ar X iv : a st ro - p h / 01 12 42 7 v 2 2 1 D ec 2 00 1 The Universal Equilibrium of CDM Halos : Making Tracks on the Cosmic Virial Plane ⋆

Dark-matter halos are the scaffolding around which galaxies and clusters are built. They form when the gravitational instability of primordial density fluctuations causes regions which are denser than average to slow their cosmic expansion, recollapse, and virialize. Objects as different in size and mass as dwarf spheroidal galaxies and galaxy clusters are predicted by the CDM model to have halos with a universal, self-similar equilibrium structure whose parameters are determined by the halo’s total mass and collapse redshift. These latter two are statistically correlated, however, since halos of the same mass form on average at the same epoch, with smallmass objects forming first and then merging hierarchically. The structural properties of dark-matter dominated halos of different masses, therefore, should reflect this statistical correlation, an imprint of the statistical properties of the primordial density fluctuations which formed them. Current data reveal these correlations, providing a fundamental test of the CDM model which probes the shape of the power spectrum of primordial density fluctuations and the cosmological background parameters. 1 The Truncated Isothermal Sphere (TIS) Model We have developed an analytical model for the postcollapse equilibrium structure of virialized objects which condense out of a cosmological background universe, either matter-dominated or flat with a cosmological constant [23,8]. The model is based upon the assumption that cosmological halos form from the collapse and virialization of “top-hat” density perturbations and are spherical, isotropic, and isothermal. This leads to a unique, nonsingular TIS, a particular solution of the Lane-Emden equation (suitably modified when Λ 6= 0). The size rt and velocity dispersion σV are unique functions of the mass and redshift of formation of the object for a given background universe. Our TIS density profile flattens to a constant central value, ρ0, which is roughly proportional to the critical density of the universe at the epoch of collapse, with a small core radius r0 ≈ rt/30 (where σ V = 4πGρ0r 2 0 and r0 ≡ rKing/3, for the “King radius” rKing, defined by [1], p. 228). The density profiles for gas and dark matter are assumed to be the same (no bias), with gas temperature T = μmpσ 2 V /kB. These TIS results differ from those of the more familiar approximations in which the virialized sphere resulting from a top-hat perturbation is assumed to be either the standard uniform sphere (SUS) or else a singular isothermal ⋆ to appear in ”The Mass of Galaxies at Low and High Redshift” (ESO Astrophysics Symposia), eds. R. Bender & A. Renzini, Springer-Verlag, Heidelberg, in press (2002) 2 Ilian T. Iliev and Paul R. Shapiro Table 1 SUS SIS TISa η/ηSUS 1 0.833 1.11;1.07 T/TSUS 1 3 2.16;2.19