An Exact Bianchi Type-V Cosmological Model in Linear f(R,T)=R+2f(T) Gravity with Hybrid Expansion: Dynamics, Anisotropy and Energy Conditions

Abstract

We construct and investigate an exact spatially homogeneous but anisotropic Bianchi type-V cosmological model in the framework of linear gravity. A perfect-fluid source with matter Lagrangian is considered. The off-diagonal field equation reduces the directional scale factors to a mean scale factor and a single anisotropy function. An exact hybrid expansion law, is adopted, which naturally produces a transition from decelerated to accelerated expansion. Exact expressions are obtained for the Hubble parameter, deceleration parameter, expansion scalar, shear scalar, mean anisotropy parameter, spatial-curvature term, Ricci scalar, statefinder parameters, energy density, pressure and equation-of-state parameter. For some representative theoretical parameters, the transition from deceleration to acceleration occurs at . The anisotropic contribution decreases rapidly and the model dynamically approaches isotropy. The null, weak and dominant energy conditions remain satisfied over the investigated interval, whereas the strong energy condition becomes violated during the late accelerating epoch. The statefinder pair approaches the CDM fixed point asymptotically. The model therefore provides a simple exact framework for studying modified-gravity effects, anisotropy, spatial curvature and late-time accelerated expansion without introducing additional phenomenological fields.

Introduction

The text presents a theoretical cosmological study of the accelerated expansion of the Universe using a modified theory of gravity and an anisotropic Bianchi type-V cosmological model.

Background

The discovery that the Universe is undergoing accelerated expansion was initially supported by observations of distant Type-Ia supernovae and later confirmed by cosmic microwave background radiation, baryon acoustic oscillations, and large-scale structure observations.

The standard ΛCDM model explains this acceleration using the cosmological constant, Λ, which behaves like a fluid with negative pressure. However, ΛCDM faces two major theoretical problems:

• The cosmological-constant problem, involving the large discrepancy between observed vacuum energy and theoretical predictions from quantum field theory.

• The cosmic-coincidence problem, concerning why matter and dark-energy densities are comparable specifically around the present cosmic epoch.

These problems motivate alternative explanations based on modified theories of gravity, where cosmic acceleration arises from changes to the gravitational sector rather than an independent dark-energy component.

Modified gravity and Bianchi type-V model

The study considers a modified gravity framework in which the gravitational Lagrangian depends on the Ricci scalar and the trace of the energy-momentum tensor. This creates a direct coupling between matter and geometry and can produce effects capable of influencing cosmic expansion.

The authors use a Bianchi type-V spacetime, an anisotropic but spatially homogeneous cosmological model. This geometry is useful because it allows different spatial directions to expand at different rates while containing the open Friedmann–Lemaître–Robertson–Walker model as its isotropic limit.

The model therefore allows the researchers to investigate:

• Cosmic acceleration

• Anisotropic expansion

• Spatial curvature

• Matter–geometry coupling

• Evolution toward isotropy

Main objective

The primary goal is to construct an exact anisotropic Bianchi type-V cosmological solution within a linear modified-gravity model.

The study assumes:

• A perfect-fluid matter source

• A linear form of the modified gravitational function

• A hybrid scale factor, combining power-law and exponential expansion

The hybrid scale factor is chosen because it can represent a transition from an early decelerating Universe to a late-time accelerating Universe.

Methodology and mathematical development

The authors begin with the Bianchi type-V metric and derive the modified gravitational field equations using the action principle.

They then:

1. Introduce a linear functional form for the modified-gravity model.

2. Use a perfect-fluid energy-momentum tensor.

3. Derive the modified Einstein field equations.

4. Integrate the directional equations to obtain the individual scale factors.

5. Introduce the hybrid scale factor.

6. Derive the directional and mean Hubble parameters.

7. Obtain expressions for the expansion, anisotropy, and curvature-related quantities.

Some of the resulting integrals are expressed using the lower incomplete gamma function, allowing exact directional scale factors to be obtained.

Physical quantities investigated

The model is used to study several important cosmological parameters, including:

• Energy density

• Isotropic pressure

• Hubble parameter

• Expansion scalar

• Shear scalar

• Anisotropy parameter

• Deceleration parameter

• Equation-of-state parameter

• Spatial-curvature contribution

• Statefinder diagnostic parameters

• Energy conditions

The study also examines whether the Universe becomes increasingly isotropic at late times by studying the asymptotic behaviour of the shear and anisotropy parameters.

Conclusion

In this work, we have constructed an exact anisotropic Bianchi type-V cosmological model in linear f(R,T) gravity with f(R,T) = R + 2?T, in units where 8?G = c = 1. A perfect-fluid matter source and the hybrid scale factor a(t) = a?t?e?? have been considered. The hybrid scale factor provides a unified description of cosmic evolution, with power-law expansion at early times and exponential expansion at late times. The exact directional scale factors, energy density, and pressure were obtained from the modified field equations. The parameter ? measures the strength of the matter–geometry coupling, while ? and m characterize the anisotropy and the negative spatial-curvature contribution, respectively. The general-relativistic limit is recovered when ? = 0. The deceleration parameter evolves from a positive value at early times to (q=-1) at late times for (0

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Copyright

Copyright © 2026 Dr. Saurabh P. Gaikwad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.