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

We perform the numerical simulation of primordial black hole formation from a nonspherical profile of the initial curvature perturbation $\zeta$. We consider the background expanding universe filled with the perfect fluid with the linear equation of state $p=w\rho$ ($w=1/3$ or $1/5$), where $p$ and $\rho$ 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 $\zeta$ and $\triangle \zeta$ at the central peak are misaligned, where $\triangle$ is the Laplacian. In this setting, since the linearized density is proportional to $\triangle \zeta$, the inertia tensor and deformation tensor $\partial_i\partial_j \zeta$ are misaligned. Thus tidal torque may act and the spin of a resultant primordial black hole would be non-zero 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 $1/5\lesssim w \lesssim 1/3$. More specifically, our results suggest that the dimensionless PBH spin $s$ is typically so small that $s\ll0.1$ for $w\gtrsim0.2$.