The post-collapse structure of objects that form by gravitational condensation out of the expanding cosmological background universe is a key element in the theory of galaxy formation. Towards this end, we have reconsidered the outcome of the non-linear 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 post-collapse object as a uniform sphere in virial equilibrium by a more self-consistent one as a truncated, non-singular, 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 post-collapse objects. The virial temperature that results is more than twice the previously used 'standard value' of the post-collapse 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 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 infinite collapse predicted by the top-hat solution. For the top-hat fractional overdensity delta(L) predicted by extrapolating the linear solution into the non-linear 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 non-linear top-hat reaches infinite density. The surface of the collapsing sphere meets that of the post-collapse 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 haloes in the cold dark matter (CDM) model. The non-singular 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 & Navarro from simulation results for the CDM model.
