Carr and Hawking showed that the proper size of a spherical overdense region
surrounded by a flat FRW universe cannot be arbitrarily large as otherwise the
region would close up on itself and become a separate universe. From this
result they derived a condition connecting size and density of the overdense
region ensuring that it is part of our universe. Carr used this condition to
obtain an upper bound for the density fluctuation amplitude with the property
that for smaller amplitudes the formation of a primordial black hole is
possible, while larger ones indicate a separate universe. In contrast, we find
that the appearance of a maximum is not a consequence of avoiding separate
universes but arises naturally from the geometry of the chosen slicing. Using
instead of density a volume fluctuation variable reveals that a fluctuation is
a separate universe iff this variable diverges on superhorizon scales. Hence
Carr's and Hawking's condition does not pose a physical constraint on density
fluctuations. The dynamics of primordial black hole formation with an initial
curvature fluctuation amplitude larger than the one corresponding to the
maximum density fluctuation amplitude was previously not considered in detail
and so we compare it to the well-known case where the amplitude is smaller by
presenting embedding and conformal diagrams of both types in dust spacetimes.Comment: Updated version corresponds to the published version
10.1103/PhysRevD.83.124025, 22 pages, 22 figure
