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Recent breakthroughs in numerical relativity enable one to examine thevalidity of the post-Newtonian expansion in the late stages of inspiral. Forthe comparison between post-Newtonian (PN) expansion and numerical simulations,the waveforms in terms of the spin-weighted spherical harmonics are more usefulthan the plus and cross polarizations, which are used for data analysis ofgravitational waves. Factorized resummed waveforms achieve better agreementwith numerical results than the conventional Taylor expanded post-Newtonianwaveforms. In this paper, we revisit the post-Newtonian expansion ofgravitational waves for a test-particle of mass $\m$ in circular orbit ofradius $r_0$ around a Schwarzschild black hole of mass $M$ and derive thespherical harmonic components associated with the gravitational wavepolarizations up to order $v^{11}$ beyond Newtonian. Using the more accurate$h_{\ell m}$'s computed in this work, we provide the more complete set ofassociated $\rho_{\ell m}$'s and $\delta_{\ell m}$'s that form important bricksin the factorized resummation of waveforms with potential applications for theconstruction of further improved waveforms for prototypical compact binarysources in the future. We also provide ready-to-use expressions of the 5.5PNgravitational waves polarizations $h_+$ and $h_\times$ in the test-particlelimit for gravitational wave data analysis applications. Additionally, weprovide closed analytical expressions for 2.5PN $h_{\ell m}$, 2PN $\rho_{\ellm}$ and 3PN $\delta_{\ell m}$, for general multipolar orders $\ell$ and $m$ inthe test-particle limit. Finally, we also examine the implications of thepresent analysis for compact binary sources in Laser Interferometer SpaceAntenna.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

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 hol