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 (2016), 104017], 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 here improved and generalized to all multipoles up to $\ell=6$. For a test-particle orbiting a Kerr black hole, each multipolar amplitude is truncated at relative 6~post Newtonian (PN) order, both for the orbital (nonspinning) and spin factors. By taking a certain Pad\'e approximant (typically the $P^4_2$ one) of the orbital factor in conjuction with the inverse Taylor (iResum) representation of the spin factor, it is possible to push the analytical/numerical agreement of the energy fluxe at the level of $5\%$ at the last-stable-orbit for a quasi-maximally 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 3PN comparable-mass terms are hybridized with higher PN test-particle terms up to 6PN relative order. The same Pad\'e 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 stronger in the resummed case than when using the standard Taylor-expanded form of Pan et al.~[Phys.~Rev.~D~83 (2011) 064003]. We finally introduce a method to consistently hybridize comparable-mass and test-particle information {\it 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.