Spin of Primordial Black Holes from Broad Power Spectrum: Radiation Dominated Universe

We perform a study to obtain the initial spin or the nondimensional Kerr parameter $a_{*}$ of primordial black holes (PBHs) created during the radiation dominated phase of the universe from not only nearly monochromatic but also broad curvature power spectra. Motivated by inflation and first-order phase transitions, we consider a power law shape for the curvature perturbation. Although we can naturally neglect the contribution from the length scales smaller than the scale of interest, that from the larger scales may potentially be significant for a broad power spectrum, for which the spin is sensitive to the width of the power spectrum. So, we introduce a width parameter $r_{k}$, the ratio of the largest scale to the length of interest. We find that the root mean square of $a_{*}$ is largest for PBHs created from locally nearly scale invariant curvature power spectra with $r_{k} \sim 3.5$. The upper limit is $\sim 1\times 10^{-4}$ for $M=10^{17}-10^{23}$ g, $\sim 1.7\times 10^{-4}$ for $M=1-100 M_{\odot}$ and $\sim 2.5\times 10^{-3}$ even for an incredibly large mass of $M=10^{50}$ g.