Energy versus Angular Momentum in Black Hole Binaries

TL;DR

The prediction of the effective one body formalism, based purely on known analytical results (without any calibration to numerical relativity), agrees strikingly well with the numerical-relativity results.

Abstract

Using accurate numerical relativity simulations of (nonspinning) black-hole binaries with mass ratios 1:1, 2:1 and 3:1 we compute the gauge invariant relation between the (reduced) binding energy $E$ and the (reduced) angular momentum $j$ of the system. We show that the relation $E(j)$ is an accurate diagnostic of the dynamics of a black-hole binary in a highly relativistic regime. By comparing the numerical-relativity $E^{\rm NR} (j)$ curve with the predictions of several analytic approximation schemes, we find that, while the usual, non-resummed post-Newtonian-expanded $E^{\rm PN} (j)$ relation exhibits large and growing deviations from $E^{\rm NR} (j)$, the prediction of the effective one-body formalism, based purely on known analytical results (without any calibration to numerical relativity), agrees strikingly well with the numerical-relativity results.

Figures (2)

• Figure 1: Equal-mass case: comparison between four $E(j)$ curves. The standard "Taylor" PN curve shows the largest deviation from NR results, especially at low $j$'s, while the two (adiabatic and nonadiabatic) 3PN-accurate, non-NR-calibrated EOB curves agree remarkably well with the NR one.
• Figure 2: Differences between seven $E^{\rm X}(j)$ curves and $E^{\rm EOB_{3PN}}(j)$, for the three mass ratios considered. From top to bottom the labelling is: ${\rm X\,=\,PN}$, ${\rm EOB_{5PN}^{wo\,NQC}}$, ${\rm EOB_{5PN}^{NQC}}$, NR, ${\rm EOB_{3PN}}$ (baseline), ${\rm EOB_{3PN}^{NQC}}$ and ${\rm EOB_{3PN}^{adiabatic}}$. While the PN curve exhibits the largest deviations, all EOB curves remain close to the NR one during the full inspiral, especially the 3PN-accurate, non-NR-calibrated one.