Cosmological long-wavelength solutions and primordial black hole formation

Abstract

We construct cosmological long-wavelength solutions without symmetry in general gauge conditions which are compatible with the long-wavelength scheme. We then specify the relationship among the solutions in different time slicings. Nonspherical long-wavelength solutions are particularly important for primordial structure formation in the epoch of very soft equations of state. Applying this general framework to spherical symmetry, we show the equivalence between longwavelength solutions in the constant mean curvature slicing with conformally flat spatial coordinates and asymptotic quasihomogeneous solutions in the comoving slicing with the comoving threading. We derive the correspondence relation between these two solutions and compare the results of numerical simulations of primordial black hole (PBH) formation in these two different approaches. To discuss the PBH formation, it is convenient and conventional to useδc, the value which the averaged density perturbation at threshold in the comoving slicing would take at horizon entry in the lowestorder long-wavelength expansion. We numerically find that within (approximately) compensated models, the sharper the transition from the overdense region to the Friedmann-Robertson-Walker universe is, the larger theδc becomes. We suggest that, for the equation of state p = (Γ − 1)ρ, we can apply the analytic formulas for the minimumδc,min ≃ [3Γ/(3Γ+2)] sin 2 π √ Γ − 1/(3Γ − 2) and the maximumδc,max ≃ 3Γ/(3Γ + 2). As for the threshold peak value of the curvature variable ψ0,c, we find that the sharper the transition is, the smaller the ψ0,c becomes. We analytically explain this intriguing feature qualitatively with a compensated top-hat density model. Using simplified models, we also analytically deduce an environmental effect that ψ0,c can be significantly larger (smaller) if the underlying density perturbation of much longer wavelength is positive (negative).

Spins of primordial black holes formed in the matter-dominated phase of the Universe

“…The relation betweenδ and the peak value of ψ (or ζ) is not one to one but largely profile dependent. In fact, δ th and ψ th even show opposite behaviors on the sharpness of the transition between the overdense region and the flat FriedmannLemaître-Robertson-Walker (FLRW) exterior, as can be seen in Tables I and II and Figs. 2 and 3 in [20]. This suggests that the black hole threshold is profile dependent because of the complexity of gravitational collapse against the pressure gradient force.…”

“…By recent numerical relativity simulations in spherical symmetry [20], the black hole threshold in the radiation-dominated era has been found and is ψ th ≃ 1.40 − 1.69 in terms of the peak value of ψ, which is equivalent to |ζ th | ≃ 0.67 − 1.05 in terms of the peak value through Eq. (5.12), depending on the profile of the perturbation.…”

“…In the lowest and second lowest orders of the long-wavelength limit, ψ is time independent and of the order of unity. The density perturbation δ in the comoving slicing and the ψ are related to each other by the following relation [20]: is time independent, where r 0 is the comoving length scale of perturbation and is identified with r 0 in the previous sections. In general relativistic numerical simulations, r 0 has been chosen as the radius of the boundary of the overdense region.…”

“…In this phase, there is a threshold δ th of black hole formation, which is governed by pressure gradient force, and the production rate of black holes is given by ∼ (δ th /σ H ) exp[−δ 2 th /(2σ 2 H )], where σ H is density fluctuation at horizon entry. The threshold δ th of density perturbation was originally estimated to ∼ 1/3 [13] and recently to ∼ 0.42 − 0.56 for relatively gentler profiles of density field [14][15][16][17][18][19][20]. On the other hand, black hole formation in a matter-dominated phase is not yet studied so much.…”

Primordial Black Holes as a Novel Probe of Primordial Gravitational Waves

“…(See also [17,18] for updated discussions of the formation condition and see also [19][20][21][22][23][24][25] for numerical simulations of the formation of PBHs.) There is no conclusive evidence for the existence of PBHs and so upper bounds on the abundance of PBHs on various mass scales have been obtained by various kinds of observations (see e.g.…”