,

Fig. 3. Concentration parameter vs. scale factor.

Fig. 4. Halo mass vs. scale factor. “Gadget” curve used

code of Springel et al. (2001) to simulate same problem.

3 < a/ac< 7, M200∝ a, consistent with self-similar

spherical infall (Bertschinger 1985). For a/ac > 7,

growth flattens due to finite mass supply. This mass

history closely resembles that for CDM halos found

by Wechsler et al. (2002) (Fig. 4).

Self-Similar Infall: The radial velocity profile,

mass, and radius are consistent with self-similar in-

fall for 3 < a/ac < 7, with λ200/λc ≃ 0.8, where

λ200= r200/rta, rta is the time-varying turnaround

radius, and λcis the radius of the outermost caustic

in the self-similar solution.

Virial Ratio: The virial ratio 2T/|W| just after

virialization is ∼ 1.35, close to that of the N-body re-

sults for CDM halos, as predicted by the TIS model

and consistent with the value expected for a viri-

alized halo in which mass infall contributes an ef-

fective surface pressure. Thereafter, the virial ratio

evolves towards the value expected for an isolated

halo, 2T/|W| ∼ 1, as the mass infall rate declines.

Acknowledgments

This work was supported in part by grants NASA

ATP NAG5-10825 and NAG5-10826 and Texas Ad-

vanced Research Program 3658-0624-1999.

REFERENCES

Bertschinger, E., 1985, ApJS, 58, 39

Navarro, J.F., Frenk, C.S., White, S.D.M., 1997, ApJ,

490, 493

Shapiro, P.R., Iliev, I.T., Raga, A., 1999, MNRAS, 307,

203

Springel, V., Yoshida, N., White, S.D.M., 2001, NewA,

6, 79

Valinia, A., Shapiro, P.R., Martel, H., Vishniac, E.T.,

1997, ApJ, 479, 46

Wechsler, R.H., Bullock, J.S., Primack, J.R., Kravtsov,

A.V., Dekel, A., 2002, ApJ, 568, 52