Love can be Tough to Measure

TL;DR

This paper investigates how neutron-star equation-of-state measurements from gravitational waves are impacted by the interplay between point-particle and finite-size (tidal) terms in the waveform phase. It shows that leading-order point-particle corrections at $4\mathrm{PN}$ and higher can meaningfully affect the phase, comparable to the tidal contributions that first appear at $5\mathrm{PN}$, and thus can bias EoS inferences if neglected. Through a Fisher-misher analysis, the authors demonstrate that systematic errors from missing high-order PP terms can dominate over statistical errors for current detectors and become even more problematic for future, more sensitive observatories, unless higher-order terms (including next-to-leading order in $\eta$) are incorporated. They argue that the solution may lie in including these terms in analytical templates or in constructing accurate numerical-relativity–informed hybrids, while also emphasizing the need for tight control of numerical waveform systematics. Overall, the work highlights the necessity of high-precision waveform modeling to robustly extract NS EoS information from gravitational-wave observations.

Abstract

The waveform phase for a neutron star binary can be split into point-particle terms and finite-size terms (characterized by the Love number) that account for equation of state effects. The latter first enter at 5 post-Newtonian (PN) order (i.e. proportional to the tenth power of the orbital velocity), but the former are only known completely to 3.5 PN order, with higher order terms only known to leading-order in the mass-ratio. We here find that not including point-particle terms at 4PN order to leading- and first-order in the mass ratio in the template model can severely deteriorate our ability to measure the equation of state. This problem can be solved if one uses numerical waveforms once their own systematic errors are under control.

Figures (3)

• Figure 1: (Color online) $N_{\hbox{\tiny useful}}$ vs. mass for point-particle phase terms to leading order in $\eta$ and for the leading-order, finite-size GW phase term. We focus on an Adv. LIGO detection, with (high-power, zero-detuned) spectral noise AdvLIGO-noise, $f_{\hbox{\tiny min}} = 10$Hz and $f_{\hbox{\tiny max}} = \min(f_{\hbox{\tiny ISCO}},f_{\hbox{\tiny cont}})$, with $f_{{\hbox{\tiny ISCO}}} = (6^{3/2} \pi M)^{-1}$ the innermost-stable circular orbit frequency for a point-particle in a Schwarzschild background and $f_{{\hbox{\tiny cont}}}$ the approximate contact frequency. The finite-size terms are modeled with two representative EoSs for realistic NSs (SLy SLy, Shen Shen1Shen2), both of which allow for stars above the PSR J0348+0432 limit 2.01NS. Other realistic EoSs APRLS lead to results that fall between those shown. We also show 1/SNR for a NS binary at luminosity distance $D_L=100$Mpc. Observe that the finite-size terms and the (incomplete) point-particle terms lead to a comparable number of useful cycles, all above the rough 1/SNR threshold.
• Figure 2: (Color online) Ratio of the estimate of the systematic to the statistical error on the averaged dimensionless deformability $\bar{\lambda}_{2,s}$ versus NS mass for Adv. LIGO. The systematic errors arise due to not including the $n$th PN order point-particle term to leading-order in $\eta$, which is currently known. The statistical error is induced by detector noise. These errors are estimated using the NS EoSs SLy and Shen, as explained in Fig. \ref{['fig:useful']}. Observe that the systematic error dominates the error budget when $n \leq 6$.
• Figure 3: (Color online) Statistical and systematic errors on $\ln \bar{\lambda}_{2,s}$ due to not including the 4PN term at next-to-leading order in $\eta$ (currently unknown) for SLy and Shen EoSs using Adv. LIGO. For the latter, we set $|\psi^{\hbox{\tiny PP}}_{{\hbox{\tiny 4PN}},1}/\psi^{\hbox{\tiny PP}}_{{\hbox{\tiny 4PN}},0}|=1$ and 5, while other choices can be obtained through a simple linear rescaling. Observe that when the coefficient of the next-to-leading order term is large enough, the systematic error dominates over the statistical one.