Exact vacuum FLRW solutions in $q$-deformed Brans-Dicke cosmology

Abstract: We study a $q$-deformed extension of Brans-Dicke gravity in a spatially flat Friedmann-Lemaître-Robertson-Walker space-time. The deformation enters through a coupling function that modifies the effective gravitational strength and leads to generalized Friedmann equations. In the matter-free sector, we obtain exact analytic solutions for the scale factor and the Brans-Dicke scalar field, and recast the scalar contribution as an effective fluid. We show that the corresponding equation-of-state parameter and the deceleration parameter are constants and depend only on the Brans-Dicke coupling $ω$ and the deformation function, allowing the scalar sector to mimic radiation-, matter-, or dark-energy-like behavior for a restricted region of parameter space.

• The paper presents exact analytic vacuum FLRW solutions in q-deformed Brans-Dicke cosmology, highlighting the impact of statistical quantum corrections on gravitational dynamics.
• It derives power-law expressions for both the scalar field and the scale factor, leading to constant equation-of-state parameters controlled by the BD parameter and the deformation function.
• The analysis shows that only the negative branch permits accelerated expansion, emphasizing tensions between the model’s predictions and observational constraints.

$q$-Deformed Brans-Dicke Cosmology: Exact Vacuum FLRW Solutions

Motivation and Background

The paper develops an explicit $q$-deformed extension of Brans-Dicke (BD) gravity in the context of spatially flat FLRW spacetime, combining scalar-tensor modifications with statistical quantum deformation. This synthesis is motivated by the emergent and thermodynamic approaches to gravitation, where microscopic (quantum/statistical) corrections to entropy-area relations and horizon thermodynamics induce macroscopic changes in cosmological dynamics. BD theory provides a prototypical scalar-tensor framework with a dynamical gravitational coupling, relevant for various cosmological and astrophysical phenomena including time-dependent $G$, inflation, and dark energy [Brans:1961sx]. The introduction of $q$-deformation, inspired by nonstandard quantum statistics, further modifies the effective gravitational strength, accessing a broader class of dynamics and entropic gravity theories [Senay:2018xaj, Kibaroglu:2018mnx, Kibaroglu:2025QBD]. Interactions between $q$-deformations and scalar-tensor sectors have not previously been analyzed at the level of exact cosmological solutions.

$q$-Deformational Framework and Modified Field Equations

The $q$-deformation enters via a coupling function $\alpha(z,q)$ derived from a statistically deformed entropy-area law in the Verlinde-type entropic gravity setup. This function alters the local Unruh temperature and the equipartition relation on the holographic screen, leading to a modified effective gravitational coupling:

$G_{\text{eff}} = [\alpha(z,q) \psi]^{-1}$

The exact form is retained to explicitly isolate deformation effects. For $q\to1$, the theory recovers standard BD gravity; for $q$0, it reduces to $q$1-deformed Einstein gravity [Senay:2018xaj], and for $q$2, $q$3, to Einstein gravity.

The field equations generalize the usual BD structure:

$q$4

with the scalar sector's energy-momentum tensor acquiring explicit dependence on $q$5 and BD parameter $q$6.

Vacuum FLRW Solutions and Scalar Dynamics

Focusing on the matter-free sector, the metric is assumed spatially flat FLRW:

$q$7

The scalar field $q$8 and scale factor $q$9 obey exact analytic expressions derived from the deformed Friedmann equations:

• The scalar field evolves via a power law in $G$0:

$G$1

with $G$2.

• The scale factor has a power-law solution in time:

$G$3

where $G$4 encodes parameters of the deformation and BD coupling.

These exact forms generalize standard BD vacuum FLRW solutions and depend nontrivially on $G$5.

Effective Fluid Interpretation: Equation of State and Expansion

The scalar sector is recast as an effective self-conserved fluid, characterized by its equation of state (EoS) parameter:

$G$6

This parameter is constant, independent of cosmic time, determined solely by $G$7 and $G$8. Depending on branch and parameter values, the scalar fluid mimics radiation ($G$9), matter ($q$0), or dark energy ($q$1) behavior. Notably, only the negative-sign branch ($q$2) accesses $q$3, required for acceleration; the positive branch ($q$4) remains non-accelerating. The deceleration parameter is similarly constant:

$q$5

Accelerated expansion ($q$6) further restricts $q$7 to regions where $q$8.

The solutions admit strong numerical claims:

• The EoS is strictly constant for each vacuum solution, contradicting the usual time-dependent behavior in cosmological models.
• The scalar sector alone can support sustained accelerated expansion only for $q$9, which is in sharp tension with standard cosmological and Solar System bounds ($q$0) [Avilez:2014CosmologicalBD, Liddle:1998Radiation].

Physical and Phenomenological Implications

The $q$1-deformation parameter acts as a tunable control allowing the scalar sector to span the standard regimes of cosmic expansion. However, the model's vacuum solutions, despite exhibiting exact analytic structure, do not by themselves capture the observed transition from deceleration to acceleration nor admit observationally viable parameter ranges for late-time acceleration in standard BD without additional ingredients (e.g., scalar potentials, matter couplings, field-dependent $q$2). The positivity and reality of $q$3 yield further constraints. The time variation of $q$4 is directly controlled by the scalar field's power-law evolution.

Connections to $q$5 and other modified gravity models are natural: $q$6-deformed BD theory can be interpreted as an effective description within the broader landscape of entropic and statistical-corrected gravities [Nojiri:2022FromNonextensive, Nojiri:2025TheCorrespondence]. Recent DESI results favor dynamical dark energy and modified gravity models over $q$7CDM, supporting the utility of such frameworks for modeling phantom-to-quintessence transitions and dynamical EoS [Odintsov:2025Modified, Odintsov:2026Dynamical, Odintsov:2026ViablefR].

Conclusion

The paper provides an explicit analytic structure for vacuum FLRW solutions in $q$8-deformed BD cosmology, demonstrating that statistical deformation of the gravitational coupling via $q$9 enables the scalar sector to mimic all standard cosmological phases in a self-conserved, constant-EoS framework. However, only the minus branch admits late-time acceleration, restricted to $q$0, in contradiction with empirical constraints. The model does not account for the cosmological transition between deceleration and acceleration, nor solve observational tensions for large positive $q$1. The $q$2-deformation's role is isolated as a free parameter in this analytic toy model, and full phenomenological viability requires additional sectors and confrontation with empirical data.

Further extensions are motivated, including incorporation of matter, scalar self-interactions, dynamical $q$3, and detailed confrontation with local and cosmological tests. The framework highlights the link between microscopic statistical corrections and macroscopic gravitational dynamics, stimulating possible avenues for unified cosmic history modeling beyond standard GR and BD theory (2606.19563).