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 δ _pk 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 σ _H ≤ 10^-3 with σ _H 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 σ _H > 10^-3. Since a̅ _* can be written as a function of ν := δ _pk/σ _H, 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 σ _H 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 σ _H = 0.1, P(a̅ _*) takes a maximum at aa̅ _* ≃ 0.25.

中文翻译：

---

基于峰值理论的物质主导时代重新审视原始黑洞的自旋


我们估计了在宇宙中早期物质主导时代形成的原始黑洞 （PBH） 自旋的概率分布。我们采用 Zel'dovich 近似，并专注于导致潮汐扭矩的宇宙学扰动的线性顺序效应。假设波动服从高斯统计，我们应用随机高斯变量的峰值理论来计算均方根 （RMS） 和无量纲 Kerr 参数 a_* 在其形成时的概率分布。a_* 的值是通过周转时间的角动量来计算的。我们发现，具有给定波动幅度的 RMS a̅_* δ_pk 随着幅度的增加而减小。这种行为允许我们设置在极值下 a̅_* < 1 下的波动幅度的阈值。然后，我们讨论了自旋和各向异性塌陷对 PBHs 生成速率的影响。我们发现，对于σ_H ≤ ^10-3，σ _H 是水平再入时波动方差的平方根，自旋效应的抑制占主导地位，而各向异性效应对_{σ H} > ^10-3 变得更加重要。由于 a̅_* 可以写为 ν ：= δ_pk/σ_H 的函数，因此我们可以通过 ν 的概率分布获得 a̅_*， P（a̅_*），其特征是波动的给定功率谱。P（a̅_*） 取决于_{σ H} 和表征功率谱宽度的参数 γ。 结果表明，在我们感兴趣的参数区域中，与辐射主导宇宙中 PBH 形成相比，预计 PBH 自旋的可观值。例如，当 γ = 0.6 且 σ_H = 0.1 时，P（a̅_*） 在 aa̅_* ≃ 0.25 处达到最大值。