Comparison of 4.5PN and 2SF gravitational energy fluxes from quasicircular compact binaries

TL;DR

This work cross-validates two perturbative approaches to gravitational-wave energy flux from quasicircular, nonspinning binaries: the analytic 4.5PN flux derived via the PN–MPM formalism and the numerical 2SF flux computed with a second-order gravitational self-force framework. By recasting fluxes in terms of the invariant waveform frequency and decomposing into mode contributions ${\cal F}_{\ell m}$, the authors demonstrate consistency between 4.5PN and 2SF results across available orders, with particularly strong agreement up to $x \approx 0.16$ in the mildly relativistic regime. Mode-by-mode comparisons and the use of invariant frequencies bolster the claim of cross-method consistency, while residuals reveal limitations due to overlapping validity ranges and numerical noise, complicating a definitive asymptotic scaling assessment. The findings underscore the complementary strengths of PN, SF, and numerical relativity in validating GW flux predictions and improving waveform models for gravitational-wave astronomy.

Abstract

Recent years have seen significant advances in models of gravitational waveforms emitted by quasicircular compact binaries in two regimes: the weak-field, post-Newtonian regime, in which the gravitational wave energy flux has now been calculated to fourth-and-a-half post-Newtonian order (4.5PN) [Phys. Rev. Lett. 131}, 121402 (2023)]; and the small-mass-ratio, gravitational self-force regime, in which the flux has now been calculated to second perturbative order in the mass ratio (2SF) [Phys. Rev. Lett. 127, 151102 (2021)]. We compare these results and find agreement, showing consistency between the two (very distinct though both first-principle) perturbative calculations.

Figures (8)

• Figure 1: Comparison between GSF and PN for the 2SF $\mathcal{O}(\nu)$ contribution to the (Newtonian-normalized) total flux \ref{['eq:normalizedflux']}. The 2SF results are shown with (blue) dots. The PN results are shown with coloured curves; integer PN orders are shown with solid curves and corresponding half-integer orders are shown with dashed curves.
• Figure 2: Detailed comparison of $\hat{\mathcal{F}}^{(2)}$, the $\mathcal{O}(\nu)$ coefficient of the (Newtonian-normalized) total flux, computed from (i) numerical GSF and (ii) PN theory. The 2SF data are shown by (blue) dots and the corresponding 3.5PN series is plotted as a solid (blue) curve. As ever higher-order PN series are subtracted from the 2SF results, the residuals are compared with the next term in the PN series at small $x$. After subtracting the 3.5PN series from the 2SF data, one gets the (yellow) squares which follow the 4PN term (yellow curve). Subtracting the 4PN series from the SF data, one gets the (green) diamonds, which are compared against the 4.5PN term (green curve). The amplitude of the residual with the 4PN series is consistent with the amplitude of the 4.5PN term, although the slope of the residual is unclear, partly due to numerical noise. Further subtracting the 4.5PN series gives the (red) triangles. For sufficiently small values of $x$, we would expect the residual to scale as $x^5$. We plot a (red) dashed reference $x^5$ curve here but the accuracy of our numerical data is insufficient to claim agreement. One reason that the agreement with 4.5PN is not clear can be seen by considering the dominant $(2,2)$ mode: over the frequency range of our numerical GSF data there is evidence that the residual with 4PN has not reached the asymptotic regime -- see Fig. \ref{['fig:flux22']}. The agreement between PN and GSF results is clearer for some individual $(\ell,m)$ modes -- see Figs. \ref{['fig:flux22']}, \ref{['fig:comparisonmodes_l3m3_l3m2']} and the figures in Appendix \ref{['app:Flm-comparison']}.
• Figure 3: (Top) Detailed comparison of $\hat{\mathcal{F}}^{(2)}_{22}$, the $\mathcal{O}(\nu)$ coefficient of the $(\ell,m)=(2,2)$ mode of the (Newtonian-normalized) flux, computed from (i) numerical GSF and (ii) PN theory. The conventions and colorings are the same as in Fig. \ref{['fig:detailed_comparison_total_flux']}, but the blue curve now shows the 3PN series and the additional upside-down (purple) triangles show the residual after subtracting the 3PN series from the 2SF data. The latter residual closely follows the 3.5PN term (purple curve). The results of comparisons with higher PN orders are more ambiguous due to numerical noise in the GSF data. For example, the residual after subtracting the 4PN series from the 2SF result (green diamonds) appears at first sight to approach the 4.5PN term [before degrading again for even smaller values of $x$, due to numerical noise], but this is in fact an illusion caused by plotting the absolute magnitude of the residual. (Bottom) A subset of the above data (without taking the absolute magnitude) on a linear-log scale. Over the range of frequency values where we have numerical data, the residual between the GSF and 4PN is negative, but the 4.5PN term is positive. Thus, in order for these to agree asymptotically, there must be a zero crossing in the residual data at smaller values of $x$ than we have access to. Only after this zero crossing would we expect the residual and the 4.5PN curve to agree. The agreement between GSF and PN for other $(\ell,m)$ modes is clearer -- see Appendix \ref{['app:Flm-comparison']}.
• Figure 4: (Left panel) Detailed comparison of $\hat{\mathcal{F}}^{(2)}_{33}$, the $\mathcal{O}(\nu)$ coefficient of the $(3,3)$ mode of the (Newtonian-normalized) flux, computed from (i) numerical GSF and (ii) PN theory. The residuals between the GSF data and the various PN series (dotted lines) do not clearly follow the next term in the PN series (full lines) in this frequency range. However, the relative difference between the dotted lines and the corresponding full lines become smaller as one goes to smaller frequencies. This suggests that at smaller frequencies, one could credibly reach the asymptotic regime where the two curves overlap. Finally, the residual with the 4.5PN term approaches a 5PN slope, as shown by the $\mathcal{O}(x^5)$ reference line. (Right panel) Same as the left panel but for the (3,2) mode. The residual with the 3PN series (purple triangles) agrees perfectly with the 3.5PN term (purple line). The residual with the 3.5PN terms (orange squares) undergoes a passage through zero, then becomes noisy for $x < 0.05$, so it is not possible to claim agreement with the 4PN term (yellow line). Nonetheless, when going to the next order, we find that the residual with the 4PN series (green diamonds) agrees fairly well with the 4.5PN term (green line). Although the asymptotic regime is not yet reached, the relative error between the two curves decreases for smaller frequencies, and suggests better agreement deeper in the weak field regime. Finally, the residual with the 4.5PN term exhibits a very clear 5PN slope, as shown by the perfect fit with a $\mathcal{O}(x^5)$ reference line.
• Figure 5: (Left Panel) Comparison between the (Newtonian-normalized) GSF and PN total flux at different orders in the mass ratio expansion for $q=1$ ($\nu=0.25$). It is interesting to note that the 4.5PN expansion, including all orders in $\nu$ (black, dashed curve), is closely approximated by the PN expansion truncated at $\mathcal{O}(\nu^2)$ (yellow, solid curve), despite the equal mass ratio. The two gray curves show the flux computed from an NR simulation by the SXS collaboration Boyle_2019, specifically SXS:BBH:1132 SXS:BBH:1132. The two gray curves are computed using, respectively, 3rd and 4th order polynomials to extrapolate the NR waveform from the edge of the computational domain to null infinity and thus the shaded gray region between them is an estimate on the numerical error in the simulation. (Right Panel) The same as the left panel but for $q=10$ ($\nu=0.0826$). The NR simulation in this plot is SXS:BBH:1107 SXS:BBH:1107.