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

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

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

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.

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

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)

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

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 )

Observational constraints Carr, Gilbert & Lidsey ‘94 Mevap

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