A model for the post-collapse equilibrium of cosmological structure: truncated isothermal spheres from top-hat density perturbations

The post-collapse structure of objects that form by gravitational condensat ion out of the expanding cosmological background universe is a key element in the theory of galaxy formation. Towards this end, we have reconsidered t he outcome of the non-linear growth of a uniform, spherical density perturb ation in an unperturbed background universe - the cosmological 'top-hat' pr oblem. 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 rapi d virialization caused by the growth of small-scale inhomogeneities in the initial perturbation. We replace the standard description of the post-colla pse object as a uniform sphere in virial equilibrium by a more self-consist ent one as a truncated, non-singular, isothermal sphere in virial and hydro static equilibrium, including for the first time a proper treatment of the finite-pressure boundary condition on the sphere. The results differ signif icantly from both the uniform sphere and the singular isothermal sphere app roximations for the post-collapse objects. The virial temperature that resu lts is more than twice the previously used 'standard value' of the post-col lapse 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 rati o of the truncation radius to the core radius is 29.4, yielding a central d ensity that is 514 times greater than at the surface and 1.8 x 10(4) times greater than that of the unperturbed background density at the epoch of inf inite collapse predicted by the top-hat solution. For the top-hat fractiona l overdensity delta(L) predicted by extrapolating the linear solution into the non-linear regime, the standard top-hat model assumes that virializatio n is instantaneous at delta(L) = delta(c) = 1.686 i.e. the epoch at which t he non-linear top-hat reaches infinite density. The surface of the collapsi ng sphere meets that of the post-collapse equilibrium sphere slightly earli er, however, when delta(L) = 1.52. These results will have a significant ef fect 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 pr ofiles for a range of cosmic structures, from dwarf galaxies to galaxy clus ters, indicated by observation and by N-body simulation of cosmological str ucture formation, including the recent suggestion of a universal density pr ofile for haloes in the cold dark matter (CDM) model. The non-singular isot hermal sphere solution presented here predicts the virial temperature and i ntegrated mass distribution of the X-ray clusters formed in the CDM model a s found by detailed, 3D, numerical gas and N-body dynamical simulations rem arkably well. This solution allows us to derive analytically the numericall y calibrated mass-temperature and radius-temperature scaling laws for X-ray clusters, which were derived empirically by Evrard, Metzler & Navarro from simulation results for the CDM model.