$I-$Love$-$Curvature: Exploring compact stars' quasi-universal relation with curvature scalars

TL;DR

This work investigates quasi-universal relations in compact stars by correlating standard observables, specifically the dimensionless tidal deformability $\Λ$ and normalized moment of inertia $I/M^3$, with normalized curvature scalars in general relativity: $\mathcal{R}$, $\mathcal{J}$, $\mathcal{K}$, and $\mathcal{W}$. Using piecewise polytropic neutron-star and color-flavor-locked quark-star equations of state, the authors compute stellar structures and curvature scalars (including central, surface, and volume-averaged quantities) and fit their relations to $I$ and $Λ$ via 5-parameter log-space polynomials, finding strong EOS-insensitive correlations, notably a near-EOS-independent maximum for $\mathcal{R}_c M^2$. They uncover new universal relations linking normalized central and volume-averaged pressure and energy density to $I$ and $Λ$, and demonstrate that $Λ$ measurements constrain curvature and interior properties of canonical-mass stars, consistent with EOS-dependent Bayesian inferences. The results broaden the reach of universal relations to spacetime geometry and microphysics, offering a robust framework to infer interior properties from gravitational-wave observations and potentially test strong-field gravity.

Abstract

We investigate quasi-universal relations in neutron stars linking standard observables, such as tidal deformability ($Λ$) and normalized moment of inertia ($\bar{I}$), with normalized curvature scalars in general relativity. These curvature scalars include the Ricci scalar ($\mathcal{R}$), the Ricci tensor contraction ($\mathcal{J}$), the Weyl scalar ($\mathcal{W}$), and the Kretschmann scalar ($\mathcal{K}$). We systematically examine both piecewise polytropic and color-flavor-locked equations of state, finding: (1) significant correlations between both local (central and surface) and global (volume-averaged) curvature scalars with $\bar{I}$ and $Λ$; (2) especially strong correlations between surface and volume-averaged curvature scalars and both $\bar{I}$ and $Λ$; (3) a near equation-of-state-independent maximum for the normalized Ricci scalar, suggesting a link to the trace anomaly; and (4) new universal relations involving normalized central and volume-averaged pressure and energy density, which also correlate strongly with $\bar{I}$ and $Λ$. Using constraints from GW170817 and low-mass X-ray binaries, we demonstrate that $Λ$ measurements directly constrain both scalar curvature quantities and the interior properties of canonical-mass neutron stars. These findings agree with the literature on equation-of-state-dependent Bayesian inference estimates. Our identified relations thus provide an equation-of-state-insensitive connection between stellar observables, spacetime geometry, and the microphysics of compact stars.

Figures (8)

• Figure 1: Mass–radius (left panel), mass–moment of inertia (middle panel), and mass–tidal deformability (right panel) relations are shown for the EoS sets used in this study. The models can be grouped as follows: (i) neutron stars with nucleonic (npem), hyperonic, mesonic, and quark matter compositions from Ref. Read2008, and (ii) quark stars in the color–flavor–locked (CFL) phase from Ref. Flores2017.
• Figure 2: The universal $I$--Love--$C$ relations are reproduced by the stellar solution of the EoS set used in this study, with data points color-coded similarly to Fig. \ref{['fig:mass_radius_I_lambda']}. The left and middle panels display the compactness $M/R$ as a function of the tidal deformability $\Lambda$ and the normalized moment of inertia $I/M^{3}$, respectively. The right panel shows the $I$--Love relation between $I/M^{3}$ and $\Lambda$. The percentages indicate the fraction of stellar solution data points within 0.2 dex of the fit, while MARE denotes the mean absolute residual error. The lower subpanels present the residuals relative to the fits for each relation. In the $I$--$\Lambda$ relation, neutron stars and quark stars overlap along a common trend.
• Figure 3: The universal central curvature $\mathcal{R}_c-\mathcal{J}_c-\mathcal{K}_c$ relations (normalized with $M$) obtained from the stellar solutions of the EoS set, with color categorization similar to Fig. \ref{['fig:mass_radius_I_lambda']}. The upper panels show the relations with tidal deformability $\Lambda$, while the lower panels show the corresponding relations with $I/M^{3}$. The percentages indicate the fraction of data points lying within 0.2 dex of the fit, MARE denotes the mean absolute residual error, and the lower subpanels present the residuals relative to the fits, similar to Fig. \ref{['fig:I_lambda_C']}. Both neutron star (NS) and quark star (QS) sequences are represented by the polynomial fits on Eq. (\ref{['EQ:fittingfunction']}) with coefficients listed in Table \ref{['tab:fitting_coeff_NS_Lam']}-\ref{['tab:fitting_coeff_QS_I']}.
• Figure 4: Similar to Fig. \ref{['fig:central_curvature_relations']}, but for the normalized surface curvature relations with tidal deformability (left) and moment of inertia (right). At the stellar surface, only the Kretschmann scalar and the Weyl scalar do not vanish, and both yield identical values. Note that the overall trend shows very similar characteristics to the $C$-- $\Lambda$--$I$ relations shown in Fig. \ref{['fig:I_lambda_C']}.
• Figure 5: Similar to Fig. \ref{['fig:central_curvature_relations']}, but for the volume-averaged curvature scalars ($\langle \mathcal{R} \rangle M^{2}$, $\langle \mathcal{J} \rangle M^{4}$, $\langle \mathcal{K} \rangle M^{4}$, and $\langle \mathcal{W} \rangle M^{4}$) relations with tidal deformability $\Lambda$ (upper) and moment of inertia $I/M^{3}$ (lower).