Extreme-value statistics of the spin of primordial black holes

Abstract

How rare are extreme-spin primordial black holes? We show how, from an underlying distribution of primordial black hole (PBH) spin, extreme-value statistics can be used to quantify the rarity of spinning PBHs with Kerr parameter close to 1. Using the peaks-over-threshold method, we show how the probability that a PBH forms with spin exceeding a sufficiently high threshold can be calculated using the generalized Pareto distribution. This allows us to estimate the average number of PBHs amongst which we can find a single PBH that formed with spin exceeding a high threshold. We found that the primordial spin distribution gives rise to exceedingly rare near-extremal spin PBHs at formation time: for typical parameter values, roughly up to one in a hundred million PBHs would be formed with spin exceeding the Thorne limit. We discuss conditions under which even more extreme-spin PBHs may be produced, including modifying the skewness and kurtosis of the spin distribution via a smooth transformation. We deduce from our calculations that, if indeed asteroid-mass PBHs above the current observational limit on evaporating PBHs of mass $\ensuremath{\sim}{10}^{17}\text{ }\text{ }\mathrm{g}$ contribute significantly to the dark matter, it is very likely that some of them at somewhat lower masses could be long-lived near-extremal PBHs.