Evolution of axial perturbations in space-time of a non-rotating uncharged primordial black hole

TL;DR

This paper derives the axial perturbation equation for a non-rotating primordial black hole in an expanding universe, revealing how its space-time dynamics differ from classical black holes under early universe conditions.

Contribution

It presents the first derivation of axial perturbation equations for a primordial black hole modeled by the generalized McVittie metric, incorporating mass variation and cosmological effects.

Findings

Derived the axial perturbation equation for generalized McVittie black holes.

Separated the perturbation equation into radial and angular parts under early universe approximations.

Identified the potential in Schrödinger-like form and provided physical insights.

Abstract

We derive the equation governing the axial-perturbations in the space-time of a non-rotating uncharged primordial black hole (PBH), produced in early Universe, whose metric is taken as the generalized McVittie metric. The generalized McVittie metric is a cosmological black hole metric, proposed by V. Faraoni and A. Jacques in 2007 [Phys. Rev. D 76, 063510 (2007)]. This describes the space-time of a Schwarzschild black hole embedded in FLRW-Universe, while allowing its mass-change. Our derivation has basic similarities with the procedure of derivation of S. Chandrasekhar, for deriving the Regge-Wheeler equation for Schwarzschild metric [S. Chandrasekhar, The Mathematical Theory of Black holes ; Oxford University Press (1983)] ; but it has some distinct differences with that due to the complexity and time-dependency of the generalized McVittie metric. We show that after applying some approximations which are very well valid in the early radiation-dominated Universe, the overall equation governing the axial perturbations can be separated into radial and angular parts, among which the radial part is the intended one, as the angular part is identical to the case of Schwarzschild metric as expected. We identify the potential from the Schr\"odinger-like format of the equation and draw some physical interpretation from it.