Recent astronomical observations, such as those of Type Ia supernovae with large redshift and measurements of cosmic microwave background (CMB) anisotropic, strongly suggest that the universe is undergoing an accelerated expansion [1]-[4]. This unexpected acceleration cannot be fully explained by Einstein’s General Relativity (GR) combined with ordinary matter and has led to the introduction of dark energy (DE), a mysterious form of energy that constitutes about 73% of the total energy content of the universe. One favorable candidate for dark energy is axion dark matter, which derived from axions and initially suggested in particle physics to fix the strong CP problem in quantum chromodynamics (QCD). When these axions are very light and move slowly, they can behave like quintessence a dynamic scalar field responsible for the negative pressure providing cosmic acceleration. In this reference, axions are modeled using a scalar field with a periodic potential, usually resembling a cosine function, which evolves slowly over cosmic time and generates repulsive pressure counteracting gravitational attraction, thus accelerating the expansion of the universe.

To better explain this acceleration and go after the restrictions of GR, several modified theories of gravity have been introduced such as the f ( R ) , f ( G ) , f ( R , G ) , f ( T ) and f ( R , T ) [5]-[9] etc. The f ( R ) theory of gravity is derived by substituting the Ricci scalar R in Einstein–Hilbert action with a general function f ( R ) . This theory first explored by Buchdahl [5]. f ( R ) theory is further expanded to f ( R , T ) theory of gravitation, which is presented by Harko et al. [8]. In this framework, the gravitational Lagrangian is a function of both the Ricci scalar R , which compact curvature due to gravity, and the trace T of the energy-momentum tensor, which contain the effect of matter and energy. This coupling between matter and geometry permit f ( R , T ) gravity to include quantum effects and matter-geometry actively more naturally than general relativity. At first, the f ( R , T ) theory universalize the well-known f ( R ) gravity and yield a more workable and practical model of the universe’s evolution. It has shown strong potential in modeling not only the late-time acceleration of the universe but also other cosmological phenomena such as transitions between different phases of expansion, the presence of introduced structures like wormholes, and the behavior of compact astrophysical objects.

In the cosmological reference, f ( R , T ) gravity has shown effective in simulating cosmic acceleration without requiring a clear dark energy component, making it very suitable for integrating scalar fields like axion dark matter. Several studies have introduced the evolution of the universe in f ( R , T ) gravity under different cosmological scenarios, including anisotropic models and Bianchi type metrics. Houndjo [10] has discussed the transition from a matter dominated phase to an accelerated phase in the f ( R , T ) gravity. The case of perfect fluid for the Bianchi type III in f ( R , T ) is investigated by Reddy et al. [11]. Moraes [12] has presented unification of the Kaluza-Klein extra-dimensional model with the f ( R , T ) gravity. Azizi [13] investigated the wormhole solutions in the background of f ( R , T ) gravity. Elizalde and Khurshudyan [14] investigated the models for static wormholes in the framework of f ( R , T ) gravity. Banerjee et al. [15] studied the static spherically symmetric wormhole structures sustained by the isotropic matter sources in f ( R , T ) gravity.

Thus, in this work, we discuss a cosmological model in f ( R , T ) gravity with flat FLRW line element by introducing axion dark matter and a linearly varying deceleration parameter (LVDP), which facilitates a smooth evolution from the deceleration to acceleration phase in the evolution of the universe. This together offers a mathematically advanced and meaningful setting to study the accelerated expansion of the universe from a mathematical point of view.

The paper is organized as follows: Section 2 deals with metric and field equations. In Sections 2.1 and 2.2, we used the linearly varying deceleration parameter to solve the f ( R , T ) gravity field equations. Section 3 is devoted to the analysis of state-finder pairs. The energy conditions are presented in Section 4. The last Section 5, is devoted to summarizing obtained results.

The flat Friedmann-Lemaitre-Robertson-Walker (FLRW) models are the best for describing the current large-scale structure of the universe. The homogeneous and isotropic nature of the FLRW model plays an important role in understanding the origin of the universe. We have considered the flat FLRW spacetime as

where a ( t ) is a scale factor. Recently, Pradhan et al. [16] have studied dark energy behavior of the flat FLRW universe with viscous fluid in f ( Q ) theory of gravity. The energy momentum tensor of perfect fluid is given as

Here μ i is the four-velocity of the fluid’s with components ( 0 , 0 , 0 , 1 ) satisfying μ i μ i = 1 , ρ t is the total energy density of the fluid, p t is the isotropic pressure of the fluid. The equation of state (EoS) of a perfect fluid is p = γ ρ with γ ∈ [ 0 , 1 ] where ρ , p are the energy density and pressure of matter respectively.

We discuss the case of modified gravity f ( R , T ) . A generalization of f ( R ) is f ( R , L m ) where coupling of matter Lagrangian and of the Ricci scalar was proposed by Bertolami et al. [17]. Coupling of matter and geometry is well described in f ( R , T ) gravity. It allows matter and geometry to be arbitrary function of T and of R .

The action of f ( R , T ) is given by Harko et al. [8]

where L m is matter Lagrangian density, and f ( R , T ) is arbitrary function of Ricci scalar R and the trace T of T i j . All other symbol have their usual meaning in f ( R , T ) theory of gravity. By varying the action (3) with respect to metric g i j , we can derive field equations of the f ( R , T ) gravity model as

where f R ( R , T ) = ∂ f ( R , T ) ∂ R , f T ( R , T ) = ∂ f ( R , T ) ∂ T , ∇ μ is operator for covariant derivative. Field equations are depends on Θ μ v . Here, we choose L m = − p t which gives Θ μ v = − 2 T μ v − p t g μ v . It should be note that f ( R , T ) is arbitrary function of R and T , therefore, different models of f ( R , T ) gravity are possible for choice of R and T . Singh and Singh [18] have chosen the functional form f ( R , T ) = R + 2 f 1 ( T ) to explore field equations. Here, we consider the following general form of the functional f ( R , T ) proposed by Harko et al. [8]:

here, f 1 ( R ) = R and f 2 ( T ) = μ T , where μ is constant.

The field equation for line element (1), using Equations (4), (5) and (2) takes the form

with

where the over dot ( . ) denotes differentiation with respect to t . Clearly we have two equations (6) and (7) in five unknown variables H , ρ , p , ρ a , and p a , where H = a ˙ a is Hubble parameter, ρ is energy density and p is pressure of

matter (including all the species of the universe) while ρ a and p a are the corresponding densities for the axion field. To obtained the exact solution of the field equations, we consider a novel generalized form of linearly varying deceleration parameter (LVDP) proposed by Akarsu and Dereli [19] that covers Berman’s law indicates that the results are consistent with cosmological data. Next we consider the most general form of LVDP is given by

where, α ≥ 0 and m ≥ 0 are constants. For α = 0 , equation (9) reduces to the law of Berman [20]. The sign of the deceleration parameter (DP) indicate whether the model is accelerating or decelerating. A positive sign of the DP corresponds to the standard decelerating model whereas the negative sign of the DP indicates an accelerating model. The model is inflationary for q = 0 .

In cosmology, the term “statefinder pair” denotes a pair of dimensionless quantities that can be utilized to differentiate among various cosmological models in terms of the evolution of the cosmic scale factor a ( t ) and its derivatives with time. Such parameters were proposed by Sahni et al. [31]. The statefinder pair { r , s } may be estimated from observational data to investigate dark energy and the expansion of the universe. Various cosmological models, such as those with various types of dark energy or modified gravity, may yield different curves in the { r , s } plane, which enables scientists to constrain and compare these models with observations. Ratra and Peebles [32], Kamenschik et al. [33], Armendariz et al. [34] and Dvali et al. [35] have investigated various studies to observe the behaviour the different DE forms, such as quintessence, Chaplygin gas, k-essence and brane world model. The statefinder parameters pair are given by:

It is primarily invented to characterized flat universe model with cold dark matter and dark matter. When r = 1 , s = 0 , it defines cosmological constant model. Below we considered subsequently the case of f ( R , T ) gravity. Therefore. for the case I, the Statefinder pair are obtained as

Figure 6. Plot of { r , s } parameter of case I for α = 0.03 , m = 0.01 , c 1 = − 0.9 , c 2 = 5 .

Figure 6 shows plot of state finder parameter. The statefinder pair { r , s } a diagnostic tool employed to differentiate various cosmological models of dark energy.In the present cosmological model, the trajectory of r and s lies in between 0 < r < 1.05 and − 0.01 < s < 0 . It is observed that when r near zero, s greater than zero which characterizes quintessence type dark energy. As the universe evolves, the trajectory approaches the ΛCDM fixed point { r , s } = { 1 , 0 } and then moves into the region r > 1 , s < 0 which is Chaplygin gas dominated. Therefore, initially model behaves like a quintessence universe and in future it will be Chaplygin gas dominated. Santhi et al. [36], Samantha and Mishra [37], Hatkar et al. [38] are some of the authors who have studied statefinder parameters in the present of cosmic time.

Also, for the case II, the Statefinder pair are found to be

In case II, for m = 0.01 , { r , s } approaches to { 1 , 0 } that is it represent ΛCDM.

Within the framework of general relativity and its generalizations, energy conditions are essential mathematical tools to place physical bounds on matter and spacetime geometry. Energy conditions are inequalities based on the energy-momentum tensor and can be used to decide the feasibility of a gravitational theory. Arora et al. [39] explore the energy conditions in f ( Q , T ) gravity generalized theory of gravity where in the gravitational Lagrangian is a function of the non-metricity scalar Q and the trace of the energy-momentum tensor T . The energy conditions are defined as follows:

i) Null Energy Condition (NEC): ⇔ ρ t + p t ≥ 0 . It ensures that energy density along light ray is non-negative.

ii) Weak Energy Condition (WEC): ⇔ ρ t + p t ≥ 0 and ρ t ≥ 0 . This indicates that the energy density is non-negative .

iii) Strong Energy Condition (SEC): ⇔ ρ t + p t ≥ 0 and ρ t + 3 p t ≥ 0 . The SEC indicates that the gravity is universally attractive. It comes into the Raychaudhuri equation and is connected with the deceleration of the universe. Violation of SEC is required for the acceleration of the universe.

iv) Dominant Energy Condition (DEC): ⇔ ρ t ± p t ≥ 0 and ρ t ≥ 0 . The DEC does not allow energy to generate superluminally and insists that pressure is dominated by energy density.