Separate universe problem: 40 years on

The claim that an overdense (positive-curvature) region in the early Universe cannot extend beyond some maximum scale and remain part of our Universe, first made 40 years ago, has recently been questioned by Kopp et al. Their analysis is elucidating and demonstrates that one cannot constrain the form of primordial density perturbations using this argument. However, the notion of a separate universe scale still applies and there is also an upper limit on the scale of a region collapsing to a black hole at any epoch, these scales being simply related. We calculate these scales for equations of state of the form p = k rho c(2) with -1 < k < infinity, refining earlier calculations on account of the Kopp et al. criticisms. For -1/3 < k < infinity, the scale is always of the order of the cosmological particle horizon size, with a numerical factor depending on k. This confirms the earlier claim that a primordial black hole cannot be much larger than the particle horizon at formation. For -1 < k < -1/3, as expected for some periods in the history of the Universe, the situation changes radically, in that a sufficiently large positive-curvature region produces a baby universe rather than a black hole. There is still a separate universe scale but the interpretation of these solutions requires care.