Determination of neutron star radius from pulse profile modeling using profile likelihood

Technical Summary: Determination of Neutron Star Radius from Pulse Profile Modeling Using Profile Likelihood

Problem Statement
The Neutron Star Interior Composition Explorer (NICER) utilizes X-ray pulse profile modeling to determine neutron star mass and radius, primarily employing the X-ray Pulse Simulation and Inference (X-PSI) package. Current standard practice relies on Bayesian inference using nested sampling algorithms (e.g., MultiNest) to estimate posterior distributions. However, discrepancies in inferred radii have been observed across different studies, potentially attributed to sampler choice or convergence issues, particularly on flat likelihood surfaces for radii greater than 11 km. Furthermore, Bayesian marginalization over large prior ranges can introduce volume effects that bias parameter estimates. While Bayesian methods are the astrophysical standard, frequentist inference remains dominant in particle physics. This work addresses the need to compare Bayesian and frequentist approaches to ensure robustness in fundamental physics constraints derived from X-PSI.

Methodology
The authors implemented a frequentist inference pipeline using the profile likelihood technique within the existing X-PSI framework, ensuring the underlying physical model and likelihood implementation remained identical to the standard Bayesian approach.

• Data Source: The analysis utilized a synthetic NICER + XMM-Newton dataset (designated syntX1), previously used in H25 to test nested samplers.
• Model Configuration: The study employed the ST-U pulse-profile model, featuring two single-temperature, simply connected circular hot spots. The neutron star atmosphere was modeled using the NSX fully ionized hydrogen model, with photon propagation calculated via general-relativistic ray tracing in an oblate Schwarzschild spacetime.
• Frequentist Implementation:
  • The parameter vector was divided into the parameter of interest (equatorial radius, $R_{eq}$) and nuisance parameters ($\alpha$).
  • The nuisance parameters (including mass, distance, inclination, hotspot geometry, temperatures, and interstellar hydrogen column density) were profiled out by maximizing the likelihood $L(R_{eq}, \alpha)$ with respect to $\alpha$ for fixed values of $R_{eq}$.
  • Optimization was performed using the Nelder-Mead simplex algorithm (scipy.optimize.minimize).
  • Confidence intervals were constructed using the $\Delta\chi^2$ statistic ($\Delta\chi^2 = -2 \ln L_{profile} + 2 \ln L_{max}$), assuming a $\chi^2$ distribution for one degree of freedom.
• Comparison Baseline: Results were compared against a standard Bayesian analysis using the MultiNest nested sampling algorithm with identical settings and low-resolution X-PSI configurations.

Key Contributions

1. Proof-of-Principle Application: This work demonstrates the first application of frequentist profile likelihood inference to determine neutron star radii using the X-PSI package.
2. Methodological Comparison: It provides a direct comparison between Bayesian and frequentist inference pipelines on identical synthetic data, isolating the inference methodology as the sole variable.
3. Open Science: The authors have publicly released their frequentist inference code modifications for X-PSI to facilitate further research.

Results
Using the synthetic dataset syntX1 with an injected true radius of 12.176 km:

• Bayesian Inference: The marginalized 68% credible interval for the radius was $R_{68,bayes} = 11.970^{+0.180}_{-0.181}$ km. This result agreed with the injected value within $1.1\sigma$.
• Frequentist Inference: The profile likelihood yielded a best-fit radius of 12.096 km. The 68% confidence interval was $r_{68} = 12.096^{+0.064}_{-0.182}$ km, and the 95% interval was $r_{95} = 12.096^{+0.316}_{-0.397}$ km. The frequentist estimate agreed with the injected value within $0.4\sigma$.
• Precision: The precision of the frequentist method was found to be comparable to the Bayesian analysis.
• Computational Efficiency: The frequentist analysis required approximately 9 CPU-core hours, whereas the Bayesian analysis required approximately 3,840 CPU-core hours (80 hours on a 48-core node). The frequentist approach is thus orders of magnitude faster.

Significance and Claims
The paper positions this work as a proof-of-principle study. The authors claim that frequentist inference via profile likelihood can successfully recover the true neutron star radius to an accuracy of $< 1\sigma$ using synthetic data. The technique offers precision comparable to Bayesian analysis but with significantly reduced computational cost. The authors conclude that this method serves as a valuable complement to Bayesian inference, offering a robust alternative for parameter estimation in X-PSI. They explicitly state that future work will involve applying this technique to real pulsar data from the NICER telescope, but no such application is presented in this manuscript.