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

Abstract

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 \(w\gtrsim0.2\).