Extreme-Value Statistics of the Spin of Primordial Black Holes

How rare are extreme-spin primordial black holes? We show how, from an underlying distribution of 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 Generalised Pareto Distribution. This allows us to estimate the average number of PBHs amongst which we can find a single PBH which 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 ~10^{17} g contribute significantly to the dark matter, it is likely that some of them could be near-extremal PBHs.