Abstract. The abundance of cosmic voids can be described by an analogue of halo mass functions for galaxy clusters. In this work, we explore a number of void mass functions: from those based on excursion-set theory to new mass functions obtained by modifying halo mass functions. We show how different void mass functions vary in their predictions for the largest void expected in an observational volume, and compare those predictions to observational data. Our extreme-value formalism is shown to be a new practical tool for testing void theories against simulation and observation.

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 ∼10 17 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.

We present a formalism for calculating the probability distribution of the most massive primordial black holes (PBHs) expected within an observational volume. We show how current observational upper bounds on the fraction of PBHs in dark matter translate to constraints on extreme masses of primordial black holes. We demonstrate the power of our formalism via a case study, and argue that our formalism can be used to produce extreme-value distributions for a wide range of PBH formation theories.