Non-spherical effects on the mass function of Primordial Black Holes

Abstract

In this letter, we investigate the impact of non-spherical effects on the Primordial Black Hole mass function, based on the ellipticity-dependent threshold calculated by performing $3+1$ relativistic numerical simulations. We consider an equation of state of radiation $w:=P/ρ=1/3$ and a softer one $w=1/10$ with $P$ and $ρ$ being the pressure and energy density, respectively. We suppose that the curvature perturbations obey Gaussian statistics with a monochromatic power spectrum and examine the most probable ellipsoidal configurations utilizing peak theory. We also suppose the critical scaling law of the PBH mass near the threshold following the known results. The simulations arXiv:2410.03452 show that the non-sphericity can easily prevent the system from black hole formation when the initial fluctuation amplitude is near the threshold (critical scaling regime). Nevertheless, we show that the non-spherical effects make the mass function just a few times smaller and are insignificant on the mass function distribution, including the power-law scaling in the small mass region.

Figures (3)

• Figure 1: Density plot of the PBH mass in the parameter space of $e$ and $\mu-\mu_{\rm c,sp}$. The dashed line indicates the mean value of $e$ with one sigma deviation specified by the dotted lines (see companion for the analytical equation). The solid colored lines indicate the lines of constant mass. The cross symbols indicate the cases with PBH formation (yellow) and non-formation (blue) from the numerical results obtained in companion. The case shown corresponds to radiation fluid $w=1/3$.
• Figure 2: Probability distribution of the prolateness $p$ and ellipticity $e$ of the shape fixing $\nu=8$ given by Eq. \ref{['eq:prob_e_p']}. For large peaks, only a small region in the parameter space $(e,p)$ has a non-negligible probability with $\langle e \rangle \approx 6 \cdot 10^{-2}, \langle p \rangle \approx 10^{-3}$.
• Figure 3: PBH mass function for the case $k_p$ in terms of $M_{k_p}$. The dashed line denotes the result assuming spherical symmetry, and the solid line shows the estimation taking into account the non-spherical configurations.