# 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 {mu} in circular orbit of radius r{sub 0} around a Schwarzschild black hole of mass M and derive the spherical harmonic components associated with the gravitational wave polarizations up to order v{sup 11} beyond Newtonian. Using the more accurate h{sub lm}'s computed in this work, we provide the more complete set of associated {rho}{sub lm}'s and {delta}{sub lm}'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{sub +} and h{sub x} in the test-particle limit for gravitational wavesmore »

- Authors:

- Raman Research Institute, Bangalore 560 080 (India)

- Publication Date:

- OSTI Identifier:
- 21432927

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 82; Journal Issue: 4; Other Information: DOI: 10.1103/PhysRevD.82.044051; (c) 2010 American Institute of Physics

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ANTENNAS; BLACK HOLES; COMPARATIVE EVALUATIONS; COMPUTERIZED SIMULATION; DATA ANALYSIS; GRAVITATIONAL WAVES; INTERFEROMETERS; LASER RADIATION; MASS; ORBITS; PARTICLES; POLARIZATION; SCHWARZSCHILD METRIC; SPACE; SPHERICAL CONFIGURATION; SPHERICAL HARMONICS; SPIN; WAVE FORMS; ANGULAR MOMENTUM; CONFIGURATION; ELECTRICAL EQUIPMENT; ELECTROMAGNETIC RADIATION; EQUIPMENT; EVALUATION; FUNCTIONS; MEASURING INSTRUMENTS; METRICS; PARTICLE PROPERTIES; RADIATIONS; SIMULATION

### Citation Formats

```
Fujita, Ryuichi, and Iyer, Bala R.
```*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*. United States: N. p., 2010.
Web. doi:10.1103/PHYSREVD.82.044051.

```
Fujita, Ryuichi, & Iyer, Bala R.
```*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*. United States. doi:10.1103/PHYSREVD.82.044051.

```
Fujita, Ryuichi, and Iyer, Bala R. Sun .
"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". United States. doi:10.1103/PHYSREVD.82.044051.
```

```
@article{osti_21432927,
```

title = {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},

author = {Fujita, Ryuichi and Iyer, Bala R.},

abstractNote = {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 {mu} in circular orbit of radius r{sub 0} around a Schwarzschild black hole of mass M and derive the spherical harmonic components associated with the gravitational wave polarizations up to order v{sup 11} beyond Newtonian. Using the more accurate h{sub lm}'s computed in this work, we provide the more complete set of associated {rho}{sub lm}'s and {delta}{sub lm}'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{sub +} and h{sub x} in the test-particle limit for gravitational waves data analysis applications. Additionally, we provide closed analytical expressions for 2.5PN h{sub lm}, 2PN {rho}{sub lm}, and 3PN {delta}{sub lm}, 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.},

doi = {10.1103/PHYSREVD.82.044051},

journal = {Physical Review. D, Particles Fields},

number = 4,

volume = 82,

place = {United States},

year = {Sun Aug 15 00:00:00 EDT 2010},

month = {Sun Aug 15 00:00:00 EDT 2010}

}