Threshold of Primordial Black Hole Formation against Velocity Dispersion in Matter-Dominated Era

We study the effects of velocity dispersion on the formation of primordial black holes~(PBHs) in a matter-dominated era. The velocity dispersion is generated through the nonlinear growth of perturbations and has the potential to impede the gravitational collapse and thereby the formation of PBHs. To make discussions clear, we consider two distinct length scales. The larger one is where gravitational collapse occurs which could lead to PBH formation, and the smaller one is where the velocity dispersion develops due to nonlinear interactions. We estimate the effect of the velocity dispersion on the PBH formation by comparing the free-fall timescale and the timescale for a particle to cross the collapsing region. As a demonstration, we consider a log-normal power spectrum for the initial density perturbation with the peak value $\sigma_0^2$ at a scale that corresponds to the larger scale. We find that the threshold value of the density perturbation $\tilde \delta_{\rm th}$ at the horizon entry for the PBH formation scales as $\tilde \delta_{\rm th}\propto \sigma_0^{2/5}$ for $\sigma_0\ll1$.