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 $\ensuremath{\zeta}$. We consider the background expanding universe filled with the perfect fluid with the linear equation of state $p=w\ensuremath{\rho}$ ($w=1/3$ or $1/5$), where $p$ and $\ensuremath{\rho}$ 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 $\ensuremath{\zeta}$ and $\mathrm{\ensuremath{\Delta}}\ensuremath{\zeta}$ at the central peak are misaligned, where $\mathrm{\ensuremath{\Delta}}$ is the Laplacian. In this setting, since the linearized density is proportional to $\mathrm{\ensuremath{\Delta}}\ensuremath{\zeta}$, the inertia tensor and deformation tensor ${\ensuremath{\partial}}_{i}{\ensuremath{\partial}}_{j}\ensuremath{\zeta}$ 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 $1/5\ensuremath{\lesssim}w\ensuremath{\lesssim}1/3$. More specifically, our results suggest that the dimensionless PBH spin $s$ is typically so small that $s\ensuremath{\ll}0.1$ for $w\ensuremath{\gtrsim}0.2$.