The Close-Limit Approximation for Black Hole Binaries with Post-Newtonian Initial Conditions

TL;DR

This paper develops a method to model the ringdown phase of black hole mergers using the close-limit approximation with initial conditions derived from second-post-Newtonian theory, enabling more accurate waveform predictions.

Contribution

It introduces a formalism that expands 2PN binary metrics into close-limit form and uses this to generate initial data for perturbation equations, improving ringdown modeling.

Findings

Successfully evolves ringdown waveforms for head-on and circular orbit mergers.

Demonstrates consistency of 2PN initial conditions with perturbative field equations.

Provides a framework for studying gravitational recoil during ringdown.

Abstract

The ringdown phase of a black hole formed from the merger of two orbiting black holes is described by means of the close-limit (CL) approximation starting from second-post-Newtonian (2PN) initial conditions. The 2PN metric of point-particle binaries is formally expanded in CL form and identified with that of a perturbed Schwarzschild black hole. The multipolar coefficients describing the even-parity (polar) and odd-parity (axial) components of the linear perturbation consistently satisfy the 2PN-accurate perturbative field equations. We use these coefficients to build initial conditions for the Regge-Wheeler and Zerilli wave equations, which we then evolve numerically. The ringdown waveform is obtained in two cases: head-on collision with zero-angular momentum, composed only of even modes, and circular orbits, for which both even and odd modes contribute. In a separate work, this formalism is applied to the study of the gravitational recoil produced during the ringdown phase of coalescing binary black holes.