The asymptotic collapsed fraction in an eternal universe

We calculate the maximum fraction of matter which is able to condense out of the expanding background universe by gravitational instability – the asymptotic collapsed fraction – for any universe which is unbound and, hence, will expand forever. We solve this problem by application of a simple, pressure-free, spherically symmetric, nonlinear model for the growth of density fluctuations in the universe. This model includes general kinds of Friedmann universes, such as the open, matter-dominated universe and those in which there is an extra, uniform background component of energy-density (e.g. the cosmological constant or so-called “quintessence”), perturbed by Gaussian random noise matter-density fluctuations. These background universes all have the property that matter-domination eventually gives way either to curvature-domination or domination by the positive energy density of the additional background component. When this happens, gravitational instability is suppressed and, with it, so is the growth of the collapsed fraction. Our results serve to identify a limitation of the well-known Press-Schechter approximation for the time-dependent mass function of cosmological structure formation. In the latter approximation, the mass function determined from the predicted collapse of positive density fluctuations is multiplied by an ad hoc correction factor of 2 based upon an assumption that every positive density fluctuation which is fated to collapse will simultaneously accrete an equal share of additional matter from nearby regions of compensating negative density fluctuation. The model presented here explicitly determines the actual value of the factor by which any positive density fluctuation which ever collapses will asymptotically increase its mass by accreting from a compensat-