Primordial black hole formation from a nonspherical density profile with a misaligned deformation tensor

We perform the numerical simulation of primordial black hole (PBH) formation from a nonspherical profile of the initial curvature perturbation ζ. We consider the background expanding universe filled with the perfect fluid with the linear equation of state p=wρ ($w=1/3$ or $1/5$), where p and ρ are the pressure and the energy density, respectively. The initial condition is set in a way such that the principal directions of the second derivatives of ζ and Δζ at the central peak are misaligned, where Δ is the Laplacian. In this setting, since the linearized density is proportional to Δζ, the inertia tensor and deformation tensor ∂ᵢ∂ⱼζ are misaligned. Thus, tidal torque may act, and the spin of a resultant primordial black hole would be nonzero, in general, although it is estimated to be very small from previous perturbative analyses. As a result, we do not find a finite value of the spin within our numerical precision, giving support for the negligibly small value of the black hole spin for 1/5w1/3. More specifically, our results suggest that the dimensionless PBH spin s is typically so small that s0.1 for w0.2.