Separate Universes Do Not Constrain Primordial Black Hole Formation

Abstract

Hawking and Carr 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 a maximum 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 and the Hamiltonian constraint. Using instead of density a volume fluctuation variable reveals that a fluctuation is a separate universe iff this variable diverges on superhorizon scales. Hence density fluctuations can never form or be separate universes and Hawking_s14s and Carr_s14s condition does not pose a physical constraint on density fluctuations. Primordial black hole formation with an initial fluctuation amplitude larger than the one corresponding to the maximum density fluctuation amplitude was previously not considered 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.