Impact on Inferred Neutron Star Equation of State due to Nonlinear Hydrodynamics, Background Spin, and Relativity

Author(s)

Bretz, Joseph, Yu, Hang

Abstract

Tidal interaction is a unique, detectable signature in gravitational wave signals from inspiraling binary neutron stars (BNSs), which can be used to constrain the neutron star (NS) equation of state (EoS). The tidal interaction is resonantly amplified as the orbital frequency approaches the NS fundamental mode (f-mode) frequency. It has been shown that the exclusion of tidal resonance in parameter estimation leads to a significant bias in the inferred NS tidal deformability and hence the NS EoS [Pratten et al. PRL 129, 081102 (2022)]. The strength and location of tidal resonance depend sensitively on the f-mode frequency, which is typically modeled using its universal relation with the tidal deformability that is derived for an isolated, non-spinning NS assuming only linear fluid perturbations. In a BNS inspiral, the f-mode frequency can be corrected by at least three known effects: nonlinear hydrodynamics, background spin, and relativity. We use Hamiltonian Monte Carlo simulations to estimate the systematic bias on tidal deformability when each frequency correction is ignored. Our study considers both loud, individual events and the stacking of a population of detections. Both scenarios are expected when the next-generation detectors are available with a sensitivity level increased by about an order of magnitude.

Figures

The impact of TRCs on the frequency-domain GW waveform phase. The difference between $\Psi_x$ -- the waveform phase with the TRC denoted in the legend -- and $\Psi_\mathrm{b}$ -- the baseline waveform phase with FF corrections but no TRC -- is plotted in radians against GW frequency in Hz up to approximate merger. The nonlinear hydrodynamics TRC (blue dashed) reduces the f-mode frequency more strongly at higher orbital frequency, resulting in a reduction in waveform phase that sharply grows at high $f_\mathrm{gw}$. The anti-aligned background spin TRC in the primary NS (red dotted) reduces the f-mode frequency by a fixed amount, resulting in a more gradual reduction in waveform phase. The aligned background spin TRC (olive dotted) does the opposite, increasing the waveform phase. The GR TRC (gray dot-dashed) increases the f-mode frequency, resulting in less resonance and a longer inspiral relative to the baseline.
Caption The impact of TRCs on the frequency-domain GW waveform phase. The difference between $\Psi_x$ -- the waveform phase with the TRC denoted in the legend -- and $\Psi_\mathrm{b}$ -- the baseline waveform phase with FF corrections but no TRC -- is plotted in radians against GW frequency in Hz up to approximate merger. The nonlinear hydrodynamics TRC (blue dashed) reduces the f-mode frequency more strongly at higher orbital frequency, resulting in a reduction in waveform phase that sharply grows at high $f_\mathrm{gw}$. The anti-aligned background spin TRC in the primary NS (red dotted) reduces the f-mode frequency by a fixed amount, resulting in a more gradual reduction in waveform phase. The aligned background spin TRC (olive dotted) does the opposite, increasing the waveform phase. The GR TRC (gray dot-dashed) increases the f-mode frequency, resulting in less resonance and a longer inspiral relative to the baseline.
Comparison of the recovered posteriors for an injected waveform with the nonlinear TRC, $\mathcal{A}_\mathrm{nl}=-0.03$ and a SNR of 500. The blue posterior is recovered from the baseline model that has the free parameters listed in the legend, which does not include the nonlinear TRC, \textit{i.e.}, with $\mathcal{A}_\mathrm{nl}=0$. The purple posterior is recovered from a model that includes the nonlinear TRC with a uniform prior on $\mathcal{A}_{\rm nl} \in [-0.06, 0.06]$. The red posterior is recovered from a model that includes the nonlinear TRC whose prior is only negative. All three recovery models impose the Love-$\omega_{a0}$ relation. The vertical gold line denotes the injected value, $\bar{\lambda}=400$. The 5, 50, and 95 percentiles are denoted by vertical lines within each posterior.
Caption Comparison of the recovered posteriors for an injected waveform with the nonlinear TRC, $\mathcal{A}_\mathrm{nl}=-0.03$ and a SNR of 500. The blue posterior is recovered from the baseline model that has the free parameters listed in the legend, which does not include the nonlinear TRC, \textit{i.e.}, with $\mathcal{A}_\mathrm{nl}=0$. The purple posterior is recovered from a model that includes the nonlinear TRC with a uniform prior on $\mathcal{A}_{\rm nl} \in [-0.06, 0.06]$. The red posterior is recovered from a model that includes the nonlinear TRC whose prior is only negative. All three recovery models impose the Love-$\omega_{a0}$ relation. The vertical gold line denotes the injected value, $\bar{\lambda}=400$. The 5, 50, and 95 percentiles are denoted by vertical lines within each posterior.
Corner plot of the recovered posterior for an injected waveform with the nonlinear TRC, $\mathcal{A}_\mathrm{nl}=-0.03$ and SNR of 500. The gray posterior is recovered with the baseline model which has the free parameters listed in the legend and does not include the nonlinear TRC, $\mathcal{A}_\mathrm{nl}=0$. Unlike Figure \ref{fig:hist-Anl}, the Love-$\omega_{a0}$ relation is not used, allowing the f-mode frequency, $\bar{\omega}_{a0}$, to vary. The gold lines are the injected values, $\bar{\lambda}=400$ and $\bar{\omega}_{a0}=0.0786$. The dashed lines denote the 5, 50, and 95 percentiles. The contours in the 2D plot enclose 50\%, 75\%, and 90\% of the samples.
Caption Corner plot of the recovered posterior for an injected waveform with the nonlinear TRC, $\mathcal{A}_\mathrm{nl}=-0.03$ and SNR of 500. The gray posterior is recovered with the baseline model which has the free parameters listed in the legend and does not include the nonlinear TRC, $\mathcal{A}_\mathrm{nl}=0$. Unlike Figure \ref{fig:hist-Anl}, the Love-$\omega_{a0}$ relation is not used, allowing the f-mode frequency, $\bar{\omega}_{a0}$, to vary. The gold lines are the injected values, $\bar{\lambda}=400$ and $\bar{\omega}_{a0}=0.0786$. The dashed lines denote the 5, 50, and 95 percentiles. The contours in the 2D plot enclose 50\%, 75\%, and 90\% of the samples.
The difference between the waveform phase of the baseline model ($\Psi_\mathrm{b}$) and the waveform phases of a.) solid gold: the injected model with the nonlinear TRC ($\mathcal{A}_\mathrm{nl}$); b.) dotted gray: the recovered model with the f-mode frequency ($\bar{\omega}_{a0}$) as a free parameter; c.) dashed blue: the recovered model with the f-mode frequency ($\bar{\omega}_{a0}$) constrained by tidal deformability ($\bar{\lambda}$) and the Love-$\omega_{a0}$ relation. Using waveform mismatch, the SNR threshold is 206 between the blue and gold waveforms and 354 between the gray and gold waveforms.
Caption The difference between the waveform phase of the baseline model ($\Psi_\mathrm{b}$) and the waveform phases of a.) solid gold: the injected model with the nonlinear TRC ($\mathcal{A}_\mathrm{nl}$); b.) dotted gray: the recovered model with the f-mode frequency ($\bar{\omega}_{a0}$) as a free parameter; c.) dashed blue: the recovered model with the f-mode frequency ($\bar{\omega}_{a0}$) constrained by tidal deformability ($\bar{\lambda}$) and the Love-$\omega_{a0}$ relation. Using waveform mismatch, the SNR threshold is 206 between the blue and gold waveforms and 354 between the gray and gold waveforms.
Comparison of the recovered posteriors, stacked over 16 events, for an injected model with the nonlinear TRC, $\mathcal{A}_\mathrm{nl}=-0.03$. Each event has a SNR of 175 but different NS masses. The blue posteriors are recovered from a model that does not include the nonlinear TRC, $\mathcal{A}_\mathrm{nl}=0$. The red posteriors are recovered from a model that includes the nonlinear TRC as a free parameter, $\mathcal{A}_\mathrm{nl}$. The thin, faint posteriors are for individual events, and the thick, shaded posteriors are the product of stacking the 16 individual event posteriors. The vertical gold lines are the injected values. (Left) Stacked posteriors for tidal deformability with an injected value of, $\bar{\lambda}=400$. (Right) Stacked posterior for the nonlinear TRC with an injected value of $\mathcal{A}_\mathrm{nl}=-0.03$.
Caption Comparison of the recovered posteriors, stacked over 16 events, for an injected model with the nonlinear TRC, $\mathcal{A}_\mathrm{nl}=-0.03$. Each event has a SNR of 175 but different NS masses. The blue posteriors are recovered from a model that does not include the nonlinear TRC, $\mathcal{A}_\mathrm{nl}=0$. The red posteriors are recovered from a model that includes the nonlinear TRC as a free parameter, $\mathcal{A}_\mathrm{nl}$. The thin, faint posteriors are for individual events, and the thick, shaded posteriors are the product of stacking the 16 individual event posteriors. The vertical gold lines are the injected values. (Left) Stacked posteriors for tidal deformability with an injected value of, $\bar{\lambda}=400$. (Right) Stacked posterior for the nonlinear TRC with an injected value of $\mathcal{A}_\mathrm{nl}=-0.03$.
Comparison of the recovered posteriors, stacked over 16 events, for an injected model with the nonlinear TRC, $\mathcal{A}_\mathrm{nl}=-0.03$. Each event has a SNR of 175 but different NS masses. The blue posteriors are recovered from a model that does not include the nonlinear TRC, $\mathcal{A}_\mathrm{nl}=0$. The red posteriors are recovered from a model that includes the nonlinear TRC as a free parameter, $\mathcal{A}_\mathrm{nl}$. The thin, faint posteriors are for individual events, and the thick, shaded posteriors are the product of stacking the 16 individual event posteriors. The vertical gold lines are the injected values. (Left) Stacked posteriors for tidal deformability with an injected value of, $\bar{\lambda}=400$. (Right) Stacked posterior for the nonlinear TRC with an injected value of $\mathcal{A}_\mathrm{nl}=-0.03$.
Caption Comparison of the recovered posteriors, stacked over 16 events, for an injected model with the nonlinear TRC, $\mathcal{A}_\mathrm{nl}=-0.03$. Each event has a SNR of 175 but different NS masses. The blue posteriors are recovered from a model that does not include the nonlinear TRC, $\mathcal{A}_\mathrm{nl}=0$. The red posteriors are recovered from a model that includes the nonlinear TRC as a free parameter, $\mathcal{A}_\mathrm{nl}$. The thin, faint posteriors are for individual events, and the thick, shaded posteriors are the product of stacking the 16 individual event posteriors. The vertical gold lines are the injected values. (Left) Stacked posteriors for tidal deformability with an injected value of, $\bar{\lambda}=400$. (Right) Stacked posterior for the nonlinear TRC with an injected value of $\mathcal{A}_\mathrm{nl}=-0.03$.
The same as Figures \ref{fig:hist-Anl} and \ref{fig:bw-Anl}, but using an injected waveform with a spinning primary NS, resulting in a background spin TRC of $\Omega_\mathrm{A}/2\pi=(-100$ Hz). The SNR for both posteriors is 1000. Both recovery models include spin in the PP contribution to the waveform via $\chi_\pm$, but neither include the background spin TRC as a free parameter. Instead, the red posterior is recovered using a model that constrains the background spin TRC ($\Omega_\mathrm{A}$) using $\chi_\mathrm{A}$ and the I-Love relation, thus enabling the model to recover the injected parameter values. The waveform mismatch between the injected and recovered waveforms implies a threshold SNR of 2900 to discern between the two.
Caption The same as Figures \ref{fig:hist-Anl} and \ref{fig:bw-Anl}, but using an injected waveform with a spinning primary NS, resulting in a background spin TRC of $\Omega_\mathrm{A}/2\pi=(-100$ Hz). The SNR for both posteriors is 1000. Both recovery models include spin in the PP contribution to the waveform via $\chi_\pm$, but neither include the background spin TRC as a free parameter. Instead, the red posterior is recovered using a model that constrains the background spin TRC ($\Omega_\mathrm{A}$) using $\chi_\mathrm{A}$ and the I-Love relation, thus enabling the model to recover the injected parameter values. The waveform mismatch between the injected and recovered waveforms implies a threshold SNR of 2900 to discern between the two.
The same as Figures \ref{fig:hist-Anl} and \ref{fig:bw-Anl}, but using an injected waveform with a spinning primary NS, resulting in a background spin TRC of $\Omega_\mathrm{A}/2\pi=(-100$ Hz). The SNR for both posteriors is 1000. Both recovery models include spin in the PP contribution to the waveform via $\chi_\pm$, but neither include the background spin TRC as a free parameter. Instead, the red posterior is recovered using a model that constrains the background spin TRC ($\Omega_\mathrm{A}$) using $\chi_\mathrm{A}$ and the I-Love relation, thus enabling the model to recover the injected parameter values. The waveform mismatch between the injected and recovered waveforms implies a threshold SNR of 2900 to discern between the two.
Caption The same as Figures \ref{fig:hist-Anl} and \ref{fig:bw-Anl}, but using an injected waveform with a spinning primary NS, resulting in a background spin TRC of $\Omega_\mathrm{A}/2\pi=(-100$ Hz). The SNR for both posteriors is 1000. Both recovery models include spin in the PP contribution to the waveform via $\chi_\pm$, but neither include the background spin TRC as a free parameter. Instead, the red posterior is recovered using a model that constrains the background spin TRC ($\Omega_\mathrm{A}$) using $\chi_\mathrm{A}$ and the I-Love relation, thus enabling the model to recover the injected parameter values. The waveform mismatch between the injected and recovered waveforms implies a threshold SNR of 2900 to discern between the two.
The same as Figures \ref{fig:hist-Anl} and \ref{fig:bw-Anl}, but using an injected waveform with the relativistic TRC. The SNR for both posteriors is 1000. Both recovery models have the same four free parameters and impose the Love-$\omega_{a0}$ relation, but the red posterior includes the relativistic TRC. The recovery model for the blue posterior does not include relativistic effects and cannot fully recover the injected waveform. The best fit waveform from the blue posterior is plotted over the injected waveform in the right panel. The mismatch between the injected and recovered waveforms implies a threshold SNR of 610.
Caption The same as Figures \ref{fig:hist-Anl} and \ref{fig:bw-Anl}, but using an injected waveform with the relativistic TRC. The SNR for both posteriors is 1000. Both recovery models have the same four free parameters and impose the Love-$\omega_{a0}$ relation, but the red posterior includes the relativistic TRC. The recovery model for the blue posterior does not include relativistic effects and cannot fully recover the injected waveform. The best fit waveform from the blue posterior is plotted over the injected waveform in the right panel. The mismatch between the injected and recovered waveforms implies a threshold SNR of 610.
The same as Figures \ref{fig:hist-Anl} and \ref{fig:bw-Anl}, but using an injected waveform with the relativistic TRC. The SNR for both posteriors is 1000. Both recovery models have the same four free parameters and impose the Love-$\omega_{a0}$ relation, but the red posterior includes the relativistic TRC. The recovery model for the blue posterior does not include relativistic effects and cannot fully recover the injected waveform. The best fit waveform from the blue posterior is plotted over the injected waveform in the right panel. The mismatch between the injected and recovered waveforms implies a threshold SNR of 610.
Caption The same as Figures \ref{fig:hist-Anl} and \ref{fig:bw-Anl}, but using an injected waveform with the relativistic TRC. The SNR for both posteriors is 1000. Both recovery models have the same four free parameters and impose the Love-$\omega_{a0}$ relation, but the red posterior includes the relativistic TRC. The recovery model for the blue posterior does not include relativistic effects and cannot fully recover the injected waveform. The best fit waveform from the blue posterior is plotted over the injected waveform in the right panel. The mismatch between the injected and recovered waveforms implies a threshold SNR of 610.
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