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 long-wavelength 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 use ${\stackrel{\texttildelow{}}{\ensuremath{\delta}}}_{c}$, 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 (approximately) compensated models, the sharper the transition from the overdense region to the Friedmann-Robertson-Walker universe is, the larger the ${\stackrel{\texttildelow{}}{\ensuremath{\delta}}}_{c}$ becomes. We suggest that, for the equation of state $p=(\mathrm{\ensuremath{\Gamma}}\ensuremath{-}1)\ensuremath{\rho}$, we can apply the analytic formulas for the minimum ${\stackrel{\texttildelow{}}{\ensuremath{\delta}}}_{c,\mathrm{min}}\ensuremath{\simeq}[3\mathrm{\ensuremath{\Gamma}}/(3\mathrm{\ensuremath{\Gamma}}+2)]{\mathrm{sin}}^{2}[\ensuremath{\pi}\sqrt{\mathrm{\ensuremath{\Gamma}}\ensuremath{-}1}/(3\mathrm{\ensuremath{\Gamma}}\ensuremath{-}2)]$ and the maximum ${\stackrel{\texttildelow{}}{\ensuremath{\delta}}}_{c,\mathrm{max}}\ensuremath{\simeq}3\mathrm{\ensuremath{\Gamma}}/(3\mathrm{\ensuremath{\Gamma}}+2)$. As for the threshold peak value of the curvature variable ${\ensuremath{\psi}}_{0,c}$, we find that the sharper the transition is, the smaller the ${\ensuremath{\psi}}_{0,c}$ 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 ${\ensuremath{\psi}}_{0,c}$ can be significantly larger (smaller) if the underlying perturbation of much longer wavelength is positive (negative).