Introduction Analytic threshold formula Primordial fluctuations Numerical simulations Conclusion . . Primordial black hole formation from cosmological fluctuations . . . . . Tomohiro Harada Department of Physics, Rikkyo University, Tokyo, Japan 11/08/2015 HTGRG2 @ ICISE, Quy Nhon This talk is based on Harada, Yoo, Nakama and Koga, arXiv:1503.03934 Harada, Yoo and Kohri, arxiv:1309.4201. . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula Primordial fluctuations Numerical simulations Conclusion . Primordial black holes and cosmology Black holes may have formed in the early universe. (Zeldovich & Novikov 1967, Hawking 1971) PBHs the source of emission due to Hawking radiation and the source of gravitational field and gravitational waves Hawking radiation: nearly black-body radiation � c 3 dE dt = − dM R g = 2 GM dt = g eff 4 π R 2 g σ T 4 T H = , H , c 2 8 π GMk B Mass accretion: important only immediately after the formation . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula Primordial fluctuations Numerical simulations Conclusion . Observational constraints on the PBH abundance PBHs of mass M is formed at the epoch when the mass contained within the Hubble length is M . Observational data can constrain the abundance of PBHs. Complementary to CMB observation. (Carr (1975), Carr et al. (2010)) . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula Primordial fluctuations Numerical simulations Conclusion . Production rate of PBHs PBHs of mass M are formed from δ ( M ) , the density perturbation of mass M . A Gaussian-like probability distribution for δ ( M ) (Carr (1975), cf. Kopp, Hofmann and Weller (2011)) PBH production rate √ δ 2 2 σ ( M ) ( ) c β 0 ( M ) δ c ( M ) exp , ≃ − 2 σ 2 ( M ) π where δ c = O ( 1 ) is the PBH threshold of δ and σ is the standard deviation of δ . . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula Primordial fluctuations Numerical simulations Conclusion . Contents . . . Analytic threshold formula 1 . . . Primordial fluctuations 2 . . . Numerical simulations 3 EOS dependence Profile dependence . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula Primordial fluctuations Numerical simulations Conclusion . Dynamics in PBH formation . . . Fluctuations in super-horizon scales are generated by 1 inflation. . . . The fluctuations enter the Hubble horizon in the 2 decelerated phase of the univese. . . . The Jeans instability sets in and the fluctuation collapses if 3 its amplitude is nonlinearly large enough. . . . A black hole apparent horizon is formed. 4 . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula Primordial fluctuations Numerical simulations Conclusion . 3-zone model Simplified model for analytic approach Background: a flat FRW ds 2 = − c 2 dt 2 + a 2 b ( t )( dr 2 + r 2 d Ω 2 ) for r > r b Overdense region: a closed FRW ds 2 = − c 2 dt 2 + a 2 ( t )( d χ 2 + sin 2 χ d Ω 2 ) for 0 ≤ χ < χ a . Compensating region in between . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula Primordial fluctuations Numerical simulations Conclusion . Jeans instability argument t ff (free-fall time) versus t sc (sound-crossing time) . Jeans criterion . . . If and only if the overdensity reaches maximum expansion before a sound wave crosses over its radius from the big bang, it collapses to a black hole. Equivalently, if and only if the overdense region ends in singularity before a sound wave crosses its diameter from the big bang, it collapses to a black hole. . . . . . . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula Primordial fluctuations Numerical simulations Conclusion . Analytic formula We assume an EOS p = (Γ − 1 ) ρ for simplicity. We define ˜ δ as the density perturbation at the horizon entry in the comoving slicing. Γ = 4 / 3 for radiation. Carr’s formula (1975): partially Newtonian estimate 3 Γ ˜ δ c = 3 Γ + 2 (Γ − 1 ) , or ˜ δ CMC , c = Γ − 1 in the constant-mean-curvature slicing. Harada, Yoo and Kohri (2013): fully GR √ ( π ) 3 Γ Γ − 1 3 Γ ˜ ˜ 3 Γ + 2 sin 2 δ c = , δ max = 3 Γ + 2 . 3 Γ − 2 . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula Primordial fluctuations Numerical simulations Conclusion . Contents . . . Analytic threshold formula 1 . . . Primordial fluctuations 2 . . . Numerical simulations 3 EOS dependence Profile dependence . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula Primordial fluctuations Numerical simulations Conclusion . Cosmological long-wavelength solutions Generic initial conditions for numerical simulations The specetime is assumed smooth in the scales larger than L = a / k , which is much longer than the local Hubble length H − 1 . By gradient expansion, the exact solution is expanded in powers of ϵ ∼ k / ( aH ) . . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula Primordial fluctuations Numerical simulations Conclusion . Construction of the CWLW solutions 3+1 and cosmological conformal decomposition ds 2 = − α 2 dt 2 + ψ 4 a 2 ( t )˜ γ ij ( dx i + β i dt )( dx j + β j dt ) , where ˜ γ = η , ˜ γ = det (˜ γ ij ) , η = det ( η ij ) , and η ij is the metric of the 3D flat space. We assume the spacetime approach the flat FRW in the limit ϵ → 0 (Lyth, Malik & Sasaki (2005)). Einstein eqs in O ( 1 ) imply the Friedmann eq. We can deduce ψ = Ψ( x i ) + O ( ϵ 2 ) for a perfect fluid with barotropic EOS. Ψ( x i ) generates the solution. . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula Primordial fluctuations Numerical simulations Conclusion . Two approaches to spherically symmetric system Shibata and Sasaki (1999) CMC slicing + conformally flat spatial coordinates ( ϖ ) Initial conditions: The CLWL soln is generated by Ψ( ϖ ) , where ψ = Ψ( ϖ ) + O ( ϵ 2 ) . Polnarev and Musco (2007) (and many others) Comoving slicing + comoving threading ( r ) Initial conditions: The metric is assumed to approach dr 2 [ ] 1 − K ( r ) r 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) ds 2 = − dt 2 + a 2 ( t ) in the limit ϵ → 0. The exact solution is expanded in powers of ϵ and generated by K ( r ) . . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula Primordial fluctuations Numerical simulations Conclusion . Equivalence of the two approaches The CLWL solutions and the Polnarev-Musco solutions are equivalent with each other through the relation r = Ψ 2 ( ϖ ) ϖ, ) 2 ( ϖ d Ψ( ϖ ) K ( r ) r 2 = 1 − 1 + 2 . Ψ( ϖ ) d ϖ One of the correspondence relations is given by 3 Γ 3 Γ + 2 δ CMC + O ( ϵ 4 ) . δ C = . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula EOS dependence Primordial fluctuations Profile dependence Numerical simulations Conclusion . Contents . . . Analytic threshold formula 1 . . . Primordial fluctuations 2 . . . Numerical simulations 3 EOS dependence Profile dependence . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula EOS dependence Primordial fluctuations Profile dependence Numerical simulations Conclusion . Amplitude of the perturbation ˜ δ : “the density perturbation at the horizon entry” ˜ ¯ δ C ( t , r 0 ) ϵ − 2 , δ := lim ϵ → 0 where ¯ δ C ( t , r 0 ) is the density in the comoving slicing averaged over r 0 , the radius of the overdense region. ˜ δ is directly calculated from Ψ( ϖ ) or K ( r ) . ψ 0 : the initial peak value of the curvature variable ψ 0 := Ψ( 0 ) Note ψ = Ψ( x i ) + O ( ϵ 2 ) . The PBH threshold is determined by numerical simulations. . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula EOS dependence Primordial fluctuations Profile dependence Numerical simulations Conclusion . Carr’s formula and numerical result 1 Musco & Miller (2012) Carr Gauged Carr 0.8 Maximum 0.6 ˜ δ 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 w Figure: The gauged Carr’s formula underestimates ˜ δ c by a factor of 2 for w (= Γ − 1 ) = 1 / 3 and by a factor of 10 for the smaller values of Γ − 1. The numerical result is taken from Musco and Miller (2013) for the Gaussian curvature profile. . . . . . . Harada PBH from fluctuations
Introduction Analytic threshold formula EOS dependence Primordial fluctuations Profile dependence Numerical simulations Conclusion . HYK formula and numerical result 1 Musco & Miller (2012) Our formula Carr 0.8 Gauged Carr Maximum 0.6 ˜ δ 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 w Figure: Harada-Yoo-Kohri formula agrees with the numerical result within 10 − 20 % accuracy for 0 . 01 ≤ Γ − 1 ≤ 0 . 6. . . . . . . Harada PBH from fluctuations
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