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

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.

• The paper demonstrates a non-conservative extension of conformal Killing gravity that naturally couples the dark sector with curvature and matter, recovering Einstein equations for specific parameter choices.
• It introduces two parameters, σ and τ, which govern deviations from standard energy-momentum conservation and yield second-order field equations suitable for cosmological analysis.
• The model predicts altered matter dilution laws and a dynamic dark energy equation of state, offering testable departures from standard ΛCDM cosmology.

Non-Conservative Conformal Killing Gravity: Coupling the Dark Sector with Curvature and Matter

Introduction

The paper "Non conservative conformal Killing gravity: coupling the dark sector with curvature and matter" (2606.08213) develops a non-conservative extension of conformal Killing gravity (CKG), yielding a framework that generalizes both Rastall gravity and Unimodular gravity. This approach introduces explicit coupling between the dark sector and the ordinary matter sector through curvature and the trace of the energy-momentum tensor. Such non-conservative formulations allow for violations of standard local energy-momentum conservation, opening up dynamical behaviors that contrast with the standard $\Lambda$CDM and conservative CKG models. The theoretical structure is validated via parametrizations that make the field equations second order and suitable for cosmological analysis.

Theoretical Framework: Non-Conservative CKG and Its Generalizations

The authors revisit the Harada gravity model, where energy-momentum non-conservation is central. The proposed generalization introduces two parameters $\sigma$ and $\tau$ governing the coupling of the non-conservative sector to curvature and the energy-momentum tensor's trace. The resulting field equations are cast as an Einstein-like structure: $$R_{kl} - \frac{1}{2} R g_{kl} = T_{kl} - \sigma R g_{kl} + \tau T g_{kl} + K_{kl}$$ Here, $K_{kl}$ is a divergence-free conformal Killing tensor (CKT), and $\sigma,\,\tau$ mediate the new couplings. Special choices of these parameters recover:

• Rastall gravity: The non-conservation arises directly from coupling to the scalar curvature gradient, connecting with phenomenology like cosmological particle creation.
• Unimodular gravity: A traceless field equation is achieved for specific $\sigma,\,\tau$, leading to standard vacuum solutions with constant scalar curvature.

In this construction, the total source for gravitation includes both matter and a term $$\Theta_{kl} = K_{kl} - \sigma R g_{kl} + \tau T g_{kl}$$, which acts as a dynamical dark sector. Key formal results include the demonstration that the usual Einstein equations are recovered when $\sigma+\tau=0$, and that vacuum solutions are essentially the spaces of constant curvature, consistent with standard results in unimodular and Rastall extensions.

Cosmological Dynamics: FRW Background and Modified Conservation Laws

Specializing to a Friedmann-Robertson-Walker (FRW) background, the authors derive the modified continuity and Friedmann equations. The matter sector is taken as pressureless dust and radiation, while the dark sector emerges from the new tensor components.

The principal modification appears in the continuity equation for the matter energy density: $(1+\tau)\dot{\mu}_M + 3H\mu_M = 0$ yielding $\sigma$0. The parameter $\sigma$1 thus directly regulates the dilution and accumulation of matter energy density with cosmic expansion, in contrast to the standard $\sigma$2 law.

Significantly, observational constraints from CMB evolution enforce $\sigma$3, so the dark sector couples only to matter and not to radiation. As a result,

$\sigma$4

is preserved, aligning with established radiation evolution.

In the flat $\sigma$5 case, the modified Friedmann equation reads: $\sigma$6 with additional contributions interpreted as geometric (dark sector) densities.

Dark Sector Fluid and Equation of State

The dark sector induced by $\sigma$7 is shown to behave as a fluid. The resulting energy density and pressure are: $\sigma$8

$\sigma$9

The equation of state evolves dynamically, exhibiting a phantom behavior ($\tau$0 for $\tau$1) at late times and reducing to a cosmological constant-like regime ($\tau$2) at early times. This produces nontrivial cosmological phenomenology, potentially accommodating observational indications of time-dependence in the equation of state of dark energy suggested in baryon acoustic oscillations (BAO) and DESI results.

Cosmological Implications and Observational Considerations

A central result is the prediction that the Hubble parameter now depends on $\tau$3, which alters the redshift scaling of matter density and thus modifies both the expansion history and the transition from cosmic deceleration to acceleration: $\tau$4 The present-day deceleration parameter is correspondingly shifted: $\tau$5 The asymptotic value as $\tau$6 also acquires explicit dependence on $\tau$7, suggesting a richer diversity for early universe behavior than in conservative models.

The authors emphasize that future statistical fits to cosmological data, including Hubble parameter evolution, BAO, SNe Ia, and CMB acoustic peaks, will be necessary to establish whether such non-conservative CKG improvements yield genuinely distinguishable signatures compared to $\tau$8CDM or conservative CKG scenarios.

Conclusion

This work positions non-conservative CKG as a coherent, well-posed framework generalizing both Rastall and Unimodular gravity, offering a natural geometric mechanism for coupling between the dark sector and the curvature/matter content. Strong claims include the robustness of second-order field equations, the explicit form of dark sector coupling, and the identification of observationally testable parameters that regulate the deviations from standard cosmological evolution. The model predicts a modified matter dilution law, a nontrivial dynamical dark energy equation of state, and altered expansion history. Further progress requires comprehensive confrontation with cosmological observations to constrain or validate the relevance of such non-conservative gravitational interactions.