Factorization and resummation: A new paradigm to improve gravitational wave amplitudes. II. The higher multipolar modes

The factorization and resummation approach of Nagar and Shah [Phys. Rev. D 94, 104017 (2016)], designed to improve the strong-field behavior of the post-Newtonian (PN) residual waveform amplitudes $f_{\ell m}$’s entering the effective-one-body, circularized, gravitational waveform for spinning coalescing binaries, is improved and generalized here to all multipoles up to $\ell =6$. For a test particle orbiting a Kerr black hole, each multipolar amplitude is truncated at relative 6 PN order, both for the orbital (nonspinning) and spin factors. By taking a certain Padé approximant (typically the $P_2^4$ one) of the orbital factor in conjunction with the inverse Taylor (iResum) representation of the spin factor, it is possible to push the analytical/numerical agreement of the energy flux at the level of 5% at the last-stable orbit for a quasimaximally spinning black hole with dimensionless spin parameter $+0.99$. When the procedure is generalized to comparable-mass binaries, each orbital factor is kept at relative $3^{+3}$ PN order; i.e., the globally 3 PN-accurate comparable-mass terms are hybridized with higher-PN test-particle terms up to 6 PN relative order in each mode. The same Padé resummation is used for continuity. By contrast, the spin factor is only kept at the highest comparable-mass PN order currently available. We illustrate that the consistency between different truncations in the spin content of the waveform amplitudes is more marked in the resummed case than when using the standard Taylor-expanded form of Pan et al. [Phys. Rev. D 83, 064003 (2011)]. We finally introduce a method to consistently hybridize comparable-mass and test-particle information also in the presence of spin (including the spin of the particle), discussing it explicitly for the $\ell =m=2$ spin-orbit and spin-square terms. The improved, factorized and resummed, multipolar waveform amplitudes presented here are expected to set a new standard for effective one body–based gravitational waveform models.

Figure 1

Comparison between the factorized and resummed analytical ${\rho }_{\ell m}$’s (colored) and the corresponding numerical (exact) functions (black) up to $\ell =6$ for values of the black-hole dimensionless spin $\widehat{a}=(-0.99,-0.5,0,+0.5,+0.99)$ (red, orange, green, cyan, blue, and purple, respectively). The filled circles mark the LSO location. This plot is obtained using relative 6 PN information for all modes except for ${\rho }_{32}$ that use 5 PN relative accuracy in ${\rho }_{32}^{\mathrm{orb}}$ and ${\rho }_{66}$ and ${\rho }_{61}$ that use 5 PN relative accuracy for ${\widehat{\rho }}_{66}^{\mathrm{S}}$ and ${\widehat{\rho }}_{61}^{\mathrm{S}}$. The Padé approximants used on ${\rho }_{\ell m}^{\mathrm{orb}}$ are listed in the second column of Table 1. The same table also lists the numerical/analytical relative difference at the LSO. The agreement remains good (except for few exceptions; see the text for details) also for $\widehat{a}=+0.99$.

Figure 2

Test-particle limit: comparison between analytical and numerical exact (see the text) fluxes for dimensionless black-hole spin $\widehat{a}=+0.99$. The functions ${\rho }_{33}^{\mathrm{orb}}$, ${\rho }_{44}^{\mathrm{orb}}$, and ${\rho }_{55}^{\mathrm{orb}}$ are either resummed using (4,2) Padé approximants (orange, dotted-dashed line) or kept in PN-expanded form up to (relative) 6 PN order. This second choice improves the agreement with the numerical curve. The dotted line represents the analytical flux where the total 6 PN-accurate ${\rho }_{\ell m}$’s are kept in the standard, nonfactorized, Taylor-expanded form.

Figure 3

Nonresummed (black) and resummed (colored) waveform amplitudes ${\rho }_{22}$ (top panels) and $f_{21}$ (bottom panels) for a few configurations. The 3 PN-accurate orbital factors are hybridized with test-particle information up to relative 6 PN order for each mode and then resummed with the Padé approximants of Table 1. The consistency between NNLO and NLO truncations of the spin dependence is dramatically improved when the factorization and resummation procedure is applied.

Figure 4

Nonresummed (black) and resummed (colored) residual waveform amplitudes for $\ell =3$ multipoles. The 3 PN-accurate orbital factors are hybridized with test-particle information up to relative 6 PN order for each mode, except for the (3,1) mode that is taken at relative 5 PN order, and then resummed with the Padé approximants of Table 1. Like in the $\ell =2$ case of Fig. 3, the consistency between NLO and NNLO truncations of the spin dependence is improved by the resummation.

Figure 5

Modification of the results of Fig. 3 when (nonspinning) test-particle terms up to 6PN are hybridized with the NNLO $\nu$-dependent waveform [see Eqs. (71)–(73) and the text]. When $\nu =0$, one is using here the same spin-orbit and spin-square analytical information used in Fig. 1. The frequency parameter approximately corresponding to the BBH merger is $x\approx 0.38$ for $(1,+0.99,+0.99)$ and $x\approx 0.32$ for $(8,+0.50,0)$. In this latter case, the effect of the additional test-particle terms is important toward merger.