Taitung Winter School 2008 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

• 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.

For radiation-dominated universe (w=1/3) on superhorizon scales. D F ··· 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)

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=(2 GH)-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 + d. M N. B. PBHs with mass < Mevap=1015 g 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 3 D 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