The Overlap of Numerical Relativity, Perturbation Theory and Post-Newtonian Theory in the Binary Black Hole Problem

TL;DR

The study synthesizes the interfaces among numerical relativity, black hole perturbation theory, and post-Newtonian theory for binary black holes, advocating coordinate-invariant diagnostics to validate and connect disparate approaches. It demonstrates that perturbation theory can extend beyond extreme mass ratios, informing a universal semi-analytical template framework that integrates NR data and PN expansions via the EOB model. Across waveforms, fluxes, redshift, spin precession, perihelion advance, and binding energy, cross-checks show strong consistency and guide the construction of robust templates for gravitational-wave detection. The work outlines future directions, including second-order self-force, Kerr geometries, eccentricity and precession, and 4PN+ improvements to unify PN, BHP, and NR in practical data-analysis pipelines.

Abstract

Inspiralling and coalescing binary black holes are promising sources of gravitational radiation. The orbital motion and gravitational-wave emission of such system can be modelled using a variety of approximation schemes and numerical methods in general relativity: the post-Newtonian formalism, black hole perturbation theory, numerical relativity simulations, and the effective one-body model. We review recent work at the multiple interfaces of these analytical and numerical techniques, emphasizing the use of coordinate-invariant relationships to perform meaningful comparisons. Such comparisons provide independent checks of the validity of the various calculations, they inform the development of a universal, semi-analytical model of the binary dynamics and gravitational-wave emission, and they help to delineate the respective domains of validity of each approximation method. For instance, several recent comparisons suggest that perturbation theory may find applications in a broader range of physical problems than previously thought, including the radiative inspiral of intermediate mass-ratio and comparable-mass black hole binaries.

Figures (9)

• Figure 1: Different analytical approximation schemes and numerical techniques are used to model the orbital dynamics and gravitational-wave emission from black hole binaries, according to the mass ratio $0 < m_1 / m_2 \leqslant 1$ and the compactness parameter $0 < M / r \lesssim 1$, where $M = m_1 + m_2$ is the total mass and $r$ the typical binary separation.
• Figure 2: The mode $\psi_{22}$ of the Weyl scalar $\Psi_4$ for the late quasi-circular inspiral of an equal-mass, non-spinning binary black hole, as computed using NR simulations (solid blue) and two PN Taylor approximants (dashed red and dahsed green). Reproduced from Ref. ?.
• Figure 3: The modes $\psi_{20}$ and $\psi_{30}$ of the Weyl scalar $\Psi_4$, rescaled by the symmetric mass ratio $\nu$, for the head-on collision of two non-spinning black holes with mass ratios $q = 1/10$ (left panel) and $1/100$ (right panel), as computed using NR simulations (dotted red and dashed-dotted blue) and BHP theory to leading order (solid black). Reproduced from Ref. ?.
• Figure 4: The recoil velocity $v = |\bm{v}|$ as a function of the symmetric mass ratio $\nu$, as computed for non-spinning black hole binaries using NR simulations (dashed blue, dashed-dotted black, magenta triangles) and linear BHP theory with a $\nu$-rescaling of the modes of the gravitational waveform. Reproduced from Ref. ?.
• Figure 5: The conservative gravitational self-force contribution $u^t_\text{GSF}$ to the redshift observable as a function of $r_\Omega \equiv (m_2 / \Omega^2)^{1/3}$, a coordinate-invariant measure of the orbit separation, as computed numerically in BHP theory and analytically in PN theory up to 4PN order. Notice that $r_\Omega = 6 m_2$ corresponds to the Schwarzschild innermost stable circular orbit (ISCO). Reproduced from Ref. ?.