Exploiting Newton-factorized, 2PN-accurate waveform multipoles in effective-one-body models for spin-aligned noncircularized binaries

We present a new approach to factorize and resum the post-Newtonian (PN) waveform for generic equatorial motion to be used within effective-one-body (EOB-)based waveform models. The new multipolar waveform factorization improves previous prescriptions in that (i) the generic Newtonian contribution is factored out from each multipole; (ii) the circular part is factored out and resummed using standard EOB methods; and (iii) the residual, 2PN-accurate, noncircular part, and in particular the tail contribution, is additionally resummed using Padé approximants. The resulting waveform is validated in the extreme-mass-ratio limit by comparisons with nine (mostly nonspinning) numerical waveforms either from eccentric inspirals, with eccentricities up to $e=0.9$, or dynamical captures. The resummation of the noncircular tail contribution is found essential to obtain excellent ($\lesssim 0.05\mathrm{rad}$ at periastron for $e=0.9$) analytical/numerical agreement and to considerably improve the prescription with just the Newtonian prefactor. In the comparable mass case, the new 2PN waveform shows only a marginal improvement over the previous Newtonian factorization, though yielding maximal unfaithfulness $\simeq {10}^{-3}$ with the 28 publicly available numerical relativity simulations with eccentricity up to $\sim 0.3$ (except for a single outlier that grazes ${10}^{-2}$). We finally use test-particle data to validate the waveform factorization proposed by Khalil et al. [Phys. Rev. 104, 024046 (2021)] and conclude that its amplitude can be considered reliable (though less accurate, $\sim 6\%$ fractional difference versus 1.5% of our method) only up to eccentricities $\sim 0.3$.

Figure 1

Comparisons with nonresummed noncircular tail factor: comparing analytical and numerical $\ell =m=2$ waveforms for the transition from inspiral to plunge of a test particle on a Kerr black hole with spin parameter $\widehat{a}$. We consider different configurations with $\nu ={10}^{-3}$. Each panel displays the numerical waveform (black, indistinguishable) and two EOB waveforms: (i) the solid-red one with noncircular information only in the Newtonian prefactor and (ii) the dashed-blue one with noncircular 2PN corrections with the nonresummed tail ${\widehat{h}}_{22}^{{\mathrm{nc}}_{\mathrm{tail}}}$ of Eq. (31). The bottom panel shows both the phase differences and the relative amplitude differences with respect to the numerical waveform. We use dashed lines for the differences corresponding to the wave with 2PN corrections. The vertical line marks the merger time, corresponding to the peak of the numerical amplitude.

Figure 2

Last part of the time evolution of $p_{\phi }^{2}u$ for different combinations of $(\widehat{a},e_0,p_0)$. Note that during the plunge $p_{\phi }^{2}u$ can grow up to $\sim 10$. This growth is mostly responsible of the unacceptably large analytical/numerical phase disagreement during the plunge, at 2PN-accuracy, seen in Fig. 1.

Figure 3

Behavior of various truncations of the ${\widehat{t}}_{p_{r_{*}}}^{22}$ polynomial of Eq. (32). The various truncations of ${\widehat{t}}_{p_{r_{*}}}^{22}$ oscillate and become very large for values of $p_{\phi }^{2}u$ of the order of those reached during the plunge; see Fig. 2. A straightforward diagonal Padé approximant tapers the behavior of the polynomials in strong field and eventually improves the behavior of the waveform there.

Figure 4

Comparisons with resummed noncircular tail factor: comparing analytical and numerical $\ell =m=2$ waveforms for the transition from inspiral to plunge of a test particle on a Kerr black hole with spin parameter $\widehat{a}$. We consider different configurations with $\nu ={10}^{-3}$. Each panel displays the numerical waveform (black, indistinguishable) and two EOB waveforms: (i) the solid-red one with noncircular information only in the Newtonian prefactor and (ii) the dashed-blue one with noncircular 2PN corrections with the tail ${\widehat{h}}_{22}^{{\mathrm{nc}}_{\mathrm{tail}}}$ of Eq. (31) resummed following the procedure discussed in Sec. 3a. The bottom panel shows both the phase differences and the relative amplitude differences with respect to the numerical waveform. We use dashed lines for the differences corresponding to the wave with 2PN corrections. The vertical line marks the merger time, corresponding to the peak of the numerical amplitude. The resummation strongly improves the analytical/numerical agreement with respect to Fig. 1.

Figure 5

Same configurations of Fig. 4, but here we focus on the phase, and we show also the analytical/numerical agreement obtained considering the resummed noncircular tail and the resummed instantaneous noncircular correction (dashed light green). The color scheme of the other differences is the same of Fig. 4: solid light blue for the wave with only the generic Newtonian prefactor and dashed blue for the wave with 2PN corrections with resummed tail and Taylor-expanded instantaneous corrections.

Figure 6

Instantaneous and hereditary noncircular 2PN corrections to the quadrupolar phase for four nonspinning geodesic cases $(e,p)=(0.3,9),(0.5,9),(0.9,9),(0.9,13)$. The instantaneous phase corrections are shown with dot-dashed blue lines, while the orange lines are for the phase contributions of the resummed eccentric tail (dashed for the expanded results and solid for the resummed ones). The corresponding sums between instantaneous and hereditary are shown in green with the same style scheme of the considered tail. The vertical dashed line marks the periastron passage. For $e=0.9$, we do not show the whole radial period in order to highlight the periastron.

Figure 7

Analogous to Fig. 6, but here we focus on the relevance of the resummation for the instantaneous part. The orange solid line is the phase contribution of the resummed eccentric tail, while the blue lines correspond to the instantaneous phase contributions: dot-dashed for the nonresummed results and solid line for the resummed ones. The corresponding sums between tail and instantaneous are shown in green with the same style scheme of the instantaneous terms. The vertical dashed line marks the periastron passage. For $e=0.9$, we do not show the whole radial period in order to highlight the periastron.

Figure 8

Same color scheme of Fig. 4, but here in each row, we consider the subdominant modes (2, 1), (3, 3), (3, 2), and (4, 4) for the configurations $(e_0,\widehat{a},p_0)=(0.1,0,6.7)$, (0.5, 0, 7.35), (0.9, 0, 8.05), (0.5, 0.4, 5.9).

Figure 9

Comparisons of the waveform modes (2, 2), (2, 1), and (3, 3) for the nonspinning case $(e_0,p_0)=(0.7,7.7)$. Top panels: numerical waveforms (black), the waveforms with only Newtonian noncircular corrections (red), and the waveforms with 2PN noncircular corrections with the resummed eccentric tail factor (dashed blue). Bottom panels: analytical/numerical phase differences (in radians) for different prescriptions: (i) only Newtonian noncircular corrections (solid light blue); (ii) only instantaneous noncircular corrections up to 2PN, i.e., without eccentric tail (dashed purple); (iii) waveform with 2PN noncircular corrections, both instantaneous and hereditary in expanded form (solid aqua-green); and (iv) waveform with 2PN noncircular corrections, both instantaneous and hereditary, with resummation applied to the latter (dashed blue).

Figure 10

Comparisons for the mode (2, 0) on nonspinning geodesic orbits with $e=(0.5,0.9)$ and $p=(9,13)$. We show the numerical waveform (black), the EOB waveform with noncircular corrections only at Newtonian level (red), and with corrections at 2PN as discussed in Sec. 2c (dashed blue).

Figure 11

Dynamical capture of a particle on a Schwarzschild black hole with $\nu ={10}^{-2}$, $p_{\phi ,0}=4.01$, and three different initial energies $E_0=(1.000711,1.000712,1.00124)$. Top panels: the trajectories, all starting from $r_0=120$, although the plot only focuses on the latest part. Middle panels: comparing the real part of the numerical waveform (black, barely indistinguishable) with two analytical waveforms: the one with only the Newtonian noncircular corrections (red) and the one with the 2PN corrections where the tail factor is resummed while the instantaneous factor is not (dashed blue). The corresponding phase differences are reported in the bottom panels (solid clear blue and dashed blue, respectively). The same panels also show the analytical/numerical phase difference obtained with the resummation applied to both the tail and the instantaneous 2PN noncircular phase ${\delta }_{22}^{{\mathrm{nc}}_{\mathrm{inst}}}$ (dashed green). The parameters of the ringdown and of the NQC corrections, as well the merger time (marked by the vertical line), are extracted from numerical data as in Ref. [19]; see the text. The closest analytical/numerical agreement is obtained resumming only the noncircular tail factor.

Figure 12

EOB/NR time-domain phasing for all the nonspinning datasets considered. The dashed vertical lines indicate the alignment window. For each configuration, we show together the NR (black) and EOB (red) real part of the waveform (top panel) and the phase difference (blue) and the relative amplitude difference (orange) (bottom panel).

Figure 13

EOB/NR time-domain phasing for all spinning datasets considered. The dashed vertical lines indicate the alignment window. For each configuration, we show together the NR (black) and EOB (red) real part of the waveform (top panel) and the phase difference (blue) and the relative amplitude difference (orange) (bottom panel).

Figure 14

EOB/NR unfaithfulness for the $\ell =m=2$ mode computed over the eccentric SXS simulations publicly available, Table 4. The horizontal lines mark the 0.03 and 0.01 values. The value of ${\overline{F}}_{\mathrm{EOB}/\mathrm{NR}}^{\max }$ does not exceed the 0.7% except for the single outlier given by SXS:BBH:1149, corresponding to $(3,+0.70,+0.60)$ with $e_{{\omega }_a}^{\mathrm{NR}}=0.037$, which is around 1%. This is consistent with the slight degradation of the model performance for large positive spins already found in the quasicircular limit, as pointed out in Ref. [25].

Figure 15

Testing the waveform factorization of Ref. [27], Eq. (62): comparisons between the $\ell =m=2$ numerical and analytical waveforms emitted by the eccentric inspiral of a test particle on a Schwarzschild black hole. The initial eccentricities and semilatera recta are $(e_0,p_0)=(0.1,6.7)$, (0.3, 7), (0.7, 7.7). Each top panel displays the numerical waveform (black, indistinguishable) and the EOB waveform with the generic Newtonian prefactor (dot-dashed red, labeled N) and the 2PN-accurate waveform with the quasicircular factorization of Eq. (62) (labeled $2{\mathrm{PN}}_{\mathrm{qc}}$). The corresponding phase differences and relative amplitude differences are shown in the bottom panels. The vertical black line marks the merger time, corresponding to the peak of the numerical waveform amplitude. For simplicity, the $2{\mathrm{PN}}^{\mathrm{qc}}$ waveform has not been completed by NQC corrections and ringdown. The analytical/numerical phase agreement is comparable for the two choices (blue lines); by contrast, the amplitude disagreement is always larger for the $2{\mathrm{PN}}_{\mathrm{qc}}$ prescription and worsens up to 30% when the eccentricity increases.

Figure 16

Comparing waveforms generated by a test-particle inspiralling and plunging into a Schwarzschild black hole with $(e_0,p_0)=(0.7,7.7)$. Top panel: the numerical waveforms (black, indistinguishable); the quasicircular EOB waveform (gray), the waveform with the Newtonian noncircular corrections (red), the waveform of Eq. (24) that adds to the previous one the 2PN corrections (blue), and the waveform of Eq. (62) with the quasicircular factorization and 2PN noncircular corrections (green, without merger and ringdown). Bottom panel: relative amplitude differences with the numerical waveforms. Apastra are marked by dotted (red) vertical lines, while periastra are marked with dot-dashed (blue) lines. See the text for discussion.

Figure 17

Contrasting different noncircular corrections for the same configuration of Fig. 16. Top and middle panels: the noncircular contributions to amplitude and phase. We consider the noncircular Newtonian prefactor ${\widehat{h}}_{\ell m}^{(\mathrm{N},\epsilon {)}_{\mathrm{nc}}}$ of Eq. (26) (red); the Newtonian-factorized instantaneous corrections up to 2PN, ${\widehat{h}}_{\ell m}^{(\mathrm{N},\epsilon {)}_{\mathrm{nc}}}{\widehat{h}}_{\ell m}^{{\mathrm{nc}}_{\mathrm{inst}}}$, (blue); and the 2PN noncircular corrections of Eq. (62) written as $1+f_{22}^{\mathrm{ecc}}/f_{22}$ (green). The bottom panel shows $\ddot{r}$, ${\mathrm{\Omega }}^2$, and $\dot{\mathrm{\Omega }}$. The correction proportional to $\ddot{r}/(r{\mathrm{\Omega }}^2)$ in Eq. (26) yields larger values at apastron than $1+f_{22}^{\mathrm{ecc}}/f_{22}$. See the text for discussion.

Figure 18

Same type of comparison of Fig. 15 but considering the dynamical capture configuration in the left panels of Fig. 11. Top panel: real part of the waveform. Middle panel: instantaneous frequency. Bottom panel: phase and fractional amplitude differences. The waveform of Ref. [27], Eq. (62), accumulates rather large amplitude differences up to the apastron of the quasielliptic orbit following the first encounter.

Figure 19

Analytical/numerical comparisons of the $\ell =m=2$ mode for different values of initial eccentricity and Kerr spin parameter. In the top panels, we show the real part of the numerical waveform (black, almost indistinguishable), the analytical waveform with resummed 2PN noncircular corrections written in terms of $(p_{r_{*}},p_{\phi })$ (solid blue), and the analytical one with corrections written in terms of $(p_{r_{*}},{\dot{p}}_{r_{*}})$ (dashed yellow). The corresponding analytical/numerical phase differences and amplitude relative differences are shown in the bottom panels (solid lines for resummed corrections with $p_{\phi }$, dashed lines for the corrections with ${\dot{p}}_{r_{*}}$).