Spins of primordial black holes formed in the matter-dominated phase of the Universe

Angular momentum plays very important roles in the formation of primordial black holes in the matter-dominated phase of the Universe if it lasts sufficiently long. In fact, most collapsing masses are bounced back due to centrifugal force, since angular momentum significantly grows before collapse. For masses with q≤qc≃2.4I1/3σH1/3, where q is a nondimensional initial quadrupole moment parameter, σH is the density fluctuation at horizon entry t=tH, and I is a parameter of the order of unity, angular momentum gives a suppression factor ∼exp(−0.15I4/3σH−2/3) to the production rate. As for masses with q>qc, the suppression factor is even stronger as ∼exp(−0.0046q4/σH2). We derive the spin distribution of primordial black holes and find that most of the primordial black holes are rapidly rotating near the extreme value a*=1, where a* is the nondimensional Kerr parameter at their formation. The smaller σH is, the stronger the tendency towards the extreme rotation. Combining this result with the effect of anisotropy, we numerically and semianalytically estimate the production rate β0 of primordial black holes. Then we find that β0≃1.9×10−6fq(qc)I6σH2exp(−0.15I4/3σH−2/3) for σH≲0.005, while β0≃0.05556σH5 for 0.005≲σH≲0.2, where fq(qc) is the fraction of masses whose q is smaller than qc and we assume fq(qc) is not too small. We argue that matter domination significantly enhances the production of primordial black holes despite the suppression factor. If the end time tend of the matter-dominated phase satisfies tend≲(0.4IσH)−1tH, the effect of the finite duration significantly suppresses primordial black hole formation and weakens the tendency towards large spins.