[Paper Review] Detectability of Mode Resonances in Coalescing Neutron Star Binaries

Prakash Balachandran, Éanna É. Flanagan|ArXiv.org|Jan 15, 2007

Pulsars and Gravitational Waves Research3 citations

TL;DR

This paper investigates the detectability of gravitational wave mode resonances in coalescing neutron star binaries using Fisher matrix analysis. It finds that for a 1.4M⊙,1.4M⊙ binary, resonance effects require phase shifts of ∼8.1, 2.9, and 1.8 radians at 16, 32, and 64 Hz, respectively, to be detectable by LIGO—implying such resonances are unlikely unless neutron stars have anomalously high spins.

ABSTRACT

Inspirals of neutron star-neutron star binaries are a promising source of gravitational waves for gravitational wave detectors like LIGO. During the inspiral, the tidal gravitational field of one of the stars can resonantly excite internal modes of the other star, resulting in a phase shift in the gravitational wave signal. We compute using a Fisher-matrix analysis how large the phase shift must be in order to be detectable. For a $1.4 M_\odot, 1.4 M_\odot$ binary the result is $\sim 8.1, 2.9$ and 1.8 radians, for resonant frequencies of $16, 32$ and 64 Hz. The measurement accuracies of the other binary parameters are degraded by inclusion of the mode resonance effect.

Motivation & Objective

• To determine the minimum detectable phase shift due to resonant excitation of neutron star internal modes in gravitational wave signals.
• To assess whether such resonances could be observed in advanced LIGO data, given realistic signal-to-noise ratios and noise spectra.
• To quantify how the inclusion of mode resonance parameters degrades the measurement accuracy of standard binary parameters.
• To evaluate the dependence of detectability on resonance frequency and binary mass configuration.

Proposed method

• Uses a simplified gravitational wave signal model with seven parameters: amplitude 𝒜, coalescence time tc, phase φc, total mass M, reduced mass μ, resonance frequency f₀, and phase shift ΔΦ.
• Applies the Fisher matrix formalism to compute measurement uncertainties, with the inner product defined using the advanced LIGO noise power spectral density Sn(f).
• Modifies the base signal phase Φ₀(f) to include a linear phase shift ΔΦ near resonance frequency f₀, as per Eq. (1), with zero bandwidth approximation.
• Numerically computes the Fisher matrix and its inverse to obtain parameter error covariances, with ΔΦ set to 1 for baseline computation.
• Determines detectability threshold as ΔΦ ≳ Δ(ΔΦ), where Δ(ΔΦ) is the rms error in ΔΦ from the inverse Fisher matrix.

Experimental results

Research questions

• RQ1 What is the minimum phase shift ΔΦ required for a mode resonance to be detectable in the gravitational wave signal of a coalescing neutron star binary?
• RQ2 How does the detectability threshold of ΔΦ depend on the resonance frequency f₀ and the binary mass configuration?
• RQ3 To what extent does including mode resonance parameters degrade the measurement accuracy of standard binary parameters like chirp mass and coalescence time?
• RQ4 Are resonant excitations of r-modes or other modes detectable under realistic neutron star spin assumptions?

Key findings

• For a 1.4M⊙,1.4M⊙ binary, the minimum detectable phase shift is approximately 8.1 radians at 16 Hz, 2.9 radians at 32 Hz, and 1.8 radians at 64 Hz.
• Resonances at higher frequencies (e.g., 64 Hz) are easier to detect due to lower measurement uncertainty in the phase shift parameter.
• The inclusion of mode resonance parameters increases the measurement errors of standard parameters such as chirp mass and coalescence time.
• The detectability criterion ΔΦ ≳ 1 is generally necessary, but for typical neutron star spin frequencies, ΔΦ remains small, making detection unlikely.
• The results suggest that detectable resonances require neutron star spin frequencies of several hundred Hz, which are considered unlikely in most NS-NS inspirals.
• The phase shift effect is most prominent in the 16–64 Hz band, with detectability thresholds decreasing with increasing f₀ due to better signal resolution in that band.