The 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 it places an important upper limit on the mass of primordial black holes forming at any epoch. We calculate this scale for equations of state of the form $p = k \rho c^2$ with $-1 <k < \infty$, refining earlier calculations on account of the Kopp et al. criticisms. For $-1/3 < k < \infty$, the scale is always of order 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.