Exploring the universal $\bar{\mathcal{I}}-\mathcal{C}$ relations for relativistic stars in $f(Q)$ gravity

TL;DR

This work probes neutron stars in covariant $f(Q)$ gravity by deriving slowly rotating TOV equations and computing angular-velocity profiles and moments of inertia for four $f(Q)$ forms across 53 realistic EoS. It finds that rotational observables, notably the MOI, exhibit larger deviations from GR than mass profiles, indicating strong-field rotational signatures of nonmetricity. The authors analyze the quasi-universal $\bar{I}-C$ relation, finding an overall $\sim10\%$ residual across EoS, with linear and quadratic $f(Q)$ models remaining within observational bounds for PSR J0737–3039A, while logarithmic and exponential forms can produce larger deviations that may help distinguish gravity models when combined with precise data. These results confirm that $f(Q)$ gravity can be tested in the strong-field regime and motivate future exploration of EoS-insensitive relations such as $\bar{I}(\Lambda)$ to break degeneracies between dense-matter physics and modified gravity in upcoming observations.

Abstract

We investigate the properties of neutron stars within the framework of $f(Q)$ gravity by incorporating rotational effects through a slowly rotating metric. We derive the modified TOV equations and calculate the angular velocity profiles and moments of inertia (MOI) for linear, quadratic, exponential, and logarithmic $f(Q)$ models. Our results show that deviations in the MOI are more pronounced than those in the stellar mass profiles, suggesting that rotational observables are highly sensitive to geometric corrections. We also calculate a quasi-universal relation between the dimensionless MOI and compactness ($\bar{I}$-$C$). The linear and quadratic models are generally consistent with observational data from PSR J0737-3039A, although the deviations are small and difficult to distinguish from General Relativity due to inherent EoS variability. On other hand, the logarithmic and exponential models show larger deviations (over 20 %), exceeding the EoS-induced uncertainty reported by Suleiman & Read (2024), highlighting the relation's sensitivity to the $f(Q)$ gravity model. These results indicate that $f(Q)$ gravity could potentially be tested in the strong-field regime and point to a direction for future studies, such as investigating EoS-insensitive quasi-universal relations, like the $\bar{I}(Λ)$ relations, within the $f(Q)$ framework. Such relations may provide a clearer pathway for exploring possible signatures in strong-field gravity when combined with more precise future observations.

Figures (6)

• Figure 1: The normalized $\bar{\omega}/\Omega$ profile as a function of radius, computed using the SLy4 EoS at a density of $\rho = 1 \times 10^{15}$ g/cm$^3$. For each model, various values of $\alpha$ and $\beta$ are considered. By applying the matching condition between the interior and exterior solutions, different initial values are determined for each parameter, highlighting the deviations of each $f(Q)$ model from GR. In this graph, the star surface radius in GR is approximately $11.7$ km.
• Figure 2: Moment of inertia as a function of mass for SLy EoS using different values of the parameter $\alpha$ and $\beta$ of each model, in units of $\text{g cm}^2$. The red and green vertical lines represent the moment of inertia constraints for PSR J0737-3039A ($M = 1.338~ M_\odot$) from Jiang:2020uvb and Miao_2021_515_5071, respectively, derived using different methods at $90\%$ Credible Interval. The blue vertical line shows the constraint for a canonical neutron star mass at $M = 1.4~ M_\odot$ from Jiang:2020uvb calculation.
• Figure 3: Universal $\bar{I}-C$ relation of various $f(Q)$ model.
• Figure 4: The best-fitting $\bar{I}-C$ relation for GR and all $f(Q)$ models with various parameters $\alpha$ and $\beta$. The percentage deviation of the best-fitting $\bar{I}-C$ relation for each model from the GR best-fitting relation in this calculation is also shown. The magenta and pink line corresponds to the $\bar{I}-C$ relation calculation for PSR J0737-3039A from Miao_2021_515_5071 with error estimation about $\sim 10\%$. The orange solid line represents the difference between our GR calculation and the results from Breu_2016_459_646, which is approximately 4%.
• Figure 5: The best-fitting $\bar{I}$--$C$ relations for all $f(Q)$ models with an estimated EoS variability of 4-10%. The shaded regions represent the corresponding $\bar{I}$--$C$ bands, showing partial overlaps among models. These results highlight the sensitivity of the $\bar{I}$--$C$ relation to both the gravitational modification parameters and the EoS uncertainty.