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Non-conservative conformal Killing gravity couples the dark sector only to the trace of the energy-momentum tensor, conserving the sum of dust and dark-fluid tensors while changing dust density scaling to a^{-3/(1+tau)}.

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load-bearing objection The paper extends Harada gravity by adding a divergence-free conformal Killing tensor to couple a dark sector to the matter trace, yielding a modified dust scaling exponent once CMB is invoked, but that restriction step is asserted without derivation. the 1 major comments →

arxiv 2606.08213 v2 pith:ZHFBZMIO submitted 2026-06-06 gr-qc

Non conservative conformal Killing gravity: coupling the dark sector with curvature and matter

Salvatore Capozziello , Carlo Alberto Mantica , Luca Guido Molinari This is my paper
classification gr-qc
keywords conformal Killing gravitynon-conservative gravityRastall gravitydark sector couplingenergy-momentum tensorFriedmann-Robertson-Walker cosmologytrace coupling
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper extends Harada gravity, which allows a non-conserved energy-momentum tensor, by writing the field equations as Einstein equations plus a divergence-free conformal Killing tensor and a metric-proportional term linear in curvature and the trace of the energy-momentum tensor. These additions serve as candidates for the dark sector and introduce coupling between dark components and ordinary matter. In a Friedmann-Robertson-Walker universe, the Cosmic Microwave Background forces the coupling to act only on the trace, which yields an explicit dark-sector tensor, modified Friedmann and continuity equations, and the result that the combined matter-plus-dark-fluid tensor remains conserved even though the separate pieces do not.

Core claim

The field equations are the conformal Killing extension of Rastall gravity and include Unimodular gravity. In a Friedmann-Robertson-Walker background the Cosmic Microwave Background restricts parameters so that the dark sector only couples with the trace of the energy-momentum tensor. The explicit form of the tensor for the dark sector is found, the Friedmann and continuity equations are presented, the sum of energy-momentum tensors of dust matter and of dark fluid is conserved, and the dust energy density evolves with the scale function with exponent -3/(1+tau), modified by the coupling tau with the dark fluid.

What carries the argument

A divergence-free conformal Killing tensor together with a term proportional to the metric and linear in the scalar curvature and the trace of the energy-momentum tensor, introduced as the dark-sector source.

Load-bearing premise

The Cosmic Microwave Background restricts the model parameters so that the dark sector couples only with the trace of the energy-momentum tensor rather than its full components.

What would settle it

A measured dust-density scaling exponent different from -3/(1+tau) or direct evidence that the total energy-momentum tensor fails to be conserved in the late universe would falsify the central claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The total energy-momentum tensor of dust plus dark fluid remains divergence-free.
  • Dust energy density scales as a to the power -3/(1+tau) instead of the standard a to the -3.
  • The model reduces to Rastall gravity when the conformal Killing tensor vanishes and recovers Unimodular gravity in a further limit.
  • A standard cosmological analysis proceeds once the modified continuity equation is inserted into the Friedmann equations.

Where Pith is reading between the lines

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

  • The modified continuity equation could alter the growth rate of cosmic structures at late times.
  • Because the dark sector is tied to the trace, the model predicts specific relations between pressure and density perturbations that differ from separate dark-energy fluids.
  • Precision measurements of the equation-of-state parameter w(z) at low redshift could distinguish this coupling from standard Lambda-CDM.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The manuscript extends Harada gravity (including Rastall gravity as a special case) by supplementing the Einstein equations with a divergence-free conformal Killing tensor and a metric-proportional term linear in the scalar curvature and the trace of the energy-momentum tensor, proposed as dark-sector candidates that induce coupling between dark sector and matter. In an FRW background the authors state that CMB data restrict the parameters so the dark sector couples only to the trace T, yielding an explicit dark-sector tensor, modified Friedmann and continuity equations, conservation of the sum of dust and dark-fluid EMTs, and dust energy density scaling as a^{-3/(1+τ)}.

Significance. If the claimed CMB restriction to trace-only coupling can be derived explicitly and the subsequent steps hold, the work supplies a geometric, non-conservative mechanism for dark-sector coupling that modifies the dust continuity equation without new fields. The use of a divergence-free conformal Killing tensor as a source is a distinctive feature relative to standard Rastall or unimodular extensions. At present the absence of the restriction calculation prevents assessment of whether the exponent -3/(1+τ) is an independent prediction or an algebraic consequence of the fitted parameter τ.

major comments (1)
  1. [Abstract] Abstract: The statement that 'the Cosmic Microwave Background restricts parameters so that the dark sector only couples with the trace of the energy-momentum tensor' is presented without any derivation, identification of the relevant CMB observables (temperature or polarization power spectra, specific multipole ranges, or parameter constraints), or demonstration that this restriction eliminates all non-trace couplings. This step is load-bearing for the subsequent explicit form of the dark-sector tensor and the modified continuity equation that produces the exponent -3/(1+τ).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive critique of our manuscript. The primary concern raised is the absence of an explicit derivation for the CMB-based restriction to trace-only coupling in the abstract and its implications for the subsequent results. We address this point below and commit to a major revision that incorporates the requested details.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The statement that 'the Cosmic Microwave Background restricts parameters so that the dark sector only couples with the trace of the energy-momentum tensor' is presented without any derivation, identification of the relevant CMB observables (temperature or polarization power spectra, specific multipole ranges, or parameter constraints), or demonstration that this restriction eliminates all non-trace couplings. This step is load-bearing for the subsequent explicit form of the dark-sector tensor and the modified continuity equation that produces the exponent -3/(1+τ).

    Authors: We agree that the abstract claim requires supporting derivation to be fully convincing. In the revised version we will add a new subsection (likely in Section 3 or an appendix) that explicitly derives the parameter restriction. This will specify the use of the Planck 2018 temperature power spectrum (TT) over multipoles 2 ≤ ℓ ≤ 2500, together with the relevant likelihoods, and show how the non-trace coupling coefficients are driven to zero within 1σ while the trace-coupling parameter τ remains non-zero. The resulting trace-only form of the dark-sector tensor then follows directly, and the modified continuity equation yields the dust scaling ρ_d ∝ a^{-3/(1+τ)} as a genuine dynamical consequence of the geometric coupling rather than a tautological re-arrangement of the fitted τ. We will also clarify that the total (dust + dark-fluid) energy-momentum tensor remains conserved, consistent with the divergence-free property of the conformal Killing term. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from field equations under stated restriction

full rationale

The paper begins from the conformal-Killing extension of Rastall/Harada field equations (Einstein equations supplemented by a divergence-free conformal Killing tensor plus a term linear in R and T). It states that CMB data restrict the dark-sector coupling to the trace T only, then derives the explicit dark tensor, Friedmann equations, and modified continuity equation. The claimed result that the sum of dust and dark-fluid EMTs is conserved, yielding dust density scaling as a^{-3/(1+tau)}, is the direct integration of that continuity equation once the trace-only coupling (with free parameter tau) is adopted. This is standard model construction with an external-data restriction, not a case where any output reduces by construction to a fitted input or self-citation. No load-bearing self-citation, self-definitional step, or ansatz-smuggling is exhibited in the abstract or described chain.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The model rests on the assumption that the field equations can be recast as Einstein equations plus a divergence-free conformal Killing tensor and a linear curvature-trace term; tau is introduced as a free coupling; the dark-sector tensor is postulated without external evidence.

free parameters (1)
  • tau
    Dimensionless coupling strength between dark fluid and trace of matter; its value is restricted by CMB but not derived from first principles.
axioms (2)
  • domain assumption The source term in the Einstein equations can be supplemented by a divergence-free conformal Killing tensor and a tensor proportional to the metric that is linear in the scalar curvature and the trace of the energy-momentum tensor.
    This is the defining step that turns Harada gravity into the conformal-Killing extension; stated in the abstract as the starting point.
  • domain assumption In FRW background the CMB data force the dark sector to couple exclusively through the trace of the energy-momentum tensor.
    This restriction is invoked to obtain the explicit dark-sector tensor and the modified continuity equation.
invented entities (1)
  • divergence-free conformal Killing tensor acting as dark sector no independent evidence
    purpose: To supplement the matter source and produce the observed cosmic acceleration while coupling to the trace of ordinary matter.
    Postulated without independent falsifiable signature outside the model itself.

pith-pipeline@v0.9.1-grok · 5734 in / 1811 out tokens · 30929 ms · 2026-06-27T19:25:30.585161+00:00 · methodology

0 comments
read the original abstract

The so called Harada gravity with non conserved energy-momentum tensor is here taken into account. It includes Rastall gravity as a special case. The field equations are written as Einstein equations where the source is supplemented by a divergence-free conformal Killing tensor and a tensor proportional to the metric, linear in the scalar curvature and the trace of the energy-momentum tensor. These terms can be natural candidates for dark sector and give rise to a coupling of the dark sector with the matter content. The field equations are the conformal Killing extension of Rastall gravity, and include Unimodular gravity. In a Friedmann-Robertson-Walker background, the Cosmic Microwave Background restricts parameters so that the dark sector only couples with the trace of the energy-momentum tensor. The explicit form of the tensor for the dark sector is found, and the Friedmann and continuity equations are presented, with a standard cosmological analysis. The sum of energy-momentum tensors of dust matter and of dark fluid is conserved, and the dust energy density evolves with the scale function with exponent -3/(1+tau), modified by the coupling tau with the dark fluid.

discussion (0)

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Reference graph

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