A Model for the Formation and Evolution of Cosmological Halos

Abstract

Adaptive SPH and N-body simulations were carried out to study the collapse and evolution of dark matter halos that result from the gravitational instability and fragmentation of cosmological pancakes. Such halos resemble those formed by hierarchical clustering in a CDM universe and 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 results for CDM halos by Navarro, Frenk, & White (NFW). This test-bed model enables us to study the evolution of individual halos. The masses of our halos evolve in three stages: an initial collapse, continuous infall, and a final stage in which infall tapers off as a result of finite mass supply. In the continuous infall stage, halo mass grows at the rate expected for self-similar spherical infall, with M(a) proportional to the scale factor a. After the end of initial collapse at a=a_0, the concentration parameter grows linearly with a, c(a)~4a/a_0. The virial ratio 2T/|W| just after virialization is about 1.35, as predicted by the truncated isothermal sphere model 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. This mass accretion history and evolution of concentration parameter are very similar to those reported recently in N-body simulations of CDM. We therefore conclude that the fundamental properties of halo formation and evolution are generic to the formation of cosmological halos by gravitational instability and are not limited to hierarchical collapse scenarios or even to Gaussian-random-noise initial conditions.

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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
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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).
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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.
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