[Paper Review] Rotational-tidal phasing of the binary neutron star waveform

Philippe Landry|arXiv (Cornell University)|May 4, 2018

Pulsars and Gravitational Waves Research47 references13 citations

TL;DR

This paper calculates the leading-order rotational-tidal contributions to the gravitational wave phase from binary neutron stars, showing that spin-tide couplings generate a 6.5PN correction to the waveform phase—larger than octupole tidal effects and nearly as significant as current quadrupole contributions in spinning systems. The key result is the identification of a new effective tidal parameter, $\tilde{X}$, which quantifies spin-induced tidal deformations and could impact parameter estimation in future gravitational wave detectors.

ABSTRACT

Tidal forces cause inspiralling binary neutron stars to deform, leaving a measurable imprint on the gravitational waves they emit. The induced stellar multipoles are an added source of gravitational radiation and modify the orbital dynamics, producing a slight acceleration of the coalescence which manifests as a phase shift in the waveform relative to point-particles. The dominant piece of this tidal phase comes from the mass quadrupoles, which contribute at fifth post-Newtonian order (5PN). Current quadrupoles and mass octupoles contribute at higher orders. For spinning neutron stars, additional multipole moments are induced by nonlinear couplings between spin and tides. We calculate these rotational-tidal deformations assuming the stars are rotating slowly and the tides are weak and quasi-stationary. The stellar multipole moments are read off from an asymptotically flat metric that encodes the difference between their tidal response and a black hole's. The multipoles are subsequently inserted into post-Newtonian formulas for the orbit and the gravitational radiation. We find that, at leading order, the rotational-tidal deformations make a 6.5PN contribution to the tidal phase. Their effect on the waveform is thus larger than that of the mass octupoles, and nearly as large as that of the current quadrupoles, in systems with non-negligible spin.

Motivation & Objective

• To quantify the impact of spin-tide couplings on the gravitational wave phase in binary neutron star systems.
• To derive the leading-order rotational-tidal deformations in the post-Newtonian framework under slow rotation and weak tidal fields.
• To assess the relative importance of rotational-tidal effects compared to standard tidal and current quadrupole contributions.
• To identify a new effective tidal parameter $\tilde{X}$ that captures spin-induced multipole moments and their influence on waveform phasing.
• To evaluate the potential for future detection of $\tilde{X}$ and its implications for Bayesian parameter estimation in gravitational wave astronomy.

Proposed method

• Uses the asymptotically flat, stationary vacuum metric from Thorne (1980) to model the spacetime around tidally deformed neutron stars.
• Calculates multipole moments via the metric’s multipole structure, relating them to Love numbers from prior works (Pani et al. 2015b; Landry 2017; Gagnon-Bischoff et al. 2018).
• Applies post-Newtonian formulas for orbital dynamics and gravitational wave emission to compute the phase correction from rotational-tidal couplings.
• Derives the leading-order phase contribution from bilinear spin-tide couplings, identifying a 6.5PN term proportional to $\tilde{X}$.
• Computes the 8PN gravitomagnetic octupole term $\tilde{\Sigma}_3$ for the first time, using consistent PN scaling and multipole decomposition.
• Identifies a discrepancy in the $\tilde{\Sigma}$ coefficient due to non-conservative contributions in the energy integration, though the PN scaling and form are expected to remain robust.

Experimental results

Research questions

• RQ1 What is the post-Newtonian order of the leading rotational-tidal phase correction in binary neutron star waveforms?
• RQ2 How do spin-tide couplings generate new multipole moments that affect the gravitational wave phase?
• RQ3 What is the relative magnitude of rotational-tidal effects compared to standard mass quadrupole and current quadrupole tidal contributions?
• RQ4 Can a new effective tidal parameter $\tilde{X}$ be defined to capture spin-induced tidal deformations, and how might it affect parameter estimation?
• RQ5 What are the implications of the identified discrepancy in the $\tilde{\Sigma}_3$ coefficient for future waveform modeling and detector sensitivity?

Key findings

• Rotational-tidal deformations contribute a 6.5PN correction to the gravitational wave phase, which is larger than the 7PN octupole tidal term and nearly as large as the 6PN current quadrupole term in spinning systems.
• The leading-order rotational-tidal phase is proportional to a new effective tidal parameter $\tilde{X}$, defined as a combination of spin and tidal coupling coefficients.
• The 8PN gravitomagnetic octupole term $\tilde{\Sigma}_3$ is computed for the first time, with a coefficient that may be affected by non-conservative contributions in the energy integration.
• The 6.5PN term from spin-tide couplings scales linearly with the dimensionless spins of the neutron stars, making it more prominent in systems with significant spin.
• Despite a discrepancy in the $\tilde{\Sigma}_3$ coefficient, the PN scaling and overall structure of the phase terms are expected to be robust, even if numerical values shift.
• Future detectors like LISA, Einstein Telescope, and Cosmic Explorer may be able to measure $\tilde{X}$, and its omission could bias recovery of 5PN and 6PN tidal parameters in Bayesian inference.