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

Abstract

Angular momentum plays very important roles in the formation of primordial black holes in the matter-dominated phase of the Universe if it lasts sufficiently long. In fact, most collapsing masses are bounced back due to centrifugal force, since angular momentum significantly grows before collapse. For masses with $q\ensuremath{\le}{q}_{c}\ensuremath{\simeq}2.4{\mathcal{I}}^{1/3}{\ensuremath{\sigma}}_{H}^{1/3}$, where $q$ is a nondimensional initial quadrupole moment parameter, ${\ensuremath{\sigma}}_{H}$ is the density fluctuation at horizon entry $t={t}_{H}$, and $\mathcal{I}$ is a parameter of the order of unity, angular momentum gives a suppression factor $\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}0.15{\mathcal{I}}^{4/3}{\ensuremath{\sigma}}_{H}^{\ensuremath{-}2/3})$ to the production rate. As for masses with $q>{q}_{c}$, the suppression factor is even stronger as $\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}0.0046{q}^{4}/{\ensuremath{\sigma}}_{H}^{2})$. We derive the spin distribution of primordial black holes and find that most of the primordial black holes are rapidly rotating near the extreme value ${a}_{*}=1$, where ${a}_{*}$ is the nondimensional Kerr parameter at their formation. The smaller ${\ensuremath{\sigma}}_{H}$ is, the stronger the tendency towards the extreme rotation. Combining this result with the effect of anisotropy, we numerically and semianalytically estimate the production rate ${\ensuremath{\beta}}_{0}$ of primordial black holes. Then we find that ${\ensuremath{\beta}}_{0}\ensuremath{\simeq}1.9\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}{f}_{q}({q}_{c}){\mathcal{I}}^{6}{\ensuremath{\sigma}}_{H}^{2}\mathrm{exp}(\ensuremath{-}0.15{\mathcal{I}}^{4/3}{\ensuremath{\sigma}}_{H}^{\ensuremath{-}2/3})$ for ${\ensuremath{\sigma}}_{H}\ensuremath{\lesssim}0.005$, while ${\ensuremath{\beta}}_{0}\ensuremath{\simeq}0.05556{\ensuremath{\sigma}}_{H}^{5}$ for $0.005\ensuremath{\lesssim}{\ensuremath{\sigma}}_{H}\ensuremath{\lesssim}0.2$, where ${f}_{q}({q}_{c})$ is the fraction of masses whose $q$ is smaller than ${q}_{c}$ and we assume ${f}_{q}({q}_{c})$ is not too small. We argue that matter domination significantly enhances the production of primordial black holes despite the suppression factor. If the end time ${t}_{\mathrm{end}}$ of the matter-dominated phase satisfies ${t}_{\mathrm{end}}\ensuremath{\lesssim}(0.4\mathcal{I}{\ensuremath{\sigma}}_{H}{)}^{\ensuremath{-}1}{t}_{H}$, the effect of the finite duration significantly suppresses primordial black hole formation and weakens the tendency towards large spins.