Revisiting spins of primordial black holes in a matter-dominated era based on peak theory

Abstract

We estimate the probability distribution for the spins of the primordial black holes (PBHs) that formed during an early matter-dominated era in the Universe. We employ the Zel'dovich approximation and focus on the linear-order effect of cosmological perturbations which causes the tidal torque. Assuming that the fluctuations obey Gaussian statistics, we apply the peak theory of random Gaussian variables to compute the root mean square (RMS) and the probability distribution of the non-dimensional Kerr parameter a* at their formation. The value of a* is evaluated through the angular momentum at the turn-around time. We find that the RMS a̅* with a given amplitude of the fluctuation δpk decreases as the amplitude increases. This behavior allows us to set the threshold value of the amplitude of the fluctuation through the under-extremal condition a̅* < 1. Then we discuss the impact of spin and anisotropic collapse on the production rate of PBHs. We find that, for σH ≤ 10-3 with σH being the square root of the variance of the fluctuation at the horizon reentry, the suppression from the spin effect is dominant, while the effect of anisotropy becomes more important for σH > 10-3. Since a̅* can be written as a function of ν := δpk/σH, we can obtain the probability distribution of a̅*, P(a̅*), through the probability distribution of ν characterized by a given power spectrum of the fluctuation. P(a̅*) depends on σH and the parameter γ that characterizes the width of the power spectrum. It is shown that, in the parameter regions of our interests, substantial values of PBH spins are expected in contrast to the PBH formation in a radiation-dominated universe. For instance, with γ = 0.6 and σH = 0.1, P(a̅*) takes a maximum at aa̅* ≃ 0.25.