Trace anomaly and interior curvature of neutron stars in energy-momentum squared gravity

Abstract: In energy-momentum squared gravity (EMSG), the spacetime inside a neutron star is sourced by effective thermodynamic variables that need not coincide with the physical fluid pressure and energy density. It is therefore an open question whether the trace anomaly of dense matter -- the QCD measure of how strongly conformal symmetry is broken -- still organizes interior profiles and curvature in the same way it does in general relativity (GR). We adopt a clear matter-geometry separation: the trace anomaly is computed from the fluid sector alone, while spacetime curvature scalars are built from the variables that actually source the modified Tolman-Oppenheimer-Volkoff equations. For five relativistic mean-field equations of state, the radial trace-anomaly profiles increase monotonically from core to surface in all accepted EMSG models, as in GR, but split systematically with the EMSG coupling strength; the splitting grows with stellar compactness. Despite this deformation, curvature invariants still fall onto organized bands when plotted against the trace anomaly, extending the GR thermodynamic-geometric correspondence. The Ricci contraction shows the tightest organization, whereas the Ricci scalar remains the most equation-of-state sensitive. EMSG effects are modest for observationally accessible stars but largest in stiff, ultracompact configurations, indicating that the trace anomaly remains a useful thermodynamic label for interior geometry even when gravity couples nonlinearly to matter.

• The paper demonstrates that the trace anomaly, computed from fluid variables, effectively organizes spacetime curvature in the EMSG framework.
• It employs numerical integration of modified TOV equations across various EOSs and coupling constants to reveal vertical shifts and band splitting in curvature profiles.
• Results show that despite non-linear modifications from EMSG, the correlation between curvature invariants and the trace anomaly remains robust, supporting strong-field gravity tests.

Trace Anomaly and Interior Curvature of Neutron Stars in Energy-Momentum Squared Gravity

Overview and Motivation

This paper addresses the thermodynamic and geometric organization of neutron star interiors within the framework of Energy-Momentum Squared Gravity (EMSG), a nonlinear extension of General Relativity (GR) where spacetime curvature responds to quadratic combinations of the stress-energy tensor. While GR ties spacetime curvature directly to physical fluid variables—energy density and pressure—EMSG introduces effective thermodynamic variables, potentially decoupling fluid-sector quantities such as the trace anomaly from geometric invariants. The trace anomaly ($A$), stemming from QCD at supranuclear densities, characterizes the degree of conformal symmetry breaking in dense matter and has recently been shown in GR to organize both local thermodynamic state and curvature profiles across global observables.

The principal question examined is whether the trace anomaly, computed exclusively from fluid variables, remains an effective organizing parameter for interior spacetime curvature invariants when gravity is sourced by modified, nonlinear terms in EMSG. This is significant for strong-field tests of gravity, given that neutron star interiors represent a unique context where matter and geometry are both extreme and experimentally constrained.

Theoretical Formulation

EMSG modifies the Einstein-Hilbert action to include a term quadratic in the energy-momentum tensor, controlled by a coupling constant $\alpha$. The field equations permit recasting the gravitational response in terms of effective energy density ($E_{\mathrm{eff}}$) and pressure ($P_{\mathrm{eff}}$), which source the modified Tolman-Oppenheimer-Volkoff (TOV) equations. The matter-geometry separation is explicit: fluid variables ($E$, $P$) are used to define the trace anomaly, whereas curvature invariants (Kretschmann, Weyl, Ricci scalar, Ricci contraction) are constructed from $(E_{\mathrm{eff}}, P_{\mathrm{eff}})$.

The five selected relativistic mean-field equations of state (EOS)—NL3, IOPB-I, G3, SINPA, GM1—span the stiffness parameter space relevant for current neutron star phenomenology. The coupling parameter $\alpha$ is varied across a range consistent with prior EMSG studies, ensuring the nonlinear corrections are maximal in dense cores and negligible in vacuum.

Numerical Analysis and Results

Radial Evolution of Thermodynamic and Geometric Quantities

For each EOS and coupling, numerical integration of the modified TOV system yields radial profiles of the trace anomaly and curvature invariants. The key findings are as follows:

• Trace anomaly profiles: $A(r)$ increases outward for all accepted EMSG integrations and remains organized by EOS structure, mirroring the GR trend of minimal core anomaly and monotonic increase toward the surface, as proven in prior GR works (2606.20203).
• Curvature invariants: Kretschmann scalar, Ricci contraction, Ricci scalar, and Weyl scalar show vertical shifts and $\alpha$-induced band splitting, most pronounced in ultracompact configurations. The Ricci contraction ($\alpha$0) exhibits the tightest correlation with the trace anomaly, whereas the Ricci scalar ($\alpha$1) is the most EOS-sensitive, particularly in negative-$\alpha$2 cores.

Profile Organization Across Global Observables

Fixing macroscopic parameters—compactness, tidal deformability, normalized moment of inertia—organizes the trace anomaly profiles within bands with controlled EOS scatter, consistent with quasi-universal relations in GR. EMSG coupling induces systematic band splitting, most notably for stiff, compact stars. For typical observational parameters (e.g., those matched to GW170817 and NICER targets), the $\alpha$3-induced deviations remain moderate.

Curvature-Trace Anomaly Correspondence

A principal numerical test evaluates whether curvature invariants plotted against the fluid-sector trace anomaly collapse onto single-parameter bands. Results demonstrate:

• Monotonic Kretschmann–$\alpha$4 correlation: The relationship persists across EOSs and couplings, supporting the thermodynamic-geometric correspondence established in GR (2606.20203), with EMSG generating vertical shifts without destroying the ordering.
• Ricci contraction tightness: $\alpha$5 maintains minimal EOS scatter and organizes radial sequences, whereas $\alpha$6 displays branch structure reflecting effective trace decoupling and EOS sensitivity.
• Nontrivial extension to modified gravity: The geometric invariants built from effective variables in EMSG continue to be tightly parameterized by the fluid-sector trace anomaly, sustaining the interpretability of $\alpha$7 as a reduced thermodynamic coordinate even outside GR.

Implications and Future Directions

The findings demonstrate that the trace anomaly remains an effective organizing label for interior spacetime curvature even when nonlinear gravity modifies the sourcing. This preserves the practical utility of $\alpha$8 for inferring interior geometric and thermodynamic properties from global observables (mass, radius, tidal deformability), facilitating multimesenger astrophysical constraints.

Practically, the modest EMSG-induced profile splitting in observationally accessible sequences indicates that the trace anomaly–curvature correspondence remains robust amidst current bounds on $\alpha$9. Theoretical implications include the ability to segregate matter and geometry sectors for interior diagnostics even in extended gravity contexts; the Ricci contraction emerges as the most stable geometric probe across EOS ensembles.

Future developments should include

• Hybrid and phase transition EOSs: Incorporation of quark-hadron, hyperonic, and phase-transition models can probe the branch structure in curvature-trace anomaly relations.
• Self-consistent perturbation theory: Extension to rotational and tidal perturbations in fully nonlinear EMSG frameworks may refine the deformation of profile bands at fixed global observables.
• Bayesian inference and multimesenger data: Combining X-ray, gravitational wave, and moment-of-inertia measurements within a global EOS and modified gravity framework can quantitatively assess the universality and residual scatter in $E_{\mathrm{eff}}$0-curvature relations.

Conclusion

The paper establishes, through systematic matter-geometry separation and multi-EOS numerical analysis, that the trace anomaly computed from fluid variables persists as a reliable organizing parameter for interior spacetime curvature in EMSG. Nonlinear gravity introduces controlled deformations and vertical splitting of profile bands as a function of coupling and compactness, but the geometric invariants remain tightly correlated with the trace anomaly, especially for the Ricci contraction. The trace anomaly thus sustains its role as a thermodynamic label encoding information about both matter sector and interior geometry, supporting its continued relevance for strong-field tests of gravity and multimesenger astrophysics.