Spherical harmonic modes of 5.5 post-Newtonian gravitational wave polarizations and associated factorized resummed waveforms for a particle in circular orbit around a Schwarzschild black hole

Abstract

Recent breakthroughs in numerical relativity enable one to examine the validity of the post-Newtonian expansion in the late stages of inspiral. For the comparison between post-Newtonian (PN) expansion and numerical simulations, the waveforms in terms of the spin-weighted spherical harmonics are more useful than the plus and cross polarizations, which are used for data analysis of gravitational waves. Factorized resummed waveforms achieve better agreement with numerical results than the conventional Taylor expanded post-Newtonian waveforms. In this paper, we revisit the post-Newtonian expansion of gravitational waves for a test particle of mass μ in circular orbit of radius r0 around a Schwarzschild black hole of mass M and derive the spherical harmonic components associated with the gravitational wave polarizations up to order v11 beyond Newtonian. Using the more accurate hℓm’s computed in this work, we provide the more complete set of associated ρℓm’s and δℓm’s that form important bricks in the factorized resummation of waveforms with potential applications for the construction of further improved waveforms for prototypical compact binary sources in the future. We also provide ready-to-use expressions of the 5.5PN gravitational waves polarizations h+ and h× in the test-particle limit for gravitational waves data analysis applications. Additionally, we provide closed analytical expressions for 2.5PN hℓm, 2PN ρℓm, and 3PN δℓm, for general multipolar orders ℓ and m in the test-particle limit. Finally, we also examine the implications of the present analysis for compact binary sources in Laser Interferometer Space Antenna.