Thermodynamic implications and observational constraints of interacting $f(Q,\mathcal{T})$ gravity in FRW Universe

Abstract: This work investigates the dynamical evolution of the universe within the framework of symmetric teleparallel $f(Q,\mathcal{T})$ gravity, where $Q$ is the non-metricity scalar and $\mathcal{T}$ is the trace of the energy-momentum tensor. We consider a spatially flat Friedmann-Robertson-Walker (FRW) metric and explore a specific functional form $f(Q,\mathcal{T}) = αQ + β\mathcal{T}$ to derive the gravitational field equations. To characterize the late-time cosmic acceleration, we utilize a model-independent approach by adopting a particular Hubble parameter $H(z)$ parametrization. The model parameters are constrained using the latest observational datasets, including the Hubble ($H(z)$) measurements and Pantheon+ samples. Our results indicate a transition from a decelerated to an accelerated expansion phase. We further examine the physical viability of the model through various cosmological diagnostics such as energy density, the equation of state parameter and thermodynamic properties. The analysis demonstrates that $f(Q,\mathcal{T})$ gravity provides a consistent alternative to the $Λ$CDM model in explaining the current accelerated expansion of the universe.

• The paper develops a linear interacting f(Q,𝒯) gravity model that combines dark-sector energy exchange, a reconstructed Hubble function, observational data, and apparent-horizon thermodynamics.
• Joint cosmic-chronometer and Pantheon+ observations constrain H₀ = 66.18⁺²·⁰⁸₋²·⁰⁵ km s⁻¹ Mpc⁻¹ and n = 1.281, yielding a deceleration-to-acceleration transition near zₜ ≈ 0.65 and a present q₀ ≈ −0.36.
• The model produces smooth positive horizon entropy, evolving dark-energy behavior, and a de Sitter-like future, but its viability remains provisional because α, β, and the interaction strength are fixed or assumed rather than jointly fitted.

This paper develops a cosmological model within symmetric teleparallel $f(Q,\mathcal{T})$ gravity, adopting the linear functional form $f(Q,\mathcal{T}) = \alpha Q + \beta \mathcal{T}$, where $Q$ is the non-metricity scalar and $\mathcal{T}$ the trace of the energy-momentum tensor (2605.16453). The authors combine a phenomenological dark-sector interaction term, a reconstructed Hubble parametrization, observational constraints from 77 cosmic-chronometer $H(z)$ points and the Pantheon+ supernova compilation, and a thermodynamic analysis of the apparent horizon. The central claim is that this interacting $f(Q,\mathcal{T})$ framework reproduces the observed transition from decelerated to accelerated expansion while remaining thermodynamically consistent, thereby offering a viable alternative to $\Lambda$CDM.

Theoretical setup

The action is $$S = \int [\frac{1}{16\pi} f(Q,\mathcal{T}) + \mathcal{L}_m]\sqrt{-g}\,d^4x$$, built on the symmetric teleparallel geometry in which curvature and torsion vanish identically and gravitation is encoded entirely in non-metricity. Variation with respect to the metric yields field equations involving $f_Q$, $f_\mathcal{T}$, and the superpotential tensor $f(Q,\mathcal{T}) = \alpha Q + \beta \mathcal{T}$0; because of the explicit matter-geometry coupling through $f(Q,\mathcal{T}) = \alpha Q + \beta \mathcal{T}$1, the effective source includes the term $f(Q,\mathcal{T}) = \alpha Q + \beta \mathcal{T}$2, which generically implies non-conservation of the matter energy-momentum tensor.

For the spatially flat FRW metric ($f(Q,\mathcal{T}) = \alpha Q + \beta \mathcal{T}$3), the non-metricity scalar reduces to $f(Q,\mathcal{T}) = \alpha Q + \beta \mathcal{T}$4, and the modified Friedmann equations take the form

$f(Q,\mathcal{T}) = \alpha Q + \beta \mathcal{T}$5

$f(Q,\mathcal{T}) = \alpha Q + \beta \mathcal{T}$6

The matter sector is split into pressureless cold dark matter and a dark energy perfect fluid with equation of state $f(Q,\mathcal{T}) = \alpha Q + \beta \mathcal{T}$7, exchanging energy via an interaction term $f(Q,\mathcal{T}) = \alpha Q + \beta \mathcal{T}$8 with dimensionless coupling $f(Q,\mathcal{T}) = \alpha Q + \beta \mathcal{T}$9. The individual continuity equations are $Q$0 and $Q$1, so only the total fluid is conserved. For the flat case, the dark energy equation of state reduces to

$Q$2

showing explicitly that both the interaction strength $Q$3 and the density parameter $Q$4 control whether the model behaves as quintessence ($Q$5), phantom ($Q$6), or crosses the phantom divide. This is the key structural result of the interacting framework: the coupling alone can drive dynamical dark energy behavior without any scalar field.

Hubble parametrization and observational constraints

The reconstruction is performed model-independently by postulating

$Q$7

At high redshift this behaves as $Q$8, recovering power-law decelerating expansion, while at late times it approaches a nearly constant expansion rate characteristic of de Sitter-like acceleration. The parametrization is finite and positive over the observable redshift range and interpolates smoothly between regimes without introducing an explicit cosmological constant.

Parameters are constrained using 77 uncorrelated OHD measurements spanning $Q$9 with the full covariance matrix from Moresco et al., and the Pantheon+ sample of 1701 light curves from 1550 spectroscopically confirmed SNe Ia over $\mathcal{T}$0, treating the absolute magnitude $\mathcal{T}$1 as a nuisance parameter. The combined analysis yields

Parameter	Best fit
$\mathcal{T}$2	$\mathcal{T}$3
$\mathcal{T}$4	$\mathcal{T}$5

The inferred $\mathcal{T}$6 is consistent with Planck-like rather than local-shackle values, though the paper does not engage with the Hubble tension directly. For the modified-gravity couplings, the authors fix $\mathcal{T}$7 and $\mathcal{T}$8 by hand rather than fitting them; this is a substantive assumption, since the observational constraints apply only to $\mathcal{T}$9 and not to the theory parameters that define the $H(z)$0 sector itself.

Dark energy dynamics

With the best-fit parameters, the reconstructed dark energy density

$H(z)$1

remains positive and smooth across the evolution, with the modified-gravity contribution dominating at low redshift. The corresponding $H(z)$2 starts in the quintessence region and evolves toward the phantom divide line $H(z)$3 as $H(z)$4, mimicking dynamical dark energy behavior. The interaction parameter $H(z)$5 shifts the late-time trajectory and can generate crossing of the divide for suitable choices, although specific values of $H(z)$6 used in the plots are not constrained observationally — another point where the analysis relies on assumed rather than fitted parameters.

Thermodynamic analysis

Taking the apparent horizon $H(z)$7 as the thermodynamic boundary, the Hawking temperature is $H(z)$8 and the modified horizon entropy is $H(z)$9 for the linear model, giving

$f(Q,\mathcal{T})$0

The horizon radius and entropy increase monotonically toward late times while the temperature decreases, all quantities remaining positive and divergence-free over the full redshift range. The authors interpret the monotonic entropy growth as consistency with the generalized second law of thermodynamics. It should be noted that this argument is qualitative: no explicit computation of the total entropy (horizon plus matter) production rate $f(Q,\mathcal{T})$1 is presented, so the GSL compliance is asserted from the smooth behavior of the horizon quantities rather than demonstrated rigorously.

Kinematical diagnostics

Deceleration parameter: $f(Q,\mathcal{T})$2 gives $f(Q,\mathcal{T})$3 at high redshift, a transition redshift

$f(Q,\mathcal{T})$4

a present value $f(Q,\mathcal{T})$5, and $f(Q,\mathcal{T})$6 as $f(Q,\mathcal{T})$7. The transition redshift lies within the observationally favored range, and the asymptotic de Sitter future follows naturally from the parametrization.

Statefinder pair: the trajectories $f(Q,\mathcal{T})$8 depart significantly from the $f(Q,\mathcal{T})$9CDM fixed point $\Lambda$0 at intermediate redshifts — with present values $\Lambda$1 and $\Lambda$2 — before converging back to $\Lambda$3 in the far future. The negative present value of $\Lambda$4 is a distinctive signature separating this model from standard $\Lambda$5CDM at current epochs, even though the two coincide asymptotically.

Om diagnostic: since $\Lambda$6 depends only on $\Lambda$7, its non-constant evolution for $\Lambda$8 signals dynamical dark energy or modified gravity effects; the slope of the trajectory encodes quintessence versus phantom character, complementing the $\Lambda$9 analysis.

Limitations and open questions

Several caveats bear on the strength of the conclusions. First, the modified gravity parameters $$S = \int [\frac{1}{16\pi} f(Q,\mathcal{T}) + \mathcal{L}_m]\sqrt{-g}\,d^4x$$0 and $$S = \int [\frac{1}{16\pi} f(Q,\mathcal{T}) + \mathcal{L}_m]\sqrt{-g}\,d^4x$$1 are fixed rather than marginalized over, so the reported confidence contours reflect only the kinematic parametrization, not the full theory space; a joint fit including $$S = \int [\frac{1}{16\pi} f(Q,\mathcal{T}) + \mathcal{L}_m]\sqrt{-g}\,d^4x$$2, $$S = \int [\frac{1}{16\pi} f(Q,\mathcal{T}) + \mathcal{L}_m]\sqrt{-g}\,d^4x$$3, and $$S = \int [\frac{1}{16\pi} f(Q,\mathcal{T}) + \mathcal{L}_m]\sqrt{-g}\,d^4x$$4 against OHD, Pantheon+, and ideally BAO and $$S = \int [\frac{1}{16\pi} f(Q,\mathcal{T}) + \mathcal{L}_m]\sqrt{-g}\,d^4x$$5 data would be needed to establish genuine observational viability of the interacting sector. Second, the generalized second law is invoked qualitatively without computing total entropy production. Third, perturbation-level observables — growth of structure, lensing, and the effective gravitational constant — are absent, yet these are precisely where $$S = \int [\frac{1}{16\pi} f(Q,\mathcal{T}) + \mathcal{L}_m]\sqrt{-g}\,d^4x$$6 models typically face their strongest constraints. Finally, the choice of linear interaction kernel $$S = \int [\frac{1}{16\pi} f(Q,\mathcal{T}) + \mathcal{L}_m]\sqrt{-g}\,d^4x$$7 is one among many phenomenological options, and the sensitivity of the results to this choice is not assessed. Whether the statefinder deviation ($$S = \int [\frac{1}{16\pi} f(Q,\mathcal{T}) + \mathcal{L}_m]\sqrt{-g}\,d^4x$$8) can be discriminated from $$S = \int [\frac{1}{16\pi} f(Q,\mathcal{T}) + \mathcal{L}_m]\sqrt{-g}\,d^4x$$9CDM with forthcoming data remains an open quantitative question.

Conclusion

The paper constructs a self-consistent interacting cosmology in linear $f_Q$0 gravity, constrained by OHD and Pantheon+ data to $f_Q$1 and $f_Q$2. The model reproduces the full qualitative expansion history — early deceleration, transition at $f_Q$3, present acceleration with $f_Q$4, and a de Sitter future — exhibits quintessence-to-$f_Q$5-like evolution of $f_Q$6 modulated by the interaction coupling, and shows smooth, positive thermodynamic quantities on the apparent horizon. Its main limitation is that the theory parameters governing the modified gravity and interaction sectors are assumed rather than observationally fitted, leaving the extent to which the framework genuinely competes with $f_Q$7CDM at the level of precision cosmology an open question.