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. Evaluating the angular momentum at turn around based on linearly extrapolated transfer functions and peak theory, we obtain the expression $\sqrt{\langle a_{*}^{2} \rangle} \simeq 4.0\times 10^{-3} (M/M_{H})^{-1/3}\sqrt{1-\gamma^{2}}[1-0.072 \log_{10}(\beta_{0}(M_{H})/(1.3\times 10^{-15}))]^{-1}$, 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 quantity 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, while 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, the new estimate has the explicit dependence on the ratio $M/M_{\rm H}$ and no direct dependence on the current dark matter density. On the other hand, it suggests that the first-order effect can be numerically comparable to the second-order one.