High-Order Post-Newtonian Fit of the Gravitational Self-Force for Circular Orbits in the Schwarzschild Geometry

TL;DR

The paper demonstrates a powerful synergy between post-Newtonian theory and gravitational self-force analysis for circular orbits in Schwarzschild geometry. By leveraging analytically known PN parameters and high-precision SF data, it derives leading 4PN and next-to-leading 5PN logarithmic contributions to the conservative dynamics, measures non-logarithmic 4PN–7PN coefficients, and confirms the 3PN coefficient. The work provides gauge-invariant observables, notably the redshift $u^T$, with explicit 4PN/5PN logarithmic corrections and a clear small-mass-ratio expansion, enabling precise waveform calibration for LISA and supporting PN-based waveform generation for LIGO/Virgo. Overall, it demonstrates how SF data can push PN expansions to higher orders, refining inspiral templates across mass ratios and improving gravitational-wave data analysis.

Abstract

We continue a previous work on the comparison between the post-Newtonian (PN) approximation and the gravitational self-force (SF) analysis of circular orbits in a Schwarzschild background. We show that the numerical SF data contain physical information corresponding to extremely high PN approximations. We find that knowing analytically determined appropriate PN parameters helps tremendously in allowing the numerical data to be used to obtain higher order PN coefficients. Using standard PN theory we compute analytically the leading 4PN and the next-to-leading 5PN logarithmic terms in the conservative part of the dynamics of a compact binary system. The numerical perturbative SF results support well the analytic PN calculations through first order in the mass ratio, and are used to accurately measure the 4PN and 5PN non-logarithmic coefficients in a particular gauge invariant observable. Furthermore we are able to give estimates of higher order contributions up to the 7PN level. We also confirm with high precision the value of the 3PN coefficient. This interplay between PN and SF efforts is important for the synthesis of template waveforms of extreme mass ratio inspirals to be analysed by the space-based gravitational wave instrument LISA. Our work will also have an impact on efforts that combine numerical results in a quantitative analytical framework so as to generate complete inspiral waveforms for the ground-based detection of gravitational waves by instruments such as LIGO and Virgo.

Figures (2)

• Figure 1: The absolute value of the contributions of the numerically determined post-Newtonian terms to $r^5{\bar{u}^\alpha \bar{u}^\beta h^{\text{R}}_{\alpha\beta}}$. Here PNL refers to just the logarithm term at the specified order. The contribution of $a_4$ is not shown but would be a horizontal line (since the 4PN terms behaves like $r^{-5}$) at approximately 121.3 . The remainder after $a_4$ and all the known coefficients are removed from $r^5{\bar{u}^\alpha \bar{u}^\beta h^{\text{R}}_{\alpha\beta}}$ is the top (red) continuous line. The lower (black) dotted line labelled "err" shows the uncertainty in $r^5{\bar{u}^\alpha \bar{u}^\beta h^{\text{R}}_{\alpha\beta}}$, namely $2{\rm E} \, r^4 \times10^{-13}$. The jagged (green) line labelled "$|$res$|$" is the absolute remainder after all of the fitted terms have been removed. The figure reveals that, with regard to the uncertainty of the calculated ${\bar{u}^\alpha \bar{u}^\beta h^{\text{R}}_{\alpha\beta}}$, the choice $E\simeq 1$ was slightly too large.
• Figure 2: The self-force contribution $u^T_\mathrm{SF}$ to $u^T$ plotted as a function of the gauge invariant variable $y^{-1}$. Note that $y^{-1}$ is an invariant measure of the orbital radius scaled by the black hole mass $m_2$ [see Eq. \ref{['y']}]. The "exact" numerical points are taken from Ref. De.08. Here, PN refers to all terms, including logarithms, up to the specified order (however recall that we did not include in our fit a log-term at 7PN order).