Gravitational-Wave Energy Flux for Compact Binaries through Second Order in the Mass Ratio

Within the framework of self-force theory, we compute the gravitational-wave energy flux through second order in the mass ratio for compact binaries in quasicircular orbits. Our results are consistent with post-Newtonian calculations in the weak field, and they agree remarkably well with numerical-relativity simulations of comparable-mass binaries in the strong field. We also find good agreement for binaries with a spinning secondary or a slowly spinning primary. Our results are key for accurately modeling extreme-mass-ratio inspirals and will be useful in modeling intermediate-mass-ratio systems.

Figure 1

The gravitational-wave flux (normalized by its leading Newtonian behavior) for a nonspinning binary as a function of the inverse orbital separation. Shown is the $(l,m)=(2,2)$ mode for a mass-ratio $10:1$ binary computed using the PN, NR, and GSF approaches. The solid, oscillating (blue) curve shows the NR flux computed from SXS:BBH:1107 [23]. The numbers along the top axis count the cycles before the peak amplitude in the NR waveform. The solid (red) curve shows the result from our second-order GSF (2SF) calculation. This agrees remarkably well with the NR result until very close to merger, where the GSF contributions diverge as the two-timescale approximation breaks down. In the weak field, the second-order self-force data agrees with the 3.5PN series [24], shown by the (orange) dash-dotted curve. We also show the first-order self-force (1SF) result with the (green) dashed curve. The vertical, dashed (gray) line marks the location of the (geodesic) innermost stable circular orbit.

Figure 2

The same as Fig. 1, but with $q=1$. Despite being a small-$\epsilon$ (small-$\nu$, large-$q$) expansion, the 2SF result agrees remarkably well with the NR flux for this equal-mass binary. The NR flux was computed from SXS:BBH:1132 [23].

Figure 3

The same as Fig. 1, but for the $(l,m)=(3,3)$ mode. We see that the 2SF flux does not agree as well with the NR result as in the case of the (2,2) mode. The simple resummation of the 2SF flux described in the main text results in a substantial improvement in the comparison with the NR flux. The relative difference between the NR and resummed 2SF flux up to five cycles before the waveform peak is below $4.5\times {10}^{-3}$, compared to $1.3\times {10}^{-2}$ for the nonresummed case. The resummation gives similar improvements for the other modes we have computed up to $l=5$. The 1SF result is not visible on the scale of the plot.

Figure 4

Comparison of NR and GSF fluxes at $\overline{x}=1/9$ for the (2,2) mode (left panel) and (3,3) mode (right panel). At leading order, both the NR flux (blue squares) and GSF flux (orange circles) scale as ${\nu }^2$. After subtracting the 1SF flux from the NR flux, we find that the residual follows the dashed (green) ${\nu }^3$ curve. After further subtracting the 2SF fluxes, we expect the residual to scale as ${\nu }^4$ (shown as a solid red curve). For the (2,2) mode, the residual broadly follows the ${\nu }^4$ trend, but the comparison is complicated by small oscillations in the NR waveform (likely from residual eccentricity and/or center-of-mass motion in the NR simulation [40, 41]). For the (3,3) mode, the residual is less subdominant and clearly follows the expected ${\nu }^4$ behavior. The SXS datasets used in this comparison are listed in the Supplemental Material [42].

Figure 5

Flux comparison for a spinning secondary with $q\simeq 6.28$. The primary has a very low spin, and the secondary has a retrograde spin with ${\chi }_2\simeq -0.8$. The NR flux is computed from SXS:BBH:1436 [23]. The PN flux [36] is shown with the dot-dashed (orange) curve. The 2SF flux without spinning flux corrections is shown by the (green) triangles. After the spining flux corrections are added, the 2SF result (red, upside-down triangles) agrees well with the NR flux.

Figure 6

The same as Fig. 5, but with a slowly spinning primary with $q\simeq 8$. The primary is spinning with ${\chi }_1\simeq 0.12$, and the secondary has ${\chi }_2\simeq 0.11$. The NR flux is computed from SXS:BBH:1460 [23].

Figure 7

The (Newtonian-normalized) second-order flux, ${\widehat{\mathcal{F}}}^{\mathrm{SF},2}$, vs the $\mathcal{O}({\nu }^3)$ contributions to the PN series, ${\widehat{\mathcal{F}}}^{\mathrm{PN},2}$, for the $(l,m)=(2,2)$ mode. The solid (blue) curve shows the $\mathcal{O}({\nu }^3)$ contribution to the 3.5PN flux, and the circles show our 2SF result. We then subtract the leading $\mathcal{O}({\nu }^3)$ PN term, $55x/21$, from both and get the (orange) dotted curve and the squares. After subtracting all the PN terms through $x^2$, we get the (green) dot-dashed curve and the diamonds. Subtracting all the PN terms through $x^3$, we get the (red) dashed curve and the triangles. Finally, after subtracting all the PN terms through $x^{7/2}$—i.e., all the known PN terms—the subdominant residual is shown with (purple) upside-down triangles. The residual appears to approach the $x^4$ reference curve shown by the long-dashed (purple) line. The gray shaded region shows the estimated error in our 2SF flux. We see similar agreement for other modes.