1. Primordial Black Hole Formation
Taitung Winter School 2008
Primordial Black Hole Formation
Misao Sasaki
Yukawa Institute
Kyoto University

2. How do PBHs form? Astrophysical BHs --- Stars with cannot become a BH.
How can a PBH form?
Friedmann equation
gravitational radius of matter within radius R
Hubble radius H-1
= gravitational radius of matter within Hubble radius

3. Causality forbids collapse on scales greater than Hubble horizon size.
BH cannot form on scales much smaller than Hubble radius
(unless perhaps in the matter-dominated universe; p=0)
Causality forbids collapse on scales greater than Hubble horizon size.
Thus BH can form only on scales slightly smaller than H-1
Total mass within Hubble radius

4. Estimate by linear theory
behavior of scalar perturbations
Growing adiabatic solutions for
comoving density perturbation D
Newton potential (Bardeen potential) Y=-F
comoving curvature perturbation Rc
for Universe with
On superhorizon scales, Rc=const.

5. D F For radiation-dominated universe (w=1/3) on superhorizon scales.
··· condition for BH formation
Carr ’75, ···, Green, Liddle, Malik & MS ‘04

6. PBH formation probability
Smoothed density contrast on scale R
W: window function
Probability distribution of D(R) (for Gaussian random fluctuations)
variance:
power spectrum:
k-space window fcn :
(for Gaussian window)

7. Dth : threshold for BH formation
For Gaussian random fluctuations of D, the fraction of
the universe that forms a PBH with mass > M is
Dth : threshold for BH formation
Dth≃ Fth(0) ~ 0.3
BH with mass M forms on scale R=H-1 when M=MH=(2GH)-1
at radiation-dominated stage
PBH density behaves as a-3. Thus WPBH increases as a until
the matter-radiation equal time, a=aeq. This gives rise to a factor
where

8. PBH density parameter today with mass M ~ M + dM
N.B. PBHs with mass < Mevap=1015g would have evaporated
through Hawking radiation by now.
The value of Mevap may become larger/smaller in the
braneworld scenario.
(Evaporation may proceed faster/slower )

9. Observational constraints
Carr, Gilbert & Lidsey ‘94
Mevap

10. Caveats Non-linearity?
Only spherically symmetric simulations have been done so far.
Shibata & MS ’99, Musco, Miller & Rezzolla ‘04
3D GR simulations are needed.
Non-Gaussianity?
Effect of nonlinear evolution → need non-linear simulations
Primordial non-Gaussianity ··· highly model-dependent
Extra dimensions?
Large extra dimensions, warped extra dimensions
cf. Sendouda, Nagataki & Sato ‘06
PBH formation on subhorizon scales?
Quantum vs classical fluctuations
cf. Lyth, Malik MS & Zaballa ‘06