Bayesian Inference of Neutron Star Properties in $f(Q)$ Gravity Using NICER Observations

Technical Summary: Bayesian Inference of Neutron Star Properties in f(Q) Gravity Using NICER Observations

Problem Statement
While General Relativity (GR) successfully describes gravitational phenomena across various scales, it faces conceptual challenges regarding singularities, dark matter, and dark energy, motivating the exploration of Modified Gravity (MG) theories. Among these, symmetric teleparallel gravity (STEGR) and its extension, $f(Q)$ gravity, offer a distinct geometric formulation where gravity is encoded in the non-metricity of spacetime rather than curvature or torsion. Although $f(Q)$ models have been studied in cosmology, their application to the strong-field regime of neutron stars (NSs) remains largely unexplored. Specifically, there is a lack of systematic Bayesian inference studies that constrain $f(Q)$ model parameters using observational data, as previous analyses often relied on phenomenological parameter choices without rigorous statistical confrontation with data.

Methodology
The authors investigate the internal structure of neutron stars within the framework of $f(Q)$ gravity by considering three representative functional forms: linear, logarithmic, and exponential models. To isolate the effects of modified gravity from uncertainties in dense matter physics, the study adopts the realistic, density-dependent relativistic mean-field DDME2 equation of state (EoS).

The methodology proceeds as follows:

1. Theoretical Framework: The authors derive the modified Tolman–Oppenheimer–Volkoff (TOV) equations for static, spherically symmetric spacetimes in $f(Q)$ gravity. These equations govern the stellar equilibrium and are solved numerically using a fourth-order Runge-Kutta (RK4) scheme.
2. Bayesian Inference: A Bayesian framework is employed to constrain the free parameters ($\alpha, \beta, Q_0$) of the three $f(Q)$ models. The analysis utilizes mass-radius posterior distributions from NICER observations of four pulsars: PSR J0030+0451, PSR J0740+6620, PSR J0437+4715, and PSR J0614+3329.
3. Statistical Analysis: The study computes the posterior distributions for model parameters and derived NS properties (mass, radius, tidal deformability, and compactness). Model comparison is performed using the Bayes factor (BF) to determine which functional form is statistically preferred by the data.

Key Contributions

• First Bayesian Constraints in $f(Q)$: This work represents the first systematic Bayesian inference analysis of $f(Q)$ gravity model parameters using neutron star observables, moving beyond cosmological tests to the strong-field regime.
• Comparative Model Analysis: The paper provides a rigorous statistical comparison of linear, logarithmic, and exponential $f(Q)$ models, evaluating their viability against high-precision NICER data.
• Parameter Degeneracy and Correlation: The study maps the correlations between model parameters and NS properties, revealing how different functional forms influence the coupling between gravitational modifications and stellar structure.

Results

• Model Preference: The Bayesian analysis, supported by Bayes factor comparisons, statistically favors the exponential $f(Q)$ model over the linear and logarithmic cases. The exponential model exhibits well-constrained parameters with nearly Gaussian, unimodal posterior distributions. In contrast, the linear model shows pronounced bimodality and broad parameter distributions, indicating strong degeneracies, while the logarithmic model displays moderate uncertainties.
• Neutron Star Properties: For the preferred exponential model, the study derives a radius and tidal deformability at $1.4 M_\odot$ of $R_{1.4} = 11.27^{+0.53}_{-0.36}$ km and $\Lambda_{1.4} = 156.95^{+84.02}_{-41.73}$, respectively. These values are consistent with current observational constraints.
• Maximum Mass and the Mass Gap: A significant finding is that all three constrained models predict maximum neutron star masses reaching $M_{\text{max}} \simeq 2.98 M_\odot$. The 95% confidence regions for the mass-radius relation extend into the lower mass gap ($\sim 2.5\text{--}5 M_\odot$). This prediction emerges naturally from the Bayesian-constrained parameter space without being imposed by construction.
• Correlations: The exponential model demonstrates a strong structural interdependence between the maximum mass and other NS properties, whereas the linear model shows statistical independence between maximum mass and other observables.

Significance
The paper claims that these results highlight the potential of neutron stars as powerful probes of symmetric teleparallel gravity in the strong-field regime. The emergence of stable configurations in the lower mass gap is presented as a clear, testable prediction of the $f(Q)$ framework, distinguishing it from standard GR scenarios which typically yield lower maximum masses for the same EoS. The study concludes that the exponential $f(Q)$ model is the most statistically supported description of neutron star structure among the three tested, suggesting that modified gravity effects can significantly enhance high-density gravitational support. The authors note that future precise mass measurements in the mass-gap region could serve as a powerful discriminator between GR and extended gravity scenarios. They also suggest that future work could involve joint Bayesian inference with flexible EoS parameterizations to further address degeneracies between strong-field gravity and microphysics.