Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 120
6 PBH formation
6.1 The condition for PBH formation
The collapse of an overdense region, forming a BH, is possible only if the root
mean square of the primordial fluctuations, averaged over a Hubble volume, is
larger than a threshold δmin. There is also an upper bound δmax corresponding
to the case for which a separate Universe will form. Thus a PBH will form when
the density contrast δ, averaged over a volume of the linear size of the Hubble
radius, satisfies (Carr, 1975)
δmin ≤ δ ≤ δmax. (244)
The lower and upper bounds of δ can be determined following analytic arguments.
Consider, for simplicity, a spherically–symmetric region with radius R
and density ρ = ρc + δρ embedded in a flat Universe with the critical density
ρc. Within spherical symmetry the inner region is not affected by matter in
the surrounding part of the Universe. The expansion of this region will come
to an halt, at some stage, followed by a collapse. In order to reach a complete
collapse, the potential energy, V , at the time of maximal expansion (e.g. Kiefer,
2003)
2 GM
V ∼
R ∼ Gρ2R 5
has to exceed the inner energy, U, given by (e.g. Kiefer, 2003)
U ∼ pR 3
(245)
(246)
where p is the pressure. In the radiation–dominated era (which is the era of
interest for PBH formation) an overdense region will collapse to a BH provided
that the size of the region, when it stops expanding, is bigger than the Jeans
Length 31 RJ (e.g. Kiefer, 2003)
1
R ≥ RJ = . (247)
3Gρ
In order to prevent the formation of a separate Universe we must ensure, also,
that the radius of the collapsing region, R, is smaller than the curvature radius
of the overdense region at the moment of collapse (e.g. Kiefer, 2003)
R< 1
√ Gρ . (248)
One then has the condition
1
1 >R≥ , (249)
3
31 The Jeans Length is the critical radius of a region where thermal energy, which causes
the region to expand, is counteracted by gravity; this causes the region to collapse.