Spin of Primordial Black Holes

Primordial black holes, formed from rare peaks in the primordial fluctuations $\zeta$, are non-rotating at zeroth order in ${\zeta }_{\mathrm{r}\mathrm{m}\mathrm{s}}$. We show that the spin also vanishes at first order in ${\zeta }_{\mathrm{r}\mathrm{m}\mathrm{s}}$, suggesting the dimensionless spin parameter $a_{\mathrm{r}\mathrm{m}\mathrm{s}}\sim {\zeta }_{\mathrm{r}\mathrm{m}\mathrm{s}}^2$. We identify one quadratic contribution to the spin by calculating (and extrapolating to the formation time) the torque on a black hole due to ambient acoustic waves. For a reasonable density of primordial black holes this implies a percent level spin parameter.