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.
