[Paper Review] New Class of Gravitational Wave Templates for Inspiralling Compact Binaries

A. Gopakumar|ArXiv.org|Dec 19, 2007

Pulsars and Gravitational Waves Research4 citations

TL;DR

This paper introduces three new time-domain post-Newtonian (PN) waveform templates—TaylorK1, TaylorK2, and TaylorEt—for non-spinning compact binary inspirals, using energy balance and gauge-invariant variables to improve accuracy and cycle accumulation. These templates provide more gravitational wave cycles in a given frequency band than TaylorT1/T2, enhancing detection sensitivity for weak, mildly eccentric signals.

ABSTRACT

Compact binaries inspiralling along quasi-circular orbits are the most plausible gravitational wave (GW) sources for the operational, planned and proposed laser interferometers. We provide new class of restricted post-Newtonian accurate GW templates for non-spinning compact binaries inspiralling along PN accurate quasi-circular orbits. Arguments based on data analysis, theoretical and astrophysical considerations are invoked to show why these time-domain Taylor approximants should be interesting to various GW data analysis communities.

Motivation & Objective

• Develop new time-domain gravitational wave templates for non-spinning compact binaries that improve cycle accumulation in the frequency band of interest.
• Address limitations in existing TaylorT1 and TaylorT2 waveforms by introducing alternative formulations based on energy balance and gauge-invariant variables.
• Enhance detection efficiency for compact binaries with small residual eccentricities, which are astrophysically plausible but not well-captured by standard templates.
• Provide waveforms compatible with data analysis pipelines like the LSC Algorithms Library (LAL) and relevant to the LISA mission.
• Ensure numerical efficiency comparable to existing methods while improving phase accuracy and cycle count.

Proposed method

• Construct waveforms using the restricted PN approximation, with amplitude proportional to $ (Gm\omega/c^3)^{2/3} \cos 2\phi(t) $, where $ \phi(t) $ is the orbital phase.
• Derive $ \omega(t) $ and $ \phi(t) $ via energy balance: $ d\phi/dt = \omega(t) $, $ d\omega/dt = -\mathcal{L}(\omega)/ (d\mathcal{E}/d\omega) $, using 3PN energy and 3.5PN luminosity.
• Introduce TaylorEt waveforms by reparameterizing time in terms of the gauge-invariant energy variable $ \zeta = \mathcal{E} $, leading to a new evolution equation for $ \zeta(t) $.
• Use PN-expanded expressions for $ d\phi/d\hat{t} $ and $ d\zeta/d\hat{t} $ up to 3.5PN order, incorporating $ \eta $-dependent corrections and logarithmic terms.
• Ensure numerical cost is comparable to TaylorT1/T2 by solving ODEs in time domain with similar computational complexity.
• Leverage mathematical structure from the Damour-Deruelle timing formula for binary pulsars to guide the construction of phase evolution.

Experimental results

Research questions

• RQ1 Can new time-domain PN waveforms be constructed that accumulate more gravitational wave cycles than TaylorT1/T2 templates within the same frequency band?
• RQ2 How does reparameterization using the gauge-invariant energy variable $ \zeta $ affect the phase evolution and cycle count in compact binary inspirals?
• RQ3 To what extent do the new templates capture signals from mildly eccentric compact binaries, which are not well-modeled by standard circular templates?
• RQ4 Can the new waveforms be efficiently implemented in existing data analysis pipelines like LAL without increasing computational cost?
• RQ5 How do the new templates compare in accuracy and cycle count to existing families like TaylorK1, TaylorK2, and TaylorT1/T2?

Key findings

• TaylorEt 3.5PN waveforms yield 1,617.4 accumulated cycles for a neutron star binary ($ m = 2.8M_\odot, \eta = 0.25 $), exceeding TaylorT1/T2 but fewer than TaylorK1/K2.
• For a black hole–neutron star binary ($ m = 11.4M_\odot, \eta = 0.108 $), TaylorEt captures 335.4 cycles, again intermediate between TaylorT and TaylorK families.
• For a stellar-mass black hole binary ($ m = 20M_\odot, \eta = 0.25 $), TaylorEt produces 54.0 cycles, confirming the trend of higher cycle count than TaylorT1/T2.
• The increased cycle count arises because the TaylorEt evolution takes longer to reach the final frequency $ \omega_f $, compared to TaylorT1/T2, due to the reparameterization in terms of energy.
• The new templates are numerically as efficient as TaylorT1/T2, maintaining low computational cost while improving phase accuracy.
• Preliminary results suggest that TaylorEt waveforms are particularly effective at capturing GWs from compact binaries with small residual eccentricities, a key astrophysical scenario.