Inferring neutron star properties through gravitational waves from r-modes and their relativistic counterparts

Technical Summary: Inferring Neutron Star Properties through Gravitational Waves from r-modes and their Relativistic Counterparts

Problem Statement
Continuous gravitational waves (CGW) from neutron stars, specifically those generated by $r$-mode oscillations and their relativistic counterparts known as axial-led hybrid (ALH) modes, remain undetected. While current searches have established upper limits on the saturation amplitude ($\alpha_s$), the potential for future detection with improved detector sensitivity necessitates a framework to understand what physical properties can be inferred from such signals. Unlike signals from "mountains" (mass quadrupoles), ALH modes offer unique information through a parameter $\kappa$ that satisfies universal relations with the star's compactness. The central problem addressed is how to quantitatively infer neutron star properties—specifically the moment of inertia ($I_{zz}$), the perpendicular magnetic dipole moment component ($m_p$), and the saturation-related parameter ($\tilde{\alpha}$)—from CGW signals, and how the inclusion of electromagnetic distance measurements or universal relations affects the accuracy of these inferences.

Methodology
The authors propose two distinct inference frameworks, both assuming the neutron star spins down due to a combination of magnetic dipole radiation and gravitational wave emission (via $r$-modes or ALH modes). The analysis employs a Fisher information matrix-based approach for quantitative error estimation, validated through Monte Carlo simulations.

• Framework 1 (Electromagnetic Distance Assumption): This approach assumes the distance to the star ($r$) is known from electromagnetic observations with 20% accuracy. Using the observed gravitational wave strain amplitude ($h_0$), frequency ($f$), and its derivatives ($\dot{f}, \ddot{f}$), the authors derive analytical expressions to solve for $I_{zz}$, $m_p$, and $\tilde{\alpha}$. The error propagation is calculated using the covariance matrix derived from the Fisher information metric.
• Framework 2 (Universal Relations for Direct Distance Measurement): This framework is designed for targeted or narrow-band searches where the rotational frequency ($f_{rot}$) is known. It leverages the fact that ALH mode frequencies deviate slightly from the Newtonian $r$-mode relation ($f = \frac{4}{3}f_{rot}$) by a factor $\kappa$. The authors utilize universal relations linking $\kappa$ to the star's compactness ($C = M/R$) and the normalized moment of inertia ($\bar{I} = I_{zz}/M^3$). By assuming an Equation of State (EOS) and using the measured $\kappa$ (derived from $f$ and $f_{rot}$), they calculate the compactness and moment of inertia directly. This allows for the direct measurement of the distance $r$ from the CGW signal itself, without relying on electromagnetic distance estimates.

Key Contributions

1. Dual Inference Frameworks: The paper establishes two methodologies for parameter estimation. Framework 1 refines the approach of previous studies (e.g., Lu et al. 2023) by incorporating ALH modes, while Framework 2 introduces a novel method to measure distance directly from GW data using universal relations, bypassing the need for electromagnetic distance priors.
2. Quantitative Error Analysis: The study provides a detailed error budget for inferring $I_{zz}$, $m_p$, and $\tilde{\alpha}$ (and $r$ in Framework 2) as a function of observation time ($T$), braking index ($n$), signal frequency ($f$), and spin-down rate ($\dot{f}$).
3. Identification of Dominant Error Sources: The analysis explicitly identifies that in Framework 1, the error in distance measurement dominates the total error budget for all parameters, even for observation times up to 4 years. Conversely, Framework 2 demonstrates that accurate parameter inference is possible without electromagnetic distance priors, provided universal relations hold.
4. EOS Dependence Analysis: The authors investigate the dependence of Framework 2 on the Equation of State. While the absolute values of inferred parameters (like distance and moment of inertia) vary significantly with the choice of EOS, the relative errors of these parameters are found to be largely independent of the EOS.

Results

• Framework 1 Performance: For a 1-year observation, the median relative error for $I_{zz}$ is approximately 32%, and for $\tilde{\alpha}$ and $m_p$, it is approximately 16%. These errors are dominated by the assumed 20% uncertainty in the electromagnetic distance measurement. The errors saturate to the distance error limit as observation time increases.
• Framework 2 Performance: This framework yields significantly higher accuracy. For a 1-year observation, the relative error for $I_{zz}$, $\tilde{\alpha}$, and $m_p$ can be as low as 10–20%, and the distance $r$ can be measured with ~32% accuracy. The accuracy is highest for braking indices $n \in [4, 6]$, low frequencies ($f \approx 100$ Hz), and high spin-down rates ($\dot{f} \approx -10^{-8}$ Hz s$^{-1}$).
• Parameter Constraints: The study notes that for very slow spin-down rates ($\dot{f} < -10^{-11}$ Hz s$^{-1}$), the errors become prohibitively large, often violating the condition $3 < n < 7$ required for the models.
• Saturation Amplitude: While $\tilde{\alpha}$ can be inferred, converting this to the physical saturation amplitude $\alpha_s$ requires knowledge of the mass, radius, and the EOS-dependent parameter $\tilde{J}$. The authors note that $\tilde{J}$ is bounded within a factor of 2, allowing for rough fiducial estimates even without a precise EOS.

Significance and Claims
The paper claims that while the detection of continuous gravitational waves from $r$-modes or ALH modes remains a future prospect, the proposed frameworks offer a roadmap for extracting macroscopic neutron star properties once such signals are detected.

• Utility of Framework 2: The authors highlight that Framework 2 is particularly significant because it enables the direct measurement of a neutron star's distance from gravitational wave data alone, a capability not available for signals from mountains or standard $r$-mode analyses without universal relations.
• Limitations: The authors modestly acknowledge that Framework 2 is restricted to slowly rotating stars and relies on the assumption of a specific Equation of State. However, they argue that even with this dependence, the framework can provide robust bounds on parameters (especially distance) across a range of realistic EOS models.
• Future Directions: The paper suggests that if a CGW signal is detected from a source with a known distance (e.g., the Crab Pulsar), Framework 2 could be inverted to constrain the Equation of State. Furthermore, for Low-mass X-ray Binaries (LXMBs) where mass measurements are available via electromagnetic observations, Framework 2 could estimate $I_{zz}$, $m_p$, and $r$ independently of the EOS.

The authors conclude that while their current analysis uses a Fisher matrix approach, future work should employ Bayesian inference to incorporate priors and handle non-linearities more robustly. They also note the need to explore more complex spin-down models and magnetic field configurations beyond the simple dipolar assumption.