Spins of primordial black holes formed in the radiation-dominated phase of the universe: first-order effect.

The standard deviation of the initial values of the nondimensional Kerr parameter $a_{*}$ of primordial black holes (PBHs) formed in the radiation-dominated phase of the universe is estimated to the first order of perturbation for the narrow power spectrum. The obtained expression is $\sqrt{\langle a_{*}^{2} \rangle} \sim 6.5 \times 10^{-4} (M/M_{H})^{-1/3}\sqrt{1-\gamma^{2}}[1-0.072 \log_{10}(\beta_{0}(M_{H})/(1.3\times 10^{-15}))]^{-1}$, where $M_{H}$, $\beta_{0}(M_{H})$, and $\gamma$ are the mass within the Hubble horizon at the horizon entry of the overdense region, the fraction of the universe which collapsed to PBHs at the scale of $M_{H}$, and a parameter which characterizes the width of the power spectrum, respectively. This implies that for $M\simeq M_{H}$, the higher the probability of the PBH formation, the larger the standard deviation of the spins. PBHs of $M\ll M_{H}$ formed through near-critical collapse may have larger spins than those of $M\simeq M_{H}$. In comparison to the previous estimate by De Luca et al. (2019) [arXiv:1903.01179], the new estimate removes an incorrect overall factor $\Omega_{\rm dm}$, where $\Omega_{\rm dm}$ is the current ratio of dark matter to the critical density, and numerically gives a smaller value by one order of magnitude approximately. On the other hand, it suggests that the first-order effect can be numerically comparable to or smaller than the second-order one.