Simulations of ellipsoidal primordial black hole formation

We perform $3+1$ relativistic numerical simulations to study primordial black hole (PBH) formation from the collapse of adiabatic superhorizon nonspherical perturbations generated from curvature fluctuations obeying random Gaussian statistics with a monochromatic power spectrum. The matter field is assumed to be a perfect fluid of an equation of state $w\ensuremath{\mathrel{:=}}P/\ensuremath{\rho}=\mathrm{const}.$ with $P$ and $\ensuremath{\rho}$ being the pressure and the energy density, respectively. The initial spatial profile of the curvature perturbation is modeled with the amplitude $\ensuremath{\mu}$ and nonspherical parameters $e$ (ellipticity) and $p$ (prolateness) according to peak theory. We focus on the dynamics and the threshold for PBH formation in terms of the nonspherical parameters $e$ and $p$. We find that the critical values (${e}_{c}$, ${p}_{c}$) with a fixed value of $\ensuremath{\mu}$ closely follow a superellipse curve. With $p=0$, for the range of amplitudes considered, we find that the critical ellipticity for nonspherical collapse follows a decaying power law as a function of $(\ensuremath{\mu}\ensuremath{-}{\ensuremath{\mu}}_{\mathrm{c},\mathrm{sp}})$ with ${\ensuremath{\mu}}_{\mathrm{c},\mathrm{sp}}$ being the threshold for the spherical case. Our results also indicate that, for both cases of $w=1/3$ and $w=1/10$, small deviations from sphericity can avoid collapsing to a black hole when the amplitude is near its critical threshold. Finally we discuss the significance of the ellipticity on the rate of the PBH production.