COSMOLOGY
(5f) A New Model for the Postcollapse Equilibrium of
Cosmological Structures
The postcollapse structure of objects which form by gravitational condensation
out of the expanding cosmological background universe is a key element in the
theory of galaxy formation. Towards this end,
Shapiro, Iliev, and Raga (1999) have reconsidered
the outcome of the nonlinear growth of a uniform, spherical density
perturbation in an unperturbed background universe -- the
cosmological "top-hat" problem.
We adopt the usual assumption that the collapse to infinite
density at a finite time predicted by the top-hat solution is interrupted by
a rapid virialization caused by the growth of small-scale inhomogeneities in
the initial perturbation. We replace the standard description of the
postcollapse object as a uniform
sphere in virial equilibrium by a more self-consistent one as a truncated,
nonsingular,
isothermal sphere in virial and hydrostatic equilibrium, including for the
first time a proper treatment of the finite-pressure boundary condition on
the sphere. The results differ significantly from both the uniform sphere
and the singular isothermal sphere approximations for the postcollapse
objects. The virial temperature which results is more than twice the
previously used "standard value" of the postcollapse uniform sphere
approximation but 1.4 times smaller than that of the singular,
truncated isothermal sphere approximation.
The truncation radius is 0.554 times the radius of the top-hat at maximum
expansion, and the ratio of the truncation radius
to the core radius is 29.4, yielding a central density which is 514
times greater than at the surface and about 1.8x10^4
times greater than that of the unperturbed background density at the epoch
of infinite collapse predicted by the top-hat solution. For the
top-hat fractional overdensity delta_L predicted by extrapolating the linear
solution into the nonlinear regime, the standard top-hat model assumes that
virialization is instantaneous at delta_L=delta_c=1.686, i.e. the epoch at which
the nonlinear top-hat reaches infinite density. The surface of the
collapsing sphere meets that of the postcollapse equilibrium sphere
slightly earlier, however, when delta_L=1.52. These results will have a
significant effect on a wide range of applications of the Press-Schechter and
other semi-analytical models to cosmology.
We discuss the density profiles obtained here in relation to the density
profiles for a range of cosmic structures, from dwarf galaxies to galaxy
clusters, indicated by observation and by N-body simulation of cosmological
structure formation, including the recent suggestion of a universal density
profile for halos in the Cold Dark Matter (CDM) model. The truncated
isothermal
sphere solution presented here predicts the virial temperature and integrated
mass distribution of the X-ray clusters formed in the CDM model as found by
detailed, 3D, numerical gas and N-body dynamical simulations remarkably well.
This solution allows us to derive analytically the numerically-calibrated
mass-temperature and radius-temperature scaling laws for X-ray clusters which
were derived empirically by Evrard, Metzler and Navarro from simulation results
for the CDM model.
Further Developments and Applications of this New Model:
- Gravitational Lensing
Determination of the Mass Profile of CL 0024+1654
(Shapiro and Iliev 2000)
- Rotation Curves of
Dark-Matter-Dominated Galaxies
(Iliev and Shapiro 2000)
- Post-Collapse Equilibrium Structure of Haloes in a Low-Density Universe
(Iliev and Shapiro 2001a)
- The Equilibrium Structure of Cosmological Halos: From Dwarf Galaxies
to X-ray Clusters
(Iliev and Shapiro 2001b)
- Central Mass and Phase-Space Densities of Dark Matter Halos
(Shapiro and Iliev 2002)
- The Universal Equilibrium of CDM Halos: Making Tracks on the Cosmic
Virial Plane
(Iliev and Shapiro 2002)