Assessment of effective-one-body radiation reactions for generic planar orbits

In this paper, we assess the performances of different analytical prescriptions for the effective-one-body (EOB) radiation reaction along generic planar orbits using exact numerical result in the test-mass limit. We consider three prescriptions put forward in the recent literature: (i) the quasicircular prescription (QC), (ii) the QC with second post-Newtonian (2PN) order noncircular corrections (QC2PN), and (iii) the QC corrected by the noncircular Newtonian prefactor (NCN). The analytical fluxes are then compared against the exact fluxes that are computed by solving the Teukolsky equation with a test-mass source in geodesic motion. We find that the NCN prescription is the most accurate for both eccentric and hyperbolic orbits, and it is in robust agreement also for large values of the eccentricity. This result carries over to the comparable masses, as we discuss for a numerical-relativity (NR) case study. We also demonstrate that, while the EOB/NR waveform unfaithfulness is a necessary check for the precision of EOB models, the direct comparison of EOB/NR fluxes is a more stringent and informative test to select the best prescription. Finally, we propose an improved radiation reaction, NCN2PN, that includes noncircular 2PN corrections, in resummed form, as a further multiplicative contribution and that is valid for any mass ratio.

Figure 1

Test-particle orbiting around a Schwarzschild black hole ($\widehat{a}=0$) on a geodesic with eccentricity $e=0.5$ and semilatus rectum $p=13$. One radial period is highlighted on the trajectory (left panel), and we show the corresponding fluxes. We contrast the numerical fluxes (black) and the three different analytical fluxes considered in this work: (i) the NCN flux (red online), as used in teobresums, obtained from Eqs. (16) and (19); (ii) the QC flux (blue online) computed from Eq. (21), that is a proxy of the fluxes of seobnrv4ehm [20]; and (iii) the QC2PN flux (green online) computed using Eqs. (22) as taken from Khalil et al. [19]. The analytical/numerical relative differences are shown in the bottom panels. The NCN analytical fluxes deliver the closer agreement with the numerical ones either at apastron and periastron. See Secs. 3 and 4 for full discussion.

Figure 2

EOB $\ell =m=2$ Regge-Wheeler-Zerilli normalized strain waveforms of a test-particle inspiraling and plunging around a Schwarzschild black hole. Initial eccentricity $e_0$ and semilatus rectum $p_0$ are $(e_0,p_0)=(0.5,7.6)$. The inspiral behind the red waveform is driven by the NCN flux, while that of the blue one is driven by the QC flux. In the bottom panels, we show the corresponding quadrupolar frequencies. The fact that the QC flux overestimates the correct answer (see Fig. 1) at periastron yields an unphysical acceleration of the inspiral related to the (small but non-negligible) circularization of the orbit.

Figure 3

Geodesic configurations with $\widehat{a}=0$, $p=13$, and $e=(0.1,0.3,0.7,0.9)$. For each one, we show the trajectories highlighting one radial period (blue) and the corresponding fluxes. We contrast the numerical fluxes (black online) with the three analytical fluxes considered in this work: the NCN flux (red) computed using Eqs. (16) and (19); the QC flux (blue) from Eq. (21), which is a proxy of the seobnrv4ehm fluxes; and the QC2PN flux with 2PN noncircular corrections (green) from Eq. (22). Each subpanel also reports the analytical/numerical relative difference. Note that, while the relative differences in the lower panels are always shown over the complete radial period, the fluxes for $e\ge 0.7$ are shown on smaller time intervals (highlighted in aqua-green on the trajectories) in order to better highlight the burst of radiation at periastron passage (marked by a dashed vertical line).

Figure 4

Analogous to Fig. 3, but for spinning configurations with different eccentricities. The semilatera recta are computed according to $p=13p_s(e,\widehat{a})/p_s(e,0)$. For each configuration, we show the trajectories highlighting one radial period (blue) and the corresponding fluxes. We contrast the numerical fluxes (black) with the three analytical fluxes considered in this work: the NCN flux (red) computed using Eqs. (16) and (19); the QC flux (blue) from Eq. (21) which is a proxy of the seobnrv4ehm fluxes; and the QC2PN flux with 2PN noncircular corrections (green) from Eq. (22). Each subpanel also reports the analytical/numerical relative difference. Note that, while the relative differences in the lower panels are always shown over the complete radial period, the fluxes for $e\ge 0.7$ are shown on smaller time intervals (highlighted in aqua-green on the trajectories) in order to show better the burst of radiation at the periastron passage (marked by a dashed vertical line).

Figure 5

Analytical/numerical relative differences for the orbital averaged fluxes (angular momentum and energy) that correspond to eccentric configurations. For each configuration, the semilatus rectum is computed according to $p=p_{\mathrm{schw}}{p}_s(e,\widehat{a})/p_s(e,0)$ where $p_{\mathrm{schw}}=9$ (squares) or $p_{\mathrm{schw}}=13$ (triangles).

Figure 6

Means of the analytical/numerical relative differences of the orbital averaged fluxes, graphical representation of Table 1. Each point represents the mean of the averaged fluxes of all the configurations with the indicated value of eccentricity, that we indicate as $⟨\dots {⟩}_e$. Note that we consider $\widehat{a}=(0,\pm 0.2,\pm 0.6,\pm 0.9)$ and two semilatera recta for each pair $(e,\widehat{a})$, so that each point in Fig. 6 is the mean value over 14 averaged fluxes. As can be seen, NCN is, on average, the most reliable radiation reaction.

Figure 7

Hyperbolic geodesics in Schwarzschild spacetime. In the trajectories, we highlight in blue the part that corresponds to $r\le 40$; the fluxes are computed on this interval. We contrast the numerical fluxes (black online) with the three analytical fluxes considered in this work: the NCN flux (red) computed using Eqs. (16) and (19); the QC flux (blue) from Eq. (21), which is a proxy of the seobnrv4ehm fluxes; and the QC2PN flux with 2PN noncircular corrections (green) from Eq. (22). Each subpanel also reports the analytical/numerical relative difference. For each analytical flux, we also show the relative difference with the numerical result. The dashed vertical line marks the closest passage to the central black hole.

Figure 8

Equal-mass case ($q=1$), initial EOB eccentricity $\sim 0.2$, teobresums [21] evolution: effect of using the circular ${\widehat{\mathcal{F}}}_{\phi }$ (QC) instead of the full one (NCN) with the Newtonian noncircular correction. Both dynamics are started with the same initial data. This BBH configuration corresponds to the SXS:BBH:1363 configuration discussed in Table IV of Ref. [21]. The qualitative behavior is analogous to the test-mass case shown in Fig. 2.

Figure 9

EOB/NR phasing comparison for SXS:BBH:1363. Left: teobresums waveform model of Refs. [13, 21] using the complete NCN expression for ${\widehat{\mathcal{F}}}_{\phi }$ with initial parameters $(e_{{\omega }_a}^{\mathrm{EOB}},{\omega }_a^{\mathrm{EOB}})=(0.30479,0.01908)$ (see Fig. 12 of Ref. [21]). Right: Modified teobresums with the quasicircular (QC) ${\widehat{\mathcal{F}}}_{\phi }$, without leading noncircular terms, and $a_6^c=-97$. The initial parameters are $(e_{{\omega }_a}^{\mathrm{EOB}},{\omega }_a^{\mathrm{EOB}})=(0.315,0.018239)$. The vertical dot-dashed lines indicate the alignment region. On such a short-length waveform, the absence of noncircular corrections in ${\widehat{\mathcal{F}}}_{\phi }$ can be (partially) reabsorbed in the initial conditions. However, the stronger orbital circularization yielded by the QC model clearly shows up in the last three orbits.

Figure 10

EOB/NR unfaithfulness computation with the Advanced LIGO noise using the standard NCN teobresums model of Ref. [21] or the QC version discussed here. The faster circularization of the QC model results in ${\overline{F}}_{\mathrm{EOB}/\mathrm{NR}}$ slightly above the 1% threshold (see Fig. 11).

Figure 11

Time domain frequency evolution. Top panel: comparison between NR and the NCN or QC version of teobresums. Bottom panel: fractional frequency difference with NR. The picture confirms that the NCN and QC initial conditions are approximately coherent among themselves and with the NR curve. The stronger circularization yielded by the QC ${\widehat{\mathcal{F}}}_{\phi }$ entails a noticeable EOB/NR disagreement toward the late inspiral.

Figure 12

Trajectories, quadrupolar angular momentum fluxes, and 2PN corrections to the flux as discussed in Sec 6. We show the standard flux from teobresums (red), the result with 2PN resummed corrections (yellow), and the numerical flux (black). The corresponding analytical/numerical relative differences reported in the bottom panel are shown over the radial period; the corresponding part of the trajectories is highlighted in blue. On the rightmost panels, we show the 2PN corrections to the flux in Taylor expanded form, with resummation of the tail factor and with resummation of both the instantaneous and hereditary parts. The latter is used for the 2PN flux of the middle panels. For the second configuration with $e=0.9$, the fluxes and the 2PN corrections are shown over a time interval that is shorter than the radial period in order to highlight the burst of radiation at periastron; the corresponding part of the trajectories is highlighted in aqua-green.

Figure 13

Means of the analytical/numerical relative differences of the averaged fluxes, graphical representation of Table 2. The mean $⟨\dots {⟩}_e$ is performed over all the simulations with the same eccentricity. Since we consider two semilatera recta for each pair $(e,\widehat{a})$ and $\widehat{a}=(0,\pm 0.2,\pm 0.6,\pm 0.9)$, each point is an average over 14 configurations.

Figure 14

Means of the analytical/numerical relative differences of the averaged fluxes. The mean $⟨\dots {⟩}_e$ is performed over all the simulations with the same eccentricity. Since we consider two semilatera recta for each pair $(e,\widehat{a})$ and $\widehat{a}=(0,\pm 0.2,\pm 0.6,\pm 0.9)$, each point is an average over 14 configurations. We report the results for the NCN and NCN2PN fluxes and also for the fluxes computed from the 2PN waveform of Ref. [21] using Eqs. (27).

Figure 15

Analytical/numerical comparisons of the Newton-normalized circular flux $\widehat{f}$ defined in Eq. (13) for four values of the Kerr spin parameter $\widehat{a}$. The flux is plotted against $v={\mathrm{\Omega }}^{1/3}$, where $\mathrm{\Omega }$ goes from zero to the orbital frequency of the last stable orbit. The two horizontal lines mark the 0.1% and the 1% thresholds. In the bottom panel, we show the relative difference $({\widehat{f}}_{\mathrm{num}}-{\widehat{f}}_{\mathrm{EOB}})/{\widehat{f}}_{\mathrm{num}}$.