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

This study demonstrates that in Energy-Momentum Squared Gravity, the trace anomaly of dense matter continues to organize the interior curvature profiles of neutron stars into systematic bands similar to General Relativity, despite the theory's nonlinear coupling between matter and geometry.

The Gist

Imagine a neutron star as a cosmic pressure cooker, a city-sized ball of matter so dense that a single teaspoon would weigh a billion tons. For decades, physicists have used Einstein's General Relativity (GR) to understand how these stars hold themselves together. But what happens if gravity works slightly differently when matter is this dense? That is the question this paper explores using a theory called Energy-Momentum Squared Gravity (EMSG).

Here is the core of their discovery, broken down into simple concepts and analogies.

1. The Two Different "Languages"

In standard Einstein gravity, the "stuff" inside the star (matter) and the "shape" of space around it (geometry) speak the same language. If you know the pressure and density of the matter, you know exactly how space curves.

In this new theory (EMSG), the authors imagine a scenario where the star's matter and the space around it speak different dialects.

• The Matter Side: The actual physical pressure and density of the star's fluid.
• The Geometry Side: The "effective" pressure and density that actually tell space how to curve. In EMSG, gravity doesn't just react to the matter; it reacts to the square of the matter's energy, creating a modified version of reality inside the star.

The authors made a strict rule: They would calculate the "anomaly" (a measure of how weird the matter is behaving) using only the real matter, but they would calculate the curvature of space using the modified, effective matter. They wanted to see if these two different languages could still tell a coherent story together.

2. The "Trace Anomaly": The Star's Internal Fingerprint

The paper focuses on something called the trace anomaly. Think of this as a "thermodynamic fingerprint" of the star's interior.

• In a perfectly balanced, simple gas, this fingerprint has a specific value.
• In the extreme, messy environment of a neutron star, this value changes. It tells us how much the matter is breaking the rules of symmetry.

The researchers asked: If we change the laws of gravity (EMSG), does this fingerprint still organize the star's shape in a predictable way?

3. The Main Discovery: A Deformed but Organized Map

The team ran simulations using five different models for how neutron star matter behaves (like different recipes for the star's "soup"). They found three main things:

A. The Fingerprint Still Moves Upward
In our normal universe (General Relativity), as you move from the center of the star to the surface, this "anomaly fingerprint" increases in a smooth, predictable line.

• The Result: Even in this new gravity theory, the fingerprint still moves up smoothly from the core to the surface. The "map" of the star's interior is still organized, just like in Einstein's theory.

B. The "Split" Effect
However, the new gravity theory adds a twist. Depending on how strong the new gravity effect is (controlled by a number called $\alpha$), the lines on the map start to split.

• Analogy: Imagine a group of hikers walking up a mountain. In normal gravity, they all walk in a single tight line. In this new gravity, they still walk up the mountain in the same direction, but the group fans out. The "stiffer" the star (the more rigid the matter), the wider the fan becomes.
• The split is small for normal stars but gets much larger for the most extreme, compact stars.

C. The Curvature Still Follows the Fingerprint
This is the most surprising part. Even though the "matter language" and the "geometry language" are different, and even though the group of hikers has fanned out, the shape of the space (curvature) still lines up perfectly with the fingerprint.

• Analogy: Imagine you have a set of keys (the fingerprint) and a set of locks (the curvature of space). In normal gravity, Key A opens Lock A. In this new theory, the keys are slightly bent and the locks are slightly warped. Yet, if you plot them against each other, they still fit into neat, organized bands.
• Specifically, the Ricci Contraction (a specific way of measuring how much matter bends space) showed the tightest, most organized relationship with the fingerprint.

4. Why This Matters

The paper concludes that even if gravity behaves in a weird, nonlinear way inside a neutron star, the thermodynamic fingerprint (the trace anomaly) remains a useful tool.

It acts like a reliable compass. Even if the terrain (gravity) changes, the compass still points in a way that helps us understand the landscape. The researchers found that for stars we can actually observe (like those measured by the NICER telescope or gravitational wave detectors), the changes are modest. The "fan-out" effect is most dramatic in theoretical, ultra-dense stars that we haven't seen yet.

Summary

In short, the authors took a theory where gravity and matter interact in a complex, squared-off way. They asked: "Does the internal structure of a neutron star fall apart?"
The answer is No. The internal structure remains surprisingly organized. The "fingerprint" of the matter still predicts the shape of space, even if the relationship is slightly stretched and split apart by the new gravity rules. The universe, it seems, is robust enough to keep its internal order even when the rules of gravity get a little weird.

Technical Summary

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

Problem Statement
In General Relativity (GR), the interior structure of neutron stars (NSs) is governed by the Tolman–Oppenheimer–Volkoff (TOV) equations, where spacetime curvature is sourced directly by the physical energy density ($E$) and pressure ($P$) of the stellar fluid. Recent GR studies have established a robust correspondence between the trace anomaly of dense matter ($\Delta$)—a dimensionless measure of conformal symmetry breaking—and spacetime curvature invariants. However, in Energy-Momentum Squared Gravity (EMSG), the field equations couple spacetime curvature to the square of the energy-momentum tensor ($T_{\mu\nu}T^{\mu\nu}$). This introduces density-dependent corrections, meaning the metric is sourced by effective thermodynamic variables ($E_{\text{eff}}, P_{\text{eff}}$) that differ from the physical fluid variables.

The central open question addressed is whether the trace anomaly, defined strictly from the physical fluid sector, retains its organizing power over interior spacetime curvature when the geometry is sourced by these modified effective variables. Specifically, does the thermodynamic-geometric correspondence observed in GR survive the decoupling of the fluid sector from the geometry sector in EMSG?

Methodology
The authors adopt a deliberate "matter–geometry separation" protocol to isolate the effects of modified gravity:

1. Matter Sector (Theory Independent): The trace anomaly $\Delta(r)$ is computed exclusively from the physical fluid variables ($E, P$) using the standard definition $\Delta = 1/3 - P/E$. No EMSG coupling parameter ($\alpha$) enters this definition.
2. Geometry Sector (Theory Dependent): Spacetime curvature invariants—the Kretschmann scalar ($K$), Weyl scalar ($W$), Ricci scalar ($R$), and Ricci contraction ($F$)—are constructed from the metric generated by the effective variables ($E_{\text{eff}}, P_{\text{eff}}$) derived from the EMSG field equations.
3. Numerical Implementation: The modified TOV equations are solved for five relativistic mean-field (RMF) equations of state (EOS): NL3, IOPB-I, G3, SINPA, and GM1. The coupling parameter $\alpha$ is varied within the range $[-6.07 \times 10^6, 6.07 \times 10^6] \, \text{m}^2$.
4. Global Constraints: Sequences are constructed at fixed compactness ($C$), tidal deformability ($\Lambda$), and normalized moment of inertia ($\bar{I}$). While the background profiles are modified by EMSG, the mapping from global observables to central conditions utilizes GR reference integrations to ensure consistency with current observational frameworks (e.g., NICER, GW170817).

Key Contributions and Results
The study yields four primary findings regarding the behavior of neutron star interiors in EMSG:

• Monotonicity of the Trace Anomaly: In all accepted EMSG models, the radial profile of the trace anomaly $\Delta(r)$ increases monotonically from the core to the surface. This reproduces the GR trend established by Cai et al. and Ren and Lin, confirming that the modified hydrostatic equilibrium preserves the outward-increasing nature of $\Delta$. However, the profiles exhibit an $\alpha$-dependent splitting that grows with stellar compactness.
• Curvature Invariant Behavior: The curvature invariants ($K, W, R, F$) evolve smoothly with baryon density and radius. The Kretschmann scalar peaks at the center, while the Weyl scalar peaks in the outer layers, preserving the GR radial ordering. Positive $\alpha$ enhances curvature, while negative $\alpha$ suppresses it, with the most significant deviations occurring in stiff, ultracompact configurations.
• Persistence of Thermodynamic-Geometric Correspondence: When curvature invariants are plotted against the fluid-sector trace anomaly $\Delta$, the radial sequences collapse onto organized, single-parameter bands. This extends the GR correspondence found by Garibay et al. into the EMSG regime.
  • The Ricci contraction ($F$) exhibits the tightest organization with $\Delta$, showing minimal equation-of-state scatter.
  • The Ricci scalar ($R$) remains the most sensitive to the EOS, particularly in the negative-$\Delta$ core region where effective pressure-to-energy ratios exceed $1/3$.
  • The Weyl scalar ($W$) shows a branching structure at intermediate $\Delta$ values, marking the transition from the Ricci-dominated core to the tidal-dominated crust.
• Magnitude of Effects: For observationally accessible stars (matched to NICER and GW170817 constraints), the EMSG-induced deformation of the trace anomaly profiles is modest. The largest deviations are confined to stiff, ultracompact configurations where the $\alpha E^2$ source term becomes dominant.

Significance and Claims
The paper claims that nonlinear gravity in EMSG deforms, but does not erase, the thermodynamic-geometric organization of neutron star interiors established in GR. The trace anomaly $\Delta$, defined solely by the fluid sector, remains a useful thermodynamic label for interior geometry even when gravity couples nonlinearly to matter.

The authors emphasize that their results are modest in scope:

• The study is limited to static, spherically symmetric, non-rotating stars with hadronic RMF EOSs, excluding phase transitions or hyperonic degrees of freedom.
• Tidal deformability and moment of inertia are calculated using GR perturbation theory applied to the EMSG background, rather than a fully self-consistent EMSG perturbation treatment.
• The observed splitting in profiles is consistent with existing cosmological and binary-pulsar bounds on $\alpha$, suggesting that current observational data cannot yet independently constrain $\alpha$ via these interior profiles alone.

Ultimately, the work provides a framework for interpreting future multi-messenger data (pulsar timing, X-ray pulse profiles, and gravitational waves) by treating the trace anomaly and curvature invariants as complementary descriptors that can distinguish between GR and modified gravity interpretations without requiring a redefinition of the thermodynamic state of the fluid.