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arxiv: 2606.20203 · v1 · pith:OMCX5WTLnew · submitted 2026-06-18 · ⚛️ nucl-th · gr-qc· hep-ph

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

Ratikanta Swain , Sayantan Ghosh , Bharat Kumar This is my paper

Pith reviewed 2026-06-26 15:15 UTC · model grok-4.3

classification ⚛️ nucl-th gr-qchep-ph
keywords neutron starstrace anomalyenergy-momentum squared gravitycurvature invariantsequation of statemodified gravityTolman-Oppenheimer-Volkoff equationQCD
0
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The pith

The trace anomaly organizes neutron star curvature invariants into bands even under energy-momentum squared gravity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether the trace anomaly of dense matter, which quantifies conformal symmetry breaking, still correlates with interior spacetime curvature when the Einstein equations are replaced by those of energy-momentum squared gravity. In this setting the geometry is sourced by effective thermodynamic variables that differ from the physical fluid, yet the authors compute the trace anomaly exclusively from the fluid sector. Across five relativistic mean-field equations of state the radial trace-anomaly profiles remain monotonic from core to surface and curvature invariants continue to form organized bands when plotted against the trace anomaly. The organization persists despite systematic splitting that grows with the modified-gravity coupling and stellar compactness, indicating that the thermodynamic label retains its geometric utility.

Core claim

In energy-momentum squared gravity the trace anomaly computed from the fluid sector organizes curvature invariants constructed from the effective variables that source the modified Tolman-Oppenheimer-Volkoff equations into organized bands, thereby extending the thermodynamic-geometric correspondence of general relativity even though profile shapes are deformed by the coupling strength.

What carries the argument

The matter-geometry separation in which the trace anomaly is evaluated from the physical fluid while curvature scalars are built from the effective pressure and density that source the spacetime.

If this is right

  • Trace-anomaly profiles increase monotonically from core to surface in all accepted models and split systematically with EMSG coupling strength.
  • The splitting grows with stellar compactness.
  • Curvature invariants fall onto organized bands versus trace anomaly, with the Ricci contraction showing the tightest organization.
  • The Ricci scalar remains the most equation-of-state sensitive invariant.
  • EMSG effects stay modest for observationally accessible stars but become largest in stiff, ultracompact configurations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the bands survive in other nonlinear gravity-matter couplings, the trace anomaly could serve as a cross-theory diagnostic for neutron-star interiors.
  • The separation of fluid and effective sectors suggests analogous tests are feasible in any theory where the matter stress-energy is rescaled before entering the metric equations.
  • Stiffer equations of state combined with higher compactness would amplify any residual EMSG signature, offering a target for future radius or moment-of-inertia measurements.

Load-bearing premise

The trace anomaly can be isolated to the fluid sector while curvature scalars are computed from the separate effective variables that enter the field equations.

What would settle it

Curvature invariants plotted against trace anomaly in a stiff, high-compactness neutron-star model fail to form bands for any value of the EMSG coupling.

Figures

Figures reproduced from arXiv: 2606.20203 by Bharat Kumar, Ratikanta Swain, Sayantan Ghosh.

Figure 1
Figure 1. Figure 1: FIG. 1. Curvature invariants in EMSG neutron stars as functions of baryon number density [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Radial profiles of the Kretschmann scalar [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Radial profiles of the Ricci scalar [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Radial profiles of the pressure-to-energy ratio [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Trace anomaly profiles [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Trace anomaly profiles [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Trace anomaly profiles [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Kretschmann scalar [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Ricci scalar [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Combined trace anomaly profiles [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Trace anomaly profiles [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper studies neutron-star interiors in energy-momentum squared gravity (EMSG). It adopts an explicit matter-geometry split in which the trace anomaly is evaluated solely on the fluid variables (ε−3p) while curvature scalars are constructed from the effective thermodynamic quantities that source the modified Tolman-Oppenheimer-Volkoff equations. For five relativistic mean-field equations of state the radial trace-anomaly profiles remain monotonically increasing from core to surface but split systematically with the EMSG coupling; despite this splitting, the curvature invariants (especially the Ricci contraction) continue to organize into tight bands when plotted against the fluid trace anomaly, thereby extending the GR thermodynamic-geometric correspondence. Effects are modest for observationally accessible stars and largest for stiff, ultracompact configurations.

Significance. If the reported band organization survives the adopted split, the result supplies a concrete, falsifiable extension of the GR correspondence to a nonlinear matter-gravity theory and demonstrates that the trace anomaly remains a useful thermodynamic label for interior geometry. The use of multiple independent EOS tables and direct numerical integration of the modified structure equations constitutes a reproducible numerical test of the claim.

major comments (2)
  1. [Abstract and §2] Abstract and §2 (modeling choice): the separation in which the trace anomaly is computed exclusively from the fluid sector while curvature scalars are built from the effective variables that source the modified TOV equations is stated as the choice that enables the GR comparison, but no derivation of this split from the EMSG field equations is supplied. Because the central claim is that the bands extend the thermodynamic-geometric correspondence, the result is sensitive to whether the trace anomaly evaluated on the effective tensor would preserve, disperse, or reorganize the bands.
  2. [Numerical results] Numerical results (profiles and bands): the abstract asserts organized bands without reporting convergence tests, resolution studies, or error estimates on the invariants; it is therefore unclear whether the reported organization is robust under changes in radial grid spacing or EOS parameter variations.
minor comments (1)
  1. [Abstract] The range of EMSG coupling strengths explored and the precise definition of the effective variables should be stated explicitly in the abstract for immediate reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [Abstract and §2] Abstract and §2 (modeling choice): the separation in which the trace anomaly is computed exclusively from the fluid sector while curvature scalars are built from the effective variables that source the modified TOV equations is stated as the choice that enables the GR comparison, but no derivation of this split from the EMSG field equations is supplied. Because the central claim is that the bands extend the thermodynamic-geometric correspondence, the result is sensitive to whether the trace anomaly evaluated on the effective tensor would preserve, disperse, or reorganize the bands.

    Authors: The separation is a deliberate modeling choice to preserve a direct comparison with the GR thermodynamic-geometric correspondence, in which the trace anomaly is defined on the physical fluid variables. In EMSG the field equations are sourced by an effective tensor, yet the trace anomaly remains a property of the microscopic matter sector. We will revise §2 to supply an explicit justification of the split, showing its consistency with the EMSG action and explaining why the alternative (evaluating the trace anomaly on the effective tensor) would mix gravitational corrections into the thermodynamic label and thereby defeat the purpose of the test. While we defend the chosen split as the appropriate one for the central claim, we will add a short remark acknowledging the sensitivity issue raised by the referee. revision: partial

  2. Referee: [Numerical results] Numerical results (profiles and bands): the abstract asserts organized bands without reporting convergence tests, resolution studies, or error estimates on the invariants; it is therefore unclear whether the reported organization is robust under changes in radial grid spacing or EOS parameter variations.

    Authors: We agree that explicit numerical validation strengthens the results. In the revised manuscript we will expand the numerical-methods section to document the radial grid resolution, present convergence tests performed with halved and doubled spacing, and provide quantitative error estimates on the curvature invariants. These additions will confirm that the reported band organization is robust against discretization and EOS-parameter variations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; numerical results from independent EOS integration

full rationale

The paper explicitly adopts a modeling choice for the matter-geometry separation (trace anomaly from fluid sector, curvature scalars from effective variables sourcing modified TOV) and then reports outcomes of numerical integration over five independent relativistic mean-field EOS tables. The organized bands in curvature invariants versus trace anomaly emerge from this computation rather than being imposed by definition, by fitting a parameter to the target quantity, or by any self-citation chain. No load-bearing step reduces to its own inputs by construction, satisfying the criteria for a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies insufficient detail to enumerate specific free parameters or axioms beyond the stated matter-geometry separation; the EMSG coupling strength appears as a variable parameter but its fitting procedure is not described.

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discussion (0)

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Reference graph

Works this paper leans on

140 extracted references · 8 canonical work pages

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