http://arxiv.org/abs/1707.03595
Angular momentum plays most important roles in the formation of PBHs in the matter-dominated phase if it lasts sufficiently long. In fact, most of collapsing masses are bounced back due to centrifugal force. For masses with $q\le q_{c}\simeq 1.7 {\cal I}^{1/3}\sigma_{H}^{1/3}$, where $q$ is a nondimensional initial quadrupole moment parameter, $\sigma_{H}$ is density fluctuation at horizon entry $t=t_{H}$, and ${\cal I}$ is a parameter of the order of unity, angular momentum gives a suppression factor $\sim \exp(-0.15 {\cal I}^{4/3} \sigma_{H}^{-2/3})$ to the production rate. As for masses with $q> q_{c}$, the suppression factor is even stronger. We find that most of the PBHs are rapidly rotating near the extreme value $a_{}=1$, where $a_{}$ is the nondimensional Kerr parameter at their formation. The smaller the $\sigma_{H}$ is, the stronger the tendency towards the extreme rotation is. Combining this result with the effect of anisotropy, we estimate the production rate $\beta_{0}$. Then we find that if $q$ is distributed around $0$ and if $\sigma_{H}\lesssim 0.005$, $\beta_{0}\simeq 1.9 \times 10^{-6}f_{q}(q_{c}){\cal I}^{6}\sigma_{H}^{2}\exp(-0.15 {\cal I}^{4/3}\sigma_{H}^{-2/3})$ with $f_{q}(q_{c})$ the fraction of masses whose $q$ is smaller than $q_{c}$, while, if $0.005\lesssim\sigma_{H}\lesssim 0.2$, $\beta_{0}\simeq 0.05556\sigma_{H}^{5}$. We argue that matter-domination significantly enhances the production of PBHs despite the suppression factor. If the end time $t_{{\rm end}}$ of the matter-dominated phase satisfies $t_{{\rm end}}\lesssim (0.4 {\cal I}\sigma_{H})^{-1}t_{H}$, the effect of the finite duration significantly suppresses PBH formation and weakens the tendency towards large spins.
T. Harada, C. Yoo, K. Kohri, et. al.
Fri, 14 Jul 17
29/55
Comments: 34 pages, 5 figures