Cosmological long-wavelength solutions and primordial black hole formation

TL;DR

This paper develops a general framework for long-wavelength solutions in cosmology, applies it to primordial black hole formation, and derives analytical and numerical results on density thresholds and curvature variables.

Contribution

It introduces a gauge-compatible long-wavelength solution framework and compares different slicing approaches for PBH formation, providing new analytical formulas and numerical insights.

Findings

Larger density thresholds $ ilde{oldsymbol{ ilde{ ext{delta}}}}_c$ for sharper transitions.

Derived analytical formulas for minimum and maximum $ ilde{oldsymbol{ ilde{ ext{delta}}}}_c$ based on the equation of state.

Environmental effects influence the threshold curvature variable $oldsymbol{oldsymbol{ ext{psi}}}_{0,c}$.

Abstract

We construct cosmological long-wavelength solutions without symmetry in general gauge conditions compatible with the long-wavelength scheme. We then specify the relationship among the solutions in different time slicings. Applying this general framework to spherical symmetry, we derive the correspondence relation between long-wavelength solutions in the constant mean curvature slicing with conformally flat spatial coordinates and asymptotic quasihomogeneous solutions in the comoving gauge and compare the numerical results of PBH formation in these two different approaches. To discuss the PBH formation, it is convenient and conventional to use $\tilde{\delta}_{c}$, the value which the averaged density perturbation at threshold in the comoving slicing would take at horizon entry in the lowest-order long-wavelength expansion. We numerically find that within compensated models, the sharper the transition from the overdense region to the FRW universe is, the larger the $\tilde{\delta}_{c}$ becomes. We suggest that, for the equation of state $p=(\Gamma-1)\rho$, we can apply the analytic formulas for the minimum $\tilde{\delta}_{c, {\rm min}}\simeq [3\Gamma/(3\Gamma+2)]\sin^{2}\left[\pi\sqrt{\Gamma-1}/(3\Gamma-2)\right]$ and the maximum $\tilde{\delta}_{c, {\rm max}}\simeq 3\Gamma/(3\Gamma+2)$. As for the threshold peak value of the curvature variable $\psi_{0,c}$, we find that the sharper the transition is, the smaller the $\psi_{0,c}$ becomes. We analytically explain this feature. Using simplified models, we also analytically deduce an environmental effect that $\psi_{0,c}$ can be significantly larger (smaller) if the underlying density perturbation of much longer wavelength is positive (negative).